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 LIBRARY 
 
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S ! lo 
 
 ELEMENTS 
 
 OF 
 
 QUATERNIONS. 
 
 BY THE LATE 
 
 SIR WILLIAM ROWAN HAMILTON, LL. D., M. R. I. A., 
 
 D. C. L. CANTAB. ; 
 
 FKLr>OW Olf THE AMERICAN SOCIETr OF ARTS AND SCIKNCKS; 
 
 OF THE SOCIETY OF AHTS FOR SCOTLAND ; OF THE ROYAL ASTRONOMrCAL SOCIKTT OF LONDON; AND OF THB 
 
 ROYAL NORTHKRN SOCtKTY OF ANTIQUAKIES AT COPKNHAGEM : 
 
 CORRESPONDING MICMBER OF THE INSTITUTK OF FRANCE ; 
 
 HONOHART OR CORRESPONDING MEMBER OF THE IMPERIAL OR ROYAL ACADEMIES OF ST. PETERSBDRGH, 
 
 BERLIN, AND TURIN ; OF THE ROYAL SOCIETIES OF EDINBURGH AND DUBLIN; 
 
 OFTHK NATIONAL ACADEMY OF THE UNITED STATES; 
 
 OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY ; THE NEW YORK HISTORICAL SOCIETY ; 
 
 THE SOCIETY OF NATURAL SCIENCES AT LAUSANNE ; THE PHILOSOPHICAL SOCIETY OF VENICB ; . 
 
 AND OF OTHER SCIENTIFIC SOCIETIES IN BRITISH AND FORKIGN COUNTRIES ; 
 
 ANDREWS' PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF DUBLIN; 
 
 AND ROYAL ASTRONOMER OF IRELAND. 
 
 EDITED BY HIS SON, 
 
 WILLIAM EDWIN HAMILTON, A.B.T.C.D., C.E. 
 
 LONDON : 
 
 LONGMANS, GREEN, & CO, 
 
 1866. 
 
ASTRONOMY UBRARY 
 
 DUBLIN: 
 
 ^rlntetJ at tl)t ©ntijersitp ^regg, 
 
 BY M. H. GILL. 
 
ASTRONOMY 
 
 UBRARV 
 
 ^0 THE 
 
 EIGHT HONOEABLE WILLIAM EAEL OF EOSSE, 
 
 CHANCELLOR OF THE UNIVERSITY OF DUBLIN, 
 
 ^\im Mximz 
 
 IS, BY PERMISSION, DEDICATED, 
 BY 
 
 THE EDITOR. 
 
 m^772ao 
 
In my late father's Will no instructions were left as 
 to the publication of his Writings, nor specially as to 
 that of the " Elements of Quaternions," which, but 
 for his late fatal illness, would have been before now, 
 in all their completeness, in the hands of the Public. 
 
 My brother, the Rev. A. H. Hamilton, who was 
 named Executor, being too much engaged in his cle- 
 rical duties to undertake the publication, deputed this 
 task to me. 
 
 It was then for me to consider how I could best 
 fulfil my triple duty in this matter — First, and chiefly, 
 to the dead ; secondly, to the present public ; and, 
 thirdly, to succeeding generations. I came to the con- 
 clusion that my duty was to publish the work as I found 
 it, adding merely proof sheets, partially corrected by 
 my late father and from which I removed a few typo- 
 graphical errors, and editing only in the literal sense 
 of giving forth. 
 
 Shortly before my father's death, I had several con- 
 versations with him on the subject of the " Elements." 
 In these he spoke of anticipated applications of Qua- 
 ternions to Electricity, and to all questions in which 
 the idea of Polarity is involved — applications which 
 he never in his own lifetime expected to be able fully 
 to develope, bows to be reserved for the hands of 
 another Ulysses. He also discussed a good deal the 
 nature of his own forthcoming Preface ; and I may 
 intimate, that after dealing with its more important 
 topics, he intended to advert to the great labour which 
 
( vi ) 
 
 the writing of the " Elements" had cost him — labour 
 both mental and mechanical; as, besides a mass of 
 subsidiary and unprinted calculations, he wrote out 
 all the manuscript, and corrected the proof sheets, 
 without assistance. 
 
 And here I must gratefully acknowledge the ge- 
 nerous act of the Board of Trinity College, Dublin, in 
 relieving us of the remaining pecuniary liability, and 
 thus incurring the main expense, of the publication of 
 this volume. The announcement of their intention to 
 do so, gratifying as it was, surprised me the less, when 
 I remembered that they had, after the publication of 
 my father's former book, " Lectures on Quaternions," 
 defrayed its entire cost ; an extension of their liberality 
 beyond what was recorded by him at the end of his 
 Preface to the " Lectures," which doubtless he would 
 have acknowledged, had he lived to complete the Pre- 
 face of the " Elements." 
 
 He intended also, I know, to express his sense of 
 the care bestowed upon the typographical correctness 
 of this volume by Mr. M. H. Gill of the University 
 Press, and upon the delineation of the figures by the 
 Engraver, Mr. Oldham. 
 
 I annex the commencement of a Preface, left in ma- 
 nuscript by my father, and which he might possibly 
 have modified or rewritten. Believing that I have 
 thus best fulfilled my part as trustee of the unpub- 
 lished " Elements," I now place them in the hands of 
 the scientific public. 
 
 William Edwin Hamilton. 
 
 January \st^ 1866. 
 
PREFACE.* 
 
 [1.] The volume now submitted to the public is founded on 
 the same principles as the " LECTURES, "^^^ which were pub- 
 lished on the same subject about ten years ago : but the plan 
 adopted is entirely new, and the present work can in no sense 
 be considered as a second edition of that former one. The 
 Table of Contents^ by collecting into one view the headings of 
 the various Chapters and Sections, may suffice to give, to 
 readers already acquainted with the subject, a notion of the 
 course pursued : but it seems proper to offer here a few intro- 
 ductory remarks, especially as regards the method of expo- 
 sition, which it has been thought convenient on this occasion 
 to adopt. 
 
 [2.] The present treatise is divided into Three Books, each 
 designed to develope one guiding conception or^view, and to 
 illustrate it by a sufficient but not excessive number of exam- 
 ples or applications. The First Book relates to the Concep- 
 tion of a Vector^ considered as a directed right line^ in space of 
 three dimensions. The Second Book introduces a First Con- 
 ception of a Quaternion^ considered as the Quotient of two such 
 Vectors. And the Third Book treats of Products and Powers 
 of Vectors^ regarded as constituting a Second Principal Form 
 of the Conception of Quaternions in Geometry. 
 
 * This fragment, by the Author, was found in one of his manuscript books 
 by the Editor. 
 
. TABLE OF CONTENTS. 
 
 BOOK I. 
 
 Pages. 
 ON VECTORS, CONSIDERED WITHOUT REFERENCE TO 
 
 ANGLES, OR TO ROTATIONS, . • . . 1-102 
 CHAPTER* I. 
 
 FUNDAMENTAL PBINCIPLES EESPECTING TECTOES, . 1-11 
 
 SECTiONf 1. — On the Conception of a Yector ; and on Equa- 
 lity of Vectors, . 1-3 
 
 Section 2. — On Differences and Sums of Yectors, taken two 
 
 by two, 3-5 
 
 Section 3. — On Sums of Three or more Yectors, .... 5-7 
 
 Section 4. — On Coefficients of Yectors, 8-11 
 
 This short First Chapter should be read with care by a beginner ; 
 any misconception of the meaning of the word "Vector" being fatal 
 to progress in the Quaternions. The Chapter contains explana- 
 tions also of the connected, but not all equally important, words 
 or phrases, " revector," " pro vector," " transvector," "actual and 
 null vectors," "opposite and successive vectors," " origin and term of 
 a vector," " equal and unequal vectors," "addition and subtraction 
 of vectors," "multiples and fractions of vectors," &c. ; with the nota- 
 tion B - A, for the Vector (or directed right line) ab : and a deduction 
 of the result, essential but not peculiarX to quaternions, that (what 
 is here called) the vector-sum^ of two co-initial sides of a parallelo- 
 gram, is the intermediate and co-initial diagonal. The term " Scalar" 
 is also introduced, in connexion with coefficients of vectors. 
 
 * This Chapter may be referred to, as I. i. ; the next as I. ii. ; the first Chap- 
 ter of the Second Book, as II. i. ; and similarly for the rest. 
 
 t This Section may be referred to, as I. i. 1 ; the next, as I. i. 2 ; the sixth 
 Section of the second Chapter of the Third Book, as III. ii. 6 ; and so on. 
 
 X Compare the second Note to page 203. 
 
 b 
 
11 CONTENTS. 
 
 Pages, 
 CHAPTER II. 
 
 APPLICATIONS TO POINTS AND LINES IN A GIVEN PLANE, 11-49 
 
 Sectfon 1. — On Linear Equations connecting two Co-initial 
 
 Vectors, 11-12 
 
 Section 2. — On Linear Equations between three Co-initial 
 
 Vectors, 12-20 
 
 After reading these two first Sections of the second Chapter, and 
 perhaps the three first Articles (31-33, pages 20-23) of the following 
 Section, a student to whom the subject is new may find it convenient 
 to pass at once, in his first perusal, to the third Chapter of the present 
 Book; and to read only the two first Articles (62, 63, pages 49-51) 
 of the first Section of that Chapter, respecting Vectors in Space, before 
 proceeding to the Second Book (pages 103, &c.), which treats of Qua- 
 ternions as Quotients of Vectors. 
 
 Section 3. — On Plane Geometrical Nets, ...*.. 20-24 
 Section 4. — On Anharmonic Co-ordinates and Equations 
 
 of Points and Lines in one Plane, 24-32 
 
 Section 5. — On Plane Geometrical !N'ets, resumed, . . . 32-35 
 Section 6. — On Anharmonic Equations and Vector Ex- 
 pressions, for Curves in a given Plane, 35-49 
 
 Among other results of this Chapter, a theorem is given in page 43, 
 which seems to offer a new geometrical generation of (plane or spheri- 
 cal) curves of the third order. The anharmonic co-ordinates and equa- 
 tions employed, for the plane and for space, were suggested to the 
 writer by some of his own vector forms ; but their geometrical inter- 
 pretations are assigned. The geometrical nets were first discussed by 
 Professor Mobius, in his Barycentric Calculus (Note B), but they are 
 treated in the present work by an entirely new analysis : and, at least 
 for space, their theory has been thereby much extended in the Chapter 
 to which we next proceed. 
 
 CHAPTER III. 
 
 APPLICATIONS OF VECTOKS TO SPACE, . . . 49-102 
 
 Section 1. — On Linear Equations between Vectors not Com- 
 
 planar, 49-56 
 
 It has already been recommended to the student to read the first 
 two Articles of this Section, even in his first perusal of the Volume ; 
 and then to pass to the Second Book. 
 
 Section 2 — On Quinary Symbols for Points and Planes in 
 
 Space, 57-62 
 
CONTENTS. iii 
 
 Pages. 
 Section 3, — On Anharmonic Co-ordinates in Space, . . 62-67 
 
 Section- 4. — On Greometrical ]S"ets in Space, 67-85 
 
 Section 5. — On Earycentres of Systems of Points ; and on 
 
 Simple and Complex Means of Vectors, 85-89 
 
 Section 6. — On Anharmonic Equations, and Yector Ex- 
 pressions, of Surfaces and Curves in Space, .... 90-97 
 
 Section 7. — On Differentials of Yectors, 98-102 
 
 An application oi finite differences^ to a question connected with ha- 
 ry centres, occurs in p. 87. The anharmonic generation of a ruled hy- 
 perboloid (or paraboloid) is employed to illustrate anharmonic equa- 
 tions ; and (among other examples) certain cones, of the second and third 
 orders, have their vector equations assigned. In the last Section, a defi- 
 nition of differentials (of vectors and scalars) is proposed, which is 
 afterwards extended to differentials of quaternions, and which is in- 
 dependent of developments and of infinitesimals, but involves the 
 conception of limits. Vectors of Velocity and Acceleration are men- 
 tioned ; and a hint of Hodographs is given. 
 
 BOOK II. 
 
 ON QUATERNIONS, CONSIDERED AS QUOTIENTS OF 
 VECTORS, AND AS INVOLVING ANGULAR RELA- 
 TIONS, • 103-300 
 
 CHAPTER I, 
 
 fundamental peinciples respecting quotients op vectors, 103-239 
 
 Very little, if any, of this Chapter II. i., should be omitted, even 
 in a first perusal ; since it contains the most essential conceptions 
 and notations of the Calculus of Quaternions, at least so far as quo- 
 tients of vectors are concerned, with numerous geometrical illustra- 
 tions. Still there are a few investigations respecting circumscribed 
 cones, imaginary intersections, and ellipsoids, in the thirteenth Sec- 
 tion, which a student may pass over, and which will be indicated in 
 the proper place in this Table. 
 
 Section 1 Introductory Remarks ; First Principles 
 
 adopted from Algebra, 103-106 
 
 Section 2. — First Motive for naming the Quotient of two 
 
 Vectors a Quaternion, 106-110 
 
 Sections. — Additional Illustrations, .110-112 
 
 It is shown, by consideration of an angle on a desk, or inclined 
 plane, that the complex relation of one vector to another, in length and 
 
IV CONTENTS. 
 
 Pages, 
 in direction, involves generally a system oifour nvmerical elements. 
 Many other motives, leading to the adoption of the name, " Quater- 
 nion," for the suhject of the present Calculus, from its fundamental 
 connexion with the number " Four," are found to present themselves 
 in the course of the work. 
 
 Section 4 On Equality of Quaternions ; and on the Plane 
 
 of a Quaternion, 112-117 
 
 Section 5. — On the Axis and Angle of a Quaternion j and 
 
 on the Index of a Eight Quotient, or Quaternion, . . 117-120 
 
 Section 6. — On the Reciprocal, Conjugate, Opposite, and 
 
 iN'orm of a Quaternion; and on Null Quaternions, , . 120-129 
 
 Section 7. — On Radial Quotients ; and on the Square of a 
 
 Quaternion, 129-133 
 
 Section 8. — On the Yersor of a Quaternion, or of a Vec- 
 tor ; and on some General Formulae of Transformation, 133-142 
 
 In the five foregoing Sections it is shown, among other things, 
 that the plane of a quaternion is generally an essential element of its 
 constitution, so that diplanar quaternions are unequal; but that the 
 tquare of every right radial (or right versor) is equal to negative unity^ 
 whatever its plane may be. The Symbol V — 1 admits then of a real in- 
 terpretation, in this as in several other systems ; but when thus treated 
 as real, it is in the present Calculus too vague to be useful : on which 
 account it is found convenient to retain the old signification of that 
 symbol, as denoting the (uninterpreted) Imaginary of Algebra, or 
 what may here be called the scalar imaginary, in investigations re- 
 specting non-real intersections, or non-real contacts, in geometry. 
 
 Section 9. — On Yector-Arcs, and Vector- Angles, consi- 
 dered as Representatives of Versors of Quaternions ; 
 and on the Multiplication and Division of any one such 
 Versor by another, 142-157 
 
 This Section is important, on account of its constructions of mul- 
 tiplication and division ; which show that the product of two diplanar 
 versors, and therefore of two such quaternions, is not independent of 
 the order of the factors. 
 
 Section 10. — On a System of Three Right Versors, in 
 Three Rectangular Planes ; and on the Laws of the 
 Symbols, ijl, 157-162 
 
 The student ought to make himself /awjt7/«r with these laws, 
 which are all included in the Fundamental Formula, 
 
CONTENTS. V 
 
 In fact, a Quaternion may be symbolically defined to be a Quadrino- 
 mial Expression of the form, 
 
 q = w-\-ix+jy + kZj (B) 
 
 in which w, x, y, z are four scalars, or ordinary algebraic quantities, 
 while i,j, k are three new symbols, obeying the laws contained in the 
 formula (A), and therefore not subject to all the usual rules of alge- 
 bra : since we have, for instance, 
 
 ij= + k, but ji=^-k; and i'^pk^ =^- ^jk)-i. 
 
 Section 1 1 . — On the Tensor of a Vector, or of a Quater- 
 nion ; and on the Product or Quotient of any two Qua- 
 ternions, 162-174 
 
 Section 12 On the Sum or Difference of any two Qua- 
 ternions ; and on the Scalar (or Scalar Part) of a Qua- 
 ternion, 175-190 
 
 Section 13. — On the Right Part (or Yector Part) of a 
 Quaternion ; and on the Distrihutive Property of the 
 Multiplication of Quaternions, 190-238 
 
 Section 14. — On the Reduction of the General Quaternion 
 to a Standard Quadrinomial Porm ; with a Pirst Proof 
 of the Associative Principle of Multiplication of Qua- 
 ternions, . . . 233-239 
 
 Articles 213-220 (with their sub-articles), in pp. 214-233, maybe 
 omitted at first reading. 
 
 CHAPTER II. 
 
 ON COMPLANAE QITATEENIONS, OE QUOTIENTS OF VECTOES IN 
 ONE PLANE ; AND ON POWEES, EOOTS, AND LOGAEITHMS OF 
 QUATEENIONS, 240-285 
 
 The first six Sections of this Chapter (II. ii.) may be passed over 
 in a first perusal. 
 
 Section 1. — On Complanar Proportion of Vectors ; Fourth 
 Proportional to Three, Third Proportional to Two, 
 Mean Proportional, Square Root ; General Reduction 
 of a Quaternion in a given Plane, to a Standard Bino- 
 mial Porm, 240-246 
 
 Section 2. — On Continued Proportion of Four or more Vec- 
 tors ; whole Powers and Roots of Quaternions ; and 
 Roots of Unity, 246-251 
 
vi CONTENTS. 
 
 Pages. 
 
 Section 3. —On the Amplitudes of Quaternions in a given 
 Plane; and on Trigonometrical Expressions for such 
 Quaternions, and for their Powers, 251-257 
 
 Section 4. — On the Ponential and Logarithm of a Quater- 
 nion ; and on Powers of Quaternions, with Quaternions 
 for their Exponents, 257-264 
 
 Section 5. — On Finite (or Polynomial) Equations of Alge- 
 braic Form, involving Complanar Quaternions ; and on 
 the Existence of n Eeal Quaternion Boots, of any such 
 Equation of the n'^ Degree, 265-275 
 
 Section 6. — On the n^ - n Imaginary (or Symbolical) 
 Roots of a Quaternion Equation of the n*'' Degree, with 
 Coefficients of the kind considered in the foregoing 
 Section, 275-279 
 
 Section 7. — On the Reciprocal of a Vector, and on Har- 
 monic Means of Vectors ; with Remarks on the Anhar- 
 monic Quaternion of a Group of Four Points, and on 
 Conditions of Concircularity, 279-285 
 
 In this last Section (II. ii. 7) the short first Article 258, and the 
 following Art. 259, as far as the formula VIII. in p. 280, should be 
 read, as a preparation for the Third Book, to which the Student may 
 next proceed. 
 
 CHAPTER III. 
 
 ON DIPLA.NAR QUATERNIONS, OR QUOTIENTS OF VECTORS IN 
 ^^ SPACE : AND ESPECIALLY ON THE ASSOCIATIVE PRINCIPLE 
 
 OF MULTIPLICATION OF SUCH QUATERNIONS, 286-300 
 
 This Chapter may be omitted, in a first perusal. 
 
 Section 1. — On some Enunciations of the Associative Pro- 
 perty, or Principle, of Multiplication of Diplanar Qua- 
 ternions, 286-293 
 
 Section 2. — On some Geometrical Proofs of the Associative 
 Property of Multiplication of Quaternions, which are 
 independent of the Distributive Principle, .... 293-297 
 
 Section 3. — On some Additional Formulae, .... 297-300 
 
CONTENTS. vii 
 
 BOOK III. 
 
 Pages. 
 ON QUATERNIONS, CONSIDERED AS PRODUCTS OR 
 
 POWERS OF VECTORS; AND ON SOME APPLICA- 
 TIONS OF QUATERNIONS, 301 to the end. 
 
 CHAPTER I. 
 
 ON THE INTEEPEETATION OF A PRODUCT OF VECTORS, OR POWER 
 
 OF A VECTOR, AS A QUATERNION, . . . 301-390 
 
 The first six Sections of this Chapter ought to be read, even in a 
 first perusal of the -work. 
 
 Section 1 . — On a First Method of Interpreting a Product 
 
 of Two Vectors as a Quaternion, 301-303 
 
 Section 2. — On some Consequences of the foregoing Inter- 
 pretation, 303-308 
 
 This^r*^ interpretation treats th.e product 13. a, as equal to the 
 quotient /3 : a-i ; where a"i (or Ra) is the previously defined Eeeiprocal 
 (II, ii. 7) of the vector a, namely a second vector.^ -which has an in- 
 verse length, and an opposite direction. Multiplication of Vectors is 
 thus proved to be (like that of Quaternions) a Bistributive, but not 
 generally a Commutative Operation. The Square of a Vector is shown 
 to be always a Negative Scalar, namely the negative of the square of 
 the tensor of that vector, or of the number which expresses its length ; 
 and some geometrical applications of this fertile principle, to spheres, 
 &c., are given. The Index of the JRight Fart of a Product of Two Co- 
 initial Vectors, OA, ob, is proved to be a right line, perpendicular to 
 the Flane of the Triangle oab, and representing by its length the 
 Double Area of that triangle ; while the Eolation round this Index, 
 from the Multiplier to the Multiplicand, is positive. This right part, 
 or vector part, Va/3, of the product vanisJies, when the factors are 
 parallel (to one common line) ; and the scalar part, Sa/3, when they 
 are rectangular. 
 
 Section 3. — On a Second Method of arriving at the same 
 
 Interpretation, of a Binary Product of Vectors, . . . 308-310 
 
 Section 4. — On the Symbolical Identification of a Eight 
 Quaternion with its own Index : and on the Construc- 
 tion of a Product of Two Rectangular Lines, by a Third 
 Line, rectangular to both, 310-313 
 
 Section 5. — On some Simplifications of N'otation, or of 
 Expression, resulting from this Identification ; and on 
 the Conception of an Unit-Line as a Right Versor, . 313-316 
 
vni CONTENTS. 
 
 Pages. 
 In this second interpretation^ which is found to agree in all its re- 
 sults with the first, but is better adapted to an extension of the theory, 
 as in the following Sections, to ternary products of vectors, a product 
 of two vectors is treated as the product of the two right quaternions, of 
 which those vectors are the indices (II. i. 5). It is shown that, on 
 the same plan, the Sum of a Scalar and a Vector is a Quaternion. 
 
 SECTioif 6. — On the Interpretation of a Product of Three 
 
 or more Vectors as a Quaternion, 316-330 
 
 This interpretation is effected by the substitution, as in recent 
 Sections, of Eight Quaternions for Vectors, without change oiorder of 
 the factors. Multiplication of Vectors, like that of Quaternions, is 
 thus proved to be an Associative Operation. A vector, generally, is 
 reduced to the Standard Trinomial Form, 
 
 p = ix-Vjy-\-Jcz; (C) 
 
 in which i,j, h are the peculiar symbols already considered (II. i. 
 10), but are regarded now as denoting Three Rectangular Vector- Units, 
 while the three scalars x, y, z are simply rectangular co-ordinates ; from 
 the known theory of which last, illustrations of results are derived. 
 The Scalar of the Product of Three coinitial Vectors, oa, ob, oc, is found 
 to represent, with a sign depending on the direction of a rotation, the 
 Volume of the Parallelepiped under those three lines ; so that it va- 
 nishes when they are complanar. Constructions are given also for ^ro- 
 ducts of successive sides of triangles, and other closed polygons, inscribed 
 in circles, or in spheres ; for example, a characteristic property of the 
 circle is contained in the theorem, that the product of the four suc- 
 cessive sides of an inscribed quadrilateral is a scalar : and an equally 
 characteristic (but less obvious) property of the sphere is included in 
 this other theorem, that the product of the ^t?^ successive sides of an 
 inscribed gauche pentagon is equal to a tangential vector, drawn from 
 the point at which the pentagon begins (or ends). Some general For- 
 mula of Transformation of Vector Expressions are given, with which 
 a student ought to render himself very familiar, as they are of con- 
 tinual occurrence in the practice of this Calculus ; especially the four 
 formulae (pp. 316, 317) : 
 
 V.yV/3a=aS/3y-)3Sya; (D) 
 
 Vy/3a = aS|3y-/3S7a + ySa/3; (E) 
 
 pSajSy = aS/3yp + /3Syap + ySa^Sp ; (F) 
 
 |0Sa/3y = VjSySap + VyaS^p + Va/3Syp ; (G) 
 
 in which a, (3, y, p are any four vectors, while S and V are signs of 
 the operations of taking separately the scalar and vector parts of a qua- 
 ternion. On the whole, this Section (III. i. 6) must be considered 
 to be (as regards the present exposition) an important one ; and if 
 it have been read with care, after a perusal of the portions previously 
 indicated, no difficulty will be experienced in passing to any subse- 
 quent applications of Quaternions, in the present or any other work. 
 
CONTENTS. ix 
 
 Pages. 
 Section 7. — On the Fourth Proportional to Three Diplanar 
 
 Vectors, 331-349 
 
 Section 8.. — On an Equivalent Interpretation of the Fourth 
 Proportional to Three Diplanar Vectors, deduced from 
 the Principles of the Second Book, 349-361 
 
 Section 9. — On a Third Method of interpreting a Product 
 or Function of Vectors as a Quaternion; and on the 
 Consistency of the Eesults of the Interpretation so ob- 
 tained, with those which have been deduced from the 
 two preceding Methods of the present Book, . . .361-364 
 
 These three Sections may be passed over, in a first reading. They 
 contain, however, theorems respecting composition of successive rota- 
 tions (pp. 334, 335, see also p. 340); expressions for the sem^are« of a 
 spherical polygon, or for half the opening of an arbitrary pyramid^ as 
 the angle of a quaternion product, with an extension, by limits, to the 
 semiarea of a spherical figure bounded by a closed curve, or to half the 
 opening of an arbitrary cone (pp. 340, 341) ; a construction (pp. 358- 
 360), for a series of spherical parallelograms, so called from a partial 
 analogy to parallelograms in o. plane ; a theorem (p. 361), connecting 
 a certain system of such (spherical) parallelograms with ih^foci of a 
 spherical conic, inscribed in a certain quadrilateral ; and the concep- 
 tion (pp. 353, 361) of a Fourth Unit in Space (?^, or + I), which is of 
 a scalar rather than a vector character, as admitting merely of change 
 of sign, through reversal of an order of rotation, although it presents 
 itself in this theory as the Fourth Troportional {if'^h;) to Three Beet- 
 angular Vector Units. 
 
 Section 10. — On the Interpretation of a Power of a Vector 
 
 as a Quaternion, 364-384 
 
 It may be well to read this Section (III. i. 10), especially for 
 the Exjjonential Connexions which it establishes, between Quaternions 
 and Splierical Trigonometry, or rather Folygonometry, by a species of 
 extetision of Moivr^s theorem, from the plane to space, or to the spliere. 
 For example, there is given (in p. 381) an equation of six terms^ 
 which holds good for every spherical j^entagon, and is deduced in this 
 way from an exfetided exponential formula. The calculations in the 
 sub-articles to Art. 312 (pp. 375-379) may however be passed over; 
 and perhaps Art. 315, with its sub-articles (pp. 383, 384). But Art 
 314, and its sub-articles, pp. 381-383, should be read, on account of 
 the exponential forms which they contain, of equations of the circle, 
 ellipse, logarithmic spirals (circular and elliptic), h^liz, a.nd screw sur- 
 face. 
 
 Section 11 — On Powers and Logarithms of Diplanar Qua- 
 ternions ; with some Additional Formulae, .... 384-390 
 
X CONTENTS. 
 
 It may suffice to read Art. 316, and its first eleven sub-articles, 
 pp. 384—386. In this Section, tlie adopted Logarithm, \q, of a Qua- 
 ternion q, is the simplest root, q\ of the transcendental equation, 
 
 and its expression is found to be, 
 
 l^ = lT^ + Z?.UVj, (H) 
 
 in which T and U are the signs of tensor and versor, while Z. $■ is the 
 angle of q, supposed usually to be between and tt. Such logarithms 
 are found to be often useful in this Calculus, although they do not gene- 
 rally possess the elementary property, that the sum of the logarithms 
 of two quaternions is equal to the logarithm of their ^ro^wc^ ; this ap- 
 parent paradox, or at least deviation from ordinary algebraic rules, 
 arising necessarily from the corresponding property of quaternion 
 multiplication, which has been already seen to be not generally a com- 
 mutative operation {q'q" not = q'q\ unless (f and j" be complanar^. 
 And here, perhaps, a student might consider his first perusal of this 
 work as closed.* 
 
 Pages. 
 
 CHAPTER II. 
 
 ON DIFFERENTIALS AND DEVELOPMENTS OF FUNCTIONS OF QUA- 
 TERNIONS ; AND ON SOME APPLICATIONS OF QUATERNIONS 
 TO GEOMETRICAL AND PHYSICAL QUESTIONS, 391-495 
 
 It has been already said, that this Chapter may be omitted in a 
 first perusal of the work. 
 
 Section 1. — On the Definition of Simultaneous Differen- 
 tials, 391-393 
 
 * If he should choose to proceed to the Differential Calculus of Quaternions in 
 the next Chapter (III. ii.), and to the Geometrical and other Applications in the 
 third Chapter (III. iii.) of the present Book, it might be useful to read at this 
 stage the last Section (I. iii. 7) of the First Book, which treats of Differentials of 
 Vectors (pp. 98-102); and perhaps the omitted parts of the Section II. i. 13, 
 namely Articles 213-220, with their subarticles (pp. 214-233), which relate, 
 among other things, to a Oonstruction of the Ellipsoid, suggested by the present 
 Calculus. But the writer will now abstain from making any further suggestions 
 of this kind, after having indicated as above what appeared to him a minimum 
 course of study, amounting to rather less than 200 pages (or parts of pages) 
 of this Volume, which will be recapitulated for the convenience of the student 
 at the end of the present Table. 
 
CONTENTS. XI 
 
 Pages. 
 Section 2. — Elementary Illustrations of the Definition, 
 
 from Algebra and Geometry, 394-398 
 
 In the view here adopted (comp. I. iii. 7), differentials are not ne- 
 cessarily, nor even generally, small. But it is shown at a later stage 
 (Art. 401, pp. 626-630), that the principles of this Calculus a^/ot^ us, 
 whenever any advantage may be thereby gained, to treat differentials 
 as infinitesimals ; and so to abridge calculation, at least in many ap- 
 plications. 
 
 Section 3 — On some general Consequences of the Defini- 
 tion, 398-409 
 
 Partial differentials and derivatives are introduced ; and differen- 
 tials of functions of functions. 
 
 Section 4 — Examples of Quaternion Differentiation, . . 409-419 
 
 One of the most important rules is, to differentiate the /ac^or* of a 
 c^dXemion. product, in situ ; thus (by p. 405), 
 
 6..qq' = diq.q'-VqAq'. (I) 
 
 The formula (p. 399), d. ^-» = - q-^^q.q-\ (J) 
 
 for the differential of the reciprocal of a quaternion (or vector), is also 
 very often useful ; and so are the equations (p. 413), 
 
 dT^ d^ dU^ d^ 
 
 Tq q Vq q 
 
 and (p. 411), ^ • "' = Y "'^^^^ ' ^^) 
 
 g being any quaternion, and a any constant vector-unit, while tisa 
 variable scalar. It is important to remember (comp. III. i. 11), that 
 we have not in quaternions the usual equation, 
 
 Q 
 unless q and d^ be complanar ; and therefore that we have not generally, 
 
 dlp = ^, 
 P 
 
 if p be a variable vector ; although we have, in this Calculus, the 
 
 scarcely less simple equation, which is useful in questions respecting 
 
 orbital motion, 
 
 dlP-=^, (M) 
 
 a p 
 
 if a be any constant vector, and if the plane of a and p be given (or 
 constant). 
 
 Section 5. — On Successive Differentials and Developments, 
 
 of Functions of Quaternions, 420-435 
 
xii CONTENTS. 
 
 Pages. 
 In this Section principles are established (pp. 423-426), respect- 
 ing qnatermon functions which vanish together ; and a form of deve- 
 lopment (pp. 427, 428) is assigned, analogous* to Taylor's Seriesy 
 and like it capable of being concisely expressed by the symbolical 
 equation^ 1 + A = £<i (p. 432). As an example of partial and succes- 
 Bive differentiation, the expression (pp. 432, 433), 
 
 p = r¥j^kj-^k~\ 
 ■which may represent any vector ^ is operated on ; and an application 
 is made, by means of definite integration (pp. 434, 435), to deduce the 
 known area and volume of a sphere, or of portions thereof ; together 
 with the theorem, that the vector sum of the directed elements of a 
 spheric segment is zero : each element of surface being represented by an 
 inward normal, proportional to the elementary area, and correspond- 
 ing in hydrostatics to \he pressure of a fluid on that element. 
 
 Section 6. — On the Differentiation of ImpKcit runctions 
 of Quaternions ; and on the General Inversion of a Li- 
 near Function, of a Yector or a Quaternion : with 
 some connected Investigations, . . . ' 435-495 
 
 In this Section it is shown, among other things, that a Linear 
 and Vector Symbol, 0, of Operation on a Vector, p, satisfies (p. 443) a 
 Symbolic and Cubic Equation, of the form, 
 
 = w - m> + m"(p^ - ^3 ; (N) 
 
 whence m(}>~^ — m'— m"<p -\- <p^=->p, (N') 
 
 = anotJier symbol of linear operation, which it is shown how to de- 
 duce otherwise from 0, as well as the three scalar constants, m, m, m'. 
 The connected algebraical cubic (pp. 460, 461), 
 
 Jlf = w + m'c + m"c2 + c3 = 0, (0) 
 
 is found to have important applications ; and it is provedf (pp. 460, 
 462) that if SX^p = Sp^X, independently of X and p, in which case 
 the function is said to be self-conjugate, then this last cubic has three 
 real roots, ci, cz, cz ; while, in the same case, the vector equation, 
 
 \p^p = 0, (P) 
 
 is satisfied by a system of Three Heal and Rectangular Directions : 
 namely (compare pp. 468, 469, and the Section III. iii. 7), those of 
 the axes of a (biconcyclic) system of surfaces of the second order, re- 
 presented by the scalar equation, 
 
 * At a later stage (Art. 375, pp. 509, 510), a neiv Enunciation of Taylor's 
 Tlieorem is given, with a new proof , but stiU in a form adapted to quaternions. 
 
 t A simplified proof, of some of the chief results for this important case of 
 self-conjugation, is given at a later stage, in the few first subarticles to Art. 415 
 (pp. 698, 699). 
 
CONTENTS. Xlll 
 
 Pages. 
 Sp(f>p = <7p2 -f C", in which C and C are constants. (Q,) 
 
 Cases are discussed; and general forms {coX^Qdi cyclic, rectangular, 
 focal, bifocal, &c., from their chief geometrical uses) are assigned, 
 for the vector and scalar functions ^p and Sp^/o : one useful pair of 
 such (cyclic) forms being, with real and constant values of ^, X, j«, 
 
 (l>p=ffp + YXpfi, Bp^p=ffp'^ + S\pnp. (R) 
 
 And finally it is shown (pp. 491, 492) that if fg be a linear and qua- 
 ternion function of a quaternion, q, then the Symbol of Operation, f 
 satisfies a certain Symbolic and Biquadratic Equation, analogous to the 
 cubic equation in ^, and capable of similar applications. 
 
 CHAPTER III. 
 
 ON SOME ADDITIONAL APPLICATIONS OF QUATERNIONS, WITH 
 
 SOME CONCLUDING REMARKS, . . 495 to the end. 
 
 This Chapter, like the one preceding it, may be omitted in a first 
 perusal of the Volume, as has indeed been already remarked. 
 
 Section 1. — Remarks Introductory to this Concluding 
 
 Chapter, 495-496 
 
 Section 2 On Tangents and Kormal Planes to Curves in 
 
 Space, 496-501 
 
 Section 3. — On J^ormals and Tangent Planes to Surfaces, 501-510 
 
 Section 4. — On Osculating Planes, and Absolute ]N"ornials, 
 
 to Curves of Double Curvature, ........ 511-515 
 
 Section 5. — On Geodetic Lines, and Families of Surfaces, 515-531 
 
 In these Sections, dp usually denotes a tangent to a curve, and v 
 a normal to a surface. Some of the theorems or constructions may 
 perhaps be new ; for instance, those connected with the cone of paral- 
 lels (pp. 498, 513, &c.) to the tangents to a curve of double curvature ; 
 and possibly the theorem (p. 525), respecting reciprocal curves in 
 space : at least, the deductions here given of these results may serve 
 as exemplifications of the Calculus employed. In treating of Families 
 of Surfaces by quaternions, a sort of analogue (pp. 629, 530) to the for- 
 mation and integration of Partial Differential Equations presents 
 itself; as indeed it had done, on a similar occasion, in the Lectures 
 (p. 674). 
 
 Section 6. — On Osculating Circles and Spheres, to Curves 
 
 in Space; with some connected Constructions, . . . 531-630 
 
 The analysis, however condensed, of this long Section (III. iii. 6), 
 cannot conveniently be performed otherwise than under the heads of 
 the respective Articles (389-401) which compose it: each Article 
 
XIV CONTENTS. 
 
 Pages, 
 being followed by several subarticles, which form with it a sort of 
 Series* 
 
 Article 389. — Osculating Circle defined, as the limit of a circle, 
 which touches a given curve (plane or of double curvature) at a given 
 point p, and cuts the curve at a near point q (see Fig. 77, p. 511). 
 Deduction and interpretation of general expressions for the vector k 
 of the centre k of the circle so defined. The reciprocal of the radius 
 KP being called the vector of curvature, we have generally, 
 
 Vector of Curvature = (p - k)-i = -=~ = — Y ~ = &c. ; (S) 
 •^ vr- y rp^i^ dp dp 
 
 and if the arc (s) of the curve be made the independent variable, then 
 
 d2p 
 Vector of Curvature = p" = Ds^p = ~j. (S') 
 
 Examples : curvatures of helix, ellipse, hyperbola, logarithmic spiral ; 
 
 locus of centres of curvature of helix, plane e volute of plane ellipse, 531-535 
 
 A.RTICLE 390 — Abridged general calculations; return from (S') 
 to (S), 535, 536 
 
 Article 391 Centre determined by three scalar equations ; 
 
 Folar Axis, Polar Developable, 537 
 
 Article 392. — Vector Equation of o^cvloXm^ civc\e, 538,539 
 
 Article 393. — Intersection (or intersections) of a circle with a 
 plane curve to which it osculates ; example, hyperbola, 539-541 
 
 Article 394. — Intersection (or intersections) of a spherical curve 
 with a small circle osculating thereto ; example, spherical conic ; con- 
 structions for the spherical centre (or pole) of the circle osculating to 
 such a curve, and for the point of m^ersec^ww above mentioned, . . 541-549 
 
 Article 395. — Osculating Sphere, to a curve of double curvature, 
 defined as the limit of a sphere, which contains the osculating circle to 
 the curve at a given point p, and cuts the same curve at a near point 
 Q (comp. Art. 389). The centre s, of the sphere so found, is (as usual) 
 the point in which th.Q polar axis (Art. 391) touches the cusp-edge of 
 tlie polar developable. Other general construction for the same centre 
 (p. 551, comp. p. 573). General expressions for the vector, a = os, 
 and for the radius, R = Wp', -K'' is the spherical curvature (comp. Art. 
 897). Condition of Sphericity {8=1), and Coefficient of Non- sphericity 
 (^S — 1), for a curve in space. When this last coefficient is positive 
 (as it is for the helix), the curve lies outside the sphere, at least in the 
 neighbourhood of the point of osculation, 549-553 
 
 Article 396. — Notations r, r, . . for D«p, Bs^p, &c. ; properties 
 of a curve depending on the square (s^) of its arc, measured from a 
 given point p ; r = unit-tangent, t' = vector of curvature, r~^ = Tr' = cur- 
 vature (oT first curvature, comp. Art. 397), v = tt' = binormal ; the 
 
 * A Table of initial Pages of all the Articles will be elsewhere given, which will 
 much facilitate reference. 
 
CONTENTS. XV 
 
 Pages. 
 three planes, respectively perpendicular to r, r', v, are the normal 
 plane, the rectifying plane, and the osculating plane ; general theory 
 of emanant lines and planes, vector of rotation, axis of displacement, oscit- 
 lating screw surface ; condition of developahility of surface of emanants, 554-559 
 
 Article 397. — Properties depending on the cube (s^) of the are ; 
 Radius r (denoted here, for distinction, by a roman letter), and Vector 
 ir^T, oi Second Curvature ; this radius r may be either positive or ne- 
 gative (whereas the radius r of first curvature is always treated as 
 positive), and its reciprocal r^ may be thus expressed (pp. 663, 669), 
 
 d^o r" 
 
 Second Curvature* = r-i = S ^,, \^ , (T), or, r-i = S — , CT') 
 
 the independent variable being the arc in (T'), while it is arbitrary in 
 (T) : but quaternions supply a vast variety of other expressions for this 
 important scalar (see, for instance, the Table in pp. 574, 675). "We 
 have also (by p. 560, comp. Arts. 389, 395, 396), 
 
 Vector of Spherical Curvature = sp~i = (p— <y)"^ = &c., (U) 
 = projection of vector (r') of (simple or first) curvature, on radius (J2) 
 of osculating sphere : and if p and P denote the linear and angular 
 elevations, of the centre (s) of this sphere above the osculating plane, 
 then (by same page 560), 
 
 p = r tan F- R&m P = r'r = rD^r. (XT') 
 
 Again (pp. 660, 561), if we write (comp. Art. 396), 
 
 \ = V — =r-ir + Tr' = Vector of Second Curvature plus Binormal, (V) 
 
 T 
 
 this line \ may be called the Rectifying Vector ; and if TL denote the 
 inclination (considered first by Lancret), of this rectifying line (\) to 
 the tangent (r) to the curve, then 
 
 tan JT=r'-i tan P = y-ir. (V') 
 
 Known right cone with rectifying line for its axis, and with H. for its 
 seniiangle, which osculates at p to the developable locus of tangents to 
 the curve (or by p. 568 to the cone of parallels already mentioned) ; 
 new right cone, with a new scmiangle, C, connected with H by the 
 relation (p. 562), 
 
 tanC=^tanjEr, (V") 
 
 which osculates to the cone of chords, drawn from the given point p 
 
 * In this Article, or Series, 397, and indeed also in 396 and 398, several re- 
 ferences are given to a very interesting Memoir by M. de Saint- Venant, " Sur 
 les lignes courbes non planes :" in which, however, that able writer objects to such 
 known phrases as second cirvature, torsion, &c., and proposes in their stead a new 
 name " cambrure," which it has not been thought necessary here to adopt. 
 {Journal de V E'cole Poly technique, Cahier xxx ) 
 
XYl CONTENTS. 
 
 Pages, 
 to other points q of the'given curve. Other osculating cones, cylinders, 
 helix, &nd parabola ; this last being (pp. 662, 566) the parabola which 
 osculates to the projection of the curve, on its own osculating plane. De- 
 viation of curve, at any near point q, from the osculating circle at p, 
 decomposed (p. 666) into two rectangular deviations, from osculating 
 helix and parabola. Additional formulae (p. 676), for the general 
 theory of emanants (Art. 396) ; case o£ normally emanant lines, or of 
 tangentially emanant planes. General auxiliary spherical curve (pp. 
 576-578, comp. p. 515) ; new proof of the second expression (V) for 
 tan H, and of the theorem that if this ratio of curvatures be constant, 
 the proposed curve is a geodetic on a cylinder : new proof that if each 
 curvature (r'l, r~i) be constant, the cylinder is right, and therefore 
 the curve a helix, . , ' 659-578 
 
 Article 398. — Properties of a curve in space, depending on the 
 fourth and Jlfth powers (si, s^) oiita arc (s), 578-612 
 
 This Series 398 is so much longer than any other in the Volume, 
 and is supposed to contain so much original matter, that it seems 
 necessary here to subdivide the analysis under several separate heads, 
 lettered as (a), (b), (c), &c. 
 
 («). Neglecting s^, we may write (p. 578, comp. Art. 396), 
 
 OP, = ps = p + 57 + -|«2 r' + XsH" + JjS^r'" ; (W) 
 
 or (comp. p. 587), ps = p + XsT + y,rr' + z^rv, (W) 
 
 with expressions (p. 588) for the coefficients (or co-ordinates) Xs, ys, Zg, 
 in terms of r', r, r", r, r', and s. If ^ be taken into account, it be- 
 comes necessary to add to the expression (W) the term, i^s^t^^ ; 
 with corresponding additions to the scalar coefficients in (W), intro- 
 ducing r'" and r" : the laws for forming which additional terms, and 
 for extending them to higher powers of the arc, are assigned in a 
 subsequent Series (399, pp. 612, 617). 
 
 (4). Analogous expressions for t", v", k", X', cr', and p', B', F, K', 
 to serve in questions in which s^ is neglected, are assigned (in p. 579) ; 
 r" v', K, X, <T, and p, R, P, H, having been previously expressed (in 
 Series 397) ; while r", v", k", \", a", &c. enter into investigations 
 which take account of s^ : the arc » being treated as the independent 
 variable in all these derivations. 
 
 (c). One of the chief results of the present Series (398), is the 
 introduction (p. 681, &c.) of a new auxiliary angle, J, analogous in 
 several respects to the known angle H (397), but belonging to a 
 higher order of theorems, respecting curves in space : because the new 
 angle / depends on the fourth (and lower) powers of the arc s, while 
 Lancret's angle H depends only on s^ (including s^ and s"^). In fact, 
 while tan jffis represented by the expressions (V), whereof one is 
 »•'-» tan P, tan /admits (with many transformations) of the following 
 analogous expression (p. 681), 
 
 tan/=:i2'-itanP; (X) 
 
CONTENTS. xvii 
 
 Pages. 
 where JR' depends* by (A) on s^, while r' and F depend (397) on no 
 higher power than s^. 
 
 (d). To give a more distinct geometrical meaning to this new angle 
 J", than can he easily gathered from such a formula as (X), respecting 
 which it may he observed, in passing, that /is in general more simply 
 defined by expressions for its cotangent (pp. 581, 588), than for its 
 tangent, we are to conceive that, at each point p of any proposed 
 curve of double curvature, there is drawn a tangent plane to the sphere ^ 
 which osculates (395) to the curve at that point ; and that then the 
 envelope of all these planes is determined, which envelope (for reasons 
 afterwards more fully explained) is called here (p. 581) the " Cir- 
 cumscribed Developable :" being a surface analogous to the ^^ Rectifying 
 Developable'^ of Lancret, but belonging (c) to a higher order of ques- 
 tions. And then, as the A'woww angle -ff denotes (Z^l) the inclina- 
 tion^ suitably measured, of the rectifying line (\), which is a genera- 
 trix of the rectifying developable, to the tangent (r) to the curve ; so 
 the new angle / represents the inclination of a generating line (^), of 
 what has just been called the circumscribed developable, to the same 
 tangent (r), measured likewise in a defined direction (p. 581), but 
 in the tangent plane to the sphere. It may be noted as another ana- 
 logy (p. 582), that while JS'is a right angle for deplane curve, so J 
 is right when the curve is spherical. For the helix (p. 585), the an- 
 gles H and / are equal ; and the rectifying and circumscribed deve- 
 lopables coincide, with each other and with the right cylinder, on 
 which the helix is a geodetic line. 
 
 (e). If the recent line be measured from the given point p, in 
 a suitable direction (as contrasted with the opposite), and with a suit- 
 able length, it becomes what may be called (comp. 396) the Vector of 
 Eolation of the Tangent Plane (d) to the Osculating Sphere ; and then 
 it satisfies, among others, the equations (pp. 579, 581, comp. (V)), 
 
 ^ = V^, T0=i2-icosec/; (X') 
 
 V 
 
 this last being an expression for the velocity of rotation of the plane 
 just mentioned, or of its normal, namely the spherical radius R, if the 
 given curve be conceived to be described by a point moving with a con- 
 
 * In other words, the calculation of r' and P introduces no differentials 
 higher than the third order ; but that of R' requires 'Cine fourth order of differen- 
 tials. In the language of modern geometry, the/on?2^r can be determined by 
 the consideration oifour consecutive points of the curve, or by that of two consecu- 
 tive osculating circles ; but the latter requires the consideration of two consecu- 
 tive osculating spheres, and therefore oifive con^QCMiive points of the curve (sup- 
 posed to be one of double curvature). Other investigations, in the present and 
 immediately following Series (398, 399), especially those connected with what 
 we shall shortly call the Osculating Twisted Cubic, will be found to involve the 
 consideration of six consecutive points of a curve. 
 
 d 
 
xviii CONTENTS. 
 
 stant velocity^ assumed = 1. And if we denote by v the point in which 
 the given radius R or PS is nearest to a consecutive radius of the same 
 kind, or to the radius of a consecutive osculating sphere^ then this point v 
 divides the line ps internally, into segments which may (ultimately) be 
 thus expressed (pp. 580, 581), 
 
 PV = -B sin2 /, vs = i2 coss /. (X") 
 
 But these and other connected results, depending on s*, have their 
 known analogues (with H for /, and r for R), in that earlier theory 
 (c) which introduces only s^ (besides s^ and s2) : and they are all m- 
 cluded in the general theory oiemanant lines and planes (396, 397), of 
 which some new geometrical illustrations (pp. 582-584) are here 
 given. 
 
 (/). New auxiliary scalar n {=p-^RR' = cot J'secP= &c.), = ve- 
 locity of centre s of osculating sphere, if the velocity of the point p of 
 the given curve be taken as unity (e) ; n vanishes with Rf, cot J", and 
 (comp. 395) the coefficient S-1 (=wn'"i) of non- sphericity, for the 
 case of a spherical curve (p. 584). Arcs, first and second curvatures, 
 and rectifying planes and lines, of the cusp-edges of the polar and 
 rectifying* developables ; these can all be expressed without going 
 beyond s\ and some without using any higher power than s\ or diffe- 
 rentials of the orders corresponding ; r\ = wr, and ri = nr, are the 
 scalar radii of first and second curvatvire oiihe former cusp-edge, r\ 
 being positive when that curve turns its concavity at s towards the 
 given curve at p : determination of the point b, in which the latter 
 cusp-edge is touched by the rectifying line X to the original curve 
 (pp. 584-587). 
 
 (^). Equation with one arbitrary constant (p. 587), of a cone of 
 the second order, which has its vertex at the given point p, and has 
 contact of the third order (or four-side contacf) with the cone of chords 
 (397) from that point; equation (p. 590) of a cylinder of the second 
 order, which has an arbitrary line pe from p as one side, and has 
 contact of \h.e fourth order (or fwe-point contact^ with the curve at p ; 
 the constant above mentioned can be so determined, that the right line 
 PE shall be a side of the cone also, and therefore apart of the intersect 
 Hon of cone and cylinder; and then the remaining or curvilinear 
 part, of the complete intersection of those two siirfaces of the second 
 
 * The rectifying plane, of the cusp-edge of the rectifying developable, is the 
 plane of \ and t', of which the formula LIV. in p. 587 is the equation ; and the 
 rectifying line rh, of the same cusp-edge, intersects the absolute normal pk to the 
 given curve, or the radius (r) of first curvature, in the point h in which that 
 radius is nearest (e) to a consecutive radius of the same kind. But this last theo- 
 rem, which is here deduced by quaternions, had been previously arrived at by 
 M. de Saint- Venant (comp, the Note to p. xv.), through an entirely different 
 analysis, confirmed by geometrical considerations. 
 
CONTENTS. xix 
 
 Pages, 
 order, is (by known principles) a gauche curve of the third order, 
 
 or what is briefly called* a Twisted Cubic : and this last curve, in 
 virtue of its construction above described, and whatever the as- 
 sumed direction of the auxiliary line pe may be, has contact of the 
 fourth order {pv Jive-point contact) with the given curve of double cur- 
 vature at p (pp. 687-590, comp. pp. 663, 672). 
 
 (Ji). Determination (p. 690) of the cow«^«w# in the equation of the 
 cone {g), so that this cone may have contact of the fourth order (or 
 Jwe-side contact) with the cone of chords from p ; the cone thus found 
 may be called the Osculating Oblique Gone (comp. 397), of the second 
 order, to that cone of chords ; and the coefficients of its equation in- 
 volve only r, r, /, r', r', x", but not r"\ although this last derivative 
 is of no higher order than r", since each depends only on s^ (and lower 
 powers), or introduces only fifth differentials. Again, the cylinder 
 (g) will have contact of the ffth order (or six-point contact) with the 
 given curve at p, if the line pe, which is by construction a side of that 
 cylinder, and has hitherto had an arbitrary direction, be now obliged 
 to be a side of a certain cubic cone, of which the equation (p. 690) in- 
 volves as constants not only rrr'rVr", like that of the osculating cone 
 just determined, but also r"'. The two cones last mentioned have the 
 tangent (r) to the given curve for a common side,f but they have also 
 three other common sides, whereof one at least is real^ since they are 
 assigned by a cubic equation (same p. 690) ; and by taking this side 
 for the line pe in (g), there results a new cylinder of the second order, 
 which cuts the osculating oblique cone, partly in that right line pe itself, 
 and partly in a gauche curve of the third order, which it is proposed to 
 call an Osculating Twisted Cubic (comp. again (y)), because it has con- 
 tact of the fifth order (or six-point contact) with the given curve at p 
 (pp. 690, 691). 
 
 (i). In general, and independently of any question of osculation, 
 a Twisted Cubic (jf), if passing through the origin o, may be repre- 
 sented by any one of the vector equations (pp. 692, 693), 
 
 * By Dr. Salmon, in his excellent Treatise on Analytic Geometry of Three 
 Dimensions (Dublin, 1862), which is several times cited in the Notes to this final 
 Chapter (III. iii.) of these Elements. The gauche curves, above mentioned, have 
 been studied with much success, of late years, by M. Chasles, Sig. Cremona, and 
 other geometers : but their existence, and some of their leading properties, ap- 
 pear to have been first perceived and published by Prof. Mobius (see his Bary^ 
 centric Calculus, Leipzig, 1827, pp. 114-122, especially p. 117). 
 
 t This side, however, counts as three (p. 614), in the system of the six lines of 
 intersection (real or imaginary) of these two cones, which have a common vertex p, 
 and are respectively of the second oxiA. third orders (or degrees). Additional light 
 will be thrown on this whole subject, in the following Series (399) ; in which also 
 it will be shown that there is only one osculating twisted cubic, at a given point, 
 to a given curve of double curvature ; and that this cubic curve can be determined^ 
 without resolving any cubic or other equation. 
 
XX CONTENTS. 
 
 Yap + Yp<pp = 0, (Y); or (^ + e)p = a, (¥') 
 
 or p = (^ + c)-^a, (Y"); or Yap + pYyp + YpY\pfi = 0, (Y'") 
 in wliicli a, y, \, fi are real and constant vectors, but <j is a variable sca- 
 lar ; while 0p denotes (comp. the Section III. ii. 6, or pp. xii., xiii.) a 
 linear and vector function, which is A^r^ generally woiJ self -conjuff ate, 
 of the variable vector p of the cubic curve. The number of the scalar 
 constants, in the form (Y'"), or in any other form of the equation, is 
 found to be ten (p. 593), with the foregoing supposition that the curve 
 passes through the origin, a restriction which it is easy to remove. 
 The curve (Y) is cut, as it ought to be, in three points (real or imagi- 
 nary), by an arbitrary secant plane ; and its three asymptotes (real or 
 imaginary) have the directions of the three vector roots /3 (see again 
 the last cited Section) of the equation (same p. 693), 
 
 V/3^/3 = 0: (Z) 
 
 so that by (P), p. xii., these three asymptotes compose a real and rect- 
 angular system, for the case of self -conjugation of the function 
 in (Y). 
 
 (/). Deviation of a near point Ps of the given curve, from the sphere 
 (395) which osculates at the given point p ; this deviation (by p. 593, 
 comp. pp. 653, 584) is 
 
 r\^ R's^ n&^ 
 
 it is ultimately equal (p. 696) to the quarter of the deviation (397) 
 of the same near point Pj from the osculating circle at p, multiplied by 
 the sine of the small angle spSs, which the small arc sss of the locus of 
 the spheric centre s (or of the cusp-edge of the polar developable) stib- 
 tends at the same point p ; and it has an outward or an inward direc- 
 tion, according as this last arc is concave or convex (/) at s, towards the 
 given curve at p (pp. 585, 695). It is also ultimately equal (p. 696) 
 to the deviation pss - TsSs, of the given point p from the near sphere, 
 which osculates at the near point p^; and likewise (p. 597) to the com- 
 ponent, in the direction of sp, of the deviation of that near point from 
 the osculating circle at p, measured in a direction parallel to the nor- 
 mal plane at that point, if this last deviation be now expressed to the 
 accuracy of the fourth order : whereas it has hitherto been considered 
 sufficient to develope this deviation from the osculating circle (397) as 
 far as the third order (or third dimension of s) ; and therefore to treat 
 it as having a direction, tangential to the osculating sphere (comp. 
 pp. 666, 694). 
 
 (k). The deviation (Ai) is also equal to the third part (p. 698) of 
 the deviation of the near point Vg from the given circle (which osculates 
 at p), if measured in the near normal plane (at p^), and decomposed in 
 the direction of the radius Rs of the near sphere; or to the third part 
 (with direction preserved) of the deviation of the new near point in 
 which the given circle is cut by the near plane, /rom the near sphere : or 
 finally to the third part (as before, and still with an unchanged direc- 
 
CONTENTS. xxi 
 
 Pages, 
 tion) of the deviation from the given sphere, of that other near point 
 
 c, in which the near circle (osculating at Ps) is cut by the given normal 
 plane (at p), and which is found to satisfy the equation, 
 
 sc = 3sps - 2sp. (Bi) 
 
 Geometrical connexions (p, 599) between these various results (/) (^), 
 illustrated by a diagram (Fig. 83). 
 
 (J). The Surface, which is the Locus of the Osculating Circle to 
 a given curve in space, may be represented rigorously by the vector 
 expression (p. 600), 
 
 Ws, u^ps-^- rsTs sin u + n^r/ vers u ; (Ci) 
 
 in which s and u are two independent scalar variables, whereof * is 
 (as before) the arc pp^ of the given curve, but is not now treated as 
 small : and u is the (small or large) angle subtended at the centre k* of 
 the circle, by the arc of that circle, measured from its point of oscula- 
 tion Ps. But the same superficial locus (comp. 392) may be repre- 
 sented also by the vector equation (p. 611), inyolvmg a2)parentlg only 
 one scalar variable (s), 
 
 Y-^ + Vs = 0, (Di) 
 
 <t>-ps 
 
 in which Vs—Tst/, and u)= (vs,u = the vector of an arbitrary point 
 of the surface. The general method (p. 501), of the Section III. iii. 
 3, shows that the normal to this surface (Ci), at any proposed point 
 thereof, has the direction of w*, « - o-j ; that is (p. 600), the direction 
 of the radius of the sphere, which contains the circle through that 
 point, and has the same point of osculation p* to the given curve. The 
 locus of the osculating circle is therefore found, by this little calculation 
 with quaternions, to be at the same time the Envelope of the Osculat- 
 ing Sphere, as was to be expected from geometrical considerations 
 (comp. the Note to p. 600). 
 
 (m). The curvilinear locus of the point c in (Jc) is one branch of 
 the section of the surface (I), made by the normal plane to the given 
 curve at p ; and if d be the projection of c on the tangent at p to this 
 new curve, which tangent pd has a direction perpendicular to the ra- 
 dius PS or H of the osculating sphere at p (see again Fig. 83, in p. 
 599), while the ordinate dc ia parallel to that radius, then (attending 
 only to principal terms, pp. 598, 599) wc have the expressions, 
 
 and therefore ultimately (p. 600), 
 
 DC3 81 w3^5r((T-p) ^ ,_. 
 
 from which it follows that p is a singular 2>oint of the section here 
 considered, but not a cusp of that section, although the curvature 
 at p is infinite : the ordinate dc varying ultimately as the power 
 with exponent ^ of the abscissa pd. Contrast (pp. 600, 601), of this 
 
xxii CONTENTS. 
 
 section, with that of the developable Locus of Tangents, made by the 
 same normal plane at p to the given curve ; the vectors analogous to 
 PD and DC are in this case nearly equal to - fs^/ and — ^s^v^v ; so 
 that the latter varies Tiltimately as the power f of the former, and the 
 point p is (as it is known to be) a cusp of this last section. 
 
 (n). A given Curve of double curvature is therefore generally a 
 Singular Line (p. 601), although not a cusp-edge, upon that Surface {T)j 
 which is at once the Locus of its osculating Circle, and the Envelope 
 of its osculating Sphere : and the new developable surface {d), as being 
 circumscribed to this superficial locus (or envelope), so as to touch it 
 along this singular line (p. 612), may naturally be called, as above, 
 ihe Circumscribed Developable (;;^. h^i). 
 
 (o). Additional light may be thrown on this whole theory of the 
 singular line (n), by considering (pp. 601-611) a problem which was 
 discussed by Monge, in two distinct Sections (xxii. xxvi.) of his well- 
 known Analyse (comp. the Notes to pp. 602, 603, 609, 610 of these 
 Elements') ; namely, to determine the envelope of a sphere with varying 
 radius R, whereof the centre s traverses a given curve in space ; or 
 briefly, to find the Envelope of a Sphere with One varying Parameter 
 (comp. p. 624) : especially for the Case of Coincidence (p. 603, &c.), of 
 what are usually two distinct branches (p. 602) of a certain Charac- 
 teristic Curve (or arete de rebroussement), namely the curvilinear enve- 
 lope (real or imaginary) of all the circles, along which the superficial 
 envelope of the spheres is touched by those spheres themselves. 
 
 {^p)' Quaternion forms (pp. 603, 604) of the condition of coinci- 
 dence (o) ; one of these can be at once translated into Monge' s equa- 
 tion of condition (p. 603), or into an equation slightly more general, 
 as leaving the independent variable arbitrary ; but a simpler and 
 more easily interpretable form is the following (p. 604), 
 
 ridr = ±MB, (Gi) 
 
 in which r is the radius of the circle of contact, of a sphere with its 
 envelope (o), while ri is the radius of (first) curvature of the curve (s), 
 which is the locus of the centj-e s of the sphere. 
 
 (^). The singular line into which the two branches of the curvi- 
 linear envelope ^refused, when this condition is satisfied, is in general 
 an orthogonal trajectory (p. 607) to the osculating planes of the curve 
 (s) ; that curve, which is noiv the given one, is therefore (comp. 391, 
 395) the cusp-edge (p. 607) of the^o^ar developable, corresponding to 
 the singular line just mentioned, or to what may be called the curve 
 (p), which was formerly the given curve. In this way there arise 
 many verifications of formulae (pp. 607, 608) ; for example, the 
 equation (Gi) is easily shown to be consistent with the results of (/). 
 
 (r). With the geometrical hints thus gained from interpretation 
 of quaternion results, there is now no difficulty in assigning the Com- 
 plete and General Integral of the Equation of Condition {p), which was 
 presented by Monge under the form (comp. p. 603) of a non-linear 
 differential equation of the second order, involving three variables 
 
 Pages. 
 
CONTENTS. xxiii 
 
 Pages. 
 (0, \jj, tt) considered as functions of a fourth (a), namely the co-or- 
 dinates of tlie centre of the sphere, regarded as varying with the ra- 
 dius, but which does not appear to have been either integrated or 
 interpreted by that illustrious analyst. The general integral here 
 found presents itself at first in a ^^wa^^rw/ow/orm (p. 609), but is easily 
 translated {^. 610) into the usual language of analysis. A less ge- 
 neral integral is also assigned, and its geometrical signification exhi- 
 bited, as answering to a case for which the singular line lately consi- 
 dered reduces itself to a singular point (pp. 610, 611). 
 
 (s). Among the verifications (jf) of this whole theory, it is shown 
 (pp. 608, 609) that although, when the two branches (o) of the general 
 curvilinear envelope of the circles of the system are real and distinct, 
 each branch is a cusp-edge (or arete de rebroussement, as Monge per- 
 ceived it to be), upon the superficial envelope of the spheres, yet in the 
 case of fusion (p) this cuspidal character is lost (as was likewise 
 seen by Monge*) : and that then a section of the surface, made by 
 a normal plane to the singular line, has precisely the form (on), ex- 
 pressed by the equation (Fi). In short, the result is in many ways 
 confirmed, by calciilation and by geometry, that when the condition of 
 coincidence (j») is satisfied, the Surface is, as in (n), at once the JEnve- 
 lope of the osculating Sphere and the Locus of the osculating Circle, to 
 that Singular Line on itself, into which by ((?) the two branches (o) 
 of its general cusp- edge are fused. 
 
 ({). Other applications of preceding formulae might be given ; 
 for instance, the formula for k" enables us to assign general ex- 
 pressions (p. 611) for the centre and radius of the circle, which oscu- 
 lates at K to the locus of the centre of the osculating circle, to a given 
 curve in space : with an elementary verification, for the case of the 
 plane evolute of the plane evolute of a plane curve. But it is time to con- 
 clude this long analysis, which however could scarcely have been 
 much abridged, of the results of Series 398, and to pass to a more 
 brief account of the investigations in the following Series. 
 
 Akticle 399. — Additional general investigations, respecting that 
 gauche curve of the third order (or degree), which has been above 
 called an Osculating Twisted Cubic (398, (A))) to any proposed curve 
 of double curvature ; with applications to the case, where the given 
 curve is a Me:r, 612-621 
 
 (a). In general (p. 614), the tangent pt to the given curve is a 
 nodal side of the cubic cone 398, (A) ; one tangent plane to that cone 
 (C3), along that side, being the osculating plane (P) to the curve, and 
 therefore touching also, along the same side, the osculating oblique cone 
 (C2) of the second order, to the cone of chords (397) from p ; while the 
 other tangent plane to the cubic cone (Ca) crosses ihsit first plane (P), 
 or the quadric cone (C2), at an angle of which the trigonometric cotan- 
 
 * Compare the first Note to p. 609 of these Elements. 
 
XXIV CONTENTS. 
 
 Pages. 
 gent (^r') is equal to half the differential of the radius (r) of second 
 curvature, divided hy the differential of the are (s). And the three 
 common sides, pe, pb', pe", of these two cones, which remain when the 
 tangent pt is excluded, and of which one at least must be real, are the 
 parallels through the given point p to the three asymptotes (398, (t)) 
 to the gauche curve sought ; being also sides of three quadric cylin- 
 ders, say (Z2), (X'2), (-^"2), which contain those asymptotes as other 
 sides (or generating lines) : and of which each contains the twisted 
 cubic sought, and is cut in it by the quadric cone ( G2). 
 
 (b). On applying this First Method to the case of a given h,elix, it 
 is found (p. 614) that the general cubic cone (^C^ breaks up into the 
 system of a new quadric cone, (jO-i), and a new plane (P') ; which lat- 
 ter is the rectifying plane (396) of the helix, or the tangent plane at p 
 to the right cylinder, whereon that given curve is traced. The two 
 quadric cones, (Co) and (C2), touch each o^Aer andthe plane (P) along 
 the tangent pt, and have no other real common side : whence tivo of 
 the sought asymptotes, and tivo of the corresponding cylinders (a), are 
 in this case imaginary, although they can still be used in calculation 
 (pp. 614, 615, 617). But the plane (P') cuts the cone (C2), not only 
 in the tangent pt, but also in a second real side pe, to which the real 
 asymptote is parallel (a) ; and which is at the same time a side of a 
 real quadric cylinder (Z2), which has that asymptote for another side 
 (p. 617), and contains the twisted cubic : this gauche curve being thus 
 the curvilinear part (p. 615) of the intersection of the real cone (C2), 
 with the real cylinder (Zo)- 
 
 (c). Transformations and verifications of this result ; fractional ex- 
 pressions (p. 616), for the co-ordinates of the twisted cubic ; expres- 
 sion (p. 615) for the deviation of the helix irom that osculating curve, 
 which deviation is directed inwards, and is of the sixth order : the 
 least distance, between the tangent pt and the real asymptote, is a right 
 line PB, which is cut internally (p. 617) by the axis of the right cylin- 
 der (h), in a point a such that pa is to ab as three to seven. 
 
 {cT). The First Method (a), which had been established in the pre- 
 ceding Series (398), succeeds then for the case of the Jielix, with a faci- 
 lity which arises chiefly from the circumstance (J)), that for this case 
 the general cubic cone (Cz) breaks up into two separate loci, whereof 
 one is a. plane (P'). But usually the foregoing method requires, as in 
 398, (Ji)), the solution of a cubic equation : an inconvenience which is 
 completely avoided, by the employment of a Second General Method, 
 as follows. 
 
 (e). This Second Method consists in taking, for a second locus of the 
 gauche osculatrix sought, a certain Cuhic Surface (63), of which 
 every point is the vertex* of a quadric cone, having six-point con- 
 
 * It is known that the locics of the vertex of a quadric cone, which passes 
 through six given points of space, a, b, c, d, e, f, whereof no four are in one 
 
CONTENTS. XXV 
 
 Pages. 
 tact with the given curve at p : so that this new surface is cut by the 
 
 plane at infinity^ in the same cubic curve as the cubic cone ipz). It is 
 found (p. 620) to be a Ruled Surface^ with the tangent pt for a Sin- 
 gular Line ; and when this right line is set aside, the remaining (that 
 is, the curvilinear') part of the intersection of the two loci, (C2) and 
 (aSs), is the Osculating Twisted Cubic sought : which gauche osculatrix 
 is thus completely and generally determined, without any such difficulty 
 or apparent variety, as might be supposed to attend the solution of a 
 cubic equation (d), and with new verifications for the case of the helix 
 (p. 621). 
 
 Article 400. — On Involutes and Evolutes in Space, .... 621-626 
 {a). The usual points of Monge's theory are deduced from the two 
 fundamental quaternion equations (p. 621), 
 
 S((r-p)p'=0, V(or-p)(T'=0, (Hi) 
 
 in which p and a are corresponding vectors of involute and evolute ; 
 together with a theorem of Prof. De Morgan (p. 622), respecting the 
 case when the involute is a spherical curve. 
 
 (b). An involute in space is generally the only real part (p. 624) of 
 the envelope of a certain variable sphere (comp. 398), which has its 
 centre on the evolute, while its radius R is the variable intercept be- 
 tween the two curves : but because we have here the relation (p. 622, 
 
 comp. p. 602), 
 
 i2'2 H- <t'2 = 0, (Hi') 
 
 the circles of contact (398, (0)) reduce themselves each to a point (or 
 rather to a pair of imaginary right lines, intersecting in a real point), 
 and the preceding theory (398), of envelopes of spheres with one 
 varying parameter, undergoes important modifications in its results, 
 the conditions of the application being different. In particular, the 
 involute is indeed, as the equations (Hi) express, an orthogonal tra- 
 jectory to the tangents of the evolute; but not to the osculating planes 
 
 plane, is generally a Surface, say {S^), of the Fourth Degree : in fact, it is cut by 
 the plane of the triangle abc in a system of four right lines, whereof three are 
 the sides of that triangle, and the fourth is the intersection of the two planes, 
 ABC and DEF. If then we investigate the intersection of this surface (^S\) with 
 the quadric cone, (a.bcdef), or say ((72), which has a for vertex, and passes 
 through the five other given points, we might expect to find (in some sense) a 
 curve of the eighth degree. But when we set aside ^efive right lines, ab, ac, ad, 
 AE, AF, which are common to the two surfaces here considered, we find that the 
 (remaining or) curvilinear part of the complete intersection is reduced to a curve 
 of the third degree, which is precisely the twisted cubic through the six given points. 
 In applying this general (and perhaps new) method, to the problem of the oscu- 
 lating twisted cubic to a curve, the osculating ^?(m« to that curve may be excluded, 
 as foreign to the question : and then the quartic surface {Si) is reduced to the 
 cubic surface {S3), above described. 
 
 e 
 
XXvi CONTENTS. 
 
 Pages, 
 of that curve, as tlie singular line (398, {q)) of the former envelope 
 was, to those of the curve which was the locus of the centres of the 
 spheres hQiovQ considered, when a certain condition of coincidence (or 
 oi fusion, 398, {p)") was satisfied. 
 
 (c). Curvature of hodograph of evolute (p. 625) ; if p, Pi, P2, • • and 
 s, Si, S2, . . he corresponding points of involute and evolute, and if we 
 draw right lines sti, st2, . . in the directions of SiPi, S2P2, • • and with 
 a common length = sp, the spherical curve PT1T2 . . will have contact 
 of the second order at p, with the involute PP1P2 • • (pp. 625, 626). 
 
 Article 401. — Calculations abridged, by the treatment of quater- 
 nion differentials (which have hitherto been finite, comp. p. xi.) as 
 infinitesimals ;* new deductions of osculating plane, circle, and sphere, 
 with the vector equation (392) of the circle ; and of the first and se- 
 cond curvatures of a curve in space, 626—630 
 
 Section 7. — On Surfaces of the Second Order; and on 
 
 Curvatures of Surfaces, 630-706 
 
 Article 402. — References to some equations of Surfaces, in earlier 
 
 parts of the Volume, 630, 631 
 
 Article 403. — Quaternion equations of the Sphere (p2 = - 1, &c.), 631-633 
 In some of these equations, the notation N for norm is employed 
 (comp. the Section II. i. 6). 
 
 Article 404. — Quaternion equations of the Ellipsoid, .... 633-635 
 One of the simplest of these forms is (pp. 307, 635) the equation, 
 
 T(tp + pfc) = »e2_t2^ (Ii) 
 
 * Although, for the sake of brevity, and even of clearness, some phrases have 
 been used in the foregoing analysis of the Series 398 and 399, such di^ four-side 
 or five-side contact between cones, and five-point or six-point contact between 
 curves, or between a curve and a surface, which are borrowed from the doctrine 
 of consecutive points and lines, and therefore from that of infinitesimals ; with a 
 few other expressions of modern geometry, such as the plane at infinity, &c. ; 
 yet the reasonings in the text of these Elements have all been rigorously reduced, 
 so far, or are all obviously reducible, to the fundamental conception of Limits : 
 compare the definitions of the osculating circle and sphere, assigned in Articles 
 389, 395. The object of Art. 401 is to make it visible how, without abandoning 
 such ultimate reference to limits, it is possible to abridge calculation, in several 
 cases, by treating (at this stage) the differential symbols, dp, d^p, &c., as if 
 they represented infinitely small differences, Ap, A'^p, &c. ; without taking the 
 trouble to write these latter symbols first, as denoting finite differences, in the 
 rigorous statement of a problem, of which statement it is not always easy to assign 
 the proper form, for the case of points, &c., at finite distances : and then having 
 the additional trouble of reducing the complex expressions so found to simpler forms, 
 in which differentials shall finally appear. In short, it is shown that in Quater- 
 nions, as in other parts of Analysis, the rigour of limits can be combined with 
 the facility of infinitesimals. 
 
CONTENTS, xxvii 
 
 Pages, 
 in which i and k are real and constant vectors, in the directions of 
 
 the ci/ch'c normals. This form (Ii) is intimately connected with, and 
 indeed served to suggest, that Construction of the Ellipsoid (II. i. 13), 
 by means of a Diacentric Sphere and a Point (p. 227, comp. Fig. 53, 
 p. 226), which was among the earliest geometrical results of the Qua- 
 ternions. The three semiaxes, a, b, c, are expressed (comp. p. 230) in 
 terms of i, k as follows : 
 
 a = THT/c; ^=r^^-:^'-y o = Tc-Tk; (!,') 
 
 whence «*-»<; = T (t - /c). (Ii") 
 
 Article 405. — General Central Surface of the Second Order (or 
 central quadric), Sp^p -fp = 1, 636-638 
 
 Article 406. — General Cone of the Second Order (or quadric cone), 
 Sp(pp^fp=0, • . . 638-643 
 
 Article 407. — Bifocal Form of the equation of a central but non- 
 conical surface of the second order : with some quaternion formulae, 
 relaiing to Confocal Surf aces, 643-663 
 
 (a). The bifocal form here adopted (comp. the Section III. ii. 6) 
 is the equation, 
 
 Cfp = (Sap)2 - 2^SapSa'p + (Sa'p)2 + (1 - e^) p2 = C, (Ji) 
 in which, C= (e« - 1) (^ + Saa') l^. (Ji') 
 
 a, a' are two (real) focal unit-lines, common to the whole system of 
 confocals ; the (real and positive) scalar I is also constant for that sys- 
 tem : but the scalar e varies, in passing from surface to surface, and 
 may be regarded as a parameter, of which the value serves to distin- 
 guish one confocal, say (<?), from another (pp. 643, 644). 
 
 (i). The squares (p. 644) of the three scalar semiaxes (real or ima- 
 ginary), arranged in algebraically descending order, are, 
 
 a2 = (e+l)^, i2=(g+Saa')/2, c^ = (e-l)P; (Ki) 
 
 whence ''=-Y~' ^"'^^^^' ^^'^ 
 
 and the three vector semiaxes corresponding are, 
 
 aU(a + a'), iUVatt', cU(a-a'). (Mi) 
 
 (c). Rectangular, unifocal, and cyclic forms (pp. 644, 648, 650), 
 
 of the scalar function fp, to each of which corresponds a form of the 
 
 vector function 0p ; deduction, by a new analysis, of several known 
 
 theorems* (pp. 644, 645, 648, 652, 653) respecting confocal surfaces, 
 
 * For example, it is proved by quaternions (pp. 652, 653), that the focal 
 lines of the focal cone, which has any proposed point p for vertex, and rests on 
 the focal hyperbola, are generating lines of the single- sheeted hyperboloid (of the 
 given confocal system), which passes through that point : and an extension of 
 this result, to the focal lines of any cone circumscribed to a confocal, is deduced 
 by a similar analysis, in a subsequent Series (408, p. 656). But such known 
 theorems respecting confocals can only be alluded to, in these Contents. 
 
XXVIU CONTENTS. 
 
 Pages, 
 and their focal conies ; the lines a, a' are asymptotes to the focal hy- 
 perbola (p. 647), whatever the species of the svirface may be : refe- 
 rences (in Notes to pp. 648, 649) to the Lectures* for the/om^ ellipse 
 of the Ellipsoid, and for several different generations of this last sur- 
 face. 
 
 (<?). General Exponential Transformation (p. 651), of the equation 
 of any central quadric ; 
 
 p = xa + yYa% (Ni), with x-^fa + y^fVYaa = 1, (Ni') 
 ^ (a - ea) JJY aa .„ ,,. 
 
 this auxiliary vector /3 is constant, for any one confocal (e) ; the expo- 
 nent, t, in (Ni), is an arbitrary or variable scalar ; and the coefficientSj 
 X and y, are two other scalar variables, which are however connected 
 with each other by the relation (Ni'). 
 
 (i). If onj fixed value be assigned to t, the equation (Ni) then re- 
 presents the section made by a plane through a (p. 651), which sec- 
 tion is an ellipse if the surface be an ellipsoid, but an hyperbola for 
 either hyperboloid ; and the cutting plane makes with the focal plane 
 of a, a', or with the plane of the focal hyperbola, an angle = J^tt. 
 
 (/). If, on the other hand, we allow t to vary, but assign to 
 X and y any constant values consistent with (Ni'), the equation (Ni) 
 then represents an ellipse (p. 651), whatever the species of the surface 
 may be ; x represents the distance of its centre from the centre o of the 
 surface, measured along the focal line a; y is the radius of a right 
 cylinder, with a for its axis, of which the ellipse is a section, or the 
 radius of a circle in a plane perpendicular to a, into which that ellipse 
 can be oxthogonallj projected : and the angle J^tt is now the excentric 
 anomaly. Such elliptic sections of a central quadric may be otherwise 
 obtained from the unifocal form (c) of the equation of the surface ; 
 they are, in some points of view, almost as interesting as the known 
 circular sections : and it is proposed (p. 649) to call them Centro- 
 Focal Ellipses. 
 
 (g). And it is obvious that, by interchanging the two focal lines 
 a, a' in ((?), a Second Exponential Transformation is obtained, with a 
 Second System of centro-focal ellipses, whereof the proposed surface is 
 the locus, as well as of the first system (/), but which have their 
 centres on the line a', and are projected into circles, on a plane per- 
 pendicular to this latter line (p. 649). 
 
 (A). Equation of Confocals (p. 652), 
 
 Vv,0v, = Yvf,v. (Oi) 
 
 Article 408. — On Circumscribed Quadric Cones; and on the 
 Umbilics of a central quadric, 653-663 
 
 * Lectures on Quaternions (by the present author), Dublin, Hodges and 
 Smith, 1853. 
 
COMTENTS. XXIX 
 
 Pages. 
 {a). Equations (p. 653) of Conjugate Points^ and of Conjugate Di- 
 rections, with respect to the surface /p = 1, 
 
 fdp, p') = 1, (Pi), and/(p, p') = ; (Pi') 
 
 Condition of Contact, of the same surface with the right line pp', 
 
 (/(p,p')-i)^ = (/p-i)(/f>'-i); (QO 
 
 this latter is also a form of the equation of the Cone, with vertex at 
 p', which is circumscribed to the same quadric (/p = 1). 
 
 {b). The condition (Qi) may also he thus transformed (p. 654), 
 FYpp' = aH^c^f<ip-p), (QO 
 
 F being a scalar function, connected with / by certain relations of 
 reciprocity (comp. p. 483) ; and a simple geometrical interpretation 
 may be assigned, for this last equation. 
 
 (c). The Reciprocal Cone, or Cone of Normals a at p',to the circum- 
 scribed cone (Q,i) or (Qi'), may be represented (p. 655) by the very 
 simple equation, 
 
 i?'((r:Sp'(T)=l; (Qi") 
 
 which likewise admits of an extremely simple interpretation. 
 
 (<?). A given right line (p. 656) is touched by two confocals, and 
 other known results are easy consequences of the present analysis ; 
 for example (pp. 658, 659), the cone circumscribed to any surface of 
 the system, from any point of either of the two real focal curves^ is a 
 cone of revolution (real or imaginary) : but a similar conclusion holds 
 good, when the vertex is on the third (or imaginary) focal, and even 
 more generally (p. 663), when that vertex is any point of the (known 
 and imaginary) developable envelope of the confocal system. 
 
 (e). A central quadric has in general Twelve Umbilics (p. 659), 
 whereof only /owr (at most) can be real, and which are its intersections 
 with the three focal curves : and these twelve points are ranged, three by 
 three, on eight imaginary right lines (p. 662), which intersect the circle 
 at infinity, and which it is proposd to call the Eight Umbilicar Ge- 
 neratrices of the surface. 
 
 (jT). These (imaginary) umbilicar generatrices of a quadric are 
 found to possess several interesting properties, especially in relation 
 to the lines of curvature : and their locus, for a confocal system, is a 
 developable surface (p. 663), namely the known envelope (d) of that 
 system. 
 
 Article 409 — Geodetic Lines on Central Surfaces of the Second 
 Order, 664-667 
 
 (a). One form of the general differential equation of geodetics on 
 an arbitrary surface being, by III. iii. 5 (p. 515), 
 
 VvdV = 0, (Ri), if Tdp=: const., (R/) 
 
 this is shown (p. 664) to conduct, for central quadrics, to the first 
 integral, 
 
 p-2^-2 = Ti.2/Udp = /i = const; (Si) 
 
 vfhexe F is the perpendicular from the centre o on the tangent plane, 
 
XXX CONTENTS. 
 
 Pages, 
 and D is tlie (real or imaginary) semidiameter of the surface, which 
 is parallel to the tangent (dp) to the curve. The known equation 
 of Joachimstal, F.B = const., is therefore proved anew ; this last 
 cotistant, however, heing hy no means necessarily real, if the surface 
 be not an ellipsoid. 
 
 (b). Deduction (p. 665) of a theorem of M. Chasles, that the tan- 
 gents to a geodetic, on any one central quadric {e), touch also a common 
 eonfocal (e,) ; and of an integral (p. 666) of the form, 
 
 e\ sin^ vi + e^ cos^ v\ = e, = const. , (Si') 
 
 which agrees with one of M. Liouville. 
 
 (c). Without the restriction (Ri'), the differential of the scalar h 
 in (Si) may be thus decomposed into factors (p. 666), 
 
 dA = d. P-22)-2 = 2Svdjvdp-i. Sj/dp-id2p ; (Si") 
 
 but, by the lately cited Section (III. iii. 5, p. 515), the differential 
 equation of the second order ^ 
 
 Sj^dpd2p=0, (Ri") 
 
 with an arbitrary scalar variable, represents the geodetic lines on any 
 surface : the theorem («) is therefore in this way reproduced. 
 
 (d). But we see, at the same time, by (Si"), that the quantity h, 
 ox P.D = h-\ is constant, not only for the geodetics on a central quadric, 
 but also for a certain other set of curves, determined by the differen- 
 tial equation of the Jirst order, Svdvdp = 0, which will be seen, in the 
 next Series, to represent the lines of curvature. 
 
 Article 41 0. — On Lines of Curvature generally ; and in particu- 
 lar on such lines, for the case of a Central Quadric, 667-674 
 
 (a). The differential equation (comp. 409, («?)), 
 
 Svdvdp = 0, (Ti) 
 
 represents (p. 667) the Zincs of Curvature, upon a,n arbitrary surface ; 
 because it is a limiting form of this other equation, 
 
 SrAi/Ap = 0, (Ti') 
 
 which is the condition of intersection (or of parallelism), of the normals 
 drawn at the extremities of the two vectors p and p + Ap. 
 
 (b). The normal vector v, in the equation (Ti), may be multiplied 
 (pp. .673, 700) by any constant or variable scalar n, without any real 
 change in that equation ; but in this whole theory, of the treatment 
 of Curvatures of Surfaces by Quaternions, it is advantageous to con- 
 sider the expression Srdp as denoting the exact differential of some 
 scalar function of p ; for then (by pp. 486, 487) we shall have an equa- 
 tion of the form, 
 
 dj/ = 0dp = a self -conjugate function of dp, (Ui) 
 
 which iwually involves p also. For instance, we may write generally 
 (p. 669, comp. (R), p. xiii), 
 
 di/ = ^dp+V\dp/^; (Ui') 
 
. CONTENTS. xxxi 
 
 Pages, 
 the scalar g, and the vectors X, fi being real, and being gemrally* func- 
 tions of p, but not involving dp. 
 
 (c). This being understood, the tivo^ directions of the tangent dp, 
 •vrhich satisfy at once the general equation (Ti) of the lines of curva- 
 ture, and the differential equation S^-dp = of the surface, are easily- 
 found to be represented by the two vector expressions (p. 669), 
 
 XJVj/X + UVj/At; (Ti") 
 
 they are therefore generally rectangular to each other, as they have 
 long been known to be. 
 
 (^). The surface itself remaining still quite arbitrary, it is found 
 useful to introduce the conception of an Auxiliary Surface of the Se- 
 cond Order (p. 670), of which the variable vector is p + p', and the 
 equation is, 
 
 Sp>p' = gp'^ -f SXpVp' = 1, (Ui") 
 
 or more generally = const. ; and it is proposed to call this surface, of 
 which the ce^itre is at the given point p, the Index Surface, partly 
 because its diametral section, made by the tangent plane to the given 
 surface at p, is a certain Index Curve (p. 668), which may be consi- , 
 dered to coincide with the known " itidicatrice" of Dupin. 
 
 (e). The expressions (Ti") show (p. 670), that whatever the given 
 surface may be, the tangents to the lines of curvature bisect the angles 
 formed by the traces of the two cyclic planes of the Index Surface (^d), 
 on the tangent plane to the given surface ; these two tangents have 
 also (as was seen by Dupin) the directions of the axes of the Index 
 Curve (p. 668) ; and they are distinguished (as he likewise saw) from 
 all otJier tangents to the given surface, at the given point p, by the 
 condition that each is perpendicular to its own conjugate, with respect ^ 
 to that indicating curve : the equation of such conjugation, of two 
 tangents r and r', being in the present notation (see again p. 668), 
 
 Sr0r' = 0, or Sr^r = 0. (Ui'") 
 
 (/). New proof (p. 669) of another theorem of Dupin, namely 
 that if a developable be circumscribed to any surface, along any curve 
 thereon, its generating lilies are everywhere conjugate, as tangents to 
 the surface, to the corresponding tangents to the curve. 
 
 {g). Case of a central quadric ; new proof (p. 671) of still another 
 theorem of Dupin, namely that the curve of orthogonal intersection 
 (p. 645), of two confocal surfaces, is a line of curvature on each. 
 
 Qi). The system of the eight umbilicar generatrices (j^(i%, (^)), of a 
 central quadi'ic, is the imaginary envelope of the lines of curvature on 
 that surface (p. 671) ; and each such generatrix is itself &.n imaginary 
 
 * For the case of a central quadric, g, X, /i are constants. 
 t Generally two ; but in some cases more. It will soon be seen, that three 
 lines of curvature pass through an wnbilic of a quadric. 
 
XXXll CONTENTS. 
 
 Pages. 
 
 line of curvature thereon : so that through each of the twelve umUlics 
 (see again 408, (e)) there pass three lines of curvature (comp. p. 677), 
 whereof however only one, at most, can be real : namely two genera- 
 trices, and a principal section of the surface. These last results, which 
 are perhaps new, will be illustrated, and otherwise proved, in the 
 following Series (411). 
 
 Article 411. — Additional illustrations and confirmations of the 
 foregoing theory, for the case of a Central* Quadric ; and especially 
 of the theorem respecting the Three Lines of Curvature through an 
 Umbilic, whereof two are always imaginary and rectilinear, .... 674-679 
 
 (a). The general equation of condition (Ti'), or Si/AvAp = 0, for 
 the intersection of two finitely distant normals, may be easily trans- 
 formed for the case of a quadric, so as to express (p. 675), that when 
 the normals at p and p' intersect (or are parallel), the chord pp' is per- 
 pendicular to its own polar. 
 
 (b). Under the same conditions, if the point p be given, the locus 
 of the chord pp' is usually (p. 676) a quadric cone, say (C) ; and there- 
 fore the locus of the point p' is usually a quartic curve, with p for a 
 double poinf, whereat two branches of the curve cut each other at right 
 angles, and touch the two lines of curvature. 
 
 (c). If the point p be one of aprincipal section of the given surface, 
 but not an umbilie, the cone (C) breaks up into a. pair of planes, whereof 
 one, say (P), is the plane of the section, and the other, {F'), is perpen- 
 dicular thereto, and is not tangential to the surface ; and thus the 
 quartic (J) breaks up into a pair of conies through p, whereof one is 
 the principal section itself, and the other is perpendicular to it. 
 
 (<?). But if the given point p be an umbilie, the second plane (P') 
 becom^es a tangent plane to the surface ; and the second conic (/) breaks 
 up, at the same time, into a pair of imaginary f right lines, namely 
 the two umbilicar generatrices through p (pp. 676, 678, 679). 
 
 (e). It follows that the normal pn at a real umbilie p (of an ellip- 
 soid, or a double-sheeted hyperboloid) is not -intersected by any other 
 real normal, except those which are in the same principal section ; but 
 that this real normal pn is intersected, in an imaginary sense, by all 
 the no7'mals p'n', which are drawn at points p' oi either of the two ima- 
 ginary generatrices through the real umbilie p ; so that each of these 
 
 * Many, indeed most, of the results apply, without modification, to the case of 
 the Paraboloids ; and the rest can easily be adapted to this latter case, by the con- 
 sideration of infinitely distant points. We shall therefore often, for conciseness, 
 omit the term central, and simply speak of quadrics, or surfaces of the second 
 order. 
 
 t It is well known that the single-sheeted hyperboloid, which (alone of 
 central quadrics) has real generating lines, has at the same time no real umbilies 
 (comp. pp. 661, 662). 
 
CONTENTS. xxxiii 
 
 Pages. 
 imaginary right lines is seen anew to be a Urn* of curvature^ on the sur- 
 face (comp. 410, (7i)), because all the normals p'n', at points of this 
 line, are situated in one common {imaginary) normal plane (p. 676) : 
 and as before, there are thus three lines of curvature through an um- 
 bilie. 
 
 (/). These geometrical results are in various ways deducible from 
 calculation with quaternions ; for example, a form of the equation of 
 the lines of ctrrvature on a quadric is seen (p. 677) to become an 
 identity at an umbilic (y || \) : while the differential of that equation 
 breaks up into two factors, whereof one represents the tangent to the 
 principal section, while the other (SXd^p = 0) assigns the directions of 
 the two generatrices. 
 
 (g). The equation of the cone (C), which has already presented 
 itself as a certain locus of chords (i), admits of many quaternion 
 transformations ; for instance (see p. 675), it may be written thus, 
 SapAp SaVAp 
 
 SaAp ^ Sa'Ap"""' ^ '^ 
 
 p being the vector of the vertex p, and p + Ap that of any other point 
 p' of the cone ; while a, a' are still, as in 407, (a), two redl focal lines, 
 of which the lengths are here arbitrary, but of which the directions 
 are constant, as before, for a whole confocal system. 
 
 (A). This cone (C), or (Vi), is also the locus (p. 678) of a system 
 
 * It might be natural to suppose, from the known general theory (410, (c)) 
 of the ttvo rectangular directions, that each such generatrix pp' is crossed perpendi- 
 cularly, at every one of its non-umbilicar points p', by a second (and distinct, 
 although imaginary') line of curvature. But it is an almost equally well known 
 and received result of modem geometry, paradoxical as it must at first appear, that 
 when a right line is directed to the circle at infinity, as (by 408, (e)) the gene- 
 ratrices in question are, then this imaginary line is everyivhere perpendicular to 
 itself. Compare the Notes to pages 459, 672. Quaternions are not at all re- 
 sponsible for the introduction of this principle into geometry, but they recognise 
 and employ it, under the following very simple form : that if a non-evanescent 
 vector be directed to the circle at infinity, it is an imaginary value of the symbol Oi 
 (comp. pp. 300, 459, 662, 671, 672) ; and conversely, that ivhen this last symbol 
 represents a vector which is not null, the vector thus denoted is an imaginary line, 
 which cuts that circle. It may be noted here, that such is the case with the reci- 
 procal polar of every chord of a quadric, connecting any two mnbilics which are not 
 in one principal plane' ; and that thus the quadratic equation (XXI., in p. 669) 
 from which the two directions (410, (cj) can usually be derived, becomes an iden- 
 tity for every umbilic, real or imaginary : as it ought to do, for consistency with 
 the foregoing theory of the three lines through that umbUic. And as an addi- 
 tional illustration of the coincidence of directions of the lines of curvature at any 
 non-umbilicar point p' of an umbilicar generatrix, it may be added that the cone 
 of chords (C), in 411, (b), is found to touch the quadric along that generatrix. 
 when its vertex is at any such point p'. 
 
 f 
 
XXXIV CONTENTS. 
 
 Pages. 
 
 of three rectangular lines ; and if it be cut by any plane perpendicular 
 to a side, and not passing through the vertex, the section is an equila- 
 teral hyperbola. 
 
 (i). The same cone (C) has, for three of its sides pp', the normals 
 (p. 677) to the three eonfocals (p. 644) of a given system which pass 
 through its vertex p ; and therefore also, by 410, (^), the tangents 
 to the three lines of curvature through that point, which are the inter- 
 sections of those three eonfocals. 
 
 (/). And because its equation (Vi) does not involve the constant 
 /, of 407, (a), (3), we arrive at the following theorem (p. 678) : — If 
 indefinitely many quadrics. with a common centre o, have their asymp' 
 totic cones biconfocal, and pass through a common point p, their normals 
 at that point have a quadric cone (G) for their locus. 
 
 Article 412 — On Centres of Curvature of Surfaces, .... 679-689 
 
 (a). If a be the vector of the centre s of curvature of a normal 
 section of an arbitrary surface^ which touches one of the two lines of 
 curvature thereon, at any given point p, we have the two fundamental 
 equations (p. 679), 
 
 a = p-{RVv, ("Wi), and i2->dp -1- dUi/ = ; (Wi') 
 
 whence 
 
 VdpdUa/ = 0, (Wi"), and ^+S^ = 0; (Wi'") 
 
 M up 
 
 the equation (Wi") being a new form of the general differential equa- 
 tion of the lines of curvature. 
 
 (J). Deduction (pp. 680, 681, &c.) of some known theorems from 
 these equations ; and of some which introduce the new and general 
 conception of the Index Surface (410, (<?)), as well as that of the 
 known Index Curve. 
 
 (c). Introducing the auxiliary scalar (p. 682), 
 
 in which r (|| dp) is a tangent to a line of curvature, while dv = ^dp, 
 as in (Ui), the two values of r, which answer to the two rectangular 
 directions (Ti") in 410, (c), are given (p. 680) by the expression, 
 
 r = ~ ^r - TX/i . cos {I ~+ L -^), (X'l) 
 
 ^ /*. 
 
 in which ^, X, [x, are, for any given point p, the constants in the equa- 
 tion (Ui") of the index surface; the difference of the tioo curvatures 
 jK"> therefore vanishes at an umbilic of the given surface, whatever the 
 form of that surface may be : that is, at a point, where v || X or || ^, 
 and where consequently the index curve is a circle. 
 
 (d). At any other point p of the given surface, which is as yet en- 
 tirely arbitrary, the values of r may be thus expressed (p. 681), 
 
 n = ar2,r2=ao-2, (Xi") 
 
 ai, &2 being the scalar semiaxes (real or imaginary) of the index curve 
 (defined, comp. 410, {d), by the equations Sp'^p' = 1, S»'p' = 0), 
 
CONTENTS. XXXV 
 
 Pages. 
 (<?). The quadratic equation, of wMch. ri and rg, or the inverse 
 squares of the two last semiaxes, are the roots, maybe ■written (p. 683) 
 under the symhoUeal form, 
 
 Sv-i (^ + r) -ij/ = ; (Yi) 
 
 ■which may be developed (same page) into this other form, 
 
 y2 + rSv-ixi' + Sv-J 1//J/-0, (Y'l) 
 
 the linear and vector functions, i// and x> being derived from the func- 
 tion <p, on the plan of the Section III. ii. 6 (pp. 440, 443). 
 
 (/). Hence, generally, the product of the two curvatures of a sur- 
 face is expressed (same p. 683) by the formula, 
 
 JSi-ii^a-i = n ^-z Ti/ -» = - S — ii/ — ; (Zi) 
 
 V V 
 
 •which -will be found useful in the foUo-wing series (413), in connexion 
 ■with the theory of the Measure of Curvature. 
 
 {g). The given surface being still quite general, if ■we ■write 
 (p. 686), 
 
 r = TJdjO, r' = U (I'dp), (A2), and therefore tt = Uv, ' (A'2) 
 so that T and r' are unit tangents to the lines of curvature, it is easily 
 proved that 
 
 dr' = rSr'dr, (B2), or that Yrdr' = 0; (B'2) 
 
 this general parallelism of dr to r being geometrically explained, by 
 observing that a line of curvature oti any surface is, at the same time, 
 a line of curvature on the developable normal surface, -which rests upon 
 that line, and to -which r' or vt is normal, if r be tangential to the 
 line. 
 
 (A). If the vector of curvature (389) of a line of curvature be 
 projected on the normal v to the given surface, the projection 
 (p. 686) is the vector of curvature of the normal section of that sur- 
 face, -which has the same tangent r ; but this result, and an analo- 
 gous one (same page) for the developable normal surface {g), are 
 virtually included in Meusnier's theorem, -which -will be proved by 
 quaternions in Series 414. 
 
 if). The vector a of a centre s of curvature of the given surface, 
 ans-wering to a given point p thereon, may (by (Wi) and (Xi)) be ex- 
 pressed by the equation, 
 
 (T = p + r-iv; (C2) 
 
 •w^hich may be regarded also as a general form of the Vector Equation 
 of the Surface of Centres, or of the locus of the centre s : the vari- 
 able vector p of the point p of the given surface being supposed (p. 501) 
 to be expressed as a vector function of two independent and scalar 
 variables, whereof therefore v, r, and <t become also functions, 
 although the two last involve an ambiguous sign, on account of the 
 Two Sheets of the surface of centres. 
 
 {j ). The normal at s, to -what may be called the First Sheet, has 
 the direction of the tangent r to -what may (on the same plan) be 
 called the First Line of Curvature at r ; and the vector v of the point 
 
XXXvi CONTENTS. 
 
 corresponding to 8, on tho corresponding sA^^^ of tho Heeiproeal (comp. 
 pp. 607, 508) of the Surface of Centres^ has (by p. 684) the expres- 
 sion, 
 
 i; = r(Spr)-i; (Dg) 
 
 which may also be considered (comp. (ff) to be a form of the Vector 
 Equation of that Reciprocal Surface. 
 
 (Jc). The vector v satisfies generally (by same page) the equations 
 of reciprocity, 
 
 Su(r = Say = l, Su5<r = 0, Sc^u = 0, (Dg') 
 
 ^(T, 5v denoting any infinitesimal variations of the vectors o and v, 
 consistent with the equations of the surface of centres audits recipro- 
 cal, or any linear and vector elements of those two surfaces, at two 
 corresponding points ; we have also the relations (pp. 684, 685), 
 
 Spv=l, Sj/v = 0, Si/v0u = O. (D2") 
 
 {t). The equation Sv (w - p) = 0, or more simply, 
 
 Svw = 1, (E2) 
 
 in which w is a variable vector, represents (p. 684) the normal plane 
 to \h.Q first line (/) of curvature at p ; or the tangent plane at s to the 
 first sheet of the surface of centres : or finally, the tangent plane to 
 that developable normal surface (y), which rests upon the second line of 
 curvature, and touches the first sheet oion^ a certain mrw, whereof we 
 shall shortly meet with an example. And if v be regarded, comp. («), 
 as a vector function of two scalar variables, the envelope of the variable 
 2)lane (E2) is a sheet of the surface of centres ; or rather, on account of 
 tho ambiguous sign {i\ it is that surface of centres itself : while, in 
 like manner, the reciprocal surface (j) is the envelope of this other 
 
 Pages. 
 
 S(Ta> = 1. (E2') 
 
 (m). The equations (Wi), (Wi) give (comp. the Note to p. 684), 
 
 d(T=di2.Uv; (F2) 
 
 combining which with (C2), we see that the equations (Hi) of p. xxv. 
 are satisfied, when the derived vectors p' and tr' are changed to the cor- 
 responding differentials, dp and d<r. The known theorem (of Monge), 
 that each Line of Curvature is generally an involute, with the corre- 
 sponding Curve of Centres for one of its evolutes (400), is therefore in 
 this way reproduced : and the connected theorem (also of Monge), 
 that tJiis evolute is a geodetic on its oicn sheet of the surface of centres, 
 follows easily from what precedes. 
 
 (n). In the foregoing paragraphs of this analysis, the given sur- 
 face has throughout been arbitrary, or general, as stated in {d) and 
 (^). But if we now consider specially the case of a central quadric, 
 several less general but interesting results arise, whereof many, but 
 perhaps not all, are known ; and of which some may be mentioned 
 here- 
 
CONTENTS. xxxvii 
 
 Pages, 
 (o). Supposing, then, that not only dj/ = ^d/o, but also v — 0p, and 
 
 Spv =fp = 1, the Index Surface (410, {d)) becomes simply (p. 670) the 
 given surface, with its centre transported from o to p ; whence many 
 simplifications foUow. 
 
 (p). For example, the semiaxes ai, 0.2 of the index curve arc now 
 equal (p. 681) to the semiaxes of the diametral section of the given 
 surface, made by a plane parallel to the tangent plane ; and Tv is, as 
 in 409, the reciprocal P-i of the perpendicular^ from the centre on this 
 latter plane ; whence (by (Xi) and Xi")) these known expressions 
 for the two* curvatures result : 
 
 iJi-i = Par* ; /?2-> = i^3-'. (G2) 
 
 (§'). Hence, by (e), if a neio surface be derived from a given cen- 
 tral quadric (of any species'), as the locus of the extremities of normals, 
 erected at the centre, to the planes of diametral sections of the given 
 surface, each such normal (when real) having the letigth of one of the 
 semiaxes of that section, the equation of this new surf ace f admits 
 (p. 683) of being written thus : 
 
 Sp(0-p-2)-ip = o. (H2) 
 
 (r). Under the conditions (o),the expression (C2) for a gives (p. 684) 
 the two converse forms, 
 
 (r = r-i(^ + r)p, (I2), p = r(0 + y)-i<T; (I2') 
 
 whence (pp. 684, 689), 
 
 v = r(^ + r)-i^(r, (J2), cr = (^-1 +^"0 1^; (J2') 
 
 and therefore (p. 689), by (<?), (p), and by the theory (407) of con- 
 focal surfaces, 
 
 <Tl = (p2'^V = 02" ' 0p, (K3) 
 
 if 02 be formed from (p by changing the semiaxes abc to ^252^2 ; it 
 being understood that the given quadric (abc) is cut by the two confo- 
 cals (^aibiCi) and {aib^e^), in the first and second lines of curvature 
 through the given point p : and that <ri is here the vector of that^rs^ 
 centre s of curvature, which answers to the first line (comp. (y ). Of 
 course, on the same plan, we have the analogous expression. 
 
 * Throughout the present Series 412, we attend only (comp. («)) to the curva- 
 tures of the two normal sections of a surface, which have the directions of the two 
 lines of curvature : these being in fact what are always regarded as the tivo princi- 
 pal curvatures (or simply as the two curvatures) of the surface. But, in a shortly 
 subsequent Series (414), the more general case will be considered, of the curva- 
 ture of any section, normal or oblique. 
 
 t When the given surface is an ellipsoid, the derived SMriQ.CQ is the celebrated 
 Wave Surface of Fresnel : which thus has (H2) for a symbolical form of its equa- 
 tion. When the given surface is an hyperboloid, and a semiaxis of a section is 
 imaginary, the (scalar and now positive) square, of the (imaginary) normal erected, 
 is still to be made equal to the square of that semiaxis. 
 
Pages. 
 
 XXXVUl CONTENTS. 
 
 (T2= ^1-1^ = ^1-1^/3, (K2') 
 
 for the vector of the second centre. 
 
 (s). These expressions for «ri, 02 include (p. C89) a theorem of Dr. 
 Salmon, namely that the centres of curvature of a given quadric at a 
 given point are ihe poles of the tangent plane, with respect to the two 
 confocals through that point ; and either of them may he regarded, 
 by admission of an ambiguous sign (comp. (?)), as a new Vector Form* 
 of the Equation of the Surface of Centres, for the case (0) of a given 
 central quadric. 
 
 (t). In connexion with the same expressions for ci, (T2, it may be 
 observed that if ri, r^ be the corresponding values of the auxiliary 
 scalar r in (c), and if r, r' stiU denote the unit tangents (g) to the 
 first and second lines of curvature, while abc, aibiCi, and a2hc2 retain 
 their recent significations (r), then (comp. pp. 686, 687, see also p. 
 
 652), 
 
 n =fT =fTJdp = (a-i - fl22)-» = &c., (L2) 
 
 and rz ^fr' =f'Uvdip = (a^ — «i2)-i = &c. ; (L2') 
 
 this association of ri and ci with 02, &c., and of r^ and 02 with ai, 
 Sec, arising from the circumstance that the tangents t andr' have re- 
 spectively the directions of the normals vz and vi, to the two confocal 
 surfaces, (ozhcz) and (aihiCi'). 
 
 (ti). By the properties of such surfaces, the scalar here called rz is 
 therefore constant, in the whole extent of o. first line of curvature ; 
 and the same constancy of r^, or the equation, 
 
 d/Ui/dp = 0, (M2) 
 
 may in various ways be proved by quaternions (p. 687). 
 
 (w). "Writing simply r and r' for ri and r^, so that r' is constant, 
 but r variable, for afrst line of curvature, while conversely r is con- 
 stant and r' variable for a second line, it is found (pp. 684, 685, 586), 
 that the scalar equation of the surface of centres (i) may be regarded 
 as the result of the elimination of r-i between the two equations, 
 I = S.er(l + r-V)-2^(T, (N2), and = S.(r (l + y-i^)-^^^; (N2O 
 whereof the latter is the derivative of the former, with respect to the 
 scalar r'K It follows (comp. p. 688), that the First Sheet of the Sur- 
 face of Centres is touched hy an Auxiliary Quadric (N2), along a Quartic 
 Curve (N2) (N'2')» which curve is the Locus of the Centres of First Cur- 
 vature, for all the points of a Line of Second Curvature ; the same 
 sheet being also touched (see again p. 688), along the same curve, by 
 the developable normal surface (I), which rests on the same second line : 
 with permission to interchange the words, frst and second, through- 
 out the whole of this enunciation. 
 
 (tv). The given surface being still a central quadric (0), the vec- 
 tors p, (T, V can be expressed as functions of v (comp. {j) (/t) (t)), 
 
 * Dr. Salmon's result, that this surface of centres is of the twelfth degree, may 
 be easily deduced from this form. 
 
CONTENTS. XXXix 
 
 Pages, 
 and conversely the latter can be expressed as a function of any one of 
 the former ; we have, for example, the reciprocal equations (p. 685), 
 ff={l+r'Hy^-% (O2), and t; = (l + y-i^)-2 ^(t ; (O2') 
 from which last the formula (N2) may be obtained anew, by observ- 
 ing (A-) that Scru = 1. Hence also, by (r), we can infer the expres- 
 sions,* 
 
 p = (^ -I + r-i) u = 02 "^ V, (P2), and v = (p^p = v%; (P2') 
 
 and in fact it is easy to see otherwise (comp. p. 645), that vg |1 r jj v, 
 and Spj/2 = 1 = Spy, whence V2=^vqs, before. 
 
 {x). More fully, the two sheets of the reciprocal (/) of the surface 
 of centres may have their separate vector equations written thus, 
 
 vi = 02 10 = v%, V2 = 0ip = vi ; (Pa") 
 
 and the scalar equationf of this reciprocal surface itself, considered 
 as including both sheets, may (by page 685) be thus written, the func- 
 tions/and i?' being related as in 408, (i), 
 
 t;4 = (i^y-l)/v, (Q2) 
 
 with several equivalent forms ; one way of obtaining this equation 
 being the elimination of r between the two following (same p. 685) : 
 
 Fu + r-^v^ = l, (Q2') ; fv + rv^ = 0. (Q2") 
 
 (y). The two last equations may also be written thus, for the^rst 
 sheet of the reciprocal surface, 
 
 F2 VI = 1, (Hz), and/Uvi = r, (RgO 
 
 in which (comp. pp. 685, 689), 
 
 Fzv = S V02 -iv = Su (0-1 +r-i) v ; (R2") 
 
 and accordingly (comp. pp. 483, 645), we have F2V2=Fv=\, oxi^ 
 /Uj/2=/r = n 
 
 (z). For a line of second curvature on the given surface, the scalar 
 r is constant, as before ; and then the two equations (0,2') » (Q-s'Oj or 
 (R2), (Il'2), represent jointly (comp. the slightly different enunciation 
 in p. 688) a certain quartic curve, in which the quadric reciprocal (^-2), 
 of the second confocal {0,2 h ^2)* intersects the first sheet (j/) of the Re- 
 ciprocal Surface (Q2) ; this quartic curve, being at the same time the 
 intersection of the quadric surface (Q2') or (R2), with the quadric cone 
 (Qa") or (R2')) which is biconcyclic with the given quadric, fp= 1. 
 
 * The equation v = V2,= the normal to the confocal (^2 h C2) at r, is not ac- 
 tually given in the text of Series 412 ; but it is easily deduced, as above, from 
 the formulge and methods of that Series. 
 
 t The equation (Q2) is one oi^Q fourth degree; and, when expanded by co- 
 ordinates, it agrees perfectly with that which was first assigned by Dr. Booth 
 (see a Note to p. 685), for the Tangential Equation of the Surface of Centres of a 
 quadric, or for the Cartesian equation of the Reciprocal Surface, 
 
xl CONTENTS. 
 
 Pages. 
 
 Article 413. — On the Measure of Curvature of a Surface, . . 689-693. 
 
 The object of this short Series 413 is the deduction by quaternions, 
 somewhat more briefly and perhaps more clearly than in the Lectures, 
 of the principal results of Gauss (comp. Note to p. G90), respecting 
 the Measure of Curvature of a Surface, and questions therewith con- 
 nected. 
 
 (a). Let p, Pi, P2be any three near points on a given but arbitrary 
 surface, and n, Ri, R3 the three corresponding points (near to each other) 
 on the U/nit spliere, which are determined by the parallelism of the radii 
 OR, ORi, 0R2 to the normals pn, piNi, P2 N2 ; then the areas of the two 
 small triangles thus formed will bear to each other the ultimate ratio 
 p. 690), 
 
 .. ARR1R2 V.dUi/^Ui/ a 1 , 1 /QN 
 
 lim. = =rr-r = _ S — ^Z — ; (S2) 
 
 APP1P2 Yapcp V V 
 
 whence, with Gauss's definition of the measure of ctirvature, as the 
 ultimate ratio of corresponding areas on surface and sphere, we have, by 
 the formula (Zi) in 412, (/), hia fundamental theorem, 
 
 Measure of Curvature — Hi "' R<i "', (Sg') 
 
 = Product of the two Principal Curvatures of Sections. 
 
 (b). If the vector p of the surface be considered as a function of 
 two scalar variables, t and u, and if derivations with respect to these 
 be denoted by upper and lower accents, this general transformation 
 results (p. 691), 
 
 Measure of Curvature =S^S^'-(S^^, (T2) 
 
 V V \ V j 
 
 in which v = Npp, ; (T2') 
 
 with a verification for the notation pqrst of Monge. 
 
 (c). The square of a linear element d«, of the given but arbitrary 
 surface, may be expressed (p. 691) as follows : 
 
 ds2 = (Tdp2 =) edif^ + 2/dMw + ^dw^ ; (U2) 
 
 and with the recent use (J) of accents, the measure (T2) is proved 
 (same page) to be an explicit function of the ten scalars, 
 
 ^yfy9\ e\f\9'\ ^.J.^g.\ and e,-1f:^g"; (U2') 
 
 the form of this function (p. 692) agreeing, in all its details, with the 
 corresponding expression assigned by Gauss. * 
 
 (<?). Hence follow at once (p. 692) two of the most important 
 results of that great mathematician on this subject; namely, that 
 every Reformation of a Surface, consistent with the conception of it as 
 an infinitely thin and flexible but inextensible solid, leaves unaltered, 
 
 * References are given, in Notes to pp. 690, «&;c. of the present Series 413, 
 to the pages of Gauss's beautiful Memoir, " Bisquisitiones generales circa Superfi- 
 cies Curvas,^^ as reprinted in the Additions to Liouvillo's Monge. 
 
CONTENTS. xK 
 
 Pages. 
 1st, the Measure of Curvature at any Point, and Ilnd, the Total 
 Curvature of any Area : this last being the area of the corresponding 
 portion (a) of the unit-sphere. 
 
 (e). By a suitable choice of t and ti, as ccTtsdn yeodetic co-ordinates, 
 the expression (Uo) naay be reduced (p. 692) to the following, 
 
 ds2 = d^2 ^ y,2(1^^2 . (-XJ2") 
 
 where ^ is the length of a geodetic arc ap, from a fixed point a to a 
 variable point p of the surface, and u is the angle bap which this 
 variable arc makes with a fixed geodetic ab : so that in the immediate * 
 neighbourhood of a, we have n=t, and n' — Dtn = 1. 
 
 (/). The general expression (c) for the measure of curvature takes 
 thus the very simple form (p. 692), 
 
 i?i-i J?2-i =r - n-^n" = - n-^Dt^n ; (V2) 
 
 and we have (comp. (^)) the equation (p. 693), 
 
 Total Curvature of Area apq, = Aw - J w'dw ; (Vg') 
 
 this area being bounded by two geodetics, ap and aq, which make with 
 each other an angle = Am, and by an arc pq, of an arbitrary curve on 
 the given surface, for which t, and therefore n, may be conceived to 
 be a given function of u. 
 
 (jl'). If this arc pq be itself a geodetic, and if we denote by v the 
 variable angle which it makes at p with ap prolonged, so that tan v 
 = ndu:dt, it is found that df = - ^/dw ; and thus the equation (V2') 
 conducts (p. 693) to another very remarkable and general theorem of 
 Gauss, for an arbitrary surface, which may be thus expressed, 
 
 Total Curvature of a Geodetic Triangle abc = a+b + c — tt, (V2") 
 
 = what may be called the Spheroidal Excess of that triangle, the total 
 area (47r) of the unit-sphere being represented by eight right angles : 
 with extensions to Geodetic Polygons, and modifications for the case of 
 what may on the same plan be called the Spheroidal Defect, when the 
 two curvatures of the surface are oppositely directed. 
 
 Article 414. — On Curvatures of Sections (Normal and Oblique) 
 of Surfaces ; and on Geodetic Curvatures, 694-698 
 
 (a). The curvatures considered in the two preceding Series hav- 
 ing been those of the principal normal sections of a surface, the present 
 Scries 414 treats briefly the more general case, where the section is 
 made by an arbitrary plane, such as the osculating plane at p to an 
 arbitrary curve upon the surface. 
 
 (J>). The vector of curvature (389) of any such curve or section 
 being (p - k)-i = T>s'^p, its normal and tangential components are found 
 to be (p. 694), 
 
 (p - (t)-i = y-^S -^ = (p - (Ti)-i cos2 V + {p - (To)-! sin2 v, (Wa) 
 
 and (p - !)-!= j/-'dp-iSj/dp-id2p ; (W2') 
 
 the former component being the Vector of Normal Curvature of the 
 
 g 
 
xlii CONTENTS. 
 
 Pages. 
 Surface, for the direction of the tangent to the curve : and the latter 
 being the Vector of Geodetic Curvature of the same Curve (or section). 
 
 (c). In the foregoing expressions, <r and ^ are the vectors of the 
 points s and x, in which the axis of the osculating circle to the curve 
 intersects respectively the normal and the tangent plane to the sur- 
 face (p. 694) ; s is also the centre of the sphere, which osculates to 
 the surface in the direction dp of the tangent ; ci, (Xz are the vectors 
 of the two centres Si, S2, of curvature of the surface, considered in Se- 
 ries ^2, which are at the same time the centres of the two osculating 
 spheres, of which the curvatures are (algebraically) the greatest and 
 least : and v is the angle at which the curve here considered crosses 
 the^rs^ line of curvature. 
 
 (d). The equation (W2) contains a theorem of Euler, under the 
 form (p. 695), 
 
 E- 1 = i?i- 1 cos2 v + i22-5 sin2 v ; (W2") 
 
 it contains also Meusnier's theorem (same page), under the form 
 (comp. 412, (7i)) that the vector of normal ctcrvature (J) of a surface, 
 for any given direction, is the projection on the normal v, of the vector 
 of oblique curvature, whatever the inclination of the plane of the sec- 
 tion to the tangent plane may be. 
 
 (e). The expression (W2'), for the vector of geodetic curvature, ad- 
 mits (p. 697) of various transformations, with corresponding expres- 
 sions for the radius T(p — ^) of geodetic curvature, which is also the 
 radius of plane curvature of the developed curve, when the developable 
 circumscribed to the given surface along the given curve is unfolded 
 into a plane : and when this radius is constant, so that the developed 
 curve is a circle, or part of one, it is proposed (p. 698) to caU the given 
 curve &Didonia (as in the Lectures), from its possession of a certain iso- 
 perimetrical property, which was first considered by M. Delaunay, 
 and is represented in quaternions by the formula (p. 697), 
 
 ^JS(U»/.dp^jo) + c^JTdp = 0; (X2) 
 
 or c-1 dp = V(U J/ . dUdp), (X'2) 
 
 by the rules of what may be called the Calculus of Variations in Qua- 
 ternions : c being a constant, which represents generally (p. 698) 
 the radius of the developed circle, and becomes infinite for geodetic 
 lines, which are thus included as a case of JDidonias. 
 
 Article 41 5. — Supplementary Remarks, 698-706 
 
 (a). Simplified proof (referred to in a Note to p. xii), of the gene- 
 ral existence of a system oi three real and rectangular directions, which 
 satisfy the vector equation Yp<pp = 0, (P), when ^ is a linear, vector, 
 and self-conjugate function ; and of a system of three real roots of the 
 cubic equation M=Q (p. xii), under the same condition (pp. 698- 
 700). 
 
 (h). It may happen (p. 701) that the differential equation, 
 
 S»'dp = 0, (Y2) 
 
CONTENTS. xliii 
 
 Pages, 
 is inteffrable, or represents a system of surfaces, without the expression 
 Svd/o being an exact differential, as it was in 410, (b). In this case, 
 there exists some scalar /ac^or, n, such that S^^vdp is the exact diffe- 
 rential of a scalar function of p, without the assumption that this vec- 
 tor p is itself 0. function of a scalar variable, t; and then if we write 
 (pp. 701, 702, comp. p. xxx), 
 
 div = ^dp, d . wv = (idp, (Y2') 
 
 this new vector function d) will be self-conjugate, although the function 
 is not such now, as it was in the equation (Ui). 
 
 (tf). In this manner it is found (p. 702), that the Condition* ofln- 
 tegrability of the equation (Y3) is expressed by the very simple for- 
 mula, 
 
 Syv=0; (Y2") 
 
 in which y is a vector function of p, not generally linear, and deduced 
 from ^ on the plan of the Section Ill.^ii. 6 (p. 442), by the relation, 
 0dp-fdp = 2Vydp; (Ya"') 
 
 0' being the conjugate of <j), but not here equal to it. 
 
 (d). Connexions (pp. 702, 703) of the Mixed Transformations in 
 the last cited Section, with the known Modular and Umbilicar Gene- 
 rations of a surface of the second order. 
 
 (/). The equation (p. 704), 
 
 T(p-V.^Vya) = T(a-V.yV/3p), (Z,) 
 
 in which a, (3, y are ant/ three vector constants, represents a central 
 quadric, and appears to offer a new mode of generation\ of such a sur- 
 face, on which there is not room to enter, at this late stage of the 
 work. 
 
 (/). The vector of the centre of the quadric, represented by the 
 equation /p - 2S£p = const., with /p = Sp^p, is generally k = ^-'f 
 = m"it//f (p. 704) ; case oi paraboloids, and of cylinders. 
 
 (g). The equation (p. 705), 
 
 ^qpq'pq'p + Sp^p + Syp + C = 0, (Z2') 
 
 represents the general surface of the third degree, or briefly the General 
 Cubic Surface ; C being a constant scalar, y a constant vector, and q, 
 q', q" three constant quaternions, while ^p is here again a linear, 
 vector, and self-conjugate function of p. 
 
 (Ji). The General Cubic Cone, with its vertex at the origin, is thus 
 represented in quaternions by the monomial equation (same page). 
 
 * It is shown, in a Note to p. 702, that this monomial equation (^"-i) be- 
 comes, when expanded, the known equation of six terms, which expresses the con- 
 dition of integrability of the differential equation ^;daJ4- g'd?/ + rdz = 0. 
 
 t In a Note to p. 649 (akeady mentioned in p. xxviii), the reader will find 
 references to the Lectures, for several different generations of the ellipsoid, derived 
 from quaternion forms of its equation. 
 
xliv CONTENTS. 
 
 Pages. 
 Sqpq'pq'p = 0. (Z-i") 
 
 (t). Scretv Surface, Screw Sections (p. 705) ; Skew Centre ofS/cew 
 Arch, with illustration by a diagram (Fig. 85, p. 708). 
 
 Section 8. — On a few Specimens of Physical Applications 
 
 of Quaternions, with some Concluding Remarks, 707 to the end. 
 
 Article 416.— On the Statics of a Rigid Body, 707-709 
 
 («). Equation of Equilibrium, 
 
 Vr2/3 = SVa^; (Ag) 
 
 each a is a vector of application ; (3 the corresponding vector of applied 
 force ; y an arbitrary/ vector : and this one quaternion formula (A3) 
 is equivalent to the system of the six usual scalar equations 
 
 (X = 0, r= 0, ^= 0, 2 = 0, M= 0, ]sr= o). 
 
 (A.) When S (2/3. SVaiS) = 0, (B3), but not ^(5 = 0, (C3) 
 the applied forces have an unique resultant = 2^3, which acts along 
 the Hne whereof (A3) is then the equation, with y for its variable 
 vector. 
 
 (c). When the condition (C3) is satisfied, the forces compound 
 themselves generally into one couple, of which the nxis='S,Ya(3, what- 
 ever may be the position of the assumed origin o of vectors. 
 
 (d). When 2 V«/3 = 0, (D3), with or without (C3), 
 the forces have no tendency to turn the body round that point o ; and 
 when the equation (A3) holds good, as in (a), for an arbitrary/ vector 
 y, the forces do not tend to produce a rotation* round anf/ point c, 
 so that they completely balance each other, as before, and both the 
 conditions (C3) and (D3) are satisfied. 
 
 (e). In the general case, when neither (C3) nor (D3) is satisfied, if g 
 be an auxiliary quaternion, such that 
 
 j2/3 = 2Va/3, (E3) 
 
 then \g is the vector perpendicular from the origin, oa the central 
 axis of the system ; and if c = S-7, then c2/3 represents, both in quan- 
 tity and in direction, the axis of the central couple. 
 
 (/). If Q be another auxiliary quaternion, such that 
 
 Q2/3 = 2fl/3, (F3) 
 
 with T2/3 > 0, then SQ = c = central moment divided by total force ; 
 
 * It is easy to prove that the moment of \!ne force (3, acting at the end of the 
 vector a from o, and estimated with respect to any unit-line i from the same ori- 
 gin, or the energy with which the force so acting tends to cause the body to turn 
 round that line t, regarded as a. fixed axis, is represented by the scalar, - Sfa/3, or 
 St"ia^; so that when the condition (D3) is satisfied, the applied forces have no 
 tendency to produce rotation round any axis through the origin : which origin 
 becomes an arbitrary ^jo in t c, when the equation of equilibrium (A3) holds good. 
 
CONTENTS. 
 
 and V^ is the vector y of a point c ujjon the central axis which does 
 not vary Math the origin o, and which there are reasons for considering 
 as the Central Foint of the system, or as the general centre of applied 
 forces : in fact, for the case of parallelism, this point c coincides with 
 what is usually called the centre of parallel forces. 
 
 (^g). Conceptions of the Total Ifoment Iia(3, regarded as being ge- 
 nerally a quaternion ; and of the Total Tension, — Sa/3, considered as 
 a scalar to which that quaternion with its sign changed reducesitself 
 for the case of equilibrium (a), and of which the value is in that 
 case independent of the origin of vectors. 
 
 (A). Frinciple of Virtual Velocities, 
 
 ^S(3Sa = 0, (G3) 
 
 Article 417. — On the Dynamics of a Eigid Body, 
 
 {a). General Eqication of Bgnamics, 
 
 2wS(Di2a-4)^a = 0; (H3) 
 
 the vector ^ representing the accelerating force, or m% the moving 
 force, acting on a particle m of which the vector at the time z! is a ; 
 and ha being any infinitesimal variation of this last vector, geometri- 
 cally compatible with the connexions between the parts of the 
 system, which need not here be a rigid one. 
 
 (5). For the case oi^free system, we may change each ^a to e + Vta, 
 £ and t being any two infinitesimal vectors, which do not change in 
 passing from one particle m to another ; and thus the general equa- 
 tion (H3) furnishes two general vector equations, namely, 
 
 2w (Di2a - ^) = 0, (I3), and 2mVa (D^^a - ?) = ; (J3) 
 which contain respectively the law of the motion of the centre of 
 gravity, and the law of description of areas, 
 
 {c). If a body be supposed to be rigid, and to have o, fixed point 
 o, then only the equation (J3) need be retained ; and we may write, 
 D<a=Vta, (K3) 
 
 t being here o. finite vector, namely the Vector Axis of Instantaneous 
 Rotation : its versor TJt denoting the direction of that axis, and its 
 tensor Tt representing the angular velocity of the body about it, at the 
 time t. 
 
 {d). "When the forces vanish, or balance each other, or compound 
 themselves into a single force acting at the fixed point, as for the case 
 of a heavy body turning freely about its centre of gravity, then 
 
 SwVa4 = 0, (L3) ; and if we write, ^i='2maYai, (M3) 
 so that (p again denotes a linear, vector, and self- conjugate function, 
 we shall have the equations, 
 
 0Dii + V*0i=O, (N3); 0t+r = O, (O3); St0t=A2; (P3) 
 whence Siy + h^ = 0, (Q3), and 0D<t = Vty; (E3) 
 
 the vector y being what we may call the Constant of Areas, and the 
 scalar h^ being the Constant of Living Force. 
 
 xlv 
 Pages. 
 
 709-713 
 
xlvi CONTENTS. 
 
 (e). One of Poinsot's representations of the motion of a body, under 
 tlie circumstances last supposed, is thus reproduced under the form, 
 that the Ellipsoid of Living Force (P3), with its centre at ilnQ fixed 
 point o, rolls witJiout gliding on the f zed plane (Q3), which is parallel 
 to the Plane of Areas (Sty = 0) ; the variable semidiameter of contact^ 
 I, being the vector-axis (c) of instantaneous rotation of the body. 
 
 (/). The Moment of Inertia, with respect to ang axis i through 0, 
 is equal to the living force (Ji^) divided by the square (Tt^) of the 
 semidiameter of the ellipsoid (P3), which has the direction of that axis ; 
 and hence may be derived, with the help of the first general construc- 
 tion of an ellipsoid, suggested by quaternions, a simple geometrical 
 representation (p. 711) of the square-root of the moment of inertia 
 of a body, with respect to any axis ad passing through a given point 
 A, as a certain right line bd, if cd = ca, with the help of two other 
 points B and c, which are likewise fixed in the body, but may be 
 chosen in more ways than one. 
 
 (y). A cone of the second degree, 
 
 Stj/=0, (S3), with V = -y^^t _ ^202t^ (T3) 
 
 ia fixed in the body, but rolls in space on that other cone, which is the 
 locKs of the instantaneous axis i ; and thus a second representation, 
 proposed by Poinsot, is found for the motion of the body, as the rolling 
 of one cone on another. 
 
 (A). Some of Mac Cullagh's results, respecting the motion here 
 considered, are obtained with equal ease by the same quaternion 
 analysis ; for example, the line y, although fxed in space, describes 
 in the body an easily assigned cone of the second degree (p. 712), which 
 cuts the reciprocal ellipsoid, 
 
 Sy0-iy = A2, (U3) 
 
 in a certain sphero-conic : and the cone of normals to the last men- 
 tioned cone (or the locus of the line t + h^y-^) rolls on the plane of areas 
 (Sty = 0). 
 
 (0- The Three {Frincipat) Axes of Inertia of the body, for the 
 given point o, have the directions (p. 712) of the three rectangular and 
 vector roots (comp. (P), p. xii., and the paragraph 415, (a), p. xlii.) 
 of the equation 
 
 Vt^i= 0, (V3), because, for each, D<t = ; (V3') 
 
 and if ^, B, C denote the three Principal Moments of inertia corre- 
 sponding, then the Symbolical Cubic in (comp. the formula (N) in 
 page xii.) may be thus written, 
 
 (0 \A) (^ + ^) (0 + C) = 0. (W3) 
 
 (». Passage (p. 713), from moments referred to axes passing 
 through a given point o, to those which correspond to respectively 
 parallel axes, through any other point Q of the body. 
 
 Pages 
 
. CONTENTS. xlvii 
 
 Pages. 
 Article 418. — On the motions of a System of Bodies, considered 
 as free particles m, m, . . whicli attract each other according to the 
 
 law of the Inverse Square 713-717 
 
 (a). Equation of motion of the system, 
 
 SmSD^Sa^a + ^P= 0, (X3), if P= 2mm'T (a - a')"' ; (Y3) 
 a is the vector, at the time t, of the mass or particle m ; P is the po- 
 tential (jav force-function) ; and the infinitesimal variations ^a are ar- 
 bitrary. 
 
 (i). Extension of the notation of derivatives, 
 
 dP= 2S (DaP. Sa). (Z3) 
 
 (<?), The differential equations of motion of the separate masses 
 m, . . become thus, • 
 
 mDt2a+DaP=0, . . ; (A4) 
 
 and the laws of the centre of gravity, of areas, and of living force, 
 are obtained under the forms, 
 
 2mD<a = /3, (B4); 2MVaD<a = y; (C4) 
 
 and r=-i5:w(D<a)2=P+^; (d/) 
 
 (3, y being two vector constants, and S a scalar constant 
 (d). Writing, 
 
 P= r (P+ T) df, (E4), and r= r 2 Pdi{ = P + tR, (Ft) 
 
 F may be called the Principal* Function, and V the Characteristic 
 Function, of the motion of the system ; each depending on the final 
 vectors of position, a, a', . . and on the initial vectors, uq, a'o, . . ; but 
 F depending also (explicitly) on the time, t, while V (= the Action^ 
 depends instead on the constant JBTof living force, in addition to those 
 final and initial vectors : the masses m, m', . . being supposed to be 
 known, or constant. 
 
 (e). We are led thus to equations of the forms, 
 
 mBta + DaP= 0, . . (G4) ; -mB^a + Da^F= 0, . . (H4) ; 
 
 (BtF) = -Sr, (I4) 
 
 whereof the system (G4) contains what may be called the Interme- 
 diate Integrals, while the system (H4) contains the Final Integrals, 
 of the differential Equations of Motion (A4), 
 
 (/). In like manner we find equations of the forms, 
 Dar=-mD<a, .. (J4); D„^r=wDoa, . . (K4); DjF=*; (L4) 
 the intermediate integrals (e) being here the result of the elimination 
 
 * References are given to two Essays by the present writer, " On a General 
 Method in Dynamics," in the Philosophical Transactions for 1834 and 1835, in 
 which the Action (V), and a certain other function (S), which is here denoted by P, 
 were called, as above, the Characteristic and Principal Fimctions. But the ana- 
 lysis here used, as being founded on the Calculus of Quaternions, is altogether 
 unlike the analysis which was employed in those former Essays. 
 
xlviii CONTENTS. 
 
 Pages. 
 
 of H, between the system (J4) and the equation (L4) ; and the final 
 integrals, of the same system of differential equations (A4), being now 
 (theoretically) obtained, by eliminating the same constant R between 
 (K4) and (L4). 
 
 {g). The functions F and V are obliged to satisfy certain Partial 
 Differential Equations in Quaternions, of which those relative to the 
 final vectors a, a\ . . are the following, 
 
 (D,i^)-i2m-i(D„jF)2=P, (M4); |2m-i(D^r)2 + P+Jf = 0; (N4) 
 
 and they are subject to certain geometrical conditions, from which 
 can be deduced, in a new way, and as new verifications, the law of mo- 
 tion of the centre of gravity, and the law of description of areas. 
 
 (A). General appro:^mate expressions (p. 717) for the functions 
 i^and V, and for their derivatives jH" and t, for the case oi o. short mo- 
 tion of the system. 
 
 Article 419. — On the Relative Motion of a Binary System ; and 
 on the Law of the Circular Hodograph, 717-72 
 
 («). The vector of one body from the other being a, and the dis- 
 tance being r (= Ta), while the sum of the masses is M, the differen- 
 tial equation of the relative motion is, with the law of the inverse 
 square, 
 
 D^a = jlfa-»r-i ; (O4; 
 
 D being here used as a characteristic of derivation, with respect to the 
 time t. 
 
 (J)). As a first integral, which holds good also for any other law 
 of central force, we have 
 
 VaDa = /8 = a constant vector ; (P4) 
 
 which includes the two usual laws, of the constant plane {-^ j3), and 
 
 of the constant areal velocity ( - = |T/3 
 
 (c). Writing r = Da = vector of relative velocity, and conceiving this 
 new vector r to be drawn from that one of the two bodies which is 
 here selected for the origin o, the locus of the extremities of the vector 
 T is (by earlier definitions) the Hodograph of the Relative Motion ; 
 and this hodograph is proved to be, for the Law of the Inverse Square, 
 a Circle. 
 
 (d). In fact, it is>hown (p. 720), that for any /««<? of central force, 
 the radius of curvature of the hodograph is equal to the force, multi- 
 plied into the square of the distance, and divided by the doubled areal 
 velocity ; or by the constant parallelogram c, under the vectors (a 
 and r) ot position and velocity, or of the orbit and the hodograph. 
 
 (e). It follows then, conversely, that the law of the inverse square 
 is the only law which renders the hodograph generally a circle ; so 
 that the law of nature may be characterized, as the Law of the Circular 
 Hodograph : from which latter law, however, it is easy to deduce 
 the form of the Orbit, as a conic section with di focus at o. 
 
CONTENTS. Xlix 
 
 Pages. 
 (/). If the semiparameter of this orbit be denoted, as usual, by 
 
 Pf and if h be the radius of the hodograph^ then (p. 719), 
 
 h = Mc-^ = cp-^ = {Mp-^yi. (Qi) 
 
 (jg). The orbital excentricity e is also the hodographic excentri- 
 city, in the sense that eh is the distance of the centre h of the hodo- 
 graph, from the point o which is here treated as the centre of force. 
 
 (A). The orbit is an ellipse^ when the point o is interior to the 
 hodographic circle (^ < 1) ; it is a parabola, when o is on the circum- 
 ference of that circle (e= 1) ; and it is an hyperbola, when o is an an- 
 terior point (e> 1). And in all these cases, if we write 
 
 a=p(l-e^y^ = ch-^(l-e^y\ (R^) 
 
 the constant a will have its usual signification, relatively to the 
 orbit. 
 
 (0- The quantity Mr-^ being here called the Potential, and de- 
 noted by P, geometrical constructions for this quantity P are assigned, 
 with the help of the hodograph (p. 723) ; and for the harmonic mean, 
 2M(r + y')-», between the two potentials, P and P', which answer to 
 the extremities t, t' of any proposed chord of that circle : all which 
 constructions are illustrated by a new diagram (Fig. 86). 
 
 ij). If u be the pole of the chord tt' ; m, m' the points in which 
 the line ou cuts the circle ; l the middle point, and n the pole, of the 
 new chord mm', one secant from which last pole is thus the line ntt' ; 
 u' the intersection of this secant with the chord mm', or the harmonic 
 conjugate of the point u, with respect to the same chord ; and nt,t/ 
 any near secant from n, while u, (on the line ou) is the pole of the 
 near chord Tjs I : then the two small arcs, Tjr and t't/, of the hodo- 
 graph, intercepted between these two secants, are proved to be xHiii- 
 maielj proportional to the ttvo potentials, P andP'; or to the two 
 ordinates tv, t'v', namely the perpendiculars let fall from t and t', on 
 what may here be called the hodographic axis ln. Also, the harmonic 
 mean between these two ordinates is obviously (by the construction) 
 the line u'l; while ux, ut', and u,t, u,t/ oxe four tangents to the 
 hodograph, so that this circle is cut orthogonally, in the two pairs of 
 points, T, t' and t,, t/, by two other circles, which have the two near 
 points TJ, u^ for their centres (pp. 724, 725). 
 
 (k). In general, for any motion of a point (absolute or relative, in 
 one plane or in space, for example, in the motion of the centre of the 
 moon about that of the earth, under the perturbations produced by the 
 attractions of the sun and planets), with a for the variable vector (418) 
 oi position of the point, the time dit which corresponds to any vector- 
 element dDa of the hodograph, or what may be called the time of ho- 
 dographically describing that element, is the quotient obtained by 
 dividing the same element of the hodograph, by the vector of accelera- 
 tion D«a in the orbit ; because we may write generally (p. 724), 
 
 J, dDa , TdDa ., , 
 
 d. = __, or d.= .jj^, .f d*>0. (S.) 
 
 h 
 
1 CONTENTS. 
 
 Q). For tlie law of the inverse square (comp. («) and (O)? the 
 measure oi the force is, 
 
 TD2a = Mr-^ = M-^P^ ; (T4) 
 
 the times d^, d^, of hodographically describing the small circular 
 arcs T,T and t't/ of the hodograph, being found by multiplying the 
 lengths (y) of those two arcs by the mass, and dividing each product 
 by the square of the potential corresponding, are therefore inversely 
 as those two potentials, P, P', or directly as the distances, r, r', in the 
 orbit : so that we have the proportion, 
 
 d^:df :di + d<'=r:/:r + r'. (U4) 
 
 (m). If we suppose that the mass, M, and ihe Jive points 0, l, m, 
 "U, u^ upon the chord mm' are given, or constant, but that the ra- 
 dius, h, of the hodograph, or the position of the centre h on the hodo- 
 graphic axis ln, is altered, it is found in this way (p. 725) that 
 although the two elements of time, d^, dd', separately vary, yet their 
 sum remains unchanged : from which it follows, that even if the two 
 circular arcs, tt, t't/, be not small, but still intercepted (/) between 
 two secants from the pole n of ihe fixed chord mm', the sum (say, M + 
 A^') of the two times is independent of the radius, h. 
 
 (n). And hence may be deduced (p. 726), by supposing one secant 
 to become a tangent, this Theorem of Sodographic Isochronism, which 
 was communicated without demonstration, several years ago, to the 
 Royal Irish Academy,* and has since been treated as a subject of 
 investigation by several able writers : 
 
 If two circular hodographs, having a common chord, which passes 
 through, or tends towards, a common centre of force, he cut perpendicu- 
 larly by a third circle, the times of hodographically describing the inter- 
 cepted arcs will be equal. 
 
 (0). This common time can easily be expressed (p. 726), under the 
 form of the definite integral, 
 
 , 2MC^ dw 
 
 Time of TMT = -^ ; ; (V4) 
 
 9^ Jo (l-e'cosw)2' ^ '^ 
 
 2g being the length oi the fixed chord mm'; e' the quotient lo : lm, 
 which reduces itself to - 1 when is at m', that is for the case of a pa- 
 rabolic orbit ; e lying between ± 1 for an ellipse, and outside those limits 
 for an hyperbola, but being, in all these cases, constant ; while w is a 
 certain auxiliary angle, of which the sine = ut : ul (p. 727), or 
 = 5 (r + r')"i, if s denote the length pp' of the chord of the orbit, cor- 
 responding to the chord tt' of the hodograph ; and w varies from to 7r, 
 when the yjhiAe periodic time 2'7rn~^ for a closed orbit is to be computed : 
 with the verification, that the integral (V4) gives, in this last case, 
 M=ahi^, as usual. (Wi) 
 
 * See the Proceedings of the 16th of March, 1847. It is understood that the 
 common centre o oi force is occupied by a common mass, M. 
 
CONTENTS. 
 
 (p). By examining the general composition of the definite inte- 
 gral (V4), or by more purely geometrical considerations, which are 
 illustrated by Fig. 87, it is found that, with the law of the inverse 
 square, the time t of describing an are pp' of the orbit (closed or un- 
 closed) is Q. function (p. 729) of the three ratios^ 
 
 a3 ,.+ / s 
 
 M' "^' ^^" ^^^ 
 
 and therefore simply a function of the chord (s, or fp') of the orbit, 
 and of the sum of the distances (r + r*, or op + op') when M and a are 
 given : which is a form of the Theorem of Lambert. 
 
 (q). The same important theorem may be otherwise deduced, 
 through a quite different analysis, by an employment of partial deri- 
 vatives, and of partial differential equations in quaternions, which is 
 analogous to that used in a recent investigation (418), respecting the 
 motions of an attracting system of any number of bodies, m, m', &c. 
 
 (r). "Writing now (comp. p. xlvii) the following expression for the 
 relative living force, or for the mass {M= m + m'), multiplied into the 
 square of the relative velocity (TDa), 
 
 2T=-ifDa2= 2(P+ J?) = if (2r-i - «-i) ; (Y4) 
 
 introducing the two new integrals (p. 729), 
 
 J5'=r(P+T)d^, (Z4), and r=[*^2TdLt = F+tH, (A5) 
 
 which have thus (comp. (E4) and (r4)) the same forms as before, but 
 with different (although analogous) significations, and may stiU be 
 called the Principal and Characteristic Functions of the motion ; and 
 denoting by a, a' (instead of ao, a) the initial and final vectors of po- 
 sition, or of the orbit, while r, r' are the two distances, and r, r' the 
 two corresponding vectors of velocity, or of the hodograph : it is found 
 that when M is given, F may be treated as a function of a, a', t, or 
 of r, r, s, t, and Fas a function of a, a, a, oxofr, r, s, andJS"; and 
 that their partial derivatives, in the first view of these two functions, 
 are (p. 729), 
 
 BaF^DaV^T, (Bo); Ba'F=J)a'V=-T'; (Cs) 
 
 (J)t)F=-H, (Ds); and D^r = — Dar=<; (E5) 
 
 while, in the second view of the same functions, they satisfy the two 
 partial differential equations (p. 730), 
 
 DrF=^Dr'F, (F5), and D,.F=D/r; (G5) 
 
 along with two other equations of the same kind, but of the second 
 degree, for each of the functions here considered, which are analogous 
 to those mentioned in p. xlviii. 
 
 (5). The equations (Fa) (G5) express, that the two distances, r 
 and /, enter into each of the two functions only by their sum ; so that, 
 if M be still treated as given, F may be regarded as a function of the 
 
lii CONTENTS. 
 
 Pages. 
 three quantities, r + Z, s, and t\ while F, and therefore also t by 
 (Es), is found in like manner to be a function of the three scalars, 
 r + r', s, and a : which last result respecting the time agrees with 
 (p), and furnishes a new proof of Lambert' s Theorem. 
 
 (0- The three partial differential equations (r) in F conduct, by 
 merely algebraical combinations, to expressions for the three partial 
 derivatives, DrF, D,' V {=J)rV), and D^F; and thus, with the help 
 of (E5), to twoneiv definite integrals* (p. 731), which express respec- 
 tively the Action and the Time, in the relative motion of a binary 
 system here considered, namely, the two following : 
 
 ]-s\r^r'-^s a j 
 whereof the latter is not to be extended, without modification, be- 
 yond the limits within which the radical is finite. 
 
 Article 420. — On the determination of the Distance of a Comet, 
 or new Planet, from the Earth, 733, 734 
 
 (a). The masses of earth and comet being neglected, and the mass 
 of the sun being denoted by M, let r and w denote the distances of 
 earth and comet from sun, and z their distance from each other, while 
 a is the heliocentric vector of the earth (Ta = r), known by the theory 
 of the sun, and p is the unit- vector, determined by observation, which 
 is directed from the earth to the comet. Then it is easily proved by 
 quaternions, that we have the equation (p. 734), 
 
 SpDpDV r[M M\ 
 
 CJ5) 
 
 SpDpUa 
 with t<;2 = r2 + 2;2 _ 2zSa|0 ; (K5) 
 
 eliminating w between these two formulae, clearing of fractions, and 
 dividing by a, we are therefore conducted in this way to an algebrai- 
 cal equation of the seventh degree^ whereof owe root is the sought dis- 
 tance, z. 
 
 (J}). The final equation, thus obtained, differs only by its notation, 
 and by the facility of its deduction, from that assigned for the same 
 purpose in the Mecanique Celeste; and the rw/^ofLaplace there given, 
 for determining, by inspection of a celestial globe, which of the two 
 
 * References are given to the First Essay, &c., by the present writer (comp. 
 the Note to p. xlvii.), in which wore assigned integrals, substantially equivalent 
 to (H5) and (I5), but deduced by a quite different analysis. It has recently been 
 remarked to him, by his friend Professor Tait of Edinburgh, that while the area 
 described, with Newton's Law, about the full focus of an orbit, has long been 
 known to be proportional to the time corresponding, so the area about the empty 
 foam represents (or is proportional to) the action. 
 
CONTENTS. liii 
 
 Pages, 
 bodies (earth and comet) is the nearer to the sun, results at sight from 
 the formula (Js)- 
 
 Article 421. — On the Development of the Disturbing Force of 
 the Sun on the Moon ; or of one Planet on another, which is nearer 
 than itself to the Sun, 734-736 
 
 («). Let a, <T be the geocentric vectors of moon and sun ; r (= Ta), 
 and s(=T(t), their geocentric distances ; JLTthe sum of the masses of 
 earth and moon ; S the mass of the sim ; and D (as in recent Series) 
 the mark of derivation with respect to the time : then the differential 
 equation of the disturbed motion of the moon about the earth is, 
 
 D2a = Jf^a4-»7, (Lg), if 0a = 0(a) = a-iTa-', (M5) 
 and rj — Vector of Disturbing Force = S {(pa - (tr — a)) ; . (N5) 
 
 denoting here a vector function, but not a linear one. 
 
 (Z»). If we neglect rj, the equation (L5) reduces itseK to the form 
 T>-a = M<pa ; which contains (comp. (O4)) the laws of undisturbed 
 elliptic motion. 
 
 (c). If we develope the disturbing vector rj, according to ascend- 
 ing powers of the quotient r : s, of the distances of moon and sun from 
 the earth, we obtain an infinite series of terms, each representing a 
 finite group oi partial disturbing forces, which may be thus denoted, 
 
 »?=»?i+»?2+»;3 + &c. ; (O5) 
 
 n\ = nh\^*lh2l »72=»72,l+»?2j2+ J?2,3, &C. ; (P5) 
 
 these partial forces increasing in number, but diminishing in intensity, 
 in the passage from any one group to the following ; and being con- 
 nected with each other, within any such group, by simple numerical 
 ratios and angular relations. 
 
 {d). For example, the two forces r\\,\, »;i,2 of the /rs^ group 
 are, rigorously, proportional to the numbers 1 and 3 ; the three forces 
 »72,i» »72,2, >72,3 of the second %xo\y^ are as the numbers 1, 2, 5; and 
 the /02<r forces of the ^Aw-<f group are proportional to 5, 9, 15, 35 : 
 while the separate intensities of i\ie first forces, in these three first 
 groups, have the expressions, 
 
 'Sr _, 3<Sr« ^ 5Sr3 
 
 , (J). All ih.QS>Q partial forces are conceived to act at the moon ; but 
 their directions may be represented by the respectively jj^mW^/ unit- 
 lines \J r]\, i, &c., drawn /rom the earth, and terminating on a great 
 circle of the celestial sphere (supposed here to have its radius equal to 
 unity), which passes through the geocentric (or apparent) places, 
 and ]), of the sun and moon in the heavens. 
 
 (/). Denoting then the geocentric elongation D oimoon from sun 
 (in the plane of the three bodies) by 4 ; and by 0i, 03, and ])i, 1)2, 
 Da, what may be called tivo fictitious suns, and three fictitious moons, 
 of which the corresponding elongations from 0, in the same great 
 
liv CONTENTS. 
 
 Pages. 
 
 circle, are +29,- 29, and -0, +B9,-39, as illustrated by Fig. 88 
 (p. 735) ; it is found that tte directions of the two forces of the Jirst 
 group are represented by the two radii of this unit-circle, which termi- 
 nate in D and ])i ; those of the three forces of the secowc? group, by the 
 three radii to 0i, 0, and 03 ; and those ot the four forces of the 
 third group, by the radii to h, D, Dij and %', with facilities for ex- 
 tending all these results (with the requisite modifications), to the 
 fourth and subsequent groups, by the same quaternion analysis. 
 
 (g). And it is important to observe, that no supposition is here 
 made respecting any smallness of excentricities or inclinations (p. 736) ; 
 so that all the formulce apply, with the necessary changes oi geocen- 
 tric to heliocentric vectoT^, &c., to the perturbations of the motion of a 
 coinet aboict the sun, produced by the attraction of a planet, which is 
 (at the time) more distant than the comet from the sun. 
 
 Article 422— On Fresnel's Wave, 736-756 
 
 (a). If p and fi be two corresponding vectors, of ray-velocity and 
 wave-slowness, or briefly Hay and Index, in a biaxal crystal, the velo- 
 city of light in a vacuum being unity ; and if dp and Sfx, be any infi- 
 nitesimal variations of these two vectors, consistent with the equa- 
 tions (supposed to be as yet unknown), of the Wave (or wave- surface), 
 and its reciprocal, the Index-Surface {or surface of ivave-sloivness) : we 
 have then first the fundamental Equations of ^Reciprocity (comp. p. 
 417), 
 
 S/ip=-l, (Ra); S/ti5p = 0, (Ss); Sp^/i = 0, (T5) 
 
 which are independent of any hypothesis respecting the vibrations of 
 the ether. 
 
 (b). If dp he next regarded as a displacement (or vibration), tan- 
 gential to the wave, and if de denote the elastic force resulting, there 
 exists then, on Fresnel's principles, a relation between these two small 
 vectors ; which relation may (with our notations) be expressed by 
 either of the two following equations, 
 
 de = r'^p, (U5), or dp = ^ds; (Vg) 
 
 the function ^ being of that linear, vector, and self- conjugate kind, 
 which has been frequently employed in these Elements. 
 
 {c). The fundamental connexion, between the functional symbol 
 <p, and the optical constants abc of the crystal, is expressed (p. 741, 
 comp. the formula (W3) in p. xlvi) by the symbolic and cubic equa- 
 tion, 
 
 i<p + «-2) (^ + i-2) (0 + c-2) = ; ( W5) 
 
 of which an extensive use is made in the present Series. 
 
 (d). The normal component, /x-iS/x^c, of the elastic force de, is in- 
 effective in Fresnel's theory, on account of the supposed incompressi- 
 bility of the ether; and the tangential component, ^-^dp~ fi-^S/xds, is 
 (in the same theory, and with present notations) to be equated to 
 
CONTENTS, Iv 
 
 Pages. 
 fi-^Sp, for the propagation of a rectilinear vibration (p. 737) ; we ob- 
 tain then thus, for such a vibration or tangential displacement, dp, the 
 expression, 
 
 ^p = (r^-/i-2)-V-»S;/5€; CX5) 
 
 and therefore by (S5) the equation, 
 
 O = S/i-K0-»-/x-2)-V-S (Y5) 
 
 which is a Symbolical Form of the scalar Equation of the Index-Sur- 
 face, and may be thus transformed, 
 
 l = S;u(/*2-^)-V. (Z5) 
 
 (e). The Wave- Surface, as being the reciprocal (a) of the index- 
 surface {d), is easily found (p. 738) to be represented by this other 
 Symbolical Equation, 
 
 O=Sp-i(0-p-2)-'p-i; (Ae) 
 
 or l = Sp(p2-^-i)-ip. (Be) 
 
 (/). In such transitions, from one of these reciprocal surfaces to 
 the other, it is found convenient to introduce two auxiliary vectors, 
 V and w(= ^v), namely the lines ou and ow of Fig. 89 ; both drawn 
 from the common centre o of the two surfaces ; but v terminating (p. 
 738) on the tangent plane to the wave, and "being parallel to the direc- 
 tion of the elastic force de ; whereas w terminates (p. 739) on the tan- 
 gent plane to the index- surface, and is parallel to the displacement dp. 
 
 {g). Besides the relation, 
 
 b) = <i>v, or V = ^"'w, (Ce) 
 
 connecting the two new vectors (/) with each other, they are con- 
 nected with p and ft by the equations (pp. 738, 739), 
 
 S^t; = -1, (De); Spi; = 0; (Ee) 
 
 Spw=-1, (Fe); S/^a; = 0; (Ge) 
 
 and generally (p. 739), the following Rule of the Interchanges holds 
 good: In any formula involving p, fi, v, w, and 0, or some of them, 
 it is permitted to exchange p with /a, v -with a>, and with 0'' ; pro- 
 vided that we at the same time interchange dp with Se, but not gene- 
 rally* Sfi with dp, when these variations, or any of them occur. 
 (A). We have also the relations (pp. 739, 740), 
 
 _ p-i = v-iVv/i = fi + v-i^; (He) 
 
 — /*-J = (o'^Ywp = p + 0)-' ; (le) 
 
 * This apparent exception arises (pp. 739, 740) from the circumstance, that 
 dp and ^6 have their directions generally fixed, in this whole investigation 
 (although subject to a common reversal by +), when p and p. are given ; whereas 
 dfi continues to be used, as in (a), to denote any infinitesimal vector, tangential to 
 the index- surface at the end of /u. 
 
Ivi CONTENTS. 
 
 with others easily deduced, whichmay all be illustrated by the above- 
 cited Fig. 89. 
 
 (i). Among such deductions, the following equations (p. 740) 
 may be mentioned, 
 
 (Yv<pvy + Sv<pv = 0, (Je); (Vw0-iw)2 + Sw^-iw = ; (Ke) 
 which show that the Zocus of each of the itvo Auxiliary Points, v and 
 w, wherein the two vectors v and w terminate (/), is a Surface of 
 the Fourth Degree, or briefly, a Quartic Surf ace ; of which two loathe 
 constructions xii9.Y\>e connected (as stated in p. 741) with those of the 
 two reciprocal ellipsoids, 
 
 Sp<pp=l, (Lg), and Sp^-ip = l; (Me) 
 
 p denoting, for each, an arbitrary semidiameter. 
 
 (y). It is, however, a much more interesting use of these two 
 ellipsoids, of which (by (W5), &c.) the scalar semiaxes are a, b, c for 
 the first, and <?"i, b~'^, c-^ for the second, to observe that they may be 
 employed (pp. 738, 739) for the Constructions of the Wave and the 
 Index- Surface, respectively, by a very simple rule, which (at least for 
 t\Q first of these two reciprocal surfaces (a)) was assigned by Fres- 
 nel himself. 
 
 (ky In fact, on comparing the symbolical form (Ae) of the equa- 
 tion of the Wave, with the form (H2) in p. xxxvii, or with the equa- 
 tion 412, XLI., in p. 683, we derive at once FresneVs Construction : 
 namely, that if the ellipsoid (abc) be cut, by an arbitrary plane 
 through its centre, and \i perpendiculars to that plane be erected at 
 that central point, which shall have the lengths of the semiaxes of 
 the section, then the locus of the extremities, of the perpendiculars so 
 erected, will be the sought Wave-Surface. 
 
 (J). A precisely similar construction applies, to the derivation of 
 the Index- Surface from the ellipsoid (a"'Z>"'c-i) : and thus the two 
 auxiliary surfaces, (Lg) and (Me), may be briefly called the Generat- 
 ing Ellipsoid, and the Reciprocal Ellipsoid. 
 
 (jn). The cubic (W5) in (j) enables us easily to express (p. 741) the 
 inverse function (^ + e)-J, where e is any scalar ; and thus, by chang- 
 ing 6 to — p-3, &c., new forms of the equation (Ac) of the wave are 
 obtained, whereof one is, 
 
 = (0-ip)2 + (p2 + «2 + j2 + c2) Sp^-'p - ame^ ; (Ne) 
 
 with an analogous equation in fx (comp. the rule in (y)), to represent 
 the index-surface : so that each of these two surfaces is of the fourth 
 degree, as indeed is otherwise known. 
 
 (n). If either Sp(p-^p or p2 be treated as constant in (Ne), the 
 degree of that equation is depressed from the fourth to the second; 
 and therefore the Wave is cut, by each of the two concentric quadrics, 
 
 Sp^-ip = AS (Oe), p2 + r2 = 0, (Po) 
 
 in a (real or imaginary) curve of the fourth degree : of which two quar- 
 
 Pages. 
 
CONTENTS. Ivii 
 
 Pages. 
 tic curves, answering to all scalar values of the constants h and r, the 
 wave is the common locus. 
 
 (o). The new ellipsoid (Oe) is similar to the ellipsoid (Me), and 
 similarly placed, while the sphere (Pe) has r for radius ; and every 
 quartic of the second system (n) is a sphero-conic, because it is, by the 
 equation (A^) of the wave, the intersection of that sphere (Pe) with 
 the concentric and quadrie coney 
 
 O = Sp(0 + r2)-ip; (Qe) 
 
 or, by (Be), with this other concentric quadrie,* 
 
 -l = Sp(0-i + y2)-ip^ (Re) 
 
 whereof the conjugate (obtained by changing - 1 to + 1 in the last 
 equation) has 
 
 fl;2_y2^ ^2_y2j c2_y2, (Se) 
 
 for the squares of its scalar semiaxes, and is therefore confocal with 
 the generating ellipsoid (Le). 
 
 (^). For any point p of the wave, or at the end of any ray p, the 
 tangents to the two curves (w) have the directions of a> and /iw ; so 
 that these two quartics cross each other at right angles, and each is a . • 
 common orthogonal in all the curves of the other system. 
 
 ((?). But the vibration dp is easily proved to be parallel to (o ; 
 hence the curves of the^rs^ system (n) are Zincs of Vibration of the 
 Wave : and the curves of the second system are the Orthogonal Trajec- 
 toriesf to those Zines. 
 
 (r). In general, the vibration dp has (on Fresnel's principles) the 
 direction of the projection of the ray p on the tangent plane to the 
 wave ; and the elastic force de has in like manner the direction of the 
 projection of the index-vector fi on the tangent plane to the index- 
 surface : so that the ray is ^ms, perpendicular to the elastic force 
 
 Article423.— Mac Cullagh's Theorem of the Polar Plane, . . 757-762 
 ******** 
 ******** 
 
 * For real curves of the second system (n), this new quadrie (Ee) is an hy- 
 perboloid, with one sheet or with two, according as the constant r lies between a 
 and b, or between b and c ; and, of course, the conjugate hyperboloid (o) has two 
 sheets or one, in the same two cases respectively. 
 
 t In a different theory of light (comp. the next Series, 423), these sphero- 
 conics on the wave are themselves the lines of vibration. 
 
Iviii 
 
 CONTENTS. 
 Table* of Initial Pages of Aeticles. 
 
 Art. 
 
 Page. 
 
 Art. 
 
 Page. 
 
 Art. 
 
 I 
 Page, t 
 
 Art. 
 
 1 
 Page. 
 
 Art. 
 
 Page. 
 
 Art. 
 
 Page. 
 
 1 
 
 1 
 
 49 
 
 37 
 
 97 
 
 1 
 88 
 
 145 
 
 i 
 126 
 
 193 
 
 173 
 
 241 
 
 260 
 
 2 
 
 2 
 
 50 
 
 38 
 
 98 
 
 90 
 
 146 
 
 129 
 
 194 
 
 174 
 
 242 
 
 262 
 
 3 
 
 8 
 
 51 
 
 39 
 
 99 
 
 95 
 
 147 
 
 130 
 
 195 
 
 175 
 
 243 
 
 264 
 
 4 
 
 
 52 
 
 )) 
 
 100 
 
 98 
 
 148 
 
 11 
 
 196 
 
 176 
 
 244 
 
 265 
 
 5 
 
 4 
 
 53 
 
 40 
 
 101 
 
 103 
 
 149 
 
 131 
 
 197 
 
 183 
 
 245 
 
 n 
 
 6 
 
 6 
 
 54 
 
 41 
 
 102 
 
 104 
 
 150 
 
 132 
 
 198 
 
 184 
 
 246 
 
 266 
 
 7 
 
 
 55 
 
 42 
 
 103 
 
 t) 
 
 151 
 
 133 
 
 199 
 
 185 
 
 247 
 
 11 
 
 8 
 
 5 
 
 66 
 
 43 
 
 104 
 
 105 
 
 152 
 
 11 
 
 200 
 
 187 ; 
 
 248 
 
 11 
 
 9 
 
 6 
 
 57 
 
 44 
 
 105 
 
 ^^ 
 
 153 
 
 134 
 
 201 
 
 190 1 
 
 249 
 
 267 
 
 10 
 
 )) 
 
 58 
 
 46 
 
 106 
 
 106 
 
 154 
 
 11 
 
 202 
 
 11 
 
 250 
 
 »» 
 
 11 
 
 7 
 
 59 
 
 j» 
 
 107 
 
 „ ; 
 
 155 
 
 135 
 
 203 
 
 192 : 
 
 251 
 
 
 12 
 
 8 
 
 60 
 
 47 
 
 108 
 
 »j 
 
 156 
 
 11 
 
 204 
 
 193 i 
 
 252 
 
 268 
 
 13 
 
 
 61 
 
 48 
 
 109 
 
 107 
 
 157 
 
 136 
 
 205 
 
 200 
 
 253 
 
 269 
 
 14 
 
 9 
 
 62 
 
 49 
 
 110 
 
 108 
 
 158 
 
 137 
 
 206 
 
 202 I 
 
 254 
 
 272 
 
 15 
 
 
 63 
 
 50 
 
 111 
 
 t) 
 
 159 
 
 138 
 
 207 
 
 203 
 
 255 
 
 274 
 
 16 
 
 10 
 
 64 
 
 51 
 
 112 
 
 109 
 
 160 
 
 139 
 
 208 
 
 204 ; 
 
 256 
 
 275 
 
 17 
 
 
 65 
 
 53 
 
 113 
 
 110 
 
 161 
 
 140 
 
 209 
 
 207 ! 
 
 257 
 
 277 
 
 18 
 
 11 
 
 66 
 
 ); 
 
 114 
 
 111 
 
 162 
 
 142 
 
 210 
 
 208 
 
 268 
 
 279 
 
 19 
 
 )) 
 
 67 
 
 54 
 
 115 
 
 )> 
 
 163 
 
 143 
 
 211 
 
 213 
 
 269 
 
 11 
 
 20 
 
 12 
 
 68 
 
 55 
 
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 217 
 
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 229 
 
 267 
 
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 28 
 
 18 
 
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 124 
 
 116 
 
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 220 
 
 232 
 
 268 
 
 292 
 
 29 
 
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 173 
 
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 221 
 
 233 
 
 269 
 
 293 
 
 30 
 
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 126 
 
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 174 
 
 151 
 
 222 
 
 234 
 
 270 
 
 11 
 
 81 
 
 20 
 
 79 
 
 62 
 
 127 
 
 117 
 
 175 
 
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 223 
 
 236 
 
 271 
 
 295 
 
 82 
 
 22 
 
 80 
 
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 128 
 
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 176 
 
 152 
 
 224 
 
 239 
 
 272 
 
 11 
 
 33 
 
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 81 
 
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 129 
 
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 177 
 
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 225 
 
 240 
 
 273 
 
 297 
 
 34 
 
 23 
 
 82 
 
 63 
 
 130 
 
 118 
 
 178 
 
 ^, 
 
 226 
 
 11 
 
 274 
 
 298 
 
 35 
 
 24 
 
 83 
 
 64 
 
 131 
 
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 179 
 
 154 
 
 227 
 
 241 
 
 275 
 
 301 
 
 36 
 
 26 
 
 84 
 
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 132 
 
 119 
 
 180 
 
 155 
 
 228 
 
 244 
 
 276 
 
 ^j 
 
 37 
 
 28 
 
 85 
 
 65 
 
 133 
 
 120 
 
 181 
 
 157 
 
 229 
 
 246 
 
 277 
 
 302 
 
 38 
 
 29 
 
 86 
 
 j» 
 
 134 
 
 11 
 
 182 
 
 158 
 
 230 
 
 11 
 
 278 
 
 11 
 
 39 
 
 30 
 
 87 
 
 66 
 
 135 
 
 121 
 
 183 
 
 159 
 
 231 
 
 247 
 
 279 
 
 303 
 
 40 
 
 ii 
 
 88 
 
 67 
 
 136 
 
 11 
 
 184 
 
 161 
 
 232 
 
 11 
 
 280 
 
 11 
 
 41 
 
 31 
 
 89 
 
 68 
 
 137 
 
 11 
 
 185 
 
 162 
 
 233 
 
 248 
 
 281 
 
 
 42 
 
 32 
 
 90 
 
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 138 
 
 122 
 
 186 
 
 163 
 
 234 
 
 250 
 
 282 
 
 305 
 
 43 
 
 83 
 
 91 
 
 69 
 
 139 
 
 11 
 
 187 
 
 166 
 
 235 
 
 251 
 
 283 
 
 308 
 
 44 
 
 5) 
 
 92 
 
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 140 
 
 123 
 
 188 
 
 167 
 
 236 
 
 253 
 
 284 
 
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 45 
 
 34 
 
 93 
 
 77 
 
 141 
 
 11 
 
 189 
 
 168 
 
 237 
 
 255 
 
 285 
 
 310 
 
 46 
 
 35 
 
 94 
 
 80 
 
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 124 
 
 190 
 
 169 
 
 238 
 
 257 
 
 286 
 
 11 
 
 47 
 
 36 
 
 95 
 
 83 
 
 143 
 
 1 
 
 191 
 
 170 
 
 239 
 
 11 
 
 287 
 
 311 
 
 48 
 
 37 
 
 96 
 
 85 
 
 144 
 
 125 
 
 192 
 
 171 
 
 240 
 
 259 
 
 288 
 
 312 
 
 * This Table was mentioned in the Note to p. xiv. of the Contents, as one 
 likely to facilitate reference. In fact, the references in the text of the Elements 
 are almost entirely to Articles (with their sub -articles), and not to pages. 
 
. CONTENTS. 
 Table of Initial Taq-es— continued. 
 
 lix 
 
 Art. 
 
 Page. 
 
 Art. 
 
 Page. 
 
 Art. 
 337 
 
 Page. 
 
 Art. 
 
 Page. 
 
 Art. 
 
 Page. 
 
 Art. Page. 
 
 289 
 
 312 
 
 313 
 
 379 
 
 417 
 
 361 
 
 482 
 
 385 
 
 524 
 
 \ 409 664 
 
 290 
 
 313 
 
 314 
 
 381 
 
 338 
 
 420 
 
 362 
 
 484 
 
 386 
 
 625 
 
 410 667 
 
 291 
 
 5) 
 
 315 
 
 383 
 
 339 
 
 421 
 
 363 
 
 485 
 
 387 
 
 527 
 
 i 411 674 
 
 292 
 
 314 
 
 316 
 
 384 
 
 340 
 
 422 
 
 364 
 
 487 
 
 388 
 
 529 
 
 ! 412 679 
 
 293 
 
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 391 
 
 341 
 
 423 ' 
 
 365 
 
 491 
 
 389 
 
 631 
 
 413 ( 
 
 389 
 
 294 
 
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 318 
 
 )) 
 
 342 
 
 427 
 
 366 
 
 495 
 
 390 
 
 535 
 
 414 694 
 
 295 
 
 321 
 
 319 
 
 
 343 
 
 429 
 
 367 
 
 496 
 
 391 
 
 637 
 
 415 698 
 
 296 
 
 324 
 
 320 
 
 292 
 
 ! 344 
 
 431 
 
 368 
 
 
 392 
 
 538 
 
 416 ' 
 
 ro7 
 
 297 
 
 331 
 
 321 
 
 393 
 
 345 
 
 432 
 
 369 
 
 )> 
 
 393 
 
 639 
 
 1 417 709 1 
 
 298 
 
 343 
 
 322 
 
 394 
 
 346 
 
 435 
 
 370 
 
 498 
 
 394 
 
 641 
 
 418 ' 
 
 ri3 
 
 299 
 
 347 
 
 323 
 
 399 
 
 347 
 
 436 
 
 371 
 
 500 
 
 395 
 
 649 
 
 i 419 ' 
 
 '17 
 
 300 
 
 349 
 
 324 
 
 400 
 
 348 
 
 439 
 
 372 
 
 501 
 
 396 
 
 664 
 
 420 ' 
 
 r33 
 
 301 
 
 351 
 
 325 
 
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 373 
 
 602 
 
 397 
 
 659 
 
 421 ' 
 
 r34 
 
 302 
 
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 326 
 
 403 
 
 350 
 
 443 
 
 374 
 
 508 
 
 398 
 
 578 
 
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 r36 
 
 303 
 
 352 
 
 327 
 
 404 
 
 351 
 
 445 
 
 375 
 
 509 
 
 399 
 
 612 
 
 423 ' 
 
 757 
 
 304 
 
 354 
 
 328 
 
 405 
 
 352 
 
 447 
 
 376 
 
 611 
 
 400 
 
 621 
 
 424 
 
 
 305 
 
 356 
 
 329 
 
 406 
 
 353 
 
 453 
 
 377 
 
 512 
 
 401 
 
 626 
 
 425 
 
 
 
 306 
 
 358 
 
 330 
 
 407 
 
 354 
 
 459 
 
 378 
 
 613 
 
 402 
 
 630 
 
 426 
 
 
 
 307 
 
 361 
 
 331 
 
 408 
 
 355 
 
 464 
 
 379 
 
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 403 
 
 631 
 
 427 
 
 
 
 308 
 
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 332 
 
 409 
 
 356 
 
 466 
 
 380 
 
 515 
 
 404 
 
 633 
 
 428 
 
 
 
 309 
 
 366 
 
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 411 
 
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 468 
 
 381 
 
 519 
 
 405 
 
 636 
 
 429 
 
 
 
 310 
 
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 412 
 
 358 
 
 470 
 
 382 
 
 620 
 
 406 
 
 638 
 
 430 
 
 
 
 311 
 
 373 
 
 335 
 
 414 
 
 359 
 
 474 
 
 383 
 
 622 
 
 1 407 
 
 649 
 
 . , 
 
 
 
 312 
 
 374 
 
 336 
 
 416 
 
 360 
 
 481 
 
 384 
 
 624 
 
 1 408 
 
 1 
 
 663 
 
 • • 
 
 
 
 Table of Pages foe the Figuees. 
 
 Figure. 
 
 Page. 
 
 Figure. 
 
 1 
 Page. 
 
 Figure. 
 
 Page. 
 
 Figure. 
 
 Page. 
 
 Figure. 
 
 Page. 
 
 1 
 
 1 
 
 21 
 
 21 i 
 
 38 
 
 119 
 
 64 
 
 247 
 
 72 
 
 348 
 
 2 
 
 2 
 
 i 22 
 
 25 1 
 
 39 
 
 129 
 
 65 
 
 269 
 
 73 
 
 369 
 
 3 
 
 
 1 23 
 
 27 
 
 40 
 
 130 
 
 66 bis 
 
 
 74 
 
 397 
 
 4 
 
 
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 33 
 
 41 
 
 
 56 
 
 
 75 
 
 426 
 
 6 
 
 3 
 
 25 
 
 36 
 
 41 bis 
 
 )i 
 
 67 
 
 274 
 
 76 
 
 499 
 
 6 
 
 
 26 
 
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 42 
 
 132 
 
 68 
 
 280 
 
 77 
 
 611 
 
 7 
 
 4 
 
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 42 bis 
 
 141 
 
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 78 
 
 517 
 
 8 
 
 5 
 
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 43 
 
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 60 
 
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 9 
 
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 80 
 
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 10 
 
 >» 
 
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 669 
 
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 7 
 
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 46 bis 
 
 
 63 
 
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 12 
 
 8 
 
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 98 
 
 46 
 
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 63 bis 
 
 325 
 
 83 
 
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 13 
 
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 84 
 
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 120 
 
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 326 
 
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 706 
 
 15 
 
 13 
 
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 48 
 
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 66 
 
 327 
 
 86 
 
 724 
 
 16 
 
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 112 
 
 49 
 
 172 
 
 67 
 
 332 
 
 87 
 
 727 
 
 17 
 
 16 
 
 i 85bis 
 
 143 
 
 60 
 
 190 
 
 68 
 
 334 
 
 88 
 
 736 
 
 18 
 
 17 
 
 ; 36 
 
 112 
 
 61 
 
 216 
 
 69 
 
 ^^ 
 
 89 
 
 740 
 
 19 
 
 20 
 
 1 B6bis 
 
 126 
 
 62 
 
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 70 
 
 343 
 
 90 
 
 . . 
 
 20 
 
 » 
 
 i " 
 
 116 
 
 63 
 
 226 
 
 71 
 
 344 
 
 91 
 
 • • 
 
 Note. — It appears by these Tables tbat tbe Author intended to have com- 
 pleted the work by the addition of Seven Articles, and Two Figures.— Ed. 
 
ELEMENTS OF QUATERNIONS. 
 
 BOOK I. 
 
 ON VECTORS, CONSIDERED WITHOUT REFERENCE TO ANGLES, 
 OR TO ROTATIONS. 
 
 CHAPTER I. 
 
 FUNDAMENTAL PRINCIPLES RESPECTING VECTORS. 
 
 Section l,— 0?i the Conception of a Vector; and on Equality 
 
 of Vectors. 
 
 Art, 1 . — A right line ab, considered as having not only length, 
 but also direction, is said to be a Vector. Its initial point a 
 is said to be its origin; and its final point b is said to be its 
 term, A vector ab is conceived to be (or to construct) the 
 differerice of its two extreme points ; or, more fully, to be the 
 result of the subtraction of its own origin from its own term ; 
 and, in conformity with this conception, it is also denoted by 
 the symbol b - a : a notation which will be found to be exten- 
 sively useful, on account of the analogies which it serves 
 to express between geometrical and algebraical operations. 
 When the extreme points a and b are distinct, the vector ab 
 or B - A is said to be an actual (or an effective) vector ; but 
 when (as a limit) those two points are conceived to coincide, 
 the vector aa or a - a, which then results, is said to be null. 
 Opposite vectors, such as ab and ba, 
 or B - a and a - b, are sometimes 
 called vector and revector. Succes- 
 sive vectors, such as ab and bc, or Kevector. 
 B - a and c - b, are occasionally said ^'S- ^• 
 to be vector and provector: the line ac, or c - a, which is 
 
 A 
 
 Vector, 
 
 
 b-'a 
 
 A 
 
 ^7^ 
 
ELEMENTS OF QUATERNIONS. 
 
 [book I, 
 
 Fig. 2. 
 
 drawn from the origin a of the first to the term c of the second, 
 being then said to be the trans- 
 vector. At a later stage, we shall 
 have to consider vector-arcs and 
 vector-angles ; but at present, our 
 only vectors are (as above) right 
 lines. 
 
 2. Two vectors are said to be equal to each other, or the 
 equation ab = CD, or b - a = d - c, is said to hold good, when 
 (and only when) the origin and term of the one can be brought 
 to coincide respectively with the corresponding points of the 
 other, by transports (or by translations) without rotation. It 
 follows that all null vectors are equal, and may therefore be 
 denoted by a common symbol, such as that used for zero ; so that 
 wemaywrite, ^_ ^ = b _b =&«. = O; 
 
 but that two actual vectors, ab and cd, are not (in the present 
 full sense) equal to each other, unless they have not merely 
 equal lengths, but also similar directions. If then they do not 
 happen to be parts of one common line, they must be opposite 
 
 sides of a parallelogram, /^ c. ^^ ,d 
 
 abdc ; the two lines ad, bc 
 becoming thus the two dia- 
 gonals of such a figure, and 
 consequently bisecting each 
 other, in some point e. 
 Conversely, if the two equa- 
 tions, 
 
 D - E = E - A, and 
 
 are satisfied, so that the two lines 
 AD and BC are commedial, or have 
 a common middle point e, then even 
 if they be parts of one right line, 
 the equation D-c=B-Ais satis- 
 fied. Two radii, ab, ac, of any 
 one circle (or sphere), can never be equal vectors ; because their 
 directions differ. 
 
 Pig. 4. 
 
CHAP. I.J FUNDAMENTAL PRINCIPLES VECTORS. 3 
 
 3. An equation between vectors^ considered as an equidif- 
 ference of points, admits of inversion and ^ ^ 
 
 alternation ; or in symbols, if 
 
 D - C = B - A, 
 
 then 
 
 c - D =A-B, 
 
 and 
 
 D - B = C - A. 
 
 Fig. 5. 
 
 Two vectors, cd and ef, which are 
 
 equal to the same third vector, ab, ^( 
 
 are also equal to each other ; and 
 
 these three equal vectors are, in 
 
 general, the three parallel edges of '^p. g 
 
 a prism. 
 
 Section 2. — On Differences and Sums of Vectors taken two 
 
 by two, 
 
 4. In order to be able to write, as in algebra, 
 
 (c' - a') - (b - a) = c - B, if c' - a' = c - a, 
 
 we next define, that when a first vector ab is subtracted from 
 a second vector ac which is co-initial with it, or from a third 
 vector a'c' which is equal to that second vector, the remainder 
 is that fourth vector bc, which is drawn from the term b of the 
 first to the term c of the second vector : so that if a vector be 
 subtracted from a transvector (Art. 1), the remainder is the 
 provector corresponding. It is evident that this geometrical 
 subtraction of vectors answers to a decomposition of vections (or 
 of motions) ; and that, by such a decomposition of a null vec- 
 tion into two opposite vections, we have the formula, 
 
 - (b - a) = (a - a) - (b - a) = A - b ; 
 
 so that, if an actual vector ab be subtracted from a null vector 
 A A, the remainder is the revector ba. If then we agree to 
 abridge, generally, an expression of the form - « to the 
 shorter form, - «, we may write briefly, - ab = ba; a and - a 
 being thus symbols of opposite vectors, while a and - (- a) are, 
 
4 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 for the same reason, symbols of one common vector : so that 
 we may write, as in algebra, the identity^ 
 
 5. Aiming still at agreement with algebra, and adopting 
 
 on that account the formula of relation between the two signs^ 
 
 + and -, 
 
 (b -a) + a = b, 
 
 in which we shall say as usual that b- ais added to «, and that 
 their sum is b, while relatively to it they may be jointly called 
 summands, we shall have the two following consequences : 
 
 I. If a vector, ab or b - a, be added to its own origin a, 
 the sum is its term b (Art. 1) ; and 
 
 II. If a provector bc be added to a vector ab, the sum is 
 the transvector ac ; or in symbols, 
 
 I. . (b - a) + A = B ; and II. . (c - b) + (b - a) = c - a. 
 
 In fact, the first equation is an immediate consequence of the 
 general formula which, as above, connects the signs + and -, 
 when combined with the conception (Art. 1 ) of a vector as a dif- 
 ference of two points ; and the second is a result of the same 
 formula, combined with the definition of the geometrical sub- 
 traction of one such vector from another, which was assigned 
 in Art. 4, and according to which we have (as in algebra) for 
 any three points^ a, b, c, the identity, 
 
 (c - a) - (b - a) = c - B. 
 
 It is clear that this geometrical addition of successive vectors 
 corresponds (comp. Art. 4) to a composition of successive vec- 
 tions, or motions ; and that the sum of 
 two opposite vectors (or of vector and 
 revector) is a null line ; so that 
 
 ba + ab = 0, or (a - b) + (b - a) = 0. 
 
 It follows also that the sums of equal 
 
 pairs of successive vectors are equal; ^,. 
 
 or more fully that 
 
 if b' - a' = b - a, and c' - b' = c - b, then c' - a' = c 
 
CHAP. I.] FUNDAMENTAL PRINCIPLES VECTORS. 5 
 
 the two triangles, abc and a'b'c', being in general the two oppo- 
 site faces of ^ prism (comp. Art. 3). 
 
 6. Again, in order to have, as in algebra, 
 
 (c' - b') + (b - a) = c - A, if c' - b' = c - B, 
 
 we shall define that if there be two successive vectors, ab, bc, 
 and if a third vector b'c' be equal to the second, but not suc- 
 cessive to the first, the sum obtained by adding the third to the 
 first is that fourth vector, ac, which is drawn from the origin 
 A of the first to the term c of the se- 
 cond. It follows that the sum of any 
 two co-initial sides, ab, ac, of 2iny paral- 
 lelogram abdc, is the intermediate and 
 co-initial diagonal ad ; or, in symbols, 
 
 (C - a) + (b - a) = D - A, if D - C = B - A ; Fig. 8. 
 
 because we have then (by 3) c-a = d-b. 
 
 7. The sum of any two given vectors has thus a value which 
 is independent of their order ; or, in symbols, a -f j3 = j3 + a. 
 If equal vectors be added to equal vectors, the sums are equal 
 vectors, even if the summands be not given as successive 
 (comp. 5) ; and if a null vector be added to an actual vector, 
 the sum is that actual vector ; or, in symbols, + a = a. If 
 then we agree to abridge generally (comp. 4) the expression 
 + « to + fl, and if a still denote a vector, then + a, and + (+ a), 
 &c., are other symbols for the same vector; and we have, as 
 in algebra, the identities, 
 
 - (- a) = + a, + (- a) = - (+ a) = - a, (+ a) + (- a) = 0, &c. 
 
 Section 3. — On Sums of three or more Vectors. 
 
 8. The sum of three given vectors, a, j3, y, is next defined 
 to be that fourth vector, 
 
 ^ = 7 + (/3 + a), or briefly, S=7 + /3 + a, 
 
 which is obtained by adding the third to the sum of the first 
 and second ; and in like manner the sum of any number of 
 vectors is formed by adding the last to the sum of all that 
 
6 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 precede it: also, for any four vectors, a, /3, 7, S, the sum 
 S + (7 + j3 + a) is denoted simply by 8 + 7 + /3 + a, without pa- 
 rentheses, and so on for any number of summands. 
 
 9. The sum of any number of successive vectors, ab, bc, 
 CD, is thus the line ad, which is 
 drawn from the origin a of the first, 
 to the term d of the last ; and be- 
 cause, when there are three such vec- 
 tors, we can draw (as in Fig. 9) the 
 two diagonals ac, bd of the (plane "^ p. 9 
 or gauche) quadrilateral abcd, and 
 may then at pleasure regard ad, either as the sum of ab, bd, 
 or as the sum of ac, cdj we are allowed to establish the follow- 
 ing general formula of association ^ for the case oi' any three 
 summand lines, a, f5, y '• 
 
 (7 + /3) + a = 7 + (j3 + a)=7 + j3 + a; 
 
 by combining which with the formula of commutation (Art. 7), 
 namely, with the equation, 
 
 a + j3 = |3 + a, 
 
 which had been previously established for the case of any two 
 such summands, it is easy to conclude that the Addition of 
 Vectors is always both an Associative and a Commutative Ope- 
 ration. In other words, the sum oYany number of given vectors 
 has a value which is independent of their order, and of the 
 mode of grouping them ; so that if the lengths and directions of 
 the summands be preserved, the length and direction of the 
 sum will also remain unchanged : except that this last direction 
 may be regarded as indeterminate, when the Zew^^A of the sum- 
 line happens to vanish, as in the case 
 which we are about to consider. 
 
 1 0. When any n summand-lines, 
 
 AB, bc, CA, or AB, bc, CD, DA, &C., 
 
 arranged in any one order, are the n 
 
 successive sides of a triangle ab c, or of f" 10 
 
 a quadrilateral abcd, or of any other 
 
 closed polygon, their sum is a 7iull line, aa ; and conversely. 
 
CHAP. I.] FUNDAMENTAL PRINCIPLES VECTORS. 
 
 when the sum of any given system of n vectors is thus equal 
 to zero, they may be made {in any order ^ hy transports without 
 rotatioTi) the n successive sides of a closed polygon (plane or 
 gauche). Hence, if there be given any such polygon (p), sup- 
 pose a pentagon abcde, it is possible to construct another 
 closed polygon (p'), such as a'b'c'd'e', with an arbitrary initial 
 point a', but with the same number of sides, a'b', . . e'a', which 
 new sides shall be equal (as vectors) to the old sides ab, . . ea, 
 taken in any arbitrary order. For example, if we draw^wr 
 successive vectors, as follows, 
 
 A B = CD, B C 
 
 AB. 
 
 CD = EA, 
 
 D E = BC, 
 
 and then complete the new pentagon by drawing the line e'a', 
 this closing side of the second figure (p') will be equal to the 
 remaining side de of the^rs^ figure (p). 
 
 11. Since a closed figure abc . . is still a closed one, when 
 all its points ^vq projected on any assumed joZawe, by any system 
 of parallel ordinates (although the 
 area of the projected figure a'b'c' . . . 
 may happen to vanish), \t follows that 
 if the sum of any number of given 
 vectors a, j3, y, . . be zero, and if we 
 project them all 07i any one plane by 
 parallel lines drawn from their extre- 
 mities, the sum of the projected vec- 
 tors a, /3') y'i . . will likeivise be null; ^' 
 so that these latter vectors, like the 
 former, can be so placed as to become the successive sides of a 
 closed polygon, even if they be not already such. (In Fig. 1 1 , 
 a"b"c" is considered as such a polygon, namely, as a triangle 
 loith evanescent area ; and we have the equation, 
 
 Fig. 11. 
 
 as well as 
 
 a"b" + b"c" + c"a" = 0, 
 
 a'b' + b'c' -f cV = 0, and ab + bc + ca = 0.) 
 
8 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 Section 4. — On Coefficients of Vectors, 
 
 12. The simple or single vector, a, is also denoted by la, 
 or by 1 . a, or by (+ 1 ) a ; and in like manner, the double vector, 
 a-\^a, is denoted by 2a, or 2 . a, or (+ 2) a, &c. ; the rule being, 
 that for any algebraical integer, m^ regarded as a coefficient by 
 which the vector a is multiplied, we have always, 
 
 \a + ma = {\ -^r m) a I 
 
 the symbol 1 + m being here interpreted as in algebra. Thus, 
 Oa = 0, the zero on the one side denoting a null coefficient, and 
 the zero on the other side denoting a null vector ; because by 
 the rule, 
 
 la -f Oa = (l + 0)a = la = a, and .'.Oa = a-a = 0. 
 
 Again, because (I) a + (- 1) a = (1 - 1 ) a = Oa = 0, we have 
 (- l)a = - a = -a = -(la); in like manner, since(l)a+ (-2)a 
 = (l-2)a = (- l)a = -a, we infer that (- 2)a = -a - a = - (2a) ; 
 and generally, (^-m) a = - (ma), whatever whole number m 
 may be : so that we may, without danger of confusion, omit 
 the parentheses in these last symbols, and write simply, - la, 
 - 2a, -ma. 
 
 13. It follows that whatever two whole numbers (positive or 
 negative, or null) may be represented by m and n, and what- 
 
 Fig. 12. 
 
 ever two vectors may be denoted by a and j3, we have always, 
 as in algebra, the formulae, 
 
 na±ma = {n± m) a, n (ma) = (nm) a =« nma, 
 
 and (compare Fig. 12), 
 
 m (/3 ± a) = /w/3 ±ma; 
 
CHAP. I.J FUNDAMENTAL PRINCIPLES VECTORS. 9 
 
 SO that the multiplication of vectors by coefficients is a doubly 
 distributive operation^ at least if the multipliers be whole 
 numbers; a restriction which, however, will soon be re- 
 moved. 
 
 14. If ma = j3, the coefficient m being still whole, the vector 
 |3 is said to be a multiple ol' a ; and conversely (at least if the 
 integer m be different from zero), the vector a is said to be a 
 sub-multiple of /3. A multiple of a sub-multiple of a vector is 
 said to be infraction of that vector ; thus, if /3 = ma, and y = na, 
 
 n 
 then y is a fraction of j3j which is denoted as follows, 7 = — jS ; 
 
 m 
 
 n 
 
 also j3 is said to be multiplied by the fractional coefficient — , 
 
 and y is said to be the product of this multiplication. It fol- 
 lows that if a; and y be any two fractions (positive or negative 
 or null, whole numbers being included), and if a and (3 be any 
 two vectors, then 
 
 ya±xa==(y±x)a, ' y{xa) = {yx)a = yxa, x(P ± a) = xj3 ±Xa ; 
 
 results which include those of Art. 1 3, and may be extended 
 to the case where x and y are incommensurable coefficients, con- 
 sidered as limits oi' fractional ones. 
 
 15. For any actual vector a, and for any coefficient x, of 
 any of the foregoing kinds, ihaproduct xa, interpreted as above, 
 represents always a vector j3, which has the same direction as 
 the multiplicand-line a, if x> 0, but has the opposite direction 
 if aj < 0, becoming null if x= 0. Conversely, if a and /3 be any 
 two actual vectors, with directions either similar or opposite, in 
 each of which two cases we shall say that they are parallel 
 vectors, and shall write j3 H a (because both are then parallel, 
 in the usual sense of the word, to one common line), we can 
 always find, or conceive as found, a coefficient x^O, which shall 
 
 satisfy the equation j3 = xa; or, as we shall also write it, 
 f3 = ax; and the positive or negative number x, so found, will 
 bear to ± 1 the same ratio, as that which the lenyth of the line 
 3 bears to the lengtli of a. 
 
10 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 16. Hence it is natural to say that this coefficient x is the 
 quotient which results, from the division of the vector j3, hy the 
 parallel vector a ; and to write, accordingly, 
 
 x = Q-7-a, orx=Q:a, or^ = ^; 
 
 a 
 
 SO that we shall have, identically, as in algebra, at least if the 
 divisor-line a be an actual vector, and if the dividend-line ^hQ 
 parallel thereto, the equations, 
 
 (j3 : a) .a = — a = j3, and Xa\a=- — = x', 
 
 which will afterwards be extended, by definition, to the case of 
 non-parallel vectors. We may write also, under the same 
 
 conditions, d = — , and may say that the vector a is the quotient 
 
 X 
 
 of the division of the other vector j3 hy the numher x ; so that 
 we shall have these other identities, 
 
 — .a3 = (aa;=)j3, and — = a. 
 
 17. The positive or negative quotient, x-=^, which is thus 
 
 obtained by the division of one of two parallel vectors by ano- 
 ther, including zero as a limit, may also be called a Scalar ; 
 because it can always be found, and in a certain sense con- 
 structed, by the comparison of positions upon one common scale 
 (or axis) ; or can be put under the form, 
 
 c - A AC 
 
 b-a~ab' 
 
 where the three points, a, b, c, are collinear (as in the figure 
 annexed). Such scalar s are, there- ^ 
 
 fore, simply the Re a ls (or real quan- ' ^, ' 
 
 tities) oi Algebra; but, in combina- 
 tion with the not less real Vectors above considered, they 
 form one of the main elements of the System, or Calculus, to 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 11 
 
 which the j)resent work relates. In fact it will be shown, at 
 a later stage, that there is an important sense in which we can 
 conceive a scalar to be added to a vector ; and that the sum 
 so obtained, or the combination, 
 
 Scalar plus Vector^'* 
 
 is a Quaternion. 
 
 CHAPTER II. 
 
 APPLICATIONS TO POINTS AND LINES IN A GIVEN PLANE. 
 
 Section 1. — On Linear Equations connecting two Co-uiitial 
 
 Vectoj's. 
 
 18. When several vectors, oa, ob, . . are all drawn from 
 one common point o, that point is said to be the Origin of the 
 System ; and each particular vector, such as oa, is said to be 
 the vector of its own term, a. In the present and future sec- 
 tions we shall always suppose, if the contrary be not expressed, 
 that all the vectors a, j3, . . which we may have occasion to 
 consider, are thus drawn from one common origin. But if it 
 be desired to change that origin o, without changing the term- 
 points a, . . we shall only have to subtract, from each of their 
 old vectors a, . . one common vector w, namely, the old vector 
 oo' of the new origin d ; since the remainders, a - w, j3 - w, • • 
 will be the new vectors a, /3', . . of the old points a, b, . . . For 
 example, we shall have 
 
 a = o'a = a - o' = (a - o) - (o' - o) = oa - oo' = a - w. 
 
 19. If tivo vectors a, /3, or oa, ob, be thus drawn from a 
 given origin o, and if their o a b 
 directions be either similar or ' "■; ~ ' 
 opposite, so that the three 
 
 points, o, A, B, are situated on one right line (as in the figure 
 
12 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 annexed), then (by 16, 17) their quotient — is some positive or 
 
 negative scalar, such as x ; and conversely, the equation 
 j3 = xa, interpreted with this reference to an origin, expresses 
 the condition of collinearity , of the points o, a, b ; the particu- 
 lar values, 03 = 0, x=\, corresponding to the particular /)052- 
 tions, o and a, of the variable point b^ whereof the indefinite 
 right line OA is the locus. 
 
 20. The linear equation, connecting the two vectors a and 
 j3, acquires a more symmetric ^or/w, wlien we write it thus : 
 
 aa + ^/3 = ; 
 
 where a and b are two scalars, of which however only the ratio 
 is important. The condition of coincidence, of the two points 
 
 A and B, answering above to a? = 1, is now -j- = 1 ; or, more 
 
 symmetrically, 
 
 « + 5 = 0. 
 
 Accordingly, when a=-b, the linear equation becomes 
 
 b{(5-a)-^0, or i3-a = 0, 
 
 since we do not suppose that both the coefficients vanish ; and 
 the equation j3 = a, or ob = oa, requires that ihepointB should 
 coincide with the point a : a case w^hich may also be conve- 
 niently expressed by the formula, 
 
 B = a; 
 
 coincident points being thus treated (in notation at least) as 
 eqy^L In general, the linear equation gives, 
 
 a . OA + 6 . OB = 0, and therefore « : 6 = bo : oa. 
 
 Section 2. — On Linear Equations between three co-initial 
 Vectors. 
 
 21. If two (actual and co-initial) vectors, a, /3, be not con- 
 nected by any equation of the form aa 4 Z>/3 = 0, with any two 
 scalar coefficients a and b whatever, their directions c^n neither 
 be similar nor opposite to each other ; they therefore determine 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 
 
 13 
 
 a plane aob, in which the (now actual) vector, represented by 
 
 the sum aa + Z>/3, is situated. For if, for the sake of symmetry, 
 
 we denote this sum by the 
 
 symbol - cy, where c is some 
 
 third scalar, and 7=00 is 
 
 some third vector, so that the 
 
 three co-initial vectors, a, )3, 
 
 7, are connected by the linear 
 
 equation, 
 
 «a -f ^>/3 + C7 = ; 
 
 and if we make 
 
 , - aa 
 
 oa = , 
 
 c 
 
 then the two auxiliary points, a' and b', will be situated (by 
 19) on the two indefinite right lines, oa, ob, respectively: 
 and we shall have the equation, 
 
 oc = oa'+ ob', 
 
 so that the figure a'ob'c is (by 6) a parallelogram, and conse- 
 quently plane. 
 
 22. Conversely, if c be any point in the plane aob, we can 
 draw from it the ordinates, ca' and cb', to the lines oa and ob, 
 and can determine the ratios of the three scalars, a, b, c, so as 
 to satisfy the two equations. 
 
 OA 
 
 oa' 
 
 OB 
 OB 
 
 after which we shall have the recent expressions for oa', ob', 
 with the relation oc = oa' + ob' as before ; and shall thus be 
 brought back to the linear equation aa + b^ + cy = 0, which 
 equation may therefore be said to express the condition ofcom- 
 plariarity of the^wr points, o, a, b, c. And if we write it under 
 the form, 
 
 Xa + 7/f5 + zy = 0, 
 
 and consider the vectors a and j3 as ^iven, but 7 as a variable 
 vector, while x, y, z are variable scalars, the locus of the va- 
 riable poirit will then be the given plane, oab. 
 
14 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book 
 
 23. It may happen that the point c is situated on the right 
 line ABj which is here considered as a given one. In that 
 
 AC 
 
 case (comp. Art. 17, Fig. 13), the quotient — must be equal 
 
 AB 
 
 to some scalar, suppose t ; so that we shall have an equation of 
 the form, 
 
 = t, or y = a + t(f5-a), or (1 - #) a + ^/3 - 7 = ; 
 
 jS-a 
 
 by comparing which last form 
 with the linear equation of Art. 
 21, we see that the condition 
 of collinearity of the three 
 points A, B, c, in the given 
 plane oab, is expressed by the 
 formula, 
 
 « + i + c = 0. 
 
 This condition may also be thus written, 
 
 Fig. 10. 
 
 -a -b 
 c c ' 
 
 OA OB 
 
 or — + — = 1 ; 
 
 OA OB 
 
 and under this last form it expresses a geometrical relation, 
 which is otherwise known to exist. 
 
 24. When we have thus the two equations, 
 
 «a + 6/3 + c-y = 0, and « + 6 + c = 0, 
 
 so that the three co-initial vectors a, /3, 7 terminate on one 
 right line, and may on that account be said to be ternwio-col- 
 linear, if we eliminate, successively and separately, each of 
 the three scalars a, b, c, we are conducted to these three other 
 equations, expressing certain ratios of segments : 
 
 b(j5-a) + c{y-a) = 0, dy - (5) + a(a - (^) = 0, 
 
 a(a-7) + i(/3-7) = 0; 
 
 or 
 
 = 6.AB 4 C.AC = C.BC + «.BA = a.CA + 6.CB. 
 
 Hence follows this proportion, between coefficients and seg- 
 ments, 
 
 « :6:c = Bc : CA : ab. 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 15 
 
 We might also have 
 
 ! observed that the 
 
 proposed equations 
 
 give, 
 
 
 
 
 
 
 bf3 + Cy 
 
 P- 
 
 cy + aa 
 
 aa + bf3 
 a + b ' 
 
 whence 
 
 
 
 
 
 
 AC_ 
 
 y -a 
 
 = ^=-*<S 
 
 ;c. 
 
 AB j3 - a a +b c 
 
 25. If we still treat a and j3 as given, but regard y and 
 
 - as variable, the equation 
 
 xa-\- yfi 
 ^~ ^ + y 
 
 will express that the variable point c is situated someivhere 
 on the indefinite right line ab, or that it has this line for its 
 locus : while it divides the Jinite line ab into segments, of which 
 the variable quotient is, 
 
 CB x' 
 
 Let c' be another point on the same line, and let its vector be, 
 
 
 
 
 > 
 1 
 
 
 
 
 
 then, in 
 
 like 
 
 manner, 
 
 we 
 
 shall have this 
 
 other 
 
 ratio 
 
 ofseg^ 
 
 ments, 
 
 
 
 
 AC _ 2/' 
 
 c'b ~ a?'* 
 
 
 
 
 If, then, we agree to employ, generally, ^o?- any group offo 
 collinear points, the notation. 
 
 ^ ab CD AB AD 
 
 (abcd) = — = — : — 
 
 ^ bc da bc dc 
 
 SO that this symbol, 
 
 (abcd), 
 
 may be said to denote the anharmonic function, or anharmonic 
 quotient, or simply the anharmonic of the group, a, b, c, d : we 
 shall have, in the present case, the equation, 
 
 „ AC Ac' yx 
 (acbc ) = — :-T- = ^. 
 ^ ^ CB CB xy 
 
16 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 26. When the anharmonic quotient h^QomQ^ equal to nega- 
 tive unity, the group becomes (as is well known) harmonic. 
 If then we have the two equations, 
 
 xa + y(^ , xa- yj5 
 
 ' x + y X -y 
 
 the two points c and c' are harmonically conjugate to each other, 
 with respect to the two given points^ a and b ; and when they 
 vary together, in consequence of the variation of the value of 
 
 -, they form (in a well-known sense), on the indefinite right 
 
 line AB, divisions in involution; the double points (ov foci) of 
 this involution, namely, the points of which each is its oion 
 conjugate, being the points a and b themselves. As a verifi- 
 cation, if we denote by p. the vector of the middle point m of 
 the given interval ab, so that ^ 
 
 A M C B C' 
 
 /3-/i=/x-a, or/i = J(a+/3), Fig. 17. 
 
 we easily find that 
 
 y - f-i _y - X P -luL MCMB^ 
 
 /3-jU y ^ X~ y' - fx MB MC'* 
 
 so that the rectangle under the distances mc, mc', of the two 
 variable but conjugate points^ c, c', from the centre m of the 
 involution, is equal to the constant square of half the interval 
 between the two double points, a, b. More generally, if we 
 write 
 
 xa+ y(5 , _ Ixa + my (5 
 
 ' X +y ^ lx + my ' 
 
 where the anharmonic quotient — = — ,- is any constant scalar, 
 
 then in another known and modern* phraseology, the points 
 c and c' will form, on the indefinite line ab, tivo homographic 
 divisions, of which a and b are still the double points. More 
 generally still, if we establish the two equations, 
 
 * See the Gtometrie Supe'rieure of M. Chasle?, p. 107. (Paris, 1852.) 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 17 
 
 xa + vQ , , lxa + my 3' 
 
 y= ^, and 7'=— ^^, 
 
 x^y lx-\- my 
 
 I , , y . 
 
 — beinof still constant, but - variable, while a = oa', 3' = ob', 
 
 and y' = oc', the two given lines, ab and a'b', are then homo- 
 graphically divided, by the two variable points, c and c', not 
 now supposed to move along one common line. 
 
 27. When the linear equation aa + bf3 + cy = subsists, 
 without the relation « -i- ^ + c = between its coefBcients, then 
 the three co-initial vectors a, /3, y are still complanar, but they 
 no longer terminate on one right line ; their term-points a, b, c 
 being now the corners of a triangle. 
 
 In this more general case, we may propose to find the vec- 
 tors a', j3', y' of th€ three points, 
 
 a' = oabc, b'=obca, 
 
 C'= OCAB ; 
 
 that is to say, of the points in 
 
 which the lines drawn from the 
 
 origin o to the three corners of 
 
 the triangle intersect the three 
 
 respectively opposite sides. The three collineations oaa', &c., 
 
 give (by 19) three expressions of the forms, 
 
 a = Xa, (5' = yj3, y' = Z.y, 
 
 where x, y, z are three scalars, which it is required to deter- 
 mine by means of the three other collineations, a'bc, &c., with 
 the help of relations derived from the principle of Art. 23. 
 Substituting therefore for a its value re 'a', in \)i\^ given linear 
 equation, and equating to zero the sum of the coefficients of 
 the new linear equation which results, namely, 
 
 and eliminating similarly j3, 7, each in its turn, from the ori- 
 ginal equation ; we find the values, 
 
 -a -h -c 
 
 X = , y = , z = 7 ; 
 
 ft + c ^c + a a^ b 
 
18 ELEMENTS OF QUATERNIONS. [boOK I. 
 
 whence the sought vectors are expressed in either of the two 
 following ways : 
 
 or 
 
 
 J , -aa 
 1. , . a =7 , 
 
 b + c 
 
 ^ c + a 
 
 '^~a + b' 
 
 II. 
 
 , bfi + Cy 
 
 C + a 
 
 , aa + b[5 
 
 ^ a + b 
 
 In fact we see, by one of these expressions for a, that a' is on 
 the line oa ; and by the other expression for the same vector 
 a', that the same point a' is on the line bc. As another veri- 
 fication, we may observe that the last expressions for a, j5', y\ 
 coincide with those which Avere found in Art. 24, for a, /3, y 
 themselves, on the particular supposition that the three points 
 a, B, c were collinear. 
 
 28. We may next propose to determine the ratios of the 
 segments of the sides of the triangle abc, made by the points 
 a', b', c'. For this purpose, we may write the last equations 
 for a', j3', y under the form, 
 
 0=^b{a'-(5)-c{y-a') = c((5'-y)-a{a-(5') = a{y'-a) 
 
 and we see that they then give the required ratios, as follows : 
 
 ba'_ c cb' a Ac'_ b 
 
 a'c b' b'a c' c'b a' 
 
 whence we obtain at once the known equation of six segments, 
 
 ba' cb' ac' 
 a'c b'a c'b ' 
 
 as the condition of concurrence of the three right lines a a', bb', 
 cc', in a common point, such as o. It is easy also to infer, from 
 the same ratios of segments, the following proportion of coeffi- 
 cients and areas, 
 
 a:b:c= OBC : oca : gab, 
 
 in which we must, in general, attend to algebraic signs ; a tri- 
 angle being conceived to pass {through zero) from positive to 
 negative, or vice versa, as compared with any give?i triangle in 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 19 
 
 its own plane, when (in the course of any continuous change) 
 its vertex crosses its base. It may be observed that with this 
 conveiition (which is, in fact, a necessary one, for the establish- 
 ment o{ general for mulce) we have, for any three points^ the 
 equation 
 
 ABC + BAC = 0, 
 
 exactly as we had (in Art. 5) for any two points, the equa- 
 tion 
 
 AB+ BA= 0. 
 
 More fully, we have, on this plan, the formula3, 
 
 ABC = - BAC = BCA = - CBA = CAB = - ACB ; 
 
 and any two complanar triangles, abc, a'b'c', bear to each other 
 a positive or a negative ratio, according as the two rotations, 
 which may be conceived to be denoted by the same symbols 
 ABC, a'b'c', are similarly or oppositely directed. 
 
 29. If a' and b' bisect respectively the sides bc and ca, 
 then 
 
 a = b = c, 
 
 and c' bisects ab ; whence the known theorem follows, that 
 the three bisectors of the sides of a triangle concur, in a point 
 which is often called the centre of gravity, but which we pre- 
 fer to call the mean point of the triangle, and which is here the 
 origiji o. At the same time, the first expressions in Art. 27 
 for a, ft', y' become, 
 
 "~~2' ^^"2' ^^"2' 
 
 whence this other known theorem results, that the three bisec- 
 tors trisect each other, 
 
 30. The linear equation between a, ft, y reduces itself, in 
 the case last considered, to the form, 
 
 a + /3 4 7 = 0, or oa + ob + oc = ; 
 
 the three vectors a, ft, y, or oa, ob, oc, are therefore, in this 
 ca^e, adapted (by Art. 10) to become the successive sides of a. 
 
20 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book I. 
 
 triangle, by transports without rotation ; and ticcordingly, if 
 we complete (as in Fig. 19) the /^c 
 
 parallelogram aobd, the triangle 
 GAD will have the property in 
 question. • It follows (by 11) 
 that if we project the four points 
 o, A, B, c, by any system of pa- 
 rallel ordinates, into four other A^ 
 points, o^, A^, B^, c , on any as- 
 sumed pZ«we, the sum of the three j^ 
 projected vectors^ a^, j3^, y^, or Fig. 19. 
 o A , &c., will be null; so that we shall have the new linear 
 equation, 
 
 or. 
 
 o A^ + o B^ + o^c^ = ; 
 
 and in fact it is evident (see 
 
 Fig. 20) that the projected 
 
 mean point o^ will be the mean 
 
 point of the projected triangle, ^'^" ^^• 
 
 A^, B^, c^. We shall have also the equation, 
 
 (a,-o) + (/3,-^) + (y,--y) = 0; 
 
 where 
 
 hence 
 
 a^- a = O^A - OA = (O^A + AA ) - (OO^ + O^a) = AA^ - 00^ ; 
 
 OO^ = ^ (aA^ -\ BB^ + CC ). 
 
 or the ordinate of the mean point of a triangle is the mean of 
 the ordinates of the three corners. 
 
 Section 3. — On Plane Geometrical Nets, 
 
 31. Resuming the more general case of Art. 27, in which 
 the coefficients «, b, c are supposed to be unequal, we may next 
 inquire, in what points a", b", c" do the lines b'c', c'a', a'b' 
 meet respectively the sides bc, ca, ab, of the triangle ; or may 
 seek to assign the vectors a\ /3", y" of the points of intersec- 
 tion (comp. 27), 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 
 
 21 
 
 A =BC*BC, B =CA*CA, C -ABAB. 
 
 The first expressions in Art. 27 for |3', 7', give the equa- 
 tions, 
 
 B" 
 
 Fig. 21. 
 
 (c -f «) j3' + ^>i3 = 0, (a + &)y + C7 = ; 
 
 whence 
 
 b[5-cy _ (a + b)y-(c+a)j5\ 
 b- c {a + b) ~ {c -\- a) 
 
 but (by 25) one member is the vector of a point on bc, and 
 the other of a point on b'c' ; each therefore is a value for the 
 vector a" of a", and similarly for j3" and 7". We may there- 
 fore write, 
 
 „_bfi- Cy ^„ Cy - aa „ tta- b[5 
 
 a = -7 , ~ 
 
 o - c 
 
 ^,.^cy-aa^ 
 
 c - a 
 
 7 = 
 
 and by comparing these expressions with the second set of 
 values of a', /3', 7' in Art. 27, we see (by 26) that the points 
 a", b", c" are, respectively, the harmonic conjugates (as they 
 are indeed known to be) of the points a', b', c', with respect 
 to the three pairs of points, b, c ; c, a ; a, b ; so that, in the 
 notation of Art. 25, we have the equations, 
 
 (baca") = (cb'ab") = (ac'bc") =- I. 
 
 And because the expressions for a", /3", 7" conduct to the fol- 
 lowing linear equation between those three vectors, 
 
22 ELEMENTS OF QUATERNIONS. [boOK I. 
 
 {b-c)a'+ (c-«)j3"+ {a - b)y"=0, 
 with the relation 
 
 (b-c)+ {c-a) + (a-b) = 
 
 between its coefficients, we arrive (by 23) at this other known 
 theorem, that the three points a", b", c" are collifiear, as indi- 
 cated by one of the dotted lines in the recent Fig. 2 1 . 
 
 32. The line a"b'c' may represent any rectili?iear transver- 
 sal, cutting the sides of a triangle abc ; and because we have 
 
 ba"_ «"-/3 ^ c 
 a"c 7 - a" b 
 
 while -7- = -, and —r- = -, as before, we arrive at this other 
 ba c cb a 
 
 equation of six segments, for any triangle cut by a right line 
 
 (comp. 28), 
 
 ba" cb' ac' _ 
 
 a"c b'a c'b 
 
 which again agrees with known results. 
 
 33. Eliminating j3 and 7 between either set of expressions 
 
 (27) for j3' and y', with the help of the given linear equation, 
 
 we arrive at this other equation, connecting the three vectors 
 
 a, /3', 7' : 
 
 O = - «a + (c + «) j3' + (a + ^) 7'. 
 
 Treating this on the same plan as the given equation between 
 a, j3, 7> we find that if (as in Fig. 21) we make, 
 
 a'" = OA • Bc', b"' = OB • c'a', C ' = DC ' a'b', 
 
 the vectors of these three new points of intersection may be ex- 
 pressed in either of the two following ways, whereof the first 
 is shorter, but the second is, for some purposes (comp. 34, 36) 
 more convenient : 
 
 '" ^ «« n.n^ bP ,„^ Cy ^ 
 
 2a + b + c ^ 2b^c + a ^ 2c + a + b' 
 
 or 
 
 „, _ 2aa + bj5 + Cy ^,„ _ 2^/3 + cy + aa 
 
 ^ 2a + b^c ' ^ ~ 26 + c + « ' 
 ,„ _ 2cy -{ aa^bf5 
 ^ 2c + « + ft 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 23 
 
 And the three equations, of which the following is one, 
 
 {h-c)a:'- (26+ c + «)/3'"+ (2c+ « + 6)7'" = 0, 
 
 with the relations between their coefficients w^hich are evident 
 on inspection, show (by 23) that we have the three additional 
 collineations, a"b'"c'", b"c"'a'", c"a"'b'", as indicated by three of 
 the dotted lines in the figure. Also, because we have the two 
 expressions, 
 
 „, (a-\-b)y+(c + a)(5' „ _(a +b)y - (c -¥ a)^' 
 
 ." ~ {a-\-b) + (€ + a) ' {a + b)-(c + a) ^ 
 
 we see (by 26) that the two points a", a'" are harmonically con- 
 jugate with respect to b' and c' ; and similarly for the two 
 other pairs of points, b", b'", and c", c'", compared with c', a', 
 and with a', b': so that, in a notation already employed (25, 
 31), we may write, 
 
 (b a'"c a") = (c b'Vb") = (a'c'"b'c") = - 1 . 
 
 34. If we beyin^ as above, with any four complanar points, 
 o, A, B, c, of which no three are collinear, we can (as in Fig. 
 18), by what may be called a First Construction, derive from 
 them six lines, connecting them two by two, and intersecting 
 each other in three new points, a', b', c' ; and then by a Second 
 Construction (represented in Fig. 21), we may connect these 
 by three new lines, which will give, by their intersections with 
 the former lines, six new points, a", . . c"\ We might pro- 
 ceed to connect these with each other, and with the given 
 points, by sixteen new lines, or lines of a Third Construction, 
 namely, the four dotted lines of Fig. 21, and twelve other 
 lines, whereof three should be drawn from each of the four 
 given points : and these would be found to determine eighty- 
 four new points of intersection, of which some may be seen, 
 although they are not marked, in the figure. 
 
 But however far these processes oi linear construction may 
 be continued, so as to form what has been called* a plane 
 
 * By Prof. A. F. INIobujs, in page 274 of his Bary centric Calculus (dcr baryrcu- 
 trische Calcul, Leipzig, 1827). 
 
24 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 geometrical net, the vectors of the points thus determined have 
 all one common property : namely, that each can be represented 
 by an expression of the form, 
 
 xaa H- yh^ -1- zcy 
 xa + yh + zc 
 
 where the coefficients x, y, z are some whole numbers. In fact 
 we see (by 27, 31, 33) that such expressions can be assigned 
 for the nine derived vectors, a', . . . y", which alone have been 
 hitherto considered ; and it is not 'difficult to perceive, from 
 the nature of the calculations employed, that a similar result 
 must hold good, for every vector subsequently deduced. But 
 this and other connected results will become more completely 
 evident, and their geometrical signification will be better un- 
 derstood, after a somewhat closer consideration of anharmonic 
 quotients, and the introduction of a certain system o^ anhar- 
 monic co-ordinates, for points and lines in one plane, to which 
 we shall next proceed : reserving, for a subsequent Chapter, 
 any applications of the same theory to space. 
 
 Section 4. — On Anharmonic Co-ordinates and Equations of 
 Points and Lines in one Plane. 
 35. If we compare the last equations of Art. 33 with the 
 corresponding equations of Art. 31, we see that the harmowc 
 group ba'ca", on the side bc of the triangle abc in Fig. 21, 
 has been simply reflected into another such group, b V'c'a", on 
 the line b'c', by a harmonic pencil of four rays, all passing 
 through the point o ; and similarly for the other groups. 
 More generally, let oa, ob, oc, od, or briefly o.abcd, be 
 any pencil, with the point o for vertex ; and let the new ray 
 OD be cut, as in Wig. 22, by the three sides of the triangle 
 ABC, in the three points Ai, Bi, Ci ; let also 
 
 yh^ + zcy 
 
 OAi = ai = — ^ ^, 
 
 yb 4- zc 
 
 so that (by 25) we shall have the anharmonic quotients, 
 
 y , ^ 
 
 (ba'cai) = -, (ca'b.\i) = -; 
 
 ^ ^ 2 y 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 
 
 25 
 
 and let us seek to express the two other vectors of intersec- 
 tion, j3i and 71, with a view to 
 determining the anharmonic ra- 
 tios of the groups on the two 
 other sides. The given equation 
 (27), 
 
 «a + 6/3 + cy = 0, 
 
 shows us at once that these two 
 vectors are. 
 
 OB 
 
 1 - Pl ; 
 
 Fig. 22. 
 
 001 = ^1 = 
 
 {z-y)h^-^zaa 
 
 {z-y)b + za * 
 whence we derive (bj 25) these two other anharmonics, 
 
 (cb'aBi) = 
 
 (bCACi) 
 
 y -2 
 
 so that we have the relations, 
 
 (CB'aBi) + (ca'bAi) = (bc'aCi) + (ba'cAi) = 1. 
 
 Bat in general, for any four collinear points a, b, c, d, it is 
 not difficult to prove that 
 
 AB AC 
 
 CD+ BD= DA 
 
 BC CB 
 
 whence by the definition (25) of the signification of the sym- 
 bol (abcd), the following identity is derived, 
 
 (abcd) + (acbd)= 1. 
 
 Comparing this, then, with the recently found relations, we 
 have, for Fig. 22, the following anharmonic equations ; 
 
 (cab'Bi) = (ca'bAi) = - ; 
 
 y 
 
 (bac'Ci) = (ba'cAi) =-; 
 
 and we see that (as was to be expected from known princi- 
 
26 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 pies) the anharmonic of the group does not change, when we 
 pass from one side of the triangle, considered as a transversal 
 of the pencil, to another such side, or transversal. We may 
 therefore speak (as usual) of such an anharmonic of a group^ 
 as being at the same time the Anharmonic of a Pencil ; and, 
 with attention to the order of the rays, and to the definition 
 (25), may denote the two last anharmonics by the two following 
 reciprocal expressions: 
 
 z y 
 
 (o.cabd) = -; (o.bacd) = -; 
 
 y ^ 
 
 with other resulting values, when the order of the rays is 
 changed ; it being understood that 
 
 (o . cabd) = (c'aVd'), 
 
 if the rays oc, oa, ob, od be cut, in the points c', a\ b\ d\ 
 by any one right line. 
 
 36. The expression (34), 
 
 xaa + yh^ + zcy 
 p- J 
 
 xa +yo + zc 
 
 may represent the vector o^ any point p in the given plane ^ by a 
 suitable choice of the coefficients x, y, x, or simply of their ra- 
 tios. For since (by 22) the three complanar vectors pa, pb, 
 PC must be connected by some linear equation, of the form 
 
 «' . PA + i' . PB -r c' . PC = 0, 
 or 
 
 aXa-p) + b'(f5-p) + c(y-p) = 0, 
 which gives 
 
 a a + b'Q + cy 
 P~ 
 
 a' + b' + c 
 
 we have only to write 
 
 a' b' d 
 
 a b " c 
 
 and the proposed expression for p will be obtained. Hence 
 it is easy to infer, on principles already explained, that if we 
 write (compare- the annexed Fig. 23), 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 27 
 
 Pi=PABC, P2 = PB'CA, P3 = PCAB, 
 
 we shall have, with the same coefficients xyz, the following 
 expressions for the vectors opj, 0P2, 
 0P3, or |0i, /02, /03, of these three points 
 of intersection, Pi,*P25 P3 : 
 
 yh^ + zcy 
 ^^~ yb + zc 
 
 
 ^2=- 
 
 zcy + xaa 
 
 J 
 
 zc\ xa ' 
 
 xaa + yhfi ^ 
 
 
 ^ xa^yh Fig. 23 
 
 which give at once the following anharmonics of pencils, or of 
 groups, 
 
 (a . BOCP) = (ba CPi) = - ; 
 
 z 
 z 
 
 (B . COAP) = (cb'aPz) = - ; 
 X 
 
 X 
 (C . AOBP) = (ac'bPs) = - ; 
 
 y 
 
 whereof we see that the product is unity. Any two of these 
 three pencils suffice to determine the position of the point P, 
 when the triangle abc, and the origin o are given ; and there- 
 fore it appears that the three coefficients x, y, z, or any scalars 
 proportional to them, of which the ^'z^o^zVw^a- thus represent the 
 anhai^monics of those pencils, may be conveniently called the 
 Anharmonic Co-ordinates of that point, p, with respect to 
 the given triangle and origin : while the point p itself may be 
 denoted by the Symbol, 
 
 p = (07, y, z). 
 
 With this notation, the thirteen points of Fig. 21 come to be 
 thus symbolized ; 
 
 a =(1,0,0), b =(0,1,0), c =(0,0,1), = (1,1,1); 
 
 a' =(0,1,1), B' =(1,0,1), €'=(1,1,0); 
 
 a" = (0,1,-1), B" = (-1,0, 1), €"=(1,-1,0); 
 
 A'"=(2, 1, 1), B'"= (1,2,1), €'"=(1,1,2). 
 
28 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 37. If Pi and Pa be any two points in the given plane, 
 
 Pi = (^H yi, zi), P2 = (^2> y2, Z2), 
 
 and if t and u be any two scalar coefficients, then the following 
 third pointy 
 
 p = (toi + UX2, tyi + uy^i tzx + uz^, 
 
 is collinear with the two former points, or (in other words) is 
 situated on the right line PiPg. For, if we make 
 
 a; = ^a!i + 11X2, y=ty\^ wyz) z = #Zi + uz^r 
 
 and 
 
 a^ifla + . . x^aa + . . xaa + . . 
 
 p\ = J Pa"" > /> = J 
 
 aJia + . . ^2« + • • a:a + . . 
 
 these vectors of the three points P1P2P are connected by the 
 linear equation, 
 
 t (xia -h . .)pi + u (x^a + . 0/02 - {xa + . .) /o = ; 
 
 in which (comp. 23), the s?im of the coefficients is zero. Con- 
 versely, the point p cannot be collinear with Pi, Pg, unless its 
 co-ordinates admit of being thus expressed in terms of theirs. 
 It follows that if a variable point p be obliged to move along a 
 given right line PiPg, or if it have such a line (in the given 
 plane) for its locusy its co-ordinates xyz must satisfy a homo- 
 geneous equation of the first degree, with constant coefficients ; 
 which, in the known notation of determinants, may be thus 
 written, 
 
 X, y, z 
 = Xu yi, z^ 
 
 «^2> y^i Z2 
 or, more fully, 
 
 = x {yxZ^ - z{y^ + y {zix^ ~ ofiZz) + z {x^y^ - y^x^) ; 
 
 or briefly, 
 
 = l.v + my + nz, 
 
 where /, m, n are three constant scalars, whereof the quotients 
 determine the position of the right line A, which is thus the 
 locus of the point p. It is natural to call the equation, which 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 29 
 
 thus connects the co-ordinates of the point p, the Anharmonic 
 Equation of the Line A ; and we shall find it convenient also 
 to speak of the coefficients /, w, n, in that equation, as being 
 the Anharmonic Co-ordinates of that Line: which line may 
 also be denoted by the Symbol^ 
 
 A = [Z, m, w] . 
 
 38. For example, the three sides bc, ca, ab of the given 
 triangle have thus for their equations, 
 
 a; = 0, y = 0, 2=0, 
 
 and for their symholsy 
 
 [1,0,0], [0,1,0], [0,0,1]. 
 
 The three additional lines oa, ob, oc, of Fig. 18, have, in hke 
 manner, for their equations and symbols, 
 
 3/-0 = O, 2-37 = 0, x-y=0, 
 
 [0,1,-1], [-1,0,1], [1,-1,0]. 
 
 The lines b'c'a", c'a'b", a'b'c", of Fig. 21, are 
 
 y + z -x = 0, z-rx-i/ = 0) x + y -z = 0, 
 or 
 
 [-1,1,1], [1,-1,1], [1,1,-1]; 
 
 the lines aV'c'", b"c'V", cV'b'', of the same figure, are in like 
 manner represented by the equations and symbols, 
 
 y + z-Sx = 0, z + x-3y=0, x-\^y-3z = 0, 
 [-3,1,1], [1,-3,1], [1,1,-3]; 
 
 and the line a"b "c" is 
 
 X -^ y + z=0, or [1, 1, 1]. 
 
 Finally, we may remark that on the same plan, the equation 
 and the symbol of what is often called the line at infinity, or 
 of the locus of all the irifinitely distant points in the given plane, 
 are respectively, 
 
 ax -v by ^ cz = 0, and [a, b, c] ; 
 
30 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 because the linear function, ax + hy + cz, of the co-ordinates 
 z, y, 2r of a point p in the plane, is the denominator of the ex- 
 pression (34, 36) for the vector p of that point : so that the 
 point p is at an infinite distance from the origin o, when, and 
 only when, this linear function vanishes. 
 
 39. These anharmonic co-ordinates of a line, although 
 above interpreted (37) with reference to the equation of that 
 line, considered as connecting the co-ordinates of a variable 
 point thereof, are capable of receiving an independent geome- 
 trical interpretation. For the three points l, m, n, in which 
 the line A, or [/, m, w], or lx\my \nz = 0, intersects the three 
 sides BC, CA, ab of the given triangle abc, or the three given 
 lines a? = 0, 7/=0, 2:=0 (38), may evidently (on the plan of 
 36) be thus denoted : 
 
 L = (0, 7i, - m) ; M = (- w, 0, /) ; n = (m, - I, 0). 
 
 But we had also (by 36), 
 
 a" = (0,1,-1); b"=(- 1,0,1); c"= (1,-1,0); 
 
 whence it is easy to infer, on the principles of recent articles, 
 that 
 
 — = (ba"cl) ; - = (cb"am) ; — = (ac'bn) ; 
 m ^ n ^ ' I ^ 
 
 with the resulting relation, 
 
 (ba"cl) . (cb"am) . (ac"bn) = 1. 
 
 40. Conversely, this last equation is easily proved, with 
 the help of the known and general relation between segments 
 (32), applied to any two transversals, a"b"c" and lmn, of any 
 triangle abc. In fact, we have thus the two equations, 
 
 ba" cb" ac"_ bl cm an 
 
 a"c b"a c"b ' LC MA NB ' 
 
 on dividing the former of which by the latter, the last formula 
 of the last article results. We might therefore in this way 
 have been led, without any consideration of a variable point p, 
 
CHAP. II.] POINTS AND LINES IN A GIVEN PLANE. 31 
 
 to introduce three auxiliary scalar s^ /, ?w, n^ defined as having 
 
 71 I Tfl 
 
 their quotients — , -, — equal respectively, as in 39, to the 
 
 three anharmonics of groups, 
 
 (ba"cl), (cb"am), (ac"bn); 
 
 and then it would have been evident that these three scalars, 
 /, m, n (or any others proportional thereto), are sufficient to 
 determine the position of the right line A, or lmn, considered 
 as a transversal of the given triangle abc : so that they might 
 naturally have been called, on this account, as above, the an-- 
 harmonic co-ordinates of that line. But although the anhar- 
 monic co-ordinates of a point and of a line may thus be inde- 
 pendently defined^ yet the geometrical utility of such definitions 
 will be found to depend mainly on their combination : or on the 
 formula Ix ^-my a- nz=0 of 37, which may at pleasure be con- 
 sidered as expressing, either that the variable point (re, y, z) is 
 situated somewhere upon the given right line [/, m, ri\ ; or else 
 that the variable line [/, tw, n\ passes, in some direction, through 
 the given point {x, y, z). 
 
 41. If Ai and As be any two right lines in the given plane, 
 
 Ai = [/i, mi, ni], Aa = [h, m^, Wo], 
 
 then any third right line A in the same plane, which passes 
 through the intersection ArA25 or (in other words) which cow- 
 curs with them (at a finite or infinite distance), may be repre- 
 sented (comp. 37) by a symbol of the form, 
 
 'A = [til + uli, tmi + um2, tn^ + uji.^, 
 
 where t and u are scalar coefficients. Or, what comes to the 
 same thing, if I, m, n be the anharmonic co-ordinates of the 
 line A, then (comp. again 37), the equation 
 
 /, m, n 
 = 1 (min-i- nimz) + &c. = h, mi, Ui 
 
 hi 'mi, ni 
 
 must be satisfied ; because, if {X, Y, Z) be the supposed point 
 common to the three lines, the three equations 
 
32 ELEMENTS OF QUATERNIONS. [boOK I. 
 
 lX+mY+nZ=0, hX + m,Y+n,Z =0, kX + m^Y+n^Z^(S, 
 
 must co-exist. Conversely, this coexistence will be possible, 
 and the three lines will have a common point (which may be 
 infinitely distant), if the recent condition of concurrence be sa- 
 tisfied. For example, because [a, J, c] has been seen (in 38) 
 to be the symbol of the line at infinity (at least if we still re- 
 tain the same significations of the scalars a, 6, c as in articles 
 27, &c.), it follows that 
 
 A = [Z, m, ri] , and A' = [/+ ua, m + ub, n + uc] , 
 
 are symbols of two parallel lines ; because they concur at infi- 
 nity. In general, all problems respecting intersections of right 
 lines, coUineations of points, &c., in the given plane, when 
 treated by this anharmonic method, conduct to easy elimina- 
 tions between linear equations (of the scalar kind), on which 
 we need not here delay : the mechanism of such calculations 
 being for the most part the same as in the known method of 
 trilinear co-ordinates : although (as we have seen) the geome- 
 trical interpretations are altogether different. 
 
 Section 5. — On Plane Geometrical Nets, resumed. 
 
 42. If we now resume, for a moment, the consideration of 
 those plane geometrical nets, which were mentioned in Art. 34 ; 
 and agree to call those points and lines, in the given plane, ra- 
 tional points and rational lines, respectively, which have their 
 anharmonic co-ordinates equal (or proportional) to whole num- 
 bers ; because then the anharmonic quotients, which were dis- 
 cussed in the last Section, are rational ; but to say that a point 
 or line is irrational, or that it is irrationally related to the 
 given system o^four initial points o, a, b, c, when its anhar- 
 monic co-ordinates are not thus all equal (or proportional) to 
 integers ; it is clear that ivhatever four points we may assume 
 as initial, and however far the construction of the net may be 
 carried, the net-points and net-lines which result will all be ra- 
 tional, in the sense just now defined. In fact, we begin with 
 such; and the subsequent eZz/w ma ^20W5 (41) oan never after- 
 
CHAP. II.] PLANE GEOMETRICAL NETS. 33 
 
 wards conduct to any, that are of the contrary kind : the right 
 line which connects two rational points being always a rational 
 line ; and the point of intersection of two rational lines being 
 necessarily a rational point. The assertion made in Art. 34 
 is therefore fully justified. 
 
 43. Conversely, every rational point of the given plane, 
 with respect to the four assumed initial points oabc, is a point 
 of the net which those four points determine. To prove this, 
 it is evidently sufficient to show that every rational point 
 Ai = (0, y, z), on any one side bc of the given triangle abc, can 
 be so constructed. Making, as in Fig. 22, 
 
 Bi = oAi • CA, and Ci = oAi • ab, 
 
 we have (by 35, 36) the expressions, 
 
 Bi = (2/, 0,2/-2r), Ci=(z, ;2-y, 0); 
 
 from which it is easy to infer (by 36, 37), that 
 
 c' Bi • BC = (0, y,z- y), b'Ci • bc = (0, 2/ - z, z) ; 
 
 and thus we can reduce the linear construction of the rational 
 point (0, 2/j 2;), in which the two whole numbers y and z may 
 be supposed to be prime to each other, to depend on that of 
 the point (0, 1, 1), which has already been constructed as a'. 
 It follows that although no irrational point Q of the plane can 
 he a net-point, jet every such point can be indefinitely approached 
 to, by continuing the linear construction; 
 so that it can be included within a quadrila- 
 teral interstice P1P2P3P4, or even within a tri- 
 angular interstice P1P2P3, which interstice of p^^ -T^^* 
 
 the net can be made as small as we may de- 
 sire. Analogous remarks apply to irrational 
 lines in the plane, which can never coincide 
 with net-lines, but may always be indefinitely approximated to 
 by such. 
 
 44. If p, Pi, P2 be any three collinear points of the net, so 
 that the formulae of 37 apply, and if p'be any ^wr^^ net-point 
 {x, y, z) upon the same line, then writing 
 
 Xxa + yj) + z^c ~ Vx, x^a + y-h + z.c = v^. 
 
34 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 we shall have two expressions of the forms, 
 
 _ tVipi + UV2P2 , t'Vipi + UV2P2 
 
 tVi + UV2 ' t'Vi + UV2 ' 
 
 in which the coefficients tut'u are rational, because the co-or- 
 dinates xyz, &c., are such, whatever the constants abc may be. 
 We have therefore (by 25) the following rational expression 
 for the anharmonic of this net-group : 
 
 "" ^ tu' {X1/2 - yXi) {x'y, - y'x,) ' 
 
 and similarly for every other group of the same kind. Hence 
 every group of four coUinear net-points, and consequently also 
 every pencil of four concurrent net-lines, has a rational value for 
 its anharmonic function ; which value depends only on the pro- 
 cesses of linear construction employed, in arriving at that group 
 or pencil, and is quite independent of the configuration or ar- 
 rangement oiihefour initial points : because the three initial 
 constants, «, b, c, disappear ^vom the expression which results. 
 It was thus that, in Fig. 21, the niiie pencils, which had the 
 nine derived points a' . . c"' for their vertices, were all harmo- 
 nic pencils, in whatever manner the four points o, a, b, c 
 might be arranged. In general, it may be said that plane 
 geometrical nets are all homo graphic figures ;* and conversely, 
 in any two such ^2,wq figures, corresponding points may be con- 
 sidered as either coinciding, or at least (by 43) as indefinitely 
 approaching to coincidence, with similarly constructed points 
 of two plane nets : that is, with points of which (in their re- 
 spective systems) the anharmonic co-ordinates (36) are equal 
 integers. 
 
 45. Without entering heref on any general theory of trans- 
 fi)rmation of anharmonic co-ordinates, we may already see that 
 if we select any fjur net-points Oi, Ai, Bi, Ci, of which no three 
 are collinear, every other point p of the same net is rationally 
 related (42) to these ; because (by 44) the three new anhar- 
 
 * Compare the Geometrie Svpe'rieure of M. Chasles, p. 362., 
 t See Note A, on Anharmonic Co-ordinates. 
 
CHAP. II.] CURVES IN A GIVEN PLANE. 35 
 
 monies of pencils, (Aj . BiOiCip) = — , &c., are rational : and 
 
 therefore (comp. 36) the new co-ordinates Xi, r/i, Zi of the point 
 p, as well its old co-ordinates xi/z, are equal or proportional to 
 whole numbers. It follows (by 43) that everi/ point p of the 
 net can be linearly constructed, if ani/ four such points be 
 ffiven (no three being collinear, as above) ; or, in other words, 
 that the whole net can be reconstructed,* \^ any one of its qua- 
 drilaterals (such as the interstice in Fig. 24) be known. As 
 an example, we may suppose that the four points oa'b'c' in 
 Fig. 21 are given, and that it is required to r^c^juer from them 
 the three points abc, which had previously been among the 
 data of the construction. For this purpose, it is only neces- 
 sary to determine first the three auxiliary points a'", b'", c"', as 
 the intersections oa' • b'c', &c. ; and next the three other auxi- 
 liary points a", b", c", as b'c' • b'"c'", &c. : after which the for- 
 mulae, A = b'b" • c'c"j &c., will enable us to return, as required, 
 to the points a, b, c, as intersections of known right lines. 
 
 Section 6. — On Anharmonic Equations, and Vector Expres- 
 sions, for Curves in a given Plane. 
 
 46. When, in the expressions 34 or 36 for a variable vec- 
 tor p = OP, the three variable scalars (or anharmonic co-ordi- 
 nates) X, y, z are connected by any given algebraic equation, 
 such as 
 
 fp{x,y, 2) = 0, 
 
 supposed to be rational and integral, and homogeneous of the 
 p^^ degree, then the locus of the term v (Art. 1) of that vector 
 is biplane curve of the jo^^ order; because (comp. 37) it is cut 
 
 * This theorem (45) of the possible reconstruction of a plane net, from any one 
 of its quadrilaterals^ and the theorem (43") respecting the possibility of indefi- 
 nitely approaching by net-lines to the points above called irrational (ii), without 
 ever reaching such points by any processes of linear constrtiction of the kind here 
 considered, have been taken, as regards their substance (although investigated by a 
 totally different analysis), from that highly original treatise of Mobius, which was 
 referred to in a former note (p. 23). Compare Note B, upon the Bai-ycentric Calcu- 
 lus ; and the remarks in the following Chapter, upon nets in space. 
 
 I 
 
36 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 in p points (distinct or coincident, and real or imaginary), by 
 any given right line, Ix -^^ my ■\- nz = 0, in the given plane. 
 For example, if we write 
 
 f^aa + u^b^ + v'^cy 
 
 where t, u, v are three new variable scalars, of which we shall 
 suppose that the sum is zero, then, by eliminating these be- 
 tween the four equations, 
 
 a; = t^, y = u\ z=v\ t + u+v = 0, 
 
 we are conducted to the following equation of the second 
 
 degree, q =^ = ^2 ^ ^2 + ^^ - 22/z - 2zx - 2xy ; 
 
 so that here p-% and the locus of p is a conic section. In fact, 
 it is the conic which touches the sides of the given triangle abc, 
 at the points above called a', b', c' ; for if we seek its inter sec- 
 tions with the side bc, by making a; = (38), we obtain a 
 quadratic with equal roots, namely, {y-zy = 0\ which shows 
 that there is contact with this side at the point (0, 1, 1), or a' 
 (36) : and similarly for the two other sides. 
 
 47. If the point o, in which the three right lines aa', bb', 
 cc' concur, be (as in Fig. 18, &c.) interior to the triangle abc, 
 the sides of that triangle are then all cut internally, by the 
 points a', b', c' of contact with the conic ; so that in this case 
 (by 28) the ratios of the constants «, h, c are all positive, and 
 the denominator of the recent expression (46) for p cannot va- 
 nish, for any real values of the va- 
 riable scalars t, u^ v, and conse- 
 quently no such values can render 
 infinite that vector p. The conic is 
 therefore generally in this case, as in 
 Fig. 25, an inscribed ellipse ; which 
 becomes however the inscribed cir- 
 cle, when 
 
 «-M &-^ : c"^ = s - a : s - b : s - c ; 
 
 a, b, c denoting here the lengths of ^'^' ^^* 
 
 the sides of the triangle, and s being their semi-sum. 
 
CHAP. II.] ANHARMONIC EQUATIONS OF PLANE CURVES. 37 
 
 48. But if the point of concourse o be exterior to the tri- 
 angle of tangents abc, so that two of its sides are cut externally^ 
 then two of the three ratios o^ segments (28) are negative; and 
 therefore one of the three constants a, h, c may be treated as 
 < 0, but each of the two others as > 0. Thus if we suppose 
 that 
 
 i>0, oO, «<0, « + J>0, a+oO, 
 
 a' will be a point on the side b itself, but the points b', c', o 
 will be on the lines Ac, ab, ka! prolonged, as in Fig. 26 ; and 
 then the conic a'b'c' will be an 
 ellipse (including the case of a 
 circle), or a parabola, or an hy- 
 perbola^ according as the roots of ^ 
 the quadratic. 
 
 Fig. 26. 
 
 {a + c) t^ + 2ctu +{b + c)u^ = 0, 
 
 obtained by equating the deno- b' 
 
 minator (46) of the vector p to 
 
 zero, are either, 1st, imaginary ; or Ilnd, real and equal; or 
 
 Ilird, real and unequal : that is, according as we have 
 
 bc + ca + ab>0, or = 0, or < ; 
 
 or (because the product abc is here negative), according as 
 
 a'^ + b-^ + c-^ < 0, or =0, or > 0. 
 
 For example, if the conic be what is often called the exscribed 
 circle, the known ratios of segments give the proportion, 
 
 a'^ : 6"^ : c'^ = - s : s - c : s - b ; 
 and 
 
 -s + s-c + s-b<0. 
 
 49. More generally, if c^ be (as in Fig. 26) a point upon 
 the side ab, or on that side prolonged, such that cc^ is parallel 
 to the chord b'c', then 
 
 c^c' : Ac' = cb' : ab' = - rt : c, and ab : ac' = « + i : 6 ; 
 
 writing then the condition (48) of ellipticity (or circularity) 
 
38 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 under the form, ^— < —7—, we see that the conic is an ellipse, 
 c 
 
 parabola, or hyperbola, according as c^c' < or = or > ab ; the 
 arrangement being stilU in other respects, that which is repre- 
 sented in Fig. 26. Or, to express the same thing more sym- 
 metrically, if we complete the parallelogram cabd, then ac- 
 cording as the point d falls, 1st, beyond the chord b'c', with 
 respect to the point a; or llnd, on that chord; or Ilird, 
 ivithin the triangle ab'c', the general arrangement of the same 
 Figure being retained, the curve is elliptic^ or parabolic, or 
 hyperbolic. In that other arrangement or configuration, which 
 answers to the system of inequalities, Z>>0, c>0, « + 5 + c<0, 
 the point a' is still upon the side bc itself, but o is on the line 
 a'a prolonged through a ; and then the inequality, 
 
 a (^ + c) + 6c < - (^>2 + 6c + c2) < 0, 
 
 shows that the conic is necessarily an hyperbola ; whereof it is 
 easily seen that one branch is touched by the side bc at a', 
 while the other branch is touched in b' and c', by the sides 
 CA and ba prolonged through a. The curve is also hyperbo- 
 lic, if either a + 6 or a + c be negative, while b and c are posi- 
 tive as before. 
 
 50. When the quadratic (48) has its roots real and un- 
 equal, so that the conic is an hyperbola, then the directions of 
 the asymptotes may be found, by substituting those roots, 
 or the values of t, u, v which correspond to them (or any 
 scalars proportional thereto), in the numerator of the expres- 
 sion (46) for p ; and similarly we can find the direction of the 
 axis of the parabola, for the case when the roots are real but 
 equal : for we shall thus obtain the directions, or direction, in 
 which a right line op must be drawn from o, so as to meet the 
 conic at infinity. And the same conditions as before, for dis- 
 tinguishing the species of the conic, maybe otherwise obtained 
 by combining the anharmonic equation, /= (46), of that 
 conic, with the corresponding equation ax + by ^■cz={) (38) of 
 the line at infinity ; so as to inquire (on known principles of 
 modern geometry) whether that line meets that curve in tivo 
 
CHAP. II.] DIFFERENTIALS — TANGENTS POLARS. 39 
 
 imaginary points^ or touches it, or cuts it, in points which (al- 
 though infinitely distant) are here to be considered as real, 
 
 51. In general, if /(a?, y, z) = be the anharmonic equa- 
 tion (46) oi any plane curve, considered as the locus of a varia- 
 ble point p ; and if the differential* of this equation be thus 
 denoted, 
 
 = d/(a?,3/, ^) = Xdar+ Ydy+^ds'; 
 
 then because, by the supposed homogeneity (46) of the func- 
 tion/, we have the relation 
 
 Xx^Yy + Zz=^fd, 
 
 we shall have also this other but analogous relation, 
 
 if , 
 
 x' - x'.y' -y \z' - z = diX',diy\<\.z\ 
 
 that is (by the principles of Art. 37), if p'=-(a;'j y\ z!) be any 
 point upon the tangent to the curve, drawn at the point 
 p = (re, y, z), and regarded as the limit of a secant. The sym- 
 hoi (37) of this tangent at p may therefore be thus written, 
 
 [X,y, ZJ, or [D,/ D,/, D,/]; 
 
 where d^, d^, d^ are known characteristics of partial deriva- 
 tion. 
 
 52. For example, whenyhas the form assigned in 46, as an- 
 swering to the conic lately considered, we have d.t/= 2{x-y-z), 
 &c. ; whence the tangent at any point (x, y, z) of this curve 
 may be denoted by the symbol, 
 
 \_x-y-z, y-z-x, z-x-y]; 
 
 in which, as usual, the co-ordinates of the line may be replaced 
 by any others proportional to them. Thus at the point a', or 
 (by 36) at (0, 1, 1), which is evidently (by the form of/) a 
 point upon the curve, the tangent is the line [- 2, 0, 0], or 
 [1, 0, 0] ; that is (by 38), the side bc of the given triangle, as 
 
 * In the theory of qziaternions, as distinguished from (although including) that 
 of vectors, it will he found necessary to introduce a new definition of differentials, on 
 account of the non- commutative property o{ quaternion-multiplication : hut, for the 
 present, the usual significations of the signs d and d are sufficient. 
 
40 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 was Otherwise found before (46). And in general it is easy to 
 see that the recent symbol denotes the right line, which is (in 
 a well known sense) the polar of the point {x, y, z), with re- 
 spect to the same given conic ; or that the line [X', F', Z''\ is 
 the polar of the point (x', y, z) : because the equation 
 
 Xx'+Yy' + Zz^O, 
 
 which for a conic may be written as X'x + Y'y + Z'z = 0, 
 expresses (by 51) the condition requisite, in order that a point 
 (x, y, z) of the curve* should belong to a tangent which passes 
 through the point {x\ y\ z). Conversely, the point {x, y, z) 
 is (in the same well-known sense) the po/^ of the line [X, Y, Z"] ; 
 so that the centre of the conic, which is (by known principles) 
 i\\Q pole of the line at infinity (38), is the point which satisfies 
 the conditions a-^X=^h-^Y=c-^Z \ it is therefore, for the pre- 
 sent conic, the point k = (6 + c, c + «, a + S), of which the 
 vector OK is easily reduced, by the help of the linear equation, 
 «a + Z>j3 + cy = (27), to the form, 
 
 2 {he + c« + ah) ' 
 
 with the verification that the denominator vanishes^ by 48, 
 when the conic is a parabola. In the more general case, when 
 this denominator is different from zero, it can be shown that 
 every chord of the curve, which is drawn through the extremity 
 K of the vector k, is bisected at that point k : which point 
 would therefore in this way be seen again to be the centre. 
 
 53. Instead of the inscribed conic (46), which has been the 
 subject of recent articles, we may, as another example, consi- 
 der that exscribed (or circumscribed) conic, which passes 
 through the three corners a, b, c of the given triangle, and 
 touches there the lines aa", bb", cc" of Fig. 21. The anhar- 
 monic equation of this new conic is easily seen to be, 
 
 yz -v zx -\^ xy = ; 
 
 * If the curve /= were of a degree higher than the second, then the two equa- 
 tions above written would represent what are called the first polar, and the last or 
 the line-polar, of the point (x', y\ z'), with respect to the given curve. 
 
CHAP. 11 ] VECTOR OF A CUBIC CURVE. 41 
 
 the vector of a variable point p of the curve may therefore be 
 expressed as follows, 
 
 with the condition ^ + m + v = 0, as before. The vector of its 
 centre k' is found to be, 
 
 ^2 _^ 52 4. c2 - 2bc - 2ca - lab ' 
 
 and it is an ellipse, a parabola, or an hyperbola, according as 
 the denominator of this last expression is negative, or null, or 
 positive. And because these two recent vectors^ jc, k, bear a 
 scalar ratio to each other, it follows (by 19) that the three 
 points o, K, k' are collinear ; or in other words, that the line 
 of centres kk', of the two conies here considered, passes through 
 the point of concourse o of the three lines aa', bb', cc'. More 
 generally, if l be the pole of any given right line A = [/, w, n] 
 (37), with respect to the inscribed conic (46), and if l' be the 
 pole of the same line A with respect to the exscribed conic of 
 the present article, it can be shown that the vectors ol, ol', or 
 A, X', of these two poles are of the forms, 
 
 \ = k (laa + mb^ + ncy)^ A' = h! {laa + mb^ + ncy), 
 
 where k and k' are scalar s ; the three points o, l, l' are there- 
 fore ranged on one right line. 
 
 54. As an example of a vector-expression for a curve of an 
 order higher than the second, the following may be taken : 
 
 t^aa + U^bQ + v^Cy 
 ^ t^a + v?b + v^c 
 
 with ^ 4- M + r = 0, as before. Making x = t^, y^u^, z = v^, we 
 find here by elimination of t, u, v the anharmonic equation^ 
 
 {x-\-y+ zy - 27 xgz--^0; 
 
 the locus of the point p is therefore, in this example, a curve of 
 the third order, or briefly a cubic curve. The mechanism (41) 
 
 G 
 
42 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book I. 
 
 Fig. 27. 
 
 of calculations with anharmonic co-ordinates is so much the 
 same as that of the known trilinear method, that it may suffice 
 to remark briefly here that the sides of the given triangle abc 
 are the three (real) tangents of inflexion; the points of inflexion 
 being those which are marked as a", b", c" in Fig. 2 1 ; and the 
 origin of vectors o being a conjugate point* lia=b = c,in which 
 case (by 29) this origin o becomes (as in Fig. 19) the mean 
 point of the trian- 
 gle, the chord of 
 inflexion a"b"c" is 
 then the line at 
 infinity, and the 
 curve takes the 
 form represented 
 in Fig. 27; hav- 
 ing three infinite 
 branches, inscribed within the angles vertically opposite to 
 those of the given triangle abc, of which the sides are the 
 three asymptotes. 
 
 55. It would be improper to enter here into any details of 
 discussion of such cubic curves, for which the reader will na- 
 turally turn to other works.f But it may be remarked, in 
 passing, that because the general cubic may be represented, on 
 the present plan, by combining the general expression of Art. 
 34 or 36 for the vector p, with the scalar equation 
 
 s^ = 27kxgz, where s = a; + y-\- z; 
 
 k denoting an arbitrary constant, which becomes equal to 
 unity, when the origin is (as in 54) a conjugate point; it fol- 
 lows that if p = (x, y, z) and p' = (a?', y', z) be any two points 
 of the curve, and if we make s' = x' + y' + z, we shall have the 
 
 relation, 
 
 x^ ys' zs 
 
 sx sy sz' 
 
 xyzs ^ = xyz s^, or — ; 
 
 * Answering to the values ^=1, m = 0, v=Q\ where is one of the imaginary 
 cube-roots of unity ; which values of t^ u, v give x — y = z, and p = 0. 
 
 t Especially the excellent Treatise on Higher Plane Curves^ by the Rev. George 
 Salmon, F. T. C D., &c. Dublin, 1852. 
 
CHAP. II.] ANHARMONIC PROPERTY OF CUBIC CURVES. 43 
 
 in which it is not difficult to prove that 
 
 •^'=(a".pbp'b"); ^,= (b".pcp'c"); — , = (c". papV); 
 sx ^ sy ^ sz 
 
 the notation (35) of anharmonics of pencils being retained. 
 We obtain therefore thus the following Theorem : — " If the 
 sides of any given plane* triangle abc he cut (as in Fig. 2\)hy 
 any given rectilinear transversal a"b"c'', and if any two points 
 p and p' in its plane be such as to satisfy the anharmonic rela- 
 tion 
 
 (a". pbpV) . (b". pcp'c") . (c". papa") = 1, 
 
 then these two points p, p' are on one common cubic curve, which 
 has the three collinear points a", b", c" for its three real points 
 of inflexion^ and has the sides bc, ca, ab of the triangle for its 
 three tangents at those points ;" a result which seems to offer 
 a new geometrical generation for curves of the third order, 
 
 5Q. Whatever the order of a plane curve may be, or what- 
 ever may be the degree p of ihQfunctionf'm. 46, we saw in 51 
 that the tangent to the curve at any point p = (a:, y, z) is the 
 right line 
 
 A = [/, m, w], if 1= Hxf, rn = Hyfi n = n^f-, 
 
 expressions which, by the supposed homogeneity off, give the 
 relation, Ix -\-my+nz^ 0, and therefore enable us to establish 
 the system of the two following differential equations, 
 
 Idx + mdy + ndz = 0, xdl + ydm + zdn = 0. 
 
 If then, by elimination of the ratios of x, y, z, we arrive at a neio 
 homogeneous equation of the form, 
 
 as one that is true for all values of x, y, z which render the 
 function /= (although it may require to be cleared of factors, 
 introduced by this elimination), we shall have the equation 
 
 F(l,m, n) = 0, 
 
 * This Theorem may be exteaded, with scarcely any modification, from plane to 
 spherical curves., of the third order. 
 
44 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 as a condition that must be satisfied by the tangent A to the 
 curve, in all the positions which can be assumed by that right line. 
 And, by comparing the two differential equations, 
 
 dr(/, 772, W) = 0, red/ + 7/d77Z + 2:d77 = 0, 
 
 we see that we may write the proportion, 
 
 x\y\z= D/F : D,rtF : d„f, and the symbol v = (d^f, d„iF, d^f), 
 
 if {x, 7/, z) be, as above, the point of contact p of the variable 
 line [/, 772, n\ in any one of its positions, with the curve which 
 is its envelope. Hence we can pass (or return) from the tan- 
 gential equation f = 0, of a curve considered as the envelope of 
 a right line A, to the local equation f= 0, of the same curve 
 considered (as in 46) as the locus of a point p : since, if we ob- 
 tain, by elimination of the ratios of /, m, n, an equation of the 
 form 
 
 0=/(dzF, d,„f, d„f), 
 
 (cleared, if it be necessary, of foreign factors) as a conse- 
 quence of the homogeneous equation f = 0, we have only to 
 substitute for these partial derivatives, D/F, &c., the anhar- 
 monic co-ordinates x, 7/, z, to which they are proportional. 
 And when the functions /"and f are not only homogeneous (as 
 we shall always suppose them to be), but also rational and 
 integral (which it is sometimes convenient not to assume them 
 as being), then, while the degree of the function^ or of the 
 local equation, marks (as before) the order of the curve, the 
 degree of the other homogeneous function f, or of the tangential 
 equation F = 0, is easily seen to denote, in this anharmonic 
 method (as, from the analogy of other and older methods, it 
 might have been expected to do), the class of the curve to 
 which that equation belongs : or the number of tangents (dis- 
 tinct or coincident, and real or imaginary), which can be drawn 
 to that curve, from an arbitrary point in its plane. 
 
 57. As an example (comp. 52), if we eliminate x, y, z be- 
 tween the equations, 
 
 l = x-y-z, m = y-z - X, n = z-x-y, Ix + my + 7iz== 0, 
 where /, in, n are the co-ordinates of the tangent to the inscribed 
 
CHAP. II.] LOCAL AND TANGENTIAL EQUATIONS. 45 
 
 conic of Art. 46, we are conducted to the following tangen- 
 tial equation of that conic, or curve of the second class, 
 
 f(1, m,n) = mn + nl+ lm = ; 
 
 with the verification that the sides [1, 0, 0], &g. (38), of the 
 triangle abc are among the lines which satisfy this equation. 
 Conversely, if this tangential equation were given, we might 
 (by 5Q) derive from it expressions for the co-ordinates of con- 
 tact X, 2/, z, as follows : 
 
 a;=D/F = 772+72, 2/ = n -^ I, Z = I -^ m ', 
 
 with the verification that the side [1, 0, 0] touches the conic, 
 considered now as an envelope, in the point (0, 1, 1), or a', as 
 before : and then, by eliminating /, m, n, we should be brought 
 back to the local equation, f= 0, of 46. In like manner, from 
 the local equation /= yz + zx-\- xy = of the exscribed conic (53), 
 we can derive by differentiation the tangential co-ordinates,* 
 
 I = T>jf^= y -^ z, rn = z-\- X, n = X + y, 
 
 and so obtain by elimination the tangential equation, namely, 
 
 f(/, 7w, n) = l^ + m^+n''- 2mn - 2nl -2lm = 0; 
 
 from which we could in turn deduce the local equation. And 
 (comp. 40), the very simple formula 
 
 Ix + my+nz = 0, 
 
 which we have so often had occasion to employ, as connecting 
 two sets of anharmonic co-ordinates, may not only be consi- 
 dered (as in 37) as the local equation of a given right line A, 
 along which a point p moves, but also as the tangential equa- 
 tion of a given point, round which a right line turns : according 
 as we suppose the set I, 7n, n, or the set x, y, z, to be given. 
 Thus, while the right line a"b"c", or [1, 1, 1], of Fig. 21, was 
 
 * This name of " tangential co-ordinates'^ appears to have been first introduced 
 by Dr. Booth in a Tract published in 1840, to which the author of the present Ele- 
 ments cannot now more particularly refer : but the system of Dr. Booth was entirely 
 dilFerent from his own. See the reference in Salmon's Higher Plane Curves, note to 
 page 16. 
 
46 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 represented in 38 by the equation a; + z/ + 2: = 0, the point o of 
 the same figure, or the point (1, 1, 1), may be represented by 
 the analogous equation^ 
 
 l + m + n = 0; 
 
 because the co-ordinates I, ni, n of every line, which passes 
 through this point o, must satisfy this equation of the first de- 
 gree, as may be seen exemphfied, in the same Art. 38, by the 
 lines OA, ob, oc. 
 
 58. To give an instance or two of the use of forms, which, 
 although homogeneous, are yet not rational and integral {pQ), 
 we may write the local equation of the inscribed conic (46) as 
 
 follows : 
 
 ai + ?/4 + 22 = ; 
 
 and then (suppressing the common numerical factor J), the 
 partial derivatives are 
 
 I = x% m = 2/"2, n = z'h; 
 so that a form of the tangential equation for this conic is, 
 /-I + ni-i ^ ^-1 = Q . 
 
 Avhich evidently, when cleared of fractions, agrees with the first 
 form of the last Article : with the verification (48), that 
 ^-1 4. ^-1 4. c-i = when the curve is a parabola ; that is, when 
 it is touched (50) by the line at infinity (38). For the ex- 
 scribed conic (53), we may write the local equation thus, 
 
 x-'^ + y^ + 2-^ = 0; 
 
 whence it is allowed to write also, 
 
 Z=a;-2, m = y-'^, n-=z-\ 
 
 and 
 
 lh + mh + n^=0 ; 
 
 a form of the tangential equation which, when cleared of radi- 
 cals, agrees again Avith 57. And it is evident that we could 
 return, with equal ease, from these tangential to these local 
 equations. 
 
 59. For the cubic curve with a conjugate point (54), the 
 local equation may be thus written,* 
 
 * Compare Salmon's Higher Plane Curves, page 172. 
 
CHAP. II.] LOCAL AND TANGENTIAL EQUATIONS. 47 
 
 we may therefore assume for its tangential co-ordinates the 
 expressions, 
 
 / = x'i, m = ?/-!, n = ^i ; 
 
 and a form of its tangential equation is thus found to be, 
 
 Conversely, if this tangential form were given, we might re- 
 turn to the local equation, by making 
 
 X = Zf , y = m"f , z = w"2, 
 
 which would give x^-vy^-^ zi= 0, as before. The tangential 
 equation just now found becomes, when it is cleared of radi- 
 cals, 
 
 = 7-2 + ^-2 ^ ^-2 _ 2m-i n' - 2n-' l' - 21' m' ; 
 
 or, when it is also cleared of fractions, 
 
 = F = m^n^ + ^2/2 4. /2^2 _ 2nl^m - 2Im^n - 2mnH ; 
 
 of which the biquadratic form shows (by 5Q) that this cubic 
 is a curve of the fourth class, as indeed it is known to be. 
 The inflexional character (54) of the points a", b", c" upon 
 this curve is here recognised by the circumstance, that when 
 we make m -n = 0, in order to find the four tangents from 
 a" =(0, 1,- 1) (36), the resulting biquadratic, = m*- Alm^, has 
 three equal roots ; so that the line [1, 0, 0], or the side Bc, 
 counts as three, and is therefore a tangent of inflexion : the fourth 
 tangent from a" being the line [1, 4, 4], which touches the 
 cubic at the point (- 8, U 1). 
 
 60. In general, the two equations {6Q), 
 
 nDj.f- lDzf= 0, nTfyf- mBzf^ 0, 
 
 may be considered as expressing that the homogeneous equa- 
 tion, ^ 
 f{nx,ny, -lx-my) = 0, 
 
 which is obtained by eliminating z with the help of the rela- 
 tion Ix + my-^nz^ 0, from f(x, y, z) = 0, and which we may 
 
48 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 denote by {x, y) = 0, has two equal roots x:y,\{ /, wi, n be 
 still the co-ordinates of a tangent to the curve/*; an equality 
 which obviously corresponds to the coincidence of two intersec- 
 tions of that line with that curve. Conversely, if we seek by 
 the usual methods the condition of equality of two roots xiy of 
 the homogeneous equation of the p^^ degree, 
 
 = ^ (a;, y) =f{nx, ny, -Ix- my), 
 
 by eliminating the ratio x : y between the two derived homo- 
 geneous equations, = Dj.^, = d,^0, we shall in general be 
 conducted to a result of the dimension 2p{p- 1) in /, m, n, 
 
 and of the ^rm, 
 
 = wP^P-i) F (/, m, n) ; 
 
 and so, by the rejection of the foreign factor nP^P-'^\ introduced 
 by this elimination,* we shall obtain the tangential equation 
 F = 0, which will be in general of the degree /?(p - 1 ) ; such being 
 generally the known class (pQ) of the curve of which the 
 order (46) is denoted by p : with (of course) a similar mode of 
 passing, reciprocally, from a tangential to a local equation. 
 
 61. As an example, when the function /has the cubic form 
 assigned in 54, we are thus led to investigate the condition for 
 the existence of two equal roots in the cubic equation, 
 
 = (p(x,y)= [(n-l)x+ (m - l)y]'^ + ''277i^xy(lx+ my), 
 
 by eliminating x : y between two derived and quadratic equa- 
 tions ; and the result presents itself, in the first instance, as of 
 the twelfth dimension in the tangential co-ordinates /, m, n ; 
 but it is found to be divisible by n^, and when this division is 
 effected, it is reduced to the sixth degree, thus appearing to 
 imply that the curve is of the sixth class, as in fact the general 
 cubic is well known to be. A. further reduction is however 
 possible in the present case, on account of the conjugate point 
 o (54), which introduces (comp. 57) the quadratic factor, 
 
 * Compare the method employed in Sahnon's Higher Plane Curves, page 98, to 
 find the equation of the reciprocal of a given curve, with respect to the imaginary 
 conic, *2 4.y3-|- j2 = 0. In general, if the function f be deduced from /as above, 
 then F(a;y?)= 0, and f(xyz) = are equations of two reciprocal curves. 
 
CHAP. III.] VECTORS OF POINTS IN SPACE. 49 
 
 (/+ m + w)2 = ; 
 
 and when this factor also is set aside, the tangential equation 
 is found to be reduced to the biquadratic form* already assigned 
 in 59 ; the algebraic division, last performed, corresponding 
 to the known geometric depression of a cubic curve with a 
 double point, from the sixth to ihQ fourth class. But it is time 
 to close this Section on Plane Curves ; and to proceed, as in 
 the next Chapter we propose to do, to the consideration and 
 comparison of vectors of points in space. 
 
 CHAPTER III. 
 
 APPLICATIONS OF VECTORS TO SPACE. 
 
 Section 1. — On Linear Equations between Vectors not Com^ 
 
 planar. 
 
 62. When three given and actual vectors oa, ob, oc, or 
 «5 i3j 7 J are not contained in any common plane, and w^hen 
 the three scalars a, b, c do not all vanish, then (by 21, 22) 
 the expression aa + b[5 + cy cannot become equal to zero ; it 
 must therefore represent 50/w^ actual vector (1), which we may, 
 for the sake of symmetry, denote by the symbol - d^ : where 
 the new (actual) vector B, or od, is not contained in any one 
 
 * If we multiply that form f = (59) by z% and then change nz to-lx- my, 
 we obtain a biquadratic equation in / : w, namely, 
 
 = ;//(;, w) = (^ - m)2 (Ix + myy^ + 2lm {I + m) {Jx -f my) z + I'^nfiz^ \ 
 
 and if we then eliminate I : m between the two derived cubics, = Dii|/, = d,„i//, 
 we are conducted to the following equation of the twelfth degree, = x^y'^z^fix, y, z), 
 where /ha3 the same cubic form as in 54. "We are therefore thus brought hack 
 (comp. 59) from the tangential to the local equation of the cubic curve (54) ; com- 
 plicated, however, as we see, with the /ac^or x^y'^z^^ which corresponds to the sys- 
 tem of the three real tangents of inflexion to that curve, each tangent being taken 
 three times. The reason why we have not here been obliged to reject also the foreign 
 factor, 2*2, as by the general theory (60) we might have expected to be, is that we 
 multiplied the biquadratic function f only by z2, and not by z'^. 
 
 H 
 
50 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book I. 
 
 of the three given and distinct planes, boc, coa, aob, unless 
 some one, at least, of the three given coefficients «, 6, c, va- 
 nishes ; and where the new scalar^ d, is either greater or less 
 than zero. We shall thus have a linear equation between four 
 vectors, 
 
 aa + b(5 + cy + dd = ; 
 which will give 
 
 g = 
 
 aa 
 
 bfi 
 
 where oa', ob', oc'. 
 
 -Cj 
 
 or 
 
 or od = oa'+ ob'+ oc' 
 
 aa 
 
 -b(5 ~Cy 
 
 Fig. 28. 
 
 —7-5 — -T-j — r, are the 
 a d d 
 
 vectors of the three points 
 
 a', b', c', into which the 
 
 point D is projected^ on the 
 
 three given lines oa, ob, oc, 
 
 by planes drawn parallel to 
 
 the three given planes, boc, 
 
 &c. ; so that they are the 
 
 three co-initial edges of a 
 
 parallelepiped, whereof the sum, od or §, is the internal 
 
 and co-initial diagonal (comp. 6). Or we may project d on 
 
 the three planes, by lines da", db", dc" parallel to the three 
 
 • . bQ + Cy 
 
 given lines, and then shall have oa" = ob' + oc'= — — — ^, &c., 
 
 - d 
 
 and 
 
 g = OD = oa' + oa" = ob' + ob" = oc' + oc". 
 
 And it is evident that this construction will apply to any ffth 
 point D of space, if the j^wr points oabc be still supposed to be 
 given, and not complanar : but that some at least of the three 
 ratios of the four scalars a, b, c, d (which last letter is not 
 here used as a mark of differentiation) will vary with the^o- 
 sition of the point d, or with the value of its vector 8. For 
 example, we shall have a = 0, if d be situated in the plane boc ; 
 and similarly for the two other given planes through o. 
 
 63. We may inquire (comp. 23), ichat relation between 
 these scalar coefficients must exist, in order that the point d 
 
CHAP. III.] VECTORS OF POINTS IN SPACE. 51 
 
 may be situated in the fourth given plane abc ; or what is the 
 condition of complanarity o^ \hQ four points, a, b, c, d. Since 
 the three vectors da, db, dc are now supposed to be complanar, 
 they must (by 22) be connected by a linear equation, of the 
 form 
 
 fl(a-g) + 6(j3-g) + c(y-g) = 0; 
 
 comparing which with the recent and more general form (62), 
 we see that the required condition is, 
 
 a + 5 + c + c?= 0. 
 
 This equation may be written (comp. again 23) as 
 
 -a -b -c , oa' ob' oc' , 
 
 d d d OA OB 00 
 
 and, under this last form, it expresses a known geometrical 
 property of a plane abcd, referred to three co-ordinate axes 
 OA, OB, oc, which are drawn from any common origin o, and 
 terminate upon the plane. We have also, in this case of com- 
 planarity (comp. 28), the following proportion of coefficients 
 and areas : 
 
 a :b: c :- d = dbc : dca : dab : abc ; 
 
 or, more symmetrically, with attention to signs of areas, 
 
 a :b: c : d = bcd : - cda : dab : - abc ; 
 
 where Fig. 1 8 may serve for illustration, if we conceive o in 
 that Figure to be replaced by d. 
 
 64. When we have thus at once the two equations, 
 
 aa-¥bf^ + cy + d^ = 0, and a + b + c + d=0, 
 
 so that the four co-initial vectors a, /3, y, S terminate (as above) 
 on one common plane, and may therefore be said (comp. 24) to 
 be termino-complanar, it is evident that the two right lines, 
 da and bc, which connect two pairs of the four complanar 
 points, must intersect each other in some point a' of the plane, 
 at a finite or infinite distance. And there i no difficulty in 
 perceiving, on the plan of 31, that the vectors of the three 
 
52 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book I. 
 
 points a', b', c' of intersection, which thus result, are the fol- 
 lowing : 
 
 for a' = bc'Da, 
 for b'= ca'DB, 
 for c' = ab • DC, 
 
 ^'= 
 
 b^c - 
 
 a + 
 
 d 
 
 cy + aa 
 
 &/3 + 
 
 dd 
 
 cv a 
 
 - b + 
 
 d 
 
 aa + b^ 
 
 Cy + 
 
 d^ 
 
 a +b 
 
 c + 
 
 d 
 
 expressions which are independent of the position of the arbi- 
 trary origin o, and which accordingly coincide with the cor- 
 responding expressions in 27, when we place that origin in the 
 point D, or make S = 0. Indeed, these last results hold good 
 (comp. 31), even when the^wr vectors a, ^, y, ^, or the Jive 
 points o, A, b, c, d, are all complanar. For, although there 
 then exist two linear equations between those four vectors, 
 which may in general be written thus, 
 
 a a + ft'j3 + Cy + d'^ = 0, a"a f 6"/3 + c'y + d"8 = 0, 
 
 without the relations, a' + &c. = 0, a" + &c. = 0, between the 
 coefficients, yet if we form from these another linear equation, 
 of the form, 
 
 (a" + ta)a + {b" + tb')fi + (c" + tc')y + (d" + td')^ = 0, 
 
 and determine t by the condition, 
 
 t = 
 
 a" + b" + c" + d" 
 a+b' + c+d'^ 
 
 we shall only have to make a = a"+ ta, &c., and the two equa- 
 tions written at the commencement of the present article will 
 then both be satisfied; and will conduct to the expressions 
 assigned above, for the three vectors of intersection : which 
 vectors may thus be found, without its being necessary to em- 
 ploy those processes of scalar elimination^ which were treated 
 of in the foregoing Chapter. 
 
 As an Example, let the two given equations be (comp. 27, 33), 
 aa + ij3 + cy = 0, (2a + fc + c)a'"- aa = ; 
 
CHAP. III.] VECTORS OF POINTS IN SPACE. 53 
 
 and let it be required to determine the vectors of the intersections of the three pairs 
 of lines bc, aa'" ; CA, ba'" ; and ab, ca"'. Forming the combination, 
 
 (2a + 6 + c)a" - aa-\- t(aa + JjS + cy) = 0, 
 
 and determining t by the condition, 
 
 (2a + 6 + c) - a + <(a + 6 + c) = 0, 
 
 which gives * = — 1, we have for the three sought vectors the expressions, 
 
 bfi + cy cy + 2aa 2aa + bjS 
 
 b + c ' c+2a ' 2a + 6 ' 
 
 whereof the first = a, by 27. Accordingly, in Fig. 21, the line aa'" intersects bc in 
 the point a' ; and although the two other points of intersection here considered, 
 which belong to what has been called (in 34) a Third Construction, are not marked 
 in that Figure, yet their anharmonic symbols (36), namely, (2, 0, 1) and (2, 1, 0), 
 might have been otherwise found by combining the equations y = and x — lz for the 
 two lines ca, ba'" ; and by combining z = 0, x = 2y for the remaining pair of lines. 
 
 Q5. In the more general case, when the four given points 
 A, B, c, D, are not in sluj common plane, let k be any fifth given 
 point of space, not situated on any one o^ the fijur faces of the 
 given pyramid abcd, nor on any such face prolonged ; and let 
 its vector oe = c. Then the/owr co-initial vectors, ea, eb, ec, 
 ED, v^hereof (by supposition) no three are complanar, and which 
 do not terminate upon one plane, must be (by 62) connected 
 by some equation of the form, 
 
 tf .EA + 6.EB + C.E0 + 6?.ED = 0; 
 
 where the^wr scalar s, a, b, c, d, and their sum, which we shall 
 denote by - e, are all different fiom zero. Hence, because 
 ea = a - £, &c., we may establish the following linear equation 
 between five co-initial vectors, a, j3, 7, S, e, whereof wo j^tt?- are 
 termlno-complanar (64), 
 
 aa + Jj3 + Cy + c?S + e£ = ; 
 
 with the relation, a+^ + c + c?+e = 0, between ih^five scalars 
 a, b, c, d, e, whereof no one now separately vanishes. Hence 
 also, £ = (aa + b(5+cy + d^) : (a+b + c+ d), &c. 
 66. Under these conditions, if we write 
 
 Di = DE*ABC, and ODi = ^i, 
 
 that is, if we denote by di the vector of the point Di in which 
 the right line de intersects the plane abc, we shall have 
 
54 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 Oi = r = — = . 
 
 a + b+ e d+ e 
 
 In fact, these two expressions are equivalent^ or represent one 
 common vector, in virtue of the given equations; but the first 
 shows (by 63) that this vector Si terminates onthe/>Z«we abc, 
 and the second shows (by 25) that it terminates on the line 
 DE ; its extremity Di must therefore be, as required, the inter- 
 section of this line with that plane. We have therefore the two 
 equations, 
 
 I. . .a(«-gi) + *(i3-^i) + c(y-S0 = 0; 
 II.. .d{d~Si) + e(e-Bi)^0; 
 
 whence (by 28 and 24) follow the two proportions, 
 
 T, . . a:b:c= DjBC : DiCA : DiAB ; 
 ir. . . d:e= EDiiDiD ; 
 
 the arrangement of the points, in the 
 
 annexed Fig. 29, answering to the case 
 
 where all the four coefficients a, b, c, d 
 
 are positive (or have one common sign), 
 
 and when therefore the remaining co- '^" ' 
 
 efficient e is negative (or has the opposite sign). 
 
 67. For the three complanar triangles, in the first propor- 
 tion, we may substitute any three pyramidal volumes, which 
 rest upon those triangles as their bases, and which have one 
 common vertex, such as D or e ; and because the collineation 
 DEDi gives DDiBc - EDiBc ~ DEBc, &c., wc may write this other 
 proportion, 
 
 F. . . a:b:c = debc : deca : deab. 
 
 Again, the same collineation gives 
 
 EDi : DDi = EABC : DABC ; 
 
 we have therefore, by IP., the proportion, 
 
 II". . . d: -e = EABC : DABC. 
 
 But 
 
 DEBC + DECA + DEAB + EABC = DABC, 
 
 and 
 
CHAP. III.] VECTORS OF POINTS IN SPACE. 55 
 
 a-^ b + c + d= -e; 
 
 we may therefore establish the following fuller formula of 
 proportion, between coefficients and volumes : 
 
 III. . . aibicid: -e = debc : deca : deab : eabc : dabc ; 
 
 the ratios of all these five pyramids to each other being consi- 
 dered as positive^ for the particular arrangement of the points 
 which is represented in the recent figure. 
 
 68. The formula III. may however be regarded as per- 
 fectly general^ if we agree to say that a pyramidal volume changes 
 sign, or rather that it changes its algebraical character, as po- 
 sitive or negative, in comparison with a given pyramid, and 
 with a given arrangement of points, in passing through zero 
 (comp. 28) ; namely when, in the course of any continuous 
 change, any one of its vertices crosses the corresponding base. 
 With this convention* we shall have, generally, 
 
 DABC = -ADBC = ABDC = - ABCD, DEBC = BCDE, DECA = CDEA ; 
 
 the proportion III. may therefore be expressed in the follow- 
 ing more symmetric, but equally general form : 
 
 Iir. . , a:b:c:d:.e = bcde : cdea : deab : eabc : abcd ; 
 
 the sum of these j^ve pyramids being always equal to zero, 
 when signs (as above) are attended to. 
 
 69. We saw (in 24) that the two equations, 
 
 aa + bfi + cy = 0, a + b + c = 0, 
 
 gave the proportion of segments, 
 
 a : b : c = BC : CA : ab, 
 
 whatever might be the position of the origin o. In like man- 
 ner we saw (in 63) that the two other equations, 
 
 ♦ Among the consequences of this convention respecting signs of volumes, which 
 has already been adopted by some modern geometers, and which indeed is necessary 
 (comp. 28) for the establishment of general formulae, one is that any two pyramids, 
 ABCD, a'b'c'd', bear to each other a positive or a negative ratio, according as the two 
 rotations, BCD and b'c'd', supposed to be seen respectively from the points A and a', 
 have similar or opposite directions, as right-handed or left-handed. 
 
56 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 aa + bfi + Cy+d^^O, a + 6 + c + c? = 0, 
 
 gave the proportion of areas, 
 
 a:b:c: d= bcd : - cda : dab : - abc ; 
 
 where again the origin is arbitrary. And we have just deduced 
 (in 68) a corresponding proportion of volumes, from the two 
 analogous equations {65), 
 
 fla + 6/3 + cy + </S + ee = 0, a + b + c-\^d+e=0, 
 
 with an equally arbitrary origin. If then we conceive these 
 segments, areas, and volumes to be replaced by the scalars to 
 which they are thus proportional, we may establish the three 
 general for mulce. : 
 
 I. OA.BC + OB.CA+ OC.AB = ; 
 II. OA.BCD - OB. CD A + 00. DAB -0D.ABC = ; 
 III. OA.BCDE + OB.CDEA+ OC.DEAB + OD . EABC+ OE . ABCD = ; 
 
 where in I., a, b, c are ani/ three collinear points ; 
 in II., A, B, c, D are any four complanar points ; 
 
 and in III., a, b, c, d, e are any five points of space ; 
 
 while o is, in each of the three formulas, an entirely arbitrary 
 point. It must, however, be remembered, that the additions 
 and subtractions are supposed to be performed according to the 
 rules of vectors, as stated in the First Chapter of the present 
 Book ; the segments, or areas, or volumes, which the equations 
 indicate, being treated as coefficients of those vectors. We 
 might still further abridge the notations, while retaining the 
 meaning of these formulae, by omitting the symbol of the arbi- 
 trary origin o ; and by thus writing,* 
 
 r. A.BC + B.CA + CAB = 0, 
 
 for any three collinear points ; with corresponding formulae II'. 
 and III'., for any four complanar points, and for any five points 
 of space. 
 
 * We should thus have some of the notations of the Barycentric Calculus (see 
 Note B), but employed here with different interpretations. 
 
CHAP. III.] QUINARY SYMBOLS FOR POINTS IN SPACE. 57 
 
 Section 2. — On Quinary Symbols far Points and Planes in 
 
 Space. 
 
 70. The equations of Art. Q6 being still supposed to hold good, 
 the vector p of any point P of space may, in indefinitely many ways, 
 be expressed (comp. 36) under the form : 
 
 xaa + yhB + zc<^ + wd^ + vee 
 I. . . op = /> = ^!-^. ^ ; 
 
 in which the ratios of the differences of ihe five coefficients^ xyzwv, de- 
 termine the position of the point. In fact, because the four points 
 ABCD are not in any common plane, there necessarily exists (comp. 
 65) a determined linear relation between the four vectors drawn to 
 them from the point P, which may be written thus, 
 a/a . PA + y'b . pb + z^c . PC + w'd . pd = 0, 
 giving the expression, 
 
 _ x'aa + y^h^ + z'c^ + w'dh 
 x'a + y'b + z'c + w'd * 
 in which the ratios of the four scalars x'y'z'w'^ depend upon, and 
 conversely determine, the position of p ; writing, then, 
 
 ic=te' + v, y = ty'^v^ z-tz'-^v^ w-tw' + Vy 
 where t and v are two new and arbitrary scalars, and remembering 
 that aa + . . + ee = 0, and « + . . + e = (65), we are conducted to the 
 form for /», assigned above. 
 
 71. When the vector p is thus expressed, the point p maybe 
 denoted by the Quinary Symbol {x, ?/, z^ Wy v) ; and we may write 
 the equation, 
 
 p = (x, y, z, w, v). 
 
 But we see that the same point p may also be denoted by this other 
 symbol, oHhe same kind, (a/, y, z\ w\ v'), provided that the follow- 
 ing /jropor^eoM between differences of coefficients (70) holds good: 
 
 x' -v' '. y' -v''.z' -v''.w' -v' = x-v'.y-v\z-v'.w-v, 
 Undei' this condition, we shall therefore write the following /orww/a 
 of congruence, 
 
 {x\ y', z', w', v') E {x, y, z, w, v), 
 
 to express that these two quinary symbols, although not identical in 
 
 composition, have yet the same geometrical signification, or denote one 
 
 common point. And we shall reserve the symbolic equation, 
 
 {x', y, z', w', v') = {x, y, z, w, v), 
 
 I 
 
58 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 to express that the Jive coefficients, x' . . . v\ of the one symbol, are 
 separately equal to the corresponding coefficients of the other, 
 a;' = flj, . . v' = v. 
 
 72. Writing also, generally, 
 
 (to, ty^ tZf tw, tv) = t (x, y, z, w, v), 
 
 {x' + a;, . . v' + v) = (x\ . . v') + (a;, . . v), &c., 
 
 and abridging the particular symbol* (1, 1, 1, 1, 1) to (Z7), while 
 
 (Q)> (Q0» • • "^^y briefly denote the quinary symbols (a;, . . v), 
 
 {x', . . v'), . . we may thus establish the congruence (71), 
 
 (Q')=(a), if (Q)=«(ao+w(£^); 
 
 in which t and u are arbitrary coefficients. For example, 
 (0,0, 0,0, 1)E (1,1, 1,1,0), and (0, 0, 0, 1, 1)E(1, 1, 1, 0, 0); 
 each symbol of the first pair denoting (fi5) the given point e; and 
 each symbol of the second pair denoting ifiQ) the derived point Di. 
 When the coefficients are so simple as in these last expressions, we 
 may occasionally omit the commas^ and thus write, still more briefly, 
 (00001) = (11110); (00011) E (1 1100). 
 
 73. If three vectors, />, /?', p"^ expressed each under the first 
 form (70), be termino-collinear (24) and if we denote their denomi- 
 
 tors, a;a + . . , rc'a + . . , x"a + . . , by m-, m\ m!\ they must then (23) be 
 connected by a linear equation, with a null sum of coefficients, which 
 may be written thus : 
 
 tmp + t'm'p' + i"m"p" = ; tm^ t'm' + t"m" + 0. 
 We have, therefore, the two equations of condition^ 
 
 t {xaa + . . + vee) + 1' {x'aa + . . + v'ee) + 1" {x"aa + . . + v"ee) = ; 
 
 t{xa + . . + ve) + 1' {x'a + . . + v'e) + f' {x"a + . . + v"e) = ; 
 
 where t, f, t" are three new scalars, while the five vectors a . . e, and 
 
 the five scalars a..e, are subject only to the two equations (65); 
 
 but these equations of condition are satisfied by supposing that 
 
 tx + t'x' + t"x" = . . = a' + t'v' + t"v" = -u, 
 where u is some new scalar, and they cannot be satisfied otherwise. 
 Hence the condition of collinearity of the three points p, p', p'', in 
 which the three vectors />, p', p" terminate, and of which the qui- 
 nary symbols are (Q), (QOi {.01% "^^y briefly be expressed by the 
 equation, 
 
 * This quinary symbol ( U) denotes no determined point, since it corresponds 
 (by 70, 71) to the indeterminate vector /o = - ; but it admits of useful combinations 
 with other quinary symbols, as above. 
 
CHAP. III.] QUINARY SYMBOLS OF PLANES. 59 
 
 t{Q) + V {Q) + t" {Q")^-u{U); 
 so that if ant/ four scalars, <, t\ t'\ u, can he found, which satisfy this 
 last symbolic equation, then, but not in any other case, those three 
 points pp'p" are ranged on one right line. For example, the three 
 points D, E, Di, which are denoted (72) by the quinary symbols, 
 (00010), (00001), (11100), are coUinear ; because the sum of these 
 three symbols is ( U). And if we have the equation, 
 
 where t, f, u are any three scalars, then {Q") is a symbol for a point 
 v", on the right line pp'. For example, the symbol (0, 0, 0, t, t') may 
 denote any point on the line de. 
 
 74. By reasonings precisely similar it may be proved, that if 
 (Q) (QO (^'0 (Q'^0 be quinary symbols for &ny four points pp^p'^p'^' 
 in any common plane, so that the four vectors pp'p^p'^' are termino- 
 complanar (64), then an equation, of the form 
 
 UQ) + i^QO + 1" (Q'O + i'" ( Q''0 = - «^( C^)» 
 must hold good; and conversely, that \i the fourth symbol can be 
 expressed as follows, 
 
 {Cl"^) = t{a)^t' {Cl')^t"{Q!') ^u{U\ 
 with any scalar values oit, t', t" , u, then the fourth point 2'^' is situ- 
 ated in the plane pp'p'^ of the other three. For example, the four 
 
 points, 
 
 (10000), (01000), (00100), (11100), 
 
 or A, B, c, Di {^^\ are complanar; and the symbol {t, t' , t", 0, 0) 
 may represent any point in the plane abc. 
 
 75. When a point p is thus complanar with three given points, 
 Po, Pi, P2, we have therefore expressions of the following forms, for 
 ih.Q five coefficients x, ..v oi its quinary symbol, in terms of the fif- 
 teen given coefficients oi their symbols, and of /owr new and arbitrary 
 scalars : 
 
 X = ^o^^o + <i^i + k^i + «^; . . . V = ^0^0 + t,Vi + kv.i + u. 
 And hence, by elimination of these four scalars, tQ..u, we are con- 
 ducted to a linear equation of the form 
 
 l{x -v) -^^ m{y - v) + n{z - v) ^- r (w -v) = 0, 
 which may be called the Quinary Equation of the Plane PqPiPo, or of 
 the supposed locus of the point p: because it expresses a common 
 property of all the points of that locus; and because the three ratios 
 of the/owr new coefficients I, m, n, r, determine the position of the plane 
 
60 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 in space. It is, however, more symmetrical, to write the quinary 
 equation of a plane 11 as follows, 
 
 Ix -h my + nz + rw + sv ~ 0, 
 ■where the ffth coefficient, s, is connected with the others by the rela- 
 tion, 
 
 /4-w + n + r+5 = 0; 
 
 and then we may say that [/, w, n, r, 5] is (comp. 37) the Quinary 
 Symbol of the Plane 11, and mtiy write the equation, 
 
 n = [I, m, w, r, s]. 
 For example, the coefficients of the symbol for a point p in the plane 
 ABC may be thus expressed (comp. 74) : 
 
 X=^tQ + U, y = ti + U, Z = t^ + U, W=U, V=U'^ 
 
 between which the only relation, independent of the four arbitrary 
 scalars to. .u, is w-v=0; this therefore is the equation of the plane 
 ABC, and the symbol of that plane is [0, 0, 0, 1, - 1]; which may 
 (comp. 72) be sometimes written more briefly, without commas, as 
 [00011]. It is evident that, in any such symbol, the coefficients may 
 all be multiplied by any common factor. 
 
 76. The symbol of the plane P0P1P2 having been thus determined, 
 we may next propose to find a symbol for the^om^, p, in which that 
 plane is intersected by a given line P3P4: or to determine the coefficients 
 a; . . «>, or at least the ratios of their differences (70), in the quinary 
 symbol of that point, 
 
 (x, y, z, w, v) = T = PoPiPg • P3P4. 
 Combining, for this purpose, the expressions, 
 
 X = ^30:3 + tiX4, + u',. . v = t^Vs + ^4^4 + u\ 
 
 (which are included in the symbolical equation (73), 
 
 {Q)=^t,{Q,)-\-t,(CL) + u^iU). 
 and express the collinearity PP3P4,) with the equations (75), 
 
 /a?+ .. +5t;=0, Z+.. + 5 = 0, 
 (which express the complanarity pPqPiP^,) we are conducted to the 
 formula, 
 
 ^3 {Ix^ + . . + svg) -I- «4 {Ix^ + . . + 5^4) = 0; 
 
 which determines the ratio t^ : ^4, and contains the solution of the 
 
 problem. For example, if p be a point on the line de, then (comp. 
 
 73), 
 
 X=:y = z-u', w^tz+u', V = «4 + ?/; 
 
CHAP. III.] QUINARY TYPES OF POINTS AND PLANES. 61 
 
 but if it be also a point in the plane abc, then w-v-0 (75), and 
 therefore ^3 - ^4 = ; hence 
 
 (Q) = ^3(00011) + w^(ll 111), or (Q) = (00011); 
 which last symbol had accordingly been found (72) to represent the 
 intersection (fi^), Dj = abc • de. 
 
 77- When the five coefficients, xyzwv, of any given quinary 
 symbol (Q) for a point p, or those of any congruent symbol (71), are 
 any whole numbers (positive or negative, or zero), we shall say 
 (comp. 42) that the point p is rationally related to the five given points, 
 A . . E ; or briefly, that it is a Kational Point of the System, which 
 those five points determine. And in like manner, when the five 
 coefficients, Imnrs, of the quinary symbol (75) of a plane 11 are either 
 equal or proportional to integers, we shall say that the plane is a Ra- 
 tional Plane of the same System; or that it is rationally related to the 
 same five points. On the contrary, when the quinary symbol of a 
 point, or of a plane, has not thus already whole coefficients, and can- 
 not be transformed (comp. 72) so as to have them, we shall say that 
 the point or plane is irrationally related to the given points; or 
 briefly, that it is irrational. A right line which connects two rational 
 points, or is the intersection of two rational planes, may be called, on 
 the same plan, a Rational Line ; and lines which cannot in either 
 of these two ways be constructed, may be said by contrast to be 
 Irrational Lines. It is evident from the nature of the eliminations 
 employed (comp. again 42), that a plane, which is determined as con- 
 taining three rational points, is necessarily a ra^eowaZ^Zawe; and in 
 like manner, that o. point, which is determined as the common inter- 
 section of three rational planes, is always a rational jwint : as is also 
 every point which is obtained by the intersection of a rational line 
 with a rational plane ; or of two rational lines with each other (when 
 they happen to be complanar). 
 
 78. Finally, when two points^ or two planes, differ only by the ar- 
 rangement (or order) of the coefficients in their qn'mar j symbols^ those 
 points or planes may be said to have one common type; or briefly 
 to be syntypicaL For example, ihefive given points, a, . . e, are thus 
 syntypical, as being represented by the quinary symbols (10000), . . 
 (00001); and the ten planes, obtained by taking all the ternary 
 combinations of those five points, have in like manner one common 
 type. Thus, the quinary symbol of the plane abc has been seen 
 (75) to be [OOOll]; and the analogous symbol [11000] represents 
 the plane cde, &c. Other examples will present themselves, in a 
 
62 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 shortly subsequent Section, on the subject of Nets in Space. But 
 it seems proper to say here a few words, respecting those Aiihar- 
 monic Co-ordinates, Equations^ Symbols, and Types, for Space, which 
 are obtained from the theory and expressions of the present Section, 
 by reducing (as we are allowed to do) the number of the coefficients^ 
 in each symbol or equation, from Jive to four. 
 
 Section 3. — On Anharmonic Co-ordinates in Space. 
 
 79. When we adopt the second form (70) for />, or suppose (as 
 we may) that the fifth coefficient in the yir5^ form vanishes, we get this 
 other general expression (comp. 34, 36), for the vector of a point in 
 space: 
 
 xaa + yh3 + zc^ + wdb 
 xa + yb-\-zc + wd 
 and may then write the symbolic equation (comp. 36, 71), 
 
 p=(a7, y, z, w), 
 and call this last the Quaternary Symbol of the Point P : although 
 we shall soon see cause for calling it also the Anharmonic Symbol of 
 that point. Meanwhile we may remark, that the only congruent 
 symbols (71), of this last form, are those which differ merely by the 
 introduction oi s. common factor : the three ratios of the /owr coeffi- 
 cients, X . ,w, being all required, in order to determine the position of 
 the point; whereof those four coefficients may accordingly be said 
 (comp. 36) to be the Anharmonic Coordinates in Space. 
 
 80. When we thus suppose that v = 0, in the quinary symbol of 
 t\ie point p, we may suppress the fifth term sv, in the quinary equation 
 of 2i plane IT, lx-\- ..+sv = (75) ; and therefore may suppress also (as 
 here unnecessary) th^ fifth coefficient, s, in the quinary symbol of that 
 plane, which is thus reduced to the quaternary form, 
 
 n = [/, m, n, r]. 
 This last may also be said (37, 79), to he the Anharmonic Symbol of 
 the Plane, of which the Anharmonic Equation is 
 
 Ix + my + nz + rw = 0', 
 the four coefficients, Imnr, which we shall call also (comp. again 37) 
 the Anharmonic Co-ordinates of that Plane 11, being not connected 
 among themselves by any general relation (such as Z+ . .+5 = 0): since 
 their three ratios (comp. 79) are all in general necessary, in order to 
 determine the position of the plane in space. 
 
 81. If we suppose that the fourth coefficient, w, also vanishes, in 
 
CHAP. III.] ANHARMONIC CO-ORDINATES IN SPACE. 63 
 
 the recent symbol of a point, thsit point p is in theplane abc ; and may- 
 then be sufficiently represented (as in 36) by the Ternary Symbol 
 (a?, y, z). And if we attend only to the points in which an arbitrary 
 plane n intersects the given plane abc, we may suppress its fourth co- 
 efficient, r, as being for such points unnecessary. In this manner, 
 then, we are reconducted to the equation, lx+my + nz= 0, and to the 
 symbol, A= [Z, m, w], for a right line (37) in the plane abc, considered 
 here as the trace, on that plane, of an arbitrary plane H in space. If 
 this plane n be given by its quinary symbol (75), we thus obtain 
 the ternary symbol for its tf^ace A, by simply suppressing the two last 
 coefficients, r and s. 
 
 82. In the more general case, when the point p is not confined 
 to the plane abc, if we denote (comp. 72) its quaternary symbol by 
 (Q), the lately established formulae of collineation and complanarity 
 (73, 74) will still hold good: provided that we now suppress the 
 symbol ( U), or suppose its coefficient to be zero. Thus, the formula, 
 
 {Q)=t'{Q)^t"{Q^)-Vt"'{Q"), 
 expresses that the point p is in the plane -j^'^f'-p'" ; and if the coeffi- 
 cient t"' vanish, the equation which then remains, namely, 
 
 signifies that p is thus complanar with the two given points p^, v", 
 and with an arbitrary third ^wint; or, in other words, that it is on 
 the right line v'v" ; whence (comp. 76) problems of intersections of 
 lines with planes can easily be resolved. In like manner, if we de- 
 note briefly by [i?] the quaternary symbol \l, m, n, r'] for a plane 
 n, the formula 
 
 [i2] = t' [i?'] + 1" IR"^ + 1"' [R"q 
 
 expresses that the plane n passes through the intersection of the thr^ 
 planes, 11', II'', W ; and if we suppose t'^' = 0, so that 
 
 [ij]=«'[fi']+«"[fi"3, 
 
 the formula thus found denotes that the plane 11 passes through 
 the point of intersection of the two planes, 11', 11", with any third 
 jilane; or (comp. 41), that this plane n contains the line of intersec- 
 tion of n', n" ; in which case the three planes, Tl, 11', 11", may be 
 said to be coUinear. Hence it appears that either of the two expres- 
 sions, 
 
 I. . . t' ( Q') + ^" ( a^O. II- • • i' [-^G + i" \.Rf'\ 
 may be used as a Symbol of a Right Line in Space : according as we 
 consider that line A either, 1st, as connecting two given points, or 
 
64 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 Ilnd, as being the intersection of two given planes. The remarks (77) 
 on rational and irrational points, planes, and lines require no modifi- 
 cation here; and those on types (78) adapt themselves as easily to 
 quaternary as to quinary symbols. 
 
 83. From the foregoing general formulee of collineation and conj- 
 planarity, it follows that the point p', in which the line ab inter- 
 sects the plane cdp through CD and any proposed point P = {xyzw) 
 of space, may be denoted thus : 
 
 p' = AB • CDP = {xy{)Q)) ; 
 for example, e = (U 1 1), and c' = ab • cde = (1100). In general, if 
 ABCDEF be any six points of space, the four collinear planes (82), abc, 
 abd, ABE, ABF, are said to form a pencil through ab; and if this be 
 cut by any rectilinear transversal, in four points, c, D, e, f', then 
 (comp. 35) the anharmonic function of this group of points (25) is 
 called also the Anharmonic of the Pencil of Planes: which may be 
 thus denoted, 
 
 (ab . cdef) = (c'dVf'). 
 
 Hence (comp. again 25, 35), by what has just been shown respect- 
 ing c' and p', we may establish the important formula: 
 
 (cD . AEBp) = (ac'bpO = - ; 
 
 so that this ratio of coefficients, in the symbol {xyzw) for a variable 
 
 point p (79), represents the anharmonic of a pencil of planes, of which 
 
 the variable plane cdp is one; the three other planes of this pencil 
 
 being given. In like manner, 
 
 • \ y 1 / \ -2^ 
 
 (ad . BECP) = -, and (bd . ceap) = - ; 
 
 ^ Z X 
 
 so that (comp. 36) the product of these three last anharmonics is 
 unity. On the same plan we have also, 
 
 (bc.aedp)=— , (ca.bedp) = — , (ab.cedp) = -; 
 w w ^ ^ w 
 
 so that the three ratios, of the three first coefficients xyz to the 
 fourth coefficient w, suffice to determine the three planes, bcp, cap, 
 ABP, whereof \h.Q point p is the common intersection, by means of the 
 anharmonics of thxe pencils of planes, to which the three planes re- 
 spectively belong. And thus we see a motive (besides that of analogy 
 to expressions already used for points in a given plane), for calling 
 the/owr coefficients, xyzw, in the quaterna/ry symbol (Jd) for 9, point in 
 space, the Anharmonic Co-ordinates of that Point. 
 
 84. In general, if there be any four collinear points, Vq, . . P3, so 
 
CHAP. III.] ANHARMONIC CO-ORDINATES IN SPACE. 65 
 
 that (comp. 82) their symbols are connected by two linear equations, 
 
 such as the following, 
 
 (Qi) = «(Qo) + u{Cl,), (as) = t'{Q,) + w'(Q2), 
 
 then the anharmonic of their group may be expressed (comp. 25, 44) 
 
 as follows : 
 
 ut' 
 (PoPiP.P3) = -,; 
 
 as appears by considering the pencil (cd . PoPiPgPa), and the transversal 
 AB (83). And in like manner, if we have (comp. again 82) the two 
 other symbolic equations, connecting /om?' collinear planes IIq . . n^, 
 
 the anharmonic of their pencil (8.3) is expressed by the precisely 
 similar formula, 
 
 ut' 
 
 (n„n,n,n,) = _; 
 
 as may be proved by supposing the pencil to be cut by the same 
 transversal line ab. 
 
 85. It follows that ii f{xyzw) and /j (a^^^it') be any two homo- 
 geneous and linear functions of ic, y, z^w\ and if we determine four 
 collinear planes IIo . . Ila (82), by the four equations, 
 
 ■/=0, /i=/, /x = 0, j\ = kf, 
 where h is any scalar ; we shall have the following value of the an- 
 harmonic function, of the pencil of planes thus determined: 
 
 f 
 
 Hence we derive this Theorem^ which is important in the application 
 of the present system of co-ordinates to space : — 
 
 " The Quotient of any two given liomogeneous and linear Functions^ 
 of the anharmonic Co-ordinates (79) of a variable Point p in space, may 
 be expressed as the Anharmonic (noninalls) of a Pencil of Planes; 
 w^hereof three are given, while the fourth passes through the variable 
 point p, and through a given right line A which is common to the three 
 former planes y 
 
 86. And in like manner may be proved this other but analogous 
 Theorem : — 
 
 " The Quotient of any two given homogeneous and linear Functions, 
 of the anharmonic Co-ordinates (80) of a variable Plane n, may be ex- 
 pressed as the Anharmonic (PoPiP^Pa) of a Group of Points; whereof 
 three are given and colliriear, and the fourth is the intersection, A ' 11, 
 of their common and given right line A, with the variable plane H," 
 
 K 
 
66 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 More fully, if the two given functions of Imnr be f and y^^ and 
 if we determine three points P0P1P2 by the equations (comp. 57) 
 F = 0, Fi = F, Fi=:0, and denote by P3 the intersection of their com- 
 mon line A with n, we shall have the quotient, 
 
 ^=(P0P,P,P3). 
 
 For example, if we suppose that 
 
 A2=(1001), B2=(010]), C2=(0011), 
 
 A'2 = (1001), B'2 = (OIOT), c'2 = (00 iT), 
 so that 
 
 A2 = DA*BCE, &c., and (dA2Aa'2) = - 1, &c., 
 
 we find that the three ratios of Z, m, n to r, in the symbol n = [/mnr], 
 may be expressed (comp. 39) under the form of anharmonics of 
 groups, as follows; 
 
 - = (da'sAQ) ; - = (db^^br) ; - = (dc'sCs) ; 
 
 where q, r, s denote the intersections of the plane n with the three 
 given right lines, da, db, dc. And thus we have a motive (comp. 
 83) besides that of analogy to lines in a given plane (37), for calling 
 (as above) the, four coefficients I, m, n, r, in the quaternary symbol (80) 
 for a, plane n, the Anharmonic Co-ordinates of that Plane in Space. 
 
 87. It may be added, that if we denote by l, m, n the points in 
 which the same plane IT is cut by the three given lines bc, ca, ab, 
 and retain the notations a'', b''', c'^ for those other points on the same 
 three lines which were so marked before (in 31, &c.), so that we may 
 now write (comp. 36) 
 
 A''= (0110), b'' = (1010), c''= (llOO), 
 
 we shall have (comp. 39, 83) these three other anharmonics of groups, 
 with their product equal to unity : 
 
 — = (ca'^bl) ; - = (ab^^cm) ; — = (bc'^an) ; 
 n V 7ft 
 
 and the six given points, a.'\ e", c", A'2, B'2, c'2, are all in one given plane 
 [e], of which the equation and symbol are: 
 
 x + y + z + w = 0\ [e] = [11111]. 
 The six groups of points, of which the anharmonic functions thus 
 represent the six ratios of the four anharmonic co-ordinates, lmm\ 
 of a variable plane n, are therefore situated on the six edges of the 
 given pyramid^ abcd; two poi7iis in each group being corners of that 
 
CHAP. III.] GEOMETRICAL NETS IN SPACE. 67 
 
 pyramid, and the tiuo others being the intersections of the edge with 
 the two planes^ [e] and n. Finally, the plane [e] is (in a known 
 modern sense) the plane of homology ^^' and the point e is the centre 
 of homology^ of the given pyramid abcd, and of an inscribed pyramid 
 AiBiCiDi, where Ai = ea*bcd, &c.; so that Di retains its recent signi- 
 fication (QQ, 76), and we may write the anharmonic symbols, 
 
 Ai = (0111), Bi = (1011), Ci=(1101), Di = (IllO). 
 
 And if we denote by a'ib'iC^d'i the harmonic conjugates to these 
 last points, with respect to the lines ea, eb, ec, ed, so that 
 
 (eaiAA'i) = . . = (eDiDD'i) = - 1, 
 
 we have the corresponding symbols, 
 
 A'i=(2111), B^ = (1211), c'i = (1121) D^ = (1112). 
 
 Many other relations of position exist, between these various points, 
 lines, and planes, of which some will come naturally to be noticed, 
 in that theory of nets in space to which in the following Section we 
 shall proceed. 
 
 Section 4. — On Geometrical Nets in Space, 
 
 88. When we have (as in Q5) five given points a . . e, whereof no 
 four are complanar, we can connect any two of them by a right line^ 
 and the three others by a plane, and determine the point in which 
 these last intersect one another: deriving t\i\\s a system oHen lines Aj, 
 ten planes Hi, and ten points Pi, from the given system oi five points 
 Po, by what may be called (comp. 34) a First Construction. We may 
 next propose to determine all the new and distinct lines, A,, and 
 planes, Ila, which connect the ten derived points Pj with the five 
 given points Fq, and with each other ; and may then inquire what 
 new and distinct points Pa arise (at this stage) as intersections of lines 
 with planes, or oHines in one plane with each other: all such new lines, 
 planes, and points being said (comp. again 34) to belong to a Second 
 Construction. And then we might proceed to a Third Construction 
 of the same kind, and so on for ever : building up thus what has 
 been calledf a Geometrical Net in Space. To express this geome- 
 trical process by quinary symbols (71, 75, 82) o^ points, planes, and 
 lines, and by quinary types (78), so far at least as to the end of the 
 second construction, will be found to be an useful exercise in the 
 
 * See Poncelet's Traite des Propriete's Projectives (Paris, 1822). 
 t By Mbbius, in p. 291 of his already cited Barycentric Calculus, 
 
68 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 application of principles lately established : and therefore ulti- 
 mately in that Method of Vectoks, which is the subject of the 
 present Book. And the quinary form will here be more convenient 
 than the quaternary^ because it will exhibit more clearly the geome- 
 trical dependence of the derived points and planes on ih^five given 
 points, and will thereby enable us, through a principle of symmetry^ 
 to reduce the number of distinct types. 
 
 89. Of the five given points, Pq, the quinary type has been seen 
 (78) to be (10000); while of the ten derived points p,, o^ first con- 
 struction, the corresponding type may be taken as (00011); in fact, 
 considered as symbols, these two represent the points a andDj. The 
 nine other points Pi are a Vc/AiBjCiAaBaCa ; and we have now (comp. 
 83, 87, 86) the symbols, 
 
 A'= BC • ADE = (01 100), Ai = EA • BCD = (10001), 
 
 A2=DA -BCE^ (10010); 
 
 also, in any symbol or equation of the present form, it is permitted 
 to change a, b, c to b, g, a, provided that we at the same time write 
 the third, first, and second co-efficients, in the places of the first, 
 second, and third: thus, b' = ca • bde = (10100), &c. The symbol 
 (a;^000) represents an arbitrary point on the line ab ; and the sym- 
 bol [OOm'5], with n + r + 5 = 0, represents an arbitrary plane through 
 that line : each therefore may be regarded (comp. 82) as a symbol also 
 of the line ab itselfi and at the same time as a type of the ten lines 
 Ai; while the symbol [000 ll], of the plane abc (75), may betaken 
 (78) as a type of the ten planes Hi. Finally, the five pyramids, 
 
 bcde, cade, abde, abce, abcd, 
 and the ten triangles, such as abc, whereof each is a common face of 
 two such pyramids, may be called pyramids i?i, and triangles T^, of 
 the First Construction. 
 
 90. Proceeding to a Second Construction (88), we soon find that 
 the lines A, may be arranged in two distinct groups; one group con- 
 sisting oi fifteen lines Aj, i, such as the line* aa''d„ whereof each coti- 
 nects two points Pi, and passes also through one point Pq, being the inter- 
 section of two planes IIi through that point, as here of abc, ade; 
 while the other group consists of thirty lines Ag, 2, such as b'c', each 
 connecting two points Pi, but not passing through any point ?„, and 
 being one of the thirty edges of five new pyramids R^, namely, 
 
 C'b'AzA,, A'c'B^B], B^A'C^C,, A.B^C^Di, AiBjCiDj : 
 
 * AB1C2, ABoCi, da'Ai, ea'Ao, are other lines of this group. 
 
CHAP. 111.] GEOMETRICAL NETS IN SPACE. 69 
 
 which pyramids i?2 may be said (comp. 87) to be inscribed homo- 
 logues of the five former pyramids i?i, the centres of homology for these 
 Jive pairs of pyramids being the five given points a . . e ; and \)i\Q. planes 
 of homology being five planes [a] . . [e], whereof the last has been 
 already mentioned (87), but which belong properly to a third con- 
 struction (88). IhQ planes lis, oi second construction, form in like 
 manner two groups; one consisting o^ fifteen planes U^, i, such as the 
 plane of the five points, AB1B3C1C2, whereof each passes through one 
 point Po, and t\iVou^\ four points Pi, and contains two lines Ag, 1, as 
 here the lines AB1C2, AC1B2, besides containing /<?wr lines A2,2, as here 
 BiB^, &c. ; while the other group is composed of twenty planes H^, 2, 
 such as AiBiCi, namely, the twenty faces of the five recent pyramids It^t 
 whereof each contains three points Pj, and three lines Agjg, but does 
 not pass through any point Pq. It is now required to express these 
 geometrical conceptions* of the forty-five lines A^ ; the thirty-five planes 
 112; and the five planes of homology of pyramids, [a] . . . [e], by qui- 
 nary symbols and types, before proceeding to determine the points P2 
 of second construction. 
 
 91. An arbitrary joom^ on the right line aa'Dj (90) may be re- 
 presented by the symbol {tuuOO); and an arhiirsiry plane through 
 that line by this other symbol, [Ommrr], where m and r are written 
 (to save commas) instead of-m and -r; hence these two symbols 
 may also (comp. 82) denote the litie aa'Di itself, and may be used as 
 types (78) to represent the g7-oup of lines Ag, 1. The particular sym- 
 bol [01111], of the last form, represents that particular plane 
 through the last-mentioned line, which contains also the line AB1C2 
 of the same group ; and may serve as a type for the group of planes 
 rig,!. The line B^c^ and the group A2,2, may be represented by 
 (stuOO) and [tttus'], if we agreef to write s = t + Uy and s--s; while 
 the plane b'c'A2, and the group rig, 2, may be denoted by [111 12]. 
 Finally, the plane [e] has for its symbol [11114]; and the four 
 other planes [a], &c., of homology of pyramids (90), have this last 
 for their common type. 
 
 92. The points -p^, of second construction (88), are more nume- 
 
 * Mbbius (in his Bary centric Calculus, p. 284, &c.) has very clearly pointed 
 out the existence and chief properties of the foregoing lines and planes ; but besides 
 that his analysis is altogether different from ours, he does not appear to have aimed 
 at enumerating, or even at classifying, all the points of what has been above called 
 (88) the second construction, as we propose shortly to do. 
 
 f With this convention, the line ab, and the group Ai, may be denoted by 
 the plane -symbol [OfXvs] their point-syrnbol being (tuOOO). 
 
70 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 rous than the lines Ag Midi planes Ilg of that construction: yet with 
 the help of types, as above, it is not difficult to classify and to 
 enumerate them. It will be sufficient here to write down these 
 types, which are found to be eighty and to oiFer some remarks re- 
 specting them ; in doing which we shall avail ourselves of the eight 
 ioWoYimgtypical points^ whereof the two first have already occurred, 
 and which are all situated in the plane of abc : 
 
 A'' = (0lT00); A^^' = (21100); a'^ =(21100); a^ =(02100); 
 A"' = (02100) ; A"" = ( 1 2 1 00) ; a^'" = (32 1 00) ; A« = (23 100) ; 
 
 the second and third of these having (10011) and (30011) for con- 
 gruent symbols (71). It is easy to see that these eight types repre- 
 sent, respectively, ten, thirty, thirty, twenty, twenty, sixty, sixty, 
 and sixty distinct points, belonging to eight groups^ which we shall 
 mark as Po, i, . . P25 8; so that the total number of the points v., is 290. 
 If then we consent (88) to close the present inquiry, at the end of 
 what we have above defined to be the Second Construction^ the total 
 number of the net points^ Pi, Pj, which are thus derived by lines 
 Midi planes from the, five given points Pq, is found to be exactly three 
 hundred: while i\iQ joint number of the net-lines, A^, A2, and of the 
 net-planes^ IIi, Ila, has been seen to be one hundred^ so far. 
 
 (1.) To the type Pq,! belong the ten points^ 
 
 a"b"c", a'2B'2C'2, a'iB'iC'iD'i, 
 
 with the quinary symbols, 
 
 A"=(0ir00),.. A'z =(10010),.. A'l = (10001),.. D'i= (00011), 
 which are the harmonic conjugates of the ten points Pi, namely, of 
 
 a'b'c', A2B2C2, AiBiCiDi, 
 
 with respect to the ten lines Ai,on which those points are situated ; so that we have 
 ten harmonic equations, (ba'ca") = — 1, &c., as already seen (31, 86, 87). Each point 
 P2, 1 is the common intersection of a line Ai with three lines A2,2 ; thus we may esta- 
 blish the four following /brwiMZcB of concurrence (equivalent, by 89, to ten such for- 
 mulae) : 
 
 a" =BC'B'c' -Bid -8202; A'2 = DA-DiArB'C2*c'B3; 
 
 A'i = EA*DiA3'b'Ci-c'Bi; d'i = DE'AiA2-BiB2*CiC2. 
 
 Each point P2, i is also situated in three planes Hi ; in three other planes, of the 
 group 112,1; and in six planes 112,2; for example, a" is a point common to the 
 twelve planes, 
 
 ABC, BCD, BCE ; AB1C2C1B2, Db'BiC'Ci, Eb'B2C'C2 ; 
 b'c'Ai, BiCiA], B2C2A2, b'c'Ao, BiCjDi, B2C2D1. 
 
 Each line, Ai or Aa,?, contains- one point P^, i; but no line Ao, i contains any. Each 
 plane, Hi or 112,2, contains f /tree such points; and each plane Uo^\ contains two, 
 
CHAP. III.] GEOMETRICAL NETS IN SPACE. 71 
 
 which are the intersections of opposite sides of a quadrilateral Q2 in that plane, 
 whereof the diagonals intersect in a point Po : for example, the diagonals BiC2, B2C1 
 of the quadrilateral B]B2C2Ci, which is (by 90) in one of the planes Ila,!, intersect* 
 each other in the point a ; while the opposite sides CiBi, B2C3 intersect in a" ; and 
 the two other opposite sides, B1B2, C2C1 have the point d'i for their intersection. 
 The ten points P2, 1 are also ranged, three hy three, on ten lines of third construction 
 As, namely, on the axes of homology, 
 
 A"b'iC'i, . . a"b'2C'2, . . a'iA'2D'i, . . A"b"c", 
 
 of ten pairs of triangles Ti, 22, which are situated in the ten planes ITi, and of 
 which the centres of homologj' are the ten points pi : for example, the dotted line 
 a"b"c", in Fig. 21, is the axis of homology of the two triangles, abc, a'b'c', whereof 
 the latter is inscribed in the former, with the point o in that figure (replaced by Di 
 in Fig. 29), to represent their centre of homology. The same ten points P2,i are 
 also ranged six hy six, and the ten last lines A3 are ranged four by four, in fve 
 planes lis, namely in the planes of homology of five pairs of pyramids, i?x, -R2J 
 already mentioned (90) : for example, the plane [e] contains (87) the six points 
 a"b"c"a'2b'2c'2, and the four right lines, 
 
 A"b'2C'2, b"c'2A'2, c"a'2B'2, A"b"c" ; 
 
 which latter are the intersections of the four faces, 
 
 DCB, DAC, DBA, ABC, 
 
 of the pyramid abcd, with the corresponding faces, 
 
 DiCiBi, DiAiOi, DiBiAi, AiBiCi, 
 
 of its inscribed homologue AiBiCiDi ; and are contained, besides, in the four other 
 planes, 
 
 A2B'c', B2C'a', C2A'b', A2B2C2 : 
 
 the three triangles, abc, AiBiCi, A2B2C2, for instance, being all homologous, although 
 in different planes, and having the line a"b"c" for their common axis of homology. 
 We may also say, that this line a"b"c" is the common trace (81) of two planes 112, 2, 
 namely of AiBiCi and A2B2C2, on the plane abc ; and in like manner, that the point 
 a" is the common trace, on that plane TIi, of two lines A2,2, namely of BiCi and B2C2 : 
 being also the common trace of the two lines b'ic'i and b'2c'2, which belong to the 
 third construction. 
 
 (2.) On the whole, these ten points, of second construction, a". . ., may be 
 considered to be already well known to geometers, in connexion with the theory 
 of transversal-]; lines and planes in space : but it is important here to observe, 
 with what simplicity and clearness their geometrical relations are expressed (88), 
 by the quinary symbols and quinary types employed. For example, the col- 
 linearity [^i) of the four planes, ABC, AiBiCi, A2B2C2, and [e], becomes evident 
 from mere inspection of their jTowr symbols, 
 
 * Compare the Note to page 68. 
 
 t The collinear, complauar, and harmonic relations between the ten points, 
 which we have above marked as P2, 1, and which have been considered by Mcibiua 
 also, in connexion with his theory of nets in space, appear to have been first noticed 
 by Carnot, in a Memoir upon transversals. 
 
72 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 [OOOllJ, [U121], [11112], [11114], 
 which represent (75) the four quinary equations^ 
 
 w-»=0, a:+y+z-2M>-u=0, a; + y + z -u)-2y= 0, x -V y + z^-w -Av = 0', 
 with this additional consequence, that the ternary symbol (81) of the common trace, 
 of the three latter on the former, is [111]: so that this trace is (by 38) the line 
 A"B"c"of Fig. 21, as above. And if we briefly denote the quinary symbols of the 
 four planes, taken in the same form and order as above, by \_Rq\ [iZi] [-Rg] [-^3], we 
 see that they are connected by the two relations, 
 
 [iJi] =- [/2o] + [i?2] ; [.Rz'\ = 2[/?o] + [Ro] ; 
 whence if we denote the planes themselves by IIi, 112, n'2, lis, we have (comp. 84) 
 the following value for the anharmonic of their pencil, 
 
 (Hinan'sHs) = - 2 ; 
 a result which can be very simply verified, for the case when abcd is a regular py- 
 ramid, and E (comp. 29) is its mean point : the plane lis, or [e], becoming in this 
 case (comp. 38) the plane at infinity, while the three other planes, abc, AiBiCi, 
 A2B2C2, axe parallel ; the second being intermediate ioei^eQn the other two, but twice 
 as near to the third as to the first. 
 
 (3.) "We must be a little more concise in our remarks on the seven other types of 
 points P2, which indeed, if not so well known,* are perhaps also, on the whole, not 
 quite so interesting : although it seems that some circumstances of their arrangement 
 in space may deserve to be noted here, especially as affording an additional exercise 
 (88), in the present system of symbols and types. The type P2, 2 represents, then, a. group 
 oi thirty points, of which a", in Fig. 21, is an example; each being the intersection 
 of a line A2,i with a line A2,2, as a'" is the point in which aa' intersects b'c' : but 
 each belonging to no other line, among those which have been hitherto considered. 
 But without aiming to describe here all ihe lines, planes, and points, of what we have 
 called the third construction, we may already see that they must be expected to be 
 numerous : and that the planes lis, and the hnes A3, of that construction, as well as 
 the pyramids Ro, and the triangles To, of the second construction, above noticed, can 
 only be regarded as specimens, which in a closer study of the subject, it becomes ne- 
 cessary to mark more fully, on the present plan, as lis, i, . . Tz,i. Accordingly it is 
 found that not only is each point P2, 2 one of the corners of a triangle T3, 1 of third 
 construction (as a'" is of a"'b"'c"' in Fig. 21), the sides of which new triangle are 
 lines A3, 2, passing each through one point P2,i and through two points P2,2 (hke 
 the dotted line a"b"'c"' of Fig. 21) ; but also each such point P2, 2 is the intersection 
 of two new lines of third construction, A3, 3, whereof each connects a point Pq with a 
 
 * It does not appear that any of these other types, or groups, of points P2, have 
 hitherto been noticed, in connexion with the net in space, except the one which we 
 have ranked as the fifth, po, 5, and which represents two points on each line Ai, as 
 the type P2, 1 has been seen to represent one point on each of those ten lines of first con- 
 struction : but thdX fifth group, which maybe exemplified by the intersections of the 
 line DE with the two planes AiBiCi and A2B2C2, has been indicated by Mobius (in 
 page 290 of his already cited work), although with a different notation^ and as the re- 
 sult of a different analysis. 
 
CHAP. III.] GEOMETRICAL NETS IN SPACE. 73 
 
 point P2,i. For example, the point a'" is the common trace (ou the plane abc) of the 
 two new lines, da'i, EA'g: because, if we adopt for this point a'" the second of its two 
 congruent symbols, we have (comp. 73, 82) the expressions, 
 
 A"'= (10011) = (d) - (A'l) = (e) - (A'2). 
 We may therefore establish the formula of concurrence (comp. the first sub-article) : 
 
 a'" = aa' • b'c' • da'i • E A'2 -, 
 which represents a system of thirty such formulae, 
 
 (4.) It has been remarked that the point a'" may be represented, not only by the 
 quinary symbol (21100), but also by the congruent symbol, (10011); if then we 
 write, 
 
 Ao = (Ii100), Bo = (iriOO), Co = (11100), 
 these three new points AqBoOo, in the plane of abc, must be considered to be syntypical, 
 in the quinary sense C78), with the three points a"'b"'c"', or to belong to the same 
 group P2,2, although they have (comp. 88) a different ternary type. It is easy to 
 see that, while the triangle a"'b"'c"' is (comp. again Fig. 21) an inscribed homo- 
 logue Ty,! of the triangle a'b'c', which is itself (com\). sub-article 1) an inscrihed 
 homologue To, 1 of a triangle Ti, namely of abc, with a"b"c" for their common a is 
 of homology, the new triangle AqBoCo is on the contrary an exscrihed homologue 
 Ti,2, with the same axis As,!, of the same given triangle Ti. But from the syuty- 
 pical relation, existing as above for space between the points a'" and Ao, we may 
 expect to find that these two points P2, 2 admit of being similarly consirucfed, when 
 the^ue points Pq are treated as entering symmetrically (or similarly), as geometri- 
 cal elements, into the constructions. The point Aq must therefore be situated, not 
 only on a line A2,i, namely, on aa', but also on a line A2,2, which is easily found to 
 be A1A2, and on two lines A3, 3, each connecting a point Pq with a point P2,i ; which 
 latter lines are soon seen to be bb" and cc". We may therefore establish the formula 
 of concurrence (comp. the last sub-article) : 
 
 Ao = aa'*AiA3*bb"-cc"; 
 and may consider the three points Aq, Bq, Co as the traces of the three lines AiAo, 
 B1B2, C1C2 : while the three new lines aa'', bb", cc", which coincide in position 
 with the sides of the exscribed triangle AqBoCIo, are the traces A3, 3 of three planes 
 1X2, 1, such as AB1C2B2C1, which pass through the three given points A, B, c, but do 
 not contain the Unes A2,i whereon the six points P2,2 in their plane ITi are situated. 
 Every other plane IIi contains, in like manner, six points P2 of the present group ; 
 every plane 1X2, 1 contains eight of them ; and every plane 112,2 contains three; each 
 line A2, 1 passing through two such points, but each line A2, 3 only through one. 
 But besides being (as above) the intersection of two lines Ao, each point of this group 
 P2,2 is common to two planes Yli, four planes 113,1, and two planes 112,2; while 
 each of these thirty points is also a common corner of two different triangles of 
 ^/aVrf construction, of the lately mentioned kinds Ts, 1 and 2^,2, situated respectively 
 in the two planes oi first construction which contain the point itself. It may be 
 added that each of the two points P2, 2, on a line A2, 1, is the harmonic conjugate of 
 one of the two points pi, with respect to the point Pq, and to the other point Pi oa 
 that line ; thus we have here the two harmonic equations, 
 
 (aa'dia'") = (adia'ao) = — 1, 
 by which the positions of the two points a'" and Ao miglit be determined. 
 
 L 
 
74 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 (5.) A third group, P2,3, oi second construction, consists (like the preceding group) 
 of thirty points, ranged two hxj two on the fifteen lines Aa^i, and six hy six on the 
 ten planes ITi, but so that each is common to two such planes ; each is also situated 
 in two planes Zlg,!, in two planes Il2,2, and on one line A3, i in which (by sub art. 1) 
 these two last planes intersect each other, and two of the five planes lis, i ; each 
 plane 112,1 contains /owr such points, and each plane 112,2 contains three of them ; 
 but no point of this group is on any line Ai, or A2,2' The six points P2,3, which 
 are in the plane abc, are represented (like the corresponding points of the last 
 group) by two ternary types, namely by (211) and (311) ; and may be exemplified 
 by the two following points, of which these last are the ternary symbols : 
 
 A'^ = AA' • a"b"c" = AA' • AiBiCi ' A2B2C2 ; 
 Ai'^ = AA' •d'iA'2A 1 = AA' •b'CiC2 •c'BiB2. 
 
 The three points of the first sub-group a'^ . . are collinear ; but the three points Ai''^ . . 
 of the second sub-group are the corners of a new triangle, T3, 3, which is homologous 
 to the triangle abc, and to all the other triangles in its plane which have been hitherto 
 considered, as well as to the two triangles AiBiCi and A2B2C2 ; the line of the three 
 former points being their common axis of homology ; and the sides of the new trian- 
 gle, Ai'^Bi'^Ci'^, being the traces of the three planes (comp, 90) of homology of pyra- 
 mids, [a], [b], [c] ; as (comp. sub-art. 2) the line a'^b^'^c'"' or a"b"c" is the com- 
 mon trace of the two other planes of the same group lis, 1, namely of [d] and [e]. We 
 may also say that the point Ai'"^ is the trace of the line a'ia'2 ; and because the lines 
 b'co, c'bo are the traces of the two planes 112,2 in which that point is contained, we 
 may write the formula of concurrence, 
 
 Ai" = A a' • a'ia'2 • b'Co • c'Bo. 
 
 (G.) It may be also remarked, that each of the two points P2, 3) on any line A2, 1, is 
 the harmonic conjugate of a point P2, 2, with respect to the point Pq, and to one of 
 the two points Pi on that line ; being also the harmonic conjugate of this last point, 
 with respect to the same point Pq, and the other point P2,2 : thus, on the line aa'dj, 
 we have the /oMr harmonic equations, which are not however all independent, since 
 two of them can be deduced from the two others, with the help of the two analogous 
 equations of the fourth sub-article : 
 
 (aa"'a'a''^) = (aa'aqA") = (aaqDiAi'^) = (adia"'ai*'^) = - 1. 
 And the three pairs of derived points Pi, P2,2, P2,3, on any such line A2, 1, will 
 be found (comp. 26) to compose an involution, with the given point Pq on the line for 
 07ie of its two double points (ov foci') : the other double point of this involution being 
 a point P3 of third construction ; namely, the point in which the line A2, 1 meets that 
 one of the five planes of homology IT3, 1, which corresponds (comp. 90) to the par- 
 ticular point Pq as centre. Thus, in the present example, if we denote by A'' the 
 point in which the line aa' meets the plane [a], of which (by 81, 91) the trace on 
 ABC is the line [411], and therefore is (as has been stated) the side Bi'^ci*^ of the 
 lately mentioned triangle T3, 3, so that 
 
 A^ = (1 22) = aa' • BC'" • Cb'" • Bi'^Ci"^, 
 
 we shall have the three harmonic equations, 
 
 (aa'a^Di) = (aa"'a^Ao) = (AA'^A^Ai'^) = - 1 ; 
 
 which express that this new point A" is the common harmonic conjvgate of the given 
 
CHAP. III.] GEOMETRICAL NETS IN SPACE. 75 
 
 point A, with respect to the three pairs of points^ a'di, a"'Ao, a'^Ai'^ ; and therefore 
 that these three pairs form (as has been said) an involution, with A and A'^ for its two 
 double points. 
 
 (7.) It will be found that we have now exhausted all the types of points of 
 second construction, which are situated upon lines A2, 1 ; there being only four 
 sach points on each such line. But there are still to be considered two new groups 
 of points P2 on lines Ai, and three others on lines A2,2- Attending first to the former 
 set of lines, we may observe that each of the two new types, P2,4, P2,5, represents 
 twenty points, situated two by two on the ten lines of first construction, but not on 
 any line A2 ; and therefore six by six in the ten planes ITi, each point however being 
 coinmon to three such planes : also each point P2,4 is common to three planes 172,2, 
 and each point P2, 5 is situated in one such plane ; while each of these last planes 
 contains three points P2, 4, but only one point P2, 5- If we attend only to points in the 
 plane abc, we can represent these two new groups by the two ternary types, (021) 
 and (021), which as symbols denote the two typical points, 
 
 A^ = BC • c'AiA2 • DlAiBi • «iA2B2 ; A^' = BC • c'BiBo = BC c'Bq ; 
 
 we have also the concurrence, 
 
 A^ = BC • o'Aq • DiC" • AB '", 
 
 It may be noted that A^ is the harmonic conjugate of c, with respect to Aq and 
 Bi'^, which last point is on the same trace c'aq, of the plane c'aiA2 ; and that a^' is 
 harmonically conjugate to Bi^, with respect to c' and Bq, on the trace of the plane 
 c'biB2, where bi^ denotes (by an analogy which will soon become more evident) the 
 intersection of that trace with the line ca : so that we have the two equations, 
 (AqC'Bi'^A^) = (boBi^o'a^'') = - 1. 
 
 (8.) Each line Ai, contains thus two points P2, of each of the two last new 
 groups, besides the point P2, 1, the point Pi, and the two points Pq, which had been 
 previously considered : it contains therefore eight points in all, if we still abstain (88) 
 from proceeding beyond the Second Construction. And it is easy to prove that these 
 eight points can, in two distinct modes, be so arranged as to form (comp. sub-art. 6) 
 an involution, with two of them for the two double points thereof. Thus, if we attend 
 only to points on the line bc, and represent them by ternary symbols, we may write, 
 
 B = (010), c=(001), A'=(011), a"=(0i1); 
 a^=(021), a^' = (021), AiV = (012), Ai^' = (012); 
 
 and the resulting harmonic equations 
 
 I. . . (ba'oa") = (BA^CA^') = (BAf CAi^O = - I, 
 II. . . (a'ba'c) = (A'AVA"Af') = (aVa"Ai^') = - I, 
 
 will then suffice to show : 1st., that the two points Pq, on any line Ai, are the double 
 points of an involution, in which the points Pi, Po,i form one pair of conjugates, 
 while the two other pairs are of the common form, P2,4, P2,5; and Ilnd., that the 
 two points Pi and P2, 1, on any such line Ai, are the double points of a second iiivo- 
 lution, obtained by pairing the two points of each of the three other groups. Also 
 each of the two points Pq, on a line Ai, is the harmonic conjugate of one of the 
 two points P2,5 on that line, with respect to the other point of the same group, and 
 to the point Pi on the same line ; thus, 
 
76 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 (ba'ai"a^O = (ca'a^Ai^O = - 1. 
 (9.) It remains to consider briefly three other groups of points P2, each group 
 containing sixty points , which are situated, two by two, on the thirty lines A2,2, and 
 six by six in the ten planes 11 1. Confining our attention to those which are in the 
 plane abc, and denoting them by their ternary symbols, we have thus, on the line 
 b'c', the three new typical points, of the three remaining groups, P2.6, P2,7, P2,8 : 
 
 A^"= (121) ; A^"' = (321) ; a« = (237) ; 
 with which may be combined these three others, of the same three types, and on the 
 same line b'c' : 
 
 Ai^" = (112); Ai^'" = (312); Ai« = (213). 
 Considered as intersections of a line A2,2 with lines A3 in the same plane IIi, or with 
 planes 112 (in which latter character alone they belong to the second construction), 
 the three points a"', &c., may be thus denoted : 
 
 A"^" = b'c' ■ BB" • Cb"' • AA^^ = b'c' ' BCiA2AiC2 ; 
 jjni _ 3'^' . j,^b" . ^"^v _ b'c' . DiCiAi • D1C2A2 ; 
 A™= b'c'* a'CoBi'^Ci"^B^i-BA*^Bi'^'Bi'^" = b'c''a'ciC2 ; 
 with the harmonic equation, 
 
 (CqA'Ci^A^^) = - 1, 
 
 and with analogous expressions for the three other points, Ai^", &c. The line b'c' thus 
 intersects one plane 112,1 (or its trace bb" on the plane abc), in the point a^" ; it 
 intersects two planes 112,2 (or their common trace Dib") in A"^°' ; and one other plane 
 112,2 (or its trace a'cq) in a'^ : and similarly for the other points, Ai"^", &c., of the same 
 three groups. Each plane li^, 1 contains twelve points P2,6, eight points P2,7, and eight 
 points P2,8; while every plane 112,2 contains six points P2,6) twelve points P2,7, 
 and nine points P2,8. Each point P2,6 is contained in one plane IIi; in three 
 planes 112,1; and in two planes n2,2. Each point P2,7 is in one plane ITi, in two 
 planes 112,1, and mfour planes 02,2. And each point P2,8 is situated in one plane ITi, 
 in two planes 112,1, and in three planes 112,2. 
 
 (10.) The points of the three last groups are situated o/j/y on lines A2,2; but, on 
 each such hne, two points of each of those three groups are situated ; which, along 
 with one point of each of the two former groups, P2, 1 and P2,2, and with the two 
 points Pi, whereby the line itself is determined, make up a system oitenpoints upon 
 that line. For example, the line b'c' contains, besides the six points mentioned in 
 the last sub -article, the^wr others: 
 
 b'=(101); c'=(110); a" = (011); a"'=(211). 
 Of these ten points, the two last mentioned, namely the points P2,i and P2,2upon the 
 line A2,2, are the double poitits (comp. sub-art. 8) of a new involution, in which the two 
 points of each of the four other groups compose a conjugate pair, as is expressed by 
 the harmonic equations, 
 
 (a"b'a"'c') = (A"A^"A"'Ar") = (A"A^"'A"'Ar"') = (a"a'*a"'Ai«) = - I. 
 
 And the analogous equations, 
 
 (b'a"c'a"') = (b'a^"c'a^'") = (b'ai^"c'ai^'") =- 1» 
 show that the two points Pi on any line A2,2 are the double points of of another invo- 
 lution (comp. again sub-art. 8), whereof the two points P2,i, P2,2 on that line form 
 
CHAP. III.] GEOMETIIICAL NETS IN SPACE. 77 
 
 one conjugate pair, while each of the two points P2,6 is paired with one of the points 
 P2,7 as its conjugate. In fact, the eight-rayed pencil (a.c'b'a'"a"a^'"'a^"Ai^"'Ai'") 
 coincides in position with the pencil ( A . bca Wa"^'Ai^Ai"^'), and maybe said to be 
 a pencil in double involution ; the third and fourth, the fifth and sixth, and the se- 
 venth and eighth rays forming one involution, whereof the first and second are the 
 two double* rays ; while the first and second, the fifth and seventh, and the sixth 
 and eighth rays compose another involution, whereof the double rays are the third 
 and fourth of the pencil. 
 
 (11.) If we proceeded to connect systematically the points P2 among themselves, 
 and with the points Pi and Pq, we should find many remarkable lines and planes of 
 third construction (88), besides those which have been incidentally noticed above ; for 
 example, we should have a group IIo,2 of twenty new planes^ exemplified by the 
 two following, 
 
 [E„] = [11103], [D^] = [11130], 
 which have the same common trace A3, 1, namely the line a"b"c", on the plane abc, 
 as the two planes AiBiCi, A2B2C2, and the two planes [d], [e], of the groups 1X2,2 and 
 113, 1, which have been considered in former sub- articles ; and each of these new planes 
 Ha, 2 would be found to contain one point Pq, three points P2,i, six points P2,25 and 
 three points P2, 3. It might be proved also that these twenty new planes are the 
 twenty faces of Jive new pyramids R3, which are the exscribed homologues of the five 
 old pyramids Ki (89), with the five given points Pq for the corresponding centres of 
 homology. But it would lead us beyond the proposed limits, to pursue this dis- 
 cussion further : although a few additional remarks may be useful, as serving to 
 establish the completeness of the enumeration above given, of the lines, planes, and 
 points oi second construction. 
 
 93. In general, if there be any n given points^ whereof no four 
 are situated in any common plane, the number N of the derived 
 points, which are immediately obtained from them, as intersections 
 A • n of line with plane (each line being drawn through two of the 
 given points, and each plane through three others), or the number of 
 points of the/orm ab'CDE, is easily seen to be, 
 
 _ n(^^-])(7^-2)(7.-3)(n-4) ^ 
 ^'•^^^~ 2.2.3 
 
 so that N - 10, as before, when 7t = 5. But if we were to apply this 
 formula to the case n= 15, we should iSnd, for that case, the value, 
 
 iVr=y(i5)=i5.i4. 13.11 = 30030; 
 and ikiVi^ fifteen given and independent points of space would conduct, 
 by what might (relatively to them) be called a First Construction 
 (comp. 88), to a system of more than thirty thousand points. Yet it 
 has been lately stated (92), that from the fifteen points above called 
 Po> Pi, there can be derived, in this way, onlu two hundred and ninety 
 
 * Compart; page i7'2 of the GJc:::. Srvc'rUure of il. Chasies. 
 
78 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 points P2, as intersections of the form* A -11; and therefore /e^^er 
 than three hundred. That this reduction of the number of derived 
 points^ at the end of what has been called (88) the Second Construc- 
 tion for the net in space, arising from the dependence of the ten points 
 Pi on thQJive points Pq, would be found to be so considerable, might 
 not perhaps have been anticipated; and although the foregoing ex- 
 amination proves that all the eight types (92) do really represent 
 points P2, it may appear possible, at this stage, that some other type 
 of such points has been omitted. A study of the manner in which 
 the types of points result, from those of the lines and planes oi which 
 they are the intersections, would indeed decide this question ; and 
 it was, in fact, in that way that the eight types, or groups, Po, 1, . .p^is, 
 of points of second construction for space, were investigated, and 
 found to be sufficient: yet it may be useful (compare the last sub- 
 art.) to verify, as below, the completeness of the foregoing enumeration. 
 
 (1.) ThQ ff teen points, V(!, Pi, admit of 105 binary^ and of 455 ternary combina- 
 tions; but these are far from determining so many distinct lines and planes. In fact, 
 those 15 points are connected by 25 collineations, represented by the 25 lines Ai, 
 A2,i; which lines therefore count as 75, among the 105 binary combinations of 
 points : and there remain only 30 combinations of this sort, which are constructed 
 by the 30 other lines, A2,2- Again, there are 25 ternary combinations of points, 
 which are represented (as above) by lines, and therefore do not determine any plane. 
 Also, in each of the ten planes IIi, there are 29 (=35 - 6) triangles Ti, Tg, because 
 each of those planes contains 7 points Pq, Pi, connected by 6 relations of coUinearity. 
 In like manner, each oi the fifteen planes 1X2,1 contains 8 (= 10-2) other triangles 
 T-z, because it contains 5 points po, Pi, connected by two collineations. There re- 
 main therefore only 20 (= 455 — 25 — 290 - 120) ternary combinations of points to 
 be accounted for; and these are represented by the 20 planes 112, 2- The complete- 
 ness of the enumeration of the lines and planes of the second construction is therefore 
 verified ; and it only remains to verify that the 305 points, Pq, Pi, P2, above consi- 
 dered, represent all the intersections A -IT, of the 55 lines A 1, A2, with the 45 planes 
 
 III, n2. 
 
 (2.) Each plane IIi contains three lines of each of the three groups, Ai, A2, 1, 
 A 2, 2; each plane 1X2,1 contains two lines A 2,1, and four lines A2,2; and each plane 
 1X2,2 contains three lines A2,2. Hence (or because each line Ai is contained in three 
 planes 11 1; each line A 2,1 in two planes IXi, and in two planes 1X2,1; and each 
 line A2, 2 in one plane ITi, in two planes 1X2, 1, and in two planes IX2, 2), it follows that, 
 without going beyond the second construction, there are 240 (= 30 i- 30 + 30 + 30 
 
 * The definition (88) of the points P2 admits, indeed, intersections A'A ofcom- 
 planar lines, when they are not already points Pq or Pi ; but all such intersections 
 are also points of the form A- XI ; so that no generality is lost, by confining ourselves 
 to this last form, as in the present discussion we propose to do. 
 
CHAP. 
 
 u..] 
 
 GEOMETRICAL NETS IN SPACE. 
 
 79 
 
 + 60 + 60) cases of coincidence of line and plane; so that the number of cases of 
 intersection is reduced, hereby, from 56 . 45 = 2475, to 2235 (= 2475 — 240). 
 
 (3.) Each point Pq represents twelve intersections of the form Ai'Hi ; because it 
 is common to four lines A\, and to six planes IIi, each plane containing two of those 
 four lines, but being intersected by the two others in that point Pq ; as the plane 
 ABC, for example, is intersected in A by the two lines, ad and ae. Again, each 
 point Po is common to three planes IIo, i, no one of which contains any of the four 
 lines Ai through that point ; it represents therefore a system of twelve other inter- 
 sections^ of the form Ai • ITa, i. Again, each point Pq is common to three lines Ai, i, 
 each of which is contained in two of the six planes IIi, but intersects the four others 
 in that point Pq ; which therefore counts as twelve intersections, of the form A2, rlli. 
 Finally, each of the points Pq represents three intersections, A2, 1 * ITo, 1 ; and it re- 
 presents no o^Aer intersection, of the form A -IT, within the limits of the present 
 inquiiy. Thus, each of the^re given points is to be considered as representing, or 
 constructing, thirty-nine (= 12 -f 12 + 12 +3) intersections of line with plane; and 
 there remain only 2040 (= 2235 — 195) other cases of such intersection A •IT, to be 
 accounted for (in the present verification) by the 300 derived points, Pi, P2. 
 
 (4.) For this purpose, the nine columns, headed as I. to IX. in the following 
 Table, contain the numbers of such intersections which belong respectively to the 
 nine forjns, 
 
 Ai'iii, Ai-n2,i, Arn2,2; A2,i-ni, A2,i-n2,i, A2,i-n2,2; 
 
 A2,2*ni, A2,2*n3, 1, A2,2"n2,2, 
 
 for each of the nine typical derived points, a' . . . A'^, of the nine groups Pi, P2, 1, . . 
 P2,8. Column X. contains, for each point, the sum of the nine numbers, thus tabu- 
 lated in the preceding columns ; and expresses therefore the entire number of inter- 
 sections, which any one such point represents. Column XI. states the number of the 
 points for each type ; and column XII. contains the product of the two last numbers, or 
 the number of intersections A . Tl which are represented (or constructed) by the group. 
 Finally, the sum of the numbers in each of the two last columns is written at its foot ; 
 and because the 300 derived points, of first and second constructions, are thus found 
 to represent the 2040 intersections Avhich were to be accounted for, the verification is 
 seen to be complete : and no new type, of points P2, remains to be discovered. 
 
 (5.) 
 
 
 
 Table 
 
 of Intersections A 
 
 n. 
 
 
 
 
 Type. 
 
 I. 
 
 11, 
 
 III. 
 
 IV. 
 
 V. 
 
 VI. 
 
 VII. 
 
 VIII. 
 
 IX. 
 
 X. 
 
 XI. 
 
 XII. 
 
 a' 
 
 1 
 
 6 
 
 6 
 
 6 
 
 12 
 
 18 
 
 18 
 
 24 
 
 24 
 
 115 
 
 10 
 
 1150 
 
 a" 
 
 
 
 3 
 
 6 
 
 
 
 
 
 
 
 6 
 
 3 
 
 12 
 
 30 
 
 10 
 
 300 
 
 a'" 
 
 
 
 
 
 
 
 
 
 2 
 
 2 
 
 1 
 
 2 
 
 
 
 7 
 
 30 
 
 210 
 
 A'^ 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 
 
 
 
 2 
 
 30 
 
 60 
 
 A' 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 20 
 
 60 
 
 A^' 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 20 
 
 20 
 
 A^" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 60 
 
 60 
 
 ^Tin 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 2 
 
 60 
 
 120 
 
 A'* 
 
 
 
 
 
 
 
 " 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 60 
 300 
 
 60 
 
 1 
 
 2040 
 
80 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 (6._) It is to be remembered tbat we have not admitted, by our definition (88), 
 any points which can only he determined hy intersections of three planes TIi, 02, 
 as belonging to the second construction : nor have we counted, as lines A2 of that 
 construction, any lines which can only be found as intersections of two such planes. 
 For example, we do not regard the traces Aa", &c., of certain pZanes A2,i considered 
 in recent sub-articles, as among the lines of second construction, although they would 
 present themselves early in an enumeration of the lines A3 of the third. And any 
 point in the plane abc, which can only be determined (at the present stage) as the 
 intersection of two such traces, is not regarded as a point P2. A student might find 
 it however to be not useless, as an exercise, to investigate the expressions for such 
 intersections ; and for that reason it may be noted here, that the ternary types (comp, 
 81) of the forty-four traces of planes ITi, IIo, on the plane abc, which are found to 
 compose a system of only twenty-two distinct lines in that plane, whereof nine are 
 lines Ai, A2, are the seven following (comp. 38) : 
 
 [100], [Oil], [111], [111], [Oil], [211], [211]; 
 
 which, as ternary symbols, represent the seven lines, 
 
 EC, aa', b'c', a"b"c", aa", Dia'', a'co- 
 
 (7.) Again, on the same principle, and with reference to the same definition, that 
 
 new point, say f, which may be denoted by either of the two congruent quinary 
 
 symbols (71), 
 
 F= (43210) E (01234), 
 
 and which, as a quinary type (78), represents a new group of sixty points of space 
 (and of no more, on account of this last congruence, whereas a quinary type, with all 
 its Jive coefiicients unequal, represents generally a group of 120 distinct points), is 
 not regarded by us as a point P2 ; although this new point f is easily seen to be the 
 intersection of three planes of second construction, namely, of the three following, 
 which all belong to the group IIo, 1 : 
 
 [OlIIl], [11011], [iilio], 
 
 or aa'diCiB3, cc'diBiA2, eb'b2c'c2. It may, however, be remarked in passing, that 
 each plane II 2, 1 contains twelve points P3 of this new group : every such point being 
 common (as is evident from what has been shown) to three such planes. 
 
 94. From the foregoing discussion it appears that the^ye given 
 points Po, and the three hundred derived points Pi, P2, are arranged in 
 space, upon the fifty-Jive lines A^, A^, and in the forty-Jive planes H^ 
 rig, as follows. Each line Aj contains eight of the 305 points, forming 
 on it what may be called (see the sub-article (8.) to 92) a double in- 
 volution. Each line A2, 1 contains seven points, whereof one, namely 
 the given point, Pq, has been seen (in the earlier sub-art. (6.)) to be 
 a double point of another involution, to which the thj^ee derived pairs 
 of points, Pi, p.^, on the same line belong. And each line Aj,jj con- 
 tains ten points, forming on it a 7iew involution; while eight of these 
 ten points, with a different order of succession, compose still another 
 
CHAP, in.] GKOxMETRICAL NETS IN SPACE. 81 
 
 involution* (92, (10.))- Again, each plane n, contains fifty -two 
 points, namely three given points, four points of first, and 45 points 
 of 5ecow<i construction. Each plane 11^, i contains /br^y-seven points, 
 whereof owe is a given point, four are points Pi, and 42 are points 
 
 * These theorems respecting the relations of involution, of given and derived 
 points on lines oi first and second constructions, for a net in space, are perhaps new ; 
 although some of the harmonic relations, above mentioned, have been noticed under 
 other forms by Mobius : to wliom, indeed, as has been stated, the conception of such 
 a net is due. Thus, if we consider (compare the Note to page 72) the two intersec- 
 tions, 
 
 Ei=DE'AiBiCi, E2 = DE • A2B2C2, 
 
 we easily find that they may be denoted by the quinary symbols, 
 
 El = (00012), E3= (00021); 
 they are, therefore, by Art. 9'2, the two points P3, 5 on the line de : and consequently, 
 by the theorem stated at the end of sub-art. 8, the harmonic conjugate of each, taken 
 with respect to the other and to the point Di, must be one of the two points d, e on 
 that line. Accordingly, we soon derive, by comparison of the symbols of these ^»e 
 points, DED1E1E2, the two following harmonic equations, which belong to the same 
 type as the two last of that sub-art. 8 : 
 
 (D1DE2E1) = — 1 ; (diEEiEj) = — 1 ; 
 
 but these two equations have been assigned (with notations slightly different) in the 
 formerly cited page 290 of the Barycentric Calculus. (Comp. again the recent Note 
 to page 72.) The geometrical meaning of the last equation may be illustrated, by 
 conceiving that abcd is a regular pyramid, and that e is its mean pohit; for then 
 (comp. 92, sub-art. (2.) ), vty is the mean point of the base abc ; DiD is the altitude 
 of the pyramid ; and the three segments DiE, DiEi, D1E2 are, respectively, the quar- 
 ter, the third part, and the half of that altitude : they compose therefore (as the for- 
 mula expresses) a Aarmowtc /jro^ressi'ow; or Di and Ei are conjugate points, with 
 respect to e and E2. But in order to exemplify the double involution of the same 
 sub-art. (8.), it would be necessary to consider three other points P2, on the same line 
 DE ; whereof one, above called d'i, belongs to a known group P2, i (92, (2.)); but 
 the two others are of the group Po, 4, and do not seem to have been previously noticed. 
 As an example of an involution on a line of third construction, it may be remarked 
 that on each line of the group A3, 3, or on each of the sides of any one of the ten tri- 
 angles T3, 2, in addition to one given point pq, and one derived point Pj, 1, there are 
 two points P2, 2i and two points P2,6; and that the two first points are the double 
 points of an involution, to which the two last pairs belong : thus, on the side 
 Aqbco of the exscribed triangle AqBoCo, or on the trace of the plane bciAzAiCj, we 
 have the two harmonic equations, 
 
 (b AoB"Co) = (BA'"B"crn) = - 1 . 
 
 Again, on the trace a'co of the plane ACiCa, (which latter trace is a line not passing 
 through any one of the given points,) Co and ei'^ are the double points of an invo- 
 lution, wherein a' is conjugate to cf and a'^ to b''*. But it wouid be tedious to 
 multiply such instances. 
 
 M 
 
82 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book I. 
 
 Pa: of which last, 38 are situated on the six lines Aj in the plane, but 
 four are irdersections of that plane n^, i with/owr other lines of second 
 construction. Finally, each plane 112,2 passes through no given 
 point, but coTii2Lms forty-three derived points, whereof 40 are points 
 of second construction. And because the planes o^ first construc- 
 tion alone contain specimens of all the ten groups of points^ Po, Pi, 
 P2»i> • • 1*25 81 given or derived, and of all the three groups of lines, A^, 
 ■^2)1) ^2,2, at the close of that second construction (since the types 
 P2»4j P2>5j Ai are not represented by any points or lines in any plane 
 112,1, nor are the types Pq, Ai, Ag,! represented in a plane 112,2), it 
 has been thought convenient to prepare the annexed diagram (Fig. 
 30), which may serve to illustrate, by some selected instances, the 
 arrangement oi th^ fifty -two points Pq, Pi, P2 in a plane 11^, namely, in 
 the plane abc; as well as the arrangement of the nine lines A„ A, 
 in that plane, and the ti^aces A3 of other planes upon it. 
 
 View of the Arrangement of the Principal Points and Lines in a Plane 
 of First Construction, 
 
 In this Figure, the triangle abc is suppposed, for simplicity, to be the equilateral 
 base of a regular pyramid abcd (comp. sub-art. (2.) to 92) ; and Di, again replaced 
 by o, is supposed to be its mean point (29). The first inscribed triangle, a'b'c', 
 therefore, bisects the three sides ; and the axis of homology a''b"c" is the line at in- 
 finity (38): the number 1, on the line c'b' prolonged, being designed to suggest that 
 
CHAP. 111.] GEOxMETRlCAL NETS IN SPACE. 83^ 
 
 the point a", to which that line tends, is of the type ?•.>, i, or belongs to the y/rs< 
 group of points of second construction. A second inscribed triangle, a"'b"'c"', for 
 which Fig. 21 may be consulted, is only indicated by the number 2 placed at the 
 middle of the side b'c', to suggest that this bisecting point a'" belongs to the second 
 group of points Pg. The same number 2, but with an accent, 2', is placed near the 
 corner Aq of the exscribed triangle AqBoCo, to remind us that this corner also belongs 
 (by a syntypical relation in space) to the group P2,2. The point a''', which is now 
 infinitely distant, is indicated by the number 3, on the dotted line at the top ; while 
 the same number with an accent, lower down, marks the position of the point Ai". 
 Finally, the ten other numbers, unaccented or accented, 4, 4', 5, 5', 6, 6', 7, 7', 
 8, 8', denote the places of the ten points, a^, Ai^, a^', Ai^', a"', Ai^« a'"', a^'" 
 A'*, Ai"^. And the principal harmonic relations, and relations of involution, above 
 mentioned, may be verified by inspection of this Diagram. 
 
 95. However far the series of construction of the net in space 
 may be continued, we may now regard it as evident, at least on com- 
 parison with the analogous property (42) of the plane net, that every 
 pointf line, or plane, to which such constructions can conduct, must 
 necessarily be rational (77); or that it must be rationally related to 
 the system o^ the f^ve given points : hecause ihm anharmojiic co-ordi- 
 nates (79, 80) of every net-point, and of every net-plane, are equal or 
 proportional to whole numbers. Conversely (comp. 43) every pointy 
 line, OT plane, in space, which is thus rationally related to the system of 
 points ABODE, is a point, line, or plane of the net, which those five points 
 determine. Hence (comp. again 43), every irrational point, line, or 
 plane (77), is indeed incapable of being rigorously constructed, by any 
 processes of the kind above described; but it admits of being inde- 
 finitely approximated to, by points, lines, or planes of the net. Every 
 anharmonic ratio, whether of a. group of net-points, or of a pencil of 
 net-lines, or of net-planes, has a rational value (comp. 44), which de- 
 pends only on the processes of linear construction employed, in the 
 generation of that group or pencil, and is entirely independent of the 
 arrangement, or configuration, of the five given points in space. Also,, 
 all relations of collineation, and of complanarity, are preserved, in the 
 passage from one net to another, by a change of the given system of 
 points: so that it may be briefly said (comp. again 44) that all geo- 
 metrical nets in space are homographic figures. Finally, any five points* 
 of such a net, of which no four are in one plane, are sufficient (comp. 
 
 * These general properties (95) of the space-net are in substance taken from 
 Mobius, although (as has been remarked before) the analysis here employed appears 
 to be new : as do also most of the theorems above given, respecting ihepoints of second 
 construction (92), at least after we pass beyond the Jirst group V2, \ of ten such points, 
 which (as already stated) have been known comparatively long. 
 
84 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 45) for the determination of the whole net: or for the linear construc- 
 tion of all its points, including the five given ones. 
 
 (1.) As an Example, let the five points AiBiCiDi and e be now supposed to be 
 given ; and let it be required to derive the four points abcd, by linear constructions, 
 from these new data. In other words, we are now required to exscrihe a pyramid 
 ABCD to a given pyramid AiBiCiDi, so that it may be homologous thereto, with the 
 point E for their given centre of homology. An obvious process is (comp. 45) to in. 
 scribe another homologous pyramid, A3B3C3D3,, so as to have A3 = eai*BiCiDi, &c ; 
 and then to determine the intersections of corresponding faces, such as AiBiCi and 
 A3B3C3 ; for these/owr lines of intersection will be in the common plane\E^, of homology 
 of the three pyramids, and will be the traces on that plane of the /owr sought planes, 
 ABC, &c., drawn through the four given points Di, &c. If it were only required to 
 construct one corner A of the exscribed pyramid, we might find the point above 
 called a'' as the common intersection of three planes, as follows, 
 
 A'^ = AiBiCi • Aid/e • A3B3C3 ; 
 
 and then should have this other formula of intersection, 
 
 A =EAi-DiA''. 
 
 Or the point A might be determined by the anharmonic equation, 
 
 (EAA1A3) = 3, 
 
 yrhich for a regular pyramid is easily verified. 
 
 (2.) As regards the general passage from one net in space to another, let the 
 symbols Pi ={xi . . vi), . . P5 = (a^s . . Pg) denote any Jive given points, wliereof no four 
 are complanar ; and let a'b'c'd'e and «' be six coefiicients, of which the five ratios are 
 such as to satisfy the symbolical equation (^comp. 71, 72), 
 
 a' (Pi) + bXFz) + c' (P3) + d'(Pi) + ^'(yd ==-u'CU): 
 
 or the five ordinary equations which it includes, namely, 
 
 a'xi + . . + e'x5 = . . = a'vi + . . 4- e'v^ = - u'. 
 
 Let p' be any sixth point of space, of which the quinary symbol satisfies the equa- 
 tion, 
 
 (p')=:ica'(Pi) + 2/5(P2)+ zc'(pi) + wd'(Fi) + ve'(V5)+u{ U) ; 
 
 then it will be found that this last point p' can be derived from the five points Pi . . P5 
 by precisely the same constructions, as those by which the point p = (^xyzwv') is de- 
 rived from the five points abcde. As an example, if w' = aj + y + « + w — 3w, then 
 the point {xyzwv) is derived from AiBiCiD]E, by the same constructions as (xyzwv) 
 from ABCDE ; thus a itself may be constructed from Ai . . E, as the point p = (30001) 
 is from a . . b ; which would conduct anew to the anharmonic equation of the last 
 sub-article. 
 
 (3.) It may be briefly added here, that instead of anharmonic ratios, as con- 
 nected with a net in space, or indeed generally in relation to spatial problems, we 
 are permitted (comp. 68) to substitute products (or quotients) of quotients of volumes 
 of pyramids; as a specimen of which substitution, it may be remarked, that the an- 
 harmonic relation, just referred to, admits of being replaced by the following equa- 
 tion, involving one such quotient of pyramids, but introducing no auxiliary point : 
 
CHAP. III.] MEANS OF VECTORS. 85 
 
 EA : AiA = 3eBiCiDi : AiBiCiDi. 
 
 In general, if xyzw be (as in 79, 83) the anharmonic co-ordinates of a point p in 
 space, yve may write, 
 
 X PBCD EBCD 
 ^ PCDA " ECDa' 
 
 with other equations of the same type, on which we cannot here delay. 
 
 Section 5. — On Barycentres of Systems of Points ; and on 
 Simple and Complex Means of Vectors, 
 
 96. In general, when the sum 2a of any number of co-initial 
 vectors, 
 
 ai = OAi, .. a^ = OA„„ 
 
 is divided (16) by their number, m, the resulting vector , 
 
 a = OM = — 2a = - 2oA, 
 m m 
 
 is said to be the Simple Mean of those m vectors; and ihQ point m, 
 in which this mean vector terminates, and of which the position 
 (comp. 18) is easily seen to be independent of the position of the 
 common origin o, is said to be the Mean Point (comp. 29), of the 
 system of the m points, Aj, . . A«. It is evident that we have the equa- 
 tion, 
 
 = (ai-^) + . .+(a^-/i) = 2(a-/t)-2MA; 
 
 or that the sum of the m vectors, drawn/row the mean point m, to the 
 points A of the system, is equal to zero. And hence (comp. 10, 11, 30), 
 it follows, 1st., that these m vectors are equal to the m successive 
 sides of a closed polygon ; Ilnd., that if the system and its mean 
 point be projected, by any parallel ordinates, on any assumed plane 
 (or line), the projection m', of the mean point m, is the mean point of 
 the projected system : and Illrd., that the ordinate mm', of the mean 
 point, is the mean of all the other ordinates, AiA'i, . . a^a'„. It fol- 
 lows, also, that if n be the mean point of another system, Bi, . . b„; 
 and if s be the mean point of the total system, Aj . . b,„ of the m + tj 
 = s points obtained by combining the two former, considered as par- 
 tial systems ; while v and a may denote the vectors, on and os, of 
 these two last mean points : then we shall have the equations, 
 7W/*-2a, wi^ = 2y3, 5ff = 2a+ 2)3 = w/i + /ii^, 
 
 miff- iJi) = n{v~ a), w.MS=n.SN; 
 
 so that the general mean point, s, is situated on the right line mn, 
 which connects the two partial mean points, m and n; and divides 
 
86 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 that line (internally), into tivo segments ms and sn, which are inversely 
 proportional to the two whole numbers^ m and n. 
 
 (1.) As an Example, let abcd be a gauche quadrilateral^ and let E be its mean 
 point ; or more fully, let 
 
 OE = ;i (OA + OB -t- DC -f Od), 
 
 or 
 
 that is to say, let o = 6 = c = rf, in the equations of Art. 65. Then, with notations 
 lately used, for certain derived points Di, &c., if we write the vector formuloe, 
 
 OAi = ai = i(i3 + y + 5), .. 5i=K« + /3 + r), 
 
 OA3=a2 = |(a + 5), . . r2 = Ky + ^). 
 
 oA' = a'=|(/3+r),.. y'=K«+/3), 
 
 we shall have seven different expressions for the mean vector^ i ; namely, the follow- 
 ing: 
 
 e = K« + 3ai) = .. = i(^+3^0 
 = K«'+«2) =.. = §(/ + 72). 
 
 And these conduct to the seven equations between segments^ 
 AE = 3eai, . . DB = 3edi ; 
 a'e = ea2, . . c'b = ec2; 
 
 which prove (what is otherwise known) that the four right lines, here denoted by 
 AAi, . . DDi, whereof each connects a corner of the pyramid abcd with the mean 
 point of the opposite face, intersect and quadrisect each other, in one common 
 point, e ; and that the three common bisectors a'as, b'b2, c'co, of pairs of opposite 
 edges, such as BO and da, intersect and bisect each other, in the same mean point : 
 so that the /our middle points, c', a', C2, A2, of the four successive sides ab, &c., of 
 the gauche quadrilateral abcd, are situated in one common plane, which bisects also 
 the common bisector, b'b2, ofthe^wo diagonals, AC and bd. 
 
 (2.) In this example, the number s of the points A . . D being j^wr, the number 
 of the derived lines, which thus cross each other in their general mean point E is seen 
 to be seven ; and the number of the derived planes through that point is nine : 
 namely, in the notation lately used for the net in space, four lines Ai, three lines A2, 1, 
 six planes Hi, and three planes 112, 1. Of these nine planes, the six former may (in 
 the present connexion) be called triple planes, because each contains three lines (as 
 the plane abe, for instance, contains the lines aai, bbi, c'c2), all passing through the 
 mean point e; and the three latter may be said, by contrast, to he non-triple planes, 
 because each contains only two lines through that point, determined on the foregoing 
 principles. 
 
 (3.) In general, let («) denote the number of the lines, through the general mean 
 point s of a total system of s given points, which is thus, in all possible ways, decom- 
 posed into partial systems ; let/(*) denote the number of the triple planes, obtained 
 by grouping the given points into three such partial systems ; let ;^ (s) denote the 
 number of non-triple planes, each determined by grouping those s points in two dif- 
 ferent ways into two partial systems ; and let f(«) =/(*) + »// (s) represent the entire 
 number of distinct planes through the point s : so that 
 
 ^(4) = 7, /(4) = 6, 4'(4) = 3, F(4) = &. 
 
CHAP. III.] MEAN POINTS OF SYSTEMS. 87 
 
 Then it is easy to perceive that if we introduce a new point c, each old line mn fur- 
 nishes two new lines, according as we group the new point with one or other of the 
 two old partial systems, (M) aud (A') ; and that there is, besides, one other new line, 
 namely cs : we have, therefore, the eqication infinite differences, 
 
 which, with the particular value above assigned for 0(4), or even with the simpler 
 and more obvious value, ^(2)= 1, conducts to the general expression, 
 
 0Cs) = 2*-i-l. 
 (4.) Again, if (Af) (iV) (P) be any three partial systems, which jointly make 
 up the old or given total system (-S") ; and if, by grouping a new point c with each 
 of these in turn, we form three new partial systems, {M') (N') (P') ; then each 
 old triple plane such as mnp, will furnish three new triple planes, 
 
 m'np, mn'p, mnp' ; 
 while each old line, kl, will give one new triple plane, Ckl ; nor can any new triple 
 plane be obtained in any other way. We have, therefore, this new equation in dif- 
 fer eiices : 
 
 /(*+l) = 3/(O + 0(*). 
 But we have seen that 
 
 0(» + l) = 20(5) + l; 
 
 if then we write, for a moment, 
 
 /(s) + 0(O=xW, 
 
 we have this other equation in finite differences, 
 
 X(« + I) = 3x(«)+1. 
 Also, 
 
 /(3)-l, 0(3) = 3, x(3) = 4: 
 therefore, 
 
 2x (s) = 3»-i - 1, 
 and 
 
 2/(«) = 3»-»-2»+l. 
 
 (5.) Finally, it is clear that we have the relation, 
 
 3/(*) + ^(*) = l0(O-(^(O-l) = (2-'-l) (2-2-1); 
 because the triple planes, each treated as three, and the non-triple planes, each treated 
 as one, must jointly represent all the binary combinations of the lines, drawn through 
 the mean point s of the whole system. Hence, 
 
 2»//(«) = 22«-2 + 3 . 2«-» - 3* - 1 ; 
 and 
 
 F(s) = 22»-3+2«-2-3«-i; 
 so that 
 
 P(» + 1) - 4f(») = 3*-» - 2«-i, 
 and 
 
 ^(* + l)-4,^(*) = 3/(.); 
 
 which last equation in finite differences admits of an independent geometrical inter- 
 pretation. 
 
 (6.) For instance, these general expressions give, 
 
 0(5) = 15; /(5) = 25; <//(5) = 30; f(5) = 55; 
 
 so that if we assume a gauche pentagon^ or a system of^i-e points in space, A . . e, 
 
88 ELEMENTS OF QUATERNIONS. [bOOK 1. 
 
 and determine the jnean point f of this system, there will in general be a set ofjif- 
 teen lines, of the kind above considered, all passing through this sixth point f : and 
 these will be arranged generally m fifty- five distinct planes, -whereof twenty-five will 
 be what we have called triple, the thirty others being of the non-triple kind. 
 
 97. More generally, if a^ . . a^ be, as before, a system of m given 
 
 and co-initial vectors^ and if osi, . . a^he any system of m given sea- 
 
 lars (17), then that new co-initial vector /S, or OB, which is deduced 
 
 from these by the formula, 
 
 a,a 4- . . + «,„«,» 2aa 2aoA 
 
 3 = = , or OB = , 
 
 «i + . • + «« 2a 2a 
 
 or by the equation 
 
 2a(a -/3) = 0, or Saba = 0, 
 may be said to be the Complex Mean of those m given vectors a, or 
 OA, considered as affected (or combined) with that system of given 
 scalars, a, as coefficients, or as multipliers (12, 14). It may also be said 
 that the derived point b, of which (comp. 96) the position is inde- 
 pendent of that of the origin o, is i\\e Barycentre (or centre of gravity) 
 of the given system of points Ai . . ., considered as loaded with the 
 given weights ai . . . ; and theorems of intersections of lines and planes 
 arise, from the comparison of these complex means, or harycentres, of 
 partial and total systems, which are entirely analogous to those lately 
 considered (96), for simple means of vectors and oi points. 
 
 (1.) As an Example, in the case of Art. 24, the point c is the barj'centre of the 
 system of the two points, a and b, with the weights a and h ; while, under the con- 
 ditions of 27, the origin o is the bary centre of the three points A, b, c, with the three 
 weights a,h,c; and if we use the formula for p, assigned in 34 or 36, the same three 
 given points A, b, c, when loaded with xa, yh, zc as weights, have the point p in 
 their plane for their bary centre. Again, with the equations of 65, e is the bary cen- 
 tre of the system of the ybwr given points. A, b, o, d, with the weights a, b, c, d; 
 and if the expression of 79 for the vector op be adopted, then xa, yh, zc, wd are 
 equal (or proportional) to the weights with which the same four points A . . D must 
 be loaded, in order that the point p of space may be their barycentre. In all these 
 cases, the weights are thus proportional (by 69) to certain segments, or areas, or 
 volumes, of kinds which have been already considered ; and what we have called the 
 anharmonic co-ordinates of a variable point p, in a plane (36), or in space (79), 
 may be said, on the same plan, to be quotients of quotients of weights. 
 
 (2.) The circumstance that the position of a barycentre (97), like that of a sim- 
 ple mean point (96), is independent of the position of the assumed origin of vectors, 
 might induce us (comp. 69) to suppress the symbol o of that arbitrary and foreign 
 point; and therefore to write' simply, under the lately supposed conditions, 
 
 * We should thus have some of the principal notations of the Barycentric Calcu- 
 lus : but used mainly with a reference to vectors. Compare the Note to page 56. 
 
CHAP. III.] BARYCENTRES OF SYSTEMS OF POINTS. 81) 
 
 B = — — or 65=20.4, if 6 = a. 
 2a 
 
 It is easy to prove (comp. 96), by principles already established, that the ordi- 
 nate of the barycentre of any given system of points is the complex mean (in 
 the sense above defined, and with the same system oi weights)^ of the ordinates of 
 the points of that system, with reference to any given plane : and that the projection 
 of the barycentre, on any such plane, is the barycentre of the projected system. 
 
 (3. ) Without any reference to ordinates, or to any foreign origin, the barycentrie 
 
 notation B = may be interpreted, by means of our fundamental convention 
 
 2a 
 
 (Art. 1) respecting the geometrical signification of the symbol b— A, considered as 
 denoting the vector from A to B : together with the rules for midtiplying such vec- 
 tors by scalars (14, 17), and for taking the sums (6, 7, 8, 9) of those (generally 
 new) vectors, which are (16) the products of such multiplications. For we have only 
 to write the formula as follows, 
 
 2a(A-B) = 0, 
 
 in order to perceive that it may be considered as signifying, that the system of the 
 vectors from the barycentre B, to the system of the given points Ai, A2, . . when mul- 
 tiplied respectively by the scalars (or coefficients) of the given system ai, 02, . . be- 
 comes (generally) a new system of vectors with a null sum : in such a manner that 
 these last vectors, ai . b Ai, 02 • BA2, . • can be made (10) the successive sides of a closed 
 polygon, by transports without rotation. 
 (4.) Thus if we meet the formula, 
 
 B = ^(Ai + A2), 
 
 we may indeed interpret it as an abridged form of the equation, 
 
 OB = |(OAi + OA2); 
 
 which implies that if o be any arbitrary point, and if o' be the point which completes 
 (comp. 6) the parallelogram AiOA20', then B is the point which bisects the diagonal 
 00', and therefore also the given line AiA2, which is here the other diagonal. But we 
 may also regard the formula as a mere symbolical transformation of the equation, 
 
 (a3-b)+(ai-b) = 0; 
 which (by the earliest principles of the present Book) expresses that the two vectors, 
 from B to the two given points Ai and A2, have a null sum; or that they are equal in 
 length, but opposite in direction : which can only be, by B bisecting A1A2, as before. 
 (5.) Again, the formula, bi = ^(ai + A2 4- A3), may be interpreted as an a&Hcf^- 
 ment of the equation, 
 
 OBi = J (OAi + OA3 + OA3) , 
 
 which expresses that the point B trisects the diagonal 00' of the parallelepiped 
 (comp. 62), which has OAi, 0A2, OA3 for three co-initial edges. But the same for- 
 mula may also be considered to express, in full consistency with the foregoing inter- 
 pretatiim, that the sum of the three vectors, from b to the three points Ai, A2, A3, va- 
 nishes : which is the characteristic property (30) of the mean point of the triangle 
 A1A2A3. And similarly in more complex cases : tlie legitimacy of such transforma- 
 tions being here regarded as a consequence of the original interpretation (1) of the 
 symbol n - A, and of the rules for operations on vectors, so far as as they have been 
 hitherto established. 
 
 N 
 
90 KLEMENTS OF QUATERNIONS. [bOOK I. 
 
 Section 6 On Anharmonic Equations, and Vector- Expres- 
 sions, of Surfaces and Curves in Space. 
 98. When, in the expression 79 for the vector /> of a variable 
 point P of space, the four variable scalars, or anharmonic co-ordi- 
 nates, xi/zw, are connected (comp. 46) by a given algebraic equation, 
 
 f,{x, y, z, w) = 0, or briefly /= 0, 
 supposed to be rational and integral, and homogeneous of the p''' 
 dimension, then the point P has for its locus a surface of the p^^ orde?', 
 whereof /= may be said (comp. 56) to be the local equation. For 
 if we substitute instead of the co ordinates x . .w, expressions of the 
 forms, 
 
 X = tXo + UXx^ .. w= tWo + UWi^ 
 
 to indicate (82) that p is collinear with two given points, Po, Pi, the 
 resulting algebraic equation int'.u is of the p*^ degree ; so that (ac- 
 cording to a received modern mode of vspeaking), the surface may be 
 said to be cut in p points (distinct or coincident, and real or imagi- 
 nary*), hy any arhitrary right line, PyPi- And in like manner, when 
 the four anharmonic co-ordinates Imnr of a variable plane 11 (80) are 
 connected by an algebraical equation, of the form, 
 F^(/, m, n, r) =0, or briefly F = 0, 
 where F denotes a rational and integral function, supposed to be ho- 
 mogeneous of the q^^ dimension, then this plane n has for its enve- 
 lope (comp. 5%) a surface of the q*'' class, with f= for its tangential 
 equation: because if we make 
 
 l = tlQ+ uli,.. . r = tro-\-uri, 
 to express (comp. 82) that the variable plane 11 passes through a given 
 right line ITo'IIi, we are conducted to an algebraical equation of the 
 q^^ degree^ which gives q (real or imaginary) values for the ratio t:u, 
 and thereby assigns q (real or imaginary!) tangent planes to the sur- 
 
 * It is to be observed, that no interpretation is here proposed, for imaginary in- 
 tersections of this kind, such as those of a sphere with a right line, which is wholly 
 external thereto. The language of modern geometry requires that snch imaginary 
 intersections should be spoken of, and even that they should be cnwrnera/ec? : exactly 
 as the language of algebra requires that we should count what are called the imagi- 
 nary roots of an equation. But it would be an error to confound geometrical imagi- 
 naries, of this sort, with those square roots of negatives, for which it will soon be seen 
 that the Calculus of Quaternions supplies, from the outset, a di finite and real in- 
 terpretation. 
 
 f As regards the uninterpreted character of such imaginary contacts in geometry, 
 the preceding Note to the present Article, resptcting imaginary intersections, may be 
 consulted. 
 
CHAP. III.] ANHARMONIC EQUATIONS OF SURFACES. 
 
 91 
 
 face^ drawn through any such given but arbitrary right line. We 
 may add (comp. 51, 56), that if the functions / and f be only ho- 
 mogeneous (without necessarily being rational and integral)^ then 
 
 is the anharmonic symbol (80) of the tangent plane to the surface 
 /= 0, at the point (xyziv) ; and that 
 
 (DjF, d,„f, d„f, d,f) 
 is in like manner, a symbol for the point of contact of the plane 
 \_lmnr'], with its enveloped surface^ f= 0; d^, . . d^, . . being charac- 
 teristics of partial derivation. 
 
 (1.) As an Example, the surface of the second order, which passes through the 
 nine points called lately 
 
 A, c', B, a', C, C2, D, A2, E, 
 
 has for its local equation, 
 
 0=f=xz-yw; 
 which gives, by differentiation, 
 
 I = T)xf— z; m = Dy/= — w ; 
 
 n=Dzf=X', r =DM,/=-y: 
 so that 
 
 lz,-w, a!,-2/] 
 
 is a symbol for the tangent plane, at the point (x, y, z, w). 
 
 (2.) In fact, the swrface here considered is the ruled (or hyper'holic) hyperboloid, 
 on which the gauche quadrilateral abcd is superscribed, and which passes also 
 through the point e. And if we write 
 
 p = (xyziv), Q = (aryOO), R = (OyzO), 
 then Qs and rt (see the annexed Figure 31), 
 namely, the lines drawn through p to intersect the 
 two pairs, ab, cd, and bc, da, of opposite sides 
 of that quadrilateral abcd, are the two generating 
 lines, or generatrices, through that point ; so that 
 their plane, qrst, is the tangent plane to the sur- 
 face, at the point p. If, then, we denote that tan- 
 gent plane by the symbol [Imnr], we have the 
 equations of condition, 
 
 = Zar + my = my + nz = nz + rw = rw+lx; 
 whence follows the proportion, 
 
 l:m:n:r = otr^ : — y~^ : z*' : — w • ; 
 
 or, because xz = yw, 
 
 I: m: n: r= z : —w: x 
 as before. 
 
 (3.) At the same time we see that 
 
 (ac'bq) = - = 
 
 = (002u;), T = {xOOw\ 
 
 Fig. 3 
 
 (ncacs) ; 
 
92 * ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 so that the variable generatrix QS divides (as is known) the two Jixed generatrices 
 AB and DC homographically* ; ad, bc, and c'cj being three of its positions. Con- 
 versely, if it were proposed to find the locus of the right liiie Q3, which thus divides 
 homographically (comp. 26) two given right lines in space, we might take ab and DC 
 for those two given lines, and ad, bc, c'c2 (with the recent meanings of the letters) 
 for three given positions of the variable line ; and then should have, for the two va- 
 riiible but corresponding (or homologous^ points % s themselves, and for any arbitrary 
 point p collinear with them, anharmonic symbols of the forms, 
 
 Q = (s, M, 0, 0), s = (0, 0, M, s), P = (st, tu, uv, vs) ; 
 because, by 82, we should have, between these three symbols, a relation of the form, 
 
 (p) = ^(q) + »(s)! 
 if then we write p= (ar, y, z, w), we have the anharmonic equation xz = yw, as before ; 
 80 that the locus, whether of the line qs, or of the point p, is (as is known) a ruled 
 surface of the second order. 
 
 (4.) As regards the known double generation of that surface, it may suflSce to 
 observe that if we write, in like manner, 
 
 K=(Of«0), T = (<00f), (p)=«(r) + «(t), 
 we shall have again the expression, 
 
 p = {st, tu, uv, vs), giving xz = yw, 
 as before : so that the same hyperboloid is also the locus of that other line rt, which 
 divides the other pair of opposite sides bc, ad of the same gauche quadrilateral abcd 
 homographically ; ba, cd, and A'Ag being three of its positions ; and the lines a'a2, 
 c'c2 being still supposed to intersect each other in the given point e. 
 
 (5.) The symbol of an arbitrary point on the variable line kt is (by sub-art. 2) 
 of the form, t(0, y, z, 0) +u(x, 0, 0, w), or (ux, ty, tz, uw) ; while the symbol of an 
 arbitrary point on the given line c'C2 is (t', f, u, u'). And these two symbols repre- 
 sent one common point (comp. Fig. 31), 
 
 p' = RT-c'c2=(y,y,2,2), 
 when we su[)pose 
 
 , , y 2 
 
 t =y, u =z, t=\, «=-=-. 
 X w 
 
 Hence the known theorem results, that a variable generatrix, kt, of one system, in- 
 tersects three fixed lines, BC, AD, c'Cg, which are generatrices of the other system. 
 Conversely, by the same comparison of symbols, for points on the two lines rt and 
 c'c2, "we should be conducted to the equation xz =yw, as the condition for their inter- 
 section ; and thus should obtain this other known theorem, that the locus of a right 
 line, which intersects three given right lines in space, is generally an hyperboloid 
 with tliose three lines for generatrices. A similar analysis shows that QS intersects 
 a'a2, in a point (comp. again Fig. 31) which may be thus denoted : 
 
 p" = QS • a'a2 = (xyyx). 
 
 (6.) As another example of the treatment of surfaces by their anharmonic and 
 local equations, we may remark that the recent symbols for p' and p'', combined with 
 
 Compare p. 298 of the Geometric Superieure. 
 
CHAP. III.] ANHAllMONIC EQUATIONS OF SURFACES. 93 
 
 those of sub-art. 2 for p, q, r, s, t; with the symbols of 83, 86 for c', a', C2, A2, e; 
 and with the equation xz = y w, give the expressions : 
 
 (p)=(q) + (8) = (r) + (t); (P') = y(c') + ^(C2)=(R)+^(T); 
 
 (E) = (c') + (C2) = (A-) + (A2) ; (p") = y{A')-^x (a^) = (q) + ^ (s) ; 
 
 whence it follows (84) that the two points p', p", and the sides of the quadrilateral 
 ABCD, divide the four generating lines through p and e in the following anharmonic 
 ratios : 
 
 (c'eCzP') = (qp"sp) = - = (bA'CR) = (AAgDT) ; 
 
 / y 
 
 (a'eA2P ') = (rp'tp) = - = (bc'Aq) = (CC2DS) J 
 
 so that (as again is known) the variable generatrices, as well as the fixed ones, of the 
 hyperboloid, are all divided homographically . 
 
 (7.) The tangential equation of the present surface is easily found, by the expres- 
 sions in sub-art. 1 for the co-ordinates Imnr of the tangent plane, to be the follow- 
 ing: 
 
 = F = /n — wir ; 
 
 which may be interpreted as expressing, that this hyperboloid is the surface of the 
 second class, which touches the nine planes, 
 
 [1000], [0100], [0010], [0001], [1100], [0110], [0011], [1001], [1111] ; 
 or with the literal symbols lately employed (comp. 86, 87), 
 
 BOD, CDA, DAB, ABC, CDc", DAa", ABc'o, BCA'2, and [e].* 
 
 Or we may interpret the same tangential equation f = as expressing (comp. again 
 86, 87, where q, l, n are now replaced by t, r, q), that the surface is the envelope of 
 a plane qrst, which satisfies either of the two connected conditions of homography : 
 
 (bc'aq) = = = (ccaDs) ; 
 
 m n 
 
 (CA Br) = = = (dA2 at) ; 
 
 n r 
 
 a double generation of the hyperboloid thus showing itself in a new way. And as re- 
 gards the. passage (or return)^ from the tangential to the local equation (comp. 66), 
 we have in the present example the formulae : 
 
 X = DiF = n ; y = d^f = — r; z = d„f = Z ; w = d^-f = — to ; 
 whence 
 
 xz — yw = 0, 
 as before. 
 
 (8.) More generally, when the surface is of the second order, and therefore also 
 of the second class, so that the two functions / and f, when presented under rational 
 and integral forms, are both homogeneous of the second dimension, then whether we 
 derive I . .r from x . .why the formulae. 
 
 * In the anharmonic symbol of Art. 87, for the plane of homology [e], the co- 
 efficient 1 occurred, through inadvertence, five times. 
 
94 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 or a; . . M7 from / . . r by the converse formulae, 
 
 X = DiF, y = DmF, Z = D„F, W = D^F, 
 
 the /)oin< p = (xyzvi) is, relatively to that surface, what is usually called (corap. 62) 
 the pole of the plane 11 = [Imnr] ; and conversely, the plane 11 is the polar of the 
 point p ; wherever in space the point P and plane 11, thus related to each other, 
 may be situated. And because the centre of a surface of the second order is known 
 to be (comp. again 52) the pole of (what is called) the plajie at infinity ; while (comp. 
 38) the equation and the symbol of this last plane are, respectively, 
 
 aa; + &y + cz -f rfw = 0, and [a, 6, c, d], 
 if the four constants aftccZ have still the same significations as in 05, 70, 79, &c., 
 with reference to the system of the five given points abode : it follows that we may 
 denote this centre by the symbol, 
 
 K=(DaFo, DfcFo, DcFq, DrfFo) ; 
 
 where Fq denotes, for abridgment, the function f (abcd)^ and d is still a scalar con- 
 stant. 
 
 (9.) In the recent example, we have YQ = ac — ld; and the anharmonic symbol 
 for the centre of the hyperboloid becomes thus, 
 
 K = (c, — d, a, — 6), 
 Accordingly if we assume (comp. sub- arts. 3, 4), 
 
 p = (.<si, tu, w», »s), p' = (s't\ — t'u, uv\ = r's'), 
 where s, ;f, «, v are any four scalars, and p' is a new point, while 
 
 &' = 6^-1- cw, <' = CM + ds, u =dv ■\^ at, v =as-\-hu; 
 
 if also we write, for abridgment, 
 
 e = ac — hd, w' = ast + htu + cuv + dvs ; 
 we shall then have the symbolic relations, 
 
 e' (p) + (P ) = w (k), e' (p) - (p') = (p"), 
 if p" = {x"y"z"w") be that new point, of which the co-ordinates are, 
 
 x" = lest — cw\ y" = 2e'tu -\- dw\ z" = 2e'uv — aw\ w" = 2e'vs + hw\ 
 
 and therefore, 
 
 ax" + by" + cz" + dw" = 0. 
 
 That is to say, if pp' be any chord of the hyperboloid, which ]^SiSses through the fixed 
 point K, and if p" be the harmonic conjugate of that fixed point, with respect to that 
 variable chord, so that (pkp'p") = - 1, then this conjugate point p" is on the infinitely 
 distant plane [abed] : or in other words, the fixed point K bisects all the chords pp' 
 which pass through it, and is therefore (as above asserted) the centre of the surface. 
 (10.) With the same meanings (65, 79) of the constants a, b, c, d, the mean 
 point (96) of the quadrilateral abcd, or of the system of its comers, may be denoted 
 by the svmbol, 
 
 M = («-!, 6 1, cS rf-i); 
 
 if then this mean point be on the surface, so that 
 
 ac-bd=0, 
 the centre K is on the plane [a, /», r, d] ; or in other words, it is infinitely distant : so 
 
CHAP. III.] VECTORS OF SURFACES AND CURVES IN SPACE. 95 
 
 that the surface becomes, in this case, a ruled (or hyperbolic) paraboloid. In gene- 
 ral (comp. sub-art. 8), if Fo = 0, the surface of the second order is a paraboloid of 
 some kind, because its centre is then at infinity^ in virtue of the equation 
 
 (aD« + bDb + cDc + dUd) Fo = ; 
 or because (comp. 60, 58) the plane [abed'] at infinity is then one of its tangent 
 planes, as satisfying its tangential equation, F = 0. 
 
 (11.) It is evident that a curve in space may be represented by a system of two 
 anharmonic and local equations ; because it may be regarded as the intersection nf 
 two surfaces. And then its order, or the number of points (real or imaginary*"), in 
 which it is cut by an arbitrary plane, is obviously the product of the orders of those 
 two surfaces; or iho. product of the degrees of their two local equati(,ns, supposed to 
 be rational and integral. 
 
 (12.) A curve of double curvature may also be considered as the edge of regres- 
 sion (or arete de rebrovssement) of a developable surface, namely of the locus of the 
 tangents to the curve ; and this surface may be supposed to be circumscribed at once 
 to two given surfaces, which are envelopes of variable planes (98), and are repre- 
 sented, as such, by their tangential equations. In this view, a ciirve of double cur- 
 vature may itself he represented by a system of two anharmonic and tangential equa- 
 tions ; and if the class of such a curve be defined to be the number of its osculating 
 planes, which pass through ah arbitrary point of space, then this class is the product 
 of the classes of the two curved surfaces just now mentioned: or (what comes to the 
 same thing) it is the product of the dimensions of the two tangential equations, by 
 which the curve is (on this plan) symbolized. But we cannot enter further into these 
 details ; the mechanism of calculation respecting which would indeed be found to be 
 the same, as that employed in the known method (comp. 41) of quadriplanar co-or- 
 dinates. 
 
 99. Instead of anharmonic co-ordinates, we may consider any 
 other system of n variable scalars, x^, .. x„, which enter into the ex- 
 pression of a variable vector, p\ for example, into an expression of 
 the form (comp. 96, 97), 
 
 p-Xa^-ir XSH + • . = Ixa. 
 
 And then, if those n scalars x be ^\\ functions of one independent and 
 variable scalar^ t, we may regard this vector p as being itself a func- 
 tion of that single scalar; and may write, 
 
 !.../>= (2(0. 
 But if the n scalars x . ,hQ functions of two independent and scalar 
 variables, t and u, then p becomes a function of those two scalars^ 
 and we may write accordingly, 
 
 II. . . /> = <|)(;, v). 
 In the 1st case, the term p (comp. 1) of the variable vector /> has 
 
 • Compare the Notes to page 90. 
 
96 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 generally for its locus a curve in space^ which may be plane or of 
 double curvature, or may even become a right line^ according to the 
 form of the vector-function cp ; and p may be said to be the vector of 
 this line, or curve. In the Ilnd case, p is the vector of a surface, plane 
 or curved, according to the form of <p (t, u) ; or to the manner in which 
 this vector p depends on the two independent scalars that enter into 
 its expression. 
 
 (1.) As Examples (comp. 25, 63), the expressions, 
 
 signify, 1st, that p is the vector of a variable point p on the right line ab ; or that 
 it is the vector of that line itself, considered as the locus of a point; and Ilnd, that 
 p is the vector of the plane abc, considered in like manner as the locus of an arbitrary 
 point P thereon. 
 
 (2.) The equations, 
 
 1. .. p = xa^-y(i, II. .. p = jca + y/3 + zy, 
 >nth 
 
 a;2 + y2 = 1 for the 1st, and a:^ + y2 + ^2 = i for the Ilnd, 
 
 signify 1st, that p is the vector of an ellipse, and Ilnd, that it is the vector of an 
 ellipsoid, with the origin o for their common centre, and with OA, OB, or OA, ob, 
 DC, for conjugate semi-diameters. 
 (3.) The equation (comp. 46), 
 
 p = t''a^ui^^(t^uyy, 
 
 expresses that p is the vector of a cone of the second order, with o for its vertex (or 
 
 centre), which is touched by the three planes obc, oca, gab ; the section of this cone, 
 
 /> made by the plane abc, being an ellipse (comp. Fig. 25), which is inscribed in the 
 
 /t"'' triangle ABO ; and the middle points A, b', c', of the sides of that triangle, being tlje 
 
 points of contact of those sides with that conic. 
 
 (4.) The equation (comp. 53), 
 
 p = r'a + «"i/3 + r-iy, with < + u + v = 0, 
 expresses that p is the vector of another cone of the second order, with o still for 
 vertex, but with OA, ob, oc for three of its sides (or rays). The section by the 
 plane abc is a new ellipse, circumscribed to the triangle abc, and having its tangents 
 at the corners of that triangle respectively parallel to the opposite sides thereof. 
 
 (5 J The equation (comp. 54), 
 
 p=t^a + m'/3 + v^y, with t +- m + « = 0, 
 signifies that p is the vector of a cone of the third order, of wliich the vertex is still 
 the origin ; its section (comp. Fig. 27) by the plane abc being a cubic curve, whereof 
 the sides of the triangle abc are at once the asymptotes, and the three (real) tangents 
 of inflexion; while the mean point (say o') of that <na«^Ze is Si conjugate point oi 
 the curve; and therefore the right line oo', from the vertex o to that mean point, 
 may be said to be a conjugate ray of the cone. 
 
 (6.) The equation (comp. 98, sub-art. (3.) ), 
 
CHAP. III.] VECTORS OF SURFACES AND CURVES IN SPACE. 97 
 
 staa + tuhfi + uvcy + vsdS 
 
 p =: ■ , 
 
 sta + tub + uvc + vsd 
 
 s t 
 
 in which - and - are two variable scalars, while o, 6, c, d are still four constant 
 u V 
 
 scalars, and a, /3, y, d are four constant vectors, but p is still a variable vector, ex- 
 presses that p is the vector of a ruled (or single-sheeted^ hyperholoid^ on which the 
 gauche quadrilateral abcd is superscribed, and which passes through the given point 
 E, whereof the vector e is assigned in 65. 
 
 (7.) If we make (comp. 98, sub-art (9.)), 
 
 , s't'aa - t'u'hfi + u'v'cy — v's'dd 
 
 P =; _ __ — . — ^ 
 
 s't'a — t'u'b + u'v'c — vsd 
 where 
 
 s'=bt + cv, t' = cu + ds, u'=dv-\-at, v' = as + bu, 
 
 then p' = op' is the vector of another point p' on the same hyperboloid ; and because 
 it is found that the sum of these two last vectors is constant, 
 
 „+.„'-2« if,. °<« + r)- K^ + ^) 
 
 p+-p-2«,,l.: 2(ac-6rf) 
 
 it follows that k is the vector o^ a, fixed point k, which bisects evert/ chord pp' that 
 passes through it : or in other words (comp. 52), that this point k is the centre of 
 the surface. 
 
 (8.) The three vectors, 
 
 a + y (3+d 
 
 "' 2 ' 2 ' 
 
 are termino-collinear (24) ; if then a gauche quadrilateral abcd be superscribed on 
 a ruled hyperboloid, the common bisector of the two diagonals, AC, bd, passes through 
 the centre K. 
 
 (9.) When ac = bd, or when we have the equation, 
 
 sta + tu(3 + uvy -f vsS 
 
 n = 
 
 st + tu + uv -{- vs 
 
 or simply, 
 
 p = sta + tuj3 + uvy + vsd, with s +u=t + v = l, 
 
 p is then the vector of a ruled paraboloid, of which the centre (comp. 52, and 98, sub- 
 art. (10.) ), is infinitely distant, but upon which the quadrilateral abcd is still super- 
 scribed. And this surface passes through the mean point M of that quadrilateral, or 
 of the system of the four given points A . . D ; because, when s = t = u = v = -^, th« 
 variable vector p takes the value (comp. 96, sub-art. (1.)), 
 )tt = i(a + /3 + y + ^). 
 (10.) In general, it is easy to prove, from the last vector-expression for p, that 
 this paraboloid is the locus of a right line, which divides similarly the two opposite 
 tides AB and DC of the same gauche quadrilateral abcd; or the other pair of oppo- 
 site sides, EC and ad. 
 
98 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 Section 7 — On Differentials of Vectors. 
 100. The equation (99, I.), 
 
 in which /> = op is generally the vector of a point p of sl curve in space, 
 
 PCI . . ., gives evidently, for the vector oq of another point Q of the 
 
 same curve, an expression of the form 
 
 p + Ap^<p(ti-At); 
 
 so that the chord pq,, regarded as being 
 itself a vector, comes thus to be repre- 
 sented (4) by the Jlnite difference, 
 PQ = A/> = A(p (t) = (p(t + At)-(p (t). 
 
 Suppose now that the other finite dif- 
 ference, A^, is the n*^ part of a new 
 scalar, u ; and that the chord A/>, or pq, is in like manner (comp. 
 Fig. 32), the n^^ part of a new vector, ff„, or pr ; so that we may 
 write, 
 
 nAt = u, and ?iA/3 = w . pq = o-,^ = pr. 
 
 Then, if we treat the two scalars, t and u, as constant, but the num- 
 ber n as variable (the, form of the vector function (f), and the origin o, 
 being given), the vector p and the;?om^ p will he fixed: but the two 
 points Qt and R, the two differences At and Ap, and the multiple vector 
 nAp, or <T„, will (in general) vary together. And if this number n 
 be indefinitely/ increased, or made to tend to infinity, then each of the 
 two differences At, Ap will in general tend to zero ; such being the 
 common limit, of n~^u, and of <|> (^ + n~^u) - ^(f)'. so that the variable 
 •point Q of the curve will tend to coincide with the fixed point p. But 
 although the chord pq will thus be indefinitely shortened, its n^^ mul- 
 tiple, PR or a,,, will tend (generally) to Vi finite liinit,* depending on 
 the supposed continuity oi the function <j>(^); namely, to a certain 
 definite vector, pt, or «t„, or (say) t, which vector pt will evidently 
 be (in general) tangential to the curve: or, in other words, the variable 
 point R will tend to a fixed position t, on thetangent to that curve at p. 
 We shall thus have a limiting equation, of the form 
 
 T = PT = lim. PR = croo = lim. 7iA0(^), if ?iA^ = w; 
 
 M = 00 
 
 t and u being, as above, two given and (generally) /wiVe scalars. And 
 
 * Compare Newton's Privcipia. 
 
CHAP. III.] DIFFERENTIALS OF VECTORS. 99 
 
 if we then agree to call the second of these two given scalars the dif- 
 ferential of the first, and to denote it by the symbol d^, we shall de- 
 fineih2i,\, the vector-limit^ r or o-», is the (corresponding) differential of 
 the vector p, and shall denote it by the corresponding symbol^ d/>; so 
 as to have, under the supposed conditions, 
 u = dt, and t = dp. 
 Or, eliminating the two symbols u and t, and not necessarily suppos- 
 ing that p is SL point of a curve, we may express our Definition"^ of the 
 Differential of a Vector />, considered as a Function ^ of a Scalar t, 
 by the following General Formula : 
 
 dp = d^{t)=\m-i.n\cl^{t+-\-^(f)\, 
 n = cc ( n J ) 
 
 in which t and d^ are two arbitrary and independent scalars, both ge- 
 nerally finite ; and dp is, in general, a new and finite vector, depending 
 on those two scalars, according to a law expressed by the formula, 
 and derived from that given law, whereby the old ov former vector, p 
 or <p (t), depends upon the single scalar, t. 
 
 (1.) As an example, let the given vector-function have the form, 
 
 p — ^(f) — ^t^a, "where a is a given vector. 
 
 u 
 Then, making Af = -, where u is any given scalar, and n is a variable whole number, 
 
 we have 
 
 (Tn = nAp = au{t + —]; a^^ — atu ; 
 
 and finally, writing dt and dp for u and (Tx, 
 
 dp=d0(O = df^U«<d^ 
 
 (2.) In general, let <p(t)=af(t), where a is still a given or constant vector, and 
 f(f) denotes a scalar function of the scalar variable, t. Then because a is a common 
 factor within the brackets { } of the recent general formula (100) for dp, we may 
 write, 
 
 dp = d0(O=d.a/(O = ad/(O; 
 provided that we now define that the differential of a scalar function, f{t), is a new 
 scalar function of two independent scalars, t and dt, determined by the precisely 
 similar formula : 
 
 d/(0 = lhn.n|/(^ + ^']-/(0}; 
 
 * Compare the Note to page 39. 
 
100 ELEMENTS OF QUATERNIONS. [bOOK 1. 
 
 which can easily be proved to agree^ in all its consequences^ with the usual rules for 
 differentiating functions of one variable. 
 
 (3.) For example, if we write dt = nh, where A is a new variable scalar, namely, 
 the »*'» part of the given and (generally) finite differential, At, we shall thus have 
 the equation, 
 
 4/*(0 ,. /(^ + / 0-/(0 . 
 
 — — - = lim. ; 
 
 dt 7^=0 h 
 
 in which the first member is here considered as the actual quotient of two finite sca- 
 lars, df(i) : d^, and not merely as a differential coefficient. We may, however, as 
 usual, consider this quotient, from the expression of which the differential dt disap- 
 pears, as a derived function of the former variable, t ; and may denote it, as such, by 
 
 either of the two usual symbols, 
 
 fit) and J)tf{t). 
 
 (4.) In like manner we may write, for the derivative of a vector-function,* ^(t), 
 
 the formula : 
 
 ,, dp d0(O 
 p' = f(<) = D<p = D<5&(0=^= -^; 
 
 these two last forms denoting that actual and finite vector, p' or ^' (t), which is 
 obtained, or deri')ed, by dividing (comp. 16) the not less actual (or finite) vector, 
 dp or d<p(t), by the finite scalar, dt. And if again we denote the n'^ part of this 
 last scalar by h, we shall thus have the equally general formula : 
 
 Dtp = Dt(}) (t) = hm. ; 
 
 A = « 
 
 with the equations, 
 
 dp = Dtp . d^ = pdt ; d0 (t) = Dt(p (t) . dt = ^'(t) . dt, 
 exactly as if the vector-function, p or ^, were a scalar function, f. 
 
 (5.) The particular value, dt — 1, gives thus dp = p'\ so that the derived vector 
 p' is (with our definitions) a particular but important case of the differential of a 
 vector. In applications to mechanics, if t denote the time, and if the term v of 
 the variable vector p be considered as a moving point, this derived vector p' may be 
 called the Vector of Velocity : because its length represents the amount, and its di- 
 rection is the direction of the velocity. And if, by setting off vectors ov = p' (comp. 
 again Fig. 32) /rom one origin, to represent thus the velocities of a point moving in 
 space according to any supposed law, expressed by the equation p = (p(t), we con- 
 struct a new curve vw . . of which the corresponding equation may be written as 
 p' = <p'(t), then this new curve has been defined to be the HoDOGKAPH,t as the old 
 curve FQ. . mav be called the orbit of the motion, or of the moving point. 
 
 * In the theory of Differentials of Functions of Quaternions, a definition of the 
 differential d^{q) will be proposed, which is expressed by an equation of precisely 
 the same form as those above assigned, for df(t), and for d<p {t) ; but it will be found 
 that, for qyafernions, the quotient d^(«7): d^' is not generally independent of dq ; 
 and consequently that it cannot properly be called a derived function, such as ^'(9), 
 of the quaternion q alone. (Compare again the Note to page 39.) 
 
 t The subject of the Hodograph will be resumed, at a subsequent stage of this 
 work. In fact, it almost requires the assistance of Quaternions, to connect it, in 
 what appears to be the best mode, with Newton's Law of Gravitation. 
 
CHAP. III.] DIFFERENTIALS OF VECTORS. 101 
 
 (6.) We may differentiate a vector-function twice (or oftener), and so obtain its 
 successive differentials. For example, if we diff^erentiate the derived vector p', we 
 obtain a result of the form, 
 
 dp' = p"dt, where p" = D(p' = D<2p, 
 by an obvious extension of notation ; and if we suppose ihat the second differential, 
 dd^ or d-^, of the scalar t is zero, then the second differential of the vector p is, 
 
 d2(0 = ddp = d. p'd^ = dp'. At^p'Afi ; 
 -where At^, as usual, denotes (d<)2 ; and where it is important to observe that, with 
 the definitions adopted, d^p is as finite a vector as dp, or as p itself. In applications 
 to motion, lit denote the time, p" may be said to be the Vector of Acceleration. 
 
 (7.) "We may also say that, in mechanics, i\\Q finite differential dp, of the Vector 
 of Position p, represents, in length and in direction, the right line (suppose pt in 
 Fig. 32) which would have been described, by a freely moving point p, in the finite 
 interval of time At, immediately /oZ/owzw^r the time t, z/at the end of this time t all 
 foreign forces had ceased to act.* 
 
 (8.) In geometry, if p = <p(t) be the equation of a curve of double curvature, re- 
 garded as the edge of regression (comp. 98, (12.) ) of a developable surface, then the 
 equation of that surface itself, considered as the locus of the tangents to the curve, 
 may be thus written (comp. 99, II.) : 
 
 p = (p(t) + u(p'(ty, or simply, p = (p(t)+ d(p(t), 
 if it be remembered that u, or d^, may be any arbitrary scalar. 
 
 (9.) If any other curved surface (comp. again 99, 11.) be represented by an equa- 
 tion of the form, p = (p(x, y), where now denotes a vector -function of two indepen- 
 dent and scalar variables, x and y, we may then differentiate this equation, or this 
 expression for p, with respect to either variable separately, and so obtain what may 
 be called two partial (hwt finite) differentials, d^p, dyp, and two partial derivatives, 
 X)xp, Dyp, whereof the former are connected with the latter, and with the two arbitrary 
 (hut finite') scalar s, dx, dy, by the relations, 
 
 dxp = D^-p . dx ; dyp = Dyp . dy. 
 
 And these two differentials (or derivatives) of the vector p of the surface denote two 
 tangential vectors, or at least two vectors parallel to two tangents to that surface at 
 the point P : so that their plane is (or is parallel to) the tangent plane at that point. 
 
 (10.) The mechanism of all such differentiations of vector-functions is, at the 
 present stage, precisely the same as in the usual processes of the Differential Calcu- 
 lus; because the most general form of such a vector- function, which has been consi- 
 dered in the present Book, is that of a sum of products (comp. 99) of the form xa, 
 where a is a constant vector, and a? is a variable scalar : so that we have only to 
 operate on these scalar coefficients a; . ., by the usual rules of the calculus, the vec- 
 tors a. . being treated as constant factors (comp. sub-art. 2). But when we shall 
 come to consider quotients or products oi vectors, or generally those new functions of 
 vectors which can only be expressed (in our system) by Quaternions, then some few 
 new rules of differentiation become necessary, although deduced from the same (or 
 nearly the same) definitions, as those which have been established in the present 
 Section. 
 
 As is well illustrated by Atwood's machine. 
 
102 ELEMENTS OF QUATERNIONS. [bOOK I. 
 
 (11.) As an example of partial differentiation (comp. sub-art. 9), of a vector 
 function (the word *' vector" being here used as an adjective) of two scalar variables, 
 let us take the equation, 
 
 p = ^(a;,y)=i{a;2a + y2/3 + (a, + y)2^}; 
 
 in which p (comp. 99, (3.) ) is the vector of a certain cone of the second order; or 
 more precisely, the vector of one sheet of such a cone, if x and y be supposed to be 
 real scalars. Here, the two partial derivatives of p are the following : 
 
 DarjO = jca + (ar + y) y ; i>yp = y/3 + (a; + .y) y ; 
 and therefore, 
 
 2p = xDxp + !/T>yp ; 
 
 so that the three vectors, p, D^p, i>,jp, if drawn (18) from one common origin, are con- 
 tained (22) in one common plane; which implies that the tangent plane to the sur- 
 face, at any point p, passes through the origin o : and thereby verifies the conical 
 character of the locus of that point p, in which the variable vector p, or op, termi- 
 nates. 
 
 (12.) If, in the same example, we make a: = 1, y = — 1, we have the values, 
 P = l(a-V^), ^xp = ct, Dyp = -/3; 
 whence it follows that the middle point, say c', of the right line ab, is one of the 
 points of the conical locus ; and that (comp. again the sub-art. 3 to Art. 99, and the 
 recent sub-art. 9) the right lines OA and ob are parallel to two of the tangents to the 
 surface at that point ; so that the cone in question is touched by the plane aob, along 
 the side (or ray) oc'. And in like manner it may be proved, that the same cone is 
 touched by the two other planes, BOC and COA, at the middle points a' and b' of the 
 two other lines BC and CA ; and therefore along the two other sides (or rays), oa' 
 and ob' : which again agrees with former results. 
 
 (13.) It will be found that a vector function of the turn of two scalar variables, 
 t and (\t, may generally be developed, by an extension of Taylor's Series, under the 
 form, 
 
 0(< + dO = ^(O+d<&(O + id2^(O + ^d'^(O + -- 
 
 d2 d3 
 
 "^^^"^^ 2 + 2:^+--^^^'^=''^^'^' 
 
 it being supposed that d'^t= 0, dH = 0, &c. (comp. sub-art. 6). Thus, if <pt=: ^at^, 
 (as in sub-art. 1), where a is a constant vector, we have d<pt = atdt, d^cpt^adt"^, 
 d^^t = 0, &c. ; and 
 
 (< + dt) = !«(< + dty = laf^ + atdt + |ad^2, 
 rigoroiisly, without any supposition that dt is small. 
 
 (14.) When we thus suppose At = dt, and develope the finite difference, A^{t) 
 = (< + dt) - ^(t), the first term of the development so obtained, or the term of first 
 dimension relatively to dt, is hence (by a theorem, which holds good for vector -func- 
 tions, as well as for scalar functions) the first dfferential d<pt of the function ; but 
 we do not choose to defi7ie that this Differential is (or means) thsii first term : be- 
 cause the Formula (100), which we prefer, does not postulate the j9ossJ6i7%, nor even 
 suppose the conception, of any such development. Many recent remarks will perhaps 
 appear more clear, when we shall come to connect them, at a later stage, with that 
 theory of Qnaternions, to which we next proceed. 
 
BOOK II. 
 
 ON QUATERNIONS, CONSIDERED AS QUOTIENTS OF VECTORS, 
 AND AS INVOLVING ANGULAR RELATIONS. 
 
 CHAPTER I. 
 
 FUNDAMENTAL PRINCIPLES RESPECTING QUOTIENTS OF VECTORS. 
 
 Section 1. — Introductory Remarks ; First Principles adopted 
 from Algebra. 
 
 Art. 101. The only angular relations^ considered in the fore- 
 going Book, have been those of parallelism between vectors 
 (Art. 2, &c.) ; and the only quotient s,\iii\iQvto employed, have 
 been of the three following kinds : 
 
 I. Scalar quotients ofscalars^ such as the arithmetical frac- 
 
 n 
 
 tion — in Art. 14; 
 m 
 
 II. Vector quotients^ of vectors divided by scalar s, as — = a 
 
 in Art. 16; 
 
 III. Scalar quotients of vectors^ with directions Qiih^r simi- 
 lar or opposite, as — = oj in the last cited Article. But we now 
 
 a 
 
 propose to treat of other geometric Quotients (or geometric 
 Fractions, as we shall also call them), such as 
 
 — =- = q, with /3wo^ II a (comp. 15); 
 OA a 
 
 for each of which the Divisor (or denominator), a or oa, and 
 the Dividend (or numerator), /3 or ob, shall not only both be 
 
104 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 Vectors^ but shall also be inclined to eacb other at an Angle, 
 distinct (in general) from zero, and from two'^ right angles, 
 
 102. In introducing this new conception, of a General Quo^ 
 tient of Vectors, with Angular Relations in a given plane, or 
 in space, it will obviously be necessary to employ some proper- 
 ties of circles and spheres, which were not wanted for the pur- 
 pose of the former Book. But, on the other hand, it will be 
 possible and useful to suppose a much less degree of acquaint- 
 ance with many important theoriesf of modern geometry, than 
 that of which the possession was assumed, in several of the 
 foregoing Sections. Indeed it is hoped that a very moderate 
 amount of geometrical, algebraical, and trigonometrical prepa- 
 ration will be found sufficient to render the present Book, as 
 well as the early parts of the preceding one, fully and easily 
 intelligible to any attentive reader. 
 
 103. It may be proper to premise a few general principles 
 respecting quotients of vectors, which are indQQ^suggestedhj 
 algebra, but are here adopted by definition. And 1st, it is 
 evident that the supposed operation of division (whatever its 
 
 full geometrical import may afterwards be found to be), by 
 which we here conceive ourselves to pass from a given divisor- 
 line a, and from a given dividend-line j3, to what we have called 
 (provisionally) their geometric quotient, q, may (or rather 
 must) be conceived to correspond to some converse act (as yet 
 not fully known) o^ geometrical multiplication : in which new 
 act the former quotient, q, becomes a Factor, and operates on 
 the line a, so as to produce (or generate) the line j3. We shall 
 therefore write, as in algebra, 
 
 (3 = q-a, or simply, j3 = qa, when f5:a = q; 
 
 * More generally speaking, from every even multiple of a right angle. 
 
 f Such as homology^ homography^ invobition, and generally whatever depends 
 on anharmonic ratio : although all that is needful to be known respecting such 
 ratio, for the applications subsequently made, may be learned, without reference to 
 any other treatise, from the definitions incidentally given, in Art. 25, &c. It was, 
 perhaps, not strictly necessary to introduce any of these modern geometrical theories, 
 in any part of the present woik ; but it was thought that it might interest one class, 
 at least, of students, to see how they could be combined with that fundamental con- 
 ception of the Vkotob, which the First Book was designed to develope. 
 
CHAP. I.] FIRST PRINCIPLES ADOPTED FROM ALGEBRA. 105 
 
 even if the two lines a and j3, or oa and ob, be supposed to 
 be inclined to each other, as in Fig. 33. And this very sim- 
 ple and n^iwroi notation (comp. 16) will then allow us to treat 
 as identities the two following formulae : 
 P \P ,, qa 
 
 a J a a 
 
 although we shall, for the present, abstain from writing also 
 such formulae* as the following : 
 
 a a 
 
 where a, /3 still denote tivo vectors, and q denotes their geo- 
 metrical quotient : because we have not yet even begun to con- 
 sider the multiplication of one vector by another, or the division 
 of a quotient by a line. 
 
 104. As a Ilnd general principle, suggested by algebra, 
 we shall next lay it down, that if 
 
 '—;=-, and a = a, then j3' = j3 ; 
 a a 
 
 or in words, and under a slightly varied form, that unequal 
 
 vectors, divided by equal vectors, give unequal quotients. The 
 
 importance of this very natural and obvious assumption will 
 
 soon be seen in its applications. 
 
 105. As a Ilird principle, which indeed may be consi- 
 dered to pervade the whole of mathematical language, and 
 without adopting which we could not usefully speak, in any 
 case, of EQUALITY as existing between any two geometrical 
 quotients, we shall next assume that two such quotients can 
 never be equal to the same third] quotient, without being at the 
 same time equal to each other: or in symbols, that 
 
 if q = q, and q" = q, then q" = q'. 
 
 * It will be seen, however, at a later stage, that these two formulae are permitted, 
 and even required, in the development of the Quaternion System, 
 
 f It is scarcely necessary to add, what is indeed included in this Ilird principle, 
 in virtue of the identity q = g, that if q' = q, then q = q' \ or in words, that we shall 
 never admit that any two geometrical quotients, q and q\ are equal to each other in 
 one order ^ without at the same time admitting that they are equal^ in the opposite 
 order also. 
 
 P 
 
106 ELEMENTS OF QUATERNIONS. [eOOK II. 
 
 106. In the lYth place, as a preparation for operations 
 on geometrical quotients^ we shall say that any two such quo- 
 tients, OY fractions (101), which have a common divisor-line, or 
 (in more familiar words) a common denominator, are added, 
 subtracted, or divided, among themselves, by adding, subtract- 
 ing, or dividing their numerators: the common denominator 
 being retained, in each of the two former of these three cases. 
 In symbols, we thus define (comp. 14) i\mt^ for any three (ac- 
 tual) vectors, a, j3, y, 
 
 7 I /^ _ 7 + ^ . 7 ^_7-/3. 
 a a a a a a ' 
 
 and 
 
 a a [5 
 
 aiming still at agreement with algebra. 
 
 107. Finally, as a Vth principle, designed (like the fore- 
 going) to assimilate, so far as can be done, the present Calculus 
 to Algebra, in its operations on geometrical quotients, we shall 
 define that the following formula holds good : 
 
 fi a J j3 a a ' 
 or that if two geometrical fractions, q and^'', he so related, that 
 the denominator, j3, of the multiplier q (here written towards 
 the left-hand) is equal to the numerator of the multiplicand q, 
 then the product, q'-q or q'q, is that third fraction, whereof 
 the numerator is the numerator y of the multiplier, and the 
 denominator is the denominator a of the nmltiplicand : all such 
 denominators, or divisor-lines, being still supposed (16) to be 
 actual (and not null) vectors. 
 
 Section 2. — First Motive fornaming the Quotient of two Vec- 
 tors a Quaternion. 
 
 108. Already we may see grounds for the application of 
 the name, Quaternion, to such a Quotient of two Vectors as 
 has been spoken of in recent articles. In the first place, such 
 a quotient cannot generally be what we have called (17) a Sca- 
 
CHAP. I.] QUOTIENT OF TWO VECTORS A QUATERNION. 107 
 
 LAR : or in other words, it cannot generally be equal to any 
 of the (so-called) reals of algebra^ whether oi ihQ positive or of 
 the negative kind. For let x denote any such (actual*) scalar, 
 and let a denote any (actual) vector; then we have seen (15) 
 that the product xa denotes another (actual) vector, say /3', 
 which is either similar or opposite in direction to a, according 
 as the scalar coefficient, or factor, x, is positive or negative ; 
 in neither case, then, can it represent any vector, such as /3, 
 which is inclined to a, at any actual angle ^ whether acute, or 
 right, or obtuse : or in other words (comp. 2), the equation 
 j3' = j3j or Xa = j3, is impossible, under the conditions here sup- 
 posed. But we have agreed (16, 103) to write, as in algebra, 
 
 '— = a; ; we must, therefore (by the Ilnd principle" of the fore- 
 a 
 
 going Section, stated in Art. 104), abstain fi-om writing also 
 
 ^ =^x, under the same conditions : x still denoting a scalar. 
 a 
 
 Whatever else a quotient of two inclined vectors may be found 
 to be, it is thus, at least, a Non-Scalar. 
 
 109. Now, in forming the conception of the scalar itself 
 as the quotient of two parallel] vectors (17), we took into ac- 
 count not only relative length, or ratio of the usual kind, but 
 also relative direction, under the form o^ similarity or opposition. 
 In passing from a to xa, we altered genevaWj (15) the length of 
 the line a, in the ratio of ± a; to 1 ; and we preserved or reversed 
 the direction of that line, according as the scalar coefficient x 
 was positive or negative. And in like manner, in proceeding to 
 form, more definitely than we have yet done, the conception of 
 the non-scalar quotient (108), q = (5: a-OB : oa, of two inclined 
 vectors, which for simplicity may be supposed (18) to be co- 
 
 * By an actual scalar, as by an actual vector (comp. 1), we mean here one that 
 is different from zero. An actual vector, multiplied by a. null scalar, has for product 
 (15) a null vector ; it is therefore unnecessary to prove that the quotient oitwo actual 
 vectors cannot be a null scalar, or zero. 
 
 f It is to be remembered that we have proposed (15) to extend the use of this 
 terra parallel, to the case of two vectors which are (in the usual sense of the word) 
 parallel to one common line, even Avhen they happen to he parts of one and the same 
 TvAit line. 
 
108 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 initial^ we have 5^2*// to take account both of the relative length, 
 and of the relative direction^ of the two lines compared. But 
 while the former element of the complex relation here consi- 
 dered, between these two lines or vectors, is still represented 
 by a simple Ratio (of the kind commonly considered in geo- 
 metry), or by a number* expressing that ratio ; the latter ele- 
 ment of the same complex relation is now represented by an 
 Angle, aob : and not simply (as it was before) by an alge- 
 braical sign, + or -. 
 
 110. Again in estimating this angle, for the purpose of 
 distinguishing one quotient of vectors from another, we must 
 consider not only its magnitude (or quantity), but also its 
 Plane : since otherwise, in violation of the principle stated 
 in Art. 104, we should have ob': oa = ob : oa, if ob and ob' 
 were two distinct rays or sides of a cone of revolution, with oa 
 for its axis; in which case (by 2) they would necessarily be 
 unequal vectors. For a similar reason, we must attend also to 
 the contrast between two opposite angles, of equal magnitudes, 
 and in one common plane. In short, for the purpose of know- 
 ing ^wZ/y the relative direction of two co-initial lines oa, ob in 
 space, we ought to know not only how many degrees, or other 
 parts of some angular unit, the angle ^^ 
 aob contains ; but also (comp. Fig. 33) 
 the direction of the rotation from oa to ^^^^^ 
 ob : including a knowledge of the plane, o- 
 in lohich the rotation is performed ; and -^'S- 33. 
 
 of the hand (as right or left, when viewed from a known side of 
 the plane), towards ichich the rotation is directed. 
 
 111. Or, if we agree to select some one fixed hand (suppose 
 the right^ hand), and to call all rotations positive when they 
 
 * This number^ which we shall presently call the tensor of the quotient, may be 
 whole or fractional^ or even incommensurable with unity ; but it may always be 
 equated, in calculation, to a poaitive scalar : although it might perhaps more pro- 
 perly be said to be a signless number, as being derived solely from comparison of 
 lengths, without any reference to directions. 
 
 t If right-handed rotation be thus considered as positive, then the positive axis 
 of the rotation aob, in Fig. 33, must be conceived to be directed downward, or below 
 the plane of the paper. 
 
CHAP. I.] QUOTIENT OF TWO VECTORS A QUATERNIOM. 109 
 
 are directed towards this selected hand, but all rotations nega- 
 tive when they are directed towards the other hand, then, for 
 any given angle aob, supposed for simplicity to be less than two 
 right angles, and considered as representing a rotation in a given 
 plane from oa to ob, we may speak oi one perpendicular oc to 
 that plane aob as being the positive axis of that rotation ; and 
 of the opposite perpendicular oc' to the same plane as being the 
 negative axis thereof; the rotation round the positive axis being 
 zY^e//" positive, and vice versa. And then the rotation aob may 
 be considered to be entirely known, if we know, 1st, its quantity, 
 or the ratio which it bears to a right rotation ; and Ilnd, the 
 direction of its positive axis, oc : but not without a knowledge 
 of these two things, or of some data equivalent to them. But 
 whether we consider the direction of an Axis, or the aspect of 
 a Plane, we find (as indeed is Avell known) that the determi- 
 nation of such a direction^ or of such an aspect, depends on two 
 polar co-ordinates'^ , or other angular elements. 
 
 112. It appears, then, from the foregoing discussion, that 
 for the complete determination, of what we have called the geo- 
 metrical Quotient of two co-initial Vectors, a System of Four 
 Elements, admitting each separately of numerical expression, 
 is generally required. Of these four elements, one serves (109) 
 to determine the relative length of the two lines compared ; 
 and the other three are in general necessary, in order to deter- 
 mine fully their relative direction. Again, of these three latter 
 elements, one represents the mutual inclination, or elongation, 
 of the two lines ; or the magnitude (or quantity) of the angle 
 between them ; while the two others serve to determine the 
 direction of the axis, perpendicular to their common plane, 
 round which a rotation through that angle is to be performed, 
 in a sense previously selected as the positive one (or tow^ards 
 a fixed and previously selected hand), for the purpose of pass- 
 ing (in the simplest way, and therefore in the plane of the two 
 lines) from the direction of the divisor-line, to the direction of 
 
 • The actual (or at least the frequent) use of such co -ordinates is foreign to the spirit 
 of the present System : but the mention of them here seems likely to assist a student, 
 by suggesting an appeal to results, with which his previous reading can scarcely fail 
 to have rendered him familiar. 
 
110 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 the dividend-line. And no more than four numerical elements 
 are necessary, for our present purpose: because the relative 
 length of two lines is not changed, when their two lengths are 
 altered proportionally^ nor is their relative direction changed, 
 when the angle which they form is merely turned about, in its 
 own plane. On account, then, of this essential connexion of 
 that complex relation (109) between two lines, which is com- 
 pounded of a relation of lengths^ and of a relation of directions, 
 and to which we have given (by an extension from the theory 
 of scalar s) the name of a geometrical quotient, with a System 
 o/'FouR numerical Elements, we have already a motive* for 
 saying, that '' the Quotient of two Vectors is generally a Qua- 
 ternion.*' 
 
 Section 3. — Additional Illustrations. 
 
 113. Some additional light may be thrown, on this first concep- 
 tion of a Quaternion, by the annexed Figure 34. In that Figure, 
 the letters cdefg are 
 designed to indicate 
 corners of a prisma- 
 tic desk, resting upon 
 a horizontal table. 
 The angle hcd (sup- 
 posed to be one of 
 thirty degrees) repre- 
 sents a (left-handed) 
 rotation, whereby the 
 horizontal ledge CD of 
 the desk is conceived 
 to be elongated (or 
 removed) from a given horizontal line ch, which may be imagined to 
 be an edge of the table. The angle gcf (supposed here to contain 
 forty degrees) represents the slopej of the desk, or tlie amount of its 
 inclination to the table. On the face cdef of the desk are drawn two si- 
 milar and similarly turned triangles, A OB and a'o'b', which are supposed 
 to be halves of two equilateral triangles ; in such a manner that each 
 
 ' Several other reasons for thus speaking will offer themselves, in the course of the 
 present work. 
 
 t These two angles, HCD and gcf, may thus be considered to correspond to lonf/i- 
 tude of node, and inclination of orbit, of a planet or comet in astronomy. 
 
 Fig. 34. 
 
CHAP. I.] ADDITIONAL ILLUSTRATIONS. Ill 
 
 rotation^ aob or a^o'b' is one of sixty degrees, and is directed towards 
 one common hand (namely the right hand in the Figure): while if 
 lengths alone be attended to, the side ob is to the side oa, in one tri- 
 angle, as the side oV is to the side o'a', in the other; or as the num- 
 ber two to one. 
 
 114. Under these conditions of construction, we consider the two 
 quotients^ or the two geometric fractions, 
 
 OB , o'b' 
 
 ob:oa and ob':oa, or — and — — -, 
 
 OA o'a' 
 
 as being equal to each other; because we regard the two lines, oa and 
 OB, as having the same relative length, and the same relative direction, 
 as the two other lines, o'a' and oV. And we consider and speak of 
 each Quotient, or Fraction^ as a Quaternion: heca,use its complete con- 
 struction (or determination) depends, for all that is essential to its 
 conception, and requisite to distinguish it from others, on a system of 
 four numerical elements (comp. 112); which are, in this Example, the 
 four numhers, 
 
 2, 60, 30, and 40. 
 
 115. Of these four eletnents (to recapitulate what has been above supposed), the 
 1st, namely the number 2, expresses that the length of the dividend-line, ob or 
 o'b', is double of the length of the divisor-line, OA or o'a'. The Ilnd numerical 
 element, namely 60, expresses here that the angle aob or a'o'b', is one of sixty de- 
 grees; while the corresponding rotation, from oa to ob, or from o'a' to o'b', is to- 
 wards a known hand (in this case the right hand, as seen by a person looking at the 
 face CDEF of the desk), which hand is the same for both of these two equal angles. 
 The Ilird element, namely 30, expresses that the horizontal ledge cd of the desk 
 makes an angle of thirty degrees with a known horizontal line ch, being removed 
 from it, by that angular quantity, in a known direction (which in this case happens 
 to be towards the left hand, as seen from above). Finally, the IVth element, 
 namely 40, expresses here that the desk has an elevation o^ forty degrees as before. 
 
 116. Now an alteration in any one of these Four Elements, such as an altera- 
 tion of the slope or aspect of the desk, would make (in the view here taken) an es- 
 sential change in the Quaternion, which is (in the same view) fAe Quotient of the two 
 Zmes compared: although (as the Figure is in part designed to suggest) no such 
 change is conceived to take place, when the triangle AOB is merely turned about, in 
 its own plane, without being turned over (comp. Fig. 36) ; or when the sides of that 
 triangle are lengthened or shortened proportionally, so as to preserve the ratio (in the 
 old sense of that word), of any one to any other of those sides. We may then briefly 
 say, in this mode of illustrating the notion of a Quaternion* in geometry, by refe- 
 
 * As to the mere word. Quaternion, it signifies primarily (as is well kncwn), like 
 its Latin original, " Quaternio," or the Greek noun TtTpaKTVQ, a 5c/ of Four : but 
 it is obviously used here, and elsewhere in the present work, in a technical sense. 
 
112 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 rence to an angle on a desk, that the Four Elements which it involves are the follow- 
 ing; 
 
 Ratio, Angle, Ledge, and Slope; 
 
 although the two latter elements are in fact themselves angles also, but are not im- 
 mediately obtained as such, from the simple comparison of the two lines, of which the 
 Quaternion is the Quotient. 
 
 Section 4 On Equality of Quaternions; and on the Plane 
 
 of a Quaternion. 
 
 117. It is an immediate consequence of the foregoing con- 
 ception of a Quaternion, that two quaternions, or tiuo quotients 
 of vectors, supposed for simplicity to be all co-initial (\8), are 
 regarded as being equal to each other, or that the equation, 
 
 d Q CD OB 
 
 -=— , or — = — , 
 y a oc oa 
 
 is by us considered and defined to hold good, ivheji the two tri- 
 angles, AOB and COD, are similar and similarly/ turned, and in 
 one common plane, as represented in the -^ 
 annexed Fig. 35 : the relative length 
 (109), and the relative direction 
 (110), of the two lines, oa, ob, being 
 then in all respects the same as the re- 
 lative length and the relative direction 
 of the two other lines, oc, on, 
 
 118. Under the same conditions, we 
 shall write the following formula of direct similitude, 
 
 A AOB a cod; 
 reserving this other formula, 
 
 A AOB oc' aob', or A a'ob a' a'ob', 
 which we shall call a formula of inverse simili- 
 tude, to denote that the two triangles, aob and 
 aob', or a'ob and a'ob', although otherwise simi- 
 lar (and even, in this case, equal,* on account 
 of their having a common side, oa or oa'), are 
 
 Fig. 35. 
 
 Fig. 36. 
 
 * That is to say, equal in absolute amount of area, but with opposite algebraic 
 
 signs (28). The two quotients OB : OA, and ob' : OA, although not equal (110), will 
 
 soon be defined to be conjugate quaternions. Under the same conditions, we shall 
 
 write also the formula, 
 
 A aob' a 'cod. 
 
CHAP. I.] CONDITIONS OF EQUALITY OF QUATERNIONS. 113 
 
 oppositely turned (comp. Fig. 36), as if one were the reflexion 
 of the other in a mirror ; or as if the one triangle were derived 
 (or generated) from the other, by a rotation of its plane through 
 two right angles. We may therefore write, 
 
 OB OD .,. ^ 
 
 — = — , II A AOB ex COD. 
 
 OA OC 
 
 119. When the vectors are thus all drawn from one com- 
 mon origin o, i\iQ plane aob oi any two of them maybe called 
 the Plane of the Quaternion (or of the Quotient), ob : oa ; and 
 of course also the plane of the inverse (or reciprocal) quater- 
 nion (or of the inverse quotient), oa : ob. And any two qua- 
 ternio7is, which have a common plane (through o), may be said 
 to be Complanar* Quaternions, or complanar quotients, or 
 fractions ; but any two quaternions (or quotients), which have 
 different planes {intersecting therefore in a right line through 
 the origin), may be said, by contrast, to be Diplanar. 
 
 120. Any two quaternions, considered as geometric frac- 
 tions (101), can be reduced to a common denominator without 
 
 OB 
 
 change of the value^ of either of them, as follows. Let — and 
 
 — be the two given fractions, or quaternions ; and if they be 
 
 complanar (119), let oe be any linem their common plane; but 
 if they be diplanar (see again 1 19), then let oe be any assumed 
 part of the line of intersection of the two planes : so that, in 
 each case, the line oe is situated at once in the plane aob, and 
 also in the plane cod. We can then always conceive two other 
 lines, OF, OG, to be determined so as to satisfy the two condi- 
 tions of direct similitude (118), 
 
 A EOF a aob, Aeoggccod; 
 
 * It is, however, convenient to extend the use of this word, complanar^ so as to 
 inchide the case of quaternions represented by angles in parallel planes. Indeed, as 
 all rectors which have equal lengths, and similar directions, are equal (2), so the 
 quaternion, which is a quotient of two such vectors, ought not to be considered as 
 undergoing any change, when either vector is merely changed in pontion, by a trans- 
 port without rotation. 
 
 •)■ That is to say, the new or transformed quaternions will be respectively equal to 
 llie old or given ones. 
 
 Q 
 
114 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 and therefore also the tioo equations between quotients (117, 
 
 118), 
 
 OF OB OG OD 
 
 OE Oa' OE OC * 
 
 and thus the required reduction is effected, oe being the com- 
 mon denominator sought, while of, og are the new or reduced 
 numerators. It may be added that if h be a new point in the 
 plane aob, such that A hoe a aob, we shall have also, 
 OE ob of 
 
 OH ~ OA ~ OE ' 
 
 and therefore, by 106, 107, 
 
 OD OB OG+OF OD OB OG OD OB OG 
 OC ~ OA OE ' OC*OA~Of' OC OA ~ OH ' 
 
 whatever tioo geometric quotients (complanar or diplanar) may 
 be represented by ob : oa and od : oc. 
 
 121. If now the two triangles aob, cod are not only com- 
 planar but directly similar (118), so that A aob oc cod, we shall 
 evidently have A eof a eog; so that we may write of = og 
 (or F = G, by 20), the two new lines of, og (or the two new 
 points F, g) in this case coinciding. The general construction 
 (120), for the reduction to a common denominator, gives there- 
 fore here only one new triangle^ eof, and one new quotient^ 
 OF : ok, to which in this case each (comp. 105) of the two given 
 equal and complanar quotients, ob : oa and od : oc, is equal. 
 
 122. But if these two latter symbols (or th^ fractional 
 forms corresponding) denote two diplanar* quotients, then the 
 two new numerator lines, of and og, have different directions, 
 as being situated iii two different planes, drawn through the new 
 denominator-line oe, without having either the direction of that 
 line itself or the direction opposite thereto ; they are therefore 
 (by 2) unequal vectors, even if they should happen to be 
 equally long; whence it follows (by 104) that the two new 
 quotients, ^[id therefore also (by 105) that the two old or given 
 quotients, are unequal, as a consequence of their diplanarity, 
 
 * And therefore non scalar (108) ; for a scalar, considered as a quotient (17), 
 has no determined plane, but must be considered as complanar with every geometric 
 quotient; since it may be represented (or constructed) by the quotient of two simi- 
 larly or oppositely directed lines, in any proposed plane whatever. 
 
CHAP. I.] PLANE OF A QUATERNION. 115 
 
 It results, then, from this analysis, that diplanar quotients of 
 vectors, and therefore that Diplanar Quaternions (119), are 
 always unequal; a new and comparatively technical process 
 thus confirming the conclusion, to which we had arrived by 
 general considerations, and in (what might be called) a popular 
 way before, and which we had sought to illustrate (comp. Fig. 
 34) by the consideration o^ angles on a desk: namely, that a 
 Quaternion, considered as the quotient oitwo mutually inclined 
 lines in space, involves generally a Plane, as an essential part 
 (comp. 110) of its constitution, and as necessary to the com- 
 pleteness of its conception. 
 
 123. We propose to use the mark 
 
 as a Sign of Complanarity, whether of lines or of quotients ; 
 thus we shall write the formula, 
 
 7lll«./3, 
 to express that the three vectors, a, /3, y, supposed to be (or to 
 be made) co-initial (18), are situated in one plane ; and the 
 analogous formula, 
 
 q\\\q, or? Ill ^, 
 
 y a 
 to express that the tioo quaternions, denoted here by q and q, 
 and therefore that i\iQ four vectors, a, /3, 7, S, are complanar 
 (119). And because we have just found (122) that diplanar 
 quotients are unequal, we see that one equation of quaternions 
 includes tivo complanar ities of vectors ; in such a manner that we 
 may write, 
 
 7|||a,/3. and 8|||«,|3. if - =^; 
 
 y a. 
 1 /. . OD OB , . . ... 7 77 
 
 the equation oj quotients, — = — , being nupossible, unless all 
 
 the four lines from o be in one common plane. We shall also 
 employ the notation 
 
 7 III?- 
 
 to express that the vector y is in (or parallel to) the plane of 
 the quaternion q» 
 
116 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 124. With the same notation for complanarity, we may 
 write generally, 
 
 ^a|||a,/3; 
 a and /3 being any two vectors, and x being ant/ scalar ; be- 
 cause, if a = OA and j3 = ob as before, then (by 15, 17) aid = oa', 
 where a' is some point on the indefinite right line through the 
 points o and a : so that the 'plane aob contains the line oa'. 
 For a similar reason, we have generally the following formula 
 oi complanarity of quotients, 
 
 whatever two scalars x and y may be ; a and /3 still denoting 
 any two vectors. 
 
 125. It is evident (comp. Fig. 35) that 
 
 if A AOB a COD, then A boa a doc, and A aoc a bod ; 
 whence it is easy to infer that for quaternions, as well as for 
 ordinary or algebraic quotients, 
 
 if - = -, then, inversely, -^=\, and alternately, ~ = t^\ 
 ay p o a JD 
 
 it being permitted now to establish the converse of the last for- 
 mula of 1 18, or to say that 
 
 . „ ob od . 
 
 II — = — , then A aob a cod. 
 oa oc 
 
 Under the same condition, by combining inversion with alter- 
 
 nation, we have also this other equation, - = ^. 
 
 126. If the sides, oa, ob, of a triangle aob, or those sides 
 either way prolonged, be cut (as in 
 Fig. 37) by dmy parallel, a'b' or a"b", 
 to the base ab, we have evidently the 
 relations o^ direct similarity (118), yf^ 
 
 A a'ob' a AOB, A a"ob" oc aob ; 
 
 whence (comp. Art. 13 and Fig. 12) 
 it follows that we may write, for qua- 
 ternions as in algebra, the general ^' 
 equation, or identity, ^'s 2^- 
 
CHAP. I.] AXIS AND ANGLE OF A QUATERNION. 117 
 
 xa a ' 
 where x is again ani/ scalar, and o, /3 are ant/ two vectors. It 
 is easy also to see, that for any quaternion q, and any scalar x, 
 we have the product (comp. 107), 
 
 xp f5 xQ (5 (5 a 
 p a a xr^a a x^a 
 
 so that, in the multiplication of a quaternion by a scalar (as in 
 the multiplication of a vector by a scalar, 15), the order of the 
 factors is indifferent. 
 
 Section 5 — On the Axis and Angle of a Quaternion ; and on 
 the Index of a Right Quotient, or Quaternion. 
 
 127. From what has been already said (HI, 112), we are 
 naturally led to define that the Axis, or more fully that the 
 positive axis, of any quaternion (or geometric quotient) ob ; oa, 
 is a right line perpendicular to the plane aob of that quaternion ; 
 and is such that the rotation round this axis, from the divisor- 
 line OA, to the dividend-line ob, is positive : or (as we shall 
 henceforth assume) directed towards the right-hand,* like the 
 motion of the hands of a watch. 
 
 128. To render still more definite this conception of the 
 axis of a quaternion, we may add, 1st, that the rotation, here 
 spoken of, is supposed (112) to be the simplest possible, and 
 therefore to be in the plane of the two lines (or of the quater- 
 nion), being also generally less than a semi-revolution in that 
 plane ; Ilnd, that the axis shall be usually supposed to be a 
 line ox drawn ^rom the assumed origin o ; and Ilird, that the 
 length of this line shall be supposed to be given, ov fixed, and 
 to be equal to some assumed unit of length : so that the term 
 X, of this axis ox, is situated (by its construction) on a given 
 spheric surface described about the origin o as centre, which 
 surface we may call the surface of the unit-sphere. 
 
 129. In this manner, for every given non-scalar quotient 
 
 * This is, of course, merely conventional, and the reader may (if lie pleases) sub- 
 stitute the /e/if-hand throughout. 
 
118 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (108), or for every given quaternion q which does not reduce 
 itself (or degenerate) to a mere positive or negative number, the 
 axis will be an entirely definite vector, which may be called an 
 UNIT-VECTOR, on account of its assumed length, and which we 
 shall denote'*, for the present, by the symbol Ax . q. Employ- 
 ing then the usual sign of perpendicularity, J_ , we may now 
 write, for any two vectors a, jS, the formula : 
 
 Ax.^±a; Ax2j_i3; or briefly, Ax.2± |^. 
 a a a [o. 
 
 130. The Angle of a quaternion, such as ob : oa, shall 
 simply be, with us, the angle aob between the tivo lines, of 
 which the quaternion is the quotient ; this angle being sup- 
 posed here to be one of the usual kind (such as are considered 
 by Euclid) : and therefore being acute, or right, or obtuse (but 
 not of any class distinct from these), when the quaternion is a 
 non-scalar (108). We shall denote this a?igle of a quaternion 
 q, by the symbol, L q ; and thus shall have, generally, the two 
 inequalities^ following : 
 
 Z5'>0; LqKiT', 
 where tt is used as a symbol for two right angles. 
 
 131. When the general quaternion, q^ degenerates into a 
 scalar, x, then the axis (like the planeX) becomes entirely in- 
 determinate in its direction ; and the angle takes, at the same 
 time, either zero or two right angles for its value, according as 
 the scalar \& positive ov negative. Denoting then, as above, any 
 such scalar by x, we have : 
 
 * At a later stage, reasons will be assigned for denoting this axis^ Ax .q, of a 
 quaternion g, by the less arbitrary (or more systematic) symbol, \^Yq ; but for the 
 present, the notation in the text may suffice. 
 
 f In some investigations respecting complanar quaternions, and powers or roots 
 of quaternions, it is convenient to consider negative angles., and angles greater than 
 two right angles; but these may then be called amplitudes ; and the word "An- 
 gle," like the word " Ilatio," may thus be restricted, at least for the present, to its 
 ordinary geometrical sense. 
 
 X Compare the Note to page 114. The angle, as well as the axis, becomes in- 
 determinate, when the quaternion reduces itself to zero ; unless we happen to know 
 a law, according to which the dividend-line tends to become null, in the transition 
 
 r ^. ° 
 from - to -. 
 
 a a 
 
CHAP. I.] CASE OF A RIGHT QUOTIENT, OR QUATERNION. 119 
 
 Ax . a; = an indeterminate unit-vector ; 
 
 Z :r = 0, if ar > ; z re = tt, if a? < 0. 
 132. Ot non-scalar quaternions, the most im- b 
 portant are those of which the angle is right, as in 
 the annexed Figure 38 ; and when we have thus, 
 
 OB , , TT 
 
 q= — , and ob_L_oa, or Lq = -, 
 
 OA 2i 
 
 >- 
 
 A 
 
 the quaternion q may then be said to be a Right Fig. 38. 
 Quotient ;* or sometimes, a Right Quaternion. 
 
 (1.) If then a = OA and p —op, where o and a are two given (ov fixed) points, 
 but P is a variable point, the equation 
 
 a 2 
 
 expresses that the locus of this point p is the plane through o, perpendicular to the 
 li?ie OA ; for it is equivalent to the formula of perpendicularity p j_ a (129). 
 (2.) More generally, if /3= ob, b being any third given point, the equation, 
 
 p (3 
 
 L- = L- 
 
 a a 
 
 expresses that the locus of p is one sheet of a cone of revolution, with o for vertex, 
 and OA for axis, and passing through the point b ; because it implies that the angles 
 AOB and AOP are equal in amount, but not necessarily in one common plane. 
 (3.) The equation (comp. 128, 129), 
 
 Ax.^ = Ax.^, 
 a a 
 
 expresses that the locus of the variable point p is the given plane aob ; or rather the 
 indefinite half-plane, which contains all the points p that are at once complanar 
 with the three given points o, A, b, and are also at the same side of the indefinite 
 right line OA, as the point B. 
 
 (4.) The system of the two equations, 
 
 a a a a ^ 
 
 expresses that the point p is situated, either on thej^mVe right linele^^, or on that line 
 prolonged through ^A, but not through o; so that the locus of p may in this case be 
 said to be the indefinite half -line, or ray, which sets out from o in the direction of the 
 vector on or /3 ; and we may write p = .r/3, x> () (x being understood to be a sca- 
 lar)^ instead of the equations assigned above. 
 
 * Reasons will afterwards be assigned, for equating such a quotient, or quater- 
 nion, to a Vector; namely to the line which will presently (133) be called the Index 
 of the Bight Quotient. 
 
120 ELEMENTS OF QUATERNIONS. [boOK II. 
 
 (5. ) This other system of two equations, 
 
 a a a a 
 
 expresses that the locus'of p is the opposite ray from o ; 
 or that p is situated on the prolongation of the revec- 
 tor BO (1) ; or that p=x(3, x<0; or that p,''' 
 
 p = x(3\ x>0, if /3' = ob' = - /3. Fig. 33, bis. 
 
 (Comp. Fig. 33, bis.) 
 
 (6.) Other notations, for representing these and other geometric loci, will be found 
 to be supplied, in great abundance, by the Calculus of Quaternions ; but it seemed 
 proper to point out these, at the present stage, as serving already to show that even 
 the two symbols of the present Section, Ax. and Z, when considered as Characteris- 
 tics of Operation on quotients of vectors, enable us to express, very simply and con- 
 cisely, several useful geometrical conceptions, 
 
 133. If a third line, oi, be drawn in the direction of the 
 axis ox of such a right quotient (and therefore perpendicular, 
 by 127, 129, to each of the two given rectangular lines, oa, 
 ob) ; and if the length of this new line oi bear to the length 
 of that axis ox (and therefore also, by 128, to the assumed 
 unit of length) the same ratio, which the length of the dividend- 
 line, OB, bears to the length of the divisor- line, oa; then the 
 line 01, thus determined, is said to be the Index of the Bight 
 Quotient. And it is evident, from this definition of such an 
 Index, combined with our general definition (117, 118) of 
 Equality between Quaternions, that tivo right quotients are 
 equal or unequal to each other, according as their two index- 
 lines (or indices) are equal or unequal vectors. 
 
 Section 6 On the Reciprocal, Conjugate, Opposite, and Norm 
 
 of a Quaternion; and on Null Quaternions. 
 
 134. The Keciprocal {ox ihQ Inverse, comp. 119) of a 
 quaternion, such as 5' = — , is that other quaternion, 
 
 which is formed by interchanging the divisor- line and the divi- 
 dend-line ; and in thus passing from any non-scalar quater- 
 nion to its reciprocal, it is evident that the angle (as lately 
 
CHAP. I.] RECIPROCAL OF A QUATERNION. 121 
 
 defined in 130) remains unchanged^ but that the axis (127, 
 1 28) is reversed in direction : so that we may write gene- 
 rally, 
 
 pa p a 
 
 135. The product of two reciprocal quaternions is always 
 equal to positive unity ; and each is equal to the quotient of 
 unity divided hy the other; because we have, by 106, 107, 
 
 1:2 = ":^ « and |.2 = f=l. 
 a a a p p a a 
 
 It is therefore unnecessary to introduce any new or peculiar 
 notation, to express the mutual relation existing between a 
 quaternion and its reciprocal; since, if one be denoted by the 
 symbol q, the other may (in the present System, as in Alge- 
 bra) be denoted by the connected symbol,* 1 : 5^, or -. We 
 have thus the two general formulae (comp. 134) : 
 
 z-=z<7; Ax.- = -Ax.o'. 
 9 9 
 
 136. Without yet entering on the general i\\QOvy of multi- 
 plication and division of quaternions, beyond what has been 
 done in Art. 120, it may be here remarked that if any two 
 quaternions q and q be (as in 134) reciprocal to each other, so 
 that q'-q^l (by 135), and if 5'" be any third quaternion, then 
 (as in algebra), we have the general formula, 
 
 . , .1 
 q :q = q ,q =9'-\ 
 
 because if (by 120) we reduce q and q' to a common denomina- 
 tor a, and denote the new numerators by j3 and 7, we shall have 
 (by the definitions in 106, 107), 
 
 „ 7^770 „ , 
 
 137. When two complanar triangles aob, aob', with a com^ 
 
 * The symbol 5-1, for the reciprocal of a quaternion q, is also permitted in the 
 present Calculus ; but we defer the use of it, until its legitimacy shall have been 
 established, in connexion with a general theory of powers of Quaternions. 
 
 R 
 
122 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 ^ 
 
 mon side OA, are (as in Fig. 36) inversely similar (\18), so that 
 the formula A aob' a' aob holds good, then the iwo unequal 
 
 quotients,* — and — , are said to be Conjugate Quater- 
 
 ^ OA OA 
 
 NiONS ; and if the ^rst of them be still denoted by q, then the 
 second, which is thus the conjugate of that^r^^, or of any other 
 quaternion which is equal thereto, is denoted by the new sym- 
 bol, K^ : in which the letter K may be said to be the Charac- 
 teristic of Conjugation. Thus, with the construction above 
 supposed (comp. again Fig. 36), we may write, 
 
 OB 
 OA 
 
 = <1 
 
 OA ^ OA 
 
 138. From this definition of conjugate quaternions, it follows, 
 
 1st, that if the equation 
 
 OB __ OB , - _ , - 17. f , 
 
 — = K — holdffood, then the line ob maybe 
 OA OA '^ "^ 
 
 called (118) the reflexion of the lineoB (and conversely, the latter line 
 the reflexion of the foi^mer), with respect to the line oa ; Ilnd, that, under 
 the same condition, the line oA (prolonged if necessary) bisects per- 
 pendicularly the line be', in some point a' (as represented in Fig. 36) ; 
 and Ilird, that any two conjugate quaternions (like any iv^o reciprocal 
 quaternions, comp. 1.34, 135) have equal angles, but opposite axes: 
 so that we may write, geujerally, 
 
 L^q=L q\ Ax . K^ = - Ax . q ; 
 
 and thereforef (by 135), 
 
 Z.K^ = Z.-; Ax.K<7 = Ax.-. 
 
 <1 9. 
 
 139. The reciprocal of a scalar, x, is simply another scalar, 
 -, or x'"^, having the same algebraic sign, and in all other re- 
 
 X 
 
 speCts related to x as in algebra. But the conjugate 'Kx, of a 
 scalar x, considered as a limit of a quaternion, is equal to that 
 scalar x itself; as may be seen by supposing the two equalhxxt 
 opposite angles, aob and aob', in Fig. 36, to tend together to 
 
 * Compare the Note to page 112. 
 
 t It will soon be seen that these two last equations (138) express, that the con- 
 jugate and the reciprocal, of any proposed quaternion 5, have always equal versors, 
 although they have in general unequal tensors. 
 
CHAP. I.] CONJUGATE AND NULL QUATERNIONS. 123 
 
 zero, or to two right angles. We may therefore write, gene- 
 rally, 
 
 Kx = x, ifx be any scalar ; 
 and conversely*, 
 
 q = 21, scalar, if Kq = q; 
 
 because then (by 104) we must have ob=ob', bb'=0; and 
 therefore each of the two (now coincident) points, b, b', must 
 be situated somewhere on the indefinite right line oa. 
 
 140. In general, by the construction represented in the 
 same Figure, the sum (comp. 6) of the two numerators (or di^ 
 vidend-Unes, ob and ob'), of the tivo conjugate fractions (or quo- 
 tients, or quaternions), q and Kq (137), is equal to the double 
 of the line oa' ; whence (by 106), the sum of those two conju- 
 gate quaternions themselves is. 
 
 Kg + g = g + Kg = • ; 
 
 ^ ^ ^ ^ OA 
 
 this sum is therefore always scalar, hemg positive if the anple 
 Z ^ be acute, but negative if that angle be obtuse. 
 
 141. In the intermediate case, when the angle aob is right, 
 the interval oa' between the origin o and the line bb' vanishes ; 
 and the two lately mentioned numerators, ob, ob', become two 
 opposite vectors^ of which the sum is null (5). Now, in gene- 
 ral, it is natural, and will be found useful, or rather necessary 
 (for consistency \fii\i former definitions), to admit that a null 
 vector, divided by an actual vector, gives always a Null Qua- 
 ternion as the quotient; and to denote this null quotient by 
 the usual symbol for Zero, In fact, we have (by 106) the 
 equation, 
 
 ? = fLZf = ^_5. 1.1 = 0; 
 a a a a 
 
 the zero in the numerator of the Z^^-hand fraction represent- 
 ing here a null line (or a null vector, 1,2); but the zero on the 
 riyht-hand side of the equation denoting a nidi quotient (or 
 quaternion). And thus we are entitled to infer that the sum, 
 
 * Somewhat later it will be seen that the equation Kq = q may also be written 
 as V^ = ; and that this last is another mode of expressing that the quaternion, j, 
 degenerates (131) into a scalar. 
 
124 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 J^q +q, or q + K.q, of a right-angled quaternion, or right quo- 
 tient (132), and of its conjugate, is always equal to zero, 
 
 142. We have, therefore, the three following formulae, 
 whereof the second exhibits a continuity in the transition from 
 the j^r5^ to the third : 
 
 I. . . ^r + K^r > 0, if Z^ < I ; 
 
 11. . . ^ + K^ = 0, if z^=|; 
 
 III. . . ^ + K^ < 0, if Lq>~. 
 
 And because a quaternion, or geometric quotient, with an ac- 
 tual and^nite divisor-line (as here oa), cannot become equal to 
 zero unless its dividend-line vanishes, because (by 104) the 
 equation 
 
 L- = = - requires the equation j3 = 0, 
 a a 
 
 if a be any actual and finite vector, we may infer, conversely, that 
 the sum q + Kq cannot oanish, without the line oa' also vanish- 
 ing ; that is, without the lines ob, ob' becoming opposite vectors^ 
 and therefore the quaternion q becoming a right quotient (132), 
 We are therefore entitled to establish the three following con- 
 verse formulae (which indeed result from the three former) : 
 
 T. , ,if q-V Kq > 0, then Aq <-; 
 II'. . . if 5' + Kq = 0, then Lq=-', 
 
 Iir. . . if 5- + Kq < 0, then Lq> -, 
 
 143. When two opposite vectors (1), as j3 and-/iJ, are both 
 divided by one common (and actual) vector, a, we shall say that 
 the two quotients, thus obtained are Opposite Quaternions; 
 so that the opposite of any quaternion q, or of any quotient 
 /3 : a, may be denoted as follows (comp. 4) : 
 
 -p 0-i3 /3 _ 
 a a a a 
 
CHAP. I.] OPPOSITE QUATERNIONS. 125 
 
 while the quaternion q itself m2ij, on the same plan, be denoted 
 (comp. 7) by the symbol + $', ov ■¥ q. The sum of any two 
 opposite quaternions is zero, and their quotient is negative 
 unity; so that we may write, as in algebra (comp. again 7), 
 
 (-^) + ^ = (+^) + (-^) = 0; (-^):^ = -i; -^ = (-1)^; 
 
 because, by 106 and 141, 
 
 a a a a a a p 
 
 The reciprocals of opposite quaternions are themselves oppo- 
 site ; or in symbols (comp. 126), 
 
 1 1 - a -a a 
 
 — = — , because —^ = -77" = - ts* 
 -q q -(5 (3 j5 
 
 Opposite quaternions have opposite axes, and supplementary 
 
 angles (comp. Fig. 33, bis) ; so that we may establish (comp. 
 
 132, (5.) ) the two following general formulse, 
 
 L{-q) = Tr- Lq\ Ax.(- 5-) = - Ax.^'. 
 
 144. We may also now write, in full consistency with the 
 
 recent formulae II. and 11'. of 142, the equation, 
 
 IF. , ,Kq = -q, if ^ ^ = I ; 
 and conversely* (comp. 138), 
 
 ir...ifK^ = -^, then zK^=z^ = ^. 
 
 In words, the conjugate of a right quotient, or of a right-angled 
 
 (or right) quaternion (132), is the right quotient opposite 
 
 thereto ; and conversely, if an actual quaternion (that is, one 
 
 which is not null) be opposite to its own conjugate, it must be 
 
 a right quotient. 
 
 (1.) If then we meet the equation, 
 
 Ke = _^, or ^ + K^ = 0, 
 a a a a 
 
 we shall know that p -i_ a ; and therefore (if a = oa, and p = op, as before), that the 
 
 * It will be seen at a later stage, that the equation Kq=-q, or g + Kg = 0, 
 may be transformed to this other equation, Sg = ; and that, under this last form, it 
 expresses that the scalar part of the quaternion q vanishes : or that this quaternion 
 is a right quotient (132). 
 
126 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 locus of the point p is the plane through o, perpendicular to the line OA (as in 132, 
 (2,) On the other hand, the equation, 
 
 K 
 
 P-kP = 
 
 0, 
 
 expresses (by 139) that the quotient p : a is a scalar ; and therefore (by 131) that 
 its angle I (^p : a) is either or tt ; so that in this case, the locus of p is the indefi- 
 nite right line through the two points o and A. 
 
 145. As the opposite of the opposite, or the reciprocal of the reci- 
 procal^ so also the conjugate of the conjugate, of any quaternion, is that 
 quaternion itself; or in symbols, 
 
 -(-?) = + ?; l:(l:g) = ^; K% = ^=1^; 
 so that, by abstracting from the subject of the operation, we may write 
 
 briefly, 
 
 K2 = KK=1. 
 
 It is easy also to prove, that the conjugates of opposite quaternions are 
 themselves opposite quaternions ; and that the conjugates of reciprocals 
 are reciprocal: or in symbols, that 
 
 I...K(-^) = -K^, or K^+K(-5) = 0; 
 and 
 
 II...Ki=l:K^, or K7.Ki=l. 
 
 (1.) The equation K(- g) = — Kg is included (comp. 143) in this more general 
 formula, Yi(xq') = xKq, where x is any scalar; and this last equation (comp. 126) 
 may be proved, by simply conceiving that the two lines ob, ob', in Fig. 36, are 
 multiplied by any common scalar ; or that they are both cut by any parallel to the 
 line bb'. 
 
 (2.) To prove that conjugates of reci- 
 procals are reciprocal, or that Kg . K - = 1, / 
 
 we may conceive that, as in the annexed / 
 Figure 36, bis, while we have still the f 
 relation of inverse similitude, \ 
 
 A aob' (xf AOB (118, 137), 
 as in the former Figure 36, a new point c 
 is determined, either on the line OA itself, 
 or on that line prolonged through A, so as / 
 
 to satisfy either of the two following con- ^ig. 36, bis. 
 
 nected conditions of direct similitude : ,^ 
 
 A boc a aob' ; A b'oc oc aob ; 
 or simply, as a relation between the /our points o, a, b, c, the formula, 
 
 A boc a' aob. 
 
 •- , P 
 
CHAP. I.] GEOMETRICAL EXAMPLES. 127 
 
 For then we shall have the transformations, 
 
 1 _ OA _ Ob' _ OB _ OA 1 
 
 q OB OC OC Ob' Kq 
 
 (3.) The two quotients, ob : OA, and ob : oc, that is to say, the quaternion q 
 itself, and the conjugate of its reciprocal, or* the reciprocal of its conjugate, have 
 the same angle, and the same axis ; we may therefore write, generally, 
 
 1 1 . 
 
 ZK-=Z.o; Ax.K- = Ax.g'. 
 
 (4.) Since oa : ob and OA : ob' have thus been proved (by sub-art. 2) to be 
 a pair of conjugate quotients, we can now infer this theorem, that any two geo- 
 metric fractions, — and — , which have a common numerator a, are conjugate qua- 
 ternions, if the denominator jS' of the second be the reflexion of the denominator (3 of 
 theirs*, with respect to that common numerator (comp. 138, I.) ; whereas it had 
 only been previously assumed, as a definition (137), that such conjugation exists, 
 uuder the same geometrical condition, between the two other (or inverse) fractions, 
 
 — and — ; the three vectors a, jS, (3' being supposed to be all co-initial (18). 
 a a 
 
 (5.) Conversely, if we meet, in any investigation, the formula 
 OA : ob' = K (oA : ob), 
 we shaU know that the point b' is the reflexion of the point b, with respect to the 
 line OA ; or that this line, OA, prolonged if necessary in either of two opposite direc- 
 tions, bisects at right angles the line bb', in some point a', as in either of the two 
 Figures 36 (comp. 138, II.). 
 
 (6.) Under the recent conditions of construction, it follows from the most ele- 
 mentary principles of geometry, that the circle, which passes through the three points 
 A, B, c, is touched at b, hij the right line OB ; and that this line is, in length, a 7nean 
 proportional between the lines oa, oc. Let then od be such a geometric mean, and 
 let it be set off from o in the common direction of the two last mentioned lines, so 
 that the point d falls between A and c ; also let the vectors oc, od be denoted by the 
 symbols, y, S', we shall then have expressions of the forms, 
 
 d = aa, y=a^a, 
 where a is some positive scalar, a > ; and the vector /3 of B will be connected 
 (comp. sub-art. 2) with this scalar a, and with the vector a, by the formula, 
 
 OB „ OA oc ,^ OB a^a ^ B 
 
 — = K— , or — = K— , or -— = K^. 
 
 oc OB OB OA (Ha 
 
 (7.) Conversely, if we still suppose that y = a^a, this last formula expresses the in- 
 verse similitude of triangles, A boc a' aob ; and it expresses nothing more: or in other 
 
 * It will be seen afterwards, that the common value of these two equal quater- 
 nions, K - and — , may be represented by either of the two new symbols, JJq : Tq, 
 q Kq 
 
 or 5 : Nj ; or in words, that it is equal to the versor divided by the tensor; and also 
 to the quaternion itself divided by the norm. 
 
128 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 words, it is satisfied by the vector (3 of every point b, which gives that inverse simili- 
 tude. But for this purpose it is only requisite that the length of ob should be (as 
 above) a geometric mean between the lengths of OA, oc ; or that the two lines, ob, 
 OD (sub-art. 6), should be equally long: or finally, that b should be situated some- 
 where on the surface of a sphere, which is described so as to pass through the point D 
 (in Fig. 36, bis), and to have the origin o for its centre. 
 (8). If then we meet an equation of the form, 
 
 ''^=Ki, or eK-P = a^ 
 pa a a 
 
 in which a = OA, p — op, and a is a scalar, as before, we shall know that the locus 
 of the point p is a spheric surface, with its centre at the point O, and with the vector 
 aa for a radius ; and also that if we determine a point c by the equation oc = a'^a, 
 this spheric locus of P is a common orthogonal to all the circles apc, which can be 
 described, so as to pass through the two fixed points, A and c : because every radius 
 OP of the sphere is a tangent, at the variable point p, to the circle apc, exactly as 
 OB is to ABC in the recent Figure. 
 
 (9.) In the same Fig. 30, ^is, the sinular triangles show (by elementary princi- 
 ples) that the length of BC is to that of AB in the sub-duplicate ratio of oc to OA ; or 
 in the simple ratio of OD to OA ; or as the scalar a to 1. If then we meet, in any re- 
 search, the recent equation in p (sub-art. 8), we shall know that 
 length of (^p — a^a) = a x length of{p — a) ; 
 while the recent interpretation of the same equation gives this other relation of the 
 
 same kind : 
 
 length of p = a x length of a. 
 
 (10.) At a subsequent stage, it will be shown that the Calculus of Quaternions 
 supplies Rules of Transformation, by which we can pass from any one to any other 
 of these last equations respecting p, without (at the time) constructing any Figure, 
 or (immediately) appealing to Geometry : but it was thought useful to point out, 
 already, how much geometrical meanirig* is contained in so simple a fonnula, as that 
 of the last sub- art. 8. 
 
 (11.) The product of two conjugate quaternions is said to be their common 
 
 NoRMjt and is denoted thus: 
 
 qKq = Ng. 
 
 * A student of ancient geometry may recognise, in the two equations of sub-art. 
 9 a sort of translation, into the language of vectors, of a celebrated local theorem of 
 Apollonius of Perga, which has been preserved through a citation made by his early 
 commentator, Eutocius, and may be thus enunciated : Given any two points (as here 
 A and c) in a plane, and any ratio of inequality (as here that of 1 to a), it is possible 
 to construct a circle in the plane (as here the circle bdb'), such that the (lengths of 
 the) two right lines (as here ab and cb, or ap and cp), which are inflected from the 
 two given points to any common point (as B or p) of the circumference, shall be to 
 each other in the given ratio. (Avo doOkvTCJv arjutiwv, k. t. X. Page 11 of Halley's 
 Edition of Apollonius, Oxford, mdccx.) 
 
 f This name. Norm, and the corresponding characteristic, N, are here adopted, 
 as suggestions from the Theory of Numbers ; but, in the present work, they will not 
 
CHAP. I.] RADIAL QUOTIENTS, RIGHT RADIALS. 129 
 
 It follows that NK^ = Ngr ; and that the norm of a quaternion is generally a positive 
 scalar: namely, the square of the quotient of the lengths of the two lines, of which 
 (as vectors) the quaternion itself is the quotient (112). In fact we have, by sub-art. 
 6, and by the definition of a norm^ the transformations : 
 
 OB Ob' _ OC OB' _ OC OB _ OO _ / OD Y , 
 OA~ OA Ob' OA OB OA OA \OA. ] 
 
 a a a \length of a J 
 As a limit, we may say that the norm of a null quaternion is zero; or in symbols, 
 N0 = 0. 
 
 (12.) With this notation, the equation of the spheric locus (sub-art. 8), which 
 has the point o for its centre, and the vector aa for one of its radii, assumes the 
 shorter form : 
 
 N^ = a2; or N-^=l. 
 
 Section 7. — On Radial Quotients; and on the Square of a 
 Quaternion. 
 
 146. It was early seen (comp. Art. 2, and Fig. 4) that ani/ 
 two radii, ab, ac, of any one circle, or sphere, are necessarily 
 unequal vectors ; because their directions differ. On the other 
 hand, when we are attending only to relative direction (110), 
 we may suppose that all the vectors compared are not merely 
 co-initial (18), but are also equally long; so that if their com- 
 mon length be taken for the unit, they are all radii, oa, ob, . . 
 of what we have called the Unit- Sphere ( 1 28), described round 
 the origin as centre; and may all be 
 said to be Unit- Vectors (129). And 
 then the quaternion, which is the 
 quotient of any one such vector divi- 
 ded bv any other, or generally the 
 
 .- \ i . 77 7 Fig. 39. 
 
 quotient oj any two equally long vec- 
 tors, may be called a Radial Quotient; or sometimes sim- 
 ply a Kadial. (Compare the annexed Figure 39.) 
 
 be often wanted, although it may occasionally be convenient to employ them. For 
 we shall soon introduce the conception, and the characteristic, of the Tensor, Tq, of 
 a quaternion, which is of greater geometrical utility than the Norm, but of which it 
 will be proved that this norm is simply the square, 
 
 qKq^-Sq^iTqy. 
 
 Compare the Note to sub -art, 3. 
 
 S 
 
130 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 Fig. 40. 
 
 147. The two Unit' Scalar s^ namely, Positive and Nega- 
 tive Unity ^ may be considered as limiting cases of radial quo- 
 tients, corresponding to the two extreme values, and tt, of the 
 angle aob, or z §' (131). In the intermediate 
 
 case, when aob is a right angle, or Lq = ^, 
 
 as in Fig. 40, the resulting quotient, or qua- 
 ternion, may be called (comp. 132) a Right 
 Radial Quotient; or simply, a Right Ra- 
 dial. The consideration of such right radials 
 will be found to be of great importance, in the whole theory 
 and practice of Quaternions. 
 
 148. The most important general 'property of the quotients 
 last mentioned is the following : that the Square of every Right 
 Radial is equal to Negative Unity ; it being understood that 
 we write generally, as in algebra, 
 
 q.q=^qq = q\ 
 
 and call this product of two equal quaternions the square of 
 each of them. For if, as in Fig. 41, we 
 describe a semicircle aba', with o for cen- 
 tre, and with ob for the bisecting radius, 
 then the two right quotients, ob : oa, 
 and oa' : ob, are equal (Qom^. 117); and 
 therefore their common square is (comp. 
 107) the product, 
 
 ^obV oa' ob oa' 
 
 ^OAy ob oa oa 
 where oa and ob may represent any 
 two equally long, but mutually rect- ^ 
 angular lines. More generally, the 
 Square of every Right Quotient 
 
 (132) is equal to a Negative Scalar; namely, to the negative of 
 the square of the number, which represents the ratio of the 
 lengths* of the two rectangular lines compared ; or to zero 
 
 Fig. 41, bis. 
 
 * Hence, by 145, (11.), q^ = -Nq, if Iq- 
 
CHAP.l.] GEOMETRICAL SQUARE ROOTS OF NEGATIVE UNITY. 131 
 
 minus the square of the wwm^^r which denotes (comp. 133) the 
 length of the Index of that Kight Quotient : as appears from 
 Fig. 41, his^ in which ob is only an ordinate, and not (as be- 
 fore) a radius, of the semicircle aba' ; for we have thus, 
 
 obV oa' (length of obV .r. 
 
 — = — = - , ^,; •;. , if OB ± OA. 
 
 OAy OA \lengtli oj oaJ 
 
 149. Thus everg Might Radial is, in the present System, 
 one of the Square Roots of Negative Unity ; and may there- 
 fore be said to be one of the Values of the Symbol \/ - 1 ; which 
 celebrated symbol has thus a certain degree of vagueness, or at 
 least 0^ in determination, oi meaning in this theory, on account 
 of which we shall not often employ it. For although it thus 
 admits o^ Si. perfectly clear and geometrically real Interpretation, 
 as denoting what has been above called a Right Radial Quo- 
 tient, yet the Plane of that Quotient is arbitrary; and therefore 
 the symbol itself must be considered to have (in the present 
 system) itidefinitely many values ; or in other words the Equa- 
 tion, 
 
 has (in the Calculus of Quaternions) mc^<?^w2Vc/y many Roots,* 
 which are all Geometrical Reals : besides any other roots, of 
 a purely symbolical character, which the same equation may be 
 conceived to possess, and which may be called Geometrical 
 Imaginaries.^ Conversely, if q be any real quaternion, which 
 
 * It will be subsequently shown, that if x, y, z be ani/ three scalars, of which 
 the sum of the squares is unity, so that 
 
 a:3 + y2+z2 = l; 
 
 and if i, j, k be any three right radials, in three mutually rectangular planes; then 
 
 the expression, 
 
 q = ix+jy + hz, 
 
 denotes another right radial, which satisfies {as such, and by symbolical laws to be 
 assigned) the equation q^ =— i; and is therefore one of the geometrically real values 
 of the symbol V— 1. 
 
 f Stich imaginaries will be found to offer themselves, in the treatment by Qua- 
 ternions (or rather by what will be called Biquaternions^, of ideal intersections, and 
 of ideal contacts, in geometry; but we confine our attention, for the present, to ^-ea- 
 metrical reals alone. Compare the Notes to page 90. 
 
132 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 satisfies the equation q"^ ^-\, it must he a right radial; for if, 
 as in Fig. 42, we suppose that A aob cx boc, 
 we shall have 
 
 ^/ObV OC OB oc 
 
 \oAy ~0B OA oa' 
 
 and this square of q cannot become equal to 
 negative unity ^ except by oc being = - oa, 
 or = oa' in Fig. 4 1 ; that is, by the line ob 
 being at right angles to the line oa, and 
 being at the same time equally long^ as in o 
 Fig. 40. 
 
 (1.) If then we meet the equation, 
 
 [it- 
 
 where a = OA, and p = op, as before, we shall know that the locits of the point p is 
 the circumference of a circle^ with o for its centre^ and with a radius which has the 
 same length as the line OA ; while the plane of the circle is perpendicular to that 
 given line. In other words, the locus of p is a great circle, on a sphere of which the 
 centre is the origin ; and the given point a, on the same spheric surface, is one of the 
 poles of that circle. 
 
 (2.) In general, the equation 5^ = — a^, where a is any (real) scalar^ requires 
 that the quaternion q (if real) should be some right quotient (132) ; the number a 
 denoting the letigth of the index (133), of that right quotient or quaternion (comp. 
 Art 148, and Fig. 41, 6is). But the plane of 5 is still entirely arbitrary ; and 
 therefore the equation 
 
 g2 = -a2, 
 
 like the equation 5'=— 1, which it includes, must be considered to have (in the 
 present system) indefinitely many geometrically real roots. 
 (3.) Hence the equation, 
 
 t;T 
 
 in which we may suppose that a > 0, expresses that the locus of the point p is a 
 (new) circular circumference, with the line oa for its axis,* and with a radius of 
 which the length = a x the length of OA. 
 
 150. It may be added that the index (133), and the axis (128), 
 of a right radial (147), are the same; and that its reciprocal (134), its 
 conjugate (137), and its opposite (143), are all equal to each other. Con- 
 versely, if the reciprocal of a given quaternion q be equal to the opposite 
 
 * It being understood, that the axis of a circle is a right line perpendicular to 
 the plane of that circle, and passing through its centre. 
 
CHAP. I.] RADIAL QUOTIENTS CONSIDERED AS VERSORS. 133 
 
 of that quaternion, then q is a right radial; because its square^ q^, 
 is then equal (comp. 136) to the quaternion itself, divided hy its op- 
 posite; and therefore (by 143) to negative unity. But the conjugate 
 of every radial quotient is equal to the reciprocal of ^Aa^ quotient ; 
 because if, in Fig. 36, we conceive that the three lines da, ob, ob' are 
 equally long, or if, in Fig. 39, ^iQ prolong the arc ba, by an equal arc 
 ab', we have the equation, 
 
 ^ ob' oa 1 
 Kg' = — = — = -. 
 
 OA ob §- 
 
 And conversely,* 
 
 if 'Kq- -, or if gK^= 1, 
 
 then the quaternion 5' is a radial quotient. 
 
 Section 8. — On the Versor of a Quaternion, or of a Vector ; 
 and on some General Fornfiulce of Transformation. 
 
 151. When a quaternion g' = /3 : a is thus a radial quotient 
 (146), or when the lengths of the two lines a and j3 are equal, 
 the effect of this quaternion q, considered as a Factor (103), 
 in the equation qa = jS, is simply the turning of the multipli- 
 cand-line a, in the plane ofq (119), and towards the hand de- 
 termined by the direction of the positive axis Ax . q (129), 
 through the angle denoted hj A q (130) ; so as to bring that 
 line a (or a revolving line which had coincided therewith) into 
 a neio direction : namely, into that of the product-line j3. And 
 with reference to this conceived operation of turning, we shall 
 now say that every Radial Quotient is a Versor. 
 
 152. A Versor has thus, in general, 2i plane, an axis, and 
 an angle ; namely, those of the Radial (146) to which it cor- 
 responds, or is equal : the onlg difference between them being 
 a difference in the points ofview'f from which they are respec- 
 tively regarded ; namely, the radial as the quotient, q, in the 
 
 * Hence, in the notation of norms (145, (11.) ), if l^q= 1, then 5 is a radial ; 
 and conversely, the norm of a radial quotient is always equal to positive unity. 
 
 f In a slightly metaphysical mode of expression it may be said, that the radial 
 quotient is the result of an analysis, wherein two radii of one sphere (or circle) are 
 compared, as regards their relative direction ; and that the equal versor is the instru- 
 ment of a corresponding synthesis, wherein owe radius is conceived to he generated, by 
 a certain rotation, from the other. 
 
134 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 formula, q = j3: a ; and the versor as the (equal) ^c^or, q, in 
 the converse formula, f5 = q.a; where it is still supposed that 
 the two vectors, a and )3j are equally long, 
 
 153. A versor, like a radial {} 4^), cannot degenerate into b. scalar, 
 except by its angle acquiring one or other of the two limit-values^ 
 and TT. In the first case, it becomes positive unity ; and in the second 
 case, it becomes negative unity : each of these two unit-scalars ( 1 47) 
 being here regarded as 2, factor (or coefficient^ comp. 12), which ope- 
 rates on a line, to preserve or to reverse its direction. In this view, we 
 may say that - 1 is an Inversor ; and that every Right Versor (or ver- 
 
 sor with an angle = - is a Semi-inversor :* because it half-inverts the 
 
 line on which it operates^ or turns it through half of two right angles 
 (comp. Fig. 41). For the'same reason, we are led to consider every 
 right versor (like every right radial, 149, from which indeed we have 
 just seen, in 152, that it differs only as factor differs from quotient), 
 as being one of the square-roots of negative unity : or as one of the va- 
 lues of the symbol y' - 1 . 
 
 154. In fact we may observe that the effect of a right versor, con- 
 sidered as operating on a line (in its own plane), is to turn that line, 
 towards a given hand, through a right angle. If then q be such a ver- 
 S07% and if qa = ft, we shall have also (comp. Fig. 41), qP = -a', so 
 that, if a be any line in the plane of a right versor q, we have the 
 equation, 
 
 q,qa = -a; 
 
 whence it is natural to write, under the same condition, 
 
 as in 149- On the other hand, no versor, which is not right-angled, 
 can he a value of y/ -\; or can satisfy the equation q^a --a, as Fig. 
 42 may serve to illustrate. For it is included in the meaning of this 
 last equation, as applied to the theory of versors, that a rotation 
 through 2 Lq, or through the double of the angle of q itself, is equi- 
 
 * This word, " semi -inversor," will not be often used ; but the introduction of it 
 here, in passing, seems adapted to throAV light on the view taken, in the present work, 
 of the symbol V — 1, when regarded as denoting a certain important class (149) of 
 Reals in Geometry. There are uses of that symbol, to denote Geometrical Imagi- 
 naries (comp. again Art. 149, and the Notes to page 90), considered as connected 
 with ideal intersections, and with ideal contacts ; but with such uses of V - 1 we 
 have, at present, nothing to do. 
 
CHAP. I.] VERSOR OF A QUATERNION, OR OF A VECTOR. 135 
 
 valent to an inversion of direction; and therefore to a rotation through 
 two right angles. 
 
 155. In general, if a be any vector^ and if a be used as a 
 temporary* symbol for the number expressing its length; so 
 that a is here a positive scalar, which bears to positive unity, 
 or to the scalar + 1, the same ratio as that which the length of 
 the line a bears to the assumed unit of length (comp. 128); 
 then the quotient a : a denotes generally (comp. 16) a new vec- 
 tor, which has the same direction as the proposed vector a, but 
 has its length equal to that assumed unit : so that it is (comp. 
 146) the Unit- Vector in the direction of a. We shall denote this 
 unit-vector by the symbol, Ua ; and so shall write, generally, 
 
 Ua = -, if a = length of a ; 
 
 that is, more fully, if a be, as above supposed, the number 
 (commensurable or incommensurable, but positive) which re- 
 presents that length, with reference to some selected standard. 
 
 156. Suppose now that 5- = j3 : a is (as at first) 2^ general 
 quaternion, or the quotient of any two vectors, a and j3, whether 
 equal or unequal in length. Such a Quaternion will not (gene- 
 rally) be a Versor (or at least 7iot simply such), according to the 
 definition lately given ; because its effect, when operating as a 
 factor (103) on a, will not in general be simply to turn that 
 line (151) : but will (generally) alter the length,^ as well as the 
 direction. But if we reduce the two proposed vectors, a and j3, 
 to the two unit-vectors Ua and Uj3 (155), and ^ovmthQ quotient 
 of these, we shall then have taken account of relative direction 
 alone : and the result Avill therefore be a versor, in the sense 
 lately defined (151). We propose to call the quotient, or the 
 versor, thus obtained, the versor-element, or briefly, the Yer- 
 soR, of the Quaternion q ; and shall find it convenient to em- 
 
 * "We shall soon propose a general notation for representing the lengths of vectors, 
 according to which the symbol Ta will denote what has been above called a ; but^ 
 are imwilling to introduce more than one new characteristic of operation, such as K, 
 or T, or U, &c., at one time. 
 
 f By what we shall soon call call an act of tension, which will lead us to the 
 consideration of the tensor of a quaternion. 
 
136 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 ploy the same* Characteristic, U, to denote the operation of 
 taking the versor of a quaternion, as that employed above to 
 denote the operation (155) of reducing a vector to the unit of 
 length, without any change of its direction. On this plan, the 
 symbol \]q will denote the versor ofq ; and the foregoing de- 
 finitions will enable us to establish the General Formula : 
 
 a xJa 
 
 in which the two unit-vectors, Ua and Uj3, may be called, by 
 analogy, and for other reasons which will afterwards appear, 
 the versor s^ of the vectors, a and j3. 
 
 157. In thus passing from a given quaternion, q, to its ver- 
 sor, \Jq, we have only changed (in general) the lengths of the 
 two lines compared, namely, by reducing each to the assumed 
 unit of length (155, 156), without making any change in their 
 directions. Hence \h.Q plane (119), the axis (127, 128), and 
 the angle (130), of the quaternion, remain unaltered in this 
 passage ; so that we may establish the two following general 
 formulae : 
 
 L\]q = Lq; Ax . U<7 = Ax . q. 
 
 More generally we may write, 
 
 * For the moment, this double use of the characteristic U, to assist in denoting 
 both the unit-vector Ua derived from a given line a, and also the versor Uy derived 
 from a quaternion q, may be regarded as estabhshed here by arbitrary definition; 
 but as permitted, because the difference of the symbols, as here a and q, which serve 
 for the present to denote vectors and quaternions, considered as the subjects of these 
 two operations U, will prevent Bwch. double use of that characteristic from giving rise 
 to any confusion. But we shall further find that several important analogies are by 
 anticipation expressed, or at least suggested, when the proposed notation is employed. 
 Thus it will be found (comp. the Note to page 119), that every vector a may usefully 
 be equated to that right quotient, of which it is (133) the index ; and that then the 
 unit-vector "[] a may be, on the same plan, equated to that right radial (14.7), which 
 is (in the sense lately defined) the versor of that right quotient. We shall also find 
 ourselves led to regard every unit-vector as the axis of a quadrantal (or right) rota- 
 tion, in a plane perpendicular to that axis; which will supply another inducement, 
 to speak of every such vector as a versor. On the whole, it appears that there will 
 be no inconvenience, but rather a prospective advantage, in our already reading the 
 symbol Ua as ^^ versor of a ;" just as we may read the analogous symbol \Jq, as 
 ^^ versor ofq." 
 
 t Compare the Note immediately preceding. 
 
CHAP. I.] EQUAL AND RECIPROCAL VERSORS, REVERSORS. 337 
 
 Z ^' = Z $', and Ax . ^' = Ax . ^, if \Jq' = JJq ; 
 the versor of a quaternion depending solely on, but conversely 
 being sufficient to determine, the relative direction (156) of the 
 two lines, of which (as vectors) the quaternion itself is the quo- 
 tient (112); or the axis and angle of the rotation, in the plane 
 of those two lines, from the divisor to the dividend (128) ; so 
 that any two quaternions, which have equal versors, must also 
 have equal angles, and equal (or coincident) axes, as is ex- 
 pressed by the last written formula. Conversely, from this 
 dependence of the versor \]q on relative direction'^ alone, it 
 follows that any two quaternions, of which the angles and the 
 axes are equal, have also equal versors; or in symbols, that 
 
 \]q'==\]q, if Lq'=-Lq, and Ax.^-' = Ax.^'. 
 For example, we saw (in 138) that the conjugate and the re- 
 ciprocal of any quaternion have thus their angles and their 
 axes the same ; it follows, therefore, that the versor of the 
 conjugate is always equal to the versor of the reciprocal; so 
 that we are permitted to establish the following general for- 
 mula,! 
 
 q 
 
 158. Again, because 
 
 it follows that the versor of the reciprocal of any quaternion is, 
 at the same time, the reciprocal of the versor ; so that we may 
 write, 
 
 * The unit-vector Ucr, which we have recently proposed (156) to call the versor 
 of the vector a, depends in like manner on the direction of that vector alone; which 
 exclusive reference^ in each of these two cases, to Direction, may serve as an addi- 
 tional motive for employing, as we have lately done, one common name^ Veesor, 
 and one common characteristic, U, to assist in describing or denoting both the Unit- 
 Vector Ua itself and the Quotient of two such Unit- Vectors, \Jq = U/3 : Ua ; all 
 danger of confusion being sufficiently guarded against (comp. the Note to Art. 156), 
 by the difference of the two symbols, a and q, employed to denote the vector and the 
 quaternion, which are respectively the subjects of the two operations U ; while those 
 two operations agree in this essential point, that each serves to eliminate the quan- 
 titative element, of absolute or relative length. 
 t Compare the Note to Art. 138. 
 
 T 
 
138 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 Ui = ^; or JJq.JJ-=l. 
 
 q Vq ^ q 
 
 Hence, by the recent result (157), we have also, generally, 
 UK^ = i-; or, U^.UK^ = l. 
 
 Also, because the versor XJq is always a radial quotient (151, 
 152), it is (by 150) the conjugate of its own reciprocal ; and 
 therefore at the same time (comp. 145), the reciprocal of its 
 own conjugate; so that the /?roc?wc^ of tic o conjugate versor s, 
 or what we have called (145, (!!•)) their common Norm, is 
 always equal io positive unity ; or in symbols (comp. 150), 
 
 NU^ = U^.KU^=1. 
 
 For the same reason, the conjugate of the versor of any qua- 
 ternion is equal to the reciprocal of that versor^ or (by what 
 has just been seen) to the versor of the reciprocal of that qua- 
 ternion; and therefore also (by 157), to the versor of the con- 
 jugate; so that we may write generally, as a summary of re- 
 cent results, the formula : 
 
 each of these four symbols denoting a new versor, which has 
 the same plane, and the same angle, as the old or given versor 
 \]q, but has an opposite axis, or an opposite direction of rota- 
 tion-, so that, with respect to that given Versor, it may na- 
 turally be called a Ke versor. 
 
 159. As regards the versor itself, whether of a vector or of 
 a quaternion, the definition (155) of Ua gives, 
 
 UiCo = + Ua, or = - Ua, according as rc> or < ; 
 because (by 15) the scalar coefjicient x preserves, in the first 
 ease, but reverses, in the second case, the direction of the vec- 
 tor a; whence also, by the definition (156) of U^', we have 
 generally (comp. 126, 143), 
 
 U^r^' = + U^', or = - \5q, according as a;> or < 0. 
 The versor of a scalar, regarded as the limit of a quaternion 
 (131, 139), is equal to positive or negative unity (comp. 147, 
 
CHAP. I.] GENERAL TRANSFORMATIONS OF A VERSOR. 139 
 
 153), according as the scalar itself is positive or negative ; or 
 in symbols, 
 
 Ua; = + 1, or = - 1 , according as a; > or < ; 
 
 the plane and axis of each of these two unit scalar s (147), con- 
 sidered as versors (153), being (as we have already seen) inde- 
 terminate. The versor of a null quaternion (141) must be re- 
 garded as wholly arbitrary^ unless we happen to know a Z«i^7,* 
 according to which the quaternion tends to zero^ before actually 
 reaching that limit ; in which latter case, the plane^ the axis, 
 and the angle of the versor] UO may all become determined, as 
 limits deduced from that law. The versor of a right quotient 
 (132), or of a right-angled quaternion (141), is always a right 
 radial (147)) or a right versor (153) ; and therefore is, as such, 
 one of the square roots of negative unity (149), or one of the 
 values of the symbol V - 1 5 while (by 150) the axis and the 
 index of such a versor coincide ; and in like manner its recipro- 
 cal, its conjugate, and its opposite are all equal to each other. 
 160. It is evident that if a proposed quaternion q be already 
 a versor (151), in the sense of being a radial (146), the ope- 
 ration o^ taking its versor (156) produces no change; and in 
 like manner that, if a given vector a be already an unit-vector, 
 it remains the same vector, when it is divided (155) by its own 
 length; that is, in this case, by the number one. For example, 
 we have assumed (128, 129), that the axis o^ every quaternion 
 is an unit-vector ; we may therefore write, generally, in the no- 
 tation of 155, the equation, 
 
 U(Ax./7) = Ax .§'. 
 
 A second operation U leaves thus the result of i)iQ first opera- 
 tion U unchanged, whether the subject of such successive ope- 
 rations be a line, or a quaternion; we have therefore the two 
 
 * Compare the Note to Art. 131. 
 
 t When the zero in this symbol^ UO, is considered as denoting a null vector (2), 
 the symbol itself denotes generally, by the foregoing principles, an indeterminate 
 unit-vector; although the direction of this unit- vector may, in certain questions, he- 
 come determined, as a limit resulting from a law. 
 
140 ELEMENTS OF QUATERNIONS. [boOK II. 
 
 following general formulae, differing only in the symbols of 
 that subject : 
 
 UUa=Ua; JJUq = Uq; 
 
 whence, by abstracting (comp. 145) from the subject of the 
 operation, we may write, briefly and symbolically, 
 
 16 1. Hence, with the help of 145, 158, 159, we easily deduce 
 the following (among other) transformations of the versor of a qua- 
 ternion : 
 
 K^- q q U^ ^ ^ ^ 
 
 TJq = Vxq, if £c> ; = - TJxq, if x<0. 
 We may also write, generally, 
 
 the parentheses being here unnecessary, because (as will soon be more 
 fully seen) the symbol JJq^ denotes one common versor ^ whether we 
 interpret it as denoting the square of the versor^ or as the versor of 
 the square^ of q. The present Calculus will be found to abound in 
 General Transformations of this sort; which all (or nearly all), like 
 the foregoing, depend ultimately on very simple geometrical concep- 
 tions ; but which, notwithstanding (or rather, perhaps, on account 
 of) this extreme simplicity of their origin, are often useful, as elements 
 of a new kind o^ Symbolical Language in Geometry: and generally, 
 as instruments of expression, in all those mathematical or physical 
 researches to which the Calculus of Quaternions can be applied. It 
 is, however, by no means necessary that a student of the subject, 
 at the present stage, should make himself familiar with all the 
 recent transformations of Ug-; although it may be well that he 
 should satisfy himself of their correctness, in doing which the fol- 
 lowing remarks will perhaps be found to assist. 
 
 (I.) To give &. geometrical illustration^ ■\vhich may also serve asa/3/oo/J of the 
 recent equation, 
 
CHAP. I.] GEOMETRICAL ILLUSTRATIONS. 
 
 141 
 
 we may employ Fig. 36, bis ; in which, by 145, (2.), we have 
 
 ^ Kq OA OB' Ob' \ODJ \ OA j 
 
 (2.) As regards the equation, Jj(q^) = (JJqY^ we have only to conceive that the 
 three lines oa, ob, oc, of Fig. 42, are cut (as in Fig. 42, bis) in 
 three new points, a', b', c', by an unit-circle (or by a circle with 
 a radius equal to the unit of length), which is described about 
 their common origin o as centre, and in their common plane ; for 
 then if these three lines be called a, ft, y , the three new lines oa', 
 ob', oc' are (by 155) the three unit-vectors denoted by the sym- 
 bols, Ua, U/3, Uy; and we have the transformations (comp. 148, 
 149), 
 
 u(,^)=u.(^y=uz=hi=2£;=(-;y=(u,)^. 
 
 ^^ ^ \a j a Ua OA V<^^ / 
 
 (3.) As regards other recent transformations (161), although 
 we have seen (135) that it is not necessary to invent any new or 
 peculiar symbol, to represent the reciprocal of a quaternion, yet 
 if, for the sake of present convenience, and as a merely temporary 
 notation^ we write 
 
 Bq=\, 
 
 O A' A 
 
 Fig. 42, bis. 
 
 employing thus, for a moment, the letter R as a characteristic of reciprocation, or 
 of the operation of taking the reciproeal, we shall then have the symbolical equations 
 (comp. 145, 158) : 
 
 R2 = K2 = 1; RK = KR; RU = UR = KU=UK; 
 
 but we have also (by 160), U2= U ; whence it easily follows that 
 
 U = RUR = RKU = RUK = KUR = KRU = KUK 
 = URK = UKR = UKUR = UKRU = (UK)2 = &c. 
 (4.) The equation 
 
 U 
 
 ^ -. US or simply, Up = U|3, 
 a a 
 
 expresses that the locus of the point p is the indefinite right line, or ray (comp. 132, 
 (4.)), which is drawn /rom o in the direction of ob,* but not in the opposite direc- 
 tion ; because it is equivalent to 
 
 ^^- 
 
 ^f- 
 
 or (0 = x(3, x>0. 
 
 (5.) On the other hand the equation, 
 
 or Up=-U/3, 
 
 a a 
 
 expresses (comp. 132, (5.)) that the locus of p is the opposite ray from o ; or that 
 it is the indefinite prolongation of the revector bo ; because it may be transformed to 
 
 * In 132, (4.), p. 119, OA and a ought to have been ob and b. 
 
142 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 U ^ = - 1 ; or Z ^ = TT ; or p = a:/3, cc < 0. 
 (6.) If a, j3, y denote (as in sub-art. 2) the three lines oa, ob, oc of Fig. 42 (or 
 of Fig. 42, his), so that (by 149) we have the equation - = f ^ J , then this other 
 equation, l^pV^^y^ 
 
 expresses generally that the locus of p is the system of the two last loci ; or that it is 
 the whole indefinite right line, both ways prolonged, through the two points o and B 
 (comp. 144, (2.)). 
 
 (7.) But if it happen that the line y, or oc, like oa' in Fig. 41 (or in Fig. 41, 
 6is), has the direction opposite to that of a, or of oa, so that the last equation takes 
 the particular form, 
 
 I n\2 
 
 1, 
 
 ("fl- 
 
 then U- must be (by 154) a right versor ; and reciprocally, every right versor, with 
 a 
 
 a plane containing a, will be (by 153) a value satisfying the equation. In this case, 
 
 therefore, the locus of the point p is (as in 132, (1.), or in 144, (1.)) the plane 
 
 through o, perpendicular to the line OA ; and the recent equation itself, if supposed 
 
 to be satified by a real* vector p, may be put under either of these two earlier but 
 
 equivalent /orm* • 
 
 Section 9. — On Vector- Arcs, and Vector- Angles, considered 
 as Representatives of Versors of Quaternions ; and on the 
 Multiplication and Division of any one such Versor hy 
 another. 
 
 162. Since every unit-vector oa (129), drawn from the 
 origin o, terminates in some point a on the surface of what we 
 have called the unit-sphere (128), that term a (1) may be 
 considered as a Representative Point, of which the position on 
 that surface determines, and may be said to represent, the 
 direction of the line oa in space ; or of that line multiplied 
 (12, 17) by any positive scalar. And then the Quaternion 
 which is the quotient (112) of any two such unit- vectors, and 
 which is in one view a Radial (146), and in another view a 
 Versor (151), may be said to have the arc of a great circle, 
 AB, upon the unit sphere, which connects the terms of the two 
 
 * Compare 149, (2.) ; also the second Note to the same Article ; and the Notes 
 to page 90. 
 
CHAP. I.] REPRESENTATIVE AND VECTOR ARCS. 143 
 
 vectors, for its Representative Arc, We may also call this 
 arc a Vector Arc, on account of its having a definite direc- 
 tion (comp. Art. 1), such as is indicated (for example) by a 
 curved arrow in Fig. 39 ; and as being thus contrasted with 
 its own opposite, or with what may be called by analogy the 
 Revector Arc ba (comp. again 1) : this latter arc represent- 
 ing, on the present plan, at once the reciprocal (134), and the 
 conjugate (137), of the former versor; because it represents 
 the corresponding Reversor (158). 
 
 163. This mode of representation, of versors of quaternions 
 by vector arcs, would obviously be very imperfect, unless 
 equals were to be represented by equals. We shall therefore 
 define, as it is otherwise natural to do, that a vector arc, ab, 
 upon the unit sphere, is equal to every other vector arc cd 
 which can be derived from it, by simply causing (or conceiv- 
 ing) it to slide* in its own great circle, icithout any change of 
 length, or reversal of direction. In fact, the two isosceles and 
 plane triangles aob, cod, which have the origin o for their 
 common vector, and rest upon the chords of these two arcs as 
 bases, are thus complanar, similar, and similarly turned ; so 
 that (by 117, 118) we may here write, 
 
 OB CD 
 
 A AOB OC COD, — = — ; 
 
 OA OC 
 
 the condition of the equality of the quotients (that is, here, of 
 the versors), represented by the two arcs, being thus satisfied. 
 We shall sometimes denote this sort of equality of two vector 
 arcs, AB and cd, by the formula, 
 o AB = /> CD; 
 
 and then it is clear (comp. 125, and the ear- 
 lier Art. 3) that we shall also have, by what 
 may be called inversion and alternation, j 
 these two other formulas of arcual equality, oi'-:~_ -'a 
 
 Fig. 35, his, 
 '^BA=/>DC; ^ AC = ^ BD. ^ ' 
 
 (Compare the annexed Figure 35, his^ 
 
 * Some aid to the conception may here be derived from the inspection of Fig 
 34 ; in which two equal angles are supposed to be traced on the suiface of one com- 
 
144 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 164. Conversely, unequal versors ought to be represented 
 (on the present plan) by unequal vector arcs; and accordingly, 
 we purpose to regard any two such arcs, as being, for the pre- 
 sent purpose, unequal (comp. 2), even when they agree in 
 quantity i or contain the same number of degrees^ provided that 
 they differ in direction : which may happen in either of two 
 principal ways, as follows. For, 1st, they may be opposite 
 arcs oi one great circle; as, for example, a vector arc ab, and 
 the corresponding revector arc ba ; and so may represent (162) 
 a versor, OB : oa, and the corresponding reversor, oa : ob, re- 
 spectively. Or, Ilnd, the two arcs may belong to different 
 great circles^ like ab and bc in Fig. 43 ; in which latter case, 
 they represent two radial quotients 
 ( 1 4 6) m different planes ; or (comp . 
 119) two diplanar versors, ob : oa, 
 and 00 : OB ; but it has been shown 
 generally (122), that diplanar qua- 
 ternions are always unequal: we 
 consider therefore, here again the 
 arcs, AB and bc, themselves^ to be 
 (as has been said) unequal vectors. 
 
 165. In this manner, then, we may be led (comp. 122) to 
 regard the conception of a plane, or o^ the position of a great 
 circle on the unit sphere, as entering, essentially, in general,* 
 into the conception of a vector-arc^ considered as the representa- 
 tive of a versor (162). But even without expressly referring 
 to versors, we may see that if, in Fig. 43, we suppose that b 
 is the middle point of an arc aa' of a great circle, so that in a 
 recent notation (163) we may establish the arcual equation, 
 
 we ought then (comp. 105) not to write also, 
 '^ AB = '^ bc; 
 
 mon desk. Or the four lines OA, ob, oc, od, of Fig. 35, may now be conceived to 
 be equally long; or to be cut by a circle with o for centre, as in the modification of 
 that Figure, which is given in Article 163, a little lower down. 
 
 * We say, in general ; for it will soon be seen that there is a sense in which all 
 great semicircles, considered as vector arcs, may be said to be eqval to each other. 
 
CHAP. I.] ARCUAL EQUATIONS, CO-ARCUALITY. 145 
 
 because the two co-initial arcs, ba and bc, which terminate 
 differently, must be considered (comp. 2) to be, as vector-arcs y 
 unequal. On the other hand, if we should refuse to admit (as 
 in 163) that any two complanar arcs, i^ equally long, and simi- 
 larly (not oppositely) directed, like ab and cd in the recent 
 Fig. 35, bis, are equal vectors^ we could not usefully speak of 
 equality between vector-arcs as existing under any circum- 
 stances. We are then thus led again to include, generally, the 
 conception of a plane, or of one great circle as distinguished 
 from another, as an element in the conception of a Vector-Arc, 
 And hence an equation between two such arcs must in general 
 be conceived to include two relations of co-arcuality. For 
 example, the equation ^ ab = '^ cd, of Art. 163, includes gene- 
 rally, as apart of its signification, the assertion (comp. 123) 
 that ihe four points a, b, c, d belong to ouq common great cir- 
 cle of the unit-sphere ; or that each of the two points, c and d, 
 is co-arcual Avith the two other points, a and b. 
 
 166. There is, however, a remarkable case o1 exception, vav^YiioSx 
 two vector arcs may be said to be equal, although situated in diffe- 
 rent planes: namely, when they are both great semicircles. In fact, 
 upon the present plan, every great semicircle, aa', considered as a 
 vector arc, represents an inversor (153); or it represents negative 
 unity (oa' : oa = - a : a = - 1), considered as one limit of a versor; 
 but we have seen (159) that such a versor has in general an indeter- 
 minate plane. Accordingly, whereas the initial and final points, or 
 (comp. 1) the origin a and the term b, of a vector arc ab, are in ge- 
 neral sufficient to determine the plane of that arc, considered as the 
 shortest or the most direct path (comp. 112, 128) from the one point 
 to the other on the sphere; in the particular case when one of the 
 two given points is diametrically opposite to the other, as a' to A, 
 the direction of this path becomes, on the contrary, indeterminate. 
 If then we only attend to the effect produced, in the way of change 
 of position of a point, by a conceived vection (or motion') upon the 
 sphere^ we are permitted to say that all great semicircles are equal 
 vector arcs; each serving simply, in the present view, to transport a 
 point from one position to the opposite; and thereby to reverse (like 
 the factor - 1, of which it is here the representative) the direction of 
 the radius which is drawn to that point of the unit sphere. 
 
 u 
 
146 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (1.) The equation, 
 
 r» aa' = o bb', 
 
 in which it is here supposed that a' is opposite to a, and b' to b, satisfies evidently 
 the general conditions of co-arcuality (165); because the /owr points aba'b' are all 
 on one great circle. It is evident that the same arcual equation admits (as in 163) 
 of inversion and alternation ; so that 
 
 r> a'a = r\ b'b, and n ab = «^ a'b'. 
 
 (2.) We may also say (comp. 2) that all null arcs are equal, as producing no 
 effect on the position of a point upon the sphere ; and thus may write generally, 
 
 n AA = n BB = 0, 
 with the alternate equation, or identity, r> ab = o ab. 
 
 (3.) Every such null vector arc AA is a representative, on the present plan, of the 
 other unit scalar, nsimely positive uniti/, considered as another limit of aversor (153) ; 
 and its plane is again indeterminate (159), unless some law be given, according to 
 which the arcual vection may be conceived to begin, from a given point A, to an in- 
 definitely near point B upon the sphere. 
 
 ' 167. The principal use of Vector Arcs, in the present 
 theory, is to assist in representing^ and (so to speak) in con- 
 structing, by means of a Spherical Triangle, the Multiplica- 
 tion and Division of any two Diplanar Versors (comp. 119, 
 164). In fact, any two such versors of quaternions (156), 
 considered as radial quotients (152), can easily be reduced (by 
 the general process of Art. 120) to the forms, 
 
 $- = j3 :a = OB : OA, g'' = 7 ; j3= oc : ob, 
 where a, b, c are corners of such a triangle on the unit sphere; 
 and then (by 107), the former quotient multiplied by the lat- 
 ter will give for product ; 
 
 q\q = ^ : a = OC'. OA. 
 If then (on the plan of Art. 1) any two successive arcs, as ab 
 and Bc in Fig. 43, be called (in relation to each other) vector 
 a^d provector ; while that third arc ac, which is drawn from 
 the initial point of the first to the final point of the second, 
 shall be called (on the same plan) the transvector : we may now 
 say that in the multiplication of any one versor (of a quater- 
 nion) by any other, if the multiplicand* q he represented (162) 
 by a vector-arc ab, and if the multiplier q be in like manner 
 
 * Here, as in 107, and elsewhere, we write the symbol of the multiplier towards 
 the left-hand, and that of the multiplicand towards the right. 
 
CHAP. I.] CONSTRUCTION OF MULTIPLICATION OF VERSORS. 147 
 
 represented by sl provector-arc bc, which mode of representa- 
 tion is always possible, by what has been already shown, then 
 the product q'. q, or q'q, is represented, at the same time, by 
 the transvector-arc ac corresponding. 
 
 168. One of the most remarkable consequences of this con- 
 struction of the multiplication ofversors is the following : that 
 the value of the product of two diplanar versors (164) depends 
 upon the order of tJie factors ; or that q'q and qq are unequal, 
 unless q be complanar (119) with q. For let aa' and cc' be 
 any two arcs of great circles, in different planes, bisecting each 
 other in the point b, as Fig. 43 is designed to suggest; so 
 that we have the two arcual equations (163), 
 
 '^ AB = ^ ba', and '^ bc = '^ c'b ; /^ 
 
 then one or other of the two following alternatives will hold 
 good. Either, 1st, the two mutually bisecting arcs will both 
 be semicircles, in which case the two new arcs, ac and cV, will 
 indeed both belong to one great circle, namely to that of which 
 B is a pole, but will have opposite directions therein ; because, 
 in this case, a' and c' will be diametrically opposite to a and c, 
 and therefore (by 166, (1.) ) the equation 
 
 '^ AC = '^ a'c', 
 but not the equation 
 
 '^ AC = '^ c'a', 
 
 will be satisfied. Or, Ilnd, the arcs aa' and cc', which are 
 supposed to bisect each other in b, will not both be semicircles, 
 even if one of them happen to be such ; and in this case, the 
 arcs AC, c'a' will belong to two distinct great circles, so that they 
 will be diplanar, and therefore unequal, when considered as 
 vectors. (Compare the 1st and Ilnd cases of Art. 164.) In 
 each case, therefore, ac and c'a' are unequal vector arcs; but the 
 former has been seen (167) to represent the product qq-, and 
 the latter represents, in like manner, the other product, qc[, of 
 the same two versors taken in the opposite order, because it is 
 the new transvector arc, when c b (= bc) is treated as the new 
 vector arc, and ba' (= ab) as the new provector arc, as is indi- 
 cated by the curved arrows in Fig. 43. The two products, 
 
148 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (iq and qq^ are therefore themselves unequal, as above asserted, 
 under the supposed condition of diplanarity, 
 
 169. On the other hand, when the two factors, q and q\ 
 are complanar versors^ it is easy to prove, in several different 
 ways, that their products, q'q and qq\ are equals as in algebra. 
 Thus we may conceive that the arc cc', in Fig. 43, is made to 
 turn round its middle point b, until the spherical angle cba' 
 vanishes; and then the two new transvector-arcs^ ac and cV, 
 will evidently become not only complanar but equal, in the 
 sense of Art. 163, as being still equally long, and being now 
 similarly directed. Or, in Fig. 35, bis, of the last cited Arti- 
 cle, we may conceive a point e, bisecting the arc bc, and there- 
 fore also the arc ad, which is commedial therewith (comp. 
 Art. 2, and the second Figure 3 of that Article) ; and then,, 
 if we represent the one versor q by either of the two equal 
 arcs, AE, ED, we may at the same time represent the other 
 versor q' by either of the two other equal arcs, eg, be ; so that 
 the one product, q'q, will be represented by the arc ac, and 
 the other product, qq', by the equal arc bd. Or, without re- 
 ference to vector arcs, we may suppose that the two factors 
 
 are, 
 
 q =(3: a = ob: oa, q' <= y : a== oc : OA, 
 
 oa, ob, oc being any three complanar and equally long right 
 lines (see again Fig. 35, bis) ; for thus we have only to deter- 
 mine a fourth line, S or od, of the same length, and in the same 
 plane, which shall satisfy the equation S:y=(5:a (117), and 
 therefore also (by 125) the alternate equation, 01/3 = 7: a; 
 and it will then immediately follow* (by 107), that 
 
 S 13 S S y 
 q ^q = ^-- = - = -'- = q'q' . 
 p a a y a 
 
 We may therefore infer, for any two versor s of quaternions, q 
 and q, the two following reciprocal relations : 
 
 * It is evident that, in this last process of reasoning, we make no use of the sup- 
 posed equality of lengths of the four lines compared ; so that we might prove, in ex- 
 actly the same way, that q'q = qq' if 9' | !| 9 (123), without assuming that these two 
 complanar factors, or quaternions, q and q', are versors. 
 
CHAP. I.] MULTIPLICATION OF RIGHT VERSORS. 149 
 
 l...gq = qq\ if q' \\\ q (123) ; 
 II. , . i£ q'q =qq\ then 5^' ||| 9- (168) ; 
 
 convertibility of factors (as regards -[heiv places in thQ product) 
 being thus at once a consequence and ?i proof of complanarity. 
 
 170. In the 1st case of Art. 168, th^ factors q and q' are both 
 right versors (153) ; and because we have seen that then their two 
 products^ q'q and qcf ^ are versors represented by equally long but op- 
 positely directed arcs of one great circle, as in the 1st case of 164, it 
 follows (comp. 162), that these two products are at once reciprocal 
 (134), and conjugate (137), to each other; or that they are related 
 as versor and reversor (158). We may therefore write, generally, 
 
 I. . . qq'=Kq'q, and II. .. m' = -fZ^ 
 
 if q anc] q be any two right versors; because the multiplication of 
 any two such versors, in two opposite orders, may always be repre- 
 sented or constructed by a Figure such as that lately numbered 
 43, in which the bisecting arcs aa' and cc' are semicircles. The Ilnd 
 formula may also be thus written (comp. 135, 154): 
 
 III. .. if 2'^ = -!, and q'^=-\., then qq-qq=-^^\ 
 
 and under this form it evidently agrees with ordinary algebra, be- 
 cause it expresses that, under the supposed conditions., 
 
 q'q.qq'^iKf', 
 but it will be found that this last equation is not an identity, in the 
 general theory of quaternions. 
 
 171. If the two bisecting semicircles cross each other at riyht 
 angles., the conjugate products are represented by two quadrants., 
 oppositely turned, of one great circle. It follows that if two right 
 versors, in two mutually rectangular planes, he multiplied together in two 
 opposite orders, the two resultiiig products will he two opposite right 
 versors, in a third plane, rectangular to the two former; or in symbols, 
 that 
 
 if ^^ = - 1, 2''^ = - 1, and Ax. q x Ax. q, 
 then 
 
 {qqy=-{qqy^-\, q'q = -qq\ 
 
 and 
 
 Ax. q'q 4- Ax. q. Ax. ^q a. Ax. q\ 
 
 In this case, therefore, we have what would be in algebra a paradox, 
 namely the equation, 
 
 {q'qy^-q'^.q\ 
 
150 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 if q and q' be any two right versors, in two rectangular planes ; but we 
 see that this result is not more paradoxical, in appearance, than the 
 equation 
 
 qq=-qq, 
 
 which exists, under the same conditions. And when we come to ex- 
 amine what, in the last analysis, may be said to be the meaning of this 
 last equation, we find it to be simply this : that any two quadrantal or 
 right rotations^ in planes perpendicular to each other^ compound them- 
 selves into a third right rotation^ as their resultant^ in a plane perpendi- 
 cular to each of them: and that this third ox resultant rotation has 
 one or other of two opposite directions^ according to the order in which 
 the two component rotations are taken, so that one shall be successive 
 to the other. 
 
 172. We propose to return, in the next Section, to the 
 consideration of such a System of Right Versors, as that which 
 we have here briefly touched upon : but desire at present to 
 remark (comp. 167) that a spherical triangle ABcmay serve to 
 construct, by means of represeritative arcs (162), not only the 
 multiplicatioiL, but also the division, of any one of two diplanar 
 versors (or radial quotients) by the other. In fact, we have 
 only to conceive (comp. Fig. 43) that the vector arc ab repre- 
 sents a given divisor, say q, or j3 : a, and that the transvector 
 arc AC (167) represents a given dividend, suppose q", or y : a; 
 for then the provector arc bc (comp. again 167) will represent, 
 on the same plan, the quotient of these two versors, namely 
 q" : 5', or 7 : j3 (106), or the versor lately called q ; since we 
 have generally, by 106, 107, 120, for quaternions, as in alge- 
 bra, the two identities : 
 
 (q":q)^q = q"; qq-q^q'- 
 
 173. It is however to be observed that, for reasons already as- 
 signed, we must not employ, for diplanar versors^ such an equation 
 as q. {q": q) = q" ', because we have found (168) that, for such ver- 
 sors, the ordinary algebraic identity, qq' — (^q, ceases to he true. In 
 fact by 169, we may now establish the two converse formulse: 
 
 I. . . q{q"'.q)=q'\ if q"\\\q {123); 
 11. . . iiq\q"'.q) = q", then ^'Mil q. 
 Accordingly, in Fig. 43, if q, q', q" be still represented by the 
 arcs AB, BC, AC, the product q {q"'.q), or qq', is not represented by 
 
CHAP. I.] REPRESENTATIVE AND VECTOR ANGLES. 151 
 
 AC, but by the different arc c'a^ (168), which as a vector arc has been 
 seen to be unequal thereto: although it is true that these two last 
 arcs, AC and c'a', are always equally long^ and therefore subtend 
 equal angles at the centre o of the unit sphere; so that we may write, 
 generally, for any two versors (or indeed for any two quaternions)* 
 q and q" , the formula, 
 
 Lq{q":q) = Lq''. 
 
 174. Another mode of Representation of Versors, or rather two 
 such new modes, although intimately connected with each other, 
 may be briefly noticed here. 
 
 1st. We may consider the angle aob, at the centre o of the unit- 
 sphere, when conceived to have not only a definite quantity, but also 
 a determined^Zawe (110), and a given direction therein (as indicated 
 by one of the curved arrows in Fig. 39, or by the arrow in Fig. 33), 
 as being what may be called by analogy a Vector- Angle ; and may 
 say that it represents, or that it is the Representative Angle of, the 
 Versor ob : oa, where oa, ob are radii of the unit- sphere. 
 
 Ilnd. Or we may replace this rectilinear angle aob at the centre, 
 by the equal Spherical Angle ac^b, at what may be * 
 called the Positive Pole of the representative arc ab ; 
 so that c^A and c^b are quadrants; and the rotation, 
 at this pole c', from the first of these two quadrants 
 to the second (as seen from a point outside the 
 sphere), has the direction which has been selected 
 (111, 127) for the positive one, as indicated in the 
 annexed Figure 44: and then we may consider this 
 spherical angle as a new Angular Representative of the same versor q, 
 or ob : OA, as before. 
 
 175. Conceive now that after employing ?k first spherical trian- 
 gle ABC, to construct (as in 167) the multiplication of any one given 
 versor q, by any other given versor q' , we form a second or polar 
 triangle, of which the corners a', b', c' shall be respectively (in the 
 sense just stated) tha positive poles of the three successive sides, bc, 
 CA, AB, of the former triangle ; and that then we pass to a third tri- 
 angle A^B^'c', as part of the same lune ^'^" with the second, by tak- 
 ing for -&" the point diametrically opposite to b' ; so that ^" shall be 
 
 * It will soon be seen that several of the formulae of the present Section, respect- 
 ing the multiplication and division of versors^ considered as radial quotients (151), 
 require little or no modification, in the passage to the corresponding operations on 
 quaternions, considered as general quotients of vectors (112). 
 
152 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 Fi-r. 45. 
 
 the negative pole of the arc CA, or the positive pole of what was lately- 
 called (167) the transvector-arc Ac: also let 
 c" be, in like manner, the point opposite 
 to c' on the unit sphere. Then we may not 
 only write (comp. 129), 
 Ax. 5' = oc^ Ax. §'' = oa', Ax. q'q = ob'\ 
 but shall also have the equations, 
 lq = b'^c^a^ Z g' = c' a'b^', Z q'q = C^'b^'a' ; 
 
 these three spherical angles^ namely the ivm 
 base-angles at c' and a\ and the external 
 vertical angle at b''', of the new or third 
 triangle a''b''V/, will therefore represent^ re- 
 spectively, on the plan of 174, II., the mul- 
 tiplicand^ q, the multiplier^ q\ and the pro- 
 duct, q'q. (Compare the annexed Figure 45.) 
 
 176. Without expressly referring to the former triangle abc, 
 we can connect this last construction of multiplication of versors (175) 
 with the general formula (107), as follows. 
 
 Let a and y3 be now conceived to be tw^o unit-tangents'^ to the 
 sphere at c', perpendicular respectively to 
 the two arcs c^b^' and c'a^ and drawn to- 
 wards the same sides of those arcs as the 
 points a' and b' respectively; and let two 
 other unit-tangents, equal to these, and 
 denoted by the same letters, be drawn (as 
 in the annexed Figure 45, his) at the points 
 B^' and a', so as to be normal there to the 
 same arcs c'b'^ and c'a', and to fall towards 
 the same sides of them as before. Let also 
 two other unit-tangents, equal to each b'/ 
 other, and each denoted by 7, be drawn at 
 the two last points b" and a', so as to be both perpendicular to the 
 arc a^b^^ and to fall towards the same side of it as the point c'. Then 
 (comp. 174,11.) the two quotients, (3 : a and 7 : /3, will be equal to the 
 two versors, q and q, which were lately represented (in Fig. 45) by the 
 
 * By an unit tangent is here meant simply an unit line (or unit vector, 129) so 
 drawn as to be tangential to the unit-sphere^ and to have its origin, or its initial 
 point (1), on the surface of that sphere, and not (as we have usually supposed) at 
 the centre thereof. 
 
 Fig. 46, bis. 
 
CHAP.T.] DEPENDENCE OF PRODUCT ON ORDER OF FACTORS. 153 
 
 two base angles, at c' and a', of the spherical triangle a'b'^'c'; the pro- 
 duct, q'q, of these two versors, is therefore (by 107) equal to the third 
 quotient, 7 : « ; and consequently it is represented, as before, by the 
 external vertical angle c"b"a.' of the same triangle, which is evidently 
 equal in quantity to the angle of this third quotient, and has the same 
 axis ob", and the same direction of rotation, as the arrows in Fig. 45, 
 his, may assist to show. 
 
 177. In each of the two last Figures, the internal vertical angle 
 at B^' is thus equal to the Supplement, tt - l q'q, of the angle of the 
 product; and it is important to observe that the corresponding ro- 
 tation at the vertex b", from the side b^'a' to the side b'^c', or (as we 
 may briefly express it) from the point k' to the point o', is, positive; a 
 result which is easily seen to be a general one, by the reasoning of 
 the foregoing Article.* We may then infer, generally, that when 
 the multiplication of any two versors is constructed hya spherical trian- 
 gle, of which the two ba^e angles represent (as in the two last Articles) 
 t\iQ factors, while the external vertical angle represents t\\Q product, 
 then the rotation round the axis (ob'O of that product q'q, from the 
 axis (oa') of the multiplier q', to the axis (oc^) of the multiplicand q, is 
 positive: whence it follows that the rotation round the axis Ax. q' 
 of the multiplier, from the axis Ax. q of the multiplicand, to the 
 axis Ax. q'q of the product, is also positive. Or, to express the 
 same thing more fully, since the only rotations hitherto considered 
 have hQQU plane ones (as in 128, &c.), we may say that if the two 
 latter axes be projected on a plane perpendicular to the former, so as 
 still to have a common origin o, then the rotation round Ax. q\ 
 from the projection of Ax. q to the projection of Ax. q'q, will be di- 
 rected (with our conventions) towards the right hand. 
 
 178. We have therefore thus a new mode of geometrically 
 exhibiting the inequality of the two products^ q'q and ^5-', o{two 
 diplanar versors (168), when taken a3 factors in two different 
 orders. For this purpose, let 
 
 Ax. 5-= OP, Ax.5'=0Q, Ax.qq = OR; 
 
 and prolong to some point s the arc PR of a great circle on the 
 unit sphere. Then, for the spherical triangle pqr, by prin- 
 
 * If a person be supposed to stand on the sphere at b", and to look towards the 
 arc a'c', it would appear to him to have a right-handed direction, which is the one 
 here adopted as positive (127). 
 
154 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 ciples lately established, we shall have (comp. 175) the follow- 
 ing values of the two internal base angles at p and q, and of 
 the external vertical angle at ii : 
 
 RPQ = Lq\ PQR = L q ', SRQ = L q'q ; 
 
 and the rotation at q, from the side qp to the side qr will be 
 right-handed. Let fall an arcual perpendi- g 
 
 cular, RT, from the vertex r on the base pq, 
 and prolong this perpendicular to r', in such 
 a manner as to have 
 
 /^ RT = '^ tr' ; 
 also prolong pr' to some point s'. We shall 
 then have a new triangle pqr', which will 
 be a sort of reflexion (comp. 138) of the old 
 one with respect to their common base pq ; 
 and this new triangle will serve to construct 
 the new product^ qq. For the rotation at p 
 
 Fig. 46. 
 
 from PQ to pr' will be right-handed, as it ought to be ; and 
 we shall have the equations, 
 
 qpr' = Z^; r'qp = Z5''; qr's' = Z^'^''; on' = Ax.qq \ 
 
 so that the new external and spherical angle, qr's', will repre- 
 sent the new versor, qq\ as the old angle srq represented the 
 old versor, q'q, obtained from a different order of the factors. 
 And although, no doubt, these two angles, at r and r', are 
 always equal in quantity, so that we may establish (comp. 1 73) 
 the general formida, 
 
 Lqq^Lqq, 
 
 yet as vector angles (174), and therefore as representatives of 
 versors, they must be considered to be unequal: because they 
 have different planes, namely, the tangent planes to the sphere 
 at the two vertices r and r'; or the two planes respectively 
 parallel to these, which are drawn through the centre o. 
 
 179. Division of Versors (comp. 172) can be constructed by 
 means oi Representative Angles (174), as well as by representative arcs 
 (162). Thus to divide q" by q, or rather to represent such division 
 geometrically, on a plan entirely similar to that last employed for 
 
CHAP. I.] CONICAL ROTATION OF AXIS OF VERSOR. 155 
 
 multiplication, we have only to determine the two points P and r, 
 in Fig. 46, by the two conditions, 
 
 and then to find a third point q by the two angular equations, 
 
 RPQ =Lq, QRP ^tr- L q", 
 
 the rotation round p from PR towards pq being positive ; after which 
 we shall have, 
 
 A-K. {q" \ q)=OQ,\ L{q" '.q) = VQ,Vi. 
 
 (1.) Instead of conceiving, in Fig. 46, that the dotted line rtk', which connects 
 the vertices of the two triangles, with pq for their common base (178), is an arc of 
 a great circle, perpendicularly bisected by that base, we may imagine it to be an arc 
 of a small circle^ described with the point p for its positive pole (comp. 174, II.). 
 And then we may say that the passage (comp. 17 B) from the versor q'% or qq, to 
 the unequal versor q(q" : 9), or qq\ is geometrically performed by a Conical Rota- 
 tion of the Axis Ax. 5", round the axis Ax. 7, through an angle ~2 Lq^ without 
 any (jjuantitative) change of the angle Lq"\ so that we have, as before, the general 
 formula (comp. again 173), 
 
 L q (9" : 9) = ^ 9". 
 
 (2.) Or if we prefer to employ the construction of multiplication and division by 
 representative arcs, which Fig. 43 was designed to illustrate, and conceive that a 
 new point c" is determined in that Figure by the condition ^ a'c" = "^ c'a', we may 
 then say that in the passage from the versor q'\ which is represented by ac, to the 
 versor q (5" : 5), represented by c'a' or by a'c", the representative arc of q" is made 
 to move, without change of length, so as to preserve a constant inclination* to the 
 representative arc AB ofq, while zYs initial point describes the double of that arc A^, 
 in passing from a to a'. 
 
 (3.) It maybe seen, by these few Examples, that if, even independently of some 
 new characteristics of operation, such as K and U, new combinations of old symbols, 
 such as q (q" : q), occur in the present Calculus, which are not wanted in Algebra, 
 they admit for the most part of geometrical interpretations, of an easy and interest - 
 ing kind ; and in fact represent conceptions, which cannot well be dispensed with, 
 and which it is useful to be able to express, with so much simplicity and conciseness. 
 (Compare the remarks in Art. 161 ; and the sub-articles to 182, 145.) 
 
 180. In connexion with the construction indicated by the 
 two Figures 45, it may be here remarked, that if abc be any 
 spherical triangle, and if a', b', c' be (as in 175) the positive 
 poles of its three successive sides, bc, ca, ab, then the rotation 
 (comp. 177, 179) round a' from b' to c', or that round b' from 
 
 * In a manner analogous to the motion of the equator on the ecliptic, by luni- 
 ioldv precession, in astronomy. 
 
156 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 c' to A, &c., IS positive. The easiest way, perhaps, of seeing 
 the truth of this assertion, is to conceive that if the rotation 
 round a from b to c be not already positive, we make it such, 
 by passing to the diametrically opposite triangle on the sphere, 
 which will not change the poles a', b', c'. Assuming then that 
 these poles are thus the near ones to the corresponding corners 
 of the given triangle, we arrive without any difficulty at the 
 conclusion stated above : which has been virtually employed 
 in our construction of multiplication (and division) of versors, 
 by means of Representative Angles (1 75, 176) ; and which may 
 be otherwise justified (as before), by the consideration of the 
 unit-tangents of Fig. 45, Ms. 
 
 (1.) Let then a, j3, y be any three given unit vectors, such that the rotation 
 round the first, from the second to the third, is positive (in the sense of Art. 177); 
 and let a', /3', y' be three otlier unit vectors, derived from these by the equations, 
 
 a'=Ax. (y:/3), /3'= Ax. (a : y), y' = Ax.(/3 : a) ; 
 then the rotation round a, from /3' to y', will be positive also; and we shall have 
 the converse formulae, 
 
 a = Ax.(y':/5'), ^ = Ax. (a: y'), y = Ax . (/3' : a')- 
 
 (2.) If the rotation round a from /3 to y were given to be negative, a', /3', y' 
 being still deduced from those three vectors by the same three equations as before, 
 then the signs of a, /3, y would all require to be changed, in the three last (or reci- 
 procal) formulae ; but the rotaticm round a', from /3' to y', would still be positive. 
 
 (3.) Before closing this Section, it may be briefly noticed, that it is sometimes 
 convenient, from motives of analogy (comp. Art. 5), to speak of the Transvector- 
 Arc (167), which has been seen to represent a. product of two versors. as being the 
 Arcual, Sum of the two successive vector-arcs, which represent (on the same plan) 
 the factors ; Provector being still said to be added to Fector : but the Order of such 
 Addition of Diplanar Arcs being not now indifferent (168), as the corresponding 
 order had been early found (in 7) to be, when the vectors to be added were right 
 lines. 
 
 (4.) We may also speak occasionally, by an extension of the same analogy, of 
 the External Vertical Angle of a spherical triangle, as being the Spherical Sum of 
 the two Base Angles of that triangle, taken in a suitable order of summation (comp. 
 Fig. 46); the Angle which represents (174) the Multiplier being then said to be 
 added (as a sort of Angular Provector) to that other Vector-Angle which represents 
 the Multiplicand; whilst what is here called the sum of these two angles (and is, 
 with respect to them, a species of Transvector- Angle) represents, as has been proved, 
 the Product. 
 
 (5.) This conception of angular transvaction becomes perhaps a little more clear, 
 when (on the plan of 174, I.) we assume the centre o as the common vertex of three 
 angles aob, boc, aoc, situated generally in three different planes. For then we may 
 
CHAP. 
 
 '•] 
 
 SYSTEM OF THREE RIGHT VERSORS. 
 
 157 
 
 conceive a revolving radius to be either carried by two successive angular motions, 
 frnm OA to OB, and thence to oc ; or to be transported immediately, by one such 
 motion, from the Ji?-st to the third position. 
 
 (6.) Finally, as regards the construction indicated by Fig. 45, bis, in which tan- 
 gents instead of radii were employed, it may be well to remark distinctly here, that 
 a'b"c', in that Figure, may be ani/ given spherical triangle, for which the rotation 
 round b" from a' to c' is positive (177); and that then, if the two factors, q and q', 
 be defined to be the two versors, of which the internal angles at c' and a' are (in the 
 sense of 174, II.) the representatives, the reasonings of Art. 176 will prove, without 
 necessarily referring, even in thought, to any other triangle (such as abc), that the 
 external angle at b" is (in the same sense) the representative of the product, q'q, as 
 before. 
 
 Section 10. — On a System of Three Right Versors^ in Three 
 Rectangular Planes ; and on the Laws of the Symbols, 
 
 181. Suppose that oi, oj, ok are any three given and co- 
 initial but rectangular unit-lines, the rotation round the first 
 from the second to the third being positive ; and let oi', oj, 
 ok' be the three unit- vectors respectively opposite to these, so 
 
 that 
 
 Ol' = -OI, Oj'-=-OJ, ok'=-ok. 
 
 Let the three new symbols i,j, k denote a system (comp, 172) 
 of three right versors, in three mutually rectangular planes, 
 with the three given lines for their respective axes; so that 
 
 Ax.i=oj, Ax.j=oj, Ax.k-OK, 
 and 
 
 i = ok:oj, J=oi:ok, A=oj:oi, 
 as Figure 47 may serve to illustrate. 
 We shall then have these other expres- 
 sions for the same three versors : 
 i = o y : OK = ok' 
 
 ^ = OK : 01 =01 
 k = oi : OJ = OJ 
 
 Fig. 47. 
 
 OJ = OJ : OK ; 
 
 ok'= ok: oi' ; 
 
 oi' = 01 : oj' ; 
 
 while the three respectively opposite versors may be thus ex- 
 pressed : 
 
 - z = oj : OK = OK : OJ = oj : ok 
 
 = ok: oj 
 
 -j = OK : 01 = oi' : OK = ok' : oi' 
 
 = 01 : ok 
 
 - A = 01 : OJ = OJ : 01 = oi' : oj' 
 
 = OJ : oi'. 
 
 /< 
 
 / 
 
158 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 And from the comparison of these different expressions seve- 
 ral important symbolical consequences follow, which it will be 
 worth while to enunciate separately here, although some of 
 them are virtually included in the results of former Sections. 
 
 182. In \hQjirst place, since 
 
 i^ = (oj' : ok) . (OK : oj) = oj' : OJ, &c., 
 we deduce (comp. 148) the following equal values for the 
 squares of the new symbols : 
 
 L..z^ = -1; / = -l; k' = -l; 
 as might indeed have been at once inferred (154), from the 
 circumstance that the three radial quotients (146), denoted here 
 by hj, ^3 are all right versors (181). 
 
 In the second place, since 
 
 ij= (oj:ok') .(ok':oi) = oj : oi, &c., 
 we have the following values for the products of the same three 
 symbols, or versors, when taken iioo hy two, and in a certain 
 order of succession (comp. 168, 171) : 
 
 II. . . ij= k] jk = i; ki =j. 
 But in the third place (comp. again 171), since 
 j .i= (ox : ok) . (ok : oj) = oi : oj, &c., 
 
 we have these other and contrasted formulae, for the binary 
 products of the same three right versors, when taken as fac- 
 tors with an opposite order : 
 
 III. . .ji=-k; kj = -i; ik = -j. 
 
 Hence, while the square of each of the three right versors, de- 
 noted by these three new symbols, ijk, is equal (154) to nega-^ 
 tive unity, the product of any two of them is 
 equal either to the third itself, or to the oppo- 
 site (171) of that third versor, according as 
 the multiplier precedes ov follows the multipli- 
 cand, in the cyclical succession, 
 
 h i, k, i, j\ . . . 
 which the annexed Figure 47, bis, may give some help towards 
 remembering. 
 
CHAP. I.] LAWS OF THE SYMBOLS, I, J, K. 159 
 
 (1.) To connect such multiplications ofi,j, k with the theory of representative 
 arcs (162), and of representative angles (174), we may regard any one of the four 
 quadrantal arcs, JK, Kj', j'k', k'j, in Fig, 47, or any one of the four spherical right 
 angles, jik, kij', j'ik', k'ij, which those arcs subtend at their common pole i, as re- 
 presenting the versor i ; and similarly for j and k, with the introduction of the point 
 i' opposite to I, which is to be conceived as being at the back of the Figure. 
 
 (2.) The squaring of i, or the equation i^ = - 1, comes thus to be geometrically 
 constructed by tbe doubling (comp. Arts. 148, 154, and Figs. 41, 42) of an arc, or of 
 an angle. Thus, we may conceive the quadrant kj' to be added to the equal arc jk, 
 their sum being the great semicircle jj', which (by 166) represents an inversor (153), 
 or negative unity considered as a, factor. Or we may add the right angle kij' to the 
 equal angle JIK, and so obtain a., rotation through two right angles at the jooZe i, or 
 at the centre o; which rotation is equivalent (comp. 154, 174) to an inversion of 
 direction, or to a passage from the radius OJ, to the opposite radius oj'. 
 
 (3.) The midtiplication ofj hy i, or the equation ij = k, may in like manner 
 be arcually constructed, by the addition of k'j, as a provector-arc (167), to ik' as 
 a vector-arc (162), giving ij, which is a representative of ^, as the transvector-arc, 
 or arcual-sum (180, (3.) ). Or the same multiplication may be angularly con- 
 structed, with the help of the spherical triangle ijk ; in which the base-angles at I 
 and J represent respectively the multiplier, i, and the multiplicand, j, the rotation 
 round l from j to k being positive : while their spherical sum (180, (4.)), or the ex- 
 ternal vertical angle at K (comp. 175, 176), represents the same product, k, as 
 before. 
 
 (4.) The contrasted multiplication of i hy j, or of J into* i, may in like manner 
 be constructed, or geometrically represented, either by the addition of the arc ki, as 
 a new provector, to the arc jk as a new vector, which new process gives Ji (instead 
 of ij) as the new transvector ; or with the aid of the new triangle ijk' (comp. Figs. 
 46, 47), in which the rotation round i from j to the new vertex k' is negative, so 
 that the angle at i represents now the multiplicand, and the resulting angle at the 
 new' pole k' represents the new and opposite product, ji = - k. 
 
 183. Since we have thus ji = - ij (as we had q'q = - qq in 
 171), we see that the laws of combination of the neio symbols^ 
 i,j, k, are not in all j^espects the same as the corresponding 
 laws in algebra; since the Commutative Property of Multipli- 
 cation, or the convertibility (169) of the places o^ \k\Q factors 
 without change of value of the product, does not here hold 
 good: which arises (168) from the circumstance, that the 
 factors to be combined are here diplanar versor s (181). It is 
 therefore important to observCj that there is a respect in which 
 
 * A multiplicand is said to be multiplied hy the multiplier ; -while, on the other 
 hand, a multiplier is said to be multiplied into the multiplicand : a distinction of this 
 sort between the tivo factors being necessary, as we have seen, for quaternions, 
 although it is not needed for algebra. 
 
160 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 the laws of i, j, k agree with usual and algebraic laws : namely, 
 in the Associative Property of Multiplication ; or in the pro- 
 perty that the new symbols always obey the associative for- 
 mula (comp. 9), 
 
 whichever of them may be substituted for z, for ic, and for X ; 
 in virtue of which equality of values we may omit the pointy in 
 any such symbol of a ternary product (whether of equal or of 
 unequal factors), and write it simply as lk\. In particular 
 we have thus, 
 
 i.jk = i,i = i'^ = ~ \ ; ij .k = k.k = k^ = - \ ; 
 
 or briefly, 
 
 ijk = -l. 
 
 We may, therefore, by 182, establish the following important 
 
 Formula : 
 
 p=f^k^ = ijk = -l ; (A) 
 
 to which we shall occasionally refer, as to " Formula A," and 
 which we shall find to contain (virtually) all the laws of the 
 symbols ijk, and therefore to be a sufficient symbolical basis 
 for the whole Calculus of Quaternions i* because it will be 
 shown that every quaternion can he reduced to the Quadrino- 
 mial Form, 
 
 q=w + ix +jy + kz, 
 
 where w, x, y, z compose a system of four scalar s, while 2, j, k 
 are the same three right versors as above. 
 
 (1.) A direct proof of the equation, ijk = — 1, may be derived from the definitions 
 of the symbols in Art. 181. In fact, we have only to remember that those defini- 
 tions were seen to give, 
 
 * This formula (A) was accordingly made the basis of that Calculus in the first 
 communication on the subject, by the present writer, to the Royal Irish Academy in 
 1843 ; and the letters, i, 7', k, continued to be, for some time, the only peculiar sym- 
 bols of the calculus in question. But it was gradually found to be useful to incor- 
 porate with these a few other notations (such as K and U, &c.), for representing 
 Operations on Quaternions. It was also thought to be instructive to establish the 
 principles of that Calculus, on a more geometrical (or less exclusively symbolicaT) 
 foundation than at first ; which was accordingly afterwards done, in the volume en- 
 titled : Lectures on Quaternions (Dublin, 1853) ; and is again attempted in the pre- 
 sent work, although with many differences in the adopted plan of exposition, and in 
 the applications brought forward, or suppressed. 
 
CHAP. I.] LAWS OF THE SYMBOLS, I, J, K. 161 
 
 t = oj' : OK, j = ok: oi', ^ = oi' : oj ; 
 and to observe that, by the general fornmla of multiplication (107), whatever four 
 lines may be denoted by a, /3, y, d, we have always, 
 
 y' (3 a y a a ^ a y/^a' 
 or briefly, as in algebra, 
 
 y /3 a a 
 the point being thus omitted without danger of confusion : so that 
 
 ijk = oj' : OJ = — 1, as before. 
 Similarly, we have these two other ternary products : 
 
 jki = (ok' : ot) (oi : oj') (oj' : ok) = ok' : ok = — 1 ; 
 kij = (oi' : oj) (oj : ok') (ok' : oi) = oi' : oi = - 1 . 
 (2,) On the other hand, 
 
 kji— (oj : oi) (oi : ok) (ok : oj) =oj : oj = + 1 ; 
 and in like manner, 
 
 ikj— + 1, and jik = + 1. 
 
 (3.) The equations in 182 give also these other ternary products, in which th» 
 law of association of factors is Still obeyed : 
 
 i . ij = ik = -j = iy = a .j\ iij =-j] 
 
 i .ji = i.-k = -ik=j = ki = ij . ?, iji = +j ; 
 i.jj=i.-l=-i = kj = ij.j, VJ = -i; 
 
 with others deducible from these, by mere cyclical permutation of the letters, on the 
 plan illustrated by Fig. 47, Ms. 
 
 (4.) In general, if the Associative Law of Combination exist for ani/ three 
 symbols whatever of a given class, and for a giiwn mode of combination, as for addi- 
 tion of lines in Art. 9, or for multiplication of ijk in the present Article, the same law 
 exists for any fotir (or more) symbols of the same class, and combinations of the same 
 kind. For example, if each of the four letters t, /c, X, /* denote some one of the three 
 symbols i, j, k (but not necessarily the same one), we have the formula, 
 I . (cX/i = t . K . XjLl = tK . X/i = tK . X . /f = ifcX . n = tjcX/A. 
 (5.) Hence, any multiple (or complex') product of the symbols ijk, in any manner 
 repeated, but taken in one given order, may be interpreted, with one definite result, 
 by any mode of association, or of reduction to partial factors, which can be performed 
 without commutation, or change of place of the given factors. For example, the 
 symbol ijkkji may be interpreted in either of the two following (among other) ways : 
 ij.kk.ji = ij.-ji = i.~j'Ki = ii = - 1; ijk.kji=-l. 1=-1. 
 
 184. The formula (a) of 183 includes obviously the three equa- 
 tions (I.) of 182. To show that it includes also the six other 
 equations, (H.)? (m*)' ^^ ^^^ ^^^^ cited Article, we may observe that 
 it gives, with the help of the associative principle of multiplication 
 (which may be suggested to the memory by the absence of the jpomi 
 in the symbol tjk), 
 
 Y 
 
162 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 ij =i-ij .kk = -ijk.k = + k', jk = -i. ijk = + i\ 
 ji =j .jk]=fk = -k', ik = i.ij = i V = -j ; 
 
 k' = V • ; = «;'^ = - « ; ^« = - ^^!/ = -P = + J- 
 
 And then it is easy to prove, without any reference to geometry/, if the 
 foregoing laws of the symbols be admitted, that we have also, 
 
 jki = kij = - 1 , kji =jik = ikj = + 1 , 
 
 as otherwise and geometrically shown in recent sub-articles. It may- 
 be added that the mere inspection of the formula (a) is sufficient to 
 show that the tkree'^ square roots of negative unity j denoted in it by 
 /, j, k, cannot be subject to all the ordinary rules of algebra : because 
 that formula gives, at sight, 
 
 Pfk'=(-iy^-l=-{ijky; 
 
 the non-commutative character (183) , of the multiplication of such roots 
 among themselves, being thus put in evidence. 
 
 Section 11. — On the Tensor of a Vector, or of a Quaternion ; 
 and on the Product or Quotient of any two Quaternions. 
 
 185. Having now sufficiently availed ourselves, in the two 
 last Sections, of the conceptions (alluded to, so early as in the 
 First Article of these Elements) of a vector-arc (162), and of 
 a vector-angle (174), in illustration^ of the laws o^ multiplica- 
 tion and division of vers or s of quaternions ; we propose to re- 
 turn to that use of the word. Vector, with which alone the 
 First Book, and the first eight Sections of this First Chapter 
 of the Second Book, have been concerned : and shall therefore 
 henceforth mean again, exclusively^ by that word " vector," a 
 Directed Right Line (as in 1). And because we have already 
 considered and expressed the Direction of any such line, by 
 
 * It is evident that — i, —j, — k are also, on the same principles, values of the 
 symbol V — 1; because they also are right versors (153); or because (- gy=q^. 
 More generally (comp. a Note to page 131), if a:, y, z be any three scalers which sa- 
 tisfy the condition x^ i- 1/"^ + z"^ = 1, it will be proved, at a later stage, that 
 
 (ix-\-jt/ + kzy = -l. 
 
 f One of the chief uses of such vectors, in connexion with those laws, has been 
 to illustrate the non-com>Hutative property (1G8) of multiplication of versors, by ex- 
 hibiting a corresponding property of what has been called, by analogy to the earlier 
 operation of the same kind on linear vectors (5), the addition of arcs and angles on 
 a sphere. Compare 180, (3.), (4.). 
 
CHAP. I.J TENSOR OF A VECTOR. 163 
 
 introducing the conception and notation (155) of the Unit- 
 Vector, Ua, which has the same direction with the line a, and 
 which we have proposed (156) to call the Versor of that Vec- 
 tor, a ; we now propose to consider and express the Length of 
 the same line a, by introducing the new name Tensor, and the 
 new symbol,* Ta; which latter symbol we shall read, as the 
 Tensor of the Vector a : and shall define it to be, or to denote, 
 the Number (comp. again 155) which represents the Length of 
 that line a, by expressing the Ratio which that length bears 
 to some assumed standard, or Unit (128). 
 
 186. To connect more closely these two conceptions, of 
 the versor and the tensor of a vector, we may remember that 
 when we employed (in 155) the letter a as a temporary sym- 
 bol for the number which thus expresses the length of the line 
 a, we had the equation, Ua = a : «, as one form of the defini- 
 tion of the unit-vector denoted by Ua. We might therefore 
 have written also these two other forms of equation (comp. 15, 
 
 16), 
 
 a-a.\Ja, a = a'.JJa, 
 
 to express the dependence of the vector, a, and of the scalar, 
 
 a, on each other, and on what has been called (156) the versor, 
 
 Ua. For example, with the construction of Fig. 42, bis (comp. 
 
 161, (2.) ), we may write the three equations, 
 
 « = OA : oa', b = OB : ob', c = oc : oc', 
 
 if «, b, c be thus the three positive scalars, which denote the 
 
 lengths of the three lines, oa, ob, oc ; and these three scalars 
 
 may then be considered as factors, or as coefficients (12), by 
 
 which the three unit-vectors Ua, Uj3, Uy, or oa', ob', oc' (in 
 
 the cited Figure), are to be respectively multiplied (15), in 
 
 order to change them into the three other vectors a, j3, y, or 
 
 OA, OB, oc, by altering their lengths, without any change in 
 
 their directions. But such an exclusive Operation, on the 
 
 Length (or on the extension) of aline, may be said to be an Act 
 
 of Tension ;t as an operation on direction alone may be called 
 
 (comp. 151) an act of version. We have then thus a motive 
 
 * Compare the Note to Art. 155. 
 
 t Compare the Note to Art. 156, in page 135. 
 
164 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 for the introduction of the name, Tensor, as applied to the 
 positive number which (as above) represents the length of a 
 line. And when the notation Ta (instead of a) is employed 
 for such a tensor, we see that we may write generally, for any 
 vector a, the equations (compare again 15, 16) : 
 
 Ua = a : Ta ; Ta = a : Ua ; a ~ Ta . Ua = Ua . Ta. 
 
 For example, if a be an unit-vector, so that Ua = a (160), 
 then Td = 1 ; and therefore, generally, whatever vector may 
 be denoted by a, we have always, 
 
 * TUa=l. 
 For the same reason, ivhatever quaternion may be denoted by 
 q, we have always (comp. again 160) the equation, 
 T(Ax.g)=l. 
 
 (1.) Hence the equation 
 
 where p = op, expresses that the locus of the variable point p is the surface of the 
 unit sphere (128). 
 
 (2.) The equation Tp = Ta expresses that the locus of p is the spheric surface 
 with o for centre, which passes through the point a. 
 
 (3.) On the other hand, for the sphere through o, which has its centre at A, we 
 have the equation, ., . 7> 
 
 T(p-a) = Ta; ■" /' r. ^^ " "' '^ 
 
 which expresses that the lengths of the two lines, ap, ao, are equal. , , ' [ ^^H^) ' 
 (4.) More generally, the equation, 
 
 T (p - a) = T (/3 - a), 7 (M\ r y.. 4-^- ^ 
 
 expresses that the locus of p is the spheric surface through b, which has its centre 
 at A. 
 
 (5.) The equation of the Apollonian* Locus, 145, (8.), (9.), may be written 
 under either of the two following forms : 
 
 T(p-a2a)=aT(p-a); Tp=aTa; \^^.^.^ „ ^ ' 
 
 from each of which we shall find ourselves able to pass to the other, at a later stage, 
 by general Rules of Transformation, without appealing to geometry (covv^. 145, (10.)), 
 (6.) The equation, 
 
 T(p + a) = T(p-a), 
 
 expresses that the locus of p is the plane through o, perpendicular to the line oa ; 
 because it expresses that if oa' = - oa, then the point p is equally distant from the 
 two points A and a'. It represents therefore the same locus as the equation, 
 
 * Compare the first Note to page 128. 
 
CHAP. I.] GEOMETRICAL EXAMPLES. 165 
 
 or as the equation, 
 
 Z^=^, of 132, (L); 
 a i 
 
 ^ + K^=0, of 144, (L); 
 a a 
 
 or as 
 
 f U^Y=-1, of 161, (7.); 
 
 or as the simple geometrical formula, p -L a (129). And in fact it will be found 
 possible, by General Rules of this Calculus, to transform any one of these /ue for- 
 mulae into any other of them ; or into this sixth form, 
 
 a 
 
 which expresses that the scalar part* of the quaternion - is ze/o, and therefore that 
 
 a 
 this quaternion is a right quotient (132). 
 (7.) In like manner, the equation 
 
 T(p-/3)=T(p-a) 
 expresses that the locus of p is the plane which perpendicularly bisects the line ab ; 
 because it expresses that p is equally distant from the two points A and b. 
 
 (8.) The tensor, T«, being generally a positive scalar, but vanishing (as a limit) 
 with a, we have, 
 
 Txa = + xTa, according as x> or < ; 
 
 thus, in particular, 
 
 T (- a) = Ta ; and TOa = TO = 0. 
 (9.) That 
 
 T(/3 + a) = T/3+Ta, if U/3 = Ua, 
 
 but not otherwise (a and fi being any two actual vectors), will be seen, at a later 
 stage, to be a symbolical consequence from the rules of the present Calculus ; but in 
 the mean time it may be geometrically proved, by conceiving that while a = OA, as 
 usual, we make (3+ a = oc, and therefore j3 = oc — OA = ao (4) ; for thus we shall 
 see that while, iyi general, the three points o, A, c are corners of a triangle, and there- 
 fore the length of the side oc is less than the sum of the lengths of the two other 
 sides OA and ac, the former length becomes, on the contrary, equal to the latter sum, 
 in the particular case when the triangle vanishes, by the point a falling on the finite 
 line OC ; in which case, OA and AC, or a and /3, have one common direction, as the 
 equation Ua = U/3 implies. 
 
 (10.) If a and (3 be any actual vectors, and if their versors be unequal (Ua not 
 = U/3), then 
 
 T(/3 + a)<T/3 + Ta; 
 
 an inequality which results at once from the consideration of the recent triangle oac ; 
 but which (as it will be found) may also be symbolically proved, by rules of the 
 calculus of quaternions. 
 
 * Compare the Note to page 125 ; and the following Section of the present 
 Chapter. 
 
166 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (11.) If U/3 = - Ua, then T(/3 + a) = + (T/3 - Ta), according as T/3 > or < Ta ; 
 but 
 
 T (i3 + a) >+ (T/3 -Ta), if U/3no*=-Ua. 
 
 187. The quotient, Uj3 : Ua, of the versors o^ \hQ two vec- 
 tors, a and j3, has been called (in 156) the Versor of the Quo- 
 tient, or quaternion, q = ^ : a ; and has been denoted, as such, 
 by the symbol, \]q. On the same plan, we i3ropose now to 
 call the quotient, T/3 : Ta, of the tensors of the same two vec- 
 tors, the Tensor* of the Quaternion q, or (5: a, and to denote 
 it by the corresponding symbol, Tq. And then, as we have 
 called the letter U (in 156) the characteristic of the operation 
 o^ taking the versor, so we may now speak of T as the Cha- 
 racteristic of the (corresponding) Operation of taking the Ten- 
 sor^ whether of a Vector, a, or of a Quaternion, q. We shall 
 thus have, generally, 
 
 T(j3 : a) = TjS : Ta, as we had U(/3 : a) = U/3 : Ua (156) ; 
 and may say that as the versor JJq depended solely on, but 
 conversely was sufficient to determine, the relative direction 
 (157), so the tensor Tq depends on and determines the relative 
 length] (109), of the two vectors, a and /3, of which the qua- 
 ternion q is the quotient (112). 
 
 (1.) Hence the equation T- = l, like T(0 = Ta, to which it is equivalent, ex- 
 presses that the locus of p is the sphere with o for centre, which passes through the 
 point A. 
 
 * Compare the Note to Art. 109, in page 108; and that to Art. 156, in page 
 135. 
 
 f It has been shown, in Art. 112, and in the Additional Illustrations of the 
 third Section of the present Chapter (113-116), that Relative Length, as well as 
 relative direction, enters as an essential element into the very Conception of a Qua- 
 ternion. Accordingly, in Art. 117, an agreement of relative lengths (as well as an 
 agreement of relative directions) was made one of the conditions of equality, between 
 any two quaternions, considered as quotients of vectors : so that we may now say, 
 that the tensors (as well as the versors) of equal quaternions are equal. Compare 
 the first Note to page 137, as regards what was there called the quantitative element, 
 of absolute or relative length, which was eliminated from a, or from q, by means of 
 the characteristic U ; whereas the new characteristic, T, of the present Section, 
 serves on the contrary to retain that element alone, and to eliminate what may be 
 called by contrast the qualitative element, of absolute or relative direction. 
 
CHAP. I.] TENSOR OF A QUATERNION. 167 
 
 (2.) The equation comp. 186, (6.) ), 
 
 T^i-e = l, 
 p- a 
 
 expresses that the locus of p is the plane through o, perpendicular to the line oa. 
 
 (3.) Other examples of the same sort may easily be derived from the sub-arti- 
 cles to 186, by introducing the notation (187) for the tensor of a quotient, or qua- 
 ternion, as additional to that for the tensor of a vector (185). 
 
 (4.) T(/3 : a) >, =, or < 1, according as T/3 >, =, or < Ta. 
 
 (5.) The tensor of a right quotient (132) is always equal to the tensor of its in- 
 dex (133). 
 
 (6.) The tensor of a radial (146) is always positive unity ; thus we haA^e, ge- 
 nerally, by 156, 
 
 TU^ = 1; 
 and in particular, by 181, 
 
 Tt = T; = TA=l. 
 
 (7.) Txq = + xHq, according as a; > or < ; 
 
 thus, in particular, T(— g') = T5', or the tensors oi opposite quaternions are equal. 
 
 (8.) Ta; = + ar, according as x> or < ; 
 
 thus, the tensor of a scalar is that scalar taken positively. 
 
 (9.) Hence, 
 
 TTa = Ta, TTq^Tq; 
 
 80 that, by abstracting from the subject of the operation T (comp. 145, 160), we 
 may establish the symbolical equation, 
 
 T^ = TT= T 
 (10.) Because the tensor of a quaternion is generally a positive scalar, such a 
 tensor is its own conjugate (139) ; its angle is zero (131) ; and its versor (159) is 
 positive unity : or in symbols, 
 
 KTq^Tq; LTq=Oi VTq=l. 
 
 (11.) T(l:5) = T(a:i8) = Ta:T/3 = l:T5; 
 
 or in words, the tensor of the reciprocal of a quaternion is equal to the reciprocal of 
 the tensor, 
 
 (12.) Again, since the two lines, ob and ob', in Fig. 36, are equally long, the de- 
 finition (137) of a conjugate gives 
 
 TKq = Tq', 
 
 or in words, the tensors of conjugate quaternions are equal. 
 
 (13.) It is scarcely necessary to remark, that any two quaternions which have 
 equal tensors, and equal versors, are themselves equal : or in symbols, that 
 g' = q, if T:q=Tq, and XJq'^Uq. 
 
 188. Since we have, generally, 
 
 a Ta.Ua Ta Ua Ua 
 
 we may establish the two following general formulae of decom- 
 
 15 T^.u^ t/3 uii u^ T^ ^ ,„^ ,_^ 
 
168 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 position of a quaternion into two factors, of the tensor and ver- 
 sor kinds : 
 
 I. .,q=Tq.\]qi II. . . ^ = U^.T^ ; 
 
 which are exactly analogous to the formulae (186) for the cor- 
 responding decomposition of a vector, mio factors of the same 
 two kinds : namely, 
 
 r. . .a = Ta.Ua; H'. . . a = Ua . Ta. 
 
 To illustrate this last decomposition of a quaternion, q, or 
 OB : oA, into factors, we may conceive that aa' and bb' are two 
 concentric and circular, but oppositely directed arcs, which 
 terminate respectively on the two 
 lines OB and oa, or rather on the 
 longer of those two lines itself, and 
 on the shorter of them prolonged, 
 as in the annexed Figure 48 ; so 
 that oa' has the length of oa, but 
 the direction of ob, while ob', on the 
 contrary, has the length of ob, but 
 
 the direction of oa ; and that therefore we may write, by what 
 has been defined respecting versors and tensors of vectors (155, 
 156, 185, 186), 
 
 OA' = Ta.U]3; 0B'=Tj3.Ua. 
 Then, by the definitions in 156, 187, of the versor and tensor 
 of a quaternion, 
 
 JJq = U(oB : oa) = oa' : oa = ob : ob' ; 
 
 Tq =T (oB : oa) = ob' : oa = ob : oa' ; 
 
 whence, by the general formula of multiplication of quotients 
 
 (107), 
 
 I. . q = 0b: o\ = (ob : oa') . (oa' : oa) = T^' . Uq ; 
 and 
 
 II. . ^ = ob : oa = (ob : ob') . (ob' : oa) = \Jq . Tq, 
 
 as above. 
 
 189. In words, if we wish to pass from the vector a to the vec- 
 tor /3, or from the line oa to the line ob, we are at liberty either, 
 1st, to begin by turning^ from oa to oa', and then to end by stretching^ 
 
CHAP. I.] TENSOR OF A QUATERNION. 169 
 
 from oa' to ob, as Fig. 48 may serve to illustrate; or, Ilnd, to begin 
 by stretching, from oa to ob^, and end by turning, from ob' to ob. 
 The act of multiplication of a line a by a quaternion q^ considered as 
 a factor (103), which affects both length and direction (109), may 
 thus be decomposed into two distinct and partial acts, of the kinds 
 which we have called Version and Tension ; and these two acts may 
 be performed, at pleasure, in either of tvjo orders of succession. And 
 although, if we attended merely to lengths, we might be led to say 
 that th.Qtensor of a quaternion was a signless number,'^ expressive of 
 a geometrical ratio of magnitudes, yet when the recent construction 
 (Fig. 48) is adopted, we see, by either of the two resulting expres- 
 sions (188) for 1q, that there is b. propriety in treating this tensor 
 as 2, positive scalar, as we have lately done, and propose systemati- 
 cally to do, 
 
 190. Since TYiq = Tq, by 187, (12.), and UK^=1:U^, by 158, 
 we may write, generally, for any quaternion and its conjugate, the 
 two connected expressions: 
 
 L. ,q = Tq.\]q', II. .. Kq^Tq'.Uq; 
 
 whence, by multiplication and division, 
 
 III. . . ^ . K(? = (T^)2 ; IV. . . 2 : K^ = (U^)^ 
 This last formula had occurred before; and we saw (161) that in it 
 thQ parentheses might be omitted, because (J^qf =^{q^)' In like 
 manner (comp. 161, (2.) ), we have also 
 
 (T?)-^=T(s^) = Tf/, 
 parentheses being again omitted ; or in words, the tensor of the square 
 of a quaternion is always equal to the square of the tensor: as ap- 
 pears (among other ways) from inspection of Fig. 42, his, in which h 1^/ 
 the lengths of oa, ob, oc form a geometrical progression ; whence 
 
 obV ^oc T.oc / T.ob V YrpOB"' 
 
 oa; ~ oa T.oA~\T.oAy \ oa 
 
 At the same time, we see again that the product qKq of two conju- 
 gate quaternions, which has been called (145, (U.) ) their common 
 Norm, and denoted by the symbol '^q, represents geometrically the 
 square of the quotient of the lengths of the two lines, of which (when 
 considered as vectors) the quaternion q is itself the quotient (112). 
 We may therefore write generally,! 
 
 V. . . qYiq = Tq^ = l^q\ VI. . . T^ = ^/^q^ v/(^/K^). 
 
 * Compare the Note in page 108, to Art. 109. 
 f Compare the Note in page 129. 
 Z 
 
170 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (1.) We have also, by II., the following other general transformations for the 
 tensor of a quateraion : 
 
 VII. . . Tg = Kg.U5; VIII. . . Tg^ Ug . % ; 
 of which the geometrical significations might easily be exhibited by a diagram, but 
 of which the validity is sufficiently proved by what precedes. 
 
 (2.) Also (comp. 158), 
 
 (3.) The reciprocal of a quaternion, and the conjugate* of that reciprocal, may 
 now be thus expressed : 
 
 1 _ Kg _ ^_ KUg_ J_ J^ _ ±.J.. 
 
 g~"f^~K^~ Tq ~ Vq' Tq~ Tq'Vq' 
 
 q % Tg2 Tg Kg* 
 (4.) We may also write, generally, 
 
 IX.. . Kg = Tg. KUg = N5:g. 
 
 191. In general, let any two quaternions, q and^'', be con- 
 sidered as multiplicand and multiplier, and let them be re- 
 duced (by 120) to the forms j3 : a and 7 : j3 ; then the tensor 
 and versor of that third quaternion, y.a, which is (by 107) 
 their product q'q^ may be thus expressed : 
 
 I...T^'^=T(y:a) = Ty:Ta = (T7:Ti3).(T/3:Ta) = T5'.T^; 
 Il...U^V = U(7:a) = U7:Ua=(U7:Uj3).(Uj3:Ua) = U^'.U^; 
 where Tq'q and \Jqq are written, for simplicity, instead of 
 T{q\q) and U (§''.$'). Hence, in any such multiplication, the 
 tensor of the product is the product of the tensor; and the ver- 
 sor of the product is the product of the versors; the order of 
 the factors being generally retained for the latter (comp. 168, 
 &c.), although it may be varied for the former^ on account of 
 the scalar character of a tensor. In like manner, for the divi- 
 sion of any one quaternion q\ by any other q, we have the 
 analogous formulae : 
 
 III. .. T (?':?) = Tj -.Tq; IV. . . U(?' : q) = \Jq' : JJq ; 
 or in words, the tensor of the quotient of any two quater- 
 nions is equal to the quotient of the tensors ; and similarly, the 
 versor of the quotient is equal to the quotient of the versors. 
 And because multiplication and division of tensors are per- 
 formed according to the rules 0^ algebra, or rather of a/^V/^/w^- 
 
 * Compare Art. 145, and the Note to page 127. 
 
CHAP. I.] PRODUCT OR QUOTIENT OF TWO QUATERNIONS. l7l 
 
 tic (a tensor being always, by what precedes, a positive num- 
 ber), we see that the difficulty (whatever it may be) of the 
 general multiplication and division of quaternions is thus re- 
 duced to that of the corresponding operations on versors : for 
 which latter operations geometrical constructions have been 
 assigned, in the ninth Section of the present Chapter. 
 
 (1.) The two products, q'q and qq', of any two quaternions taken as factors in 
 two different orders, are equal or unequal, according as those two factors are compla- 
 nar or diplanar ; because such equality (169), or inequality (168), has been already 
 proved to exist, for the case* when each tensor is unity : but we have always 
 (comp. 178), 
 
 Hqq = Tgq\ and lq'q=l qq. 
 
 (2.) If Lq = Lq =—i then qq' = Kq'q (170) ', SO that the products of two right 
 
 quotients, or right quaternions (132), taken in opposite orders, are always conju- 
 gate quaternions. 
 
 ,(3.) If lq = /.g'='~, and Ax.^'-i- Ax.gr, then qq=-q'q, 
 
 Lqq'=Lq'q = ^, Ax. q'q -i- Ax . q, Ax . q' q -I- Ax . q' {17 1) ', 
 
 so that the product of two right quaternions, in two rectangular planes, is a third 
 right quaternion, in a plane rectangular to both ; and is changed to its oivn opposite, 
 when the order of the factors is reversed : as we had ijz=k=-ji (182). 
 
 (4.) In general, if q and q' be any two diplanar quaternions, the rotation round 
 Ax . q', from Ax . 5 to Ax . q'q, is positive (177). 
 
 (6.) Under the same condition, q\{q' : g-) is a quaternion with the same tensor, 
 and same angle, as q', but with a different axis; and this new axis. Ax .g(q' : g), 
 may be derived (179, (1.)) from the old axis. Ax . q', by a conical rotation (in the 
 positive direction) round Ax . q, through an angle = 2 Lq. 
 
 (6.) The product or quotient of two complanar quaternions is, in general, a third 
 quaternion complanar with both ; but if they be both scalar, or both right, then this 
 product or quotient degenerates (131) into a scalar. 
 
 (7.) Whether q and q' be complanar or diplanar, we have always as in algebra 
 (comp. 106, 107, 136) the two identical equations: 
 
 V. . . (g' : g) . g = ?' ; VI. . . (9' . ?) : g = q'. 
 
 (8.) Also, by 190, V., and 191, I., we have this other general formula : 
 VII. . .Ng'g = Ng'.Ng; 
 or in words, the norm of the product is equal to the product of the norms. 
 
 192. Let ^ = j3 : a, and 5'' = 7 : j3, as before ; then 
 1 : ^'^= 1 : (7 : a) =a : 7 = (« : /3) . (/3 : 7) = (1 : g) . (1 :^'); 
 so that the reciprocal of the product of any two quaternions is 
 
 * Compare the Notes to pages 148, 151. 
 
172 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 equal to the product of the reciprocals, taken in an inverted 
 
 order : or briefly, 
 
 I. . . R^''^' = R.^ . ^q\ 
 
 if R be again used (as in 161, (3.)) as a (temporary) charac- 
 teristic of reciprocation. And because we have then (by the 
 same sub-article) the symbolical equation, KU = UR, or in 
 words, the conjugate of the versor of any quaternion q is equal 
 (158) to the versor of the reciprocal of that quaternion ; while 
 the versor of a product is equal (191) to the product of the 
 versors : we see that 
 
 KU^'^ = UR^'^ = UR^ . UR^' = KU^ . KU^'. 
 But 
 
 Kq^Tq. KU(7, by 190, IX. ; and Tq'q = Tq .Tq = T^.T^', 
 
 by 191; we arrive then thus at the following other important 
 and general formula : 
 
 II. ..K|7'^ = K^.K^'; 
 or in words, the conjugate of the product of any two quater- 
 nions is equal to the product of the conjugates, taken (still) 
 in an inverted order. 
 
 (1.) These two results, I., II., may be illustrated, for versors (Tg = T$' = 1), by 
 the consideration of a spherical triangle abc (comp. Fig. 43) ; in which the sides 
 AB and BC (comp. 167) may represent q and q', the arc Ac then representing q'q. 
 For then the new multiplier 'Rq = Kq (158) is represented (162) by ba, and the new 
 multiplicand Kg' = Kg' by CB ; whence the new product, Rg.Rg'= Kg^.K^', is re- 
 presented by the inverse arc CA, and is therefore at once the reciprocal Kg'g, and the 
 conjugate Kq'q, of the old product q'q. 
 
 (2.) If q and q' be right quaternions, then Kq = -q, Kg' = — g' (by 144) ; and 
 the recent formula II. becomes, Kg'g = gg', as in 170. 
 
 (3.) In general, that formula II. (of 192) may be thus -written : 
 
 III. .. k^ = k^.k2:; 
 
 a a /3' 
 where a, j8, y may denote anj/ three vectors. 
 
 (4.) Suppose then that, as in the annexed 
 Fig. 49, we have the two following relations of in- 
 verse similitude of triangles (118), 
 
 A AOB a' BOC, A BOE a' DOB ; 
 
 and therefore (by 137) the two equations, 
 
 /3~ a' S l3' Fig. 49. 
 
CHAP. I.] CASE OF TWO RIGHT QUATERNIONS. 173 
 
 we shall have, by III., 
 
 ^=K-, or ADOCa'AOE; 
 a 
 
 so that this third formula of inverse similitude is a consequence from the other two. 
 
 (5.) If then (comp, 145, (6.) ) any two circles, -whether in one plane or in space, 
 touch one another at a point b ; and if from any point o, on the common tangent bo, 
 two secants OAC, oed be drawn, to these two circles ; the four points of section, 
 A, c, D, E, will be on one common circle : for such concircularity is an easy conse- 
 quence (through equal angles, &c.), from the last inverse similitude. 
 
 (6.) The same conclusion (respecting concircularity, &c.) may be otherwise and 
 geometrically drawn, from the equality of the two rectangles, AOC and doe^ each 
 being equal to the square of the tangent ob ; which may serve as an instructive «*k^ 
 verification of the recent formula III., and as an example of the consistency of the 
 results, to which calculations with quaternions conduct. 
 
 (7.) It may be noticed that the construction would in general give three circles, 
 although only one is drawn in the Figure ; but that if the two triangles abc and 
 DBE be situated m different planes, then these three circles, and of course ih.& five 
 points ABODE, are situated on one common sphere. 
 
 193. An important application of the foregoing general 
 theory of Multiplication and Division, is to the case of Right 
 Quaternions (132), taken in connexion with i\iQ\Y Index- Vec- 
 tors, or Indices (133). 
 
 Considering division first, and employing the general for- 
 mula of 1 06, let /3 and y be each _L a ; and let /3' and -y' be the 
 respective indices of the two right quotients, q = j3 : a, and 
 «/' = y : a. We shall thus have the two complanarities, /3' 1 1| /3, 7, 
 and 7'||| j3, 7 (comp. 123), because the four lines /3, 7, /3', y 
 are all perpendicular to a ; and within their common plane it 
 is easy to see, from definitions already given, that these four 
 lines form a proportion of vectors, in the same sense in which 
 a, (5, y, d did so, in the fourth Section of the present Chapter : 
 so that we may write the equation of quotients. 
 
 In fact, we have (by 133, 185, 187) the following relations of 
 
 length, 
 
 TjS' = Tp : Ta, T7' = T7 : Ta, and .-. T (7' : jS') = T (7 : |3) ; 
 
 while the relation of directions, expressed by the formula, 
 
 U(y:/3') = U(y:j3), or Uy : U/3' = Uy : U/3, 
 is easily established by means of the equations, 
 
174 ELEMENTS OF QUATERNIONS. [boOK II. 
 
 Z(y:y)=Z(/3':i3) = ^; Ax . (y' : 7) = Ax . (/3' :/3) = Ua. 
 
 We arrive, then, at this general Theorem (comp. again 133): 
 that ^^the Quotient of any two Right Quaternions is equal to 
 the Quotient of their Indices.''* 
 
 (1.) For example (comp. 150, 159, 181), the indices of the right versors t, j, k 
 are the axes of those three versors, namely, the lines 01, oj, ok ; and we have the 
 equal quotients, 
 
 j: » = 01 : oj' = A = OJ : oi, &c. 
 
 (2.) In like manner, the indices of - z, —J, —k are 01', oj', ok' ; and 
 
 1 : —j = oj' : 01' = A = 01 : Oj', &c. 
 
 (3.) In general the quotient of any two right versors is equal to the quotient of 
 their axes ; as the theory of representative arcs, and of their poles, may easily 
 serve to illustrate. 
 
 1 94. As regards the multiplication of two right quaternions, 
 in connexion with their indices, it may here suffice to observe 
 that, by 106 and 107, the product 7 : a = (y : j3) . (j3 : a) is equal 
 (comp. 136) to the quotient, (7 : i3) : (a : /3) ; whence it is easy 
 to infer that ''the Product, q'q, of any two Right Quaternions, 
 is equal to the Quotient of the Index of the Multiplier, q, di- 
 vided by the Index of the Reciprocal of the Multiplicand, q" 
 
 It follows that the plane, whether of the product or of the 
 quotient of two right quaternions, coincides with the plane of 
 their indices ; and therefore also with the plane of their axes ; 
 because we have, generally, by principles already established, 
 the transformation, 
 
 if Z 5' = -, then Index of q = T5' . Ax . q, 
 
 * We have thus a new point of agreement, or of connexion, between right qua- 
 ternions, and their index-vectors, tending to justify the ultimate assumption (not yet 
 made), of equality betAveen the former and the latter. In fact, we shall soon prove 
 that the index of the sum (or difference), of any two right quotients (132), is equal to 
 the sum (or difference) of their indices ; and shall find it convenient subsequently to 
 interpret ilvQ product (5a of any two vectors, as being the quaternion-product (194) 
 of the two right quaternions, of which those two lines are the indices (133): after 
 which, the above-mentioned assumption of equality will appear natural, and be found 
 to be useful. (Compare the Notes to pages 119, 136.) 
 
CHAP. I.] SUM OF TWO QUATERNIONS. 175 
 
 Section 12. — Oii the Sum or Difference of any tico Quater- 
 nions ; and on the Scalar (or Scalar Part) of a Quater- 
 nion. 
 
 195. The Addition of any given quaternion q^ considered 
 as a geometrical quotient ov fraction (101), to any other given 
 quaternion q^ considered also as a fraction, can always be ac- 
 complished by the first general formula of Art. 106, when these 
 two fractions have a common denominator ; and if they be not 
 already ^iven as having such, they can always be reduced so as 
 to have one, by the process of Art. 120. And because the ad- 
 dition of any two lines was early seen to be a commutative ope- 
 ration (7, 9), so that we have always y + /3 = )3 + y, it follow^s 
 (by 106) that the addition of any two quaternions is likewise a 
 commutative operation, or in symbols, that 
 
 I. . . ^ + ^' = ^' + (7 ; 
 so that the Sum of any tivo* Quaternions has a Value, which 
 is independent of their Order : and which (by what precedes) 
 must be considered to be given, or at least known, or definite, 
 when the two summand quaternions are given. It is easy also 
 to see that the conjugate of any such sum is equal to the sum 
 of the conjugates, or in symbols, that 
 
 11. ..K(^'-K7)=%' + K^. 
 
 (1 .) The important formula last -written becomes geometrically evident, when it 
 is presented under the following form. Let obdc be any parallelogram, and let OA 
 be any right line, drawn from one comer of it, but not generally in its plane. Let 
 the three other comers, b, c, d, be reflected (in the sense of 145, (5.) ) with respect 
 to that line OA, into three new points, b', c', d' ; or let the three lines ob, oc, od be 
 reflected (in the sense of 138) with respect to the same line oa; which thus bisects 
 at right angles the three joining lines, bb', cc', dd', as it does bb' in Fig. 36. Then 
 each of the lines OB, oc, od, and therefore also the ^\io\q plane figure ohdc, may be 
 considered to have simply revolved round the line oa as an axis, by a conical rota- 
 tion through two right angles ; and consequently the new figure ob'd'c', like that old 
 one obdc, must be a, parallelogram. Thus (comp. 106, 137), we have 
 
 od' = oc' + ob', 5' = -y' + /3', 5': a=(y' : a)+ (/3': a); 
 and the recent formula II. is justified. 
 
 * It will be found that this result admits of being extended to the case of three 
 (or more) quaternions ; but, for the moment, we content ourselves with two. 
 
176 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (2.) Simple as this last reasoning is, and unnecessary as it appears to be to draw- 
 any new Diagram to illustrate it, the reader's attention may be once more invited to 
 the great simplicity of expression, with which many important ^reome^ncaZ concep- 
 tions, respecting space of three dimensions, are stated in the present Calculus : and 
 are thereby kept ready for future application, and for easy combination with other 
 results of the same kind. Compare the remarks already made in 132, (6.) ; 145, 
 (10.); 161; 179,(3.); 192,(6.); and some of the shortly following sub-articles to 
 196, respecting properties of an oblique cone with circular base. 
 
 196. One of the most important cases o^ addition y is that 
 of two conjugate summands^ q and K^ ; of which it has been 
 seen (in 140) that the sum is always a scalar. We propose 
 now to denote the ^^Zf of this sum by the symbol. 
 
 Is , , ^9'-> 
 
 hus writing generally, 
 
 I. . . ^ + Kg = Ky + ^=2S^; 
 or defining the new symbol S^' by the formula, 
 f^ II. ..S^ = i(^ + K^); or briefly, 11'. . . S = i (1 + K). 
 
 For reasons which will soon more fully appear, we shall also 
 call this new quantity, Sg', the scalar part, or simply the Sca- 
 lar, of the Quaternion, q ; and shall therefore call the letter 
 S, thus used, the Characteristic of the Operation of taking the 
 6'caZ«r of a quaternion. (Comp. 132, (6.) ; 137; 156; 187.) 
 It follows that not only equal quaternions, but also conjugate 
 quaternions, have equal scalars ; or in symbols, 
 
 III. . . S^'=S^, if q^q-, and IV. ..SK9 = S^; 
 
 or briefly, 
 
 IV'. ..SK=S. 
 
 And because we have seen that Kg- = + ^, if 5' be a scalar ( 1 39), 
 
 but that li^q^-q, if 5' be a right quotient (144), we find that 
 
 the scalar of a scalar (considered as a degenerate quaternion, 
 
 131) is equal to that scalar itself, \>\xi that the scalar of a right 
 
 quaternion is zero. We may therefore now write (comp. 160): 
 
 V. . Sa; = X, if ic be a scalar ; VI. . .SSg = Sg', 8^ = 88 = 8; 
 
 and Vll. ..8^ = 0, if z^ = |. 
 
 Again, because oa' in Fig. 36 is multiplied by x, when ob is 
 multiplied thereby, we may write, generally, 
 
CHAP. I.] SCALAR OF A QUATERNION. 177 
 
 VIII. • . Sxq = xSq, if oj be any scalar; 
 and therefore in particular (by 188), 
 
 IX. ..S^ = S(T^.U^) = T^.SU^. 
 
 Also because SK^=S^, by IV., while KU^ = U-, by 158, 
 we have the general equation, 
 
 X. ..SUq = SJJ-; or X'. . . SU^ = SU ^ ; 
 whence, by IX., 
 
 XI. . .S^ = T^.Sui; or XI'. . . S^ = T@. SU^ ; 
 ^ ^ q a a p 
 
 and therefore also, by 190, (V.), since T^.T- = 1, 
 
 XII.. .Sq = TqKS-=^^q.S-; XIF. . . S ^ = N^ • S " 
 ^ ^ q ^ q a a (5 
 
 The results of 142, combined with the recent definition I. or 
 II., enable us to extend the recent formula VII., by writing, 
 
 XIII. . . S^' >, =, or < 0, according as Lq <, =, or > - ; 
 and conversely, 
 
 XIV. . . Z ^ <, =, or > -, according as S^- >, =, or <q. 
 
 In fact, if we compare that definition I. with the formula of 
 140, and with Fig. 36, we see at once that because, in that 
 Figure, 
 
 S(ob: oa) = oa': OA, 
 
 we may write, generally, 
 
 XV. . . S^ = T^.cosz^; or XVI. . . SU^= cos Z ^; 
 
 equations which will be found of great importance, as serving 
 
 to connect quaternions with trigonometry ; and which show 
 
 that 
 
 XVII. ,.Lq=^Lq, if SU^' = SU^, 
 
 the angle Lq being still taken (as in 130), so as not to fall 
 outside the limits and tt ; whence also, 
 
 2 A 
 
178 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 XVIII. .. Lq'^Lq, if S^' = S^, and Tq' = T^, 
 the angle of a quaternion being thus given, when the scalar 
 and the tensor of that quaternion are given, or known. Fi- 
 nally because, in the same Figure 36 (comp. 15, 103), the 
 line, 
 
 oa' = (oa' : oa) . oa = oa . S (ob : da), 
 
 may be said to be the projection of ob on oa, since a' is the 
 foot of the perpendicular let fall from the point b upon this 
 latter line oa, we may establish this other general formula : 
 
 XIX. . . aS - = S — • a = projection of^ on a ; 
 a a 
 
 a result which will be found to be of great utility, in investi- 
 gations respecting geometrical loci, and which may be also 
 written thus : 
 
 XX. . . Projection o/ j3 o?2 a = Ua . T/3 . SU ^ ; 
 
 with other transformations deducible from principles stated 
 above. It is scarcely necessary to remark that, on account 
 of the scalar character of Sq, we have, generally, by 159, and 
 187, (8.), the expressions, 
 
 XXL . . US^ = ±1; XXII. . .TS^ = ±S^; 
 while, for the same reason, we have always, by 139, the equa- 
 tion (comp. IV.), 
 
 XX III. . . KS^ = S^ ; or XXIII'. . . KS = S ; 
 and, by 131, 
 
 XXIV. . . iSq^O, or = TT, unless Lq = -; 
 
 in which last case S^' = 0, by VII., and therefore L Sq is inde- 
 terminate :* IJSq becoming at the same time indeterminate, 
 by 159, but TS^ vanishing, by 186, 187. 
 
 8-^ = 0, 
 a 
 
 (1.) The equation, 
 
 is now seen to be equivalent to the formula, p -^ a ; and therefore to denote the 
 * Compare the Note in page 118, to Art 131. 
 
CHAP. I.] GEOMETRICAL EXAMPLES. 1^^ 
 
 same plane locus for p, as that which is represented by any one of the four other 
 equations of 186, (6.) ; or by the ecjuation, 
 
 T^-t^ = l, of 187, (2.). 
 p~a 
 (2.) The equation, 
 
 S£IJ = 0, or Se=S^, 
 a a a 
 
 expresses that bp j_ oa ; or that the points B and p have the same projection on oa j 
 or that the locus of p is the plane through b, perpendicular to the line OA. 
 (3.) The equation, 
 
 a a 
 
 expresses (comp. 132, (2.) ) that p is on one sheet of a cone of revolution, with o for 
 vertex, and OA for axis, and passing through the point b. 
 
 (4.) The other jsheet of the same cone is represented by this other equation, 
 
 a a 
 
 and hath sheets jointly by the equation, I 
 
 (6.) The equation, 
 
 S- = l, or SU^ = T-, 
 a a p 
 
 expresses that the locus of p is the plane through A, perpendicular to the line OA ; 
 because it expresses (comp. XIX J that the projection of op on oa is the line oa it- 
 
 p — a 
 self; or that the angle oap is right ; or that S =0. P 
 
 (6.) On the other hand the equation, 
 
 S^=l, or Sug=Tg,. .. \ — ' 
 
 expresses that the projection of ob on op is op itself ; or that the angle opb is right ; 
 or that the locus of p is that spheric surface, which has the line ob for a diameter. 
 (7.) Hence the system of the two equations, 
 
 sP = i, S^=l. 
 
 a p 
 
 represents the circle, in which the sphere (6.), with ob for a diameter, is cut by the 
 plane (5.), with oa for the perpendicular let fall on it from o. 
 (8.) And therefore this new equation, 
 
 S^.S^ = 1, 
 a p 
 
 obtained by multiplying the two last, represents the Cyclic* Cone (or cone of the 
 
 * Historically speaking, the oblique cone with circular base may deserve to be 
 named the Apollonian Cone, from Apollonius of Perga, in whose great work on Co- 
 
 6. 
 
 p 
 
 A 
 
 Ar^- 
 
 h'? 
 
180 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 second order, but not generally of revolution), ■which rests on this last circle (7.) as 
 its lase, and has the point o for its vertex. In fact, the equation (8.) is evidently 
 satisfied, when the two equations (7.) are so; and therefore every point of the circu- 
 lar circumference, denoted by those two equations, must be a point of the locus, re- 
 presented by the equation (8.). But the latter equation remains unchanged, at least 
 essentially, when p is changed to xp, x being any scalar ; the locus (8.) is, there- 
 fore, some conical surface, with its vertex at the origin, o ; and consequently it can 
 be none other than that particular cone (both ways prolonged), which rests (as 
 above) on the given circular base (7.). 
 (9.) The system of the two equations, 
 
 a p y 
 
 (in writing the first of which the point may be omitted,) represents a conic section ; 
 namely that section, in which the cone (8.) is cut by the new plane, which has oc 
 for the perpendicular let fall upon it, from the origin of vectors O. 
 
 (10.) Conversely, every plane ellipse (or other conic section) in space, of which 
 the plane does not pass through the origin, may be represented by a system of two 
 equations, of this last /orm (9.) ; because the cone which rests on any such conic as 
 its base, and has its vertex at any given point O, is known to be a cyclic cone. 
 
 (11.) The curve (or rather the pair of curves), in which an oblique but cyclic 
 cone (8.) is cut by a concentric sphere (that is to say, a cone resting on a circular 
 base by a sphere which has its centre at the vertex of that cone), lias come, in mo- 
 dem times, to be called a Spherical Conic. And an}- such conic may, on the fore- 
 going plan, be represented by the system of the two equations, 
 
 S^ S^=l, Tp=l; 
 a p 
 
 the length of the radius of the sphere being here, for simplicity, supposed to be the 
 unit of length. But, by writing Tp — a, where a may denote any constant and posi- 
 tive scalar, we can at once remove this last restriction, if it be thought useful or con- 
 venient to do so. 
 
 (12.) The equation (8.) may be written, by XII. or Xll'., under the form (comp. 
 191, VII.): 
 
 or br' fly, 
 
 'I'r^rWv 
 
 p a 
 
 nics (tc(tJviK'7iv), already referred Lo in a Note to page 128, the properties of such a 
 cone appear to have been first treated systematically; although the cone of revolu- 
 tion had been studied by Euclid. But the designation " cyclic cone''' is shorter ; and 
 it seems more natural, in geometry, to speak of the above-mentioned oblique cone 
 thus, for the purpose of marking its connexion with the circle, than to call it, as is 
 now usually done, a cone of the second order, or of the second degree : although 
 these phrases also have their advantages. 
 
CHAP. I.] GEOMETRICAL EXAMPLES. 181 
 
 if a' = /3T^ = Ta.U/3, and /3' = aT^= T^S-Ua ; 
 
 so that a and j3' are here the lines oa' and ob', of Art. 188, and Fig. 48. 
 
 (13.) Hence the cone (8.) is cut, not only by the plane (5.) in the circle (7.), 
 
 which is on the sphere (6.), but also by the (generally) new plane^ S -,= 1, in the 
 
 (generally) new circle, in which this new plane cuts the (generally) new sphere, 
 
 8' 
 S — = 1 ; or in the circle which is represented by the system of the two equations, 
 
 S-%1, S^'=l. 
 a p 
 
 (14.) In the particular case when (3 \\ a (15), so that the quotient /3 : a is a sca- 
 lar, Avhich must be positive and greater than unity, in order that the plane (5.) may 
 (jeally) cut the sphere (6.), and therefore that the circle (7.) and the cone (8.) may 
 be real, we may write 
 
 ]3=a2a, a>l, T(|3:a) = a2, „'=„, /3' = /3; 
 
 and the circle (13.) coincides with the circle (7.). 
 
 (15.) In the same case, the cone is one of revolution ; every point p of its circu- 
 lar 6a*e (that is, of the circumference thereof) being ai one constant distance from 
 the vertex o, namely at a distance = aTa. For, in the case supposed, the equations 
 (7.) give, by XII., 
 
 N^ = S^:S-=l:S- = a2:S^=a2; or To = aTa. 
 a a p p p 
 
 (Compare 145, (12.), and 186, (5.). ) 
 
 (16.) Conversely, if the cone be one of revolution, the equations (7.) must con- 
 duct to a result of the form, 
 
 a2=N" = S- :S- = S-:S-,or (comp. (2.) ), S' =0: 
 
 a a p p p "^ p 
 
 which can only be by the line /3 — a'^a vanishing,' or by our having (5= a^a, as in 
 (] 4.) ; since otherwise we should have, by XIV., p -i- (3- a^a, and all the points of 
 the base would be situated in one plane passing through the vertex o, which (for any 
 actual cone) would be absurd. 
 
 (17.) Supposing, then, that we have not (3 || a, and therefore not a =a, /3' = (3, 
 as in (14.), nor even a' \\ a, (3' \\ (3, we see that the cone (8.) is not a cone of revolu- 
 tion (or what is often called a right cone) ; but that it is, on the contrary, an oblique 
 (or scalene) cone, although still a cyclic one. And we see that such a cone is cut in 
 two distinct series* of circular sections, by planes parallel to the two distinct (and 
 mutually non-parallel) planes, (5.) and (13.) ; or to two new planes, drawn through 
 the vertex o, which have been calledf the two Cyclic Planes of the cone, namely, the 
 two following : 
 
 * ThGSQ two series o{ sub- contrary (or antiparallel) hut circular sections of a 
 cyclic cone, appear to have been first discovered by Apollonius : see the Fifth Propo- 
 sition of his First Book, in which he says, KuXihOuj dk )? Toiavrr} To/xtj v-rrevavria 
 (page 22 of Halley's Edition). 
 
 t By M. Chasles. 
 
182 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 a p 
 
 ■while the two lines from the vertex, OA and ob, which are perpendicular to these two 
 planes respectively, may be said to be the two Cyclic Normals. 
 
 (18.) Of these two lines, a and /3, the second has been seen to be a diameter of 
 the sphere (6.)? which may be said to be circumscribed to the cone (8.), when that 
 cone is considered as having the circle (7.) for its base ; the second cyclic plane (17.) 
 is therefore the tangent plane at the vertex of the cone, to thatj^rs* circumscribed 
 sphere (6.). 
 
 (19.) The sphere (13.) may in like manner be said to be circumscribed to the 
 cone, if the latter be considered as resting on the new circle (13.), or as terminated by 
 that circle as its new base ; and the diameter of this new sphere is the line ob', or j8', 
 which has by (12.) the direction of the line a, or of thQ first cyclic normal (17.) ; so 
 that (comp. (18.)) th^. first cyclic plane is the tangent plane at the vertex, to the 
 second circumscribed sphere (13.). 
 
 (20.) Any other sphere through the vertex, which touches the first cyclic plane, 
 and which therefore has its diameter from the vertex =b'(3% where b' is some scalar 
 co-efficient, is represented by the equation, 
 
 S^'=l,- or S^'=i; 
 
 P P 
 
 it therefore cut$ the cone in a circle, of which (by (12.) ) the equation of the plane is 
 
 S^, = 6', or S-^,= l, 
 a b a 
 
 so that the perpendicular from the vertex is b'a' \\ (3 (comp. (5.) ) ; and consequently 
 thisp/a«e of section of sphere and cone is parallel to the second cyclic plane (17.). 
 (21.) In like manner any sphere, such as 
 
 S — = 1, where b ia any scalar, 
 P 
 w^hich touches the second cyclic plane at the vertex, intersects the cone (8.) in a cir- 
 cle, of which the plane has for equation, 
 
 and is therefore /)araZZeZ to the first cyclic plane. 
 
 (22.) The equation of the cone (by IX., X., XVI,) may also be thus written : 
 
 SU^.SU^ = T^; or, cos ^^ . cos /|= T ^; 
 a p (S a (3 (3 
 
 it expresses, therefore, that the product of the cosines of the inclinations, of any va- 
 riable side (p) of an oblique cyclic cone, to two fixed lines (a and f3), namely to the 
 two cyclic normals (17.), is constant ; or that the product of the sines of the inclina- 
 tions, of the same variable side (or ray, p) of the cone, to two fixed planes, namely to 
 the two cyclic planes, is thus a constant quantity. 
 
 (23.) The two great circles, in which the concentric sphere Tp = 1 is cut by the two 
 cyclic planes, have been called the two Cyclic Arcs* of the Spherical Conic (11.), in 
 
 Bv M. Cbasles. 
 
CHAP. I.J SCALAR OF A SUM OR DIFFERENCE. 183 
 
 which that sphere is cut by the cone. It follows (by (22.) ) that the product of the 
 sines of the (arcuaV) perpendiculars, let fall from any point v of a given spherical 
 conic, on its two cyclic arcs, is constant. 
 
 (24.) These properties of cyclic cones, and of spherical conies, are not put for- 
 ward as new ; but they are of importance enough, and have been here deduced with 
 sufficient facihty, to show that we are already in possession of a Calculus, with its 
 own Rules* of Transformation, whereby one enunciation of a geometrical theorem, or 
 problem, or construction, can be translated into several others, of which some may 
 be clearer, or simpler, or more elegant, than the one first proposed. 
 
 197. Let a, /3, 7 be any three co-initial vectors, oa, &c., 
 and let 00 = ^ = 74-/3, so that obdc is a parallelogram (6);^ 
 then, if we write i^ 
 
 [5:a = q, y:a = q', and S : a = q" = q' + q (106)^ 
 
 and suppose that b', c', d' are the feet of perpendiculars let 
 fall from the points b, c, d on the line oa, we shall have, by 
 196, XIX., the expressions, 
 
 (ob' =) ^' = aSq, y' = aSq', S' = aSq" = aS (q' + q). 
 
 But also OB = CD, and therefore ob'= c'd', the similar projec- 
 tions of equal lines being equal ; hence (comp. 11) the sum of 
 the projections of the lines j3, 7 must be equal to the projec- 
 tion of the sum, or in symbols, 
 
 od' = oc'+ob', g' = y-fj3', S': a = (7 :a) + (/3': a). 
 
 Hence, generally, for any tioo quaternions, q and q, we have 
 the formula : 
 
 I. . . S(^'+9) = S^' + Sg; 
 
 or in words, the scalar of the sum is equal to the sum of the 
 scalar s. It is easy to extend this result to the case of any three 
 (or more) quaternions, with their respective scalars ; thus, if 
 q be a third arbitrary quaternion, we may write 
 
 S { ?" + (3' + ?) ) = Sj" + S (j + ?) = S/ + (Sj'+ S?) ; 
 where, on account of the scalar character of the summands, the 
 last parentheses may be omitted. We may therefore write, 
 generally, 
 
 II. . . SS^ = 2S^, or briefly, SS = 2S ; 
 where 2 is used as a sign of Summation : and may say that 
 
 * Comp. 146, (10.), &c. 
 
184 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 the Operation of tailing the Scalar of a Quaternion is a Dis- 
 tributive Operation (comp. 13). As to the general Siibtrac- 
 tion of any one quaternion from any other, there is no difficulty 
 in reducing it, by the method of Art. 120, to the second gene- 
 ral formula of 106 ; nor in proving that the Scalar oftheDiffe- 
 rence* is always equal to the Difference of the Scalars. In 
 
 symbols, 
 
 III. . . S(^'-^) = S^'-S^; 
 or briefly, 
 
 IV... SA^ = AS^, SA=AS; 
 
 when A is used as the characteristic of the operation of taking 
 a difference, by subtracting one quaternion, or one scalar, from 
 another. 
 
 (1.) It has not yet been proved (comp. 195), that the Addition oi any number 
 of Quaternions, q, q\ q" , . . is an associative and a commutative operation (comp. 9). 
 But we see, already, that the scalar of the sum of any such set of quaternions has 
 a value, which is independent of their order, and of the mode oi grouping them. 
 
 (2.) If the summands be all right quaternions (132), the scalar oieach separately 
 vanishes, by 196, VI I. ; wherefore the scalar of their sum vanishes also, and that 
 sum is consequently itself, by 196, XIV., a right quaternion : a result which it is 
 easy to verify. In fact, if /3 -i- a and y -^ a, then y + /3 -J- a, because a is then per- 
 pendicular to the plane of /3 and y ; hence, by 106, the sum of any two right qua- 
 ternions is a right quaternion, and therefore also the sum of any number of such qua- 
 ternions. 
 
 (3.) Whatever two quaternions q and q' may be, we have always, as in algebra, 
 the two identities (comp. 191, (7.) ) : 
 
 V. ..(?'-g) + 5 = g'; VI. ..(9' + 5) -9 = ^'. 
 
 198. Without yet entering on the general theory o^ scalars of 
 products or quotients of quaternions, we may observe here that be- 
 cause, by 196, XV., the scalar of a quaternion depends only on the 
 tensor and the angle, and is independent of the axis, we are at liberty 
 to write generally (comp. 173, 178, and 191, (1.), (5.)), 
 l...Sqq^=Sq'q; 11. , . S . q (q^: q) = Sq' ; 
 
 the two products^ qq' and q'q, having thus always equal scalars, 
 although they have been seen to have unequal axes, for the general 
 case of diplanarity (168, 191). It may also be noticed, that in vir- 
 tue of what was shown in 193, respecting the quotient, and in 194 
 
 ' Examples have already occurred in 196, (2.), (5.), (16.). 
 
CHAP. I.] SCALAR OF A PRODUCT, QUOTIENT, OR SQUARE. 185 
 
 respecting the product, of any two right quaternions (132), in con- 
 nexion with their indices (133), we may now establish, for any 
 such quaternions, the formulae : 
 
 III. . . S (^' : 5) = S (I^' : I?) = T {q' : q) . cos Z.(Ax. q^ : Ax. q) ; 
 
 IV. . . Sq'q =^S{q' .q) = s(lq':l-j = - Tq'q.cos L (Ax. ^': Ax. q)\ 
 
 where the new symbol \q is used, as a temporary abridgment, to 
 denote the Index of the quaternion q^ supposed here (as above) to be 
 a right one. With the same supposition, we have therefore also 
 these other and shorter formulae : 
 
 V. . . SU(g':^)=+ cosz(Ax. ^': Ax.^); 
 VI. . . SU'^'2' = - cos Z (Ax. q^ : Ax. q) ; 
 
 which may, by 196, XVI., be interpreted as expressing that, under 
 the same condition of rectangularity of q and q\ 
 
 VII. . . L{q'.q)=^L (Ax. q^: Ax. q) ; 
 
 VIII. , . Lq'q = 7r-L (Ax. q' : Ax. q). 
 
 In words, the Angle of the Quotient of two Right Quaternions is equal 
 to the Angle of their Axes; but the Angle of the Product^ of two such 
 quaternions, is equal to the Supplement of the Angle of the Axes, 
 There is no difficulty in proving these results otherwise, by con- 
 structions such as that employed in Art. 193; nor in illustrating 
 them by the consideration of isosceles quadrantal triangles, upon the 
 surface of a sphere. 
 
 199. Another important case of the scalar of a product, is 
 the case of the scalar of the square of a quaternion. On refer- 
 ring to Art. 149, and to Fig. 42, we see that while we have 
 always T (q') = {Tq)\ as in 190, and \]{q') = U(^)% as in 161, 
 we have also, 
 
 I. . .Z(g)^ = 2z^, and Ax. (q') = Ax. q, if Zg<|; /^^ 
 
 but, by the adopted definitions of ^^'(130), and of Ax. 5'^^<C[2^ 
 (127, 128), ^^[j 
 
 II. ..z(^^) = 2(7r-z^), Ax.(^0=-Ax.^, if z^>^. / 
 
 In each case, however, by 196, XVI., we may write, i^ 
 
 lU. . .S\J(q') = C0SL{q')=C0s2lq; 
 
 2 B 
 
186 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 a formula which holds even when z 5^ is 0, or -, or tt, and 
 
 which gives, 
 
 IV.. . S\](q') = 2{S\]qy-l. 
 
 Hence, generally, the scalar of q"^ may be put under either of 
 the two following forms : 
 
 V. . . S(q') = TqKcos2z.q; YL . . S C^^) = 2 (S^)^ - T^'' ; 
 
 where we see that it would not be safe to omit the parentheses, 
 without some convention previously made, and to write simply 
 Sq\ without first deciding whether this last symbol shall be 
 understood to signify the scalar of the square, or the square of 
 the scalar of q: these two things being generally unequal. 
 The latter of them, however, occurring rather oftener than the 
 former, it appears convenient to fix on it as that which is to 
 be understood by Sq^, while the other may occasionally be 
 written with a point thus, S.q^; and then, with these conven- 
 tions respecting notation* we may write : 
 
 VII. . . Sq' =-. (Sqy ; VIII. . .S.q'=S {q% 
 
 But the square of the conjugate of any quaternion is easily seen 
 to be the conjugate of the square ; so that we have generally 
 (comp. 190, II.) the formula: 
 
 IX. . . K^^ = K {q^) = {KqY = Tq^ : J]q\ 
 
 (1.) A quaternion, like a positive scalar, may be said to have in general two oppo- 
 site square roots ; because the squares of opposite quaternions are always equal 
 (comp. (3.) ). But of these two roots the principal (or simpler') one, and that which 
 we shall denote by the symbol V9, or Vg-, and shall call by eminence the Square Root 
 of q^ is that which has its angle acute, and not obtuse. We shall therefore write, 
 generally, _ 
 
 ^. . . LMq=^ Lq'; Ax. Vg'= Ax. q ; 
 
 * As, in the Differential Calculus, it is usual to write da;2 instead of (dx)2 . 
 while d(x2) is sometimes written as d.x^. But as d^a; denotes a, second differential, 
 so it seems safest not to denote the square of Sq by the symbol S^q, which properly/ 
 signifies SSg, or Sq, as in 196, VI. ; the second scalar (like the second tensor, 187, 
 (9,), or the second versor, 160) being equal to the^r*^ Still everj'^ calculator will 
 of course use his own discretion ; and the employment of the notation S^q for (87)^, 
 as cos ^x is often written for (cos x)^, may sometimes cause a saving of space. 
 
CHAP. I.] TENSOR AND NORM OF A SUM. 187 
 
 with the reservation that, when lq = 0, or = tt, this common axis of q and Vg be- 
 comes (by 131, 149) an indeterminate unit-line. 
 (2.) Hence, 
 
 XI. ..SVg'>0, if Lq<TT; 
 
 while this scalar of the square root of a quaternion may, by VI., be thus trans- ^ ^ 
 formed : 
 
 XII. ..SV9 = V{K'r? + Sg)}; \ 4 
 
 a formula which holds good, even at the limit Lq—Tr. 
 
 (3.) The principle* (1.), that in quaternions, as in algebra, the equation, 
 
 XIII... (-9)2 = 92, ^ /^ 
 
 is an identity^ may be illustrated by conceiving that, in Fig. 42, a point b' is deter- 
 mined by the equation ob' =bo ; for then we shall have (comp. Fig. 33, his\ L / 
 
 (- 0)2 = I — ) = — = 7^, because A aob' a b'oc. 
 ^ ''^ VoA y OA ^ ' 
 
 200. Another useful connexion between scalars and tensors (or 
 norms) of quaternions may be derived as follows. In any plane tri- 
 angle AOB, we havef the relation, 
 
 (T. ab)2= (T. oa)2 - 2(T.oa) . (T. ob) . cos aob + (T. ob)2; 
 in which the symbols T. oa, &c., denote (by 185, 186) the lengths of 
 the sides oa, &c. ; but if we still write q = 0B: oa, we have q-l 
 = ab: oa; dividing therefore by (T. oa)^, the formula becomes (by 
 196, &c.), 
 
 I.. . T{q-iy = l-2Tq.SUq + Tq' = Tf-2Sq+l', 
 or 
 
 II.. .N(^-1)=%-2S^+1. 
 
 But q is here a perfectly general quaternion; we may therefore 
 change its sign, and write, 
 
 III. ..T (1 + ^)^=1 + 2S?+T^^ IV. ..N(l+j)=I + 2S^ + %. 
 And since it is easy to prove (by 106, 107) that 
 
 + 1 
 
 )^=''"^' .r>-c f 
 
 whatever two quaternions q and q^ may be, while . 
 
 we easily infer this other general formula, 
 
 VII. . . N (^' + ?) =N^' + 2S . qKq' f %; 
 which gives, if x be any scalar, 
 
 VIII. . . N (^ + a;) = N^ + 2x^q -f x\ 
 
 * Compare the first Note to page 162. 
 
 t By the Second Book of Euclid, or by plane trigonometry 
 
188 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (1.) We are now prepared to effect, hy rules* of transformation^ some other pas- 
 sages from one mode of expression to another, of the kind which has been alluded to, 
 and partly exemplified, in former sub-articles. Take, for example, the formula, 
 
 T^^!^=l, of 187, (2.); 
 p-a 
 
 or the equivalent formula, 
 
 T(p + a) = T(p-a), of 186, (6.) ; 
 
 ■which has been seen, on geometrical grounds, to represent a certain locus, namely the 
 
 plane through o, perpendicular to the line oa ; and therefor the same locus as that 
 
 which is represented by the equation, 
 
 S- = 0, of 196, (1.). 
 a 
 
 To pass now from the former equations to the latter, by calculation, we have only 
 
 to denote the quotient p: ahy q, and to observe that the first or second form, as just 
 
 now cited, becomes then, 
 
 T(^ + l) = T(g-l); or N(9 + 1) = N (^ - 1) ; 
 
 or finally, by II. and IV., 
 
 S^=0, 
 
 which gives the third form of equation, as required. 
 
 (2.) Conversely, from S - = 0, we can return, by the same general formulas II. 
 a * 
 
 andlV., to the equation n[^-1J=: Nl^+l\ or by I. and III. to Tf^-1 J 
 
 = T -+l\ ortoT(p- a") = T(|0 + a), orto T*^ — -= 1, as above; and gene- 
 \a J p-a 
 
 rally, 
 
 Sq = gives T(9-1) = T(^+1), or T^=l; 
 
 while the latter equations, in turn, involve, as has been seen, the former. 
 
 (3.) Again, if we take the Apollonian Locus, 145, (8.), (9.), and employ the Jirst 
 
 of the two forms 186, (5.) of its equation, namely, 
 
 T(p-a2a)=aT(p-a), 
 
 where a is a given positive scalar difierent from unity, we may write it as 
 
 T(g-a2) = aT(5-l), or as N (q - a^) = a-^^ {q - 1) ; 
 
 or by VIII., 
 
 % - 2a^Sq + a* = a2 (Ng - 2Sq + 1) ; 
 
 or, after suppressing - 2a^Sq, transposing, and dividing by a^ _ i^ 
 
 Ny = a2; or, Np = a2Na; or, Tp = aTa ; 
 which last is the second form 186, (5.), and is thus deduced from the first, hy calcu- 
 lation alone, without any immediate appeal to geometry, or the construction of any 
 dia^am. 
 
 * Compare 145, (10.) ; and several subsequent sub-articles. 
 
CHAP. I.] GEOMETRICAL EXAMPLES. 189 
 
 (4.) Conversely if we take the equation, , ""^ ' ^ /Vl^ 
 
 N^ = a2, of 145, (12.), ^ 
 
 a 
 
 which was there seen to represent the same locus, considered as a spheric surface, 
 with o for centre, and aa for one of its radii, and write it as Ng^ = a2, we can then 
 hy calculation return to the form 
 
 N(g-a2) = a2N(9-l), or T (q-a^) = aT (q -1), 
 or finally, 
 
 T (p - a2a) = aT (p - a), as in 186, (5.) ; 
 
 this /rsf/orm of that sub- article being thus deduced from the second, namely from 
 Tp=aTa, or T- = a. ^ 
 
 (5.) It is far from being the intention of the foregoing remarks, to discourage 
 attention to i^xe geometrical interpretation of the various /orws of expression^ and ,^ x 
 
 general rules of transformation, which thus offer themselves in working with qua- '^ ^IT 
 ternions ; on the contrary, one main object of the present Chapter has been to es- 
 tablish a firm geometrical basis, for all such forms and rules. But when such a. foun- 
 dation has once been laid, it is, as we see, not necessary that we should continually 
 recur to the examination of it, in building up the superstructure. That each of the 
 two forms, in 186, (5.), involves the other, may he proved, as above, by calculation ; 
 but it is interesting to inquire what is the meaning of this result : and in seeking to 
 interpret it, we should be led anew to the theorem of the Apollonian Locus. 
 
 (6.) The result (4.) of calculation, that 
 
 N (g - a2) = a2N (g - 1), if N^ = a2, 
 may be expressed imder the form of an identity, as follows : 
 IX. . .N(g-N5) = %.N(g-l); 
 in which q may be any quaternion. 
 
 (7.) Or, by 191, VII., because it will soon be seen that 
 q(jq-i) = q^ — q, as in algebra, 
 we may write it as this other identity : 
 
 X. . . N(g-Ng) = N(52-5). 
 
 (8.) If T (9 - 1) = 1, then S - = - ; and conversely, the former equation follows 
 q 2 
 
 from the latter; because each may be put under the form (comp. 196, XII.), 
 
 Ng = 2Sg. 
 
 2a 
 (9.) Hence, if T (p - a) = Ta, then S — = 1, and reciprocally. In fact (comp. 
 
 196, (6.) ), each of these two equations expresses that the locus of p is the sphere 
 which passes through o, and has its centre at a ; or which has on = 2a for a dia- 
 meter. 
 
 (10.) By changing 7 to 7 + 1 in (8), we find that 
 
 if Tq=\, then S - — - = 0, and reciprocally. 
 
 
190 ELEMENTS OF QUATERNIONS. [boOK II. 
 
 (11.) Hence if T|0=Trt, then S^ — ^ = 0, and reciprocally ; because (by 106) 
 |0 + a 
 
 -a p—ap+a 
 p+a 
 
 (12.) Each of these two equations (11.) expresses that the locus of pis the 
 sphere through a, which has its centre at o ; and their proved agreement is a recog- 
 nition, by quaternions, of the elementary geometrical theorem, that the angle in a 
 semicircle is a right angle. 
 
 ^■■'-i'-ii-Mi^^ 
 
 Section 13. — On the Right Part (or Vector Part) of a Qua- 
 ternion ; and on the Distributive Property of the Multipli- 
 cation of Quaternions. 
 
 201. A given vector ob can always be decomposed, in one 
 but in only one way, into two component vectors, of which it 
 is the sum (6) ; and of which one, as ob' in Fig. 50, is parallel 
 (15) to another given vector oa, while ,, 
 
 the other, as ob" in the same Figure, is i . 
 
 perpendicular to that given line oa; j ^^ 
 
 namely, by letting fall the perpendicu- j ^^^ 
 
 lar bb' on oa, and drawing ob" = b'b, so \y^ 
 
 that ob'bb" shall be a rectangle. In p-j g^ 
 
 other words, if a and j3 be any two given, 
 
 actual, and co-initial vectors, it is always possible to deduce 
 
 from them, in one definite way, two other co-initial vectors, 
 
 /3' and j3", which need not however both be actual (I); and 
 
 which shall satisfy (comp. 6, 15, 129) the conditions, 
 
 j3' vanishing, when j3 _L a ; and /3" being null, when j3 || a ; 
 but both being (what we may call) determinate vector func- 
 tions of a and /3. And of these two functions, it is evident 
 that j3' is the orthographic projection of j5 on the line a ; and 
 that j3" is the corresponding j^ro/ec^/ow o/j3 on the plane through 
 o, which is perpendicular to a. 
 
 202. Hence it is easy to infer, that there is always one, 
 but only one way, of decomposing a given quaternion^ 
 
 q = 0B : 0A = /3 : a, 
 
 into two parts or summands (195), of which one shall be, as in 
 
CHAP. I.] RIGHT PART OF A QUATERNION. 191 
 
 196, a scalar, while the other shall be a right quotient (132). 
 Of these two parts, the. former has been already called (196) 
 the scalar part, or simply the Scalar of the Quaternion, and 
 has been denoted by the symbol ^q ; so that, with reference 
 to the recent Figure 50, we have 
 
 I. . . S3' = S(oB : oa) = ob': OA ; or, S (j3 : a) =/3': a. 
 And we now propose to call the latter part the Eight Part* 
 of the same quaternion, and to denote it by the new symbol 
 
 writing thus, in connexion with the same Figure, 
 
 11. . . V^ = V(ob:oa) = ob":oa; or, V(i3 : a) = i3": a. 
 The System of Notations, peculiar to the present Calculus, 
 will thus have been completed ; and we shall have the follow- 
 ing general Formula of Decomposition of a Quaternion into tivo 
 Summands (comp. 188), of the Scalar and Right kinds : 
 
 III. ..^=S^ + V^ = V^ + S^, 
 or, briefly and symbolically, 
 
 IV. . . 1 = S + Y = V+S. 
 
 (1.) In connexion with the same Fig. 50, we may write also, 
 
 
 V(ob: 
 
 OA) = 
 
 b'b : OA, 
 
 
 be( 
 
 cause, by construction, b'b = ob". 
 
 
 
 I 
 
 
 C2.) In like manner, for Fig. 36, we have the equation, 
 
 P !>^ 
 
 
 V(ob: 
 
 OA) = 
 
 a'b : OA. 
 
 
 
 (3.) Under the recent conditions, 
 
 
 
 
 
 V(j3':a)=0, 
 
 and 
 
 S(/3":a) = 0. 
 
 
 
 (4.) In general, it is evident that 
 
 
 
 
 
 V. ..g=0, if S^=0, 
 
 and 
 
 V5'=0; and] 
 
 reciprocally. 
 
 
 (5.) More generally, 
 
 
 
 
 
 VI. ..9' =9, if ^q=Sq, 
 
 and 
 
 Yq = Yq ; with the converse. 
 
 
 (6.) Also VII. ..Y9 = 0, 
 
 if 
 
 lq = 0, or = 
 
 tt; 
 
 or 
 
 VIII. .. V(^:a) = 
 
 = 0, if /3i|a; 
 
 
 the right part of a scalar being zero. 
 
 
 
 
 * This Eight Part, Yq, will come to be also called the Vector Part, or simply 
 the Vector, of the Quaternion ; because it will be found possible and useful to iden- 
 tify such part with its own Index- Vector (133). Compare the Notes to pages 119, 
 136, 174. 
 
192 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (7.) On the other hand, 
 
 X...Vn = o. if /« = 
 
 IX...Yq = q, if iq^'^; 
 
 a right quaternion behig its own right part. 
 
 203. We had (196, XIX.) a formula which may now be 
 written thus, 
 
 I. . . ob'= S(oB : oa). OA, or /3' = S--a, 
 
 to express the projection o/ob on oa, or of the vector /3 on a ; 
 and we have evidently, by the definition of the new symbol 
 V^-, the analogous formula, 
 
 II. . . ob" = V (oB : oa) . oA, or /3" = V - • a, 
 
 a 
 
 to express the projection of (5 on the plane (through o), which 
 is drawn so as to be perpendicular to a ; and which has been 
 considered in several former sub-articles (comp. 186, (6.), and 
 196, (1.) ). It follows (by 186, &c.) that 
 
 III. . . Tj^" = TY— Ta= perpendicular distance of a from oa; 
 
 this perpendicular being here considered with reference to its 
 length alone, as the characteristic T of the tensor implies. It 
 
 is to be observed that because the factor, V — , in the recent 
 
 a 
 
 formula II. for the projection jS", is not a scalar, we must write 
 
 that factor as a multiplier, and not as a, multiplicand ; althougli 
 
 we were at liberty, in consequence of a general convention 
 
 (15), respecting the multiplication of vectors and scalars, to 
 
 denote the other projection j3' under the form, 
 
 r. ..i3' = aS2(196,XIX.). 
 
 (1.) The equation, 
 
 V^ = 0, 
 
 expresses that the locus of p is the indefinite right line oa. , V 
 
 (2.) The equation, 
 
 ve:i^=o, or ve = v^, 
 
 a a a 
 
 ."■f 
 
 '4^ 
 
 
 
CHAP. I.] GEOMETRICAL EXAMPLES. 193" 
 
 expresses that the locus of p is the mdefinite right line bb", in Fig. 60, which is 
 drawn through the point B, parallel to the line oa. 
 (3.) The equation 
 
 S^Z^ = 0, or S^ = s2, ofl96, (2.), 
 a a a 
 
 has been seen to express that the locus of p is the plane through b, perpendicular 
 to the line oa ; if then we combine it with the recent equation (2.), we shall express 
 that the point p is situated at the intersection of the two last mentioned loci ; or that 
 it coincides with the point b, 
 
 (4.) Accordingly, whether we take the two first or the two last of these recent 
 forms (2.), (3.), namely, 
 
 ve^=o, st^=o, or ve=v^, se=s^, 
 
 a a a a a a 
 
 we can infer this position of the point p: in the first case by inferring, through 202, 
 
 v., that = 0, whence p- (3=0, by 142 ; and in the second case by inferring, 
 
 a 
 
 through 202, VI., that - = — ; so that we have in each case (comp. 104), or as a 
 
 consequence from each system, the equality p = /3, or op = on ; or finally (comp. 20) 
 the coincidence, P = B. 
 
 (5.) The equation, ^^ p ^ ^^ ^ 
 
 a a 
 
 expresses that the locus of the point P is the cylindric surface of revolution, which 
 passes through the point b, and has the line oa for its axis ; for it expresses, by III., 
 that the perpendicular distances of P and B, from this latter line, are equal. 
 
 (6.) The system of the two equations, 
 
 TV^=TV^, S^ = 0, 
 a a y 
 
 expresses that the locus of p is the (generally) elliptic section of the cylinder (5.), 
 made by the plane through o, which is perpendicular to the line oc. 
 
 (7.) If we employ an analogous decomposition of p, by supposing that 
 p=p' + p", p'\\a, p"-^a, 
 the three rectilinear or plane loci, (1.), (2.), (8.), may have their equations thus 
 briefly written : 
 
 p" = 0; p" = /3"; p' = /3': 
 
 while the combination of the two last of these gives p = |3, as in (4.). 
 
 (8.) The equation of the cylindric locus, (5.), takes at the same time the form, 
 Tp" = T/3"; 
 which last equation expresses that the projection p" of the point p, on the plane through 
 o perpendicular to OA, falls somewhere on the circumference of a circle, with o for 
 centre, and ob" for radius : and this circle may 'accordingly be considered as the hast 
 of the right cylinder, in the sub-article last cited. 
 
 204. From the mere circumstance that V^ is always a 
 right qvotient (132), whenceUV^' is a right versor (153), of 
 
 2 c 
 
194 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 which the plane (119), and the axis (127), coincide with those 
 of §', several general consequences easily follow. Thus we have 
 generally, by principles already established, the relations : 
 
 I. . .ZV^ = ^; 11. . . Ax.V^ = Ax.tJV^ = Ax.ry; 
 
 III. . . KV^ = - V^, or KV = - V (144) ; 
 IV. ..SV^ = 0, or SV=0(196, VII.); 
 V. . .(UV^)2 = -1 (153,159); 
 and therefore, 
 
 VI. . . (V^)2 = -(TV^)^ = -NV^,* 
 
 because, by the general decomposition (188) of a quaternion 
 mio factors^ we have 
 
 VII. .. V^ = TV^.UV^. 
 We have also (comp. 196, VI.), 
 
 VIII. . . VS^ = 0, or VS = (202, VII.) ; 
 IX. . . VV^ = V^, or V^ = VV = V (202, IX.) ; 
 and X. .. VK^=-V(?, or VK = - V, 
 
 because conjugate quaternions have opposite right parts, by the 
 definitions in 137, 202, and by the construction of Fig. 36. 
 For the same reason, we have this other general formula, 
 
 XI. . . K^ = S^-V^, or K = S-V; 
 but we had 
 
 ^ = S^ + V^, or I = S + V, by 202, III., IV.; 
 hence not only, by addition, 
 
 q + Kq = 2Sqy or 1 + K= 2S, as in 196, I., 
 but also, by subtraction, 
 
 XIL ..^-K^ = 2V^, or I-K = 2V; 
 whence the Characteristic, V, of the Operation of taking the 
 RightPartofa Quaternion (comp. 132, (6.); 137; 156; 187; 
 196), may be dejined hj either of the two following symbolical 
 equations : 
 
 XIII. .. v = i-S(202, IV.); XIV. . . v = i(i-K); 
 
 whereof the former connects it with the characteristic S, and 
 
 * Compare the Note to page 130. 
 
CHAP* I.] PROPERTIES OF THE RIGHT (OR VECTOR) PART. 195 
 
 the latter with the characteristic K ; while the dependence of 
 K on S and V is expressed by the recent formula XI. ; and 
 that of S on K by 196, 11'. Again, if the line ob, in Fig. 50, 
 be multiplied (15) by any scalar coefficient, the perpendicular 
 bb' is evidently multiplied by the same ; hence, generally, 
 
 XV. . . Nxq = rcV^, if x be any scalar ; 
 and therefore, by 188, 191, 
 XVI. .,Yq = Tq . VU^, and XVII. . . TV^ = Tq .TVU^. 
 
 But the consideration of the right-angled triangle, ob'b, in the 
 same Figure, shows that 
 
 XVIII. . .TV^ = T^.sinz^, 
 because, by 202, II., we have 
 
 TV^ = T(ob":oa) = T.ob":T.oa, 
 and 
 
 T.ob"= T.ob . sin aob ; 
 
 we arrive then thus at the following general and useful for- 
 mula, connecting quaternions with trigonometry anew : 
 
 XIX. . .TVU^ = sinz^; 
 by combining which with the formula, 
 
 SU^ = co3Z^(196, XVL), 
 we arrive at the general relation : 
 
 XX. ..(SU^)2 + (TVUy)2 = l; 
 
 which may also (by XVII., and by 196, IX.) be written thus : 
 
 XXI. ..(S^)^-f(TV^)^=(T^)^; 
 
 and might have been immediately deduced, without sines and 
 cosines, from the right-angled triangle, by the property of the 
 square of the hypotenuse, under the form, 
 
 (T.ob')2+ (T.b'b)'^ = (T.ob)^ 
 The same important relation may be expressed in various other 
 ways ; for example, we may write, 
 
 XXII. . . % = T^2 = S^^ - Yq\ 
 
 where it is assumed, as an abridgment oi notation (comp. 199, 
 
 VII., VIII.), that 
 
 XXIII. . . V^^ = {Yq)\ but that XXIV. . . V. j^ = V(f ), 
 
196 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 the import of this last symbol remaining to be examined. 
 And because, by the definition of a norm, and by the proper- 
 ties of S^ and V^', 
 
 XXV. . . NS^ = Sf , but XXVI. . . NVy = - Yq\ 
 we may write also, 
 
 XXVII. . . % = N(S^ + Yq) = NS^ + NV^ ; 
 a result which is indeed included in the formula 200, VIII., 
 since that equation gives, generally, 
 
 XXVIII. . .N(y + rr) = % + Naj, if z^ = ^; 
 
 X being, as usual, any scalar. It may be added that because 
 (by 106, 143) we have, as in algebra, the identity, 
 
 XXIX. ..-(?'+?) = -?'- y, 
 the opposite of the sum of any two quaternions being thus equal 
 to the sum of the opposites, we may (by XL) establish this 
 other general formula : 
 
 XXX. ..-K^ = V^-S^; 
 the opposite of the conjugate of any quaternion q having thus 
 the same right part as that quaternion, but an opposite scalar 
 part. 
 
 (1.) From the last formula it may be inferred, that 
 
 if q' = -Kq, then Yq' = + Yg, but Sq' = -Sq; 
 and therefore that 
 
 Tq'=Tg, and Ax. 5'= Ax. g, but L<i =^tt— Lq\ 
 
 which two last relations might have been deduced from 138 and 143, without the 
 introduction of the characteristics S and V. 
 (2.) The equation, 
 
 (v^Y=fv^V, or(byXXVL), NV^ = NV^, 
 \ a \ \ a j a a 
 
 like the equation of 203, (5.), expresses that the locus of p is the right cylinder, or 
 cylinder of revolution, with oa for its axis, which passes through the point b. 
 (3.) The system of the two equations, 
 
 V2 
 
 [^'^-[^l\ «^»- 
 
 like the corresponding system in 203, (6.), represents generally an elliptic section of 
 the same right cylinder ; but if it happen that y H a, the section then becomes cir- 
 cular. 
 
CHAP. I.] GEOMETRICAL EXAMPLES. 197 
 
 (4.) The system of the two equations, 
 
 S- = x, (v^]=£c2_i, with x>-l, x<J, 
 
 represents the circle,* in which the cylinder of revolution, with OAfor axis, and with 
 (1 - x^)iTa for radius, is perpendicularly cut by a plane at a distance = + xTa from 
 o ; the vector of the centre of this circular section being xa. 
 
 (5.) While the scalar x increases (algebraically) from — 1 to 0, and thence to 
 + 1, the connected scalar VCl - x^) at first increases from to 1, and then decreases 
 from 1 to ; the radius of the circle (4.) at the same time enlarging from zero to a 
 maximum =Ta, and then again diminishing to zero ; while the position of the centre 
 of the circle varies continuously, in one constant direction, from ajirst limit-point a', 
 if oa' = — a, to the point A, as a second limit. 
 
 (6.) The locus of all such circles is the sphere, with aa' for a diameter, and there- 
 fore with o for centre ; namely, the sphere which has already been represented by the 
 
 equation Tp = Ta of 186, (2.) ; or by T ^ = 1, of 187, (1.) ; or by 
 
 S^^^ = 0, of 200, (11.); 
 ' p + a 
 
 but which now presents itself under the new form, 
 
 1. 
 
 (»;y-('n"- 
 
 obtained by eliminating x between the two recent equations (4). 
 
 (7.) It is easy, however, to return from the last form to the second, and thence 
 to the first, or to the third, by rules of calculation already estabhshed, or by the ge- 
 neral relations between the symbols used. In fact, the last equation (6.) may be 
 written, by XXII., under the form, 
 
 a 
 
 whence 
 
 T^=l, by 190, VI.; 
 a 
 
 and therefore also Tp = Ta, by 187, and S ^^ = 0, by 200, (11.). 
 
 p -\- ci 
 
 (8.) Conversely, the sphere through a, with o for centre, might already have 
 been seen, by the first definition and property of a norm, stated in 145, (ll.)> to ad- 
 mit (comp. 145, (12.) ) of being represented by the equation N - = 1 ; and there- 
 
 a 
 
 fore, by XXII., under the recent form (6.) ; in which if we write x to denote the 
 variable scalar S -, as in the first of the two equations (4.), we recover the second of 
 those equations : and thus might be led to consider, as in (6.), the sphere in question 
 
 * By the word " circle," in these pages, is usually meant a circumference, and 
 not an area ; and in like manner, the words " sphere," *' cylinder," " cone," &c., are 
 usually here employed to denote surfaces, and not volumes. 
 
198 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 aa the locus of a variable circle^ which is (as above) the intersection of a variable 
 cylinder^ with a variable plane perpendicular to its axis. 
 
 (9.) The same sphere may also, by XXVII., have its equation written thus. 
 
 Nfs^ + V^Vl; or Tfs^-fV^V 
 
 (10.) If, in each variable plane represented^by the first equation (4.), we conceive 
 the radius of the circle, or that of the variable cylinder, to be multiplied by any con- 
 stant and positive scalar a, the centre of the circle and the axis of the cylinder re - 
 maining unchanged, we shall pass thus to a new system of circles, represented by this 
 new system of equations, 
 
 se= 
 
 "' [^L] -""-'■ 
 
 (11.) The locus of these new circles will evidently be a Spheroid of Revolution ; 
 the centre of this new surface being the centre o, and the axis of the same surface 
 being the diameter aa', of the sphere lately considered : which sphere is therefore 
 either inscribed or circumscribed to the spheroid, according as the constant a > or 
 < 1 ; because the radii of the new circles are in the first case greater, but in the se- 
 cond case less, than the radii of the old circles ; or because the radius of the equator 
 of the spheroid = aTa, while the radius of the sphere = Ta. 
 
 (12.) The equations of the two co-axal cylinders of revolution, which envelope 
 respectively the sphere and spheroid (or are circumscribed thereto) are : 
 
 (v-:y=-- (^£1=- 
 
 NV^=l, NV^=i 
 
 TV^=1, TV^=a. 
 a a 
 
 (13.) The system of the two equations, 
 
 S-=ir, (v|j=a;2_i, with j3 no< II a, 
 
 represents (comp. (3.) ) a variable ellipse, if the scalar x be still treated as a va- 
 riable. 
 
 (14.) The result of the elimination of x between the two last equations, namely 
 this new equation, 
 
 or 
 
 NS ^ + NV §= 1, by XXV., XXVI. ; 
 
 a p 
 
 or 
 
 Nfs^ + v|Ul, by XXVII.; 
 or finally, 
 
 Tfs^ + V|^=l, by 190, VI., 
 
CHAP. I.] QUATERNION EQUATION OF THE ELLIPSOID. 199 
 
 represents the locus of all such ellipses (13.), and will be found to be an adequate 
 
 representation, through quaternions, of the general Ellipsoid (with three unequal 
 
 axes) : that celebrated surface being here referred to its centre, as the origin o of 
 
 vectors to its points ; and the six scalar (or algebraic) constants, which enter into ^/ 
 
 the usual algebraic equation (by co-ordinates) of such a central ellipsoid, being here / «-^ 
 
 virtually included in the two independent vectors, a and (3, which may be called its 
 
 two Vector- Constants * 
 
 (15.) The equation (comp. (12.) ), 
 
 (il= 
 
 NV|=1, or TV^=1, 
 
 represents a cylinder of revolution, circumscribed to the ellipsoid, and touching it 
 along the ellipse which answers to the value a: = 0, in (13.) ; so that the plane of 
 this ellipse of contact is represented by the equation, 
 
 a 
 
 the normal to this pZane being thus (comp. 196, (17.) ) the vector a, or oa; while 
 the axis of the lately mentioned enveloping cylinder is (3, or ob. 
 
 (16.) Postponing any further discussion of the recent quaternion equation of the 
 ellipsoid (14.), it may be noted here that we have generally, by XXII., the two fol- 
 lowing useful transformations for the squares, of the scalar Sq, and of the right part 
 Yq, of any quaternion q : 
 
 XXXI. ..852 = T52 f V52 ; XXXII. ..¥52= Sq^ - Tq^. 
 
 (17.) In refei-ring briefly to these, and to the connected formula XXII., upon 
 occasion, it may be somewhat safer to write,' 
 
 (S)2 = (T)2 + (V)2, (Vy = (S)2 - (T)2, (T)2 = (S)2 - (V)2, 
 
 than S2 = T2 + V2, &c. ; because these last forms of notation, S2, &c., have been 
 otherwise interpreted already, in analogy to the known Functional Notation, or No- 
 tation of the Calculus of Functions, or of Operations (comp. 187, (9.); 196, VI. ; 
 and 204, IX.). 
 
 (18.) In pursuance of the same analogy, any scalar may be denoted by the gene- 
 ral symbol, 
 
 V-'O; 
 
 because scalars are the only quaternions of which the right parts vanish. 
 
 (19.) In like manner, a right quaternion, generally, maybe denoted by the sym- 
 bol, 
 
 S-'O; 
 and since this includes (comp. 204, I.) the right part of any quaternion, we may 
 establish this general symbolic transformation of a Quaternion : 
 
 5 = v-io + s-io. 
 
 (20.) With this form of notation, we should have generally, at least for realf 
 quaternions, the inequalities, 
 
 • It will be found, however, that other pairs of vector-constants, for the central 
 ellipsoid, may occasionally be used with advantage. 
 
 t Compare Art. 149 ; and the Notes to pages 90, 134. 
 
200 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (V-i0)2>0; (S-»0)2<0; 
 so that a (geometrically real) Quaternion is generally of the form : 
 
 Square-root of a Positive^ plus Square-root of a Negative. 
 (21.) The equations 196, XVI. and 204, XIX. give, as a new link between qua- 
 ternions and trigonometry, the formula : 
 
 XXXIII. . . tan Z 5 = TVUg : SUg = TV? : S?. 
 (22.) It may not be entirely in accordance with the theory of that Functional 
 (or Operational) Notation, to which allusion has lately been made, but it will be 
 found to be convenient in practice, to write this last result under one or other of the 
 abridged forms : * 
 
 TV 
 XXXIV. . . tan z: 9 = — - . 5 ; or XXXIV. . . tan Z 9 = (TV : S) 9 ; 
 
 o 
 
 which have the advantage oi saving the repetition of the symbol of the quaternion , 
 when that symbol happens to be a complex expression, and not, as here, a single let- 
 ter, q. 
 
 (23.) The transformation 194, for the index of a right quotient, gives generally, 
 by II., for any quaternion q, the formulae : 
 
 XXXV. . . IVg = TV? . Ax. ? ; XXXVI. . . IUV9 = Ax. q ; 
 so that we may establish generally the symbolicalf equation, 
 
 xxxvr. . . iuv = Ax. 
 
 (24.) And because Ax. (1 : Yq) = - Ax. Vg-, by 135, and therefore = - Ax. q, by 
 II., we may write also, by XXXV., 
 
 XXXV. . . I (1 : Vg) = - Ax. 5 : TV?. 
 
 205. If any parallelogram obdc (comp. 197) be projected 
 on the plane through o, which is perpendicular to oa, the pro- 
 jected figure obV'c" (comp. 11) is still a parallelogram; so 
 
 that 
 
 od" = oc" + ob" (6), or S" = 7" + /3" ; 
 
 and therefore, by 106, 
 
 g":a=(7":a) + (i3":a). 
 Hence, by 120, 202, for any two quaternions, q and q\ we have 
 the general formula, 
 
 • Compare the Note to Art. 199. 
 
 t At a later stage it will be found possible (comp. the Note to page 174, &c.), 
 to write, generally, 
 
 IV? = V?, lUV? = UV? ; 
 
 and then (comp. the Note in page 118 to Art. 129) the recent equations, XXXVI., 
 xxxvr., will take these shorter forms : 
 
 Ax. ? = UV? ; Ax. = UV. 
 
CHAP. I.] RIGHT PART OF A SUM OR DIFFERENCE. 201 
 
 with which it is easy to connect this other, 
 
 IL..y(q'-q) = Yq-yq. 
 Hence also, for any three quaternions, q, q\ q\ 
 
 V(?"+ (y' + !?)) = Vy"+ V(j' + 5) = V/+(V?' + V?) ; 
 and similarly for any greater number of summands : so that 
 we may write generally (comp. 197, II.), 
 
 III. . . VS.7 = SV^, or briefly III'. . . VS = SY ; 
 while the formula II. (comp. 197, IV.) may, in like manner, 
 be thus written, 
 
 IV. ..VA^ = AV^, or IV'. ..VA = AV; 
 
 the order of the terms added, and the mode Oti grouping them, 
 in III., being as yet supposed to remain unaltered, although 
 both those restrictions will soon be removed. We conclude 
 then, that the characteristic V, of the operation of taking the 
 right part (202, 204) of a quaternion, like the characteristic S 
 of taking the scalar (196, 197), and the characteristic K of 
 taking the conjugate (137, 195*), is a Distributive Symbol, or 
 represents a distributive operation: whereas the characteris- 
 tics, Ax., z, N, U, T, of the operations of taking respectively 
 theaa;25(128, 129), the«?z^/e(130), the?zorm (145, (11.) ), the 
 versor (156), and the tensor (187), are not thus distributive 
 symbols (comp. 186, (10.), and 200, VII.) ; or do not operate 
 upon a lohole (or sum)^ by operating on its parts (or sum- 
 mands). 
 
 (1.) We may now recover the sjiKibolical equation K^ = 1 (145), under the form 
 (comp. 196, VI.; 202, IV, ; and 204, IV. VIII. IX. XL): 
 
 V. . . K2 = (S-V)2 = S2-SV-VS + V2 = S + V=1. 
 (2.) In like manner we can recover eacli of the expressions for S^, V^ from the 
 other, under the forms (comp. again 202, IV.) : 
 
 VI. . . S2 = (1-V)2 = 1-2V + V2=1-V = S, as in 196, VI.; 
 VII.. . V2 = (1-S)3=1-2S+S2 = 1-S = V, as in 204, IX.; 
 or thus (comp. 196, II'., and 204, XIV.), from the expressions for S and V in terms 
 ofK: 
 
 * Indeed, it has only been proved as yet (comp. 195, (1.)), that KSj = SKj, 
 for the case of two summands ; but this result will soon be extended. 
 
 2 D 
 
202 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 VIII.. .S2 = i(l+K:)2 = i(l + 2K + K2) = i(l + K) = S; 
 IX. . . V2 = ^(l-K)2=:i(l-2K + K2; = i(l-K) = V. 
 (3.) Similarly, 
 
 X.. . SV = i(l + K)(l-K) = i(l-K2)=0, as in 204, IV.; 
 and XI. . .VS = K1-K) (1 + K) = i (1-^0=0, as in 204, VIII. 
 
 206. As regards the addition {ov subtraction) of such n^^^ 
 parts, Yq, V^-', or generally of any two right quaternions 
 (132), we may connect it with the addition (or subtraction) of 
 their indices (133), as follows. Let obdc be again any paral- 
 lelogram (197, 205), but let oa be now an unit-vector (129) 
 perpendicular to its plane ; so that 
 
 Ta=l, z(/3:a) = Z(7:a) = Z(S:a)=^, S = 7 + /3. 
 
 Let ob'd'c' be another parallelogram in the same plane, ob- 
 tained by a positive rotation of the former, through a right 
 angle, round oa as an axis ; so that 
 
 Z(i3':/3)=A(y:7) = ^(^':S)=|; 
 Ax. (j3' : ^) = Ax. (y : 7) = Ax. (S' : g) = a. 
 Then the three right quotients, /3 : a, 7 : a, and ^ : a, may re- 
 present any two right quaternions, q, q\ and their sum, q -\- q, 
 w^hich is always (by 197, (2.) ) itself o, right quaternion; and 
 the indices of these three right quotients are (comp. 133, 193) 
 the three lines j3', y\ S', so that we may write, under the fore- 
 going conditions of construction, 
 
 /3'=I(i3:a), y = I(7:a), S' = I(g:a). 
 But this third index is (by the second parallelogram) the sum 
 of the two former indices, or in symbols, ^' = 7' + /3' ; we may 
 therefore write, 
 
 I. ..!{(][ ^q) = lq +lq, if Z^ = Zg=|; 
 
 or in words the Index of the Sum* of any two Right Quater- 
 nions is equal to the Sum of their Indices, Hence, generally, 
 for any two quaternions, q and q\ we have the formula, 
 IL. .\Y{q-^q) = lYq^lYq, 
 
 * Compare the Note to page 174. 
 
CHAP. I.] GENERAL ADDITION OF QUATERNIONS. 203 
 
 because V^-, Yq are aliuays right quotients (202, 204), and 
 V {q' + q) is always their sum (205, I.) ; so that the index of 
 the right part of the sum of any two quaternions is the sum of 
 the indices of the right parts. In like manner, there is no diffi- 
 culty in proving that 
 
 m...l{q'-q)^lq-lq, if Zj = /y = |; 
 
 and generally, that 
 
 IV. ..IV(^'-^)=IV^'-IV^; 
 the Index of the Difference of any two right quotients, or of 
 the right parts of any two quaternions, being thus equal to the 
 Difference of the Indices* We may then reduce the addition 
 or subtraction of any two such quotients, or parts, to the addi- 
 tion or subtraction of their indices ; a right quaternion being 
 always (by 133) determined, when its index is given, or 
 known. 
 
 207. We see, then, that as the Multiplication of any 
 tico Quaternions was (in 191) reduced to (1st) the arithmetical 
 operation of multiplying their tensors, and (Ilnd) the geometri- 
 cal operation of multiplying their versors, which latter Avas con^ 
 structed by a certain composition of rotations^ and was repre- 
 sented (in either of two distinct but connected ways, 167, 175) 
 by sides or angles of a spherical triangle: so the Addition of 
 any two Quaternions maybe reduced (by 197, 1., and 206, II.) 
 to, 1st, the algebraical addition of their scalar parts ^ considered 
 as two positive or negative numbers (16) ; and, Ilnd, the geo- 
 metrical addition of the indices of their right parts, considered 
 as certain vectors (1) : this latter Addition of Lines being per- 
 formed according to the Rule of the Parallelogram (6.).t In 
 
 * Compare again the Note to page 174. 
 
 t It does not fall within the plan of these Notes to allude often to the history of 
 the subject ; but it ought to be distinctly stated that this celebrated Mule, for what 
 may be called Geometrical Addition of right lines, considered as analogous to compo- 
 sition of motions (or of forces), had occurred to several writers, before the invention 
 of the quaternions : although the method adopted, in the present and in a former 
 ■work, of deducing that rule, by algebraical analogies, from the symbol b — A (1) 
 for the line ab, may possibly not have been anticipated. The reader may com- 
 pare the Notes to the Preface to the author's Volume of Lectures on Quaternions 
 (Dublin, 1853). 
 
204 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 like manner, as the general Division of Quaternions was seen (in 
 191) to admit of being reduced to an arithmetical division of 
 tensors, and 2^ geometrical division ofversors, so we may now 
 (by 197, III., and 206, IV.) reduce, generally, the Subtrac- 
 tion of Quaternions to (1st) an algebraical subtraction of sea- 
 larsy and (Ilndj Sk geometrical subtraction of vectors: this last 
 operation being again constructed by a parallelogram, or even 
 by a plane triangle (comp. Art. 4, and Fig. 2). And because 
 the sum of any given set of vectors was early seen to have a 
 value (9), which is independent of their order, and of the mode 
 of grouping them, we may now infer that the Stim of any num- 
 ber of given Quaternions has, in like manner, a Value (comp. 
 197, (l'))» which is independent of the Order, and of the 
 Grouping of the Summands: or in other words, that the general 
 Addition of Quaternions is a Commutative* and an Associative 
 Operation. 
 
 (1.) The formula, 
 
 Y^q=-2Yq, of 205, III., 
 
 is now seen to hold good, for any number of quaternions, independently of the arrange- 
 ment of the terms in each of the two sums, and of the manner in which they may be 
 associated. 
 
 (2.) We can infer anew that 
 
 K (q' + q) = K^' 4 Kg-, as in 195, II., 
 under the form of the equation or identity, 
 
 S (7' + 9) - V (q +q)= {Sq - Yq) + QSq - Yq). 
 
 (3.) More generally, it may be proved, in the same way, that 
 K2g = 2 Kg, or briefly, K2 = SK, 
 whatever the number of the summands may be. 
 
 208. As regards the quotient or product of the right paHs, Yq and 
 Yq', of any two quaternions, let t and f denote the tensors of those 
 two parts, and let x denote the angle of their indices, or of their axes, 
 or the mutual inclination of the axes, or of the planes,] .of the two 
 quaternions q and q' themselves, so that (by 204, XVIII.), 
 
 * Compare the Note to page 175. 
 
 f Two planes, of course, make with each other, in general, two unequal and sup- 
 plementary angles ; but we here suppose that these are mutually distinguished, by 
 taking account of the aspect of each plane, as distinguished from the opposite aspect : 
 which is most easily done (HI-)) ''}' considering the axes as above. 
 
CHAP. I.] QUOTIENT OR PRODUCT OF RIGHT PARTS. 205 
 
 t = TVq = Tq. sin Lq, f = TYq' =Tq\ sin /.q\ 
 and 
 
 x = /. {lYq' : lYq) = L (Ax. q' : Ax. q). 
 Then, by 193, 194, and by 204, XXXV., XXXV'., 
 
 I. . .Yq':Yq = lYq' :lYq = + (TYq' : TYq) . (Ax. q' : Ax. g) ; 
 
 II. . . V^^ V^ = IV^' : I ^ = - (T V^' . TYq) . (Ax. q'-.Ax.q)-, 
 
 and therefore (comp. 198), with the temporary abridgments pro- 
 posed above, 
 III. . . S ( V^' : V^) = ft' cos X ; IV. . . SU (Yq' : V^) = + cos x ; 
 V. . . S{Yq'.Yq)=-t'tcosx- VI. . . ^U {Yq\Yq) = - cos x; 
 VII. ..L{Yq':Yq) = x; VIII. . . L{Yq' . Yq)=7r-x. 
 
 We have also generally (comp. 204, XVIII., XIX.), 
 IX. . . TV (Yq' : Yq) = ft' sin a; ; X. . . T VU ( V^' : Yq) = sin a; ; 
 XI... TV(Vg'.V^)=i'^sina;; XII. . . TY\J (Yq' .Yq)= sin x; 
 
 and in particular, 
 
 XIII. . . V ( V^' : V^) = 0, and XIV. . . V ( V^' . V^) = 0, 
 if/|||.i(123); 
 because (comp. 191, (6.), and 204, VI.) the quotient or product of 
 the right parts of two complanar quaternions (supposed here to be 
 both 7ion-scalar (108), so that t audi' are each >0) degenerates (131) 
 into a scalar, which may be thus expressed : 
 
 XV. . . V^' : V^ = + tt\ and XVI. . .Yq\Yq = - t% if a; = ; 
 but 
 
 XVII. ..V^':V^ = -«'<-', and XYIU. . . Yq\Yq = + t% ifx = 7r; 
 the first case being that of coinciderd, and the second case that of 
 opposite axes. In the more general case oi diplanarity (119), if we 
 denote by B the unit-line which is perpendicular to both their axes, 
 and therefore common to their two planes, or in which those planes 
 intersect, and which is so directed that the rotation round it from 
 Ax. q to Ax. q' is positive (comp. 127, 128), the recent formulae I., 
 II. give easily, 
 
 XIX. . . Ax. (V^': Vg) =+ a; XX. . . Ax. {Yq' ,Yq)=-h', 
 and therefore (by IX., XI., and by 204, XXXV.), the indices of the 
 right parts, of the quotient and product of the right parts of any two 
 diplanar quaternions, may be expressed as follows: 
 
 XXI. . . IV ( V^' : V<7) = + a . ft' sin x ; 
 
 XXII. . . IV {Yq'. Yq) = -S.fi sin x. 
 
206 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book ir. 
 
 (1.) Let ABC be any triangle upon the unit-sphere (128), of which the spheri- 
 cal angles and the corners may be denoted by the same letters A, b, c, while the sides 
 shall as usual be denoted by a^h^ c\ and let it be supposed that the rotation (comp. 
 177) round A from c to b, and therefore that round b from A to c, &c., U positive, 
 as in Fig. 43. Then writing, as we have often done, 
 
 q = (3: a, and q' = y ■ (3, where a = OA, &c., 
 we easily obtain the the following expressions for the three scalars t, t', x, and for 
 the vector d : 
 
 i? = sin c ; if ' = sin a ; a; = tt — b ; d = - (3. 
 
 (2.) In fact we have here, 
 
 Tq = Tq=l, Lq = c, Lq=a\ 
 
 whence t and <' are as just stated. Also if a', b', c' be (as in 175) the positive poles 
 of the three successive sides bc, ca, ab, of the given triangle, and therefore the points 
 A, b, c the negative poles (comp. 180, (2.)) of the new arcs b'c', c'a', a'b', then 
 
 Ax. q = oc'. Ax. q' = Oa' ; 
 
 but X and d are the angle and the axis of the quotient of these two axes, or of the 
 quaternion which is represented (162) by the arc c'a'; therefore x is, as above 
 stated, the supplement of the angle b, and d is directed to the point upon the sphere, 
 which is diametrically opposite to the point b. 
 
 (3.) Hence, by III. V. VII. VIII. IX. XI., for any triangle abc on the unit- 
 sphere, with a =OA, &c., we have the formulae: 
 
 XXIII. 
 
 XXV. 
 
 XXIV. . . s 
 
 ^ V- 
 
 ^v^ 
 
 (4.) Also, by XIX. XX. XXI. XXII, 
 still positive, 
 
 XXXL 
 
 ) 
 
 sin a cosec c cos b : 
 
 = + sin a sm c < 
 
 XXVI. . . L 
 
 H-^lh- 
 
 XXVII. 
 
 XXVIIL. . TV 
 
 + sm a cosec c sm b ; 
 
 + sin a sin c sin b. 
 if the rotation round b from a to c be 
 
 XXIX. . . Ax. 
 
 XXX. . . Ax. 
 
 ('r'a 
 
 = + ^; 
 
 xxxn. . .ivi v^.v 
 
 V — 1 = — /3 sin a cosec c sin b 
 a j 
 
 /3' 
 
 + (3sma sin c sin b. 
 
 (5.) If, on the other hand, the rotation round b from a to c were negative, then 
 writing for a moment ai= — a, /3i = — /?, yi = — y, we should have a new and ojo/jo- 
 site triangle, AiBiCi, in which the rotation round Bi from Ai to Ci would be positive, 
 but the angle at bi equal in magnitude to that at b ; so that by treating (as usual) 
 all the angles of a spherical triangle as positive, we should have Bi = b, as well as 
 Ci, = c, and ai — a; and therefore, for example, by XXXI. 
 
CHAP. I.] COLLINEAR QUATERNIONS. 207 
 
 IV V ^ : V— ) = - /3i sin ai cosec ci sin bi, 
 V Pi ai I 
 
 or IV I V ^ : V - 1 = + j8 sin a cosec c sin b ; 
 \ (3 a] 
 
 the four formulae of (4.) would therefore still subsist, provided that, for this new 
 direction of rotation in the given triangle, we were to change the sign of [3, in the 
 second member of each. 
 
 (6.) Abridging, generally IVg' : ^q to (IV: S)^, as TVg: Sg- was abridged, in 
 204, XXXIV'., to (TV: S)*?, we have by (5.), and by XXIV., XXXII., this other 
 general formula, for any three unit- vectors a, /3, y, considered still as terminating 
 at the corners of a spherical triangle abc : 
 
 XXXIII. .. (IV:S)f v|.V^^ = ±j6tan 
 
 the upper or the lower sign being taken, according as the rotation round b from a to 
 
 A 
 
 c, or that round /3 from a to y, which might perhaps be denoted by the symbol rtj8y, 
 and which in quantity is equal to the spherical angle b, is positive or negative. 
 
 209. When the planes of any three quaternions q, q'^ q'\ consi- 
 dered as all passing through the origin o (119), contain any co7iimon 
 line, those three may then be said to be Collinear^- Quaternions ; and 
 because the axis of each is then perpendicular to that line, it follows 
 that the Axes of ColUnear Quaternions are Complanar : while con- 
 versely, the complanarity of the axes insures the collinearity of the 
 quaternions, because the perpendicular to the plane of the axes is a line 
 common to the planes of the quaternions. 
 
 (1.) Complanar quaternions are always collinear ; but the converse proposition 
 does not hold good, collinear quaternions being not necessarily complanar. 
 
 (2.) Collinear quaternions, considered as fractions (101), can always be reduced 
 to a common denominator (120) ; and conversely, if three or more quaternions can be 
 so reduced, as to appear under the form of fractions with a common denominator e, 
 those quaternions must be collinear : because the line e is then common to all their 
 planes. 
 
 (3.) Any two quaternions are collinear with any scalar ; the plane of a scalar 
 being indeterminate^ (I'^l)- 
 
 (4.) Hence the scalar and right parts, Sg, Sg', Vg, Vg', of any two quaternions, 
 are always collinear with each other. 
 
 (5.) The conjugates of collinear quaternions are themselves collinear. 
 
 * Quaternions of which the planes are parallel to any common line may also be 
 said to be collinear. Compare the first Note to page 113. 
 t Compare the Note to page 114. 
 
208 ELEMENTS OF QUATERNIONS. [boOK II. 
 
 210. Let $', 5', ql' be any three collinear quaternions; and let a 
 denote a line common to their planes. Then we may determine 
 (comp. 120) three other lines y8, 7, ^, such that 
 
 ^ a' ^ "a' ^ a' 
 
 and thus may conclude that (as in algebra), 
 
 because, by 106, 107, 
 
 ^y .. ^y _ 7 + /^ g _ 7 + /3 _ 7 ^ /3 ^ 7 « ^ « 
 
 a a jh a ^ b S d a d a S 
 
 In like manner, at least under the same condition of collinearity,* it 
 may be proved that 
 
 II. . . {q'-q)q" = q'q"-qq''. 
 Operating by the characteristic K upon these two equations, and 
 attending to 192, II., and 195, II., we find that 
 
 III. . . K2^^(%'+%) = K$'^K/+K^^K^; 
 
 IV. . . K^'^(%'-K^) = K2'^K5'-K^'^K^; 
 
 where (by 209, (5.) ) the three conjugates of arbitrary collinears, 
 K5, K(2^ ^q"-> may represent any three collinear quaternions. We 
 have, therefore, with the same degree of generality as before, 
 
 V. . . q" {q' + g) = q"q' + q"q ; VI. . . q'^ {q' -q)= q"q' - q"q. 
 
 If, then, q^ q', q", q'"hQ any four collinear quatet-mons, we may esta- 
 blish the formula (again agreeing with algebra) : 
 
 VII. . . (q'^' + q") {q' + q) =- q'"q' + q"q' + q'"q + q'^q ; 
 and similarly for any greater number, so that we may write briefly, 
 
 VIII. .. ^q',^qr=:2q'q, 
 where 
 
 ^q' = qy + q2+"-\-qm> ^q' = q'i + q2 + ' •+q'ny 
 
 and 
 
 -Eq'q = q\q, + . . q^'q^ -Yq'-iqx + . . . + q'^q^^, 
 
 m and n being any positive whole numbers. In words (comp. 13), 
 the Multiplication of Collinearf Quaternions is a Doitbli/ Distributive 
 Operation. 
 
 * It will soon be seen, however, that this condition is unnecessary. 
 
 t This distributive property of multiplication will soon be found (compare the last 
 Note) to extend to the more general case, in which the quaternions are not collie 
 near. 
 
CHAP. I.] DISTRIBUTIVE MULTIPLICATION OF COLLINEARS. 209 
 
 (1.) Hence, by 209, (4.), and 202, III., we have this general transformation, 
 for the product of any two quaternions : 
 
 IX. .. qq = Sq. Sq + Yq\ Sq + Sq'.Yq + Yq'.Yq. 
 (2.) Hence also, for the square of any quaternion, we have the transformation 
 (comp. 126 ; 199, VII. ; and 204, XXHI.) : 
 
 X. . . q^=Sq^ + 2Sq.Yq + Yq^. 
 (3.) Separating the scalar and right par^s of this last expression, we find these 
 other general formulae : 
 
 XL . . S . 52 = S52 + Vg3 ; XII. . . V . 92 = 2Sg . V? ; 
 whence also, dividing by Tq^, we have 
 
 XIII. . . SU((?2) = (SU5)2 + (YUg)2; XIV. . . Y\JCq^) = 2S\Jq.YUq. 
 (4.) By supposing q' = Kq, in IX. , and therefore Sg' = Sg, Vg-' = — Yq, and trans- 
 posing the two conjugate and therefore complanar factors (corap, 191, (1.) ), we ob'- 
 tain this general transformation for a norm, or for the square of a tensor (comp. 190, 
 V. ; 202, III. ; and 204, XI.) : 
 
 XV. . . Tg2 = Ng = qKq = (Sg + Vg) (Sg - Vg) = Sg2 - Vg2 ; 
 
 which had indeed presented itself before (in 204, XXII.) but is now obtained in a 
 new way, and without any employment of sines, or cosines, or even of the well-known 
 theorem respecting the square of the hypotenuse. 
 
 (5.) Eliminating Vg2, by XV., from XI., and dividing by Tg2, we find that 
 
 XVI. . . S . 92 = 2Sg2 - Tg2 ; XVH. . . SU(g2) = 2 (SUg)2 - 1 ; 
 
 agreeing with 199, VI. and IV., but obtained here without any use of the known 
 formula for the cosine of the double of an angle. 
 
 (6.) Taking the scalar and right parts of the expression IX., we obtain these other 
 
 general expressions : 
 
 XVIII. . . Sg'g = Sg'. Sg + S(Vg'. Vg) ; 
 XIX. . . Yq'q = Yq'. Sq + Yq.Sq' + Y (Yq'.Yq) ; 
 in the latter of which we may (by 126) transpose the two factors, Vg', Sg, or Vg, 
 Sg'. We may also (by 206, 207) write, instead of XIX., this other formula : 
 XIX'. . . IVg'g = IVg'. Sg + IVg . Sg' + IV(Vg'. Vg). 
 (7.) If we suppose, in VII., that g" = Kg, g"' = Kg', and transpose (comp. (4.) ) 
 the two complanar (because conjugate) factors, q' + q and K(g'+g), we obtain the 
 following general expression for the norm of a sum : 
 
 (g + g) K (g' + g) = g'Kg' + gKg' + g'Kg + gKg ; 
 or briefly, 
 
 XX. . . N (g' + g) = Ng' + 2S . gKg' + Ng, as in 200, VII. ; 
 because 
 
 g'Kg = K. gKg', by 192, II., and (1 + K).gKg'= 2S.gKg', by 196, II'. 
 (8.) By changing g' to x in XX., or by forming the product of g + a? and 
 Kg + X, where x is any scalar, we find that 
 
 XXI.. .N(g + a;) = ]Srg + 2a;Sg + a;2, as in 200, VIII. ; 
 whence, in particular, 
 
 XXr. . . N(g - 1) = Ng - 2Sg -|- 1, as in 200, II. 
 2 E 
 
210 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (9.) Changing q to fi: a, and multiplying by the square of Ta, we get, for any 
 two vectors, a and /3, the formula, 
 
 XXII. . . T(/3 - a)2 = T/32 - 2T/3 . Ta . SU ^ + Ta\ 
 in which Ta2 denotes* (Ta)2; because (by 190, and by 196, IX.), 
 
 N(5-ll = Nt5=(I(^Y, and S^ = ^^Sne 
 \a J a \ Ta I a la a 
 
 (10.) In any plane triangle, abc, with sides of which the lengths are as usual 
 denoted by a, &, c, let the vertex c be taken as the origin o of vectors ; then 
 
 o = CA, /3 = CB, j3-a = AB, Ta = 6, T/3 = a, T(j3-a) = c, SU- = cosc; 
 
 a 
 
 we recover therefore, from XXII., the fundamental formula of plane trigonometry, 
 
 under the form, 
 
 XXIIl. . . c2 = a2 - 2ab cos c -i- b^. 
 
 (11.) It is important to observe that we have not here been arguing in a circle ; 
 because although, in Art. 200, we assumed, for the convenience of the student, a pre- 
 vious knowledge of the last written formula, in order to arrive more rapidly at certain 
 applications, yet in these recent deductions from the distributive property YIU. of 
 multiplication of (at least) collinear quaternions, we have founded nothing on the re- 
 sults of that former Article ; and have made no use of any properties of oblique-an- 
 gled triangles, or even of right-angled ones, since the theorem of the square of the 
 hypotenuse has been virtually proved anew in (4.) : nor is it necessary to the argu- 
 ment, that any properties of trigonometric functions should be known, beyond the 
 mere definition of a cosine, as a certain projecting factor, from which the formula 
 196, XVI. was derived, and which justifies us in writing cose in the last equation 
 (10.). The geometrical Examples, in the sub-articles to 200, may therefore be read 
 again, and their validity be seen anew, without any appeal to even plane trigonometry 
 being now supposed.' 
 
 (12.) The formula XV. gives Sg2 = T52 + V52, as in 204, XXXI. ; and we know 
 that V52, as being generally the square of a right quaternion, is equal to a negative 
 scalar (comp. 204, VI.), so that 
 
 XXIV . . Vg2 < 0, unless Lq = 0, or = tt, 
 
 in each of which two cases V9 = 0, by 202, (0.), and therefore its square vanishes ; 
 
 XXV. . . Sg2 < Tg2, (SU9)2 < 1, 
 in every other case. 
 
 * We are not yet at liberty to interpret the symbol Ta2 as denoting also T(a2) ; 
 because we have not yet assigned any meaning to the square of a vector, or generally 
 to the product of two vectors. In the Third Book of these Elements it will be shown, 
 that such a square or product can be interpreted as being a quaternion : and then it 
 will be found (comp, 190), that 
 
 T(a2) = (Ta)2 = Ta2, 
 whatever vector a may be. 
 
CHAP. I.] APPLICATIONS TO SPHERICAL TRIGONOMETRY. 211 
 
 (13.) It might therefore have been thus proved, without any use of the transfor- 
 mation SUg = cos Z. 5- (196, XVI.), that (for any real quaternion q) we have the in- 
 equalities, 
 
 XXVI. . . SU9<+1, S>\Jq>-l, and S5<+Tg, S>q>-T:q, 
 unless it happen that Z g = 0, or = tt ; &\Jq being = + 1, and 85- = + Tg-, in the first 
 case ; whereas SUg = - 1, and Sg = — Tg, in the second case. 
 
 (14.) Since Tg2 = Ng, and Tq . Tq = T. gKg' = T . q'Kq = Ng . T (g' : g), while 
 S . gKg' = S . g'Kg = Ng . S (g' : g), the formula XX. gives, by XXVI., 
 
 XXVII. . . (Tg' + Tg)2-T(g' + g)2 = 2(T-S)gKg' = 2Ng.(T-S) (g':g)>0, 
 if we adopt the abridged notation, 
 
 XXVIIL . . Tg - Sg = (T - S) g, 
 and suppose that the quotient g' : g is not a positive scalar ; hence, 
 
 XXIX. . . Tg' + Tg>T(g' + g), unless q=xq, and x>0; 
 in which excepted case, each member of this last inequality becomes = (1 + aj)Tg. 
 
 (15.) "Writing g = j3 : a, g'= 7 : a, and multiplying by Ta, the formula XXIX. 
 becomes 
 
 XXX. . . Ty + T/3>T(y + /3), unless y=a;/3, a;>0; 
 
 in which latter case, but not in any other, we have Uy = U/3 (155). We therefore 
 arrive anew at the results of 186, (9.), (10.), but without its having been necessary 
 to consider any triangle, as was done in those former sub-articles, 
 
 (16.) On the other hand, with a corresponding abridgment of notation, we have, 
 by XXVI., 
 
 XXXI. . . Tg + Sg=(T+S)g>0, unless Z.g=7r; 
 
 also, by XX., &c., 
 
 XXXII. . . T(g'+ g)^ - (Tg' -Tg)2= 2(T + S)gKg' = 2Ng.(T + S) (g' : g) ; 
 
 hence, 
 
 XXXIII. . . T (g' + g) > + (Tg' - Tg), unless g' = - a;g, a: > ; 
 where either sign may be taken. 
 
 (17.) And hence, on the plan of (15.), for any two vectors ]3, y, 
 
 XXXIV. . . T (y + 18) > + (Ty - T/3), unless Uy = - Uj3, 
 whichever sign be adopted ; but, on the contrary, 
 
 XXXV. ..T(y + /3) = ±(Ty-T/3), if Uy = -U/3, 
 
 the upper or the lower sign being taken, according as Ty > or < T/3 : all which 
 agrees with what was inferred, in 186, (11.), from ^eome^ncaZ considerations alone, 
 combined with the definition of Ta. In fact, if we make j3 = ob, y = oc, and - y 
 = oc', then obc' will be in general a plane triangle, in which the length of the side 
 BC' exceeds the difference of the lengths of the two other sides ; but if it happen that 
 the directions of the two lines ob, oc' coincide, or in other words that the lines OB, 
 oc have opposite directions, then the difference of lengths of these two lines becomes 
 equal to the length of the line bc'. 
 
 (18.) With the representations of g and g', assigned in 208, (1.), by two sides of 
 a spherical triangle abc, we have the values, 
 
 Sg = cosc, Sg' = cosa, Sg'g = S(y : a) = cos t ; 
 
212 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 the equation XVIII. gives therefore, by 208, XXIV., the fundamental formula of 
 spherical trigonometry (comp. (10.) ), as follows : 
 
 XXXVI. . . cos 6 = cos a cos c + sin a sin c cos b. 
 (19.) To interpret, with reference to the same spherical triangle, the connected 
 equation XIX., or XIX'., let it be now supposed, as in 208, (6.), that the rotation 
 round b from c to a is positive, so that b and b' are situated at the same side of the 
 arc CA, if b' be still, as in 208, (2.), the positive pole of that arc. Then writing 
 a' = oa', &c., we have 
 
 \Yq — y sin c ; IV^'' = a' sin a ; IVg-'^ = — /3' sin 6 ; 
 and IV (Vg'. Yq) = — /3 sin a sin c sin b (comp. 208, (5.) ), 
 
 with the recent values (18.), for Sg and Sj'; thus the formula XIX'. becomes, by 
 transposition of the two terms last written : 
 
 XXXVII. . . j8 sin a sin c sin b = a sin a cos c + /3' sin h-\-y' sin c cos a. 
 (20.) Let jO =op be any unit-vector; then, dividing each term of the last equa- 
 tion by jO, and taking the scalar of each of the four quotients, we have, by 196, XVI., 
 this new equation : 
 
 XXXVIII. . . sin a sin c sin b cos pb = sin a cos c cos pa' + sin h cos pb' 
 
 + sin c cos a cos pc' ; 
 where a, 6, c are as usual the sides of the spherical triangle abc, and a', b', c' are 
 still, as in 208, (2.), the positive poles of those sides; but p is an arbitrary point, 
 upon the surface of the sphere. Also cos pa', cos pb', cos pc', are evidently the sines 
 of the arcual perpendiculars, let fall from that point upon those sides ; being positive 
 when p is, relatively to them, in the same hemispheres as the opposite corners of the 
 triangle, but negative in the contrary case ; so that cos aa', &c., are positive, and 
 are the sines of the three altitudes of the triangle. 
 
 (21.) If we place p at b, two of these perpendiculars vanish, and the last formula 
 becomes, by 208, XXVIIL, 
 
 XXXIX. . . sin6cosBB' = sinasincsinB = TVt V^.V- 1; 
 
 \ ^ aj 
 
 such then is the quaternion expression for the product of the sine of the side ca, mul- 
 tiplied by the sine of the perpendicular let fall upon that side, from the opposite ver- 
 tex B. 
 
 (22.) Placing p at A, dividing by sin a cos c, and then interchanging b and c, we 
 get this other fundamental formula of spherical trigonometry, 
 XL. . . cos aa'= sin c sin b = sin 6 sin c ; 
 and we see that this is included in the interpretation of the quaternion equation 
 XIX., or XIX'., as the formula XXXVI. was seen in (18.) to be the interpretation 
 of the connected equation XVIII. 
 
 (23.) By assigning other positions to p, other formulae of spherical trigonometry 
 may be deduced, from the recent equation XXXVIII. Thus if we suppose p to co- 
 incide with b', and observe that (by the supplementary* triangle), 
 
 * No previous knowledge of spherical trigonometry, properly so called, is here 
 supposed ; the supplementary relations of two polar triangles to each other forming 
 rather a part, and a very elementary one, of spherical geometry. 
 
CHAP. I.] GENERAL DISTRIBUTIVE PROPERTY. 213 
 
 b'c' = tt — a, c'a' = tt — b, a'b'= tt — c, 
 while 
 
 cos bb' = sin a sin c = sin c sin a, by XL., 
 
 we easily deduce the formula, 
 
 XLI. . . sin a sin c sin A sin b sin c = sin b — cos c cos c sin A - cos a cos A sin c ; 
 
 which obviously agrees, at the plane limit, with the elementary relation, 
 
 A + B + C = TT. 
 
 (24.) Again, by placing p at a', the general equation becomes, 
 
 XLII. . . sin a cos c = sin 6 cos c + sin c cos a cos b ; 
 
 with the verification that, at the plane limit, 
 
 a = 6 cos c + c cos b. 
 
 But we cannot here delay on such deductions, or verifications : although it appeared 
 to be worth while to point out, that the whole of spherical trigonometry may thus be 
 developed, from the fundamental equation of multiplication of quaternions (107), when 
 that equation is operated on by the two characteristics S and V, and the results 
 interpreted as above. 
 
 211. It may next be proved, as follows, that the distributive for- 
 mula I. of the last Article holds good, when the three quaternions, 
 ^, 5', q"^ which enter into it, without being now necessarily colli- 
 7iem\ are right; in which case \h^\x reciprocals (135), and their swrns 
 (197, (2.) ), will be right also. Let then 
 
 and therefore, 
 
 We shall then have, by 106, 194, 206, 
 
 W+q')q=^l{q"+qy.lq, 
 = W:lq,) + W:lq) = q"g + q'q; 
 and the distributive property in question is proved. 
 
 (1.) By taking conjugates, as in 210, it is easy hence to infer, that the oMer dis- 
 tributive formula, 210, V., holds good for any three right quaternions ; or that 
 
 g(iq" + q') = 9q'+qq, if Lq = Lq= Lq'=-. 
 
 (2.) For any three quaternions, we have therefore the two equations: 
 (V^" + Yq') . Yq = Yq" . Yq + Yq' . Yq ; 
 Yq . (Yq" + Yq') = Yq . Yq" + Yq . Vg'. 
 
 (3.) The quaternions g, 7', q" being still arbitrary, we have thus, by 210, IX., 
 
214 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 {q" +9')'i = (S?" + S?') . S^ + (Vq" + Yq') . Sg + V(? . (Sg" + Sq') + ( Vg" + Yq') . Yq 
 = (Sq".Sq + Yq".Sq+Yq.Sq"+Yq".Yq) + {Sq'.Sq + Yq'.Sq + Yq.Sq' + Yq'.Yq) 
 
 = q"9 + QQ ; 
 
 so that the formula 210, I., and therefore also (by conjugates) the formula 210, V., 
 is valid generally. 
 
 212. The General* Mtiltiplication of Quaternions is there- 
 fore (comp. 13,210) 2i Doubly Distributive Operation; so that 
 we may extend, to quaternions generally, the formula (comp, 
 
 210, VIII.), 
 
 I. . . ^q'.^q^^g'q: 
 
 however many the summands of each set may be, and whe- 
 ther they be, or be not, coUinear (209), or right (211). 
 
 (1.) Hence, as an extension of 210, XX., we have now, 
 11. . . KSg = 2% + 2 2S gKg' ; 
 where the second sign of summation refers to all possible binary combinations of the 
 quaternions g, q\ . . 
 
 (2.) And, as an extension of 210, XXIX., we have the inequality, 
 III. . . STg>T2g, 
 unless all the quaternions g, q', . . bear scalar and positive ratios to each other, in 
 which case the two members of this inequality become equal : so that the sum of the 
 tensors, of any set of quaternions, is greater than the tensor of the sum, in every 
 other case. 
 
 (3.) In general, as an extension of 210, XXVII,, 
 
 IV. . . (STg)2 - (T2g)2 = 22 (T - S) qKq. 
 (4.) The formulae, 210, XVIII., XIX., admit easily of analogous extensions. 
 (5.) We have also (comp. 168) the general equation, 
 
 V...(2y)2_2(g2) = 2(gg' + 5'5); 
 in which, by 210, IX., 
 
 VI. . . qq' + q'q=2(iSq.Sq' + Yq.Sq +Yq'.Sq -^ S(Yq'.Yg)); 
 because, by 208, we have generally 
 
 VII. . . Y(Yq'.Yq) = -Y(Yq.Yq); 
 or VIII. . . Yq'q = - Yqq, if /.q=lq^'^. 
 
 (Comp. 191, (2.), and 204, X.) 
 
 213. Besides the advantage which the Calculus of Quaternions 
 gains, from the general establishment (212) oi the Distributive Prin- 
 ciple, or Distributive Property of Multiplication, by being, so far, 
 
 * Compare the Notes to page 208. 
 
CHAP. I.] INTERSECTIONS OF RIGHT LINES AND SPHERES. 215 
 
 assimilated to Algebra^ in processes which are of continual occur- 
 rence, this principle or property will be found to be of great im- 
 portance, in applications of that calculus to Geometry; and especially 
 in questions respecting the (real or ideal*) intersections of right 
 lines ivith spheres^ or other surfaces of the second order, including 
 contacts (real or ideal), as limits of such intersections. The follow- 
 ing Examples may serve to give some notion, how the general dis- 
 tributive principle admits of being applied to such questions : in 
 some of which however the less general principle (210), respecting 
 the multiplication of collinear quaternions (209), would be sufficient. 
 And first we shall take the case of chords of a sphere^ drawn from a 
 given point upon its surface. 
 
 (1.) From a point a, of a sphere with o for centre, let it be required to draw a 
 chord AP, which shall be parallel to a given 
 line OB ; or more fully, to assign the vector, 
 p = OP, of the extremity of the chord so drawn, 
 as a function of the two given vectors, a = OA, 
 and /3 = OB ; or rather of a and IJ(3, since it 
 is evident that the length of the line j3 cannot 
 affect the result of the construction, which Fig. 
 51 may serve to illustrate. 
 
 (2.) Since AP || ob, or p — a || /3, we may 
 begin by writing the expression, 
 
 p = a + x(3(15), 
 
 which may be considered (corap. 23, 99) as a form of the equation of the right line 
 AP ; and in which it remains to determine the scalar coefficient x, so as to satisfy the 
 equation of the sphere, 
 
 Tp=:Ta(186,(2.)). 
 In short, we are to seek to satisfy the equation, 
 
 T(a + a;/3) = Ta, 
 
 by some scalar x which shall be (in general) different from zero ; and then to sub - 
 Stitute this scalar in the expression p = a + x^, in order to determine the required 
 vector p. ^/Vo^ -t 
 
 (3.) For this purpose, an obvious process is, after dividing both sides by T/3, to 
 square, and to employ the formula 210, XXI., which had indeed occurred before, as 
 200, VIII., but not then as a consequence of the distributive property of multiplica- 
 tion. In this manner we are conducted to a quadratic equation, which admits of 
 division by x, and gives then, 
 
 2xS-. ^^ 
 
 ''*• 
 fi 
 
 2S 
 
 /3' 
 
 p = a-2(3S- 
 
 * Compare the Notes to page 90, &c. 
 
216 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 the problem (1.) being thus resolved, with the verification that /3 may be replaced 
 by U/3, in the resulting expression for p. 
 
 (4.) As a mere exercise of calculation, we may vary the last process (3.), by 
 dividing the last equation (2.) by Ta, instead of T/3, and then going on as before. 
 This last procedure gives. 
 
 a a 
 
 and therefore, 
 
 -(-^')= 
 
 2S-:N^ = - 2S^ (by 196, XII'.), as before. 
 a a (5 
 
 (5.) In general, by 196, II'., 
 
 1-2S = -K; 
 
 hence, by (3.), 
 
 and finally, 
 
 ? = -K? 
 
 P = -k|./3, 
 
 a new expression for p, in which it is not permitted generally, as it was in (3.), to 
 treat the vector /3 as the multiplier,* instead of the multiplicand. 
 
 (6.) It is now easy to see that the second equation of (2.) is satisfied ; for the 
 expression (6.) for p gives (by 186, 187, &c.), 
 
 Tp = T^.T/3 = Ta, 
 
 as was required. 
 
 (7.) To interpret the solution (3.), let c in Fig. 51 be the middle point of the 
 chord AP, and let D be the foot of the perpendicular let fall from a on ob ; then the 
 expression (3.) for p gives, by 196, XIX., 
 
 CA=i(a-p) = /3s|=OD; 
 
 and accordingly, ocad is a parallelogram. 
 
 (8.) To interpret the expression (5.), which gives 
 
 — P ^« op' ^OA .. 
 
 -f = K-, or — =K— , if op' = PO, 
 
 (3 (3 OB ob' 
 
 we have only to observe (comp. 138) that the angle aop' is bisected internally, or 
 the supplementary angle aop externally, by the indefinite right line ob (see again 
 Fig. 51). 
 
 (9.) Conversely, the geometrical considerations which have thus served in (7.) 
 and (8.) to interpret or to verifi/ the two forms of solution (3.), (5.), might have 
 been employed to deduce those two forms, if we had not seen how to obtain them, 
 by rules of calculation, from the proposed conditions"^ of the question. (Comp. 145, 
 (10.), &c.) 
 
 (10.) It is evident, from the nature of that question, that a ought to be deduci- 
 
 Compare the Note to page 159. 
 
CHAP. I.] IMAGINARY INTERSECTIONS. 2lT 
 
 ble from (3 and p, by exactly the same processes as those which have served us to de- 
 duce p from (3 and a. Accordingly, the form (3.) of p gives, 
 
 and the form (5.) gives, 
 
 K|=-|, »=-Ke.,. 
 
 And since the first form can be recovered from the second, we see that each leads us 
 back to the parallelism, p — a\\(3 (2.). 
 
 (11.) The solution (3.) for x shows that 
 
 a; = 0, p = a, p = A, if S(a:/3) = 0, or if /3 -U a. 
 And the geometrical meaning of this result is obvious ; namely, that a right line 
 drawn at the extremity of a radius OA of a sphere, so as to be perpendicular to that 
 radius, does not (in strictness) intersect the sphere, but touches it : its second point 
 of meeting the surface coinciding, in this case, as a limit, with the first. 
 
 (12.) Hence we may infer that the plane represented by the equation, 
 
 stZ^^O, or 8^=1, 
 a a 
 
 is the tangent plane (comp. 196, (5.)) to the sphere here considered, at the point a. 
 (13.) Since /3 may be replaced by any vector parallel thereto, we may substitute 
 for it y — a, if y = oc be the vector of any given point c upon the chord ap, whether 
 (as in Fig, 61) the middle point, or not; we may therefore write, by (3.) and (5.), 
 
 p = a-2(y-a)S-^ = -K-^.(y-a). :. _, /^ ^ . M 
 
 y-a y-a ^ ^^ 
 
 214. In the Examples of the foregoing Article, there was no 
 room for the occurrence of imaginary roots of an equation, or for 
 ideal intersections of line and surface. To give now a case in which 
 such imaginary intersections may occur, we shall proceed to con- 
 sider the question of drawing a secant to a sphere, in a given direc- 
 tion, from a given external point ; the recent Figure 51 still serving 
 us for illustration. 
 
 (1.) Suppose then that 6 is the vector of any given point e, through which it is 
 required to draw a chord or secant epqPi, parallel to the same given line /3 as before. 
 We have now, if po = opo, 
 
 po = £ + ^oA Ta = Tpo = T (£ + Xq^), 
 
 x„2 4-2a;oSi+Ni-N^ = 0, 
 
 ,. being a new scalar ; and similarly, if |0i = OPi, 
 
 2 F 
 
 vv - ff- "^I 
 
^18 ELEMENTS OF QUATERNIONS. [boOK II. 
 
 by transformations* which will easily occur to any one who has read recent articles 
 with attention. And the points Po, pi will be together real, or together imaginary^ 
 according as the quantity under the radical sign is positive or negative ; that is, ac- 
 cording as we have one or other of the two following inequalities, 
 
 T|> or <TV|. 
 
 (2.) The equation (comp. 203, (6.) ), 
 
 represents a cylinder of revolution, with ob for its axis, and with Ta for the radius 
 of its base. If e be a point of this cylindric surface, the quantity under the radical 
 sign in (1.) vanishes ; and the two roots xq, x\ of the quadratic become equal. In 
 this case, then, the line through e, which is parallel to on, touches the given sphere ; 
 as is otherwise evident geometrically, since the cylinder envelopes the sphere (comp. 
 204, (12.) ), and the line is one of its generatrices. If e be internal to the cylinder, 
 the intersections Po, pi are real ; but if E be external to the same surface, those in- 
 tersections are ideal, or imaginary. ^ 
 (3.) In this last case, if we make, for abridgment. 
 
 «i- - '=>/{(--;r-(^'iT}' 
 
 9 and t being thus two given and real scalars, we may write, 
 «ro = a-^V-l; Xi = s+tV -l; 
 where V — 1 is the old and ordinary imaginary symbol of Algebra, and is not in- 
 vested here with any sort of Geometrical Intei-pretation.f We merely express thus 
 the fact of calculation, that (with these meanings of the symbols a, /3, 6, * and t) 
 the formula Ta = T(e +x(S), (1.), when treated by the rules of quaternions, conducts 
 to the quadratic equation, 
 
 (X - S)2 +(2=0, 
 
 which has no real root ; the reason being that the right line through E is, in the 
 present case, wholly external to the sphere, and therefore does not really intersect it 
 at all ; although, for the sake of generalization of language, we may agree to say, 
 as usual, that the line intersects the sphere in two imaginary points. 
 
 (4.) We must however agree, then, for consistency of symbolical expression, to 
 consider these two ideal points as having determinate but imaginary vectors, namely, 
 the two following : 
 
 in which it is easy to prove, 1st, that the real part c + s/3 is the vector t' of the foot 
 e' of the perpendicular let fall from the centre o on the line through E which is drawn 
 (as above) parallel to on ; and Ilnd, that the real tensor tT/S of the coefficient of 
 
 * It does not seem to be necessary, at the present stage, to supply so many refe- 
 rences to former Articles, or Sub-articles, as it has hitherto been thought useful ta 
 give ; but such may still, from time to time, be given. 
 
 t Compare again the Notes to page 90, and Art. 149. 
 
CHAP. I.] 
 
 CIRCUMSCRIBED CONES. 
 
 219 
 
 V - 1 in tha ijnaginary part of each expression, represents the length of a tangent 
 e'e" to the sphere, drawn from that external point, or foot, e'. 
 (6.) In fact, if we write oe' = «' = £ -f «j3, we shall have 
 
 e'e = £ - 6' = - »j3 = /3S — = projection of oe on ob ; 
 
 which proves the 1st assertion (4.), whether the points Po, Pi be real or imaginary. 
 And because 
 
 /f(> 
 
 + «^ 
 
 we have, for the case of imaginary intersections, 
 
 «T^ = V(T£'2 - Ta2) = T . E'E", 
 and the Ilnd assertion (4.) is justified. 
 
 (6.) An expression of the form (4.), or of the following, 
 
 p' = /3 + V-ly, 
 in which /3 and y are two real vectors^ while V - 1 is the (scalar) imaginary of al- 
 gebra, and not a symbol for &. geometrically real right versor (149, 153), may be said 
 to be a BiVECTOR. 
 
 (7.) In like manner, an expression of the form (3.), ora:' = s+<V — 1, where » 
 and t are two real scalars, but V - 1 is still the ordinary imaginary of algebra, may 
 be said by analogy to be a Biscalar. Imaginary roofs of algebraic equations aro 
 thus, in general, biscalars. 
 
 (8.) And if a bivector (6.) be divided by a (real) vector, the quotient, such as 
 
 H 
 
 a a 
 
 1 = ?o + ^1 V - 1, 
 
 in which go and qi are two real quaternions, but V — 1 is, as before, imaginary, may 
 be said to be a Biquaternion. * 
 
 
 215. The same distributive principle (212) may be employed in 
 investigations respecting circumscribed cones^ and the tangents (real 
 or ideal), which can be drawn to a given sphere from a given point. 
 
 (1.) Instead of conceiving that o, a, b are three given points, and that limits of 
 position of the point e are sought, as in 214, (2.), which shall allow the points of in- 
 tersection Po, Pi to be real, we may suppose that o, a, e (which may be assumed to 
 be coUinear, without loss of generality, since a enters only by its tensor) are now the 
 data of the question ; and that limits of direction of the line ob are to be assigned, 
 which shall permit the same reality : epoPi being still drawn parallel to ob, as in 
 214, (1.). 
 
 (2.) Dividing the equation Ta = T(€ + xfi) by Tf, and squaring, we have 
 
 Compare the second Note to page 131. 
 
220 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 N" = ^Nfl + a;^'\='jn-2xS^ + a;2N|; 
 the quarlratic in x may therefore be thua written, 
 
 and its roots are real and unequal, or real and equal, or imaginary, according as 
 TVU^< or= or>T-; 
 
 C 6 
 
 that is, according as 
 
 sinEOB< or = or >T.oa: T.oe. 
 
 (3.) If E be interior to the sphere, then Tc < Ta, T(a : e) > 1 ; but TVUg can 
 never exceed unity (by 204, XIX., or by 210, XV., &c.) ; we have, therefore, in 
 this case, theirs* of the three recent alternatives, and the two roots of the quadratic 
 are necessarily real and unequal, whatever the direction of /3 may be. Accordingly 
 it is evident, geometrically, that every indefinite right line, drawn through an inter- 
 nal point, must cut the spheric surface in two distinct and real points. 
 
 (4.) If the point E be SM/jer/cia?, so that Tf = Ta, T(a:6) = l, then the first 
 alternative (2.) still exists, except at the limit for which (3 -^ e, and therefore 
 TVU (j3 : f) = 1, in which case we have the second alternative. One root of the qua- 
 dratic in a; is now = 0, for every direction of (3 ; and the other root, namely 
 a: = — 2S(c:/3), is likewise always j-eal, but vanishes for the case when the angle 
 Eon is right. In short, we have here the same system of chords and of tangents, 
 from a point upon the surface, as in 213 ; the only difference being, that we noAV 
 write E for a, or £ for a. 
 
 (5.) But finally, if e be an external point, so that Tc >Ta, and T(a : c) < 1, 
 then TVU (/3 : t) may either fall short of this last tensor, or equal, or exceed it ; so 
 that any one of the three alternatives (2.) may come to exist, according to the vary- 
 ing direction of (3. 
 
 (6.) To illustrate geometrically 
 the law of passage from one such 
 alternative to another, we may ob- 
 serve that the equation, 
 
 TVU^ = T-, 
 
 « £ 
 
 or 
 
 sinEOP = T.oA: T.oe, 
 
 represents (when e is thus external) 
 a real cone of revolution, with its 
 vertex at the centre o of the sphere ; 
 and according as the line on lies in- 
 side this cone, or on it, or outside it, 
 the first or the second or the third of 
 the three alternatives (2.) is to be ^^S- 52. 
 
 adopted ; or in other words, the line 
 
 through E, drawn parallel (as before) to on, either cuts the sphere, or touches it, or 
 does not (really) meet it at all. (Compare the annexed Fig. 52.) 
 
CHAP, I.] POLAR PLANES, CONJUGATE POINTS. 221 
 
 (7.) IfEbe still an external point, the cone of tangents which can be drawn 
 from it to the sphere is real ; and the equation of this enveloping or circumscribed 
 cone, with its vertex at E, may be obtained from that of the recent cone (6.), by 
 simply changing p to p — c ; it is, therefore, or at least one form of it is, 
 
 TVU^^=T-: or sinoEP = T. oa : T.oe. 
 
 € £ 
 
 (8.) In general, if q be any quaternion, and x any scalar, 
 VU(gr + ^) = V5:T(g + £c); 
 the recent equation (7.) may thcHjfore be thus written : 
 
 p-« e ' 
 
 or 
 
 T.p'p:T.ep==T.oa: T.OE, 
 
 if p' be the foot of the perpendicular let fall from p on oe ; and in fact the first quo- 
 tient is evidently = sin oep. 
 (9.) We may also write, 
 
 Tve = T2.T(e-l'j; or = (s^y-N? + N^(Ne- 2Se + , j, 
 or 
 
 as another form of the equation of the circumscribed cone. 
 (10.) If then we make also 
 
 N^ = l, or N^ = N^, 
 a e e 
 
 to express that the point p is on the enveloped sphere, as well as on the enveloping 
 cone, we find the following equation of the plane of contact, or of what is called the 
 polar plane of the point b, with respect to the given sphere : 
 
 s£-N^Y = Oi or Se-N2 = 0, -^^^ ^ - A^^-^ 
 
 £ 
 
 -J=Oi or S.--N. = 0, --^^ , /^^^ 
 
 while the fact that it is a plane of contact" is exhibited by the occurrence of the ex- 
 ponent 2, or by its equation entering through its square. 
 
 (11.) The vector, 
 
 „ jO ^^ a , 
 
 e' = f S - = cN - = OE, 
 
 is that of the point e' in which the polar plane (10.) of e cuts perpendicularly the 
 right line oe ; and we see that 
 
 Tc.T6' = Ta2, or T.oe.T.oe' = (T.oa)2, 
 
 as was to be expected from elementary theorems, of spherical or even of plane geo- 
 metry. 
 
 * In fact a modern geometer would say, that we have here a case of two coinci- 
 dent planes of intersection, merged into a single plane of contact. 
 
222 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (12.) The equation (10.), of the polar plane of e, may easily be thus trans- 
 formed : 
 
 si=[s£.Nl = V" or Si-N^ = 0; 
 P \ ^ P J P P P 
 
 it continues therefore to hold good, when e and p are interchanged. If then we take, 
 as the vertex of a new enveloping cone, any point o external to the sphere, and 
 situated on the polar plane ff' . . of the former external point b, the new plane of 
 contact, or the polar plane dd' . . of the new point c, will pass through the former 
 vertex e : a geometrical relation of reciprocity, or of conjugation, between the two 
 points c and e, which is indeed well-known, but which it appeared useful for our pur- 
 pose to prove by quaternions* anew. 
 
 (13.) In general, each of the two connected equations, 
 
 P P P' 9 
 
 which may also be thus written, 
 
 \^ ap a j a a a a 
 
 may be said to be a form of the Equation of Conjugation between any two points p and 
 p' (not those so marked in Fig. 52), of which the vectors satisfy it : because it ex- 
 presses that those two points ai-e, in a well-known sense, conjugate to each other, with 
 respect to the given sphere, Tp = Ta. 
 
 (14.) If one of the two points, as p', be given by its vector p', while tlie other 
 point p and vector p are variable, the equation then represents a plane locus; 
 namely, what is still called the polar plane of the given point, whether that point be 
 external or internal, or on the surface of the sphere. 
 
 (15.) Let P, p' be thus two conjugate points; and let it be proposed to find the 
 points s, 8', in which the right line pp' intersects the sphere. Assuming (comp. 25) 
 that 
 
 OS = <T = xp+i/p', x + i/ = l, T(T = Ta, 
 
 and attending to the equation of conj ugation (13.), we have, by 210, XX., or by 
 200, VII., the following quadratic equation in y : a;, 
 
 (a; + y)2 = N(a;^ + y^'^ = a;2N^-f2a;y + y2N^; 
 \ a a ) a a 
 
 which gives, 
 
 (16.) Hence it is evident that, if the points of intersection s, s' are to be real, one 
 of the two points p, p' must be interior, and the other must be exterior to the sphere ; 
 because, of the two norms here occurring, one must be greater and the other less than 
 linity. And because the two roots of the quadratic, or the two values of y : a;, differ 
 
 * In fact, it will easily be seen that the investigations in recent sub-articles are 
 put forward, almost entirely, as exercises in the Language and Calculus of Quaternions, 
 and not as offering any geometrical novelty of result. 
 
CHAP. I.] EQUATION OF ELLIPSOID, RESUMED. 223 
 
 only by their signs, it follows (by 26) that the right line pp' is harmonically divided 
 (as indeed it is well known to be), at the two points s, s' at which it meets the sphere : 
 or that in a notation already several times employed (25, 31, &c.), we have the har- 
 monic formula, 
 
 (pspV)=~ 1. 
 
 (17.) From a real but internal point p, we can still speak of a cone of tangents, 
 as bemg drawn to the sphere : but if so, we must say that those tangents are ideal, 
 or imaginary ;*^ and must consider them as terminating on an imaginary circle of 
 contact : of which the real but wholly external plane is, by quaternions, as by mo- 
 dern geometry, recognised as being (comp. (14.) ) the polar plane of the supposed 
 internal point. 
 
 216. Some readers may find it useful, or at least interest- 
 ing, to see here a few examples of the application of the General 
 Distributive Principle (212) of multiplication to the Ellipsoid, 
 of which some forms of the Quaternion Equation were lately 
 assigned (in 204, (14.) ); especially as those forms have been 
 found to conductf to a Geometrical Construction, previously 
 unknown, for that celebrated and important Surface : or ra- 
 ther to several such constructions. ,In what follows, it will 
 be supposed that any such reader has made himself already 
 sufficiently familiar with the chief formulae of the preceding 
 Articles ; and therefore comparatively few references J will be 
 given, at least upon the present subject. 
 
 (1.) To prove, first, that the locus of the variable ellipse, 
 
 I. ..S^=a;, (v^Y=a;2-l, 204,(13.) 
 
 « V Pi 
 
 which locus is represented by the equation, 
 
 the two constant vectors a, /3 being supposed to be real, and to be inclined to each 
 other at some acute or obtuse (but not right§) angle, is a surface of the second order, 
 
 * Compare again the second Note to page 90, and others formerly referred to. 
 
 f See the Proceedings of the Royal Irish Academy, for the year 1846. 
 
 X Compare the Note to page 218. 
 
 § If /3 -l-a, the system I. represents (not an ellipse but) a pair of right lines, 
 real or ideal, in which the cylinder of revolution, denoted by the second equation of 
 that system, is cut by a, plane parallel to its axis, and represented by the first equa- 
 tion. 
 
224 ELEMENTS OF QUATERNIONS. [dOOK II. 
 
 in the sense that it is cut by an arbitrary rectilinear transversal in two (real or ima- 
 ginary) points, and in no more than two, let us assume two points l, m, or their 
 vectors \ = ol, /* = om, as given ; and let us seek to determine the points p (real or 
 imaginary), in which the indefinite right line lm intersects the locus II. ; or rather 
 the number of such intersections, which will be suflScient for the present purpose. 
 
 (2.) Making then p =^-- — (26), we have, for y : 2, the following quadratic 
 
 y "I" 2; 
 
 equation, 
 
 without proceeding to resolve which, we see already, by its mere degree, that the num- 
 ber sought is two ; and therefore that the locus II. is, as above stated, a surface of 
 the second order. 
 
 (3.) The equation II. remains unchanged, when - p is substituted for p ; the 
 surface has therefore a centre, and this centre is at the origin o of vectors. 
 
 (4.) It has been seen that the equation of the surface may also be thus written : 
 
 IV. ..Tfs^-[-V^'\=l; 204,(14.) 
 
 it gives therefore, for the reciprocal of the radius vector from the centre, the expres- 
 sion. 
 
 -•i=<4-^)^ 
 
 and this expression has a real value, which never vanishes,* whatever real value may 
 be assigned to the versor Up, that is, whatever direction may be assigned to p : the 
 surface is therefore closed, a,ndi finite. 
 
 (5.) Introducing two new constant and auxiliary vectors, determined by the two 
 expressions, 
 
 2/3 
 
 6=- • . a, 
 
 
 ' /3-i-a 
 
 ft-a 
 
 
 which give (by 125) these other expressions, 
 
 
 
 we have 
 
 y V 
 
 
 
 VII. ..^ + ^ = 2, 
 a /3 
 
 -a-r'-- 
 
 \r^cMi^ 
 
 VII'...^+^=:1, 
 
 7 ^ 
 
 
 and under these conditions, y is said to be the harmonic mean between the two for- 
 mer vectors, a and /3 ; and in like manner, 5 is the harmonic mean between a and 
 — /3 ; while 2a is the corresponding mean between y, ^ ; and 2/3 is so, between y 
 and - d. 
 
 * It is to be remembered that we. have excluded in (1.) the case where /3 -t- a 
 in which case it can be shown that the equation II. represents an elliptic cylinder. 
 
(i 
 
 uX^ 
 
 CHAP. I.] CIRCULAR SECTIONS, CYCLIC PLANES. 225 
 
 (6.) Under the same conditions, for any arbitraiy vector p, wo have the trans- 
 formations, 
 
 VIII .e=i('e + eV p=Je_eV ,,f .•, , -7i 
 ix...e+K|t=se+v£i ^ 
 
 the equation IV. of the surface may therefore be thus written : 
 
 X...T^e + K^]=l; orthus, X'...t(^ + K^') = 1; {ki^ ^ 
 
 the geometrical meaning of wliich new forms will soon be seen. ' - ' , 
 
 (7.) The system of the two planes through the origin, which are respectively P^f^*''^^^^ f 
 perpendicular to the new vectors y and 5, is represented by the equation, /^ / tjL^ 
 
 xi...sese=o, 0. xii...(sey = (sey, ^(^^ 
 
 combining which with the equation II. we get 
 
 XIII...l = (^S^y-(^V^J=N^; or, XIV. . . Tp = T/3. 
 
 These two diametral plau«s therefore cut the surface in <«?o circular sections^ with T/3 
 for their common radius ; and the normals y and ^, to the same two planes, may be 
 called (comp, 196, (17.) ) the cyclic normals of the surface; while the planes them- 
 selves may be called its cyclic planes. 
 
 (8.) Conversely, if we seek the intersection of the surface with the concentric 
 sphere XIV., of which the radius is T/3, we are conducted to the equation XII. of 
 the system of the two cyclic planes, and therefore to the two circular sections (7.) ; 
 so that every radius vector of the surface, which is not drawn in one or other of these 
 two planes, has a length either greater or less than the radius T/3 of the sphere. 
 
 (9.) By all these marks, it is clear that the locus II., or 204, (14.), is (as above 
 asserted) an Ellipsoid; its centre being at the origin (3.), and its mean semiazis 
 being = T/3 ; while U/3 has, by 204, (15.), the direction of the axis of a circum- 
 scribed cylinder of revolution, of which cylinder the radius is T/3 ; and a is, by the 
 last cited sub- article, perpendicular to the plane of the ellipse of contact. 
 
 (10.) Those who are familiar with modem geometrj^, and who have caught the 
 notations of quaternions, will easily see that this ellipsoid II., or IV., is a deforma- 
 tion of what may be called the mean sphere XIV., and is homologous thereto ; the 
 infinitely distant point in the direction of /3 being a centre of homology, and either 
 of the two planes XL or XII. being a plane of homology corresponding. 
 
 217. The recent form, X. or X'., of the quaternion equa- 
 tion of the ellipsoid, admits of being interpreted, in such a way 
 as to conduct (comp. 216) to a simple construction of that sur- 
 face ; which we shall first investigate by calculation, and then 
 illustrate by geometry. 
 
 2 G 
 
226 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 (1.) Carrying on the Roman numerals from the sub-articles to 216, and observ- 
 ing that (by 190, &c.), 
 
 
 t=.K^-.NP, and K?=l4, 
 
 y p y p d 
 
 the equation X. 
 
 takes the form, 
 
 Ar 
 
 — {U^-^^vl^xW 
 
 or 
 
 if we make 
 
 Xyi...^=T(.H-K^.p), 
 
 
 ^^"•••4 = 1 -^ T>7^' 
 
 when I and k are two new constant vectors, and < is a new constant scalar, which we 
 shall suppose to be positive, but of which the value may be chosen at pleasure. 
 
 (2.) The comparison of the forms X. and X'. shoAvs that y and 3 may be inter- 
 changed, or that they enter symmetrically into the equation of the ellipsoid, although 
 they may not at first seem to do so ; it is therefore allowed to assume that 
 XVIir. . . Ty > T^, and therefore that XVIII'. . . Tt > Tk ; 
 for the supposition Ty = T^ would give, by VI., 
 
 T(/3 + a) = T(/3-a), and .'. (by 186, (6.) &c.) 
 which latter case was excluded in 216, (1.). 
 (3.) We have thus, 
 
 XIX. . . Ut = U5; 
 
 XX. 
 
 /3' 
 
 Tt 
 
 XXI. 
 
 Tl2 - T/c2 
 
 UK: = Uy 
 
 (to) ily) 
 
 (4.) Let ABC be a plane triangle, 
 such that 
 
 XXII. . . CB = t, CA = k;; 
 let also 
 
 AE = p. 
 
 Then if a sphere, which we shall call the 
 diacentric sphere, be described round the 
 point c as centre, with a radius = Tk, and 
 therefore so as to pass through the centre 
 A (here written instead of o) of the ellip- 
 soid, and if D be the point in which the 
 line AE meets this sphere again, we shall 
 have, by 213, (5.), (18.), 
 
 XXIII. 
 
 and therefore 
 
 CD = -K-.p, 
 
 P . 
 
 .'btit^ 
 
 Fig. 53, 
 
 xxiir. . . DB 
 
 t+K-.p; 
 
 P 
 
 - rf • 
 
 '3 
 
 
 unA.^-<^ 
 
 /t^fc 
 
 
 jfyi^ff 
 
 ty 
 
CHAP. I.] CONSTRUCTION OF THE ELLIPSOID. 227 
 
 so that the equation XVI. becomes, 
 
 XXIV. . . <2=T.AE.T.DB. 
 
 (5.) The point b is external to the diacentric sphere (4.), by the assumption (2.) ; 
 a real tangent (or rather cone of tangents) to this sphere can therefore be drawn from 
 that point ; and if we select the length of such a tangent as the value (1.) of the sca- 
 lar *, that is to say, if we make each member of the formula XXI. equal to unity^ 
 and denote by d' the second intersection of the right line bd with the sphere, as in 
 Fig. 53, we shall have (by Euclid III.) the elementary relation, 
 
 XXV. . .<2=:T.db.T.bd'; 
 whence follows this Geometrical Equation of the Ellipsoid, 
 
 XXVI. .. T.AB = T.BD'; 
 or in a somewhat more familiar notation, 
 
 XXVII. . . AE = ^; 
 where ae denotes the length of the line ae, and similarly for bd'. 
 
 (6.) The following very simple Rule of Construction (corap. the recent Fig. 53) 
 results therefore^from our quaternion analysis : — 
 
 From a fixed point A, on the surface of a given sphere, draw any chord ad ; let 
 d' he the second point of intersection of the same spheric surface with the secant bd, 
 drawn from a fixed external* point b ; and take a radius vector ae, equal in 
 length to the line bd', and in direction either coincident with, or opposite to, the chord 
 ad : the locus of the point E will he an ellipsoid, with A for its centre, and with Bfor 
 a point of its surface. 
 
 (7.) Or thus: — 
 
 If, of a plane hut variable quadrilateral abed', of which one side ab is given in 
 length and in position, the two diagonals ae, bd' he equal to each other in length, and 
 if their intersection D he always situated upon the surface of a given sphere, whereof 
 the side ad' of the quadrilateral is a chord, then the opposite side be is a chord of 
 a given ellipsoid, 
 
 218. From either of the two foregoing statements, of the 
 Rule of Construction for the Ellipsoid to wliich quaternions 
 have conducted, many geometrical consequences can easily be 
 inferred, a few of which may be mentioned here, with then: 
 proofs by calculation annexed : the present Calculus being, of 
 course, still employed. 
 
 (1.) That the corner b, of what may be called the Generating Triangle abc, is 
 in fact a point of the generated surface, with the construction 217, (6.), may be 
 
 * It is merely to fix the conceptions, that the point b is here supposed to be exter- 
 nal(5.) ; the calculations and the construction would be almost the same, if we as- 
 sumed B to be an internal point, or Ti < T/c, Ty < Td. 
 
228 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 proved, by conceiving the variable chord ad of the given dia centric sphere to take the 
 position AG; where g is the second intersection of the line ab with that spheric sur- 
 face. 
 
 (2.) Kobe conceived to approach to a (instead ofo), and therefore d' to g 
 (instead of a), the direction of ae (or of ad) then tends to become tangential to the 
 sphere at A, while the length of ae (or of bd') tends, by the construction, to become 
 equal to the length of bg ; the surface has therefore a diametral and circular section, 
 in a plane which touches the diacentric sphere at A, and with a radius = bg. 
 
 (3.) Conceive a circular section of the sphere through A, made by a plane perpen 
 
 dicular to bc ; if d move along this circle, d' will move along a parallel circle through 
 
 ,Sa^^ g, and the length of bd', or that of ae, will again be equal to bg fsuch then is the 
 
 radius of a second diametral and circular section of the ellipsoid, made by the lately 
 
 f mentioned plane. 
 
 (4.) The construction gives us thus two cyclic planes through A ; the perpendi- 
 culars to which planes, or the two cyclic normals (216, (7.)) of the ellipsoid, are 
 seen to have the directions of the two sides, ca, cb, of the generating triangle abc 
 (1.). 
 
 (5.) Again, since the rectangle 
 
 ba . BG = bd . bd' = bd . ab = double area of triangle abe : sin bde, 
 we have the equation, 
 
 XXVIII. . . perpendicular distance of e from ab = bg • sin bde ; 
 
 the third side, ab, of the generating triangle (1.), is therefore the axis of revolution 
 of a cylinder, which envelopes the ellipsoid, and of which the radius has the same 
 length, bg, as the radius of each of the two diametral and circular sections. 
 
 (6.) For the points of contact of ellipsoid and cylinder, we have the geometrical 
 relation, 
 
 XXIX. . . bdb = a right angle ; or XXIX'. . . adb = a right angle ; 
 
 the point d is therefore situated on a second spheric surface, which has the line ab 
 for a diameter, and intersects the diacentric sphere in a circle, Avhereof the plane passes 
 through A, and cuts the enveloping cylinder in an ellipse of contact (comp. 204, 
 (15.), and 216, (9.) ), of that cylinder with the ellipsoid. 
 
 (7.) Let AC meet the diacentric sphere again in f, and let bf meet it again in p' 
 (as in Fig. 53) ; the common plane of the last-mentioned circle and ellipse (6.) can 
 then be easily proved to cut perpendicularly the plane of the generating triangle abc 
 in the line af'; so that the line f'b is normal to this plane of contact; and there- 
 fore also (by conjugate diameters, &c.) to the ellipsoid, at b. 
 
 (8.) These geometrical consequences of the construction (217), to which many 
 others might be added, can all be shoAvn to be consistent with, and confirmed by, the 
 quaternion analysis from which that construction itself was derived. Thus, the two 
 circular sections (2.) (3.) had presented themselves in 216, (7.) ; and their two cy- 
 clic normals (4.), or the sides CA, cb of the triangle, being (by 217, (4.) ) the two 
 vectors k, t, have (by 217, (1 .) or (8.) ) the directions of the two former vectors y, 5 ; 
 which again agrees with 216, (7.). 
 
 (9.) Again, it will be found that the assumed relations between the three pairs of 
 constant vectors, a, j3 ; y, d ; and j, *•, any one of which pairs is sufficient to deter- 
 
CHAP. 1.] CONSEQUENCES OF THE CONSTRUCTION. 229 
 
 mine the ellipsoid, conduct to the following expressions (of which the investigation is 
 left to the student, as an exercise) : 
 
 XXX. ..a = ~ r = T^ ^=7FT ;U(i + k) = f'b; 
 
 XXXI. ../3 = ^y = /-5 = =rP^xU(t-K) = BG; 
 — y —y i {i- k) 
 
 the letters B, f', g referring here to Fig. 53, while a/3y^ retain their former mean- 
 ings (216), and are not interpreted as vectors of the points abcd in that Figure. 
 Hence the recent geometrical inferences, that ab (or bg) is the axis of revolution of 
 an enveloping cylinder (5.), and that f'b is normal to the plane of the ellipse of con- 
 tact (7.), agree with the former conclusions (216, (9.), or 204, (15.) ), that j3 is 
 such an axis, and that a is such a normal. 
 (10.) It is easy to prove, generally, that 
 
 c9-i_q (g-i)(Kg+i) ^ %- i g + 1^ yg-1 . 
 
 9 + 1 (9+i)(k:3+i) KCz + i)' 9-1 N(^-l)' 
 
 whence 
 
 t + K T (l + k)* l-K 1 (t - k)2 
 
 whatever two vectors t and k may be. But Ave have here, 
 
 XXXIII. . . <3 = Ti2 - Tk2, by 217, (5.) ; 
 the recent expressions (9.) for a and /3 become, therefore, 
 
 XXXIV. . . a=;+(i + fc)S*-— ^; i(S = -(i-K:) S— . 
 
 1 + K l-K 
 
 The last form 204, (14.), of the equation of the ellipsoid, may therefore be now thus 
 written : 
 
 XXXY. ..TiS-^:S'— ^-V-^:S— 1=1 
 
 l~K I- 
 
 \ i + K 1 + 
 
 in which the sign of the right part may be changed. And thus we verify by calcu- 
 lation the recent result (1.) of the construction, namely that b is a point of the sur- 
 face ; for we see that the last equation is satisfied, when we suppose 
 
 XXXVI. . . p = AB = t-K = /3:s2; 
 
 a 
 
 a value of p which evidently satisfies also the form 216, IV. 
 
 (11.) From the form 216, II., combined with the value XXXIV. of otitis easy 
 to infer that the plane, 
 
 XXXVII. . .s^ = i, or xxxvir. . .S-^ = S^-^, 
 
 a 1+ K 1 + K 
 
 which corresponds to the value a;= 1 in 216, I., touches the ellipnoid at the point B, 
 of which the vector p has been thus determined (10) ; the normal to the surface^ at 
 that point, has therefore the direction of t + ic, or of a, that is, of fb, or of f'b : so 
 that the last geometrical inference (7.) is thus confirmed, by calculation with quater- 
 
 219. A few other consequences of the construction (217) may 
 be here noted; especially as regards the geometrical determination 
 
230 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 of the three principal semiaxes of the ellipsoid, and the major and 
 minor semiaxes of any elliptic and diametral section ; together with 
 the assigning of a certain system of spherical conies^ of -which the 
 surface may be considered to be the locus. 
 
 (1.) Let a, 6, c denote the lengths of the greatest, the mean, and the least semi- 
 axes of the ellipsoid, respectively ; then if the side bc of the generating triangle cut 
 the diacentric sphere in the points h and h', the former lying (as in Fig. 53) between 
 the points b and c, -we have the values, 
 
 XXXVIII. ..a = BH'; 6 = bg; c = bh; 
 
 so that the lengths of the sides of the triangle abc may be thns expressed, in terms 
 
 of these semiaxes, 
 
 — a -^ c — a — c — cic 
 
 XXXIX. . .BC=Te = -|-; ca = Tk=-— -; ab =T(i - «) = — ; 
 
 and we may write, 
 
 Ti3 — Tk2 
 
 XL. . . a = Ti + T/c; h==—- -; c=Ti-T/c. 
 
 T (i - k) 
 
 (2.) If, in the respective directions of the two supplementary chords ah, ah' of the 
 sphere, or in the opposite directions, we set off lines al, an, with the lengths of bh', 
 BH, the points L, N, thus obtained, will be respectively a major and a minor summit 
 of the surface. And if we erect, at the centre a of that surface, a perpendicular am 
 to the plane of the triangle, with a length = bg, the point m (which will be common 
 to the two circular sections, and will be situated on the enveloping cylinder) will be a 
 mean summit thereof. 
 
 (3.) Conceive that the sphere and ellipsoid are both cut by a plane through a, on 
 which the points b' and c' shall be supposed to be the projections of b and c ; then c' 
 will be the centre of the circular section of the sphere ; and if the line b'c' cut this 
 new circle in the points Di, »2, of which di may be supposed to be the nearer to b', 
 the two supplementary chords adi, ad2 of the circle have the directions of the major 
 and minor semiaxes of the elliptic section of the ellipsoid ; while the lengths of those 
 semiaxes are, respectively, ba.bg: bdi, and ba. bg : BD2; or bd'i and BD'2, if the 
 secants bdi and BD2 meet the sphere again in Di' and D2'. 
 
 (4.) If these two semiaxes of the section be called a, and c„ and if we still de- 
 note by t the tangent from b to the sphere, we have thus, 
 
 XLI. . . BDi = <2 : a = oca -1 ; BD2 = *2 ; c = acc'^ ; 
 
 but if we denote by pi and p2 the inclinations of the plane of the section to the two 
 cyclic planes of the ellipsoid, whereto CA and cb are perpendicular, so that the pro- 
 jections of these two sides of the triangle are 
 
 |o'a = CA . sinpi = ^(a — c) sin pi, 
 
 XLII. 
 
 [c'b =CB.smp2 = i{a + c)s'mp2, 
 we have 
 
 XLIII. . . BD33 - BDi2 = b'd22 -b'di2 = 4b'c' . c'a = (a^ - c2) sin pi sin p> 
 
 whence follows the important formula, 
 
 XLIV. . . c,-2 - a, 2 = (c 2 _ a 2) sin pi smpz ; 
 
CHAP. I.] SEMIAXES, SPHERICAL CONICS. 231 
 
 or in words, the known and useful theorem, that " the difference of the inverse 
 squares of the semiaxes, of a plane and diametral section of an ellipsoid, varies as 
 the product of the sines of the inclinations of the cutting plane, to the two planes of 
 circular section. 
 
 (5.) As verifications, if the plane be that of the generating triangle abc, we 
 have 
 
 pi=p2= -, and a^ = a, c^ = c', 
 
 but if the plane be perpendicular to either of the two sides, ca, cb, then either pi or 
 P2 = 0, and c, = a^. 
 
 (6.) If the ellipsoid be cut by any concentric sphere, distinct from the mean 
 sphere XIV., so that 
 
 XLV. . . AE = Tp = r ^ 6, where r is a given positive scalar ; 
 then 
 
 XL VI. . . BD = «2r-i ^ acb-^j that is, ^ ba ; 
 
 so that the locus of what may be called the guide-point D, through which, by the 
 construction, the variable semidiameter ab of the ellipsoid (or one of its prolongations) 
 passes, and which is still at a constant distance from the given external point b, is 
 now again a circle of the diacentric sphere, but one of which the plane does not pass 
 (as it did in 218, (3.) ) through the centre A of the ellipsoid. The point b has there- 
 fore here, for one locus, the cyclic cone which has A for vertex, and rests on the last- 
 mentioned circle as its base; and since it is also on the concentric sphere XLV., it 
 must be on one or other of the two spherical conies, in which (comp. 196, (11.) ) the 
 cone and sphere last mentioned intersect. 
 
 (7.) The intersection of an ellipsoid with a concentric sphere is therefore, gene- 
 rally, a system of two such conies, varying with the value of the radius r, and be- 
 coming, as a limit, the system of the two circular sections, for the particular value 
 r = 6 ; and the ellipsoid itself may be considered as the locu» of all such spherical co- 
 nies, including those two circles. 
 
 (8.) And we see, by (6.), that the two cyclic planes (comp. 196, (17.), &c.) of 
 any one of the concentric cones, which rest on any such conic, coincide with the two 
 cyclic planes of the ellipsoid : all this resulting, with the greatest ease, from the con- 
 struction (217) to which quaternions had conducted. 
 
 (9.) With respect to the Figure 53, which was designed to illustrate that con- 
 struction, the signification of the letters abcdd'efk'ghh'ln has been already ex- 
 plained. But as regards the other letters we may here add, 1st, that n' is a second 
 minor summit of the surface, so that an' = na ; Ilnd, that k is a point in which the 
 chord af', of what we may here call the diacentric circle agf, intersects what may 
 be called the principal ellipse, * or the section nblen' of the ellipsoid, made by the 
 plane of the greatest and least axes, that is by the plane of the generating triangle 
 ABC, so that the lengths of AK and bf are equal; Ilird, that the tangent, vKv', to 
 this ellipse at this point, is parallel to the side ab of the triangle, or to the axis of 
 
 * In the plane of what is called, by many modern geometers, i\\Q focal hyper- 
 bola of the ellipsoid. 
 
232 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 revolution of the enveloping cylinder 218, (5.), being in fact one fide (or generatrix) 
 of that cylinder ; IVtb, that ak, ab are thus two conjugate semidiameters of the 
 ellipse, and therefore the tangent tbt', at the point b of tbat ellipse, is parallel to 
 the line akf', or perpendicular to the line bff' ; Vth, that this latter line is thus the 
 normal (comp. 218, (7.), (11.) ) to thesame elliptic section, and therefore also to the 
 ellipsoid, at b ; Vlth, that the least distance kk' between the parallels ab, kv, being 
 = the radius b of the cylinder, is equal in length to the line bg, and also to each of 
 the two semidiameters, as, as', of the ellipse, which are radii of the two circular 
 sections of the ellipsoid, in planes perpendicular to the plane of the Figure ; Vllth, 
 that AS touches the circle at A ; and Vlllth, that the point s' is on the chord Ai of 
 that circle, which is drawn at right angles to the side bc of the triangle. 
 
 220. The reader will easily conceive that the quaternion equa- 
 tion of the ellipsoid admits of being put under several other forms; 
 among which, however, it may here suffice to mention one, and to 
 assign its geometrical interpretation. 
 
 (I.) For any three vectors, t, k, p, we have the transformations, 
 
 XLVIL..N[l + K^UNi-fN^+2S-i^ 0^^ '^ 
 
 \p p ) p p p p 
 
 = NiN- + N-N- + 2S--T-T- 
 
 K p >■ p p p I K 
 
 \9 I P KJ \p K pi] 
 
 Tk ^Vk .Ti\ [JJk.Ti . _Ut.T/c 
 
 + K =N +K 
 
 P P } \ 9 P 
 
 whence follows this other general transformation : 
 
 XLVIir. ..Tfi + K-.p^ = TfuK.Tt + K Hil^!^ . p \ 
 
 (2.) If then we introduce two new auxiliary and constant vectors, i and k\ de- 
 fined by the equations, 
 
 XLIX. . . t' = - Uk . Ti, K' = -Ut.TK, 
 
 which give, 
 
 L. . . Tt' = Tt, Tfc' = Tk, T (i' - ^') = T (t - k), Tt'2 - Tk'2 = t\ 
 we may write the equation XVI. (in 217) of the ellipsoid under the following pre- 
 cisely similar form : 
 
 U...il=T(.'.Kl.,) 
 
 in which i and k have simply taken the places of t and k. 
 
 (3.) Retaining then the centre A of the ellipsoid, construct a new diaceniric 
 sphere^ with a new centre o', and a new generating triangle ab'c', where b' is a new 
 fixed external point, but the lengths of the sides are the same, by the conditions, 
 
 LII. . Ac' = — k', c'b' = + t', and therefore ab' — i -k \ 
 
 draw any secant b'd"d"' (instead of bdd'), and set off a line ae in the direction of 
 
CHAP. I.] STANDARD QUADRINOMIAL FORM. 233 
 
 ad", or in the opposite direction, with a length equal to that of bd'"; the locus of 
 the point E will be the same ellipsoid as before. 
 
 (4.) The only inference which we shall here* draw from this new construction 
 is, that there exists (as is known) a second enveloping cylinder of revolution, and that 
 its axis is the side ab' of the new triangle ab'c' ; but that the radius of this second 
 cylinder is equal to that of the first, namely to the mean semiaxis, 6, of the ellipsoid ; 
 and that the major semiaxis, a, or the line al in Fig. 53, bisects the angle bab', 
 between the two axes of revolution of these two circumscribed cylinders : the plane 
 of the new ellipse of contact being geometrically determined by a process exactly 
 similar to that employed in 218, (7.); and being perpendicular to the new vector, 
 c' + k\ as the old plane of contact was (by 218, (11.)) to t + k. 
 
 Section 14. — On the Reduction of the General Quaternion 
 to a Standard Quadrinomial Form ; icith a First Proof of 
 the Associative Principle of Multiplication of Quaternions, 
 
 221. Retaining the significations (181) of the three rect- 
 angular unit-lines oi, oj, ok, as the axes, and therefore also 
 the indices (159), of three given right versors 2, J, k, in three 
 mutually rectangular planes, we can express the index oq of 
 any other right quaternion, such as Yq^ under the trinomial 
 form (comp. 62), 
 
 I. . . IV$' = 0Q = a;.oi+y.0J + Z.OK; 
 
 where xyz are some three scalar coeflScients, namely, the three 
 rectangular co-ordinates of the extremity q of the index, with 
 respect to the three axes oi, oj, ok. Hence we may write 
 also generally, by 206 and 126, 
 
 II. . . \q = xi + yj + zk = ix +jy + kz ; 
 
 and this last form, ix +jy + kz^ may be said to be a Standard 
 Trinomial Form, to which every right quaternion, or the right 
 part Yq of any proposed quaternion q, can be (as above) re- 
 duced. If then we denote by w the scalar part, Sq, of the same 
 general quaternion q, we shall have, by 202, the following 
 General Reduction of a Quaternion to a Standard Quadri- 
 nomial Form (183) : 
 
 * If room shall allow, a few additional remarks may be made, on the relations 
 
 of the constant vectors t, k, &c., to the ellipsoid, and on some other constructions of 
 
 that surface, when, in the following Book, its equation shall come to be put under the 
 
 new form, 
 
 T(tp+pK) = /c2-t2. 
 
 2 H 
 
234 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 III. . . 2' = (Sq + V*^ =)w + ix ^jy + kz ; 
 
 in which the four scalars, wxyz^ may be said to be the Four 
 Constituents of the Quaternion. And it is evident (comp. 202, 
 (5.), and 133), that if we write in like manner, 
 
 IV. . . q =w \ ix -vji/ + kz\ 
 
 where ijk denote the same three given right versors (181) as 
 before, then the equation 
 
 between these two quaternions, q and q\ includes the Jour follow- 
 ing scalar equations between the constituents : 
 
 VI. . . w' = w, x ~x, y "^y^ z' = z\ 
 
 which is a new justification (comp. 112, 116) of the propriety 
 of naming, as we have done throughout the present Chapter, 
 the General Quotient oftioo Vectors (101) a Quaternion. 
 
 222. When the Standard Quadrinomial Form (221) is 
 adopted, we have then not only 
 
 1. . . ^q = w, and V^ = ix ^jy + kz, 
 
 as before, but also, by 204, XI., 
 
 II. . . K^ = (Sg - Yq =) 10 - ix ~jy - kz. 
 
 And because the distributive property of multiplication of qua- 
 ternions (212), combined with the laws of of the symbols ijk 
 (182), or with the General and Fundamental Formula of this 
 whole Calculus (183), namely with the formula, 
 
 P=f = k^=^ijk = -\, (A) 
 
 gives the transformation, 
 
 III. . . {ix +jy 4- kzY = - (a;2 + 2/2 + z% 
 
 we have, by 204, &c., the following new expressions : 
 
 IV. . . NVg=(TV(?)2 = -V22^a;2 + ?/2+r2. 
 
 V. . . TV2= V(^' + 3/' + -2'); 
 
 VI. . .\]Yq = {ix^jy-\-kz)'. ^/ {x^ -^ y"" ^ z"^) ; 
 
 VII. . . % = T^'' = Sy'^-V^2 = w;'^+a;^ + z/2 + 2:^ 
 
 VIII. . . T^ = V i^o'' + a;2 + 3/2 + z") ; 
 
 IX. . . U$' = (w? + ix ^jy -^kz): y/ (w^ + a;^ + z/^ ^ ^2^ . 
 
CHAP. I.] LAW OF THE NORMS. 235 
 
 X. . . SU^ = w: s/(w^ + x'^ + 2/2 + z^) ; 
 XI. . . VU^- = (ix +jy + kz): yj (yo' ^ x' + y''^ z"^) ; 
 
 xii...Tvug=) -'rr . 
 
 ^ \ 2v^ + x^ -\- y^ + z^ 
 
 (1.) To prove the recent formula III., we may arrange as follows the steps of 
 the multiplication (comp. again 182) : 
 
 Yq = ix ■\-jy + hz, 
 Yq — ix -\-jy + kz ; 
 ix .Yq = — x'^-Y kxy —jxz ; 
 jy-Yq^-y^- kyx + iyz, 
 
 kz.Yq = — z^ +jzx — izy ; 
 
 Yq^ = Yq.Yq==-x^-y^-z^. 
 (2.) We have, therefore, 
 
 XIII. . . {ix -\-jy + kzy = - 1, if x^+y^+z^ = 1, 
 a result to which we have already alluded,* in connexion with the partial indeter- 
 minateness of signification, in the present calculus, of the symbol V — 1, when consi- 
 dered as denoting a right radial (149), or a right versor (153), of which the plane 
 or the axis is arbitrary. 
 
 (3.) If q" = qq, then N/'=Ng'.%, by 191, (8.); but if g = m; + &c., 
 q =z w' ■{ &t,c., (2'"= u;"+ &c,, then 
 
 ■ w" = w'w — {x'x+y'y + z'z), 
 x" = (w'x + x'w) + {y'z - z'y), 
 
 y" = (w'y + y'w) + (z'^c — a?'*), 
 
 z" = (w'z 4- z'w) + {xy — y'x') ; 
 
 and conversely these four scalar equations are jointly equivalent to, and may be 
 summed up in, the quaternion formula, 
 
 XV. . . u?" + ix" +J7j" + kz" = (w' + ix' +jy' + kz') (w + ix +jy + kz) ; 
 we ought therefore, under these conditions XIV., to have the equation, 
 
 XVI. . . w"2 + ar"2 + y"2 -I- z"2 = (a,'2 + ^'2 + y'2 + a'2) (^j-i ^ ^'^ + y^ + z^) ', 
 
 which can in fact be verified by so easy an algebraical calculation, that its truth 
 may be said to be obvious upon mere inspection, at least when the terms in the four 
 quadrinomial expressions w" . . z' are arrangedf as above. 
 
 * Compare the first Note to page 131 ; and that to page 162. 
 
 f From having somewhat otherwise arranged those terms, the author had some 
 little trouble at first, in verifying that the twenty-four double products, in the ex- 
 pansion of w'"^ + &c., destroy each other, leaving only the sixteen /)roc?Mcfs of squares, 
 or that XVI. follows from XIV,, when he was led to anticipate that result through 
 quaternions, in the year 1843. He believes, however, that the algebraic theorem 
 XVI., as distinguished from the quaternion formula XV., with which it is here con- 
 nected, had been discovered by the celebrated Euler. 
 
 XIV. 
 
236 ELEMENTS OF QUATERNIONS. [bOOK II, 
 
 223. The principal use which we shall here make of the 
 standard quadrinomial form (221), is to prove by it the gene- 
 ral associative property of multiplication of quaternions ; which 
 can now with great ease be done, the distributive* property 
 (212) of such multiplication having been already proved. In 
 fact, if we write, as in 222, (3.)j 
 
 [ q = w + ix +jy + kz, 
 L . . ^ g' = w + ix +jy' + kz\ 
 j^/ = w" + IX ' -^jy" .+ kz% 
 
 without now assuming that the relation q" ^qq^ or any other 
 relation, exists between the three quaternions q^ q\ q\ and 
 inquire whether it be true that the associative formula^ 
 
 II. . -qq^q^q-qq, 
 holds good, we see, by the distributive principle, that we have 
 only to try whether this last formula is valid when the three 
 quaternion factors q, 5'', q are replaced, in any one common 
 order on both sides of the equation, and with or without repe- 
 tition, by the three given right versors ijk ; but this has al- 
 ready been proved, in Art. 183. We arrive then, thus, at the 
 important conclusion, that the GeJieral Multiplication 0/ Qua- 
 ternions is an Associative Operation^ as it had been previously 
 seen (2 1 2) to be a Distributive one : although we had also 
 found (168, 183, 191) that such Multiplication is not (in ge- 
 neral) Commutative : or that the two products^ q'q and qq\ are 
 generally unequal. We may therefore omit the point (as in 
 183), and may denote each member of the equation II. by the 
 symbol q'q'q- 
 
 (1.) Let v = Vq, v' = Yq', v" = Yq" \ SO that v, v', v" are any three right qua- 
 ternions, and therefore, by 191, (2.), and 196, 204, 
 
 f^, Kv'u = vv)\ Sf't? = \ (v'v + vv")j Yv'v = ~ (w'w — vv'). 
 
 Let this last right quaternion be called w„ and let Sv'v = s„ so that v'v = s^ + v/, we 
 shall then have the equations, 
 
 • At a later stage, a sketch will be given of at least one proof of this Associative 
 Principle of Multiplication^ which will not pj-esuppose the Distributive Principle. 
 
f- tA^ 
 
 CHAP. I.] ASSOCIATIVE PRINCIPLE OF MULTIPLICATION. 237 
 
 2Vv"w, = v'v, — vv" ; = v"a\ — sv" ; 
 
 whence, by addition, 
 
 2 V»"t7^ = v". v'v — v'v . v" 
 
 — (v"v' + v'v")v - v'{v"v + ry") 
 = 2wSw'y" — 2©'Su"u ; 
 and therefore generally, if r, v', t>" be still n^/t<, as above, 
 
 in. . . V. v"Yv'v = v^vv" - «'Sr"« ; 
 a formula with which the student ought to make himself completely familiar, on ac- 
 count of its extensive utility. 
 
 (2.) With the recent notations, 
 
 V . v'^v'v = Nv"s^ = v"s^ = v"S«i;'; 
 
 we have therefore this other very useful formula, ■ / ^ 
 
 IV. . . V . v"vv = v^v'v"- v'%v"v + v'^vv, ^ ^"/r ' 
 
 where the point in the first member may often for simplicity be dispensed with ; and 
 in which it is still supposed that 
 
 TT 
 
 Lv = Lv = Lv = -. 
 
 (3.) The formula IIL gives (by 206), 
 
 V. . . IV, v"Yv'v = lv. SvV- lu'. St?"»; 
 hence this last vector, which is evidently complanar with the two indices Iv and Iw', 
 is at the same time (by 208) perpendicular to the third index Iv", and therefore (by 
 (1.) ) complanar with the third quaternion q". 
 
 (4.) With the recent notations, the vector, 
 
 VI. ..lv, = l\v'v = lV(Vg'.V9), 
 
 is (by 208, XXII.) a line perpendicular to both It; and Iw'; or common to the planes 
 of q and q' ; being also such that the rotation round it from Iv' to \v is positive : 
 while its length, 
 
 TIv,, or Tu,, or TY.v'v, or TV(Vg'.Vg), 
 hears to the unit of length the same ratio, as that which the parallelogram under the 
 indices, Iv and Iv', bears to the unit of area. 
 
 (6.) To interpret (comp. IV.) the scalar expression, 
 
 VII. . . Sv'v'v = Sp"», = S.v"Yv'v, 
 
 (because S»"5,= 0), we may employ the formula 208, V. ; which gives the the trans- 
 formation, 
 
 VIII. . . Sv'v'v = Tv". Tw . cos (tt-x); 
 
 where Tv" denotes the length of the line Iv", and Tv, represents by (4.) the area 
 (positively taken) of the parallelogram under Iv' and Iv ; while x is (by 208), the 
 angle between the two indices Iv", Iv,. Tliis angle will be obtuse, and therefore the 
 cosine of its supplement will he positive, and equal to the sine of the inclination of 
 the line Iv' to the plane oflv and Iv, if the rotation round Iv" from Iv' to Iv be 
 negative, that is, if the rotation round Iv from Iv' to Iv" be positive ; but that cosine 
 will be equal the negative of this sine, if the direction of this rotation be reversed. 
 We have therefore the important interpretation : 
 
 IX. . . S«"i''v = + volume of parallelepiped under Iv, Iv, \v" ; 
 
238 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 the upper or the lower sign being taken, according as the rotation round Ir, from 
 \v' to lv\ is positively or negatively directed. 
 
 (6.) For example, we saw that the ternary products ijk and kji have scalar va- 
 lues, namely, 
 
 ijk=^-U kji = +l, by 183, (1,), (2.); 
 and accordingly the /jara^/ff/epipec? of indices becomes, in this case, a.n unit-cube ; 
 while the rotation round the index oft, from that ofj to that of ^, is positive (181). 
 
 (7.) In general, for any three right quaternions vv'v", we have the formula, 
 X. . . 8vv'v" = — Sv"v'v ; 
 and when the three indices are complanar, so that the volume mentioned in IX. va- 
 nishes, then each of these two last scalars becomes zero ; so that we may write, as a 
 new Formula of Complanarity ; 
 
 XI. . . St;"»'« = 0, if Iv" \\\\v', Iv (123) : 
 while, on the other hand, this scalar cannot vanish in any other case, if the quater- 
 nions (or their indices) be still supposed to be actual (1, 144); because it then re- 
 presents an actual volume. 
 
 (8.) Hence also we may establish the following Formula of Collinearity, for any 
 three quaternions : 
 
 XII. . . S (Yq" . Yq, Yq) = 0, if lYq" \ \ \ lYq', lYq ; 
 that is, by 209, if the planes of q, q, q" have any common line. 
 
 (9.) In general, if we employ the standard trinomial form 221, II., namely, 
 v = Yq = ix +jy + kz, v' = ix' + &c. , v"= ix" + &c. , 
 
 the laws (182, 183) of the symbols i,j, k give the transformation, 
 
 XIII. . . S^''^'^ = x"{z'y — y'z) + y'\x'z - zx) + z"{rf'x — x'y') \ 
 and accordingly this is the known expression for the volume (with a suitable sign) 
 of the parallelepiped, which has the three lines op, op', op" for three co-initial 
 edges, if the rectangular co-ordinates* of the four corners, o, p, p', p" be 000, xyz, 
 x'y'z', x"y"z". 
 
 (10.) Again, as another important consequence of the general associative pro- 
 perty of multiplication, it may be here observed, that although products oimorethan 
 two quaternions have not generally equal scalars, for all possible permutations of th« 
 factors, since we have just seen a case X. in which such a change of arrangement 
 produces a change of sign in the result, yet cyclical permutation is permitted, under 
 the sign S ; or in symbols, that for any three quaternions (and the result is easily ex- 
 tended to any greater number of such factors) the following formula holds good : 
 
 XIV. . . Sq'q'q = Bqq'q'. 
 In fact, to prove this equality, we have only to write it thus, 
 
 XIV'...S(9'V-9) = S(g.9'Y), 
 and to remember that the scalar of the product of any two quaternions remains unal- 
 tered (198, I.), when the order of those two factors is changed. 
 
 * This result may serve as an example of the manner in which quaternions, 
 although not based on any usual doctrine of co-ordinates, may yet be employed to 
 deduce, or to recover, and often with great ease, important co-ordinate expressions. 
 
CHAP. l.J COMPLANAR QUATERNIONS. 239 
 
 (11.) In like manner, by 192, II., it may be inferred that 
 
 XV. . . K'qq'q =^{q". q'q) = Kq'q . Kq" = Kq . Kq' . Kq", 
 with a corresponding result for any greater number of factors; whence by 192, I., 
 if Uq and Il'g' denote the products of any one set of quaternions taken in two op- 
 posite orders, we may write, 
 
 XVI. . . KUq = n'Kq ; XVII. . . RUq = U'Rq. 
 
 (12.) But if V be right, as above, then Ku = - v, by 144 ; hence, 
 
 XVIII. .. Knc=± n't?; XIX. . . srio = + sn'«; xx. . . vnu =+vn'w; 
 
 upper or lower signs being taken, according as the number of the right factors is 
 even or odd; and under the same conditions, 
 
 XXL . . snr = I (uv ± n'v) ;. xxii. . . vn« = i(Uv + Wv) ; 
 
 as was lately exemplified (1.), for the c&se where the number is two. 
 
 (13.) For the case where that number is three, the four last formulae give, 
 
 XXIIT. . . Sv'v'v = — Svv'v" = ~ (v"v'v — vv'v") ; 
 
 XXIV. . . Yv'v'v =-\-Yvv'v" = I (y"v'v + vv'v") ; 
 results which obviously agree with X. and IV. 
 
 224. For the case of Complanar Quaternions (119), the power of 
 reducing each (120) to the form of a fraction (101) which shall have, 
 at pleasure, for its denominator or for its numerator, any arbitrary 
 line in the given plane, furnishes some peculiar facilities for proving 
 the commutative and associative properties oi Addition (207), and the 
 distributive and associative properties oi Multiplication (212, 223); 
 while, for this case of multiplication of quaternions, we have already 
 seen (191, (I-)) *^^^ *^® commutative property also holds good, as 
 it does in algebraic multiplication. It may therefore be not irrele- 
 vant nor useless to insert here a short Second Chapter on the subject 
 oi ^UQh complanars : in treating briefly of which, while assuming as 
 proved the existence of all the foregoing properties, we shall have an 
 opportunity to say something of Powers and Roots and Logarithms ; 
 and of the connexion of Quaternions with Plane Trigonometry, and 
 with Algebraical Equations. After which, in the Third and last 
 Chapter of this Second Book, we propose to resume, for a short time, 
 the consideration oi Diplanar Quaternions; and especially to show 
 how the Associative Principle of Multiplication can be established, 
 for them, without* employing the Distributive Principle, 
 
 * Compare the Note to page 236. 
 
240 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 CHAPTER II. 
 
 ON COMPLANAR QUATERNIONS, OR QUOTIENTS OF VECTORS IN 
 
 ONE PLANE ; AND ON POWERS, ROOTS, AND 
 
 LOGARITHMS OF QUATERNIONS. 
 
 Section 1. — On Complanar Proportion of Vectors; Fourth 
 Proportional to Three, Third Proportional to Two, Mean 
 Proportional, Square Root; General Reduction of a Qua- 
 ternion in a given Plane, to a Standard Binomial Form. 
 
 225. The Quaternions of the present Chapter shall all be 
 supposed to be complanar (119); their common plane being 
 assumed to coincide.with that of the given right versor t ( 1 8 1 ). 
 And all the lines, or vectors, such as a, j3, 7, &c., or ao> oi, 02) 
 &c., to be here employed, shall be conceived to be in that 
 given plane of 2; so that we may write (by 123), for the pur- 
 poses of this Chapter, thejbrmulce of complanarity : 
 
 ?lll?'lll/---llh'; «llh-> /3||iz, ««|||i,&c. 
 
 226. Under th^se conditions, we can always (by 103, 117) 
 interpret any symbol of the form (j3 : a) .7, as denoting a line 
 8 in the given plane; which line may also be denoted (125) 
 by the symbol (7 ; a) .j3, but nof^ (comp. 103) by either of the 
 two apparently equivalent symbols, (J3.7) : a, {y.^):a\ so 
 that we may write, 
 
 I... 8 = ^7 = ^/3, 
 
 a a 
 
 and may say that this line 8 is the Fourth Proportional to the 
 
 * In fact the symbols /3 . y, y . j3, or /3y, y/3, have not as yet received -with us 
 any interpretation ; and even when they shall come to be interpreted as represent- 
 ing certain quaternions, it will be found (comp. 168) that the two combinations, 
 
 - y and — , have generally different significations. 
 a a 
 
CHAP. II.] COMPLANAR PROPORTION OF VECTORS. 241 
 
 three lines a, P, 7 ; or to the three lines a, 7, /3 ; the two 
 Means, /3 and 7, of any such Complanar Proportion of Four 
 Vectors, admitting thus of being interchanged, as in algebra. 
 Under the same conditions we may write also (by 125), 
 
 II...a = -g7 = g0. /3 = -g = -a; 7 = ^a=^S, 
 
 so that (still as in algebra) the two Extremes, a and S, of any 
 such proportion of four lines a, jS, 7, d, may likewise change 
 places among themselves : while we may also make the means 
 become the extremes, if we at the same time change the ex- 
 tremes to means. More generally, if a, /3, 7, ^, e . . . be «wy 
 odd number of vectors in the given plane, we can always find 
 another vector p in that plane, which shall satisfy the equa- 
 tion, 
 
 "I Vr-^' - "^'••- •••ii-=i' 
 
 and when such a formula holds good, for any 07ie arrangement 
 of the numerator-lines a, 7, e, . . . and of the denominator-lines 
 /o, j3, S . . . it can easily be proved to hold good also for any 
 other arrangement of the numerators, and any other arrange- 
 ment of the denominators. For example, whatever four (com- 
 planar) vectors may be denoted by ^yde, we have the trans- 
 formations, 
 
 the two numerators being thus interchanged. Again, 
 
 so that the two denominators also may change places. 
 
 227. An interesting case of such proportion (226) is that 
 in which the means coincide; so that only three distinct lines, 
 such as a, j3, 7, are involved : and that we have (comp. Art. 
 149, and Fig. 42) an equation of the form, 
 
 I. ..7 = ^^, or a=^i3, 
 a 7 
 
 2 I 
 
242 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 but nof^ 7 = ]3j3 : a, nor a = j3/3 : y. In this case, it is said that 
 the three lines afiy form a Continued Proportion; of which a 
 and y are now the Extremes, and j3 is the Memi : this line j3 
 being also said to be af Mea7i Proportional between the two 
 others, a and y ; while y is the Third Proportional to the two 
 lines a and j3 ; and d is, at the same time, the third propor- 
 tional to y and j3. Under the same conditions, we have 
 
 1I.../3 = ^, = I„; 
 
 SO that this mean, /3, between a and 7, is also the fourth pro- 
 portional (226) to itself, as first, and to those two other lines. 
 We have also (comp. again 149), 
 
 III. l^\-y fP' 
 
 ■J 
 a \y 
 
 whence it is natural to write, 
 and therefore (by 103), 
 
 although we are not here to write j3 = (ya)i, nor j3 = (ay)^. 
 But because we have always, as in algebra (comp. 199, (3.) ), 
 the equation or identity, (- qy = g\ we are equally well enti- 
 tled to write. 
 
 fi-? -^■e^-e^ 
 
 the symbol gh denoting thus, in general, either of two opposite 
 quaternions, whereof however one, namely that one of which 
 the angle is acute, has been already selectedm 199, (1.), as that 
 which shall be called by us the Square Root of the quaternion 
 
 * Compare the Note to the foregoing Article. 
 
 f "We say, a mean proportional ; because we shall shortly see that the opposite 
 line, — j3, is in the same sense another mean; although a rule will presently be given, 
 for distinguishing between them, and for selecting one, as that which may be called, 
 by eminence, the mean proportional. 
 
CHAP. II.] CONTINUED PROPORTION, MEAN PROPORTIONAL. 243 
 
 q^ and denoted by 'sj q. We may therefore establish the for- 
 mula, 
 
 if a, jS, 7 form, as above, a continued proportion ; the upper 
 signs being taken when (as in Fig. 42) the angle aoc, between 
 the extreme lines a, y, is bisected by the line ob, or /3, itself; 
 but the lower signs, when that angle is bisected by the opposite 
 line, -/B, or when j3 bisects the vertically opposite angle (comp. 
 again 199, (3.) ): but tho, proportion of tensors, 
 
 VIII. ..Ty:Tj3 = Tj3:Ta, 
 and the resulting formula3, 
 
 IX. . . T/3^ = Ta .Ty, Tj3 = v/ (Ta .Ty), 
 in ^aeA case holding good. And when we shall speak simply 
 of the Mean Proportional between two vectors, a and y, which 
 make any acute, or right, or obtuse angle with each other, we 
 shall always henceforth understand the former of these two 
 bisectors ; namely, the bisector ob of that angle aoc itself, and 
 not that of the opposite angle : thus taking upper signs, in the 
 recent formula VII. 
 
 (1.) At the limit wheu the angle aoc vanishes, so that Uy = Ua, then U/3 — 
 each of these two unit-lines; and the mean proportional /3 has the same common 
 direction as each of the two given extremes. This comes to our agreeing to write, 
 X. . . VI = + 1, and generally, X'. . . V(a2) =+ a, 
 
 if a be any positive scalar. 
 
 (2.) At the other limit, when A0C = 7r, or Uy =— Ua, the length of the mean 
 proportional /3 is still determined by IX., as the geometric mean (in the usual sense) 
 between the lengths of the two given extremes (comp. the two Figures 41); but, 
 even with the supposed restriction (225) on the plane in which all the lines are 
 situated, an ambiguity arises in this case, from the doubt which of the two opposite 
 perpendiculars at o, to the line AOC, is to be taken as the direction of the mean vec- 
 tor. To remove this ambiguity, we shall suppose that the rotation round the axis 
 of i (to which axis all the lines considered in this Chapter are, by 225, perpendicu- 
 lar), from the first line oa to the second line ob, is in this case positive ; which 
 supposition is equivalent to writing, for present purposes, 
 
 XI.* . . V-l = + i; and XI'. . . V(- a^) = la, if a>0. 
 
 * It is to be carefully observed that this square root of negative unity is not, in 
 any sense, imaginary, nor even ambiguous, in its geometrical interpretation, but 
 denotes a real and given right versor (181). 
 
244 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 And thus the mean proportional between two vectors (^in the given plane) becomes, 
 in all cases, determined : at least if their order (as first and third) be given. 
 
 (3.) If the restriction (225) on the common plane of the lines, were removed, we 
 might then, on the recent plan (227), fix definitely the direction, as well as the 
 length, of the mean OB, in every case hut one: this excepted case being that in 
 which, as in (2.), the tvio given extremes, OA, oc, have exactly opposite directions ; so 
 that the angle (aoc = tt) between them has no one definite bisector. In this case, the 
 sought point b would have no one determined position, but only a locus : namely the 
 circumference of a circle, with o for centre, and with a radius equal to the geome- 
 tric mean between oa, oc, while its plane would be perpendicular to the given right 
 line AOC. (Comp. again the Figures 41 ; and the remarks in 148, 149, 153, 154, 
 on the square of a right radial, or versor, and on the partially indeterminate cha- 
 racter of the square root of a negative scalar, when interpreted, on the plan of this 
 Calculus, as a real in geometry.) 
 
 228. The quotient of any two complanar and right quater- 
 nions has been seen (191, (6.) ) to be a scalar ; since then we 
 here suppose (225) that q\\\h we are at liberty to write, 
 
 I. . . Sg = aj ; V^ 'i=y', y^q - yi = iy ; 
 and consequently may establish the following Reduction of a 
 Quaternion in the given Plane (of i) to a Standard Binomial 
 Form* (comp. 221) : 
 
 II. . . q^x^iy, if q\\\i', 
 X and y being some two scalars, which may be called the two 
 constituents (comp. again 221) of this binomial. And then an 
 equation between two quaternions, considered as binomials of 
 this form, such as the equation, 
 
 III, ' ' q' =q, or III'. . , od ■\- iy = x + iy^ 
 breaks up (by 202, (5.) ) into two scalar equations between 
 their respective constituents^ namely, 
 
 IV. . . x=^x, y=y, 
 notwithstanding the geometrical reality of the right versor, i. 
 
 (1.) On comparing the recent equations II., III., IV., with those marked as III., 
 v., VI., in 221, we see that, in thus passing from general to com/)7anar quaternions, 
 we have merely suppressed the coefficients ofj and k, as being for our present purpose, 
 null ; and have then written x and y, instead of w and x. 
 
 * It \& permitted, by 227, XI., to write this expression as aj + y V — 1 ; but the 
 form a; + ty is shorter, and perhaps less liable to any ambiguity of interpretation. 
 
CHAP. II.] STANDARD BINOMIAL FORM, COUPLE. 245 
 
 (2.) As the word " binomial" has other meanings in algebra, it may be conve- 
 nient to call the form II. a Couple ; and the two constituent scalars x and y, of 
 which the values serve to distinguish one such couple from another, may not unna- 
 turally be said to be the Co-ordinates of that Couple, for a reason which it may be 
 useful to state. 
 
 (3.) Conceive, then, that the plane of Fig. 60 coincides with that of i, and that 
 positive rotation round Ax.i is, in that Figure, directed towards the left-hand; 
 which may be reconciled with our general convention (127), by imagining that this 
 axis of i is directed from o towards the back of the Figure ; or below* it, if horizon- 
 tal. This being assumed, and perpendiculars bb', bb" being let fall (as in the Fi- 
 gure) on the indefinite line oa itself, and on a normal to that line at o, which nor- 
 mal we may call oa', and may suppose it to have a length equal to that of oa, with 
 a left-handed rotation aoa', so that 
 
 V. . . 0A' = i.0A, or briefly, V. . . a' = ia, 
 while j3' = ob', and /3"= ob", as in 201, and q = (3:a, as in 202 ; 
 
 then, on whichever side of the indefinite right line oa the point b may be situated, 
 a comparison of the quaternion q with the binomial form II. will give the two equa- 
 tions, 
 
 VI. . . iK (= S5) = j8' : a ; y (= Yq : i = /3" : ia) = /3" : a ; 
 
 so that these two scalars, x and y, are precisely the two rectangular co-ordinates of 
 the point B, referred to the two lines OA and oa', as ttbo rectangular unit-axes, of 
 the ordinary (or Cartesian) kind. And since evert/ other quaternion, g'z=x' + iy\ 
 in the given plane, can be reduced to the form y : a, or 00 : OA, where c is a point 
 in that plane, which can be projected into c' and c" in the same way (comp. 197, 
 205), we see that the two new scalars, or constituents, x' and y', are simply (for 
 the same reason) the co-ordinates of the new point c, referred to the same pair of 
 axes. 
 
 (4.) It is evident (from the principles of the foregoing Chapter), that if we thus 
 express as couples (2.) any two complanar quaternions, q and q, we shall have the 
 following general transformations for their sum, difference, and product : 
 
 Nil.. . q±q = {x'±x) + i(jy'±y); 
 VIII. . . q,q = (x'x - y'y) + i {x'y + yx). 
 (6.) Again, for any one such couple, q, we have (comp. 222) not only Sg = x, and 
 V5 = iy, as above, but also, 
 
 IX. . .Kg = a;-z>; X. . . N9 = x2 +y2 . XL . . T5=V(a;2 +y3); 
 
 XII... U, = -^,; XIII...i=4^^;&c. 
 
 V(-x'2+y^) q a;2-fy2' 
 
 (6.) Hence, for the quotient of any two such couples, we have, 
 f 9' _ x + it/ _ x" + iy' 
 XIV. . . \'^~ x + iy ~ a;2+y 
 
 [_ x" = x'x + y'y, y" = yx - x'y. 
 
 2, x" -I- iy = g'K^, 
 
 * Compare the second Note to page 108. 
 
246 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (7.) The law of the norms (191, (8.) ), or the formula, N^'g- = N^' . Nj, is ex- 
 pressed here (comp. 222, (3.) ) by the well-known algebraic equation, or identity, 
 
 XV. . . (af^ + y^) {x^ +y^) = {,xx -y'y)^ ^{x'y + y'xy ', 
 
 in which xyx'y' may be any four scalars. 
 
 Section 2. — On Continued Proportion of Four or more Vec- 
 tors ; Whole Powers and Roots of Quaternions ; and Roots 
 of Unity, 
 
 229. The conception of continued proportion {211) may 
 easily be extended from the case o^ three to that of four or 
 more (com planar) vectors ; and thus a theory may be formed 
 oi cubes and higher whole powers of quaternions ^ with a corre- 
 spondingly extended theory of roots of quaternions, including 
 roots of scalars^ and in particular of unity. Thus if we sup- 
 pose that the four vectors a^y^ form a continued proportion, 
 expressed by the formulae. 
 
 I. . . - = 75 = -, whence II. . . - = - ^ ^ ' "^ ^^ 
 7 p a a y p a 
 
 (by an obvious extension of usual algebraic notation,) we may 
 say that the quaternion S : a is the cube^ or the third power, of 
 j3 : a ; and that the latter quaternion is, conversely, a cube- 
 root (or third root) of the former ; which last relation may na- 
 turally be denoted by writing, 
 
 III. . . ^ = ('^Y, or Iir. ../3 = ^^Ya(comp.227,IV.,V.). 
 
 230. But it is important to observe that as the equation 
 q"^ = Q, in which «/ is a sought and Q is a given quaternion, 
 was found to be satisfied by two opposite quaternions q, of the 
 form ± \/ Q (comp. 227, VII.), so the slightly less simple 
 equation q^= Q is satisfied by three distinct and real quater- 
 nions, if Q be actual and real ; whereof each, divided by either 
 of the other two, gives for quotient a real quaternion, which 
 is equal to one of the cube-roots of positive unity. In fact, if 
 we conceive (comp. the annexed Fig. 54) that /3' and /3" are 
 two other but equally long vectors in the given plane, ob- 
 
CHAP. II.J CUBE-ROOTS OF A QUATERNION, AND OF UNITY. 247 
 
 tained from j3 by two successive and positive rotations, each 
 through the third part of a circumference, 
 so that 
 
 fi' 15" 13' 
 
 IV. 
 
 or 
 
 IV'. 
 
 and therefore 
 
 V... (|)- = (|)-=,,*„v....f =(!)•, l-d 
 
 we shall have 
 
 -■.(?)--(fK!)'=!.--.^.(e 
 
 SO that we are equally entitled, at this stage, to write, instead 
 of III. or III'., these other equations : 
 
 vii...&'=f^Y, li'M' 
 
 or 
 
 Yll'...^-J'-l (5"-(^]K. 
 
 231. A (real and actual) quaternion Q may thus be said 
 to have three (real, actual, and) distinct cube-roots ; of which 
 however only one can have an angle less than sixty degrees ; 
 while none can have an angle equal to sixty degrees, unless the 
 proposed quaternion Q degenerates into a negative scalar. In 
 every other case, one of the three cube-roots of Q, or one of the 
 three values of the symbol Q^, may be considered as simpler 
 than either of the other two, because it has a smaller angle 
 (comp. 199, (!•))» ^^^ ^f w^j for the present, denote this one, 
 which we shall call the Principal Cube-Hoot of the quaternion 
 Q, by the symbol ^ Q, we shall thus be enabled to estabhsh 
 the formula of inequality, 
 
 VIII. ..Z^Q<|, if zQ<7r. 
 
 232. At the limit, when Q degenerates, as above, into a negative 
 scalar, one of its cube-roots is itself a negative scalar, and has there- 
 
248 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 fore its angle = w ; while each of the two other roots has its angle 
 
 = -. In this case, among these two roots of which the angles are 
 o 
 
 equal to each other, and are less than that of the third, we shall 
 consider as simpler^ and therefore as principal^ the one which an- 
 swers (comp. 227, (2.) ) to a positive rotation through sixty degrees ; 
 and so shall be led to write, 
 
 IX...y-l=lii^; and X...^y-l=|; 
 
 using thus the positive sign for the radical ^ 3, by which i is multi- 
 plied in the expression IX. for 2^- 1 ; with the connected for- 
 mula, 
 
 IX'. ..y(-a3) = ^(l4-^V3), if a>0; 
 
 although it might at first have seemed more natural to adopt as 
 principal the scalar value, and to write thus, 
 
 3/-l=-l; 
 which latter is in fact one value of the symbol, (- 1)*. 
 
 (1.) "We have, however, on the present plan, as in arithmetic, 
 
 XI. ,.^1 = 1; and XI'. . . ^(a3) = a, if a>0. 
 (2.) The equations, 
 
 XII...(^-^] =-1, and XIIL..|^— ^j= + l, 
 
 can be verified in calculation^ by actual cubing^ exactly as in algebra ; the only dif- 
 ference being, as regards the conception of the subject, that although i satisfies the 
 equation i^ = — 1, it is regarded here as altogether real; namely, as a real right ver- 
 sor* (181). 
 
 233. There is no difficulty in conceiving how the same general 
 principles may be extended (comp. 229) to a continued proportion 
 of 71 + 1 complanar vectors, 
 
 I. . . a, ai, aa, . . . a„, 
 
 * This conception differs fundamentally from one which had occurred to seve- 
 ral able writers, before the invention of the quaternions ; and according to which 
 the symbols 1 and V — 1 were interpreted as representing a pair of equally long and 
 mutually rectangular right lines, in a given plane. In Qtiaternions, no line is repre- 
 sented by the number, One, except as regards its length ; the reason being, mainly, 
 that we require, in the present Calculus, to be able to deal with all possible planes ; 
 and that no one right line is common to all such. 
 
CHAP. II.] FRACTI0NALPOWERS,GENERALR0OTSOFUNITY. 249 
 
 when n is a whole number greater than three ; nor in interpreting, 
 in connexion therewith, the equations, 
 
 II...^ = f^'r; III...-'=f2^\^; IV. 
 
 a \ a 
 
 •••"=(7)""- 
 
 Denoting, for the moment, what we shall call the principal n*^ root 
 of a quaternion Q by the symbol !y/Q, we have, on this plan (comp. 
 231, VIII.), 
 
 V. ..zyQ<-, if za<'^; 
 
 VI. . . ,1 (y- 1) = -; VII. . . Y(y- l):e>0; 
 
 To 
 
 this last condition, namely that there shall be a positive (scalar) co- 
 efficient y of 2, in the binomial (or couple) form x-\-iy (228), for the 
 quaternion^- 1, thus serving to complete the determination of 
 that principal fi*^ root of negative unity ; or of any other negative sca- 
 lar, since ~ 1 may be changed to -a, if «>0, in each of the two last 
 formulae. And as to the general n*^ root of a quaternion, we may 
 write, on the same principles, 
 
 VIIL.. Q^=l^. VQ; 
 
 where the factor 1», representing the general n*^ root of positive 
 unity, has n different values, depending on the division of the cir- 
 cumference of a circle into n equal parts, in the way lately illus- 
 trated, for the case ?z = 3, by Figure 54 ; and only differing from 
 ordinary algebra by the reality here attributed to i. In fact, each 
 of these n*^ roots of unity is with us a real versor; namely the quo- 
 tient of two radii of a circle, which make with each other an angle, 
 equal to the n*^ part of some whole number of circumferences. 
 
 X 
 
 (1.) "We propose, however, to interpret the particular symbol i^, as always de- 
 noting the principal value of the n*^ root of i ; thus writing, 
 
 i n/ 
 IX. . . t« = \/i; 
 
 whence it will follow that when this root is expressed under the form of a couple 
 (228), the two constituents x and y shall both be positive, and the quotient y: x 
 shall have a smaller value than for any other couple x + iy (with constituents thus 
 positive), of which the n*^ power equals i. 
 (2 ) For example, although the equation 
 
 52 = (ar + ty)2 = i, 
 
 vi satisfied by the two values, ± (1 + : V2, we shall write definitely, 
 
 2 K 
 
250 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 x....-.=.v.-=i±i. 
 
 (3.) And although the equation, 
 
 is satisfied by the three distinct and real couples, (i ± V3) : 2, and - 1, we shall adopt 
 only the one value, 
 
 XI. . . il-V t= — r— . 
 
 (4.) In general, we shall thus have the expression, 
 
 XII. . . t** = cos -- + 1 sm — - ; 
 2n 2n 
 
 which we shall occasionally abridge to the following : 
 
 i TT 
 
 Xir. . . i« = cis — : 
 2n' 
 
 and this root^ i", thus interpreted, denotes a versor, which turns any line on which it 
 operates, through an angle equal to the n*'* part of a right angle, in the positive di- 
 rection of rotation, round the given axis of i. 
 
 234. If m and n be anj/ two positive whole numbers, and q 
 any quaternion, the definition contained in the formula 233, 
 II., of the whole power, q^, enables us to write, as in algebra, 
 the two equations : 
 
 I. . . y'"^« = ^»»^ ; II. . . (^")'" = ^™" ; 
 
 and we propose to extend the former to the case of mill and 
 negative whole exponents, writing therefore, 
 
 III. . . ^°= 1 ; IV. . . q^ri-n^^m.gn . 
 
 and in particular, 
 
 Y. . . q-^ = l :q = - = reciprocal* (134) of q. 
 
 We shall also extend the formula II., by writing 
 
 VI. . . (^")'" = q^, 
 
 whether m be positive or negative ; so that this last symbol, 
 ifm and n be still whole numbers, whereof w may be supposed 
 to be positive, has as many distinct values as there are units in 
 the denominator of li^ fractional exponent, when reduced to its 
 
 * Compare the Note to page 121. 
 
CHAP. II.J AMPLITUDE OF A QUATERNION. 251 
 
 m 
 
 least terms ; among which values of q~\ we shall naturally 
 consider as the principal one, that which is the m^^ power of 
 the principal n*^ root (233) of q. 
 
 (1.) For example, the symbol gi denotes, on this plan, the square of any cube- 
 root of 9 ; it has therefore three distinct values, namely, the three values of the cube- 
 root of the square of the same quaternion q ; but among these we regard as principal, 
 the square of the principal cube-root (231) of that proposed quaternion. 
 
 (2.) Again, the symbol q'^ is interpreted, on the same plan, as denoting the 
 square of any fourth root of 5 ; but because (li)2 =z li = + 1, this square has only 
 two distinct values, namely those of the square root q^, the fractional exponent | 
 being thus reduced to its least terms; and among these the principal value is the 
 square of the principal fourth root, which square is, at the same time, the principal 
 square root (199, (l.)> ^^ 227) of the quaternion q. 
 
 (3.) The symbol q-^ denotes, as in algebra, the reciprocal of a square-root of q ; 
 while g'2 denotes the reciprocal of the square, &c. 
 
 (4.) If the exponent #, in a symbol of the form q^, be still a scalar, but a surd (or 
 incommensurable), we may consider this surd exponent, t, as a limit, towards which 
 a variable fraction tends : and the symbol itself may then be interpreted as the corre- 
 sponding limit oi a, fractional power of a quaternion, which has however (in this case) 
 indefinitely many values, and can therefore be of little or no use, until a selection 
 shall have been made, of one value of this surd power &.& principal, according to a law 
 which will be best understood by the introduction of the conception of the amplitude 
 of a quaternion, to which in the next Section we shall proceed. 
 
 (5.) Meanwhile (comp. 233), (4.) ), we may already definitely interpret the sym- 
 bol V' as denoting a versor, which turns any line in the given plane, through t right 
 angles, round Ax.i, in the positive or negative direction, according as this scalar ex- 
 ponent, t, whether rational or irrational, is itself positive or negative ; and thus may 
 establish the formula, 
 
 -TTxr w 'tt , . tir 
 
 VII. . . I* = cos — - -f I sm — ; 
 2 it I 
 
 or briefly (comp. 238, XII'.), 
 
 VIII.. . i' = cis— . 
 2 
 
 Section 3.— ^Ow the Amplitudes of Quaternions in a given 
 Plane; and on Trigonometric Expressions for such Quater- 
 nions, and for their Powers, 
 
 235. Using the binomial or couple form (228) for a qua- 
 ternion in the plane of/ (225), if we introduce two new and 
 real scalars, r and z, whereof the former shall be supposed to 
 be positive, and which are connected with the two former sca- 
 lars X and y by the equations, 
 
 I. . . x-r cos z, y =^r sin ^, r > 0, 
 
252 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 we shall then evidently have the formulae (comp. 228, (5.) ) : 
 
 n. . .Tq = T(x + ii/) = r; 
 
 III. . . TJq = U (aj + iy) = cos z + i8in.z; 
 
 which last expression may be conveniently abridged (comp. 
 
 233, Xir., and 234, VIII.) to the following : 
 
 IV. . , U<^ =cisz ; so that V. . . g==r cisz. 
 
 And the arcual or angular quantity, z, maybe called the Am- 
 pUtude* of the quaternion q ; this name being here preferred 
 by us to " Angle" because we have already appropriated 
 the latter name, and the corresponding symbol Z ^, to denote 
 (130) an angle of the Euclidean kind, or at least one not ex- 
 ceeding, in either direction, the limits and tt ; whereas the 
 amplitude, z, considered as obliged only to satisfy the equa- 
 tions I., may have any real and scalar value. We shall denote 
 this amplitude, at least for the present, by XhQ symbol,^ am.y, 
 or simply, am q ; and thus shall have the following formula, 
 of connexion between amplitude and angle, 
 
 VI. . . (2: =) am . 5^ = 2w7r ± z $» ; 
 
 * Compare the Note to Art. 130. 
 
 t The symbol V was spoken of, in 202, as completing the system of notations 
 peculiar to the present Calculus ; and in fact, besides the three letters^ i, j, k, of which 
 the laws are expressed by thQ fundamental formula (A) of Art. 183, and which were 
 originally (namely in the year 1843, and in the two following years) the only pecu- 
 liar symbols of quaternions (see Note to page 160), that Calculus does not habi- 
 tually employ, with peculiar significations, any more than the^ue characteristics of 
 operation, K, S, T, U, V, for conjugate, scalar, tensor, versor, and vector (or right 
 part) : although perhaps the mark N for norm, which in the present work has been 
 adopted from the Theory of Numbers, will gradually come more into use than 
 it has yet done, in connexion with quaternions also. As to the marks, Z, Ax., I, R, 
 and now am . (or am,,), for angle, axis, index, reciprocal, and amplitude, they are to 
 be considered as chiefly available for the present exposition of the system, and as not 
 often wanted, nor employed, in the subsequentprac^ice thereof ; and the same remark 
 applies to the recent abridgment cis, for cos + i sin ; to some notations in the present 
 Section for powers and roots, serving to express the conception of one «'^ root, &c., 
 as distinguished from another ; and to the characteristic P, of what we shall call in the 
 next section the ponential of a quaternion, though not requiring that notation after- 
 wards. No apology need be made for employing the purely geometrical signs, -i-, 
 II, III, for perpendicularity, parallelism, and complanarity : although the last of 
 them was perhaps first introduced by the present writer, who has found it frequently 
 useful. 
 
CHAP. II.] ADDITION AND SUBTRACTION OF AMPLITUDES. 253 
 
 the upper or the lower sign being taken, according as Ax. q 
 - ± Ax. i ; and n being any whole number, positive or negative 
 or null. We may then write also (for any quaternion 5' ||1 
 the general transformations following : 
 
 VII. . . \Jq = cis am q ; VIII. . . 5' = T^ . cis am q. 
 
 (1.) Writing q = f3: a, the amplitude am. g', or am (/3 : a), is thus a scalar quan- 
 tity, expressing (with its proper sign) the amount of rotation^ round Ax. i, from the 
 line a to the line /3 ; and admitting, in general, of being increased or diminished by 
 any whole number of circumferences, or oi entire revolutions, when only the direc- 
 tions of the two lines, a and /3, in the given plane of i, are given. 
 
 (2.) But the particular quaternion, or right versor, i itself, shall be considered 
 
 as having definitely/, for its amplitude, one right angle; so that we shall establish the 
 
 particular formula, 
 
 . . '"' 
 IX. . . am.t = /i 1 = -. 
 
 (3.) When, for any other given quaternion q, the generally arbitrary integer 
 n in VI. receives any one determined value, the corresponding value of the ampli- 
 tude may be denoted by either of the two following temporary symbols,* which we 
 here treat as equivalent to each other, 
 
 am„ .q, or Zn 9 ; 
 
 so that (with the same rule of signs as before) we may write, as a more definite for- 
 mula than VI., the equation : 
 
 X. . . am,, . 9 = Zm 9 = 2«7r ± Z. 9 ; 
 
 and may say that this last quantity is the n^^ value of the amplitude of q ; while the 
 zero-value, amoj, may be called the principal amplitude (or the principal value of 
 the amplitude). 
 
 (4.) With these notations, and with the convention, amo(— l) = + 7r, we may 
 write, 
 
 XI. . . amo q = loq = ±lq', 
 
 XII. . . am„ a = am,, 1 = Zn 1 = 2n7r, if a > ; 
 and 
 
 XIII. . . am„ (- a) = am„(- 1) = Z„ (- 1) = (2« + 1) tt, 
 
 if a be still a positive scalar. 
 
 236. From the foregoing definition of amplitude, and from 
 the formerly established connexion of multiplication ofversors 
 with composition of rotations (207), it is obvious that (within 
 the given plane, and with abstraction made of tensors) multi- 
 plication and division of quaternions answer respectively to 
 
 * Compare the recent Note, respecting the notations employed. 
 
254 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (algebraical) addition and subtraction of amplitudes : so that, 
 if the symbol sna.q be interpreted in the general (or indefinite) 
 sense of the equation 235, VI., we may write : 
 I. . . am {q'. q) = am q' + am q ; II. . . am (q'l q) = am q'- am q ; 
 implying hereby that, in each formula, o?ie of the values, of the 
 first member is among the values of the second member ; but 
 not here specifying which value. With the same generality 
 of signification, it follows evidently that, for a product of ani/ 
 number of (complanar) quaternions, and for a whole power of any 
 one quaternion, we have the analogous formulae : 
 
 III. . . am rig = S am 5' ; IV. . . am.qP =p.2Lmq ; 
 
 where the exponent p may be any positive or negative integer, 
 or zero. 
 
 (1.) It was proved, in 191, II., that for an7/ two quaternions, the formula Vq'q 
 = XJq'.Vq holds good; a result which, by the associative principle of multiplication 
 (223), is easily extended to ani/ number of quaternion factors (complanar or dipla- 
 nar), with an analogous result for tensors : so that we may write, generally, 
 V. . . Un^ = U\Jq ; VI. . . TUq = UTq. 
 
 (2.) Confining ourselves to the first of these two equations, and combining it with 
 III., and with 235, VII., we arrive at the important formula : 
 
 VII. . . n cis am 5 (= UVq = UII5' = cis am 119) = cis 2 am g ; 
 whence in particular (corap. IV.), 
 
 VIII. . . (cis am q)p=cis(p . am q), 
 at least if the exponent p be still any whole number. 
 
 (3.) In these last formula), the amplitudes am. 5-, am. 5^', &c., may represent a?i^ 
 angular quantities, z, 2', &c. ; we may therefore write them thus, 
 
 IX. . . n cis 2 = cis Sz ; X. . . (cis z)p = cis pz ; 
 
 including thus, under abridged forms, some known and useful theorems, respecting 
 cosines and sines of sums and multiples of arcs. 
 
 (4.) For example, if the number of factors of the form cis z be two, we have 
 
 thus, 
 
 IX'. . . cis z' . cis z = cis (z' + 2) ; X'. . . (cis 2)2 = cis 2z ; 
 whence 
 
 cos (z' + z) = S (cis z' . cis 2) = cos s;' cos z - sin z' sin 2; ; 
 
 sin(2' + z) = i-iV(cisz'.cisz) = cos 2' sin 2 + sin z' cosz ; 
 
 cos 2z = (cos zy — (sin 2)2 ; sin 22 = 2 cos z sin z ; 
 
 with similar results for more factors than two. 
 
 (5.) Without expressly introducing the conception, or at least the notation of 
 amplitude, we may derive the recent formula) IX. and X., from the consideration of 
 the power V (234), as foUoAVS. That pozrer ofi, with a scalar exponent, t, has been 
 
CHAP. II.] POWERS WITH SCALAR EXPONENTS. 255 
 
 interpreted in 234, (5.)j as a symbol satisfying an equation which may be written 
 thus: 
 
 XI. . . V — cis z, if z = ^tTT ; 
 
 or geometrically as a versor, which turns a line through t right angles, where t may 
 be any scalar. We see then at once, from this interpretation, that if*' be either the 
 same or any other scalar, the formula, 
 
 XII. . . iHt'= ii^f, or XIII, . . n . i< = i^\ 
 must hold good, as in algebra. And because the number of the factors t* is easily 
 seen to be arbitrary in this last formula, we may write also, 
 
 XIV. . .(it)p=ipf,' 
 
 if p be any whole* number. But the two last formulae may be changed by XI., to 
 the equations IX, and X., which are therefore thus again obtained ; although the 
 later forms, namely XIII. and XIV., are perhaps somewhat simpler: having in- 
 deed the appearance of being mere algebraical identities, although we see that their 
 geometrical interpretations, as given above, are important. 
 
 (6.) In connexion with the same interpretation XI. of the same useful symbol i*, 
 it may be noticed here that 
 
 XV. .. K.it=i-i', 
 and that therefore, 
 
 XVI. . . cos — = S. i' = i(z' -f i-t) ; 
 
 t'jr 
 
 XVII. . . sin — =. i-i V. i* = i i-i (it - i-ty 
 
 (7.) Hence, by raising the double of each member of XVI. to any positive whole 
 power p, halving, and substituting z for ^tir, we get the equation, 
 
 XVIII. . . 2p-» (cos z)p= I (it+ i-t)P = | (iP*+ i-p*) + Ip (i(p-2)t + i(2-/>)«) + &c. 
 
 = cos pz+p coa(p - 2)z +?-^^-^—^ cos (p - A) z+ 8ic., 
 
 with the usual rule for halving the coefficient of cos Oz, ifp be an even integer ; and 
 with analogous processes for obtaining the known expansions of 2^"^ (sin z)p, for any 
 positive whole value, even or odd, of p ; and many other known results of the same 
 kind. 
 
 237. Ifp be still a whole number, we have thus the transforma- 
 tion, 
 
 I. . . qp = (r cis zy = ?'P cis pz = (TqY cis (/> . ato q) ; 
 
 in which (comp. 190, 161) the two factors, of the tensor and versor 
 kinds, may be thus written : 
 
 II. . . T (qY = {Tqy = T^'' ; III. . . U (q^) = (U^)^ = Vq^ ; 
 
 and any value (235) of the amplitude nm.q may be taken, since all 
 
 • It will soon be seen that there is a sense, although one not quite so definite, in 
 which this formula holds good, even when the exponent p is fractional, or surd ; 
 namely, that the second member is then one of the values of the first. 
 
256 ELEMENTS OF QUATERNIONS. [bOOK II.' 
 
 will conduct to one common value of this whole power q^. And if, 
 for I., we substitute this slightly different formula (comp. 235, 
 (3-)), 
 
 IV. . . (qP)n = TqP . cis (p . am„ q\ with i? = ~, n'>0, 
 
 m^, n', n being whole numbers whereof the first is supposed to be 
 prime to the second, so that the exponent p is here a fraction in its 
 least terms, with a positive denominator n\ while the factor Tq^ is 
 interpreted as expositive scalar (of which the positive or negative 
 logarithm, in any given system, is equal io px the logarithm of T^-), 
 then the expression in the second member admits of n' distinct va- 
 lues, answering to different values of n ; which are precisely the n' 
 values (comp. 234) of the fractional power q^, on principles already 
 established : the principal value of that power corresponding to the 
 value n=0. 
 
 (1.) For any value of the integer w, we may say that the symbol (qp),i, defined 
 by the formula IV., represents the n'^ value of the power qv ; such values, however, 
 recurring periodically, when p is, as above, o. fraction. 
 
 (2.) Abridging (1p)„ to 1^,,, we have thus, generally, by 235, XII., 
 V, . . lP„ = cis Ipnir, if /j be any fraction, 
 a restriction which however we shall soon remove ; and in particular, 
 VI. . . Principal value oflP= 1Po= 1. 
 (3.) Thus, making successively jp = |, /> = ^, we have 
 
 VII. . . li„ = cis mr, Ik = + 1, l^i = - 1, 1^3 = + 1, &c. ; 
 -I7TTT -,1 • 2«7r ,. ^ ^. -l + tV3 ^. -l-tV3 ^. ^ - 
 
 VIII. . . Un = CIS — , Uo = 1, 1*1 = , 1*3 = , 1*3 = 1, &C. 
 
 (4.) Denoting in like manner the n^^ value of (- 1)p by the abridged symbol 
 (- l^w, we have, on the same plan (comp. 235, XIII.), for any fractional* value 
 of/?, 
 
 IX. . . (- iyn = clsp(2n+ l)7r; whence (comp. 232), 
 
 X. ..(-l)io = cis-=+t-, (-l)ii = ci8-2- = -i, (-l)i2 = + t, &c.; 
 and 
 
 XI...(-l)lo = iJ^^ (-1)..=-!, (-l)., = il^%c., 
 
 these three values of (- l)i recurring periodically. 
 
 (5.) The formula IV. gives, generally, by V., the transformation, 
 XII. . . (qp)n = (qP)o cis 2pmr = lP«(gP)o 5 
 so that the n*'» value of qP is equal to the principal value of that power of y, multi- 
 
 As before, this restriction is only a temporary one. 
 
CHAP. II.] PONENTIAL OF A QUATERNION. 257 
 
 plied by the corresponding value of the same power of positive unity ; and it may be 
 remarked, that if the base a be any positive scalar^ the principal p^^ power ^ (^)o) 
 is simply, by our definitions, the arithmetical value of aP. 
 
 (6.) The n*^ value of the p^^ power of any negative scalar, — a, is in like man- 
 ner equal to the arithmetical p^^^ power of the positive opposite, +a, multiplied by 
 the corresponding value of the same power of negative unity; or in symbols, 
 XIII. . . (- a)Pn={- l)Pn (aP)o = (aP)oci8i)(2n+ l)7r. 
 
 (7.) The formula IV., with its consequences V. VI. IX. XII. XIII., may be 
 extended so as to include, as a limit, the case when the exponent p being still scalar, 
 becomes incommensurable, or surd; and although the number of values of the power 
 qp becomes thus unlimited (comp. 234, (4.)), yet we can still consider one of them 
 as the principal value of this (now) surd power : namely the value, 
 
 XIV. . . (5^)0 = TqP . cis {p amo q), 
 which answers to i\xQ principal amplitude (235, (3.) ) of the proposed quaternion q. 
 
 238. We may therefore consider the symbol^ 
 
 ^^ 
 
 in which the base, q^ is any quaternion, while the exponent, p^ 
 is any scalar^ as being now fully interpreted; but no interpre- 
 tation has been as yet assigned to this other symbol of the 
 same kind, qq'^ 
 
 in which both the base q, and the exponent q, are supposed 
 to be (generally) quaternions, although for the purposes of this 
 Chapter complanar (225). To do this, in a way which shall 
 be completely consistent w^th the foregoing conventions and 
 conclusions, or rather which shall include and reproduce them, 
 for the case where the new quaternion exponent, q, degenerates 
 (131) into a scalar, will be one main object of the following 
 Section : which however will also contain a theory of loga- 
 rithms of quaternions, and of the connexion of both logarithms 
 and powers with the properties of a certain function, which 
 we shall call the ponential of a quaternion, and to consider 
 which we next proceed. 
 
 Section 4. — On the Ponential and Logarithm of a Quater- 
 ternion; and on Powers of Quaternions, with Quaternions 
 for their Exponents. 
 
 239. If we consider the polynomial function, 
 
 I. . . P(^, m)=\^q,^q^^..q,,,, 
 2 L 
 
258 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 in which q is any quaternion, and m is any positive whole number, 
 while it is supposed (for conciseness) that 
 
 11. ••^-=i.2.3..mV"r(m+l)/ 
 y^^Ai^' *^6n it is not difficult to prove that however great, hut finite and 
 C ; ^t given, the tensor Tq may be, a finite number m can be assigned, for 
 ^ off ' which the inequality 
 
 III. . . T (P(g, m+n)-F (q, m)) < a, if a > 0, 
 
 shall be satisfied, however large the (positive whole) number n may 
 be, and however small the (positive) scalar a, provided that this last 
 is given. In other words, if we write (comp. 228), 
 
 IV. . . q = x + ii/, F(q, m) = X^ + iT^, 
 a finite value of the number m can always be assigned, such that the 
 following inequality, 
 
 V. . . (X^,,-X„,y + (Y^.„-T,^y<a^ 
 shall hold good, however large the number w, and however small 
 (but given and > 0) the scalar a may be. It follows evidently that 
 each of the two scalar series, or succession of scalar functions, 
 
 VI...Xo=l, X,= \+x, X,= l+x+''-^,.. X^,.. 
 VII...ro = 0, Yr==7/, T, = y+xy,.. Y^, . . . 
 
 converges ultimately to a fixed and finite limit, whereof the one may be 
 called Xoo, or simply X, and the latter Foo, or F, and of which each 
 is a certain function of the two scalars, x and y. Writing then 
 
 VIII. . . Q = Xoo+iFoo=X+er, 
 Ave must consider this quaternion Q (namely the limit to which the 
 following series of quaternions, 
 
 IX...P(g,0)=l, P(?, 1) = 1+^, P(^,2)=l + ^ + |',.. P(^,m),... 
 
 converges ultimately) as being in like manner a Q,Qiid.m function, which 
 we shall call the ponential function, or simply the Ponential of q, in 
 consequence of its possessing certain exponential properties; and 
 which may be denoted by any one of the three symbols, 
 
 P (?» oo), or P {q), or simply P^. 
 We have therefore the equation, 
 
 X. . . Ponential of q=Q==Vq=\-{-qy + q.i-\- . . + qccy 
 with the signification II. of the term q^. 
 
CHAP. II.] EXPONENTIAL PROPERTY. 259 
 
 (1.) In connexion with the convergence of this ponential series, or with the in- 
 equality pi., it may be remarked that if we write (comp, 235) r = T^', and r^ = Tg-^, 
 we shall have, by 212, (2.), 
 
 XI. . . T (P(gr, m + n) - P (gr, m) ) < P (r, m + n) - P (r, m) ; 
 
 it is sufficient then to prove that this last difference, or the sum of the n positive 
 terms, r»i+i, . • ^w+w, can be made < a. Now if we take a number p>2r -1, we 
 shall have r^i <|rp, rp+2< |^p+i» &c,, so that a finite number m>p>2r- 1 can 
 be assigned, such that>»r<ja ; and then, ^ ,^ ft-— 
 
 XII. ..P(r,w+«)-P(r,7n)<a(2-i + 2-2 + ..'+2-«)<a; /"^' h 
 
 the asserted inequality is therefore proved to exist. ^ ^ , 
 
 (2.) In general, if an ascending series with positive coefficients, such as ^^ % 
 
 XIII. . . Ao + Aig' + A2g2 + &c., where Ao> o, Ai>o, &c., 
 
 be convergent when q is changed to a positive scalar, it will ^fortiori converge, 
 when g- is a quaternion. ' " 
 
 ^ ^ 
 
 240. Let q and q^ be any two complanar quaternions, and let q^^ 
 be their sum, so that 
 
 I...5" = S' + g, 2"|||2'|ll?; 
 then, as in algebra, with the signification 239, II. of ^,„, and with 
 corresponding significations of q'm and q'^^j we have 
 
 II. . . qJ' = 1,2.^3!. ^ " ^'"^° "^ ^'""'^' "^ ^''"■'^' "^ • • ^ ^'"^"' 
 
 where ^o = ?'o = l- Hence, writing again r = T^, r,„ = T2'„„ and in 
 like manner r' = T^', r^^=Tq'\ &c., the two differences, _ 
 
 III. . . P (r', 77z) . P (r, 7/1) - P (r'^ m), *^ ^, (^^ f)^(HUi-^ 
 
 and s. /^-^' I -vx 
 IV. ..P(r",2m)-P(r',m).P(r,m), . . ^ ^ < --1- 
 
 can be expanded as sums of positive terms of the form r'p..rp (one^^"^^*^ y? 
 sum containing ^m(m+ 1), and the other containing m(m+ 1) such ^/ 
 terms); but, by 239, HI-, the sum of these two positive differences ^ 
 
 can be made less than any given small positive scalar a, since s ^*^ (*^ "^ 
 
 V. . . P (r'^ 2m) - P (r'', m)<a, if a> 0, 
 
 provided that the number m is taken large enough ; each difference, 
 therefore, separately tends to 0, as m tends to 00 ; a tendency which 
 must exist a fortiori, when the tensors, r, r', r", are replaced by the 
 quaternions., q, q', q'^. The function Vq is therefore subject to the 
 Exponential Law, 
 
 \l...V{q'^q) = Vq'.Vq:=Vq.Vq\ if q' \\\ q. // 
 
260 ELEMENTS OF QUATERNIONS. [boOK II. 
 
 (1.) If we write (comp. 237, (5.) ), 
 
 VII. . . PI = c, then VIII. . . Par = (£*)o = arithmetical value oft" ; 
 where e is the known base of the natural system of logarithms, and x is any scalar. 
 We shall henceforth write simply £«^ to denote this principal (or arithmetical) value of 
 the x*^ power of t , and so shall have the simplified equation, 
 VIII'. . . Pa;;=£*. 
 (2.) Already we have thus a motive for writing, generally, 
 IX. . . Vq = i1', 
 but this formula is here to be considered merely as a definition of the sense in which 
 we interpret this exponential symbol, (9 ; namely as what we have lately called the 
 ponential function, Fq, considered as the sum of the infinite but converging series, 
 239, X. It will however be soon seen to be included in a more general definition 
 (comp. 238) of the symbol g-?'. 
 
 (3.) For any scalar x, we have by VIII. the transformation : 
 
 X. . . x = \'Px = natural logarithm of ponential of x. 
 
 241. The exponential law (240) gives the following general de- 
 composition of a ponential into factors, 
 
 I. . . P^ = P(a;4-e» = P^.P?>; 
 
 in which we have just seen that the factor Vx is a positive scalar. 
 The other factor, Viy, is easily proved to be a versor, and therefore 
 to be the versor ofFq, while Fx is the tensor of the same ponen- . 
 tial; because we have in general, 
 
 11. . .P^.P(-g) = PO=I, and III. ..PK^ = KP^, 
 since IV. . . (K^)- =K(q^) = {say) Kq^ (comp. 199, IX.); 
 
 and therefore, in particular (comp. 150, 158), ^ ^J £ "s- ^"^"t ' 
 
 V. ..l:P^> = P(-^» = KP^3/, or VI.". . NPz> = 1. '"^^^ 
 
 I ■ jfc 
 
 We may therefore write (comp. 240, IX., X.), " ^^,3 ^ 
 
 VII. . . TFq = VSq = Fx=^; VIII, . . x=Sq = lTFq; 
 IX. . . UP5 = PVg' = Piy = 6»>=cis?/ (comp. 235, IV.); 
 this last transformation being obtained from the two series, 
 
 X. . . SPz>=l-^ + &c. = cos^; 
 
 XI. . . r> VFiy = y - ^ + &c. = sin y. 
 
 Hence the ponential P^' may be thus transformed : 
 XII. . . P^ = P (x + iy) = e'' cis 7/. 
 
CHAP. II.] CONNEXION WITH TRIGONOMETRY. 261 
 
 (1) If we had not chosen to assume as known the series for cosine and sine^ nor 
 to select (at first) any one unit of angle, such as that known one on which their va- 
 lidity depends, we might then have proceeded as follows. Writing 
 
 xiil. ..Piy=/y + %, /(-y)-+/y, 0(-y) = -^y, ^ *^ 
 
 we should have, by the exponential law (240), 
 
 XIV. . ./(y + y') = S(Piy.Piy')=/y./y'-0y.0/; 
 
 XV. . .f{y-y)= fy.fy'+<l>y'^y'-. 
 
 and then the functional equation, which results, namely, 
 
 XVI. . . /(y + y') +/(y -yl = 2/y .//, 
 
 would show that 
 
 XVII. . . fy = cos,\ - X a right angle 
 
 whatever unit of angle may be adopted, provided that we determine the constant c 
 by the condition, 
 
 XVIII. . . c = least positive root of the equation fy(= SFiy) = ; 
 or nearly, 
 
 XVIII'. . . c= 1'5708, as the study of the series* would show. 
 (2.) A motive would thus arise for representing a right angle by this numerical 
 constant, c; or for so selecting the angular unit, as to have the equation (tt still de- 
 noting two right angles), 
 
 XIX. . . TT = 2c = least positive root of tke equation fy = — 1 ; 
 
 giving nearly, 
 
 XIX'. . . 7r = 314159, as usual; 
 
 for thus we should reduce XVII. to the simpler form, 
 
 XX. . .fy = cosy. 
 (3.) As to the function (py, since 
 
 XXL . . (fyy + (cpyy='Piy-'Pi-iy) = h 
 it is evident that 0y = + sin y ; and it is easy to prove that the upper sign is to be 
 taken. In fact, it can be shown (without supposing any previous knowledge of co- 
 sines or sines) that (pc is positive, and therefore that 
 
 XXII. . .<pc = + l, or XXIII. . . P«c= t ; 
 whence 
 
 XXIV. . . (py = S.i-^Fiy = SPi(y-c)=f(y-c), 
 and 
 
 XXV. ..Viy=fy + if{y-c). 
 
 If then we replace c by -, we have 
 
 * In fact, the value of the constant c may be obtained to this degree of accuracy, 
 by simple interpolation between the two approximate values of the function/, 
 
 - /(l-5)=+ 0-070737, /(l-6) = -0'029200; 
 
 and of course there arc artifices, not necessary to be mentioned here, by which a far 
 more accurate value can be found. 
 
 
262 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 XXVI. . . 0y = COS [ y - -^ ] = sin 3^ ; and XXVII. . . Viy = cis y, as in IX. 
 
 '; (4.) The series X. XL for cosine and sine might thus be deduced^ instead of being 
 assumed as known : and since we have the limiting value, 
 
 XXIX. . . lim. y-i sin y = lim. y-i i^ YFiy = 1, 
 
 it follows that the unit of angle, which thus gives Pty = cisy, is (as usual) the angle 
 subtended at the centre by the arc equal to radius ; or that the number tt (or 2c) is 
 to 1, as the circumference is to the diameter of a circle. 
 
 (5.) If any other angular unit had been, for any reason, chosen, then a right 
 angle would of course be represented by a different number, and not by 1 '6708 nearly ; 
 but we should still have the transformation, 
 
 XXX. . . Piy = cis ( - X a right angle j, 
 
 though not the same series as before, for cos y and sin y. 
 
 242. The usual unit being retained, we see, by 241, XII., that 
 
 I. . . P. 2m7r = 1, and II. . . P(^ + 2?W) = P^, 
 
 if n be any whole number; it follows, then, that the inverse ponen- 
 tial function, "P'^q, or what we may call the Imponential, of a given 
 quaternion q, has indefinitely many values, which may all be repre- 
 sented by the formula, 
 
 III. . .P„-'^ = lT^f 2am„^; ^ ~- -^ ^^ 
 
 and of which eac^ satisfies the equation, i' f ^ -^ P ["^^ ■* 
 
 IV. . . PP -1^ = ^- '«^ 
 
 while the one which corresponds to w = may be called the Princi- 
 pal Imponential. It will be found that when the exponent p is any 
 scalar, the definition already given (237, IV., XII.) for the n^^ value 
 of the p*^ power of q enables us to establish the formula, 
 
 v...(20.=P(pP„-V); 
 
 and we now propose to extend this last formula, by a new defiiiition, 
 to the more general case (238), when the exponent is a quaternion q': 
 thus writing generally, for any two complanar quaternions, q and q, 
 the General Exponential Formula, 
 
 VI...(g^„ = P(2'P„-ff); 
 
 the principal value of q'^' being still conceived to correspond to n = 0, 
 or to the principal amplitude of q (comp. 235, (3.) ). 
 
CHAP. II.] LOGARITHM OF A QUATERNION. 263 
 
 (1.) For example, 
 
 VII. . . (£9)o = T(qVo-h) = Fq, because Po-if = k = 1 ; 
 
 the ponential Fq, which we agreed, in 240, (2.), to denote simply by 6?, is therefore 
 now seen to be in fact, by our general definition, the principal value of that power, 
 or exponential. 
 
 (2.) With the same notations, 
 
 VIII. . . £»y = cis y, cos y = ^ (c'V + e-^v), sin 7/ =— (e'V - £-♦>) ; 
 
 these two last only differing from the usual imaginary expressions for cosine and sine, 
 by the geometrical reality* of the versor i. 
 
 (3.) The cosine and sine of a quaternion (in the given plane) may now be defined 
 by the equations : 
 
 IX. . . cos 5 = I (£»■« + £"»■«) ; X. . . sin 5^ = — (£»3 - r'l) ; 
 
 and we may write (comp. 241, IX.), 
 
 XI. . . cis 5 = £»■« = Fiq. 
 
 (4.) With this interpretation of cis q, the exponential properties, 236, IX., X., 
 
 continue to hold good ; and we may write, 
 
 XII. . . (59')« = P C^'IT?). P OV amn 5) = (Tq^ cis (5' am,, 5) ; 
 
 a formula which evidently includes the corresponding one, 237, IV., for the n*^ value 
 of the p*^ power of g, when p is scalar. 
 
 (5.) The definitions III. and VI., combined with 235, XII., give generally, 
 
 XIII. . . 1„5' = (19')« = P . 2in7rq' ; XIV. . . {qi')n = !««'. (q^'^O ; 
 
 this last equation including the formula 237, XII. 
 (6.) The same definitions give, 
 
 XV. . . Fo-H = — ; XVI. . . (iOo = £~2- ; 
 
 which last equation agrees with a known interpretation of the symbol, 
 
 -/-I 
 
 considered as denoting in algebra a real quantity. 
 
 (7.) The formula VI. may even be extended to the case where the exponent q' is 
 a quaternion, which is not in the given plane ofi, and therefore not complanar with 
 the base q ; thus we may write, 
 
 XVII.. . (i.> = P(iPo-H-) = P^-^^ = -A; 
 
 but it would be foreign (225) to the plan of this Chapter to enter into any further de- 
 tails, on the subject of the interpretation of the exponential symbol qi', for this case 
 of diplanar quaternions, though we see that there would be no difficulty in treating 
 it, after what has been shown respecting complanars. 
 
 * Compare 232, (2.), and the Notes to pages 243, 248. 
 
264 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 243. As regards the general logarithm q of a quaternion q (in the 
 given plane), we may regard it as any quaternion which satisfies the 
 equation, 
 
 I. . . ei' = Vq' = q', 
 
 and in this view it is simply the Imponential V'^q, of which the n^^ 
 value is expressed by the formula 242, III. But the principal impo- 
 nential, which answers (as above) to w = 0, may be said to be the prin- 
 cipal logarithm^ or simply the Logarithm, of the quaternion q^ and may 
 be denoted by the symbol, 
 
 so that we may write, 
 
 I. . . Ig = Po"'2' = ITg' + i amo g'; 
 or still more simply, 
 
 II. ..1^ = 1(T2.U^) = 1T^+1U^, 
 because 1TU2' = 11 =0, and therefore, 
 
 III. . . lU^ = i amo q. 
 We have thus the two general equations, 
 
 IV... % = lTg; V. .. V1(? = 1U^; 
 in which YTq is still the scalar and natural logarithm of the positive 
 scalar T^'. 
 
 (1.) As examples (comp. 235, (2.) and (4.) ), 
 
 VI. . . It = ifTT ; VII. . . 1(- 1) = iV. 
 
 (2.) The general logarithm of q may be denoted by any one of the symbols, 
 
 log . q, or log q, or (log q\, 
 
 this last denoting the «*^ value ; and then we shall have, 
 
 VIII. . . (log 9)n = 1^ + 2imr. 
 (3.) The formula, 
 
 IX. . . log . 99= log q' + log g-, if q \\\ q, 
 
 holds good, in the sense that every value of the first member is one of the values of 
 
 the second (comp. 236). 
 
 (4.) Principal value ofq'i'= tS'l? ; and one value of log . q9' = q'lq. 
 
 (6.) The quotient of two general logarithms, 
 
 X...(.og,VK.og,),= '0|^. 
 
 may be said to be the ^eweraZ logarithm of the quaternion, q', to the complanar qua- 
 ternion base, q ; and we see that its expression involves* two arbitrary and indepen- 
 dent integers, while its principal value may be defined to be Iq' : \q. 
 
 As the corresponding expression in algebra, according to Graves and Ohm. 
 
CHAP. II.] EQUATIONS OF ALGEBRAIC FORM. 265 
 
 Section 5. — On Finite"^ {or Polynomial) Equations of Alge^ 
 braic Form, involving Complanar Quaternions ; and on the 
 Existence ofn Real Quaternion Roots, of any such Equa- 
 tion of the n*'' Degree, 
 
 244. We have seen (233) that an equation of the form, 
 
 I. . .^"-Q = 0, 
 where n Is any given positive integer, and Q is anyj given, 
 real, and actual quaternion (144), has always n real, actual, 
 and unequal quaternion roots, q, complanar with Q ; namely, 
 
 the n distinct and real values of the symbol Q" (233, VIII.), 
 determined on a plan lately laid down. This result is, how- 
 ever, included in a much more general Theorem, respecting 
 Quaternion Equations of A Igebraic Form ; namely, that if 
 qy, q2i . . qn be any n given, real, and complanar quaternions, 
 then the equation, 
 
 II. . . ^" + q,q^-^ + qiq"-"" -f . . + ^n = 0, 
 has always n real quaternion roots, q, q", . . q^^\ and no more 
 in the given plane ; of which roots it is possible however that 
 some, or all may become equal, in consequence of certain 
 relations existing between the n given coefficients. 
 
 245. As another statement of the same Theorem, if we 
 
 write, 
 
 I. . . Fnq = q"" + qiq''~' + • -^ qm 
 
 the coefficients q^. . qn being as before, we may say that every 
 such polynomial function, Ynq, is equal to a product ofn real, 
 complanar, and linear {or binomial) factors, of the form q-q'; 
 or that an equation of the form, 
 
 lL..Fnq=={q-q'){q-q")--(q-q'''), 
 can be proved in all cases to exist : although we may not be 
 
 * By saying finite equations, we merely intend to exclude here equations with 
 infinitely many terms, such as Fq= 1, which has been seen (242) to have infinitely 
 many roots, represented by the expression q = 2imr, where n may be any whole 
 number. 
 
 t It is true that we have supposed Q ||| t (225) ; but nothing hinders us, in any 
 other case, from substituting for i the versor UVQ, and then proceeding as before. 
 
 2 M 
 
266 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 able, with our present methods, to assign expressions for the 
 roots, q\ . . q'^^\ in terms of the coefficients ^i, . • . qn- 
 
 246. Or we may say that there is always a certain system 
 ofn real quaternions, q\ &c., ||| 2, which satisfies the system of 
 equations, of known algebraic form, 
 
 Ill . . J qq" + qq" + qq" + . . = + 52 ; 
 
 UW"+-- = -^3; &C. 
 
 247. Or because the difference f„5' - "Enq is divisible by 
 q - q, as in algebra, under the supposed conditions of compla- 
 narity (224), it is sufficient to say that at least one real quater- 
 nion q always exists (whether we can assign it or not), which 
 satisfies the equation, 
 
 IV. ..F„^' = 0, 
 
 with the foregoing form (245, 1.) of the polynomial function f. 
 
 248. Or finally, because the theorem is evidently true for 
 the case n=\, while the case 244, 1., has been considered, and 
 the case 9'n = is satisfied by the supposition §' = 0, we may, 
 without essential loss of generality, reduce the enunciation to 
 the following: 
 
 Every equation of the form,* 
 
 l>^.q{q-q){q-q")..{q-q^"-'^) = Q, 
 
 in which q', q'\ . . and Q are any n real and given quaternions 
 in the given plane, whereof at least Q and g'' may be supposed 
 actual (144), is satisfied by at least one real, actual, and com- 
 planar quaterniDn, q. 
 
 * The corresponding ybrm, of the algebraical equation of the n*^ degree, was pro- 
 posed by Mourey, in his very ingenious and original little work, entitled La vraie 
 theorie des Quantites Negatives, et des Quant ites pretendues Tmaginaires (Paris, 
 1828). Suggestions also, towards the ^'eome^ricaZfjroo/ of the theorem in the text 
 have been taken from the same work ; in which, however, the curve here called (in 
 251) an oval is not perhaps defined with sufficient precision : the inequality, here 
 numbered as 251, XII., being not employed. It is to be observed that Mourey's 
 book contains no hint of the present calculus, being confined, like the Double Alge^ 
 bra of Prof. De Morgan (London, 1849), and like the earher work of Mr. Warren 
 (Cambridge, 1828), to questions within theplajie : whereas the very conception of the 
 Quaternion involves, as we have seen, a reference to Tridimensional Space. 
 
CHAP. II.] GEOMETRICAL EXISTENCE OF REAL ROOTS. 267 
 
 249. Supposing that the m-l last of the n-l given quater- 
 nions q' . . g-^""^^ vanish, but that the n-m first of them are actual, 
 where m may be any whole number from 1 to w - 1, and introduc- 
 ing a new real, known, complanar, and actual quaternion Qq, which 
 satisfies the condition, 
 
 Q 
 
 we may write thus the recent equation I., 
 
 and may (by 187, 159, 235) decompose it into the two following: 
 IV. ..'17^=1; and Y...Vfq=], or Yl...Simfq-=2p7r', 
 
 in which p is some whole number (negatives and zero included). 
 
 250. To give a Taoro, geometrical form to the equation, let A, be 
 any given or assumed line j|| z, and let it be supposed that a, ^, . . 
 and p, ff, or OA, ob, . . . and op, os, are n - m + 2 other lines in the 
 same planes, and that ^p is a known scalar function of /o, such that 
 
 VII. . . a = 2''X, ^ = q'%.. p = q\ <r = qo\, 
 
 and 
 
 VIII... <fp=/? ^''^"' "-"-''-^---j'^^y ^^-^' 
 
 <Tj a (J y^OSy OA OB 
 
 the theorem to be proved may then be said to be, that whatever sys- 
 tem of real points, o, a, b, . . and s, in a given plane, and whatever 
 positive whole number m, may be assumed, or given, thei^e is always at 
 least one real point p, in the same plane, which satisfies the two condi- 
 tions: 
 
 IX. . . T^P = 1 ; X. . . am ^p = 2p7r. 
 
 251. Whatever value t\\\i we may assume for the versor (or 
 unit-vector) JJp, there always exists at least one value of the tensor 
 T/9, which satisfies the condition IX. ; because the function T^p va- 
 nishes with T/3, and becomes infinite when T/o = oo, having varied 
 continuously (although perhaps with fluctuations) in the interval. 
 Attending then only to the least value (if there be more than one) 
 of T/>, which thus renders T^p equal to unity, we can conceive a real, 
 unambiguous, and scalar function Y^t, which shall have the two fol- 
 lowing properties : 
 
 XI. .. T0(tfO = l; XII. . . T^(a;ti^O<l» if a;>0, < 1. 
 And in this way the equation, or system of equations, 
 
268 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 XIII. ..p = tft, or XIV. . . Up = t, Tp = yjrc, 
 may be conceived to determine a real, finite^ and plane closed curve, 
 which we shall call generally an Oval^ and which shall have the two 
 following properties: 1st, every right line, or ray, drawn/rom the ori- 
 gin o, in any arbitrary direction within the plane, meets the curve 
 once, but once only; and Ilnd, no one of the n-m other given points 
 A, B, . . is on the oval, because ^a = ^/3 = . . = 0. 
 
 252. This being laid down, let us conceive a point p to perform 
 one circiiit of the oval, moving in thepositive direction relatively to the 
 given interior point O; so that, whatever the given direction of the 
 line OS may be, the amplitude 2im{p'.a), if supposed to vary conti- 
 nuously,* will have increased hy four right angles, or by 27r, in the 
 couTSQ oi this one positive circuit; and consequently, the amplitude 
 of the left-hand factor (/> : <r)*", of 0p, will have increased, at the same 
 time, by ^mir. Then, if the point a be also interior to the oval, so 
 that the line OA must be prolonged to meet that curve, the ray ap will 
 have likewise made one positive revolution, and the amplitude of the 
 factor (/> - a)\ a will have increased by 27r. But if a be an exterior 
 point, so that the finite line oa intersects the curve in a point m, and 
 therefore never meets it again if prolonged, although the prolonga- 
 tion of the opposite line ao must meet it once in some point n, then 
 while thQ point p performs first what we may call the positive half- 
 circuit from M to N, and afterwards the other positive half-circuit 
 from N to M again, the ray ap has only oscillated about its initial and 
 final direction, namely that of the line Ao, without ever attaining the 
 opposite direction ; in this case, therefore, the amplitude am(AP: oa), 
 if still supposed to vary continuously, has only fluctuated in lis X2i\\xe, 
 and has (upon the whole) undergone no change at alh And since 
 precisely similar remarks apply to the other given points, b, &c., 
 it follows that the amplitude, am 0p, of the product (VIJI.) of all 
 these factors, has (by 236) received a total increment =2{m + t)7r, if 
 t be the number (perhaps zero) of given internal points, a, b, . . ; 
 while the number m is (by 249) at least = 1. Thus, while p per- 
 forms (as above) 07ie positive circuit, the amplitude am >pp has passed 
 at least m times, and therefore at least once, through a value of the 
 form 2p'7r; and consequently the condition X. has been at least once 
 satisfied, Biit the other condition, IX., is satisfied throughout, by the 
 
 * That 13, so as not to receive any sudden increment, or decrement, of one or 
 more whole circumferences (comp. 235, (1.)). 
 
CHAP. II.] GEOMETRICAL ILLUSTRATIONS, QUADRATICS. 269 
 
 supposed construction of the oval : there is therefore at least one real 
 position P, upon that curve, for which <pp or fq = 1 ; so that, /or this 
 position of that point, the equation 249, HI., and therefore also the 
 equation 248, I., is satisfied. The theorem of Art. 248, and conse- 
 quently also, by 247, the theorem of 244, with its transformations 
 245 and 246, is therefore in this manner proved. 
 
 253. This conclusion is so important, that it may be use- 
 ful to illustrate the general reasoning, by applying it to the 
 case of a quadratic equation, of the form, 
 
 •^^ ^oV^ ; ^^ cr\a J OS OA 
 
 We have now to prove (comp. 250, VIII.) that a (real) point p 
 
 exists, which renders the fourth 
 
 proportional (226) to the three ^ 
 
 lines OA, op, ap equal to a 
 
 given line os, or ab, if this lat- p- ^^ 
 
 ter be drawn = os ; or which 
 
 satisfies the following condition of similarity of triangles 
 
 (118), 
 
 III. . . A aop a PAB ; 
 which includes the equation of rectangles, 
 
 I V . . . OP.AP = OA-AB. Nt 
 
 (Compare the annexed Figures, 55, and 
 55, bis.) Conceive, then, that a conti- 
 nuous curve* is described as a locus (or 
 as part of the locus) of p, by means of this equality IV., with 
 the additional condition 
 when necessary, that o 
 shall be within it; in such 
 a manner that when (as in 
 Fig. 56) a right line from 
 o meets the general or total 
 locus in several points, m. 
 
 Fig. 55, bis. 
 
 Fig. 56. 
 
 * This curve of the fourth degree is the well-known Cassinian; but when it 
 breaks up, as in Fig. 56, into two separate ovals, we here retain, as the oval of the 
 proof only the one round o, rejecting for the present that round A. 
 
270 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 m', n', we reject all but the point m which is nearest to o, as not 
 belonging (comp. 2.5 1 , XII.) to the oval here considered. Then 
 while p moves upon that oval, in the positive direction rela- 
 tively to o, from M to N, and from n to m again, so that the 
 ray op performs one positive revolution, and the amplitude of 
 the factor op : os increases continuously by 27r, the ray ap 
 performs in like manner one positive revolution, or (on the 
 whole) does not revolve at all, and the amplitude of the factor 
 AP : OA increases by 27r or by 0, according as the point a is in- 
 terior or exterior to the oval. In the one case, therefore, the 
 amplitude am (pp of the product increases by Air (as in Fig. 56^ 
 bis) ; and in the other case, it increases by 27r (as in Fig. 6Q) ; 
 so that in each case, it passes at least once through a value of 
 the^r?w 2p7r, whatever its initial value may have been. Hence, 
 for at least one real position, p, upon the oval, we have 
 V. . . am 0p = 1 , and therefore VI. . . U^/o = 1 ; 
 but VII. ..T^^=l, 
 
 throughout, by the construction, or by the equation of the locus 
 IV. ; the geometrical condition (pp=l (II) is therefore satisfied 
 by at least one real vector p ; and consequently the quadratic 
 equation fq = 1 (I.) is satisfied by at least one real quaternion 
 root, q = p:X (250, VII.). But the recent form I. has the same 
 generality as the earlier form, 
 
 VIII. . . Fgg' = q^ + qiq + q2 = (comp. 245), 
 where qi and q^ are any two given, real, actual, and complanar 
 quaternions ; thus there is always a real quaternion q' in the 
 given plane, which satisfies the equation, 
 
 Vlir. . . F^q' = q'' + qxq +^2 = (comp. 247) ; 
 subtracting, therefore, and dividing by g - q^, as in algebra 
 (comp. 224), we obtain the following depressed or Ihiear equa- 
 tion q, 
 
 IX. . . 5' + 5''+ ^i = 0, or IX. . . 5' = 9'" = - g'-5'i (comp. 246). 
 The quadratic VIII. has therefore a second real quaternion root, 
 q, related in this manner to \k\.^ first ; and because the qua- 
 dratic function Y%q (comp. again 245) is thus decomposable 
 into two linear factors, or can be put under the form, 
 
CHAP. II.] RELATIONS BETWEEN THE ROOTS. 271 
 
 it cannot vanish for any third real quaternion^ q ; so that 
 (comp. 244) the quadratic equation has no more than two such 
 real roots. 
 
 (1.) The cubic equation may therefore be put under the ybrm (comp. 248), 
 
 X. . . Vzq = q^ -]r q\q^ + 9^29' + 53 = 9 (^ - ?') (? -?") + 93= ; 
 
 it has therefore one real root, say g', hy ihQ general proof (2b2^i which has been 
 above illustrated by the case of the quadratic equation ; subtracting therefore (com- 
 pare 247) the equation 'Fzq^ = 0, and dividing hy q— q\ we can depress the cubic to 
 a quadratic, which will have two new real roots, 5" and g"' ; and thus the cubic 
 function may be put under the form, 
 
 XI. . . F39' = (? - q) (q - q') {q - q''), 
 
 which cannot vanish for any fourth real value of q ; the cubic equation X. has there- 
 fore no more than three real quaternion roots (comp. 244) : and similarly for equa- 
 tions of higher degrees. 
 
 (2.) The existence of two real roots 9 of the quadratic I., or of two real vectors, 
 p and p', which satisfy the equation II., might have been geometrically anticipated, 
 from the recently proved increase = 47r of amplitude ^p, in the course of one circuit, 
 for the case of Fig. .55, bis, in consequence of which there must be two real positions, 
 V and p', on the one oval of that Figure, of which each satisfies the condition of si- 
 milarity III. ; and for the case of Fig. 56, from the consideration that the second (or 
 lighter^ oval, which in this case exists, although not employed above, is related to A 
 exactly as HhQ first (or dark) oval of the Figure is related to o ; so that, to the real 
 position p on the first, there must correspond another real position p', upon the se- 
 cond. 
 
 (3.) As regards the law of this correspondence, if the equation II. be put under 
 the form, 
 
 and if we now write 
 
 XIII. ..p = 5a, we may write XIV. . . 9-1 = — 1, q-i = -(T:a, 
 
 for comparison with the form VIII. ; and then the recent relation IX'. (or 246) be- 
 tween the two roots will take the form of the following relation between vectors, 
 
 XV. . . p + p' = a ; or XV'. . . op' = p' = a - p = pa ; 
 
 so that the point p' completes (as in the cited Figures) the parallelogram opap', and 
 the line pp' is bisected by the middle point c of OA. Accordingly, with this position 
 of p', we have (comp. III.) the similarity, and (comp. II. and 226) the equation, 
 XVI. . . A AOP* a P'AB ; XVII. . . 0p'= ^(a - p) = 0p = 1. 
 (4.) The other relation between the two roots of the quadratic VIII., namely 
 (comp. 246), 
 
 XVIII . . . q'q" = 92, gives XIX. . . ^ p' = - (7 ; 
 
272 ELEMENTS OF QUATERNIONS. [boOK II. 
 
 and accordingly, the line <t, or os, is a fourth proportional to the three lines oa, op, 
 and AP, or a, p, and - p'. 
 
 (5.) The actual solution^ by calculation, of the quadratic eqvationYlll. in com- 
 planar quaternions, is performed exactly as in algebra ; the formula being, 
 
 XK...q^-lq,±V(iqi'-q2), 
 in which, however, the square root is to be interpreted as a real quaternion, on prin- 
 ciples already laid down. 
 
 (6.) Oubic and biquadratic equations, with quaternion coefficients of the kind 
 considered in 244, are in like manner reso^red by the known /ormMZ« of algebra; 
 but we have now (as has been proved) three real (quaternion) roots for the former, 
 and four such real roots for the latter. 
 
 254. The following is another mode of presenting the geometri- 
 cal reasonings of the foregoing Article, without expressly intro- 
 ducing the notation or conception of amplitude. The equation 
 0io= I of 253 being written as follows, 
 
 I. . . ^^^p^fi(^p^a\ or II. . . To- = Tx/>, and III. . . Uo- = Ux/>, 
 
 a 
 
 we may thus regard the vector o- as a known function of the vector /?, 
 or the point s as di. function of the point p; in the sense that, while o 
 and A s^TQ fixed, p and s vary together : although it may (and does) hap- 
 pen, that s may return to a former position without p having similarly 
 returned. Now the essential property of the oval (253) may be said 
 to be this: that it is the locus of the points p nearest to o, for which the 
 tensor Txp has a given value, say h; namely the given value o/To-, or 
 of OS, when the^om^ s, like o and a, is given. If then we conceive 
 the point p to move, as before, along the oval, and the point s also to 
 move, according to the law expressed by the recent formula I., this 
 latter point must move (by II.) on the circumference of a given circle 
 (comp. again Fig. 56), with the given origin o for centre ; and the 
 theorem is, that in so moving, s will pass, at least once, through every 
 position on that circle, while p performs one circuit of the oval. And 
 this may be proved by observing that (by III.) the angular motion of 
 the radius os is equal to the sum of the angular motions of the two rays, 
 OP and ap; but this latter sum amounts to eight right angles for the 
 case of Fig. 55-, his,, and to four right angles for the case of Fig. 56; 
 the radius os, and the point s, must therefore have revolved twice in 
 the first case, and once in the second case, which proves the theorem 
 in question. 
 
 (1.) In the first of these two cases, namely when a is an interior point, each of 
 the three angular velocities is positive throughout, and the mean angular velocity of 
 
CHAP. II.] CASSINIAN OVALS, LEMNISCATA. 273 
 
 the radius OS is double of that of each of the two rays op, AP. But in the second case, 
 when A is exterior, the mean angular velocity of the ray ap is zero; and we might 
 for a moment doubt, whether the sometimes negative velocity of that ray might not, 
 for parts of the circuit, exceed the always positive velocity of the ray op, and so 
 cause the radius os to move backwards, for a while. This cannot be, however ; for 
 if we conceive p to describe, like p', a circuit of the other (or lighter) oval, in Fig. 56, 
 the point s (if still dependent on it by the law I.) would again traverse the whole of 
 the same circumference as before ; if then it could ewer fluctuate in its motion, it 
 would pass more than twice through some given series of real positions on that circle, 
 during the successive description of the two ovals bj" p ; and thus, within certain 
 limiting values of the coefficients, the quadratic equation would have more than two 
 real roots : a result which has been proved to be impossible. 
 
 (2.) While 8 thus describes a circle round o, we may conceive the connected point 
 B to describe an equal circle round a ; and in the case at least of Fig. 56, it is easy 
 to prove geometrically, from the constant equality (253, IV.) of the rectangles OP'AP 
 and OA. AB, that these two circles (with t'u and xV as diameters), and the two ovals 
 (with MN and mV as axes), have two common tangents, parallel to the line OAj 
 which connects what we may call the two given foci (or focal points), o and a : the 
 new or third circle, which is described on this focal interval OA as diameter, passing 
 through the four points of contact on the ovals, as the Figure may serve to exhibit. 
 
 (3.) To prove the same things by quaternions, we shall find it convenient to 
 change the origin (18), for the sake of symmetry, to the central point c; and thus 
 to denote noio cp by p, and ca by a, writing also CA = Ta = a, and representing still 
 the radius of each of the two equal circles by b. We shall then have, as the joint 
 equation of the system of the two ovals, the following : 
 
 lY. . .T(p + a).TCp-a)=2ab; 
 or 
 
 V. . . T(52-l)=2c, if q = ^ and c = -. 
 a, a 
 
 But because we h&ve generally (by 199, 204, &c.) the transformations, 
 
 VI. . . S . ^2 = 2S52 _ T52 = Tq^ + 2V92 = 2NS9 - % = N^ - 2NV^, 
 the square of the equation V. may (by 210, (8.) ) be written under either of the tAvo 
 following forms : 
 
 VII. . . (N^ - 1)2 + 4NV5 = 4c2 ; VIII. . . (Ng + 1)2 _ 4NS5 = 4c2 ; 
 whereof the first shows that the maximum value of TYq is c, at least if 2c < 1, as 
 happens for this case of Fig. 56; and that this maximum corresponds to the value 
 Tq=l, or Tp = a : results which, when interpreted, reproduce those of the preceding 
 sub-article. 
 
 (4.) When 2c > 1, it is permitted to suppose S9 = 0, N V9 = Ng = 2c - 1 ; and 
 then we have only one continuous oval, as in the case of Fig. 55, bis; but if c < 1, 
 though > I, there exists a certain undulation in the form of the curve (not represented 
 in that Figure), TYq being a minimum for S^= 0, or for p -i- a, but becoming (as 
 before) a maximum when Tq = l, and vanishing when 8(72 = 2c + 1, namely at the 
 two summits M, N, where the oval meets the axis. 
 
 (5.) In the intermediate caie, when 2c = 1, the Cassinian curve IV. becomes (as 
 is known) a lemniscata; of which the quaternion equation may, by V., be written 
 (comp. 200, (8.) ) under any one of the following forms: 
 
 2 N 
 
274 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 IX. . .T(92-l) = 
 or finally, 
 
 or X. . . N93 = 2S. ^2 . or xi. . . T92 = 2SU . 92 
 
 XII. . . Tp2 = 2Ta2 cos 2 ^ 
 
 Avhich last, when written as 
 
 Xir. . . cp2 = 2ca2 . cos 2acp, 
 agrees evidently with known results. 
 
 (6.) This corresponds to the case when 
 
 XIII. 
 
 (T = -— , and 
 4 
 
 XIV. 
 
 P = P 
 
 + -, in 253, XII., 
 
 that quadratic equation having thus its roots equal; and in general, iov all degrees, 
 cases of equal roots answer to some interesting peculiarities of form of the ovah, on 
 which we cannot here delay. 
 
 (7.) It may, however, be remarked, in passing, that if we remove the restriction 
 that the vector p, or cp, shall be in a given plane (225), drawn through the line 
 which connects the two foci, o and a, the recent equation V. will then represent the 
 surface (or surfaces') generated by the revolution of theora/ (or ovals), orleraniscata, 
 about that line oa as an axis. 
 
 255. If we look back, for a moment, on the formula oi similarity, 
 253, III., we shall see that it involves not merely an equality of rect- 
 angles, 253, IV., but also an equality of angles, aop and pab; so that 
 the angle oab represents (in the Figures 55) a given difference of the 
 base angles aop, pao of the triangle oap: but to construct a triangle^ 
 by means of such a given difference, combined with a given base, and 
 a given rectangle of sides, is a known problem of elementary geome- 
 try. To solve it briefly, as an exercise, by quaternions^ let the given 
 base be the line aa', with for its middle point, as in the annexed 
 Figure 57 ; let baa' represent the given diffe- 
 rence of base angles, paa' - aa'p ; and let oa . ab 
 be equal to the given rectangle of sides, ap • a^. 
 We shall then have the similarity and equa- 
 tion, 
 
 p +a /3- a 
 
 A OA'P a PAB ; 
 
 II. 
 
 a p — a 
 
 whence it follows by the simplest calculations, 
 that 
 
 III. 
 
 -i'-.lf- 
 
 1+1 = 
 
 ^ 
 
 + 1 = 
 
 or that /> is a mean proportional (227) between a and /3. Draw, 
 therefore, a line op, which shall be in length a geometric mean be- 
 tween the two given lines, oa, ob, and shall also bisect their angle 
 
CHAP. II.] IMAGINARY QUATERNION ROOTS. 275 
 
 AOB ; its extremity will be the required vertex, p, of the sought tri- 
 angle aa'p: a result of the quaternion analysis, vflaioh geometrical syn- 
 thesis* easily confirms. 
 
 (1.) The equation III. is however satisfied also (comp. 227) by the opposite vec- 
 tor, op' = PC, or p' = - (0 ; and because /3 = (p : a) . p, we have 
 
 lV...t±i=.t=^ = t^ or IV'. ..^=-= - = ^. 
 p-\^ a a p d^ p'a oa op oa' ' 
 
 so that the^bttr following triangles are similar (the two first of them indeed being 
 equal) : 
 
 V. . . A a'op' a AOP <x FOB aAP'B ; 
 
 as geometry again would confirm. 
 
 (2.) The angles ap'b, bpa, are therefore supplementary, their sum being equal to 
 the sum of the angles in the triangle oap ; whence it follows that the four points A, 
 P, b, p' are concircular :f or in other words, the quadrilateral Apbp' is inscriptible 
 in a circle, which (we may add) passes through the centre c of the circle oab (see 
 again Fig. 57), because the angle aob is double of the angle ap'b, by what has been 
 already proved. 
 
 (3.) Quadratic equations in quaternions may also be employed in the solution 
 of many other geometrical problems; for example, to decompose a given vector into 
 two others, which shall have a given geometrical mean, &c. 
 
 Section 6 — On the n^-n Imaginary {or Symbolical) Roots 
 of a Quaternion Equation of the n'^' Degree^ with Coeffi- 
 cients of the kind considered in the foregoing Section, 
 
 256. The polynomial function F,,q (245), like the quaternions 
 q, qi, . . qn on which it depends, may always be reduced to the form of 
 a couple (228) ; and thus we may establish the transformation (comp. 
 239), 
 
 I. . . Fnq = F„ (x + iy) = X„ + i F,, = Gu {x, y) + iH,, {x, y), 
 
 Xn and y„, or Gn and Hn, being two known, real, finite, and scalar 
 functions of the two sought scalars, x and y\ which functions, rela- 
 
 * In fact, the two triangles I. are similar, as required, because their angles at o 
 and p are equal, and the sides about them are proportional. 
 
 t Geometrically, the construction gives at once the similarity, 
 A AOP oc fob, whence L bpa = opa + pad = poa' ; 
 and if we complete the parallelogram apa'p', the new similarity, 
 
 A oa'p a op'b, gives L ap'b = oa'p + a'po = aop ; 
 thus the opposite angles bpa, ap'b are supplementary, and the quadrilateral ai'bp' is 
 inscriptible. It will be shown, in a shortly subsequent Section, that these four 
 points, A, p, P, p', form a harmonic group upon their common circle. 
 
276 ELEMENTS OF QUATERNIONS. [boOK II. 
 
 tively to them, are each of the w^'' dimension, but which involve also, 
 though only in ihe first dimension, the 2n given and real scalarsy 
 rci, 2/i, . . . X,,, y„. And since the one quaternion (or couple) equation, 
 F,,q = 0, is equivalent (by 228, IV.) to the system of the two scalar 
 equations, 
 
 II. . . J:„=0, F„ = 0, or III. . . . Gn{x,y) = 0, Hn{x,y)=0, 
 we see (by what has been stated in 244, and proved in 252) that 
 suxih a system, of two equations of the n"* dimension, can always be 
 satisfied by n systems (or pairs) of real scalars, and by not more than 
 n, such as 
 
 IV. . .x^,y'\ x", y" ; . . icW, y('»^ ; 
 
 although it may happen that two or more of these systems shall coin- 
 cide with (or become equal to) each other. 
 
 (1.) \ix and y be treated as co-ordinates (comp. 228, (3.) ), the two equations 
 II. or III. represent a system of two curves, in the given plane ; and then the theo- 
 rem is, that these two curves intersect each other {generally*^ in n real points^ and 
 in no more : although two or more of these n points may happen to coincide with 
 each other. 
 
 (2.) Let h denote, as a temporary abridgment, the old or ordinary imaginary, 
 
 V— 1, oi algebra, considered as an uninterpreted symbol, and as not equal to any 
 
 real versor, such as t (comp. 181, and 214, (3.) ), but as following the rules ofsca- 
 
 lars, especially as regards the commutative property oi mvXil^WcaXXon (\2Q) ; so that 
 
 V. . . ^2 + 1 = 0, and VI. . . W = ih, but VII. . . A «o* = + i. 
 
 (3.) Let q denote still a real quaternion, or real couple, x + iy ; and with the 
 meaning just now proposed of h, let [(j\ denote the connected but imaginary alge- 
 braic quantity, or bi-scalar (214, (7.) ), x + hy ; so that 
 
 Ylll. q=x + iy, but IX. . . lq'] = x + hy', 
 and let any biqiiaternion (214), (8.), or (as we may here call it) bi-couple, of the 
 form [7'] + i\_q"'\i be said to be complanar with »; with the old notation (123) of 
 complanarity. 
 
 (4.) Then, for the polynomial equation in real and complanar quaternions, 
 JJ'^g = (244, 245), we may be led to substitute the following connected algebraical 
 equation, of the same degree, n, and involving real scalars similarly : 
 X. . . [F„g] = lq-]» + [91] [qy + • • + [^n] = ; 
 
 * Cases of equal roots may cause points of intersection, which are generally ima- 
 ginary, to become real, but coincident with each other, and vi'ith. former real roots : 
 for instance the hyperbola, x^ -y"^ = a, is intersected in two real and distinct points, 
 by the pair of right lines xy = 0, if the scalar a > or < ; but for the case a = 0, the 
 two pairs of lines, x^ — y"- =■ and xy = 0, may be considered to havofour coincident 
 intersections at the origin. 
 
CHAP. II.] NEW SYMBOLICAL ROOTS OF UNITY. 277 
 
 which, after the reductions depending on the substitution V. of - 1 for h^, receives 
 the form, 
 
 where Xn and Vn are the same real and scalar Junctions as in I. 
 
 (5.) But we have seen in II., that these two real functions can be made to va- 
 nish together, by selecting any one ofn real pairs IV. of scalar values, x and y ; the 
 General Algebraical Equation X., of the n*^ Degree, has therefore n Real or Imagi- 
 nary/ Roots,* of the Form ar + 2/ V — 1 ; and it has no more than n such roots. 
 
 (6.) Elimination of y, between the two equations IT. or III., conducts generally 
 to an algebraic equation in x, of the degree n^ ; which equation has therefore n^ alge- 
 braic roots (5.), real or imaginary ; namely, by what has been lately proved, n real 
 and scalar roots, x', . . a;("), with real and scalar values y , . .y(") (comp. IV.) of y 
 to correspond; and «(«—!) other roots, with the same number of corresponding 
 values of y, which may be thus denoted, 
 
 XII. . . [x(«+i), . . [a;(«'^] ; XIII. . . [yf"+i)], . . [y(«2)] ; 
 and which are either themselves imaginary (or bi-scalar, 214, (7.)), or at least cor- 
 respond, by the supposed elimination, to imaginary or bi-scalar values ofy; since if 
 a;(w+i) and y("+^), for example, could both be real, the quaternion equation Fnq=0 
 would then have an (w 4-l)st real root, of the form, ^(w+i) = a;(w+i) + ^^(n+i)^ contraiy 
 to what has been proved (252). 
 
 257- On the whole, then, it results that the equation F„q = in 
 complanar quaternions, of the w^'' degree, with real coefficients, 
 while it admits of only n real quaternion roots, 
 
 L..^^^^..2W(244,&c.), 
 is symbolically satisfied also (corap. 214, (3.)) by n(n- \) imaginary 
 quaternion roots, or hy n^ -n bi-quaternions (214, (8.) ), or bi-couples 
 (256, (3.) ), which may be thus denoted, 
 
 and of which the first, for example, has the /orm, 
 
 III. . . [^'"^^^] = [a;("^^)] + 2[?/^"^^T = a;/"^») + ^a;//«^') + 2(y/^"'^^ + %//^"''0 ; 
 
 where a;/"^'\ XjI''*^\ y/''^^\ and y,/"^'^^ are four real scalars, but h is 
 the imaginary of algebra (256, (2.) ). 
 
 (L) There must, for instance, be n(n - 1) imaginary n*^ roots of unity, in the 
 given plane of i (comp. 256, (3.) ), besides the n real roots already determined (233, 
 
 * This celebrated Theorem of Algebra has long been known, and has been proved 
 in other ways ; but it seemed necessary, or at least useful, for the purpose of the pre- 
 sent work, to prove it anew, in connexion with Quaternions : or rather to establish 
 the theorem (244, 252), to which in the present Calculus it corresponds. Compare 
 the Note to page 266. 
 
278 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 237); and accordingly in the case n = 2, we have the four foWowmg sqvare-roois 
 ofl \\\i, two real and two imaginary : 
 
 IV, . . +1, -1 ; +hi, -hi; 
 
 for, by 256, (2.), we have 
 
 V. . . (± hiy = hH^ = (- 1) (- 1 ) = + 1. 
 And the two imaginary roots of the quadratic equation F^q = 0, which generally 
 exist, at least as symbols (214, (3.) ), may be obtained by multiplying the square- 
 root in the formula 253, XX. by hi ; so that in the particular case, when that radi- 
 cal vanishes, the four roots of the equation become real and equal : zero having thus 
 only itself for a square-root. 
 
 (2.) Again, if we write (comp. 237, (3.)), 
 
 -1 + »V3 -l-iV3 
 Yl...q=lh= , g^ = \h= , 
 
 so that 1, q, qi are the three real cube-roots of positive unity, in the given plane ; 
 and if we write also, 
 
 v.i..,e=M=zi±i^, e^ = hp = zizA^, 
 
 so that 9 and 02 are (as usual) the two ordinary (or algebraical) imaginary cube- 
 roots of unity ; then the nine cube-roots o/ 1 (| 1 1 1) are the following : 
 VIII. . . 1 ; 9, 52 ; 0, 02 ; Qq^ e^ ; 9^q, Q^q^ ; 
 whereof the first is a real scalar ; the two next are real couples, or quaternions \\\i ; 
 the two following are imaginary scalars, or biscalars; and the four that remain are 
 imaginary couples, or bi-couples, or biquaternions. 
 
 (3.) The sixteen fourth roots of unity (|[| i) are: 
 
 IX. ..+1; ±i; +/*; ±hi; ±|(1±/0(1±0; 
 the three ambiguous signs in the last expression being all independent of each other. 
 
 (4.) Imaginary roots, of this sort, are sometimes useful, or rather necessary, in 
 calculations respecting ideal intersections,* and ideal contacts, in geometry: although 
 in what remains of the present Volume, we shall have little or no occasion to employ 
 them. 
 
 (5.) We may, however, here observe, that when the restriction (225) on the 
 plane of the quaternion q is removed, the General Quaternion Equation of the n*^ 
 Degree admits, by the foregoing principles, no fewer than «* Hoots, real or imagi- 
 nary : because, when that general equation is reduced, by 221, to the Standard 
 Quadrinomial Form, 
 
 X...Fnq= Wn + iXn +j Vn + hZn = 0, 
 
 it breaks up (comp. 221, VI.) into a System of Four Scalar Equations, each (gene- 
 rally) of the «*'» dimension, in w, x, y, z-, namely, 
 
 XI. ..r,»=0, X„=0, Yn=0, Zn = 0; 
 
 and if x, y, z be eliminated between these four, the restilt is (generally) a scalar (or 
 algebraical) equation of the degree n*, relatively to the remaining constituent, w; 
 
 Comp. Art. 214, and the Notes there referred to. 
 
CHAP. II.] RECIPROCAL OF A VECTOR. 279 
 
 which therefore has n^ (algebraical) values, real or imaginary : and similarly for the 
 three other constituents, x, y, z, of the sought quaternion q. 
 
 (6.) It may even happen, when no plane is given, that the number of roots (or 
 solutions) of a ^raiYe* equation in quaternions shall become infinite; as has been 
 seen to be the case for the equation q^ =—1 (149, 154), even when we confine our- 
 selves to what we have considered as real roots. li imaginary roots he admitted, 
 we may write, still more generally, besides the two biscalar values, + h, the expres- 
 sion, 
 
 XII. . . (-l)i = «+ Ar', S«=Sw'=S«w' = 0, Nw-Nzj'=l; 
 
 V and v' being thus any two real and right quaternions, in rectangular planes, pro- 
 vided that the norm of ih.Q first exceeds that of the secondhy unity. 
 
 (7.) And in like manner, besides the two real and scalar values, + 1, we have 
 this general symbolical expression for a square root of positive unity, with merely 
 the difference of the norms reversed : 
 
 XIII. . . li=y + Ay', Sy=S«' = Sw' = 0, N«'-Nr = l. 
 
 Section 7. — On the Reciprocal of a Vector^ and on Harmo- 
 nic Means of Vectors; with Remarks on the Anharmonic 
 Quaternion of a Group of Four Points, and on Conditions 
 of Concircularity. 
 
 258. When two vectors, a and a', are so related that 
 
 I. . . a = - Ua : Ta, and therefore 11. . . a = - Ua : Ta, 
 or that 
 
 III. . . Ta . Ta' = 1, and IV. . . Ud + Ua = 0, 
 we shall say that each of these two vectors is the Reciprocal^ 
 of the other ; and shall (at least for the present) denote this 
 relation between them, by writing 
 
 V. ..a=Ea, or VI. ..a = Ea'; 
 so that for every vector a, and every right quotient v, 
 
 VII.. .Ra = -Ua:To; VIII. . .R^a = RRa = a; 
 
 and 
 
 IX. . . EIv = IRi; (comp. 161, (3.), and 204, XXXV'.). 
 
 259. One of the most important properties of such reci- 
 procals is contained in the following theorem : 
 
 * Compare the Note to page 265. 
 
 t Accordingly, under these conditions, we shall afterwards denote this recipro- 
 cal of a vector a by the symbol a"' ; but we postpone the use of this notation, until 
 we shall be prepared to connect it with a general theory of products and poivers of 
 vectors. Compare 234, V., and the Note to page 121. And as regards the tempo- 
 rary use of the characteristic R, compare the second Note to page 252. 
 
280 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 If any two vectors oa, ob, have oa', ob' for their recipro- 
 cals, then (comp. Fig. 58) the right line a b' 
 is parallel to the tangent od, at the origin o, 
 to ' the circle oab ; and the two triangles, 
 gab, obV, are inversely similar (118). Or 
 in symbols, 
 
 I. . . if oa =R.OA, and ob' = R.ob, 
 
 then 
 
 A oab a' ob'a'. 
 
 (1.) Of course, under the same conditions, the tangent at o to the circle oa'b' is 
 parallel to the line ab. 
 
 (2.) The angles bao and ob'a' or bod being equal, the fourth proportional (226) 
 to AB, AO, and ob, or to ba, oa, and ob, has the direction of od, or the direction op- 
 posite to that of a'b' ; and its length is easily proved to be the reciprocal (or inverse) 
 of the length of the same line a'b', because the similar triangles give, 
 
 II. . . (oa : ba) . ob = (ob' : a'b'). ob = 1 : aV, 
 
 it being remembered that 
 
 III. . . OA . oa' = OB . ob' = 1 ; 
 we may therefore write, 
 
 IV. . .(oa:ba).ob = R.a'b', or V. . . — ^i3 = R(Ri3 - Ra), 
 
 a — p 
 whatever two vectors a and /3 may be. 
 
 (3.) Changing a and /3 to their reciprocals, the last formula becomes, 
 
 VI. . . R(/3-a) = - — ^.R/3; or VII. . . (oa': b'a').ob' = R. ab. 
 Ka —Up 
 
 yiii...K2=5?. 
 
 (4.) The inverse similarity I. gives also, generally, the relation, 
 
 Ra 
 R^* 
 
 (5.) Since, then, by 195, II., or 207, (2.), 
 
 IX, . . K-+1 = K'-^^, we have X... - ^- 
 
 a ' R/3 R()3±a)' 
 
 the lower signs agreeing with VI. 
 
 (6.) In general, the reciprocals of opposite vectors are themselves opposite ; or 
 
 in symbols, 
 
 XI. . . R(-a) = -Ra. 
 (7.) More generally, 
 
 XII. . . Rxa = x-^B,a, 
 if X be any scalar. 
 
 (8.) Taking lower signs in X., changing a to y, dividing, and taking conjugates, 
 we find for any three vectors a, /3, y (complanar or diplanar') the formula : 
 
 Ry:^^,J_Ry_ R(/3-a) \ a r^_oA bc 
 ■^^"••^ Ra-R/3 VK(i8-y)" Ra / /3- a ' - y "ab* co' 
 if a = OA, j3 = OB, and y = oc, as usual. 
 
CHAP. II.] ANHARMONIC AND EVOLUTIONARY QUATERNIONS. 281 
 
 (9.) If then we extend, to any four points ofspace^ the notation (25), 
 
 ,,,,, . ^ AB CD 
 
 XIV. . . Cabcd) = — .— , 
 
 ^ ' BC DA 
 
 interpreting esich. of these two factor-quotients as a quaternion, and defining that 
 t\\eiv product (in this order^ is the anharmonic quaternion function, or simply the 
 Anharmonic, of the Group of four points A, B, C, D, or oi the (^plane or gauche^ Quw 
 drilateral ABCD, we shall have the following general and useful ^rmw/a of transfor- 
 mation : 
 
 XV.. (0ABc) = KgI-^ = K_„ 
 
 where oa', ob', ob' are supposed to be reciprocals of oa, ob, oc. 
 
 (10.) With this notation XIV., we have generally, and not merely for coUinear 
 groups (35), the relations : 
 
 XVI. . . (abcd) + (acbd) = 1 ; XVII. . . (abcd). (adcb) = 1. 
 
 (11.) Let o, A, B, c, D be any five points, and oa', . . od' the reciprocals of OA, . . 
 od ; we shall then have, by XV., 
 
 XVIII. . . ^ = K (OCBA), ^ = K (oadc) ; 
 
 bo' ^ ^ DA ^ 
 
 and therefore, 
 
 XIX. . . K (a'b'c'd') = (oADc) (ocba) = - (oadcba), 
 if we agree to write generally, for any six points, the formula,* 
 
 ,„ , ^ AB CD EF 
 
 XX. . . Cabcdef) = — . — . — . 
 
 EC DE fa 
 
 (12.) If then the five points o . . d be complanar (225), we have, by 226, and 
 by XIV., 
 
 XXI. . . K (a'b'c'd') = (abcd), or XXI'. . . (a'b'c'd') = K (abcd) ; 
 the anharmonic quaternion (abcd) being thus changed to its conjugate, when the 
 four rays OA, . . od are changed to their reciprocals, 
 
 260. Another very important consequence from the defi- 
 nition (258) of reciprocals of vectors, or from the recent theo- 
 rem (259), may be expressed as follows: 
 
 If any three coinitial vectors^ oa, ob, oc, be chords of one 
 common circle^ then (see again Fig. 58) their three coinitial re- 
 
 * There is a convenience in calling, generally, this /iroc/Mc^ of three quotients, 
 (abcdef), the evolutionary quaternion, or simply the Evolutionary, of the Group 
 of Six Points, A . . F, or (if they be not collinear) of the plane or gauche Hexagon 
 abcdef : because the equation, 
 
 (abca'b'c') = - 1, 
 expresses either 1st, that the three pairs of points, aa', bb', cc', form a collinear in- 
 volution (26) of a well-known kind ; or Ilnd, that those threepairs, or the three cor- 
 responding diagonals of the hexagon, compose a complanar or a homospheric Involu- 
 tion, of a new kind suggested by quaternions (comp. 261, (11.) ). 
 
 2 
 
282 ELEMENTS OF QUATERNIONS. [BOOK 11. 
 
 ciprocals, oa', ob', oc', are terminO'ColUnear (24) : of, in other 
 words, \S ihe four points o, a, b, c be concircular, then the three 
 points a', b'j c' are situated on one right line. 
 
 And conversely, if three coinitial vectors^ oa', ob', oc', thus 
 terminate on one right line, then their three coinitial recipro- 
 cals, oa, ob, oc, are chords of one circle; the tangent to which 
 circle, at the origin, is parallel to the right line; while the 
 anharmonic function (259, (9.) )? of the inscribed quadrilateral 
 OABC, reduces itself to a scalar quotient of segments of that line 
 (which therefore is its own conjugate, by 139) : namely, 
 
 I. . . (oABc) = b'c' : bV = (oo a'b'c') = (o . oabc), 
 if the symbol oo be used here to denote the point at infinity on 
 the right line a'b'c' ; and if, in thus employing the notation 
 (35) for the anharmonic of a plane pencil, we consider the null 
 chord, 00, as having the direction^ of the tangent, od. 
 
 (1.) If p = OP be the variable vector of a point p upon the circle oab, the qua- 
 ternion equation of that circle may be thus written : 
 
 II. . . Ep = E/3 + a;(Ea - Ej3), where III. . . a; = (oabp) ; 
 the coefficient x being thus a variable scalar (comp. 99, I.), which depends on the 
 variable position of the point p on the circumference. 
 
 (2.) Or we may write, 
 
 IV...Ep = 2^±i^, 
 ^ t+u ^ 
 
 as another form of the equation of the same circle oab ; with which may usefully be 
 contrasted the earlier form (comp. 25), of the equation of the line ab, 
 
 ^ t+u 
 (3.) Or, dividing the second member of IV. by the first, and taking conjugates, 
 we have for the circle, 
 
 to up ... _^_T ta uj3 
 
 VI. . .-i-+ -^=< + «; while VII. . . - + -^ = f + m, 
 a (i P ' P 
 
 for the right line. 
 
 (4.) Or we may write, by II., 
 
 this latter symbol, by 204, (18.), denoting any scalar. 
 
 * Compare the remarks in the second Note to page 139, respecting the possible 
 determinateness of signification of the symbol UO, when the zero denotes a line, 
 which vanishes according to a law. 
 
CHAP. II.] CIRCULAR AND HARMONIC GROUP. 283 
 
 (5.) Or still more briefly, 
 
 IX. . . V(OABP) = ; or IX'. . . (oabp) = V-i 0. 
 
 (6.) If the four points o, A, b, o be still concircular, and if p be any fifth point 
 in their plane, while POi, . . PCi are the reciprocals of po, . . PO, thea by 259, XXI., 
 we have the relation, 
 
 X. , . (OiAiBiCi) = K(OABC) = (OABC) = V"! ; 
 
 the^wr new points Oi. . Ci are therefore generally concircular. 
 
 (7.) If, however, the point p be again placed on the circle oabc, those four new 
 points are (by the present Article) collinear; being the intersections of i\iQ pencil 
 p.oabo with a, parallel to the tangent at p. In this case, therefore, we have the 
 equation, 
 
 XI. . . (p. oabc) = (oiAiBiCi) = (oabc) ; 
 
 so that the constant anharmonic of the pencil (35) is thus seen to be equal to what 
 we have defined (259, (9.) ) to be the anharmonic of the group. 
 
 (8.) And because the anharmonic of a circular group is a scalar, it Is equal (by 
 187, (8.) ) to its own tensor, either positively or negatively taken : we may therefore 
 write, for any inscribed quadrilateral oABC, the formula, 
 
 XII. . . (OABc) = + T (OABc) = + (OA . BC) : (aB . CO), 
 
 = + & quotient of rectangles of opposite sides; the upper or the lower sign being 
 taken, according as the point b' falls, or does not fall, between the points a' and c' : 
 that is, according as the quadrilateral oabc is an uncrossed or a crossed one. 
 
 I; (9.) Hence it is easy to infer that /or any circular group o, A, b, c, we have the 
 equation, 
 
 XIII...U^ = + U^; 
 
 AB - CB 
 
 the upper sign being taken when the succession oabo is a direct one, that is, when 
 the quadrilateral oabc is uncrossed; and the lower sign, in the contrary case, 
 namely, when the succession is (what may be called) indirect, or when the quadri- 
 lateral is crossed: while conversely this equation XIII, is sufficient to prove, when- 
 ever it occurs, that the anharmonic (oabc) is a negative or a positive scalar, and 
 therefore by (5.) that the gro^ip is circular (if not linear^, as above. 
 
 (10.) If A, b, c, d, e be any five homospheric points (or points upon the surface 
 of owe sphere), and if o be any sixth point of space, while oa', . . oe' are the reciprocals 
 of OA, . . OE, then the five new points a'. . e' are generally homospheric (with each 
 other) ; but if o happens to be on the sphere abcde, then a' . . e' are complanar, 
 their common plane being parallel to the tangent plane to the given sphere at o : 
 with resulting anharmonic relations, on which we cannot here delay. 
 
 26 1 . An interesting case of the foregoing theory is that 
 when the generally scalar anharmonic of a circular group be- 
 comes equal to negative unity ; in which case (comp. 26), the 
 group is said to be harmonic, A few remarks upon such czV- 
 ctdar and harmonic groups may here be briefly made : the stu- 
 
284 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 dent being left to fill up hints for himself, as what must be 
 now to him an easy exercise of calculation. 
 
 (1.) For such a group (comp. again Fig. 58), we have thus the equation, 
 I. . . (oABc) = - 1 ; and therefore II. . . a'b' = b'c' ; 
 or III. ..R/3=KKa + R7); 
 
 and under this condition, we shall say (comp. 216, (5.) that the Vector /3 is the Har- 
 monic Mean between the two vectors, a and y. 
 
 (2.) Dividing, and taking conjugates (comp. 260, (3.), and 216, (5.) ), we thus 
 obtain the equation, 
 
 IV... ^ + ^=2; or V. . . /3 = -?i- y = ^a; 
 or 
 
 VI. . ./3 = -y = ^a, if VII. . . £ = |(y-f a); 
 
 £ thus denoting here the vector oe (Fig. 68) of the middle point of the chord ao. 
 We may then say that the harmonic mean between any two lines is (as in algebra) 
 the fourth proportional to their semisum^ and to themselves. 
 (3.) Geometrically, we have thus the similar triangles, 
 
 VIII. . . A AOB a EOC ; VIII'. . . A aoe a boc ; 
 
 whence, either because the angles oba and oca, or because the angles oac and obc 
 are equal, we may infer (comp. 260, (5.) ) that, when the equation I. is satisfied, 
 the four points o, a, b, c, if not coUinear^ are coneircular. 
 (4.) We have also the similarities, 
 
 IX. . . A OEC a ceb, and IX'. . . A oea a aeb ; 
 or the equations, 
 
 X...^ = I^', and X'.,.t.'=2Zi, 
 y — c -c a- 1 — c 
 
 in fact we have, by VI. and VII., 
 
 XI. 
 
 £ 
 
 .1-.; xn...^(=iL&". = x-I«] = (.-^y 
 
 (5.) Hence the line ec, in Fig. 58, is the mean proportional (227) between the 
 lines EO and eb ; or in words, the semisum (oe), the semidifference (ec), and the 
 excess (be) of the semisum over the harmonic mean (ob), form (as in algebra) a 
 continued proportion (227). 
 
 (6.) Conversely, if any three coinitial vectors, eo, ec, eb, form thus a continued 
 proportion, and if we take ea = ce, then the four points oabc will compose a circu- 
 lar and harmonic group ; for example, the points apbp' of Fig. 67 are arranged so 
 as to form such a group.* 
 
 (7.) It is easy to prove that, for the inscribed quadrilateral oabc of Fig. 58, 
 the rectangles under opposite sides are each equal to half of the rectangle under the 
 
 * Compare the Note to 255, (2.). In that sub-article, the text should have run 
 thus : of which (we may add) the centre c is on the circle oab, &c. In Fig. 68, the 
 centre of the circle oabc is coneircular with the three points o, E, b. 
 
CHAP. II.] INVOLUTION IN A PLANE, OR IN SPACE. 285 
 
 diagonals; which geometrical relation answers to either of the two anharmonic 
 equations (comp. 259, (10.)) : 
 
 XIII. . . (0BAC) = + 2; Xlir. . . (ocab) =+ ^. 
 
 (8.) Hence, or in other ways, it may be inferred that these diagonals, ob, ac, are 
 conjugate chords of the circle to which they belong : in the sense that each passes 
 through the pole of the other^ and that thus the line db is the second tangent from 
 the point d, in which the chord ac prolonged intersects the tangent at o. 
 
 (9.) Under the same conditions, it is easy to prove, either by quaternions or by 
 geometry, that we have the harmonic equations : 
 
 XIV. . . (abco) = (bcoa) = (coab) = - 1 ; 
 so that AC is the harmonic mean between ab and ao ; bo is such a mean between 
 BC and BA ; and ca between co and cb. 
 
 (10.) In any such group, any two opposite points (or opposite corners of the qua- 
 drilateral), as for example o and b, may be said to be harmonically conjugate to each 
 other, with respect to the two other points^ a and c ; and we see that when these two 
 points A and c are given, then to every third point o (whether in a given plane, or 
 in space) there always corresponds a. fourth point b, which is in this sense conju- 
 gate to that third point : this fourth point being always complanar with the three 
 points A, c, o, and being even concircular with them, unless they happen to be colli- 
 near with each other ; in which extreme (or limiting') case, the fourth point b is still 
 determined, but is now coUinear with the others (as in 26, &c.). 
 
 (11.) When, after thus selecting two* points, A and c, or treating them as given 
 or fixed, we determine (10.) the harmonic conjugates b, b', b", with respect to them, 
 of any three assumed points, o, o', o", then the three pairs of points, O, B ; o', b' ; 
 o", b", may be said to form an Involution,f either on the right line AC, (in which 
 case it will only be one of an already well-known kind), or zw a plane through that 
 line, or even generally in space : and the two points A, c may in all these cases be 
 said to be the two Double Points (or Foci^ of this Involution. But the field thus 
 opened, for geometrical investigation by Quaternions, is far too extensive to be more 
 than mentioned here. 
 
 (12.) We shall therefore only at present add, that the conception of the Aarmonic 
 mean between two vectors may easily be extended to any number of such, and need 
 not be limited to the plane : since we may define that ij is the harmonic mean of the 
 n arbitrary vectors ai, . . an, when it satisfies the equation, 
 
 XV. . . Rj; = i (Rai + . . + Ra„) ; or XVI. . . nB.r) = SRa. 
 n 
 
 (13.) Finally, as regards the notation Ra, and the definition (258) of the recipro- 
 cal of a vector, it may be observed that if we had chosen to define reciprocal vectors as 
 having similar (instead of opposite') directions, we should indeed have had the posi- 
 tive sign in the equation 258, VII. ; but should have been obliged to write, instead of 
 258, IX., the much less simple formula, 
 
 RIt> = -IRr. 
 
 * There is a sense in which the geometrical process here spoken of can be applied, 
 even when the two fixed points, or foci, are imaginary. Compare the Geomctrie 
 Superieure of M. Chasles, page 136. 
 
 t Compare the Note to 259, (11.). 
 
286 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 CHAPTER III. 
 
 ON DIPLANAR QUATERNIONS, OR QUOTIENTS OF VECTORS IN 
 SPACE : AND ESPECIALLY ON THE ASSOCIATIVE PRINCI- 
 PLE OF MULTIPLICATION OF SUCH QUATERNIONS. 
 
 Section 1. — On some Enunciatio7is of the Associative Pro- 
 perty, or Principle, of Multiplication of Diplanar Quater- 
 nions. 
 
 262. In the preceding Chapter we have confined ourselves 
 almost entirely, as had been proposed (224, 225), to the con- 
 sideration of quaternions in a y iv en plane (that of i) ; alluding 
 only, in some instances, to possible extensions* of results so 
 obtained. But we must now return to consider, as in the 
 First Chapter of this Second Book, the subject of General 
 Quotients of Vectors : and especially their Associative Multi- 
 plication (223), which has hitherto been only proved in con- 
 nexion with the Distributive Principle (212), and with the 
 Laws of the Symbols i,j\ k (183j. And first we shall give a 
 iQW geometrical enunciations of that associative principle, which 
 shall be independent of the distributive one, and in which it 
 will be sufficient to consider (corap. 191) the multiplication of 
 versors; because the multiplication of tensors is, evidently an 
 associative operation, as corresponding simply to arithmetical 
 multiplication, or to the composition of ratios in geometry.f 
 We shall therefore suppose, throughout the present Chapter, 
 that </, r, s are some three given but arbitrary versors, in three 
 given and distinct planes ;% and our object will be to throw 
 
 * As in 227, (3.); 242, (7.); 254, (7.); 257, (6.) and (7.) ; 259, (8.), (9.), 
 (10.), (11-); 2G0, (10.); and 2G1, (11.) and (12.). 
 
 f Or, move generally, for any tliree pairs of magnitudes, each pair separately- 
 being lioraogeneous. 
 
 X If the factors q, r, a were complanar, we could always (by 120) put them 
 
CHAP. HI.] ASSOCIATIVE PRINCIPLE, SYSTEM OF SIX PLANES. 287 
 
 some additional light, by new enunciations in this Section, 
 and by new demonstrations in the next, on the very impor- 
 tant, although very simple, Associative Formula (223, II.), 
 w^hich may be written thus : 
 
 I. . . sr.g = s.rq; 
 or thus, more fully, 
 
 II. ■ ' qg = t, if q' - 5r, s' = rq, and t = ss' ; 
 q\ s\ and t being here three new and derived versors, in three 
 neio and derived planes. 
 
 263. Already we may see that this Associative Theorem 
 of Multiplication^ in all its forms, has an essential reference to 
 a System of Six Planes, namely the planes of these six ver- 
 sors, 
 
 IV. . . q, r, s, rq, sr, srq, or IV. . . q, r, s, s, q', t; 
 on the judicious selection and arrangement of which, the clear- 
 ness and elegance of every geometrical statement or proof of 
 the theorem must very much depend : while the versor cha- 
 racter of the factors (in the only part of the theorem for which 
 proof is required) suggests a reference to a Sphere, namely to 
 what we have called the unit-sphere (128). And the three 
 following arrangements of the six planes appear to be the most 
 natural and simple that can be considered : namely, 1st, the 
 arrangement in which the planes all pass through the centre of 
 the sphere ; Ilnd, that in which they all touch its surface ; 
 and IlIrd, that in which they are the six faces of an inscribed 
 solid. We proceed to consider successively these three ar- 
 rangements. 
 
 264. When the Jirst arrangement (263) is adopted, it is natural 
 to employ a7'cs of great circles, as representatives of the versors, on the 
 
 under the forms, 
 
 (3 y d 
 
 and then should have (comp. 183, (1.) ) the two equal ternart/ products, 
 d 3 d dy 
 ^ (3 a a ya ^' 
 
 so that in this case (comp. 224) the associative property would be proved without any 
 difficulty. 
 
288 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 plan of Art. 162. Representing thus the factor q by the arc ab, 
 and r by the successive arc bc, we represent (167) their product rq^ 
 or 5^, by AC; or by any equal arc (165), such as de, in Fig. 59, may 
 be supposed to be. Again, representing s by ef, we shall have df 
 as the representative of the ternary iU /v 
 
 product s.rq, or ss^, or t. taken in ^^"--^ — J^ r- 
 
 one order of association. To repre- y^ /' /\ "\ //^^ 
 sent the other ternary product, \^ ( ( \ J^ y 
 
 sr. q, or q'q, we may first determine ^\^ Jzic=<r^^^$^ ^^^^ 
 three new points, g, h, i, by arcual c'^^^-~\_^/l-----g'''^ 
 
 B A 
 
 equations (165), between gh, bc, Fig. 69. 
 
 and between hi, ef, so that bc, ef 
 
 intersect in h, as the arcs representing &' and s had intersected in e; 
 and then, after thus finding an arc Gi which represents 5r, or q^, may 
 determine three other points, k, l, m, by equations between kl, ab, 
 and between lm, gi, so that these two new arcs, kl, lm, represent q 
 and g-', and that ab, gi intersect in l ; for in this way we shall have 
 an arc, namely km, which represents q^q as required. And the theo- 
 rem then is, that this last arc km is equal to the former arc df, in the 
 full sense of Art. 165; or that when (as under the foregoing condi- 
 tions of construction) the five arcual equations, 
 
 I. . . n AB = '^ KL, nBC = '^GH, <^ EF = n HI, '^ AC = O DE, nGI = '^LM, 
 
 exists then this sixth equation of the same kind is satisfied also, 
 
 II. . . '^ DF = '^ KM : 
 
 the two points, K and m, being both on the same great circle as the two 
 previously determined points, d and f; or d and m being on the 
 great circle through f and k: and the two arcs, df and km, of that 
 great circle, or the two dotted arcs, dk, fm in the Figure, being 
 equally long, and similarly directed (165). 
 
 (1.) Or, after determining the nine points a . . i so as to satisfy the three middle 
 equations I., we might determine the three other points, k, l, m, without any other 
 arcual equations^ as intersections of the three pairs of arcs ab, df ; ab, gi ; df, gi ; 
 and then the theorem would be, that (if these three last points be suitably distin- 
 guished from their own opposites upon the sphere) the two extreme equations I., and 
 the equation II., are satisfied. 
 
 (2.) The same geometrical theorem may also be thus enunciated : If the first, 
 third, and fifth sides (kl, gh, ed) of a spherical hexagon klghed be respectively 
 and arcunlhj equal (165)^0 the first, second, and third sides (ab, bc, CA) of a. sphe- 
 rical triangle ABC, then the second, fourth, and sixth sides (lg, he, dk) of the same 
 hexagon are equal to the three successive sides (mi, if, fm) of another spherical tri- 
 angle, MIF. 
 
CHAP. III.] FIRST AND SECOND ARRANGEMENTS OF PLANES. 289 
 
 (3.) It may also be said, that if five successive sides (kl, . . ed) of one spherical 
 hexagon be respectively and arcually equal to the^t^e successive diagonals (ab, mi, 
 BC, IF, ca) of another such hexagon (ambicf), then the sixth side (dk) of the^rs^ 
 is equal to the sixth diagonal (km) of the second. 
 
 (4.) Or, if we adopt the conception mentioned in 180, (3.), of an arcualsum, and 
 denote such a sum by inserting + between the symbols of the two summands, that of 
 the added arc being written to the left-hand, we may state the theorem, in connexion 
 with the recent Fig. 59, by the formula : 
 
 III... '^DF + '^BA=nEF+ OBO, if " DA = o EC ; 
 
 where b and f may denote any two points upon the sphere. 
 
 (5.) We may also express* the same principle, although somewhat less simply 
 as follows (see again Fig. 69, and compare sub-art. (2.) ) : 
 
 IV. . . if <-> ED + n GH + " KL= 0, then o DK + « HE + -^ LG= 0. 
 
 (6.) If, for a moment, we agree to write (comp. Art. 1), 
 
 V. . . '^ ab = B - A, 
 
 we may then express the recent statement IV. a little more lucidly thus : 
 
 VI. ..ifD-E + H-G + L-K = 0, then k-d + e-h + g-l, = 0. 
 
 (7 ) Or still more simply, if '^, o', r," be supposed to denote any three dipla- 
 nar arcs, which are to be added according to the rule (180, (3.) ) above referred to, 
 the theorem may be said to be, that 
 
 VII.. .(o"+o')+^ = n" + (n'+o); 
 
 or in words, that Addition ofArcsi on a Sphere is an Associative Operation. 
 
 (8.) Conversely, if any independent demonstration be given, of the truth of any 
 one of the foregoing statements, considered as expressing a theorem of spherical geo- 
 metry, f a new proof Avill thereby be furnished, of the associative property of multi- 
 plication of quaternions. 
 
 265. In the second arrangement (263) of the six planes, instead 
 of representing the three given versors, and their partial or total 
 products, by arcs, it is natural to represent them (174, 11.) by an- 
 gles on the sphere. Conceive then that the two versors, q and r, 
 are represented, in Fig. 60, by the two spherical angles, eab and 
 ABE; and therefore (175) that their product, rq or s% is represented 
 by the external vertical angle at e, of the triangle abe. Let the 
 
 * Some of these formulae and figures, in connexion with the associative principle, 
 are taken, though for the most part with modifications, from the author's Sixth Lec- 
 ture on Quaternions, in which that whole subject is very fully treated. Comp. the 
 Note to page 160. 
 
 t Such a demonstration, namely a deduction of the equation II. from the five 
 equations I., by known properties of spherical conies, will be briefly given in the en- 
 suing Section. 
 
 2 p 
 
290 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book II. 
 
 second versor r be also represented by the angle fbc, and the third 
 versor s by bcf; then the 
 other binary product, sr or 
 3', will be represented by 
 the external angle at f, of 
 the new triangle bcf. Again, 
 to represent the^r^^ ternary 
 product, t=ss' = s.rq, we have 
 only to take the external an- 
 gle at D of the triangle ecd, 
 if D be a point determined 
 
 Fig. 60. 
 
 by the two conditions, that the angle ecd shall be equal to bcf, 
 and DEC supplementary to bea. On the other hand, if we conceive 
 a point d' determined by the conditions that d'af shall be equal to eab, 
 and afd' supplementary tocFB, then the external angle at t>\ of the 
 triangle afd^, will represent the second ternary product, q^q = sr. q, 
 •which (by the associative principle) must be equal to the first. 
 Conceiving then that ed is prolonged to G, and fd' to h, the 
 two spherical angles, gdc and ad'h, must be equal in all respects ; their 
 vertices d and d' coinciding, and the rotations (174, IT 7) which they 
 represent being not only equal in amount, but also similarly/ directed. 
 Or, to express the same thing otherwise, we may enunciate (262) the 
 Associative Principle by saying, that when the three angular equations, 
 
 I. . . ABE = FBC, BCP = ECD, DEC = TT - BEA, 
 
 are satisfied, then these three other equations^ 
 
 II, . . DAF = EAB, FDA = CDE, AFD - TT - CFB, 
 
 are satisfied also. For not only is this theorem of spherical geometry a 
 consequence of the associative principle oi multiplication of quaternions , 
 but conversely any independent demonstration* of the theorem is, 
 at the same time, a proof of the principle. 
 266. The third arrangement (263) of 
 the six planes may be illustrated by con- 
 ceiving a gauche hexagon, ab'ca^bc^ to be 
 inscribed in a sphere, in such a manner that 
 the intersection d of the three planes, c'ab', 
 b'ca', a'bc', is on the surface; and there- 
 fore that the three small circles, denoted by 
 these three last triliteral symbols, concur -p. g. 
 
 * Such as we shall sketch, in the following Section, with the help of the known 
 properties of the spherical conies. Compare the Note to the foregoing Article. 
 
CHAP. III.] THIRD ARRANGEMENT, SPHERICAL HEXAGON. 291 
 
 in one point d ; while the second intersection of the two other small 
 circles, ab'c, ca'b, may be denoted by the letter d', as in the annexed 
 Fig. 61. Let it be also for simplicity at first supposed, that (as in 
 the Figure) the Jive circular successions^ 
 
 I. . . c'ab'd, ab'cd', b'ca'd, ca'bd', a'bc'd, 
 are all direct ; or that the Jive iTiscrihed quadrilaterals, denoted by 
 these symbols I., are all uncrossed ones. Then (by 260, (9.) ) it is 
 allowed to introduce three versors, q, r, 5, each having two expres- 
 sions, as follows : 
 
 __ _._b'd __ab' -^da' „ca' 
 
 ^ DC' AC/ B'D Cb' 
 
 ^^ CD' „ BD' 
 
 ca' a'b 
 
 although (by the cited sub-article) the last members of these three 
 formulae should receive the negative sign, if the first, third, and 
 fourth of the successions I. were to become indirect, or if the corre- 
 sponding quadrilaterals were crossed ones. We have thus (by 191) 
 the derived expressions, 
 
 III. . . s' = rq = TJ — • =U — ;; o' = 5r=U — - = U — ■; 
 
 ^ DC' BC' ^ cb' AB' 
 
 whereof, however, the two versors in the first formula would differ 
 in their signs, if the fifth succession I. were indirect; and those in 
 the second formula, if the second succession were such. Hence, 
 
 IV.. .t = ss^ = s.rq = V — ', q'q = sr.q = \J — ; 
 
 and since, by the associative principle, these two last versors are to 
 be equal, it follows that, under the supposed conditions of construc- 
 tion, the four points, b, c', a, d', compose a circular and dij'ect suc- 
 cession ; or that the quadrilateral, bc'ad', is plane, inscriptible* and 
 uncrossed. 
 
 267. It is easy, by suitable changes of sign, to adapt 
 the recent reasoning to the case where some or all of the suc- 
 cessions I. are indirect ; and thus to infer, from the associa- 
 tive principle, this theorem of spherical geometry : 7/*ab'ca'bc' 
 
 * Of course, siuce the four points bc'ad' are known to be homospheric (comp. 
 260. (10.)), the inseriptihility of the quadrilateral in a circle would follow from its 
 being plane, if the latter were otherwise proved : but it is here deduced from the 
 equality of the two versors IV., on the plan of 260, (9.J. 
 
292 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 he a spherical hexagon, such that the three small circles c'ab', 
 b'ca', a'bc' concur in one point d, then, 1st, the three other small 
 circles, ab'c, ca'b, bc'k, concur in another point, d'; and Ilnd, 
 of the six circular successions, 266, I., and bc'ad', the number 
 of those which are indirect is always even (including zero). 
 And conversely, any independent demonstration* of this geo- 
 metrical theorem will be a new proof oi the associative prin- 
 ciple. 
 
 268. The same fertile principle of associative multiplication may 
 be enunciated in other ways, without limiting the factors to be ver- 
 sors, and without introducing the conception of a sphere. Thus we 
 may say (comp. 264, (2.) ), that if o . abcdef (comp. 35) be any 
 pencil of six rays in space, and o.a'b^c' any pencil of three rays, and 
 if the three angles aob, cod, eof of the first pencil be respectively 
 equal to the angles b'oc', c'oa', a'ob^ of the second, then another 
 pencil of three rays, o . a'^b^'o''', can be assigned, such that the three 
 other angles boc, doe, foa oith.Q first pencil shall be equal to the 
 angles b'^oc''', c'^oa'', a'^ob'^ of the third: equality of angles (with 
 one vertex) being here understood (comp. 165) to include complana' 
 rity, and similarity of direction of rotations. 
 
 (1.) Again (comp. 264, (4.)), we may establish the following formula, in which 
 the four vectors a/3y5 form a complanar proportion (226), but e and Z, are any two 
 lines in space : 
 
 T ^^-^^ if ^_^. 
 
 ye at 7 « 
 
 for, under this last condition, we have (comp. 125), 
 
 II £? = ?^ ? = ? ^? 
 
 * " y e aye. a' d e' 
 (2.) Another enunciation of the associative principle is the following : 
 
 III. . . if -- = -, then -- = -; 
 y a e ay o 
 
 for if we determine (120) six new vectors, r]9i, and kX/i, so that 
 
 = -, - = — , whence - = -, 
 y I a It 
 
 IV. . . ^ and 
 
 I ^_« f _/^ 
 
 I. K a fi y 
 
 * An elementary proof, by stereographic projection, will be proposed in the fol- 
 lowing Section. 
 
CHAP. III.] PBOOFS BY SPHERICAL CONICS. 293 
 
 we shall have the transformations, 
 
 V - = -- = ^ -1 = -L ^ = -l = f^ or VI - = ^ 
 
 (3.) Conversely, the assertion that this last equation or proportion VI. is true, 
 •whenever the twelve vectors a . . fx are connected by the five proportions IV., is a 
 form of enunciation of the associative principle ; for it conducts (comp. IV. and V.) 
 to the equation, 
 
 VII. , .-.ij = --.^, atleastif e\\\i,0; 
 
 but, even with this last restriction, the three factor-quotients in VII. may represent 
 any three quaternions. 
 
 Section 2. — On some Geometrical Proofs of the Associative 
 Property of Multiplication of Quaternions, which are inde- 
 pendent of the Distributive* Principle. 
 
 269. We propose, in this Section, to furnish three geome- 
 trical Demonstrations of the Associative Principle, in con- 
 nexion with the three Figures (59-61) which were employed 
 in the last Section for its Enunciation ; and with the three ar- 
 rangements oi six planes, which were described in Art. 263. 
 The two first of these proofs will suppose the knowledge of a 
 few properties oi spherical conies (196, (11.)); but the third 
 will only employ the doctrine of stereographic projection, and 
 will therefore be of a more strictly elementary character. The 
 Principle itself is, however, of such great importance in this 
 Calculus, that its nature and its evidence can scarcely be put 
 in too many different points of view. 
 
 270. The only properties of a spherical conic, which we shall in 
 this Article assume as known, f are the three following: 1st, that 
 through any three given points on a given sphere, which are not on a 
 great circle, a conic can be described (consisting generally oitwo oppo- 
 site ovals), which shall have a given great circle for one of its two cyclic 
 arcs; Ilnd, that if a transversal arc cut hath these arcs, and the conic, 
 the intercepts (suitably measured) on this transversal are equal; and 
 Ilird, that if the vertex of a spherical angle move along the conic, 
 while its legs pass always through two fixed points thereof, those legs 
 
 * Compare 224 and 262 ; and the Note to page 236. 
 
 t The reader may consult the Translation (Dublin, 1841, pp. 46, 50, 55) by the 
 present Dean Graves, of two Memoirs by M. Chasles, on Cones of the Second De- 
 gree^ and Spherical Conies. 
 
294 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 intercept a constant interval^ upon each cyclic arc, separately taken. 
 Admitting these three properties, we see that if, in Fig. 59, we con- 
 ceive a spherical conic to be described, so as to pass through the 
 three points b, f, h, and to have the great circle daec for one cyclic 
 arc, the second and third equations I. of 264 will prove that the arc 
 GLIM is the other cyclic arc for this conic; the first equation I. proves 
 next that the conic passes through k ; and if the arcual chord fk be 
 drawn and prolonged, the two remaining equations prove that it 
 meets the cyclic arcs in d and m ; after which, the equation 11. of 
 the same Art. 264 immediately results, at least with the arrange- 
 ment* adopted in the Figure. 
 
 (1.) The 1st property is easily seen to correspond to the possibility of circum- 
 scribing a circle about a given plane triangle, namely that of which the comers are 
 the intersections of a plane parallel to the plane of the given cyclic arc, with the 
 three radii drawn to the three given points upon the sphere : but it may be worth 
 while, as an exercise, to prove here the Ilnd property by quaternions. 
 
 (2.) Take then the equation of a cyclic cone, 196, (8.), which may (by 196, 
 XII.) be written thus : 
 
 I...S^S^ = N^; andlet II. . . S^' S^' = K^', 
 
 p and p' being thus two rays (or sides) of the cone, which may also be considered to 
 be the vectors of two points p and p' of a spherical conic, by supposing that their 
 lengths are each unity. Let r and r' be the vectors of the two points t and t' on 
 the two cyclic arcs, in which the arcual chord pp' of the conic cuts them ; so that 
 
 III. ..S- = 0, S^=0, and IV. . . Tr = Tr' = 1. 
 
 a (5 
 
 The theorem may then be stated thus : that 
 
 V. . . if jO = a;r + xt', then VI. . . p' = aV + xt ; 
 or that this expression VI. satisfies II., if the equations I. III. IV. V. be satisfied. 
 Now, by III. V. VI., we have 
 
 a a X a j3 ^ x' (i 
 
 whence it follows that the first members of I. and II. are equal, and it only remains 
 to prove that their second members are equal also, or that Tp' = Tp, if Tr' = Tr. 
 Accordingly we have, by V. and VI., 
 
 VIII. . . ^-Ili' = ^^.^^^ = S-iO, by 200, (11.), and 204, (19.); 
 
 p' + p X +X T+T ^ ^ '^ ^' 
 
 and the property in question is proved. 
 
 * Modifications of that arrangement may be conceived, to which however it would 
 be easy to adapt the reasoning. 
 
CHAP. III.] PROOF BY STEREOGRAPHIC PROJECTION. 295 
 
 271. To prove the associative principle, with the help of Fig. 60, 
 three other properties of a spherical conic shall be supposed known :* 
 1st, that for every such curve two focal points exist, ipossessing seve- 
 ral important relations to it, one of which is, that if these two foci 
 and one tangent arc he given, the conic can be constructed; Ilnd, 
 that if, from any point upon the sphere, two tangents be drawn to the 
 conic, and also two arcs to the foci, then one focal arc makes with one 
 tangent the same angle as the other focal arc with the other tangent ; 
 and Ilird, that if a spherical quadrilateral be circumscribed to such 
 a conic (supposed here for simplicity to be a spherical ellipse, or the 
 opposite ellipse being neglected), opposite sides subtend supplementary/ 
 angles, at either of the two (interior) foci. Admitting these known 
 properties, and supposing the arrangement to be as in Fig. 60, we 
 may conceive a conic described, which shall have e and f for its two 
 focal points, and shall touch the arc bc ; and then the two first of the 
 equations I., in 265, will prove that it touches also the arcs ab and 
 CD, while the third of those equations proves that it touches ad, so 
 that ABCD is a circumscribedf quadrilateral: after which the three 
 equations II., of the same article, are consequences of the same pro- 
 perties of the curve. 
 
 272. Finally, to prove the same important Principle in a 
 more completely elementary way, by means of the arrangement 
 represented in Fig. 61, or to prove the theorem of spherical 
 geometry enunciated in Art. 267, we may assume the point d 
 as the pole of a stereograpjhic projection^ in which the three 
 small circles through that point shall be represented by right 
 lines ^hui the three others by czVc/ei", 
 
 iall being in one common plane. And 
 then (interchanging accents) the 
 theorem comes to be thus stated : 
 7/* a', b', c' be any three points 
 (comp. Fig. 62) on the sides bc, 
 CA, AB of any plane triangle^ or on 
 
 those sides prolonged, then^ 1st, ^ ^Y\si. 62 
 
 the three circles^ 
 
 * The reader may again consult pages 46 and 50 of the Translation lately cited. 
 In strictness, there are of course /owr /oa, opposite two by two. 
 
 t The writer has elsewhere proposed the notation, ef(. .) abcd, to denote the 
 relation of the focal points e, f to this circumscribed quadrilateral. 
 
296 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 I. . . c'ab', a'bc', b'ca', 
 will meet in one point d ; and Ilnd, an even number (if any) 
 of the six (linear or circular) successions, 
 
 II. . . ab'c, bc'a, cab, and 11'. . . c'ab'd, a'bcd, b'ca'd, 
 will be direct; an even number therefore also (if any) being 
 indirect. But, under i\\\Qform* the theorem can be proved 
 by very elementary considerations, and still without any em- 
 ployment of the distributive principle (224, 262). 
 
 (1.) 1h.B first part of the theorem, as thus stated, is evident from the Third Book 
 of Euclid ; but to prove both parts together, it may be useful to proceed as follows, 
 admitting the conception (235) oi amplitudes, or of angles as representing ro«a«ions, 
 ■which may have any values, positive or negative, and are to be added with attention 
 to their signs. 
 
 (2.) We may thus write the three equations, 
 
 III. . . ab'c = nTT, bc'a = w'tt, ca'b = n"7r, 
 to express the three coUineations, ab'c, &c. of Fig. 62 ; the integer, n, being odd or 
 even, according as the point b' is on the finite line AC, or on a prolongation of that 
 line ; or in other words, according as the first succession II. is direct or indirect : 
 and similarly for the two other coefficients, n' and n". 
 
 (3.) Again, if opqr be any four points in one plane, we may establish the for- 
 mula, 
 
 IV. . . POQ 4- QOR = POR + 2m7r, 
 
 with the same conception of addition of amplitudes ; if then d be any point in the 
 plane of the triangle abc, we may write, 
 
 V. . . ab'd + db'c = n7r, bc'd + dc'a = nV, ca'd + da'b = w'V ; 
 and therefore, 
 
 VI. . . (ab'd + dc'a) + (bc'd + da'b) 4 (ca'd + db'c) — (» + w' 4 »") TT. 
 
 (4.) Again, if any four points opqr bo not merely complanar but concircular, 
 we have the general formula, 
 
 VII. . . CPQ4QRO=/J7r, 
 the integer p being odd or even, according as the succession opqr is direct or indi' 
 
 * The Associative Principle of Multiplication was stated nearly under this ^brm, 
 and was illustrated by the same simple diagram, in paragraph XXII. of a commu- 
 nication by the present author, which was entitled Letters on Quaternions, and has 
 been printed in the First and Second Editions of the late Dr. Nichol's Cyclopcedia of 
 the Physical Sciences (London and Glasgow, 1857 and 1860). The same commu- 
 nication contained other illustrations and consequences of the same principle, which it 
 has not been thought necessary here to reproduce (compare however Note C) ; and 
 others may be found in the Sixth of the author's already cited Lectures on Quater- 
 nions (Dublin, 1863), from which (as already observed) some of the formulae and 
 figures of this Chapter have been taken. 
 
CHAP. III.] ADDITIONAL FORMULA, NORM OF A VECTOR. 297 
 
 red ; if then we denote by d the second intersection of the first and second circles I. , 
 whereof c' is & first intersection, we shall have 
 
 VIII. . . ab'd + dc'a =/>7r, bc'd + da'b =p'7r, 
 p and p' being odd, when the two first successions II'. are direct, but even in the con- 
 trary case. 
 
 (5.) Hence, by VI., we have, 
 
 IX. . . ca'd + db'c =/>"7r, where X. . . jo + />' + p" = « + »' f n" ; 
 the third succession II'. is therefore always circular, or the third circle I. passes 
 through the intersection D o^ ih.Q two first ; and it is direct or indirect, that is to 
 say, p" is odd or even, according as the number of even coefficients, among thej^re 
 previously considered, is itself even or odd ; or in other words, according as the 
 number of indirect successions, among the five previously considered, is even (includ- 
 ing zero), or odd. 
 
 (6.) In every case, therefore, the total number of successions of each kind is even, 
 and both parts of the theorem are proved : the importance of the second part of it 
 (respecting the even partition, if any, of the six successions II. 11'.) arising from 
 the necessity of proving that we have always, as in algebra, 
 
 XI . . sr.q = -\-s.rq, and never Xll. . , sr. q = — s.rq, 
 if q, r, s be any three actual quaternions. 
 
 (7.) The associative principle of multiplication may also be proved, without the 
 distributive principle, by certain considerations of rotations of a system, on which we 
 cannot enter here. 
 
 Section 3. — On some Additional FormulcB. 
 
 273. Before concluding the Second Book, a few additional re- 
 marks may be made, as regards some of the notations and transfor- 
 mations which have already occurred, or others analogous to them. 
 And first as to notation, although we have reserved for the Third 
 Book the interpretation of such expressions as /3a, or a^ yet we have 
 agreed, in 210, (9.), to abridge the frequently occurring symbol (Ta)^ 
 to Ta^; and we now propose to abridge it still further to Na, and to 
 call this square of the tensor (or of the length) of a vector, a, the Norm 
 of that Vector: as we had (in 190, &c.), the equation Tg'^ = N5', and 
 called N^- the norm of the quaternion q (in 145, (11.) ). We shall 
 therefore now write generally, for any vector a, the formula, 
 I. . .(Ta)2 = Ta2 = Na. 
 
 (1.) The equations (comp. 186, (1.) (2.) (3.) (4.) ), 
 
 II. ..Np = l; III. ..Np = Na; IV- . . N(p -«) = Na ; 
 
 V. ..N(p-a) = N(/3-a), 
 
 represent, respectively, the unit-sphere; the sphere through a, with o for centre ; 
 
 the sphere through o, with a for centre ; and the sphere through b, with the same 
 
 centre a. 
 
 2q 
 
298 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 (2.) The equations (comp. 186, (6.) (7.) ), 
 
 VI. ..N(p + a) = N(p-a); VII. . . N(p-i8) = N(p- a), 
 represent, respective!}'', the plane through o, perpendicular to the line oa ; and the 
 plane which perpendicularly bisects the line Ab. 
 
 274, As regards transformations, the few following may here be 
 added, which relate partly to the quaternion forms (204, 216, &c.) 
 of the Equation"^ of the Ellipsoid. 
 
 (1.) Changing K(k: p) to Ep : Rk, by 259, VIII., in the equation 217, XVI. 
 of the ellipsoid, and observing that the three vectors p, Rp, and Rk are complanar, 
 while 1 : Tp = TRp by 258, that equation becomes, when divided by TRp, and when 
 the value 217, (5.) for t^ is taken, and the notation 273 is employed : 
 
 I. .. Tf-i-+-?-VNt-m-; 
 
 V Rp ^ Rk / 
 
 of which the first member will soon be seen to admit of being written f asT(ip + p^), 
 and the second member as /c^ - i^. 
 
 (2.) If, in connexion with the earlier forms (204, 216) of the equation of the 
 same surface, we introduce a new auxiliary vector^ a or os, such that (comp. 2 1 6, 
 VIII.) • 
 
 the equation may, by 204, (14.), be reduced to the following extremely simple form : 
 
 III. .. T(T=T/3; 
 which expresses that the locus of the new auxiliary point s is what we have called 
 the mean sphere, 216, XIV. ; while the line PS, or (t — p., which connects any two 
 corresponding points, p and s, on the ellipsoid and sphere, is seen to be parallel to 
 the fixed line /3; which is one element of the homology, mentioned in 216, (10.). 
 (3.) It is easy to prove that 
 
 IV. . .S^ = S^ S?, and therefore V. . . S ^': S^ = S^' : S^, 
 a c 6 
 
 if p' and <t' be the vectors of two new but corresponding points, p' and s', on the 
 ellipsoid and sphere ; whence it is easy to infer this other element of the homology, 
 that any two corresponding chords, pp' and ss', of the two surfaces, intersect each 
 other on the cyclic plane which has d for its cyclic normal (comp. 216, (7.) ) : in 
 fact, they intersect in the point t of which the vector is, 
 
 .,,_ iPp + X'p' X(T + x'o' p' p 
 
 VI. . . r = -^--— 7- = — r-, if x = S^, and a;' = -S^; 
 
 x + x x-\-x d d 
 
 * In the verification 216, (2.) of the equation 216, (1.), considered as repre- 
 senting a surface of the second order, V— and V^ ought to have been printed, in- 
 stead of V - and V - : but this does not affect the reasoning. 
 a a 
 
 t Compare the Note to page 233. 
 
CHAP. III.] HOMOLOGIES OF ELLIPSOID AND SPHERE. 299 
 
 and this point is on the plane just mentioned (comp. 216, XI.), because 
 
 VII. . . S^=0. 
 
 
 
 (4.) Quite similar results would have followed, if we had assumed 
 
 VIII. . . cr = (- S^-f V^^/3 = p-2i8S^, 
 \ a (5 j y 
 
 which would have given again, as in III., 
 
 IX. ..T<T = T^, but with X. ..S-=-S^ S^; 
 
 r « y 
 
 the other cyclic plane, with y instead of d for its normal, might therefore have been 
 taken (as asserted in 216, (10.) ), as another plane of homology of ellipsoid and 
 sphere, with the same centre of homology as before : namely, the poin^ at infinity 07i 
 the line /3, or on the axis (204, (15.) ) of one of the two circumscribed cylinders of 
 revolution (comp. 220, (4.)). 
 
 (5.) The same ellipsoid is, in two other ways, homologous to the same mean 
 sphere, with the same two cyclic planes as /> /an es of homology, but with a new centre 
 of homology, which is the infinitely distant point on the axis of the second circum- 
 scribed cylinder (or on the line ab' of the sub-article last cited). 
 
 (6.) Although not specially connected with the ellipsoid, the following general 
 transformations may be noted here (comp. 199, XII., and 204, XXXIV.) : 
 
 XL..TVV7=V{KTry-S7)}; XIL . • tan iZ(? = (TV: S) V7 = ^I|^^. 
 
 (7.) The equations 204, XVI. and XXXV., give easily, 
 XIII. . . UYq = UVU«7 ; XIV. . . VlYq = AK.q; XV. . . TlYq = TVq ; 
 or the more symbolical forms, 
 
 Xlir. . . UVU = UV ; XIV'. . . UIV = Ax. ; XV'. . . TIV = TV ; 
 and the identity 200, IX. becomes more evident, when we observe that 
 XVI. . .5-N"5=7(l-K5). 
 (8.) We have also generally (comp. 200, (10.) and 218, (10.)), 
 
 XVII ^^ = (g-l)(Kg4l) ^ Ng-1 + 2V(7 
 '"q + 1 (q + l)(Kq+l) Nj + 1 + 28? ' 
 (9.) The formula,* 
 
 XVIII. . .V(rq + Kqr) = U(Sr. S^ + Yr.Yq) = r"! (r^-i^ q-\ 
 in which q and r may be any two quaternions, is not perhaps of any great importance 
 in itself, but will be found to furnish a student Avith several useful exercises in trans- 
 formation. 
 
 (10.) When it was said, in 257, (1.), that zero had only itself iox a square-root, 
 the meaning was (comp. 225), that no binomial expression of the form x-\- »y (228) 
 could satisfy the equation, 
 
 XIX. . . = 52 = (x + ty)8 = (x^ - y2) 4- 2ixy, 
 
 * This formula was given, but in like manner without proof, in page 587 of the 
 author's Lectures on Quaternions. 
 
300 ELEMENTS OF QUATERNIONS. [bOOK II. 
 
 for any real or imaginary values of the two scalar coefficients x and y, diflFerent 
 from zero ;* for if biquaternions (214, (8.) ) be admitted, and if h again denote, as 
 in 256, (2.), the imaginary of algebra, then (comp. 257, (6.) and (7.)) we may 
 write, generally, besides the real value Qi =■ 0, the imaginary expression^ 
 
 XX. . . Qi=v-{ hv', if S» = S»' = SW=:Ntj'-N» = 0; 
 V and v' being thus any two real right quaternions, with equal norms (or with equal 
 tensors), in planes perperpendicular to each other. 
 
 (11.) For example, by 256, (2.) andby the laws (183) of y A, we have the trans- 
 formations, 
 
 XXI. . . {i+hj)i=i^-f -Vh{ij^ji) = + A0 = 0; 
 
 so that the bi-quaternion i 4- hj is one of the imaginary values of the symbol 0^. 
 
 (12.) In general, when bi-quaternions are admitted into calculation, not only the 
 
 square of one, but the product of two such factors may vanish, without either of them 
 
 separately vanishing : a circumstance which may throw some light on the existence 
 
 of those imaginary (or symbolical) roots of equations, which were treated of in 257. 
 
 (13.) For example, although the equation 
 
 XXII. . . g2-l = (g-l) (9-t-l) = 
 has no real roots except ± 1, and therefore cannot be verified by the substitution of 
 any other real scalar, or real quaternion, for q, yet if we substitute for q the bi-qua- 
 ternionf v + hv', with the conditions 257, XIII., this equation XXII. is verified. 
 
 (14.) It will be found, however, that when two imaginary but non-evanescent 
 factors give thus a null product, the norm of each is zero; provided that we agree 
 to extend to bi-quaternions the formula Ng^= Sq'^—Yq^ (204, XXII.) ; or to define 
 that the Norm of a Biquaternion (like that of an ordinary or real quaternion) is 
 equal to the Square of the Scalar Fait, minus the Square of the Right Part : each 
 of these two parts being generally imaginary, and the former being what we have 
 called a Bi-scalar. 
 
 (15.) With this definition, if q and q' be any two real quaternions, and if h be, 
 as above, the ordinary imaginaiy of algebra, we may establish the formula : 
 
 XXIII. . . N(9 + hq) = (Sq + hSqy - (Vq + hYq'^ ; 
 or (comp. 200, VII., and 210, XX.), 
 
 XXIV. . . N(9 + A5') = N5-Ng'+2^S.5K9'. 
 (16.) As regards the norm of the sum of any two real quaternions, or real vec- 
 tors (273), the following transformations are occasionally useful (comp. 220, (2.) ); 
 XXV. . . N (5' + g) = N (Tq. Vq + Tq . Vq') ; 
 XXVI. . . N(/3 + a)=N(T/3.Ua + Ta.U/3); 
 in each of which it is permitted to change the norms to the tensors of which they are 
 the squares, or to write T for N. 
 
 * Compare the Note to page 276. 
 
 t This includes the expression + hi, of 257, (1.), for a symbolical square-root of 
 positive unity. Other such roots are + hJ, and + hk. 
 
BOOK III. 
 
 ON QUATERNIONS, CONSIDERED AS PRODUCTS OR POWERS OF 
 VECTORS ; AND ON SOME APPLICATIONS OF QUATERNIONS. 
 
 CHAPTER I. 
 
 ON THE INTERPRETATION OF A PRODUCT OF VECTORS, OR 
 POWER OF A VECTOR, AS A QUATERNION. 
 
 Section 1. — On a First Method of interpreting a Product of 
 Two Vectors as a Quaternion. 
 
 Art. 275. In the First Book of these Elements we inter- 
 preted, 1st, the difference of any two directed right lines in 
 space (4) ; Ilnd, the sum of two or more such lines (5-9) ; Ilird, 
 the product of one such line, multiplied by or into a positive 
 or negative number (15) ; IVth, the quotient of such a line, 
 divided by such a number (16), or by what we have called 
 generally a Scalar (17); and Vth, the sum of a system of 
 such lines, each affected (97) with a scalar coefficient (99), as 
 being in each case zY^e//" (generally) o. Directed Line'^ in Space ^ 
 or what we have called a Vector (1). 
 
 276. In the Second Book, the fundamental principle or 
 pervading conception has been, that the Quotient of two such 
 Vectors is, generally, a Quaternion (112, 116). It is how- 
 ever to be remembered, that we have included under this ge- 
 neral conception, which usually relates to what may be called 
 an Oblique Quotient, or the quotient of two lines in space 
 making either an acute or an obtuse angle with each other 
 
 * The Fourth Proportional to any three complanar lines has also been iaince in- 
 terpreted (226), as being another line in the same plane. 
 
302 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (130), the three following particular cases: Ist, the limiting 
 case, when the angle becomes null, or when the two lines are 
 similarly directed, in which case the quotient degenerates (131) 
 into 2i positive scalar; Ilnd, the other limiting case, w^henthe 
 angle is equal to two right angles, or when the lines are oppo- 
 sitely directed, and when in consequence the quotient again 
 degenerates, but now into a negative scalar ; and Ilird, the 
 intermediate case, when the angle is right, or when the two 
 lines are perpendicular (132), instead of being parallel (15), 
 and when therefore their quotient becomes what we have 
 called (132) a Right Quotient, or a Eight Quaternion: 
 which has been seen to be a case not less important than the 
 two former ones. 
 
 277. But no Interpretation has been assigned, in either of 
 the two foregoing Books, for a Product of two or more Vec- 
 tors ; or for the Square, or other Power of a Vector: so that 
 the Symbols, 
 
 I. . . /3a, 7j3aj . . and II. . . a% a^ . . a"S ... a*, 
 
 in which a, j3, 7 . . denote vectors, but t denotes a scalar, re- 
 main as yet entirely uninterpreted; and we are therefore /re^ 
 to assign, at this stage, any meanings to these new symbols, or 
 new combinations of symbols, which shall not contradict each 
 othei\ and shall appear to be consistent with convenience and 
 analogy. And to do so will be the chief object of this First 
 Chapter of the Third (and last) Book oi' these Elements : which 
 is designed to be a much shorter one than either of the fore- 
 going. 
 
 278. As a commencement o£ such. Interpretation we shall 
 here define, that a vector a is multiplied by another vector j3, 
 or that the latter vector is multiplied into* the former, or 
 that the product j3a is obtained, ivhen the multiplier-line j3 
 is divided by the reciprocal^a (258) of the multiplicand-line a ; 
 as we had proved ( 1 36) that one quaternion is multiplied into 
 another, when it is divided by the reciprocal thereof. In sym- 
 bols, we shall therefore write, as a first definition, the for- 
 mula: 
 
 * Compare the Notes to pages 14G, 159. 
 
CHAP. I.J INTERPRETATION OF A PRODUCT OF TWO VECTORS. 303 
 
 I. . ./3a=j3:Ra; where II. . . Ra = - Ua : Ta (258, VII.). 
 And we proceed to consider, in the following Section, some of 
 the general consequences of this definition, or interpretation, of 
 a Product of two Vectors, as being equal to a certain Quotient^ 
 or Quaternion. 
 
 Section 2. — On some Consequences of the foregoing Inter- 
 pretation, 
 
 279. The definition (278) gives the formula : 
 
 I. . . |3a = :^ ; and similarly, T. . . a/3 = ^ ; 
 
 it gives therefore, by 259, VIII., the general relation, 
 
 II. . . /3a = Ka/3 ; or 11'. . . a/3 = Kj3a. 
 The Products of two Vectors, taken in two opposite orders, are 
 therefore Conjugate Quaternions; and the Multiplication of 
 Vectors, like that of Quaternions (168), is (generally) a Non- 
 Commutative Operation. 
 
 (1.) It follows from II. (by 196, comp. 223, (1.) ), that 
 
 III. . . S/3a = + Sa/3 = i(/3a + a/3). 
 (2.) It follows also (by 204, corap. again 223, (I.) ), that 
 
 IV. . . V^a = - Ya^ =^\(pa- a(3). 
 
 280. Again, by the same general formula 259, VI II., we 
 have the transformations, 
 
 ' R{a-va) K/3 E/3 R/3 Ra IV 
 it follows, then, from the definition (278), that 
 
 II. . ./3(a + a')=/3a+/3a'; 
 whence also, by taking conjugates (279), we have this other 
 general equation, 
 
 III. . . (a + a) /3 = a/3 -f a'/3. 
 Multiplication of Vectors is, therefore, like that of Quaternions 
 (212), a Doubly Distributive Operation. 
 
 281. As we have not yet assigned any signification for a 
 ternary product of vectors, such as yfia, Ave are not yet pre- 
 
304 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 pared to pronounce, whether the Associative Principle (223) 
 o^ Multiplication of Quaternions does or does not extend to 
 Vector-Multiplication. But we can already derive several other 
 consequences from the definition (278) ofsibinari/ product, j3a ; 
 among which, attention may be called to the Scalar character 
 of a Product of two Parallel Vectors; and to the Right cha- 
 racter of a Product of two Perpendicular Vectors, or of two 
 lines at right angles with each other. 
 
 (1.) The definition (278) may be thus written, 
 
 I. ../3a = -T/3.Ta.U(/3:a); 
 it gives, therefore, 
 
 II. ..T/3a=T/3.Ta; III.. . U/3a = -U(i3 : a) = U/3.Ua ; 
 the tensor and versor of the product of two vectors being thus equal (as for quater- 
 nions, 191) to the product of the tensors^ and to the product of the versors, re- 
 spectively. 
 
 {2.) Writing for abridgment (comp. 208), 
 
 IV. ..a = Ta, 6 = T/3, y=Ax.(/3:a), a; = Z(/3:a), 
 we have thus, 
 
 V. . . T(3a = ba ; VI. . . S(3a = Saj3 = - 6a cos a; ; 
 VII. ..SU/3a = SUai3 = -cos^; VIII... L(3a = 7r-x; 
 
 so that (comp. 198) the angle of the product of any two vectors is the supplement of 
 the angle of the quotient. 
 
 (3.) We have next the transformations (comp. again 208), 
 
 IX. . . TV/3a = TVaj3 = 6a sin a; ; X. . . TVUj3a = TVUaj3 = sin a: ; 
 
 XI. . . I Vj3a = - y6a sin x ; XI'. . . I Va/3 = + yab sin x ; 
 XII. . . IUV/3a=Ax.|3a = -y; XII'. . . IUVa/3 = Ax. a/3 = + y ; 
 
 so that the rotation round the axis of a product of two vectors, from the multiplier to 
 the multiplicand, is positive. 
 
 (4.) It follows also, by IX., that the tensor of the right part of such a product, 
 (3a, is equal to the parallelogram under the factors; or to the double of the area of 
 the triangle OAB, whereof those two factors a, (3, or OA, OB, are two coinitial sides : 
 so that if we denote here this last-mentioned area by the symbol 
 
 A OAB, 
 
 we may write the equation, 
 
 XIII. . . TY(3a = parallelogram under a, (3, = 2A OAB ; 
 and the index, lY (3a, is a right line perpendicular to the plane of this parallelogram, 
 of which line the length represents its area, in the sense that they bear equal ratios 
 to their respective units (of length and of area). 
 (6.) Hence, by 279, IV., 
 
 XIV. . . T((3a - a (3) = 2 X parallelogram = 4 A oab. 
 (6.) For any two vectors, «, (3, 
 
CHAP. I.] PRODUCTS OF PARALLELS AND PERPENDICULARS. 305 
 
 XV. ..S/3a = -Na.S(|3:a); XVI. . . V^a=-Na . V(|3 : a); 
 
 or briefly,* 
 
 XVII. ../3a = -Na.(/3:a), 
 
 with the signification (273) of Na, as denoting (Ta)2. 
 
 (7.) If the two factor-lines be perpendicular to each other, so that a; is a right 
 angle, then the parallelogram (4.) becomes a rectangle, and the product (3a becomes 
 a right quaternion (132) ; so that we may write, 
 
 XVIII. . . S(3a = Sa/3 = 0, if /3 -J- a, and reciprocally. 
 
 (8.) Under the same condition of perpendicularity, 
 
 XIX. . . Z)3a=Za/3 = |; XX. . . I^a = - y6a ; XXI. . . la(3 = + yab. 
 
 (9.) On the other hand, if the two factor-lines he parallel, theright part of their 
 product vanishes, or that product reduces itself to a scalar, which is negative or po^ 
 sitive according as the two vectors multiplied have similar or opposite directions ; for 
 we may establish the formula, 
 
 XXII. . . if /3 II a, then V/3a = 0, Va/3 - ; 
 and, under the same condition oi parallelism, 
 
 XXin. ., pa=a^ = S(3a = Sa(3 = + ba, 
 the upper or the lower sign being taken, according as a; = 0, or = tt. 
 
 (10.) We may also write (by 279, (1.) and (2.) ) the following ybrmM?a of per- 
 pendicularity, and formula of parallelism : 
 
 XXIV. . . if /3 4- a, then (3a =- a(3f and reciprocally ; 
 
 XXV. . . if j8 II a, then /3a = + a/3, with the converse. 
 
 (11.) If a, (3, y be any three unit-lines, considered as vectors of the comers 
 A, B, c of a spherical triangle, with sides equal to three new positive scalars, a, b, c, 
 then because, by XVII,, (3a = - (3: a, and y/B = - y : /3, the sub-articles to 208 allow 
 us to write, 
 
 XXVI. . . S (Vy/3 . V/3a) = sin a sin c cos b ; 
 
 XXVII. . . IV(Vy(3.V/3a) = ±/3sinasincsinB; 
 
 XXVIII. . . (IV: S) (Vy/3.V/3a) = + ^3 tan b ; 
 
 upper or lower signs being taken, in the two last formulae, according as the rotation 
 round (3 from a to y, or that round b from A to c, is positive or negative. 
 (12.) The equation 274, I., of the Ellipsoid, may now be written thus : 
 XXIX. . . T(«p + pfc) = Ti2-TK2; or XXX. . . T(tp + pK)=Nt-N'K. 
 
 282. Under the general head o£ sl product of two parallel 
 vectors, two interesting cases occur, which furnish two first 
 examples of Powers of Vectors : namely, 1st, the case when 
 
 * All the consequences of the interpretation (278), of the product (3a of two vec- 
 tors, might be deduced from this formula XVII. ; which, however, it would not have 
 been so natural to have assumed for a definition of that symbol, as it was to assume 
 the formula 278, I. 
 
 2 R 
 
306 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 the two factors are equal, which gives this remarkable result, 
 that the Square of a Vector is always equal to a Negative Sca- 
 lar; and Ilnd, the case when the factors are (in the sense 
 already defined, 258) reciprocal to each other, in which case 
 it follows from the definition (278) that i\iQ\v product is equal 
 to Positive Unity : so that each may, in this case, be consi- 
 dered as equal to unity divided by the other, or to the Potver 
 of that other which has Negative Unity for its Exponent, 
 
 (I.) When (5 = a, the product (3a reduces itself to what we may call the square 
 of a, and may denote by a^; and thus we may write, as a particular but important 
 case of 281, XXIIL, the formula (comp. 273), 
 
 I. . . a2=-a2 = -(Ta)2 = -Na; 
 
 so that the square of any vector a is equal to the negative of the norm (273) of that 
 
 vector; or to the negative of the square of the number Ta, which expresses (185) 
 
 the length of the same vector. 
 
 (2.) More immediately, the definition (278) gives, 
 
 II. .. a2 = aa = a : Ra = - (Ta)« = - Na, as before. 
 
 (3.) Hence (compare the notations 161, 190, 199, 204), 
 
 III. . . S.a2 = -Na; IV. ..V.a2=0; 
 and 
 
 V. . . T.a2 = T(a2) = + Na = (Ta)2 = Ta2; 
 
 the omission of i\ie parentheses, or of the point, in this last symbol of a tensor,* for 
 the square of a vector, as well as for the square of a quaternion (190), being thus 
 justified : and in like manner we may write, 
 
 VI. ..U.a2 = U(a3) = -l=(Ua)2 = Ua2; 
 
 the square of an unit-vector (129) being always equal to negative unity, and paren- 
 theses (or points) being again omitted. 
 (4.) The equation 
 
 VII. . . p2 = a\ gives VII'. . . Np = Na, or VII". . . Tp = Ta ; 
 
 it represents therefore, by 186, (2.), the sphere with o for centre, which passes 
 through the point a. 
 
 (6.) The more general equation, 
 
 VIII. . . (p - a)2 = ((3 - a)«, (comp.f 186, (4.), ) 
 
 represents the sphere with a for centre, which passes through the point b. 
 (6.) For example, the equation, 
 
 IX. . . (p - a)2 = a2, (comp. 186, (3.), ) 
 
 represents the sphere with a for centre, which passes through the origin o. 
 
 * Compare the Note to page 210. 
 
 t Compare also the sub-articles to 275. 
 
CHAP. I.] SQUARE AND RECIPROCAL OF A VECTOR. 307 
 
 (7.) The equations (comp. 18G, (6.), (7.)), 
 
 X. . . (p + a)2 = (p-a)2; XI. . . (p - /3)2 = (p- a)^, 
 represent, respectively, the plane through o, perpendicular to the line OA ; and the 
 plane which perpendicularly bisects the line ab. 
 
 (8.) The distributive principle oi veetor'tnultiplication (280), and the formula 
 279, III., enable us to establish generally (comp. 210, (9.) ) the formula, 
 
 XII. . . (|3±a)2 = /3-2+2S/3a + a3; 
 the recent equations IX. and X. may therefore be thus transformed : 
 IX'. . . p^- = 2Sap ; and X', . . Sap = 0. 
 (9.) The equations, 
 
 XIII. . . p2+a2 = 0; XIV. .. p2 + 1=10, 
 
 represent the spheres with o for centre, which have a and 1 for their respective radii ; 
 so that this very simple formula, p'+ 1 = 0, is (comp. 186, (1.) ) a form of the Equa- 
 tion of the Unit- Sphere (128), and is, as such, of great importance in the present 
 Calculus. 
 
 (10.) The equation, 
 
 XV. . . p«-2Sap + c = 0, 
 
 may be transformed to the following, 
 
 XVI. . . N(p-a) = -(p-a)2 = c-a2 = c + Na; 
 or XVr. . . T(p-a) = V(c-a2) = V(c + Na); 
 
 it represents therefore a (real or imaginary) sphere, with a for centre, and with this 
 last radical (if real) for radius. 
 
 (11.) This sphere is therefore necessarily real, if c be a positive scalar ; or if this 
 scalar constant, c, though negative^ be (algebraically) greater than a*, or than — Na : 
 but it becomes imaginary, if c + Na < 0. 
 
 (12.) The radical plane of the two spheres, 
 
 XVII. . . p2 - 2Sap + c = 0, p2 - 2Sa'p + c' = 0, 
 
 has for equation, 
 
 XVIII. . . 2S(a'-a)p = c'-c; 
 
 it is therefore always real, if the given vectors a, a and the given scalars c, c be 
 such, even if one or both of the spheres themselves be imaginary. 
 
 (13.) The equation 281, XXIX., or XXX., of the Central Ellipsoid {ox of the 
 ellipsoid with its centre taken for the origin of vectors), may now be still further sim- 
 plified,* as follows : 
 
 XIX.. .T(tp + pK:)=/c^~i2. 
 
 (14.) The definition (278) gives also, 
 
 XX. . . aRa = a : a = 1 ; or XX'. . . Ra . a = Ra : Ra = 1 ; 
 whence it is natural to write, f 
 
 * Compare the Note to page 233. 
 
 t Compare the second Note to page 279. 
 
308 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XXL . . Ra = l:i = a-S 
 
 a 
 
 if we so far anticipate here the general theory oi powers of vectors^ above alluded to 
 (277), as to use this last symbol to denote the quotient^ of unity divided by the vector 
 a ; so as to have identically, or for every vector, the equation, 
 XXII. . . a.a-i = a-'.a=l. 
 
 (15.) It follows, by 258, VII., that 
 
 XXIII. . . a-i = - Ua : Ta ; and XXIV. . . (Sa = ft : aK 
 
 (16.) If we had adopted the equation XXIII. as a definition* otthesymbol a"', 
 then the formula XXIV. might have been used, as a formula of interpretation for 
 the symbol (3a. But we proceed to consider an entirely different method, of arriving 
 at the same (or an equivalent) Interpretation of this latter symbol : or of a Binary 
 Product of Vectors, considered as equal to a Quaternion. 
 
 Section 3. — On a Second Method of arriving at the same In- 
 terpretation, of a Binary Product of Vectors. 
 
 283. It cannot fail to have been observed by any attentive 
 reader of the Second Book, how close and intimate a connexion'\ 
 has been found to exist, between a Right Quaternion (132), and 
 its Index, or Index- Vector (133). Thus, if u and v' denote (as 
 in 223, (1.), &c.) any two right quaternions, andif lu, Iv de- 
 note, as usual, their indices, we have already seen that 
 
 I. . . Iv' = Iv, if v'=v, and conversely (133); 
 
 IL . . l(v'±v)=^Iv'±lv (206); 
 
 111. . . Iv: lv=v:v (193); 
 
 to which may be added the more recent formula, 
 
 IV. . .EI?;=mi;(258, IX.). 
 
 284. It could not therefore have appeared strange, if we 
 had proposed to establish this new formula of the same kind, 
 
 I. . . lv',Iv = v'.v = vv, 
 as a definition (supposing that the recent definition 278 had 
 not occurred to us), whereby to interpret the product of any two 
 indices of right quaternions, as being equal to thQ product of 
 those tivo quaternions themselves. And then, to interpret the 
 product /3a, of any two given vectors, taken in a given order, 
 
 * Compare the Note to page 305. 
 t Compare the Note to page 174. 
 
CHAP. I.] SECOND INTERPRETATION OF A PRODUCT. 309 
 
 we should only have had to conceive (as we always may), that 
 the two proposed ^c^6>r5, a and j3j are the indices of two right 
 quaternions, v and v, and to multiply these latter, in the same 
 order. For thus we should have been led to establish the for- 
 mula, 
 
 II. . . j3a = vv, if u = Iv, and /3 = Iv ; 
 
 or we should have this slightly more symbolical equation, 
 
 III. . .j3a = j3.a = r^i3.Fa; 
 in which the symbols, 
 
 I'a and T^jS, 
 are understood to denote the two right quaternions, whereof 
 the two lines a and /3 are the indices. 
 
 (1.) To establish now the substantial fc?ew^zVy of these two interpretations, 278 and 
 284, of a binary product of vectors (3a, notwithstanding the difference of form of 
 the definitional equations by which they have been expressed, we have only to ob- 
 serve that it has been found, as a theorem (194), that 
 
 IV. . .v'v = It)': I (1 : i;) = Iv: IRr ; 
 but the definition (258) of Ra gave us the lately cited equation, RIu = IRv ; we have 
 therefore, by the recent formula II., the equation, 
 
 V. . . Iy'.Iy = Iw':RIt?; or VI. . . i3.a = j3 : Ra, 
 as in 278, I. ; a and /3 still denoting any two vectors. The two interpretations 
 therefore coincide, at least in their results, although they have been obtained by dif- 
 ferent processes, or suggestions, and are expressed by two different /br?wwte. 
 
 (2.) The result 279, II., respecting conjugate products of vectors, corresponds 
 thus to the result 191, (2.), or to the first formula of 223, (1.)- 
 
 (3.) The two formulae of 279, (1.) and (2.), respecting the scalar and right 
 parts of the product (3a, answer to the two other formulas of the same sub-article, 
 223, (1.), respecting the corresponding parts ofv'v. 
 
 (4.) The doubly distributive property (280), oi vector-multiplication, is on this 
 plan seen to be included in the corresponding but more general property (212), of 
 multiplication of quaternions. 
 
 (5.) By changing YVq, YVq', t, t' , and o, to a, (3, a, b, and y, in those formula) 
 of Art. 208 which are previous to its sub-articles, we should obtain, with the recent 
 definition (or interpretation) II. of (3a, several of the consequences lately given (in 
 sub-arts, to 281), as resulting from the former definition, 278, I. Thus, the equa- 
 tions, 
 
 VI., VII., VIII,, IX., X., XL, XII., XXIL, and XXIII., 
 
 of 281, correspond to, and may (with our last definition) be deduced from, the for- 
 mulaa, 
 
 v., VI., VIII., XL, XIL, XXII., XX., XIV., and XVI., XVIIL, 
 of 208. (Some of the consequences from the sub- articles to 208 have been already 
 considered, in 281, (11.) ) 
 
310 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (6.) T\\& geometrical properties of the line IV/3a, deduced from ihQ first defini- 
 tion (278) of /3a in 281, (3.) and (4.), (namely, t\xQ positive rotation round that line, 
 from /3 to a ; '\U perpendicularity to their plane ; and the representation by the same 
 line of the paralellogram under those two factors^ regard being had to units oi length 
 and of area,^ might also have been deduced from 223, (4.), by means of the second 
 definition (284), of the same product^ (3a. 
 
 Section 4. — On the Symbolical Identification of a Right Qua- 
 ternion with its own Index: and on the Construction of a 
 Product of Two Rectangular Lines, by a Third Line, rect- 
 angular to both. 
 
 285. It has been seen, then, that the recent formula 284, 
 II. or III., mag replace the formula 278, 1., as a .second definition 
 of a product of two vectors, which conducts to the same conse- 
 quences, and therefore ultimately to the same interpretation 
 of such a product, as the^r^^. Now, in the ^ecowc? formula, 
 we have interpreted that product, /3a, by changing the two fac- 
 tor-lines, a and j3, to the two right quaternions, v and v, or 
 r^a and I"^j3, of which they are the indices; and by then de- 
 fining that the sought product j3a is equal to the product v'v, 
 
 of those two right quaternions. It becomes, therefore, impor- 
 tant to inquire, at this stage, how far such substitution, of I"^a 
 for a, or of v for lu, together with the converse substitution, is 
 permitted in this Calculus, consistently with principles already 
 established. For it is evident that if such substitutions can 
 be shown to be generally legitimate, or allowable, we shall 
 thereby be enabled to enlarge greatly the existing field of inter- 
 pretation: and to treat, in «// cases, Functions of Vectors, as 
 being, at the same time. Functions of Right Quaternions. 
 
 286. We have first, by 133 (comp. 283, I.), the equality, 
 
 L..r>/3 = rx if ^=a. 
 
 In the next place, by 206 (comp. 283, II.), we have the formula of 
 addition or subtraction, 
 
 11. . . r'()3±a)=I-'^ir'a; 
 
 with these more general results of the same kind (comp. 207 
 
 and 99), 
 
 III. . . I^2a = sr'fl : IV. . . l-'2xa = 2a;r^a. 
 
CHAP. I.] RIGHT QUATERNION EQUAL TO ITS INDEX. 311 
 
 In the third place, by 193 (comp. 283, III.), we have, for division, 
 
 the formula, 
 
 V... r»/3:r'a = ^:a; 
 
 while the second definition (284) oi multiplication of vectors, which has 
 been proved to be consistent with the first definition (278), has given 
 us the analogous equation, 
 
 VI. . . I-'^.I-'a = ^.a = /3a. 
 
 It would seem, then, that we might at once proceed to define, for the 
 purpose of interpreting any proposed Function of Vectors as a Quater- 
 ternion, that the following general Equation exists : 
 
 VII. ..ria=:a; or VIII. . . I«7 = V, if V = -; 
 
 or still more briefly and symholically, if it be understood that the 
 subject of the operation I is always a right quaternion, 
 
 IX. ..1=1. 
 
 But, before finally adopting this conclusion, there is a case (or rather 
 a class of cases), which it is necessary to examine, in order to be cer- 
 tain that no contradiction to former results can ever be thereby caused. 
 287. The most general form of a vector -function, or of a vector 
 regarded as a function of other vectors and of scalars, which was 
 considered in the First Book, was the form (99, comp. 275), 
 
 1. . . p = l^xa ; 
 
 and we have seen that if we change, in this form, each vector a to the 
 corresponding right quaternion I'^a, and then take the index of the 
 new right quaternion which results, we shall thus be conducted to 
 precisely the same vector p, as that which had been otherwise ob- 
 tained before; or in symbols, that 
 
 II. . . -Ixa^l^xl-'a (comp. 286, IV.). 
 
 But another form of a vector-function has been considered in the Se- 
 cond Book ; namely, the form, 
 
 III. ..^ = ...^^a(226,III.); 
 
 in which o, /3, 7, ^, e . . . are any odd number of complanar vectors. 
 And before we accept, as general, the equation VII. or VIII. or IX. 
 of 286, we must inquire whether we are at liberty to write, under 
 the same conditions of complanarity, and with the same signification 
 of the vector p, the equation, 
 
312 ELEMENTS OF QUATERNIONS. [BOOK III. 
 
 ■—(••■S-B-'-) 
 
 288. To examine this, let there be at first only three given com- 
 planar vectors, 7|||a, /3; in which case there will always be (by 
 226) 2, fourth vector />, in the same plane, which will represent or 
 construct the function (7: /3).a; namely, thQ fourth proportional to 
 /3, 7, a. Taking then what we may call the Inverse Index- Functions^ 
 or operating on these four vectors a, y3, 7, p by the characteristic I"^ 
 we obtain/owr collinear and right quaternions (209), which may be 
 denoted by v, v'^ v'\ v'" ; and we shall have the equation, 
 
 V. . . v"'\v--{p\a=r^\^^)v"\v'\ 
 or VI. . . v'"--{v"'.v').v\ 
 
 which proves what was required. Or, more symbolically, 
 
 VII ^=^=:^=Ii)f. 
 viiL..^.a = />=i(i-V)=i(J;Jj.rH 
 
 And it is so easy to extend this reasoning to the case of any greater 
 odd number of given vectors in one plane, that we may now consi- 
 der the recent formula IV. as proved. 
 
 289. We shall therefore adopts as general^ the symbolical 
 equations VII. VIII. IX. of 286; and shall thus be enabled, 
 in a shortly subsequent Section, to interpret ternary (and other) 
 products of vectors, as well as powers and other Functions of 
 
 Vectors, as hQing generally Quaternions; although they may, 
 in particular cases, degenerate (131) into scalars, or may be- 
 come right quaternions ( 132) : in which latter event they may, 
 in virtue of the same principle, be represented by, and equated 
 to, their own indices (133), and so be treated as vectors. In 
 symbols, we shall wnte generally, for any set of vectors a, j3, 
 y, . . . and any function f the equation, 
 
 I. ../(a,p,7,...)=/a-^«»I"/3,I-^y,--) = ?, 
 q being some quaternion; while in the particular case when 
 this quaternion is right, or when 
 
 q = v=S-^0 = l-'p, 
 
CHAP. I.] PRODUCT OF TWO RECTANGULAR LINES A LINK. 313 
 
 we shall write also, and usually by preference (for that case), 
 the formula, 
 
 n. . ./(a, /3, r, . . .) = i/(i-'<«. i-'/3. i-'7> • • •) =P. 
 
 jO being a vector. 
 
 290. For example, instead of saying (as in 281) that the 
 Product of any two Rectangular Vectors is a Right Quaternion, 
 with certain properties of its Index^ already pointed out (284, 
 (6.) ), we may now say that such a product is equal to that in- 
 dex. And hence will follow the important consequence, that 
 the Product of any Two Rectangular Lines in Space is equal 
 to (or may be constructed by) a Third Line, rectangular to 
 both ; the Rotation round this Product-Line, from the Multi- 
 plier-Line to the Multiplicand' Line, being Positive : and the 
 Length of the Product being equal to the Product of the 
 Lengths of the Factors, or representing (with a suitable refe- 
 rence to units) the Area of the Rectangle under them. And 
 generally we may now, for all purposes of calculation and ex- 
 pression, identify* a Right Quaternion with its own Index. 
 
 Section 5 On some Simplifications of Notation, or of Ex- 
 pression, resulting from this Identification ; and on the Con- 
 ception of an Unit-Line as a Right Versor. 
 
 29 1 . An immediate consequence of the symbolical equa- 
 tion 286, IX., is that we may now suppress the Characteristic 
 I, of the Index of a Right Quaternion, in all the formulas into 
 which it has entered ; and so may simplify the Notation. Thus, 
 instead of writing, 
 
 Ax. q = lUV^, or Ax. = lUV, as in 204, (23.), 
 or Ax. q = JJlYq, Ax. = UIV, as in 274, (7.), 
 
 we may now Avrite simplyj, 
 
 L..Ax.^=UV^; or II. ..Ax.= UV. 
 
 The Characteristic Ax., of the Operation of taking the Axis of 
 a Quaternion (132, (6.) ), may therefore henceforth be replaced 
 
 * Compare the Notes to pages 119, 136, 174, 191, 200. 
 
 t Compare tbe first Note to page 118, and the second Note to page 200. 
 
 2 s 
 
314 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 whenever we may think fit to dispense with it, by this combina- 
 tion of two other characteristics, U and V, which are of greater 
 and more ^ew^r«/ utility, and indeed cannot'* be dispensed with, 
 in the practice of the present Calculus. 
 
 292. We are now enabled also to diminish, to some extent, 
 the number of technical terms^ which have been employed in 
 the foregoing Book. Thus, whereas we defined, in 202, that 
 the right quaternion V^ was the Right Part of the Quater- 
 nion g, or of the sum Sq + Yq, we may now, by 290, identify 
 that part with its own index-vector lYq, and so may be led to 
 call it the vector part, or simply ^Ae Vector,-}- of that Quater- 
 nion q, without henceforth speaking of the right part: although 
 the plan of exposition, adopted in the Second Book, required 
 that we should do so for some time. And thus an enuncia- 
 tion, which was put forward at an early stage of the present 
 work, namely, at the end of the First Chapter of the First 
 Book, or the assertion (17) that 
 
 ^^ Scalar plus Vector equals Quaternion" 
 
 becomes entirely intelligible, and acquires a perfectly definite 
 signification. For we are in this manner led to conceive a 
 Number (positive or negative) as being added to a Li7ie,% 
 when it is added (according to rules already established) to 
 that right quotient (132), of which the line is the Index, In 
 symbols, we are thus led to establish the formula, 
 1. . . q = a-^a, when II. . . </ = a + I'^a ; 
 
 * Of course, any one who chooses may invent new symbols^ to denote the same 
 operations on qvaternions, as those which are denoted in these Elements, and in the 
 elsewhere cited Lectures, by the letters U and V ; but, under some form, such sym- 
 bols must be used: and it appears to have been hitherto thought expedient, by other 
 writers, not hastily to innovate on notations which have been already employed in 
 several published researches, and have been found to answer their purpose. As to the 
 type used for these, and for the analogous characteristics K, S, T, that must evidently 
 be a mere affair of taste and convenience : and in fact they have all been printed 
 as small italic capitals, in some examination-papers by the author. 
 
 f Compare the Note to page 191. 
 
 X On account of this possibility of conceiving a quaternion to be the sum of a 
 number and a line, it was at one time suggested by the present author, that a Qua- 
 ternion might also be called a Grammarithm, by a combination of the two Greek 
 words, ypafifit] and dpiOfiog, vhich signify respectively a Line and a Number. 
 
CHAP. I.] CONCEPTIONOF ANUNIT-LINE ASA RIGHT VERSOR. 315 
 
 lohatever scalar^ and whatever vector, may be denoted by a 
 and a. And because either of these two parts, or summands, 
 may vanish separately, we are entitled to say, that both Sca- 
 lars and Vectors, or Numbers and Lines, are included in the 
 Conception of a Quaternion, as now enlarged or modified. 
 
 293. Again, the same symbolical identification of Iv with 
 v (286, VIII.) leads to the forming of a new conception of an 
 Unit-Line, or Unit-Vector (129), as being also a Riyht Versor 
 (153) ; or an Operator, of which the effect is to tur7i a line, in 
 a plane perpendicular to itself, through a positive quadrant of 
 rotation : and thereby to oblige the Operand-Line to take a 
 neiv direction^ at right angles to its old direction, but without 
 any change of length. And then the remarks (154) on the 
 equation q'^==-\, where q was a right versor in \\iQ former 
 sense (which is still a permitted one) of its being a right ra- 
 dial quotient (147), or the quotient of two equally long hut mu- 
 tually rectangular lines, become immediately applicable to the 
 interpretation of the equation, 
 
 10^ = - I, or ^2 + 1 ^ (282, XIV.) ; 
 where p is still an unit-vector, 
 
 (1.) Thus (comp. Fig. 41), if a be any line perpendicular to such a vector p, 
 we have the equations, 
 
 I. ..pa = |8; II. . . |02a = p/3 = a'=-a; 
 
 j8 being another line perpendicular to jO, which is, at the same time, at right angles 
 to a, and of the same length with it ; and from which a third line a\ or — a, oppo- 
 site to the line a, but still equally long, is formed by a repetition of the operation, 
 denoted by (what we may here call) the characteristic p ; or having that unit-vec- 
 tor p for the operator, or instrument employed, as a sort oi handle, or axis* of ro- 
 tation. 
 
 (2.) More generally (comp. 290), if a, (3, y beany three lines at right angles to 
 each other, and if the length of y be numerically equal to the product of the lengths 
 of a and (3, then (by what precedes) the line y represents, or constructs, or is equal 
 to, the product of the two other lines, at least if a certain order of the factors 
 (comp. 279) be observed: so that we may write the equation (comp. 281, XXI.), 
 
 III. ..a/3 = y, if IV. . . /8 -J- a, y J- rt, y -i- f3, and V. . . Ta, T/3 = Ty, 
 
 * Compare the first Note to page 136. 
 
316 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 provided that the rotation round a, from /3 to y, or that round y from a to (3, &c., 
 has the direction taken as the positive one. 
 
 (3.) In this more general case, we may still conceive that the multiplier- line 
 a has operated on the multiplicand-line j3, so as to produce (or generate} the pro- 
 duct-line y ; hut not now by an operation of version alone, since the tensor of j3 is 
 (generally) multiplied by that of a, in order to form, by V., the tensor of the pro- 
 duct y. 
 
 (4.) And if (comp. Fig. 41, his, in which a was first changed to (3, and then to 
 
 a') we repeat this compound operation, of tension and version combined (comp. 189), 
 
 or if we multiply again hy a, we obtain a, fourth line (3', in the plane of /3, y, but 
 
 with a direction opposite to that of /3, and with a length generally different : namely 
 
 the line, 
 
 VI. . . ay=aaP = a'^j3=^' = - a"^^, if a = Ta. 
 
 (5.) The operator a^, or aa, is therefore equivalent, in its effect on (3, to the ne- 
 gative scalar, — a?, or — (Ta)2, or — Na, considered as a coefficient, or as a (scalar) 
 multiplier (15) : whence the equation, 
 
 VII. .. a2 = -Na(282, L), 
 
 may be again deduced, but now with a new interpretation, which is, however, as we 
 see, completely consistent, in all its consequences, with the one first proposed (282). 
 
 Section 6. — (^w the Interpretation of a Product of Three or 
 more Vectors, as a Quaternion. 
 
 294. There is now no difficulty in interpreting a ternary 
 product of vectors (comp. 277, I.), or a product of more vec- 
 tors than three, taken always in some piven order ; namely, as 
 the result (289, I.) of the substitution of the corresponding 
 right quaternions in that product: which result is generally 
 what we have lately called (276) an Oblique Quotient, or a 
 Quaternion with either an acute or an obtuse angle (130) ; but 
 maj degenerate (131) into a scalar, or may become itself a 
 right quaternion (132), and so be constructed (289, II.) by a 
 new vector. It follows (comp. 28 1), that Multiplication of Vec- 
 tors, like that of Quatetmions (223), in which indeed we now 
 see that it is included, is an Associative Operation: or that 
 we may write generally (comp. 223, II.), for ang three vec- 
 tors, a, j3, 7, the Formula, 
 
 I. . . yj5a = y »(5a. 
 
 (1.) The formulae 223, III. and IV., are now replaced by the following : 
 II. . . V.yV/3a = aS/3y -/3Sya; 
 III. . . Vy/3a= aS)3y ~^Sya + ySa/3 ; 
 
CHAP. I.] TERNARY PRODUCTS OF VECTORS. 317 
 
 in which Yy (3a is written, for simplicity, instead ofV(yj3a), or V. yj3a; and with 
 which, as with the earlier equations referred to, a student of this Calculus will find 
 it useful to render himself verj/ familiar. 
 
 (2.) Another useful form of the equation II. is the following : 
 
 IV. . . V(Va/5.y) = aS/3y-/3Sya. 
 
 (3.) The equations IX. X. XIV. of 223 enable us now to write, for any three 
 vectors, the formula : 
 
 V. . . Sy/Sa = - Sa|3y = Say (3 = - S/3ya = S(3ay = - Sya/3 
 = + volume of parallelepiped under a, /3, y, 
 = + 6 X volume of pyramid oabc ; 
 
 upper or lower signs being taken, according as the rotation round a from /3 to y is 
 positive or negative : or in other words, the scalar Sy/3a, of the ternary product of 
 vectors yj3a, being positive in the first case, but negative in the second. 
 
 (4.) The condition of complanarity of three vectors, a, (3, y, is therefore ex- 
 pressed by the equation (comp. 223, XI.) : 
 
 VI. ..Sy/3a = 0; or VI'. . . Sa/3y = ; &c. 
 
 (5.) If a, (3, y be any three vectors, complanar or diplanar, the expression, 
 
 VII. .. 5 = aS/3y-/3Sya, 
 
 gives ^VIII. . . Sy5=0, and IX. . . Sa/3^ = 0; 
 
 it represents therefore (comp. II. and IV.) a. fourth vector S, which is perpendicular 
 
 to y, but complanar with a and (3: or in symbols, 
 
 X. ..^_Ly, and XL . . d \\\ a, (3. 
 (Compare the notations 123, 129.) 
 
 (6.) For any four vectors, we have by II. and IV. the transformations, 
 
 XII. . . V(Vaj3 . Vy5) = dSa^y - ySa[3d ; 
 XIII. . . V (Ya(3 . Vy ^) = asl3yd- /3Say 5 ; 
 
 and each of these three equivalent expressions represents a. fifth vector t, which is at 
 once complanar with a, (3, and with y, ^; or a line oe, which is in the intersection 
 of the two planes, OAB and ocd. 
 
 (7.) Comparing them, we see that any arbitrary vector p may be expressed as 
 a linear function of any three given diplanar vectors, a, (3, y, by the formula : 
 
 XIV. . . pSajSy = aS(3yp + /SSyap + ySa/3p ; 
 which is found to be one of extensive utility. 
 
 (8.) Another very useful formula, of the same kind, is the following: 
 XV. . . pSa/3y=V/3y.Sap+Vya.S/3p+Vaj3.Sy|0; 
 in the second member of which, the points may be omitted. 
 
 (9.) One mode of proving the correctness of this last formula XV., is to operate 
 on both members of it, by the three symbols, or characteristics of operation, 
 
 XVI. ..S. a, S./3, S.y; 
 the common results on both sides being respectively the three scalar products, 
 
 XVII. . .Sap. Sa(3y, S(3p . Sa(3y, Syp . Sa/3y ; 
 where again the points may be omitted. 
 
318 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (10.) We here employ the principle, that if the three vectors a, /3, y he actual 
 and diplanar, then no actual vector \ can satisfy at once the three scalar equations^ 
 
 XVIII. . . SaX = 0, S/3X = 0, SyX = ; 
 because it cannot he perpendicular at once to those three diplanar vectors. 
 
 (11.) If, then, in any investigation with quaternions, we meet a system of this 
 form XVIII., we can at once infer that 
 
 XIX. ..X = 0, if XX. . . Sa;3y^0; 
 
 while, conversely, if X he an actual vector, then a, /3, y must be complanar vectors, 
 or Sa/3y = 0, as in VI'. 
 
 (12.) Hence also, under the same condition XX., the three scalar equations, 
 XXI. . . SaX = Saju, S/3X = S/3/«, SyX = Sy/z, 
 give XXII. . . X = /ii. 
 
 (13.) Operating (comp. (9.)) on the equation XV. by the symbol, or charac- 
 teristic, S . ^, in which d is any new vector, we find a result which may be written 
 thus (with or without the points) : 
 
 XXIII. . . = Sap . S/3y^ - S/3p . Sy ^a + Syp . S^a/3 - S^p . Sa/3y ; 
 
 where a, /3, y, ^, p may denote any five vectors. 
 
 (14.) In drawing this last inference, we assume that the equation XV. holds 
 good, even when the three vectors a, /3, y are complanar : which in fact must be true, 
 as a limit, since the equation has been proved, by (9.) and (12.), to be valid, if y be 
 ever so little out of the plane of a and /3. 
 
 (15.) We have therefore this new formula : 
 
 XXIV. . . V/3y Sap + Vy a S,3p + Va/3Syp = 0, if Sa/3y = ; 
 in which p may denote any fourth vector, whether in, or out of, the common plane 
 of a, /3, y. 
 
 (16.) If p ha perpendicular to that plane, the last formula is evidently true, each 
 term of the first member vanishing separate!}^, by 281, (7.) ; and if we change p to 
 a vector d in the plane of a, /3, y, we are conducted to the following equation, as an 
 interpretation of the same formula XXIV., which expresses a known theorem of 
 plane trigonometry, including several others under it ; 
 
 XXV. . . sin Boc . cos aod + sin coa . cos bod + sin aob . cos cod = 0, 
 for any four complanar and co-initial lines, OA, OB, oc, OD. 
 
 (17.) By passing from od to a line perpendicular thereto, but in their common 
 plane, we have this other known* equation : 
 
 XXVI. . . sin BOC sin aod + sin coA sin bod + sin aob sin cod = ; 
 
 which, like the former, admits of many transformations, but is only mentioned here 
 as offering itself naturally to our notice, when we seek to interpret the formula 
 XXIV. obtained as above by quaternions. 
 
 (18.) Operating on that formula by S.^, and changing p to c, we have this new 
 equation : 
 
 * Compare page 20 of the Oeometrie Snperieure of M. Chasles. 
 
CHAP. 1.] ELIMINATION OF A VECTOR. 319 
 
 XXVII. . . = SaeSfiyd + Si^eSyaS + SyeBal3d, if Sa/3y = ; 
 which might indeed have been at once deduced from XXIII. 
 
 (19.) The equation XIV., as well as XV., must hold good at the limit, when o, 
 /3, y are complanar ; hence 
 
 XXVIII. . . aS/3yp + (3Syap + ySafSp = 0, if Sa(3y = 0. 
 (20.) This last formula is evidently true, by (4.), if p be in the common plane 
 of the three other vectors ; and if we suppose it to be perpendicular to that plane, 
 so that 
 
 XXIX. . . p II Yj3y 11 Yya \\ YajS, 
 
 and therefore, by 281, (9.), since S (S/Sy. p) = 0, 
 
 XXX. . . S/3yp = S(V/3y.p) = V/3y.p, &c., 
 we may divide each term hy p, and so obtain this other formula, 
 
 XXXI. . . aV/3y + /3Vya + y VajS = 0, if Sa^Sy = 0. 
 (21.) In general, the vec/or (292) of this last expression vanishes by II. ; the 
 expression is therefore equal to its own scalar, and we may write, 
 
 XXXTI. . . aV/3y + /3Vya + y Va/3 = 3Sa/3y, 
 whatever three vectors may be denoted by a, /3, y. 
 
 (22.) For the case of complanar ity, if we suppose that the three vectors are 
 equally long, we have the proportion, 
 
 XXXIII. . . V]3y : Vya : Va/3 = sin boc : sin COA : sin aob ; 
 and the formula XXXI. becomes thus, 
 
 XXXIV. . . OA . sin BOC 4- ob . sin coa + oc . sin aob = ; 
 where oa, ob, oc are any three radii of one circle, and the equation is interpreted as 
 in Articles 10, 11, &c. 
 
 (23.) The equation XXIII. might have been deduced from XIV., instead of 
 XV., by first operating with S.^, and then interchanging d and p. 
 
 (24.) A vector p may in general be considered (221) as depending on three sca- 
 lars (the co-ordinates of its term) ; it cannot then be determined hy fewer than three 
 scalar equations ; nor can it be eliminated between /ewer than four. 
 
 (25.) As an example of such determination of a vector, let a, P, y be again any 
 three given and diplanar vectors ; and let the three given equations be, 
 
 XXXV. . . Sap = a, S/3p = ^ Syp = c; 
 in which a, b, c are supposed to denote three given scalars. Then the sought vector 
 p has for its expression, by XV., 
 
 ^XXXVI. . . p = e-i(aV/3y + 6Vya + cVa/3), if XXXVII. . . e = Sa^y. 
 (26.) As another example, let the three equations be, 
 
 XXXVIII. . . S/3yp = a, Syap = 6', Sa/3p = c ; 
 then, with the same signification of the scalar e, we have, by XIV, 
 XXXIX. . . p = e-i (aa + 6'/3 + c'y). 
 
 (27.) As an example of elimination of a vector, let there be the four scalar 
 
 equations, 
 
 XL. ..Sap=a, S/3p = &, Syp=r, S^p = d; 
 
320 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 then, by XXIII., we have this resulting equation^ into which p does not enter, but 
 only the /o«r vectors, a . . d, and the^wr scalars, a ..d: 
 
 XLI. . . a . SjSyd -b.SySa+c. SSafi -d. Sa/3y = 0. 
 (28.) This last equation may therefore be considered as the condition of concur- 
 rence of the four planes, represented by the four scalar equations XL., in one com- 
 mon point; for, although it has not been expressly stated before, it follows evidently 
 from thQ definition 278 of a binary product of vectors, combined with 196, (5.), 
 that every scalar equation of the linear form (comp. 282, XVIII.), 
 
 XLII. . . Sap = a, or Spa = a, 
 in which a = OA, and p = op, as usual, represents a plane locus of the point P ; the 
 vector of the foot s, of i\xQ perpendicular on that plane from the origin, being 
 XLIIL . . OS = <T=aRa = aa-i (282, XXL). 
 (29.) If we conceive a pyramidal volume (68) as having an algebraical (or sca- 
 lar^ character, so as to be capable of bearing either a positive or a negative ratio to 
 the volume of a given pyramid, with a given order of its points, we may then omit 
 the ambiguous sign, in the last expression (3.) for the scalar of a ternary product of 
 vectors : and so may write, generally, oabc denoting such a volume, tbe formula, 
 
 XLIV. . . Sa/3y = 6 . OABC, 
 = a positive or a negative scalar, according as the rotation round OA from ob to oc is 
 negative or positive. 
 
 (30.) More generally, changing o to d, and oa or a to a - d, &c., we have thus 
 the formula : 
 
 XLV. . . 6 . DABC = S(a - ^) (i3 - 5) (y - ^) = Sa/3y - S(3yd + Sy^a - S^a/3 ; 
 
 in which it may be observed, that the expression is changed to its own opposite, or 
 negative, or is multiplied by — 1, when any two of the four vectors, a, (3, y, d, or when 
 any two of the four points, A, B, c, D, change places with each other; and therefore 
 is restored to its former value, by a second such binary interchange. 
 
 (31.) Denoting then the new origin of a, (3, y, d by E, we have first, by XLIV., 
 XLV., the equation, 
 
 XLVI. . . DABC = EABC — EBCD + ECDA — EDAB ; 
 
 and may then write the result (comp. 68) under the more symmetric form (because 
 
 — EBCD = BECD = &C.) : 
 
 XLVII. . . BCDE 4 CDEA + DEAB + EABC + ABCD = ; 
 
 in which A, B, c, d, e may denote any five points of space. 
 
 (32.) And an analogous formula (69, III.) of the First Book, for any six points 
 OABCDE, namely the equation (comp. 65, 70), 
 
 XL VIII. . . OA.BCDE + OB.CDEA+ OC. DEAB + OD. EABC + OB. ABCD = 0, 
 
 in which the additions are performed according to the rules of vectors, the volumes 
 being treated as scalar coefficients, is easily recovered from the foregoing principles 
 and results. In fact, by XLVII., this last formula may be written as 
 
 XLIX. . . ED. EABC = EA . EBCD + EB . ECAD + EC, EABD ; 
 
 or, substituting a, ft, y, S for ea, eb, ec, ed, as 
 
CHAP. I.] STANDARD TRINOMIAL FORM FOR A VECTOR. 321 
 
 L. . . SSaBy = aSjSyd + (5Syad + ySa/3^ ; 
 
 which is only another form of XIY., and onght to hQ familiar to the student. 
 
 (33.) The formula 69, II. may be deduced from XXXI., by observing that, when 
 
 the three vectors a, j8, y are complanar, we have the proportion, 
 
 LI. . . Y(3y : Vya : Ya(S : V (/3y + ya + a/3) = OBC : oca : oab : abc, 
 
 i{ signs {or algebraic ox scalar ratios) of areas be attended to (28, 63); and the 
 
 formula 69, I., for the case of three collinear points A, b, c, may now be written as 
 
 follows : 
 
 LII. . . a (|3 - y) + /3 (y - a) 4- y (a - /3) = 2 V(iSy + y a + a/3) 
 
 = 2V(/S-a)(y-a) = 0. 
 
 if the three coinitial vectors a, /3, y be termino-collinear (24). 
 
 (84.) The case when four coinitial vectors a, (3, y, d are termino-complanar (64)^ 
 
 or when they terminate in /owr complanar points A, b, c, p, is expressed by equating 
 
 to zero the second or the third member of the formula XLV. 
 
 (35 ) Finally, for ternary products of vectors in general, we have the formula: 
 
 LIIL . . a2/32y2 + (Sai8y)2 = (Va/3y)2 = (aS;8y - /3Sya + ySa/3)2 
 
 = a? (S/3y)2 + /33 (Sya)2 + y2 (Sa/3)2 - 2S|3y Sya Sa/3. 
 
 295. The identity (290) of a right quaternion with its in- 
 dex, and the conception (293) of an unit-line as a ?'/^A^ versor, 
 allow us now to treat the three important versors, i,j, k, as 
 constructed by, and even as (in our present view) identical 
 with, their own axes ; or with the three lines ox, oj, ok of 181, 
 considered as being each a certain instrument, or operator, or 
 agent in a right rotation (293, (1.) ), which causes any line, in 
 a plane perpendicular to itself, to turn in that plane, through 
 a positive quadrant, without any change of its length. With 
 this conception, or construction, the Laxcs of the Symbols ijk 
 are still included in the Fundamental Formula of 183, namely, 
 i^=f = k'^=ijk = - 1; (A) 
 
 and if we now, in conformity with the same conception, transfer 
 the Standard Trinomial Form (221) from Right Quaternions 
 to Vectors, so as to write generally an expression of the form, 
 
 I. . , p =ix +jy + kz, or T. . . a = ia +jb + he, &c., 
 where xyz and abc are scalars (namely, rectangular co-ordi- 
 nates), w^e can recover many of the foregoing results with ease : 
 and can, if we think fit, connect them with co-ordinates, 
 
 (1.) As to the laws (182), included in the Fundamental Formula A, the law 
 j2 __ 1^ &c., may be interpreted on the plan of 293, (1.), as representing the rever- 
 sal which results from two successive quadrantal rotations. 
 
 2 T 
 
322 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (2.) The two contrasted laws, or formulag, 
 
 ij = Jrk, ji = - k, (182, II. and III.) 
 
 may now be interpreted as expressing, that although a positive rotation through a 
 right angle, round the line i as an axis, brings a revolving line from the position j to 
 the position k, or + k, yet, on the contrary, a positive quadrantal rotation round the 
 line j, as a new axis, brings a new revolving line from a new initial position, i, to a 
 new final position, denoted by — k, or opposite* to the old final position, + k. 
 
 (3.) Finally, the law ijk = — 1 (183) may be interpreted by conceiving, that we 
 operate on a line a, which has at first the direction of +j, by the three lines, k,j, i, 
 in succession ; which gives three new but equally long lines, (3, y, d, in the direc- 
 tions of - i, + k, —j, and so conducts at last to a line — a, which has a direction op- 
 posite to the initial one. 
 
 (4.) The foregoing laws of ijk, which are all (as has been said) included (184) 
 in the Formula A, when combined with the recent expression I. for p, give (comp. 
 222, (1.) ) for the square of that vector the value : 
 
 1 1. . . p2 = (ia; + j> + Ar)« = - (.r2 + y' + 22) ; 
 this square of the line p is therefore equal to the negative of the square of its length 
 Tp (185), or to the negative of its norm Np (273), which agrees with the former 
 resultf 282, (1.) or (2.). 
 
 (5.) The condition of perpendicularity of the two lines p and a, when they are 
 represented by the two trinomials I. and I'., may be expressed (281, XVIII.) by the 
 fonnula, 
 
 III. . . = Sap = -(^ax + bt/+ cz) ; 
 
 which agrees with a well-known theorem of rectangular co-ordinates. 
 
 (6.) The condition of complanarity of three lines, p, p', p", represented by the 
 trinomial forms, 
 
 IV. . . p = ix +jy + kz, p' = ix' + &c., p" = ix" + &c., 
 is (by 294, VI.) expressed by the formula (comp. 223, XIII.), 
 
 V. . . = Sp'V'p = x" (z'y - y'z) + y'\x'z - zx') + z'ijy'x - x'y) ; 
 
 agreeing again with known results. 
 
 (7.) "When the three lines p, p', p", or op, op', op", are not in one plane, the 
 recent expression for Sp"p'p gives, by 294, (3.), the volume of the parallelepiped 
 
 * In the Lectures, the three rectangular unit-lines, i, j, k, were supposed (in 
 order to fix the conceptions, and with a reference to northern latitudes) to be directed, 
 respectively, towards the south, the west, and the zenith ; and then the contrast of 
 the two formulae, ij = -\- k,ji = — k, came to be illustrated by conceiving, that we at 
 one time turn a moveable line, which is at first directed westward, round an axis 
 (or handle) directed towards the south, with a right-handed (or screwing) motion, 
 through a right angle, which causes the line to take an upward position, as its fnal 
 one ; and that at another time we operate, in a precisely similar manner, on a line 
 directed at first southward, with an axis directed to the west, which obliges this new 
 line to take finally a downward (instead of, as before, an upward) direction. 
 
 t Compare also 222, IV. 
 
CHAP. I.] PRODUCT OF ANY NUMBER OF VECTORS. 323 
 
 (comp. 223, (9.) ) of which they are edges ; and this volume, thus expressed, is a 
 positive or a negative scalar, according as the rotation round p from p' to p" is itself 
 positive or negative : that is, according as it has the same direction as that round 
 + X from +y to +z (or round i from j to k), or the direction opposite thereto. 
 
 (8.) It may be noticed here (comp. 223, (13.) ), that if a, (3, y be ang three 
 vectors, then (by 294, III. and V.) we have : 
 
 VI. . . SaySy = - 8y(3a = i (a/^y - yfta) ; 
 VII. . . Va/3y = + V7|3a = |(a/3y + y|8a). 
 (9.) More generally (comp. 223, (12.) ), since a vector, considered as represent- 
 ing a right quaternion (290), is always (by 144) the opposite of its own conjugate, so 
 that we have the important formula, * 
 
 VIII. . . Ka = - a, and therefore IX. . . KTIa = + Wa, 
 we may write for ang number of vectors, the transformations, 
 
 X. . . sna = + sn'a=Kri«±n'a), 
 XI. . . vna = + vn'a = |(na +n'a), 
 
 upper or lower signs being taken, according as that number is even or odd : it being 
 understood that 
 
 XII. . . n'a = ...yj3a, if Ua = a(3y... 
 
 (10.) The relations of rectangularity, 
 
 XIII. . . Ax. i-i- Ax.j; Ax.y -i- Ax. A ; Ax. A 4- Ax. i, 
 
 which result at once from the definitions (181), may now be written more briefly, as 
 follows : 
 
 XIV. . . i-i-y-, j-i-k, A-i-i; 
 
 and similarly in other cases, where the axes, or the planes, of any two right quater- 
 nions are at right angles to each other. 
 
 (11.) But, with the notations of the Second Book, we might also have writtten, 
 by 123, 181, such formulae oi complanarity as the following, Ax.^ \\\i, to express 
 (comp. 225) that the axis of j was a line in the plane of i ; and it might cause some 
 confusion, if we were now to abridge that formula tojT ||| i. In general, it seems 
 convenient that we should not henceforth employ the sign \\\, except as connecting 
 either symbols of three lines, considered still as complanar ; or else symbols of three 
 right quaternions, considered as being collinear (209), because their indices (or axes') 
 are complanar : or finally, any two complanar quaternions (123). 
 
 (12.) On the other hand, no inconvenience will result, if we now insert the sign of 
 parallelism, between the symbols of two right quaternions which are, in the former 
 sense (123), complanar : for example, we may write, on our present plan, 
 
 XY...xi\\i, yjWj, zk\\k, 
 if xyz be any three scalars. 
 
 * If, in like manner, we interpret, on our present plan, the symbols Ua, Ta, Na 
 as equivalent to Ul"ia, Tl'a, NI''a, we are reconducted (compare the Notes to 
 page 136) to the same significations of those symbols as before (155, 185, 273) ; and 
 it is evident that on the same plan we have now, 
 Sa = 0, Va = a. 
 
324 ELEMENTS OF QUATERNIONS. [boOK III. 
 
 296. There are a few particular but remarkable cases^ of ternary 
 and oihQx products of vectors^ which it may be well to mention here, 
 and of which some may be worth a student's while to remember: 
 especially as regards the products of successive sides of closed polygons ^ 
 inscribed in circles, or in spheres. 
 
 (1.) If A, B, c, D be any four concircular points, we know, by the sub-articles to 
 260, that their anharmonic function (abcd), as defined in 259, (9.), \s scalar; being 
 a\m positive or negative, according to a law of arrangement of those four points, 
 which has been already stated. 
 
 (2.) But, by that definition, and by the scalar (though negative) character of the 
 square of a vector (282), we have generally, for any plane or gauche quadrilateral 
 ABCD, the formula : 
 
 I. . . e2(ABCD) = AB.BC.CD.DA= </ie continued product of the four sides; 
 in which the coefficient e^ is a positive scalar, namely the product of two negative 
 or of two positive squares, as follows : 
 
 II. . . e3 = BC2 . DA2 = BC2. DA^ > 0. 
 
 (3.) If then abcd be deplane and inscribed quadrilateral, we have, by 260, (8.), 
 the formula, 
 
 III. . . ab.bc.cd.da = a positive or negative scalar, 
 
 according as this quadrilateral in a circle is a crossed or an uncrossed one. 
 
 (4.) The product a(3y of any three complanar vectors is a vector, because its 
 scalar part Sa(3y vanishes, by 294, (3.) and (4.); and if the factors be three suc- 
 cessive sides AB, BC, CD of a quadrilateral thus inscribed in a circle, their product has 
 either the direction of the fourth successive side, DA, or else the opposite direction, 
 
 or in symbols, 
 
 IV. . . AB.BC.CD : DA > or < 0, 
 
 according as the quadrilateral abcd is an uncrossed or a crossed one. 
 
 (5.) By conceiving the fourth point d to approach, continuously and indefinitely, 
 to the first point A, we find that the product of the 
 
 three successive sides of any plane triangle, abc, is /""'^ ^^\C 
 
 given by an equation of the form : / ^--'""''"^iX 
 
 V. . . AB . BC . CA = AT ; -^Lc^::^— —— -4p 
 
 at being a line (comp. Fig. 63) which touches the \ \ / /' / 
 
 circumscribed circle, or (more fully) which touches \ \ //'V,/^ 
 
 the segment ABC of that circle, at the point A ; or re- \,.J\^^^> ;^X'^ 
 
 presents the initial direction of motion, along the cir- ^ IJ A 
 
 cumference, from A through B to C : while the length ^^S- ^^• 
 
 of this tangential product-line, AT, is equal to, or 
 
 represents, with the usual reference to an unit of length, the product of the lengths 
 
 of the three sides, of the same inscribed triangle abc 
 
 (6.) Conversely, if this theorem respecting the product of the sides of an inscribed 
 triangle be supposed to have been otherwise proved, and if it be remembered, then 
 since it will give in like manner the equation, 
 
CHAP. I.] PRODUCTS OF SIDES OF INSCRIBED POLYGONS. 325 
 
 A 
 Fig. 63, bis. 
 
 VI. . . AC.CD.DA=AU, 
 
 if D be any fourth pointy concircular with A, B, c, -while AU is, as in the annexed 
 
 Figures 63, a tangent to the new segment ACD, we can 
 
 recover easily the theorem (3.), respecting the product j. 
 
 of the sides of an inscribed quadrilateral ; and thence 
 
 can return to the corresponding theorem (260, (8.) ), 
 
 respecting the anharmonic function of any such figure gl 
 
 abcd: for we shall thus have, by V. and VI., the 
 
 equation, 
 
 VII. . . AB.BC.CD.DA= (at. Au) : (CA.Ac), 
 
 in which the divisor CA, AC or N. Ao, or Jc^ is always 
 positive (282, (1.) ), but the dividend at. AU is nega- 
 tive (281, (9.)) for the case of an ttwcrosse<i quadrilateral (Fig. 63), being on the 
 contrary posiiife for the other case of a crossed one (Fig. 63, bis), 
 
 (7.) If P be any point on the circle through a given point A, which touches at a 
 given origin o a given line OT = r, as represented in Fig. 64, we shall then have by 
 (5.) an equation of the form, 
 
 VIII. . . OA.AP.PO = a;.OT, 
 in which x is some scalar coetficient, which 
 varies with the position of p. Making then 
 OA= a, and op= p, as usual, we shall have 
 
 IX. . . a(p — a)p = ~ XT, 
 or 
 
 IX'. . . p-^ - a-^ = XT : a^p^, 
 or 
 
 IX". . . Vrp-i = Vra-i ; 
 
 and any one of these may be considered as a 'S* 
 
 form of the equation of the circle, determined by the given conditions. 
 
 (8.) Geometrically, the last formula IX." expresses, that the line p-i-a-\ or 
 Kp - Ra, or a'p' (see again Fig. 64), if oa' = a"' = Ra = R. OA, and op' = p-i = R. op, 
 is parallel to the given tangent t at o ', which agrees with Fig. 58, and with Art. 
 260. 
 
 (9.) If B be the point opposite to o upon the circle, then the diameter ob, or (3, 
 as being J- r, so that t(3-^ is a vector, is given by the formula, 
 
 X. . . rj3-i = Vra-i ; or X'. . . )3 = - r : Vra'i; 
 
 in which the tangent r admits, as it ought to do, of being multiplied by any scalar, 
 without the value of /3 being changed, 
 
 (10.) As another verification, the last formula gives, 
 
 XI. . . OB = T^ = Ta : TVUra"! = OA : sin act. 
 
 (11.) If a quadrilateral oabc be not inscriptihle in a circle, then, whether it be 
 plane or gauche^ we can always circumscribe (as in Fig. 65) two circles, cab and obc 
 about the two triangles, formed by drawing the diagonal OB; and then, on the plan 
 of (6.), we can draw two tangents or, ou, to the two segments CAB, obc, so as to repre- 
 sent the two ternary products. 
 
326 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 OA.AB.BO, and ob.bc.co; 
 after which we shall have the quaternary product^ 
 
 XII. . . OA.AB.BC.CO = OT.OU : 0B« ; 
 
 where the divisor, oB^, or bo . ob, or N . ob, is a 
 positive scalar, but the dividend OT.ov, and there- 
 fore also the quotient in the second member, or the 
 product in the first member, is a quaternion. 
 
 (12.) The axis of this quaternion is perpen- 
 dicular to the plane Tou of the two tangents ; and 
 therefore to the plane itself of the quadrilateral 
 oabc, if that be a plane figure ; but if it be gauche, 
 then the axis is normal to the circumscribed sphere 
 
 at the point o : being also in all cases such, that the rotation round it, from ox to 
 OU, is positive. 
 
 (13.) The angle of the same quaternion is the supplement of the angle tou be- 
 tween the two tangents above mentioned ; it is therefore equal to the angle u'ot, if 
 ou' touch the new segment ocb, or proceed in a new and opposite direction from o 
 (see again Fig. 65) ; it may therefore be said to be the angle between the two arcs, 
 oab and ocb, along which a point should move, in order to go from o, on the two 
 circumferences, to the opposite corner b of the quadrilateral OABO, through the two 
 other corners, A and c, respectively : or the angle between the arcs ocb, oab. 
 
 (14.) These results, respecting the axis and angle of the product of the four suc- 
 cessive sides, of any quadrilateral oabc, or abcd, apply without any modification to 
 the anharmonic quaternion (259, (9.)) of the same quadrilateral; and although, 
 for the case of a quadrilateral in a circle, the axis becomes indeterminate, because 
 the quaternary product and the anharmonic function degenerate together into sca- 
 lars, or because the figure may then be conceived to be inscribed, in indefinitely many 
 spheres, yet the angle may still be determined by the same rule as in the general 
 case : this angle being ■= tt, for the inscribed and uncrossed quadrilateral (Fig. 63) ; 
 but =0, for the inscribed and crossed one (Fig. 63, bis). 
 
 (15.) For the gauche quadrilateral oabc, which may ahvays be conceived to be 
 inscribed in a determined sphere, we may say, by (13.), that the angle of the qua- 
 ternion product, /. (oA. AB.BC.co), is equal to the angle of the lunule, bounded 
 (generally) by the two arcs of small circles oab, ocb ; with the same construction 
 for the equal angle of the anharmonic^ 
 
 L (oabc), or L (oa : ab. bc : co). 
 
 (16.) It is evident that the general principle 223, (10.), of the permissibility of 
 cyclical permutation of quaternion factors under the sign S, must hold good for 
 the case when those quaternions degenerate (294) into vectors ; and it is still more 
 obvious, that every permutation of factors is allowed, under the sign T : whence 
 cyclical permutation is again allowed, under this other sign SU ; and consequently 
 also (comp. 196, XVI.) under the sign L. 
 
 (17.) Hence generally, for any four vectors, we have the three equations, 
 XIII. . . SajSy^ = SiSy^a ; XIV. . . SUa/Syo = SU/3y^a ; 
 XV. . . Z. a/3y^ = L (3ySa ; 
 
CHAP. I.] PENTAGON IN A SPHERE. 327 
 
 and in particular, for the successive sides of any plane or gauche quadrilateral abcd, 
 we have ih.efour equal angles^ 
 
 XVI. . . L (ab . bc . CD . da) = Z. (bc . CD . da . ab) =r &c. ; 
 with the corresponding equality of the angles of the four anharmonics, 
 XVII. . . L (abcd) = L (bcda) = L (cdab) = L (dabc) ; 
 or of those of the four reciprocal anharmonics (259, XVII.), 
 
 XVII'. . . L (adcb) = L (badc) = L (cbad) = L (dcba). 
 *■ (18.) Interpreting now, by (13.) and (15.), these last equations, we derive from 
 them the following theorem, for the plane, or for space : — 
 
 Let abcd be any four points, connected hy four circles, each 
 passing through three of the points : then, not only is the angle 
 at A, between the arcs abc, adc, equal to the angle at c, be- 
 tween CDA and cba, but also it is equal (comp. Fig. 66) to the 
 angle at B, between the two other arcs BCD and bad, and to 
 the angle at D, between the arcs dab, dcb. 
 
 (19.) Again, let abode be any pentagon, inscribed in a 
 sphere ; and conceive that the two diagonals AC, ad are drawn. 
 We shall then have three equations, of the forms, 
 
 XVIII. . . ab.bc.ca = at; ac.cd.da = au; 
 
 AD.DE.EA=AV; 
 
 where at, au, av are three tangents to the sphere at a, so that their product is a 
 fourth tangent at that point. But the equations XVIII. give 
 
 XIX. - . AB.BC . CD . DE . EA = (at . AU . Av) : (ac^ . AD^) 
 
 = AW = a new vector, which touches the sphere at A. 
 
 We have therefore this Theorem, which includes several others'under it :-^ 
 
 " The product of the five successive sides, of any {generally gauche) pentagon 
 
 inscribed in a sphere, is equal to a tangential vector, drawn from the point at which 
 
 the pentagon begins and ends^ 
 
 (20.) Let then p be a point on the sphere which passes through o, and through 
 
 three given points A, b, c ; we shall have the equation, 
 
 XX. .. = S(oA.AB.BC.CP.Po) = Sa(|3-a)(y-/8) (p-y)(_p) 
 = a2S)3yp + /32Syap + y^^a^p - p2Sa/3y. 
 
 (21.) Comparing with 294, XIV., we see that the condition for the four co-ini- 
 tial vectors a, (3, y, p thus terminating on one spheric surface, which passes through 
 their common origin o, may be thus expressed : 
 
 XXL . .if p = xa+yj3 + zy, then p^ = xa^ + y(3^ + zy^. 
 
 (22.) If then y^e project (comp. 62) the variable point p into points a', b\ c' on 
 the three given chords OA, OB, oc, by three planes through that point p, respectively 
 parallel to the planes BOC, COA, aob, we shall have the equation : 
 
 XXII. . . op2 = OA . oa' 4- OB . ob' + oc . oc\ 
 
 (23.) That the equation XX. does in fact represent a spheric locus for the point 
 p, is evident from its mere /orm (comp. 282, (10.)); and that this sphere passes 
 
328 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 through the four given points, O, A, B, c, may be proved by observing that the equa- 
 tion is satisfied, when we change p to any one of the four vectors, 0, a, j3, y. 
 (24.) Introducing an auxiliary vector, OD or ^, determined by tlie equation, 
 XXIII. . . ^Sa/3y = a«Vi3y + /32Vya+7^Va/3, 
 or by the system of the three scalar equations (comp. 294, (25.) ), 
 
 XXIV. . . a2 = S^a, (S^ = S^/3, y2 = S^y, 
 or XXIV. . . S^a-» = S^/3-i = Soy-i = 1, 
 
 the equation XX. of the sphere becomes simply, 
 
 XXV. . . p2 = s^p, or XXV'. . . S^p-i = 1 ; 
 so that D is the point of the sphere opposite to o, and 5 is a diameter (comp. 282, 
 IX'.; and 196, (6.)). 
 
 (25.) The formula XXIII., which determines this diameter, may be written, in 
 this other way : 
 
 XXVI. . . ^Sa/3y = Va (;S - a) (y - /3) y ; 
 
 or XXVr. . . 6.0ABC.0D = - V(OA.AB.BC.CO) ; 
 
 where the symbol oabc, considered as a coefficient, is interpreted as in 294, XLIV. ; 
 namely, as denoting the volume of the pyramid oabc, which is here an inscribed 
 one. 
 
 (26.) This result of calculation, so far as it regards the direction of the axis of 
 the quaternion OA. ab.bc.co, agrees with, and may be used to confirm, the theorem 
 (12.), respecting theproduct of the successive sides of a gauche quadrilateral, oabc ; 
 including the rule of rotation, which distinguishes that axis from its opposite. 
 
 (27.) The formula XXIII. for the diameter S may also be thus written : 
 
 XXVII. .. o.Sa-i/3-iy-i = V(/3-iy-i + y-'a-i+a-i/3-0 
 = V(/3-i-a-i)(y-i-a-i); 
 
 and the equation XX. of the sphere may be transformed to the following : 
 
 XXVIII. . . = S (|S-1 - a-l) (y-i - a-i) (p'l - a"') ; 
 which expresses (by 294, (34.), comp. 260, (10.) ), that the four reciprocal vec- 
 tors, 
 
 XXIX. . . oa' = a' = a-J, ob' = ^' = /3-i, oc' = y' = y"i, of' = p'=p~^, 
 are termino-complanar (64) ; the plane a'b'cV, in which they all terminate, being 
 parallel to the tangent plane to the sphere at o : because the perpendicular let fall 
 on this plane from o is 
 
 XXX. . .d' = S'i, 
 
 as appears from the three scalar equations, 
 
 XXXI. . . Sa'd = s(5'5 = sys = 1. 
 
 (28.) In general, if d be the foot of the perpendicular from o, on the plane abc, 
 then 
 
 XXXII. . . 5 = Sa(3y :Y((3y + ya + a(3) ; 
 
 because this expression satisfies, and may be deduced from, the three equations, 
 
 XXXIII. . . Sa^-i = S/3^l = Sy^-i = 1. 
 As a verification, the formula shows that the length TS, of this perpendicular, or 
 altitude, OD, is equal to the sextuple volume of the pyramid oabc, divided by the dou- 
 ble area of the triangular base ABC. (Compare 281, (4.), and 294, (3.), (33.).) 
 
CHAP. I.] EQUATION OF HOMOSPHERICITY. 329 
 
 (29.) The equation XX., of the sphere oabc, might have been obtained by the 
 elimination of the vector ^, between the four scalar equations XXIV. and XXV., on 
 the plan of 294, (27.). 
 
 (30.) And another form of equation of the same sphere, answering to the deve- 
 lopment of XXVIII., may be obtained by the analogous elimination of the same vec- 
 tor ^, between the four other equations , XXIV. and XXV'. 
 
 (31.) The product of any even number of complanar vectors is generally a qua- 
 ternion with an axis perpendicular to their plane ; but the product of the successive 
 sides of a hexagon abcdep, or any other even-sided figure, inscribed in a circle, is 
 a scalar : because by drawing diagonals AC, ad, ae from the first (or last) point a 
 of the polygon, we find Us in (6.) that it differs only by a scalar coefficient, or divisor, 
 from the product of an e^>en number of tangents, at the first point. 
 
 (32.) On the other hand, the product oi any odd number of complanar vectors is 
 always a line, in the same plane; and in particular (comp. (19.)), the product of 
 the successive sides of a pentagon, or heptagon, &c., inscribed in a circle, is equal to 
 a tangential vector, drawn from the first point of that inscribed and odd-sided poly- 
 gon : because it differs only by a scalar coefficient from the product of an odd num- 
 ber of such tangents. 
 
 (33.) The product of any number oi lines in space is generally a quaternion 
 (289) ; and if they be the successive sides of a hexagon, or other even-sided polygon, 
 inscribed in a sphere, the axis of this quaternion (comp. (12.) ) is normal to that 
 sphere, at the initial (or final) point of the polygon. 
 
 (34.) But the product of the successive sides of a heptagon, or other odd-sided 
 polygon in a sphere, is equal (comp. (19.) ) to a vector, which touches the sphere at 
 the initial or final point ; because it bears a scalar ratio to the product of an odd 
 number of vectors, in the tangent plane at that point. 
 
 (35.) The equation XX., or its transformation XXVIII., may be called the con- 
 dition or equation of homo sphericity (comp. 260, (10.)) oi the five points o. A, B, 
 c, P ; and the analogous equation for the five points abode, with vectors afiydt 
 from any arbitrary origin o, may be written thus : 
 
 XXXIV.. . = S(a-/3) {(3-y) (y- 5) (5- f) (t - a); 
 or thus, XXXV. . . = aa* + 6/32 + cy2 + dd^ + ee^, 
 
 six times the second member of this last formula being found to be equal to the se- 
 cond member of the one i)receding it, if 
 
 XXXVI. .. a = BODE, 6 = CDEA, C = DEAB, rf = EABC, e = ABCD, 
 
 or more fully, 
 
 XXXVII. . . 6a = S (y - 18) (^ - /3) (€ - /3) = S {yh - Stf5 + sjSy - (Syd), &c. ; 
 so that, by 294, XLVIII. and XLVII., we have also (comp. 65, 70) the equation, 
 
 XXXVIII. . . = aa + bl3 + cy + d8 + ee, 
 with the relation between the coefficients, 
 
 XXXIX. . . = a + b + c + d + e, 
 which allows (as above) the origin of vectors to be arbitrary. 
 
 (36.) The equation or condition XXXV. may be obtained as the result of an 
 elimination (294, (27.) ), of a vector k, and of a scalar g, between ^ve scalar equa- 
 tions of the form 282, (10.), namely the five following, 
 
 2 u 
 
330 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XL. . . a2-2SKa + ^ = 0, /32- 2Sk/3 + ^ = 0, . . f2_2SK£4^=0; 
 K being the vector of the centre K of the sphere Abcd, of which the equation may be 
 written as 
 
 XLI. . . p2_2S/cp + 5' = 0, 
 
 ff being some scalar constant ; and on which, by the condition referred to, the Jifth 
 point E is situated. 
 
 (37.) By treating this fifth point, or its vector e, as arbitrary, we recover the 
 condition or equation of concircularily (3.), of the four points A, B, c, D ; or the 
 formula, 
 
 XLII. .. = V(a- /3)(i3-y)(y-^)(^-a). 
 
 (38.) The equation of the circle ABC, and the equation o^the sphere abcd, may 
 in general be written thus : 
 
 XLIII. ..0 = V(a-^)(/3-y)(y-p)(p-«); 
 XLIV. ..0 = S(a-/3)(/3-y)(y-^)((^-p)(p-a); 
 p being as usual the vector of a variable point p, on the one or the other locus. 
 
 (39.) The equations of the tangent to the circle abc, and of the tangent plane 
 to the sphere abcd, at the point A, are respectively, 
 
 XLV...O=V(a-^)(^-y)(y-«)(p-a), 
 and XLVI. . . = S(a -^8) (/3-y) (y-^) (^-a) (p- a). 
 
 (40.) Accordingly, whether we combine the two equations XLIII. and XLV., 
 or XLIV. and XLVI., we find in each case the equation, 
 
 XLVIL . . (p - a)2 = 0, giving p = «, or p = a(20); 
 it being supposed that the three points a, b, c are not collinear, and that the four 
 points, a, b, c, d are not complanar. 
 
 (41.) If the centre of the sphere abcd be taken for the origin o, so that 
 XLVIIL . . a2=/32 = y2=^2 = _r2, or XLIX. . . Ta = T/3 = Ty = T^ = r, 
 the positive scalar r denoting the radius, then after some reductions we obtain the 
 transformation, 
 
 L...V(a-/3)(/3-y)(y-^)(5-a) = 2aS(/3-a)(y-a)(^-«). 
 (42.) Hence, generally, if k be, as in (36.), the centre of the sphere, we have the 
 equation (comp. XXV I'.), 
 
 LI. . . V(ab.bc.cd.da) = 12ka.aecd. 
 (43.) "We may therefore enunciate this theorem : — 
 
 " The vector part of the product of four successive sides, of a gauche quadrila- 
 teral inscribed in a sphere, is equal to the diameter drawn to the initial point of the 
 polygon, multiplied by the sextuple volume of the pyramid, which its four points de- 
 termine.^^ 
 
 (44.) In effecting the reductions (41.), the following general formulce of trans- 
 formation have been employed, which may be useful on other occasions : 
 
 LIL . . aq + qa = '2{a^q + Sga) ; LII'. . . aqa = a^Kq + "la^qa ; 
 where a may be any vector, and q may be any quaternion. 
 
CHAP. I.] FOURTH PROPORTIONAL TO DIPLANAR VECTORS. 31^1 
 
 Section 7. — On the Fourth Proportional to Three Diplanar 
 
 Vectors, 
 
 297. In general, when a.nj four quaternions, q, q', q"^ q"\ satisfy 
 the equation of quotients, 
 
 I. . . q"':q"=^q':q, 
 or the equivalent formula, 
 
 II. . . q'"={q':q).q" = q'q-'q", 
 
 we shall say that they form a Proportion ; and that the fourth, 
 
 namely q'", is the Fourth Proportional to iho, first, second, and third 
 
 quaternions, namely to q, q', and q", taken in this given order. 
 
 This definition will include (by 288) the one which was assigned in 
 
 226, for the fourth proportional to three complanar vectors, a, yS, 7, 
 
 namely ih^i fourth vector in the same plane, 8= I3a^<y, which has been 
 
 already considered; and it will enable us to interpret (comp. 289) 
 
 the symbol 
 
 III. . . ;3a-i7, when ^ not\\\a, {3, 
 
 as denoting not indeed a Vector, in this new case, but at least a Qua- 
 tej-nion, which may be called (on the present general plan) the Fourth 
 Proportional to these Three Diplanar Vectors, a, /3, 7. Such fourth 
 proportionals possess some interesting properties, especially with re- 
 ference to their vector parts, which it will be useful briefly to consi- 
 der, and to illustrate by showing their connexion with spherical 
 trigonometry, and generally with spherical geometry. 
 
 (1.) Let a, (3, y be (as in 208, (1.), &c.) the vectors of the corners of a triangle 
 ABC on the unit-sphere, whereof the sides are a, b, c ; and let us write, 
 
 (I = cos a = Sy/3-i = - S^Sy, 
 IV. . . I m = cos 6 = Say"^ = — Sya, 
 [n = cos c = S/3a~^ = - Sa/3; 
 where it is understood that 
 
 V. .. a2 = /32 = 72^-1, or VI. . . Ta = T/3 = Ty = l; 
 
 it being also at first supposed, for the sake of fixing the conceptions, that each of these 
 three cosines, /, m, n, is greater than zero, or that each side of the triangle abc is 
 less than a quadrant. 
 
 (2.) Then, introducing three new vectors, S, i, ^, defined by the equations, 
 
 VII. . . jc =Vy/3-ia = Va|3-iy = ny + la - m(3^ 
 (^ =Vay- 1/3 = V]8y-' a = Zrt +m[3-ny, 
 
332 ELEMENTS OF QUATERKIONS. [bOOK III. 
 
 we find that these three derived vectors have all one common lengthy say r, because 
 they have one common norm ; namely, 
 
 VIII. . . N^ = N£ = N?=:^2^.;„2^„2_2Zmn = r2; 
 so that IX. . . T^ = Te = T? = r = V(/2 + m^ + n^ - 2lmn'). 
 
 (S.) This common length, r, is less than uniiy ; for if we write, 
 
 X. . . Sa)3y = S^a-V = e, 
 we shall have the relation, 
 
 XL . . e2 + r2=:N/3a-^y = l; 
 
 and the scalar e is different from zero, because the vectors a, (3, y are diplanar. 
 
 (4.) Dividing the three lines ^, £, ? by their lengthy r, we change them to their 
 versors (155, 156); and so obtain a new triangle, def, on the unit-sphere, of which 
 the corners are determined by the three new unit-vectors, 
 XII. . . OD = U5 = r->^ ; OE = Ue = r-h ; 
 
 (5.) The sides opposite to d, e, f, in this new or de- 
 rived triangle, are bisected, as in Fig. 67, by the corners 
 A, B, c of the old or given triangle ; because we have the d~ 
 three equations, 
 
 XIII. . .c + ^ = 2Za; ^ + ^=2»i/3; ^+e = 2«y. 
 (G.) Denoting the halves of the new sides by a', b', c' (so that the arc Er = 2a', 
 &c.), the equations XIII. show also, by IV. and IX., that 
 
 XIV. . . cos a = r cos a', cos b — r cos b', cos c = r cos e • 
 the cosines of the half-sides of the new (or bisected) triangle, def, are therefore /jro - 
 portional to the cosines of the sides of the old (or bisecting) triangle ABC. 
 (7.) The equations IV. give, by 279, (1.), 
 
 XV. .. 2Z = -(^y + y/3), 2m = -(ya + ay), 2n = - (a/3 + /3a) ; 
 we have therefore, by VII., the three following equations between quaternions, 
 
 XVI. . . af = ^a, f3K = S(3, yd = ey; 
 which may also be. thus written, 
 
 XVr. . . ea = aK, K(3 = ^d, dy = yf , 
 and express in a new way the relations of bisection (5.). 
 (8.) We have therefore the equations between vectors, 
 
 XVII. . . c = a?a-i, K = /3^/3-i, d = yty^^ ; . 
 or XVir. . . ^ = a£a-i, d = l30-\ £ = y^y-i. 
 
 (9.) Hence also, by V., or because a, j3, y are unit-vectors, 
 
 XVIII... c = -a^a, K = ~I3^P, ^ = -y£y; 
 or XVIir. . . ? = - asa, d = - /3^/3, e = - y ^y. 
 
 (10.) In general, whatever the length of the vector a mag be, the first equation 
 XVII. expresses that the line s is (comp. 138) thereflexion of the line ^, with respect 
 to that vector a ; because it may be put (comp. 279) under the form, 
 
 XIX. . . ^a-»=a-»£ = K£a-i, or XIX'. . . fa-i =K^a-'. 
 (11.) Another mode of arriving at the same interpretation of the equation 
 
CHAP. I.] EXPRESSIONS FOR CONICAL ROTATION. 333 
 
 £ = rt^a-J, is to conceive ^ decomposed into two suramand vectors, ^' and ^", one pa- 
 rallel and the other perpendicular to a, in such a manner that 
 
 XX. ..^=r+r, riia, r^a; 
 
 for then we shall have, by 281, (10.), the transformations, 
 
 XXI. . . £ = a^'a-i + aCa-^ = I'aa-^ - V'aa-^ = ^' - Z," ; 
 the parallel part of Z, being thus preserved^ but thQ perpendicular part being reversed, 
 hy the operation a (^ )a-^ 
 
 (12.) Or we may return from e = a^a"' to the form ea — a?, that is, to the first 
 equation XVI'. ; and then this equation between quaternions will show, as suggested 
 in (7.), that whatever may be the length of a, we must have, 
 
 XXII. ..T£ = T?, Ax.*£a = Ax.a^, Lta==Lal-, 
 so that the two lines s, ^ are equally long, and the rotation from £ to a is equal to 
 that from a to ^ ; these two rotations being similarly directed, and in one common 
 plane. 
 
 (13.) We may also write the equations XVII. XVII'. under the forms, 
 XXIII. . . e=a-Ka, Sec, XXIII'. . . Z=a-ha, &c. 
 
 (14.) Substituting this last expression for ^ in the second equation XVII'., we 
 derive this new equation, 
 
 XXIV. . .d = /3a-^f aj3-i ; or XXIV. . . t = a/3-i^/3a-i ; 
 that is, more briefly, 
 
 XXY. ..d = qeq-\ and XXY'. . . e = q-^dq, if XXYl. . . q = (3a-K 
 
 (15.) .A.n expression of this form, namely one with such a symbol as 
 XXVII. . . 9 ( ) g-i 
 for an operator, occurred before, in 179, (1.), and in 191, (5.) ; and was seen to in- 
 dicate a conical rotation of the axis of the operand quaternion (of which the symbol 
 is to be conceived as being written within the parentheses'), round the axis of q, 
 through an angle =2 Lq, without any change of the angle, or of the tensor, of that 
 operand; so that a vector must remain a vector, after any operation of this sort, as 
 bting still a right-angled quaternion (290) ; or (comp. 223, (10.) ) because 
 
 XXVIII. . . S9P5-1 = S9-I5P = Sjo = 0. 
 
 (1 6.) If then we conceive two opposite points, p' and p, to be determined on the 
 unit-sphere, by the conditions of being respectively ihe positive poles of the two op- 
 posite arcs, ab and ba, so that 
 
 XXIX. . . op' = Ax. /3a-' = Ax. g, and op = p'o = Ax. a/3-' = Ax. 9-', 
 we can infer from XXIV. that the line od may be derived from the tine OE, by a co- 
 nical rotation round the line op' as an axis, through an angle equal to the double of 
 the angle aob (if o be still the centre of the sphere). 
 
 (17.) And in like manner we can infer from XXIV'., that the line oe admits 
 
 * It was remarked in 291, that this characteristic Ax. can be dispensed with, 
 because it admits of being replaced by UV ; but there may still be a convenience in 
 employing it occasionally. 
 
334 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book III. 
 
 of being derived from od, by an equal but opposite conical rotation, round the line 
 OP as a new positive axis, through an angle equal to twice the angle boa. 
 
 (18.) To illustrate these and other connected results, the annexed Figure 68 ia 
 drawn ; in which p represents, as above, 
 the positive pole of the arc ba, and arcs are 
 drawn from it to D, e, f, meeting the great 
 circle through A and b in the points R, s, T. 
 (The other letters in the Figure are not, for 
 the moment, required, but their significa- 
 tions will soon be explained.) 
 
 (19.) This being understood, we see, 
 first, that because the arcs ef and fd are 
 bisected (5.) at A and b, the three arcual 
 perpendiculars, Es, FT, dr, let fall from E, 
 F, D, on the great circle through A and b, 
 are equally long; and that therefore the 
 point P is the interior pole of the small cir- 
 cle def', if f' be the point diametrically op- 
 posite ioF: so that a conical rotation round 
 
 this pole p, or round the axis op, would in fact bring the point D, or the line OD, to 
 the position E, or OE, which is one part of the theorem (17.). 
 
 (20.) Again, the quantity of this conical rotation, is evidently measured by the 
 arc RS of the great circle with p for pole ; but the bisections above mentioned give 
 (comp. 165) the two arcual equations, 
 
 XXX. . . r, rb= « bt, r,ix = ^ as; whcnce XXXI. . . '^ rs = 2 <-> ba, 
 and the other part of the same theorem (17.) is proved. 
 
 (21.) The point F may be said to be the reflexion, on the sphere, of the point D, 
 with respect to the point b, which Insects the interval between them ; and thus we 
 may say that two successive reflexions of an arbitrary point upon a sphere (as here 
 fromD to F, and then from f to e), with respect to two given points (b and a) of a 
 given great circle, are jointly equivalent to one conical rotation, round the pole (p) of 
 that great circle ; or to the description of an arc of a small circle, round that j9o/e, or 
 parallel to that great circle : and that the angular quantity (dpe) of this rotation 
 is double of that represented by the arc (ba) connecting the two given points ; or is 
 the double of the angle (bpa), which that given arc subtends, at the same pole (p)^ 
 
 (22.) There is, as we see, no difficulty in geometrically proving this theorem of 
 
 rotation : but it is remarkable how simply quaternions express it : namely by the 
 
 formula, 
 
 XXXII. . . a. i8- V|3. a- i=a|3V p. j3rt-i, 
 
 in which a, j3, p may denote any three vectors ; and which, as we see by the points^ 
 involves essentially the associative principle of multiplication. 
 
 (23.) Instead of conceiving that the point d, or the v/' ""\ 
 
 line OD, has been reflected into the position f, or of, /'' /fx. 
 with respect to the point b, or to the line ob, with a simi- / r b/ I ^XA S > 
 
 lar successive reflexion from F to E, we may conceive that \ / 
 
 a point has moved along a small semicircle, with B for 
 pole, from d to f, as indicated in Fig. 69, and then along 
 
 Fig. 09. 
 
CHAP. I.] CONSTIIUCTION OF A FOURTH PROPORTIONAL. 335 
 
 another small semicircle, with A for pole, from f to e ; and we see that the result, or 
 effect, of these two successive and semicirctdar motions is equivalent to a motion along 
 an arc de of a third small circle, which is parallel (as before) to the great circle 
 through B and A, and has a projection rs thereon, which (still as before) is double of 
 the given arc ba. 
 
 (24.) And instead of thus conceiving two successive arcual motions of a point D 
 upon a sphere, or two successive conical rotations of a radius OD, considered as cotn- 
 ponnding themselves into one resultant motion of that point , or rotation of that ra- 
 dius, we may conceive an analogous composition of two successive rotations of a 
 solid body (or rigid system^, round axes passing through a point o, which \& fixed in 
 space (and in the body) : and so obtain a theorem respecting such rotation, which 
 easily suggests itself from what precedes, and on which we may perhaps return. 
 
 (25.) But to draw some additional consequences from the equations VII., &c., and 
 from the recent Fig. 68, especially as regards the Construction of the Fourth Pro- 
 portional to three diplanar vectors, let us first remark, generally, that when we have 
 (as in 62) a linear equation, of the form 
 
 aa -f 6/3 -r cy 4 rf^ = 0, 
 connecting /oMr co-initial vectors a . . d, whereof no three are complanar, then this 
 fifth vector, 
 
 e=aai bl3= - cy - dS, 
 
 is evidently complanar (22) with a, (3, and also with y, d (comp. 294, (6.) ) ; it is 
 
 therefore part of the indefinite liiie of intersection of the plane aob, cod, of these 
 
 two pairs of vectors. 
 
 (26.) And if we divide this fifth vector e by the two (generally unequal) sca- 
 
 lars, 
 
 a + 6, and — c ~ d, 
 
 the two (generally unequal) vectors, 
 
 (aa + */3) : (a + 6), and {cy + rf^) : (c + d), 
 which are obtained as the quotients of these two divisions, are (comp. 25, 64) the 
 vectors of two (generally distinct) points of intersection, oilines yf'iih planes, namely 
 the two following : 
 
 ABOCD, and cdoab. 
 
 (27.) When the two lines, ab and cd, happen to intersect each other, the two 
 last-mentioned points coincide ; and thus we recover, in a new way, the condition 
 (63), for the complanarity of thQ four points o, A, b, c, or for the termino-compla- 
 narity of the four vectors a, j3, y, d ; namely the equation 
 
 ai-b + c + d=0, 
 which may be compared with 294, XLV. and L. 
 
 (28.) Resuming now the recent equations VII., and introducing the new vector, 
 
 XXXIII. . . X = Za-m/3-^(c-5), 
 which gives, 
 
 XXXIV. . . SyX = 0, and XXXV. . . T\ = V(r« -n^)=r sin c\ 
 
 we see that the two arcs ba, de, prolonged, meet in a point l (comp. Fig. 68), for 
 which OL= UX, and which is distant by a quadrant from o : a result which may be 
 confirmed by elementary considerations, because (by a well-kno fy-n theorem respect- 
 
336 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 iiig transversal arcs) the common bitector ba of the two sides, de and ef, must meet 
 the third side in a point i^, for which 
 
 sinDL= sin el, 
 (29.) To prove by quaternions this last equality of sines, and to assign their 
 common value, we have only to observe that by XXXIII., 
 
 XXXVL . . Va = Vf \ = AVc^« ; 
 in which, 
 
 T5\ = TfX = r2 sin c', and TV^t = r' sin 2c' ; 
 
 the sines in question are therefore (by 204, XIX.), 
 
 XXXVr. . . TVUa = TVU6X = ^r-i sin 2c : r' sin c' = cos c'. 
 
 (30.) On similar principles, we may interpret the two vector-equations y 
 
 XXXVII. . . V/3\ = lY(ia, YaX = mY(3a, 
 in which 
 
 XXXVIII. . . TX : TV/3a = r sin c' : sin c = tan c': tan c, 
 
 an equivalent to the trigonometric equations, 
 
 tan CD cosBC cos AC 
 
 XXXIX. 
 
 tan AB sm bl sin al 
 (31.) Accordingly, if we let fall the perpendicular OQ on ab (see again Fig. 68), 
 so that Q bisects rs, and if we determine two new points m, n by the arcual equa- 
 tions, 
 
 XL. . . rt I.M = -^ ab = '^ QR, r> LN = r> CD, 
 
 the arcs mr, kd will be quadrants ; and because the angle at r is right by construc- 
 tion (18.), M is the pole of dr, and dm is a quadrant ; whence d is the pole of mn 
 and the angle lnm is right : conceiving then that the arcs CA and cb are drawn, we 
 have three triangles, right-angled at Q and n, which show, by elementary principles, 
 that the three trigonometric quotients in XXXIX. have in fact a common value, 
 namely cos cq, or cos l. 
 
 (32.) To prove this last result by quaternions, and without employing the auxi- 
 liary points M, N, Q, R, we have the transformations, 
 
 XLI. . . COSL=bU -— — =SU — r- = i :^jr^- b — = 1 — — 
 Yde yX Y(3a yX V/3a 
 
 because 
 
 XLII. . . ^ = ny-X, e = ny+\, Ydt=2ny\, UV^e = UyX, 
 
 and 
 
 XLIII. . . S^ = ?^=-S/3a-'yX-' =-S5X-» =1, 
 yX (yX)2 
 
 it being remembered that X -J- y, whence 
 
 VyX = yX = - Xy, (yX)2 = - y2X« = X2, SyX'l = 0. 
 
 (33.) At the same time we see that if P be (as before) the positive pole of ba, 
 and if k, k' be the negative and positive poles of de, while l' is the negative (as l. 
 is the positive) pole of cq, whereby all the letters in Fig. 68 have their Bignification* 
 determined, we may write, 
 
 XL! V. . . OP = TJYfSa ; ok' = yUX ; ok = - yUX ; ol' = - UX ; 
 while oi< = + UX, as before. 
 
CHAP. I.] ANGLE OF A FOURTH PROPORTIONAL. 337 
 
 (34.) Writing also, 
 
 XLV. . . K = - y\, or \ = yK, and fi = (3a-^ X, 
 so that XLV. . . OK = U/c, and om = U/z, 
 
 we have XLVI. . . /3a-i.y = /t\-».\«;-» =/m»c-i ; 
 
 this fourth proportional, to the three equally long hut diplanar vectors, a, /3, y, ia 
 therefore a versor, of which the representative arc (162) is km, and the representa- 
 tive angle (174) is kdm, or l'dr, or edp 5 and we may write for this versor, or qua- 
 ternion, the expression : 
 
 XLVII. . /3a"iy = cos l'dr + od . sin l'dr. 
 
 (35.) The double of this representative angle is the sum of the two base-angles of 
 the isosceles triangle dpe ; and because the two other triangles, epf', f'pd, are also 
 isosceles (19.), the lune ff' shows that this sum is what remains, when we subtract 
 the vertical angle F, of the triangle def, from the sum of the supplements of the two 
 base-angles d and e of that triangle ; or when we subtract the sum of the three an- 
 gles of the same triangle /row four right angles. We have therefore this very simple 
 expression for the Angle of the Fourth Proportional : 
 
 XL VIII. . . L /3a-iy = l'dr = 7r - |(d + e + f). 
 
 (36.) Or, if we introduce the area, or the spherical excess, say 2, of the triangle 
 def, writing thus 
 
 XLIX. . . 2 = d + e+f- TT, 
 
 we have these other expressions : 
 
 L. . . Z./3a-^y = i7r-|S; LI. . . /3a-»y = sin|2 1- r'^o cos i2 ; 
 because 
 
 OD = U^ = r-io, by XIL 
 
 (37.) Having thus expressed (3a-^y, we require no new appeal to the Figure, in 
 order to express this other fourth proportional, ya' 1/3, which is the negative of its 
 conjugate, or has an opposite scalar, but an eqiial vector part (comp. 2U4, (1.), and 
 295, (9.) ) : the geometrical diflference being merely this, that because the rotation 
 round a from /3 to y has been supposed to be negative, the rotation round a from y 
 to j3 must be, on the contrary, positive. 
 
 (38.) We may thus write, at once, 
 
 LIL . . ya-i/3 = - K/3a-i y = - sin |2 + ri^ cos |2 ; 
 
 and we have, for the angle of this new fourth proportional, to the same three vectors 
 a, (3, y, of which the second and third have merely changed places with each other, 
 the formula : 
 
 LIII, . . Z.ya-ij3 = RDL = :i(D + E + F) = i7r + i2. 
 
 (39.) But the common vector part of these <t<JO fourth proportionals is d, by VII ; 
 we have therefore, by XI., 
 
 LIV. . . r = cos|2; c = ±sini2; 
 
 the upper sign being taken, when the rotation round a from ^ to y is negative, as 
 above supposed. 
 
 (40.) It follows by (6.) that when the sides 2a', 1b\ 2c', of a spherical triangle 
 
 2 X 
 
338 ELEMENTS OF QUATERNIONS. [bOOK IH. 
 
 DEF, of which the area is 2, are bisected by the corners A, E, c of another spherical 
 triangle, of which the sides* are a, b, c, then. 
 
 LV. . . cos a : cos a' = cosb : cos b' = cos c ; cos c' = cos i^S. 
 
 (41.) It follows also, from what has been recently shown, that the angle rdk, or 
 MDN, or the arc mx in Fig. 68, represents the semi-area of the bisected triangle def; 
 ■whence, by the right-angled triangle lmn, we can infer that the sine of this semi-area 
 is equal to the sine of a side of the bisecting triangle abc, multiplied into the sine of 
 the perpendicular, let fall upon that side from the opposite corner of the latter trian- 
 gle ; because we have 
 
 LVI. . . sin IS = sin mn = sin lm . sin l = sin ab . sin CQ. 
 
 (42.) Tlie same conclusion can be drawn immediately, by quaternions, from the 
 expression, 
 
 LVII. . . sin IS = e = Sa/3y = S(V/3a. y->) = TV/3a. SU(V/3a : y); 
 in which one factor is the sine of ab, and tlie other factor is the cosine of op, or the 
 sine of cq. 
 
 (43.) Under the same conditions, since 
 
 LVIII. . . a = U(£ + = F*(c + 0, &c., 
 ■we may write also, 
 
 LIX. . .8iniS=SU(« + ^)(?+^) (^ + = S^6? : 4//jm ; 
 
 in which, by IV. and XIII., 
 
 LX. . . 4Zm« =- 8(5 + (e + = »•= -S(t? +KS + St). 
 
 (44.) Hence also, by LIV,, 
 
 LXI. . . cos is = r = (r3 - rS (e^ + ?5 + St) ) : Umn ; 
 
 TYTT t.niT=i= S^^^ ^ SU5.^ 
 
 ^'^ ^ r r3_rS^£$ + ^5+50 1 - SUf^- SU^5 - Smc ' 
 
 and under this last form, we have & general expression for the tangent of half the 
 spherical opening at o, of any triangular pyramid odef, whatever the lengths Td, 
 Tf, T^ of the edges at o may be. 
 (45.) As a verification, we have 
 
 LXIII. . . (4/mn)3 = -i.(f + ^2 (^4 ^)2 (a + e)» 
 = 2 (r2 - SfO (^2 - SS5) (r2 - Sdt) ; 
 but the elimination of ^S between LIX. LXI. gives, 
 
 LXIV. . . (Almny = (SdeKy + (rS - r(StK + S^5 + Sds) )2 ; 
 •we ought then to find that 
 
 LXV. . . {SSeK)^ = r^-r^(SeK)^ + {BZSy+iSSey'}-2StKSKSSSe, 
 if 5* = «2 = ^3 = — r2 ; and in fact this equality results immediately from the general 
 formula 294, LIU. 
 
 (46.) Under the same condition, respecting the equal lengths of S, f, ^, we have 
 also the formula, 
 
 * These sides abc, of the bisecting triangle ABC, have been hitherto supposed for 
 simplicity (1.) to be each less than a quadrant, but it will be found that the for- 
 mula LV. holds good, without any such restriction. 
 
CHAP. I.] CONNEXION WITH SPHERICAL AREA. 339 
 
 LXVI. . . - V(^ + £) (£ + (^ + ^) = 25 (r2 - SeK - S^d - SSe) = SlmnS ; 
 whence other verifications may be derived. 
 
 (47.) If (7 denote the area* of the bisecting triangle ABC, the general principle 
 LXII. enables us to infer that 
 
 LXVII. . . tan ^ = ^-^^ = ! 
 
 2 1 - S/3y - Sya - Sa/3 l-^Z+m + w 
 
 sin c sin p , 
 
 1 + cos a -t- cos 6 + cos c 
 
 if p denote the perpendicular cq from c on ab, so that 
 
 e = sin c sin/> = sin b sine sin a = &c. (comp. 210, (21.) ). 
 (48.) But, by (IX.) and (XL), 
 
 LXVIII. . . e2 + (H-/ + m + «)2=2(l + (1 + m) (l+n) 
 
 I . a b c 
 
 = 1 4 cos - cos - cos - 
 
 \ 2 2 2 
 
 hence the cosine and sine of the 7iew semi-area are, 
 
 <7 1 + cos a + cos b 4 cos c 
 
 LXIX. 
 
 2 a b c 
 
 4cos - cos - cos - 
 
 2 2 2 
 
 a b 
 
 siu - sin - sin c 
 
 Tvv • '^ 2 2 , 
 
 LXX. . . sm - = ————— = &c. 
 2 c 
 
 cos - 
 2 
 
 (49,) Returning to the bisected triangle^ def, the last formula gives, 
 
 ^^^^T^ . 1^ sin a' sin i' sin F . , . 
 
 LXXI. . . sm ^2 = '. = sui » sm c sec c , 
 
 ^ cose ' 
 
 if />' denote the perpendicular from F on the bisecting arc ab, or ft in Fig. 68; 
 but cos ^2 = cos c sec c, by LV. ; hence 
 
 LXXII. . . tan 1 2 = sinp' tan c = sin ft . tan ab. 
 Accordingly, in Fig. 68, we have, by spherical trigonometry, 
 
 sin FT = sin es = sin le sin l = cos ln sin mn cosec lm = tan mn cot ab. 
 (50.) The arc MX, which thus represents in quantity the semiarea of def, has its 
 pole at the point d, and may be considered as the representative arc (162) of a certain 
 new quaternion^ Q, or of its versor, of which the axis is the radius OD, or U^ ; and 
 this new quaternion may be thus expressed : 
 
 LXXIII. .. Q = dya(3 = -S^+ dSaiSy = r^-^ ed; 
 its tensor and versor being, respectively, 
 
 LXXIV. . . TQ = r = cos|2; LXXV. . . UQ = cos^2 +0D.sin^2. 
 (51.) An important transformation of this last versor maybe obtained as fol- 
 lows : 
 
 * The reader will observe that the more usual symbol 2, for this area of abc, 
 in here employed (36.) to denote the area of the exscribed triangle def. 
 
340 ELEMENTS OF QUATERNIONS. [bOOK ill. 
 
 LXXVI. . . UQ = U(V.ar».?j3-0=(^OK«^0K^^-'>i 
 so that 
 
 LXXVII. . . iS = A Q= A dya[3=L (^£->> (f^O' (^^0* ? 
 these powers of quaternions, with exponents each = |, being interpreted as square 
 roots (199, (1.) ), or as equivalent to the symbols V(^£-i), &c. 
 
 (52.) The conjugate (or reciprocal) versor, UQ"i, which has nm for its repre- 
 tentative arc, may be deduced from UQ by simply interchanging /3 and y, or c and 
 ^ ; the corresponding quaternion is, 
 
 LXXVIII. . . Of = KQ=S(3ay = r« - e^ ; 
 and we have 
 
 LXXIX. . . UQ' = cos IS - OD . sin 12 = (5^i> (^f-i)' (f ^')* ; 
 the rotation round d, from e to f, being still supposed to be negative. 
 
 (53.) Let H be any other point upon the sphere, and let oh = rj; also let 2' be 
 the area of the new spherical triangle, dfh ; then the same reasoning shows that 
 
 LXXX. . . cos |S' + OD.sin p'= (^^-i)' (.W^y^ (»?5'0s 
 if the rotation round d from f to h be negative ; and therefore, by multiplication of 
 the two co-axal versors, LXXVI. and LXXX., we have by LXXV. the analogous 
 formula : 
 
 LXXXL . . cos 1(2 + 2') + oD.sin |(2 + 20 = (^£"0' (f^O^' iKrj-^y {no'')'; 
 where 2 + 2' denotes the area of the spherical quadrilateral, defh. 
 
 (54.) It is easy to extend this result to the area of ang spherical polygon, or to 
 the spherical opening (44.) oi any pyramid; and we may even conceive an exten- 
 sion of it, as a limit, to the area of any closed curve upon the sphere, considered as 
 decomposed into an indefinite number of indefinitely small triangles, with some cofn- 
 mon vertex, such as the point d, on the spheric surftice, and with indefinitely small 
 arcs EP, FH, . . of the curve, for their respective bases : or to the spherical opening 
 of any cone, expressed thus as the Angle of a Quaternion, which is the limit* of the 
 product of indefinitely many factors, each equal to the square-root of a quaternion, 
 lohich differs indefinitely little from unity. 
 
 (55.) To assist the recollection of this result, it may be stated as follows (comp. 
 180, (3.) for the definition of an arcual sum) : — 
 
 " The Arcual Sum of the Halves of the successive Sides, of any Spherical Poly- 
 gon, is equal to an arc of a Great Circle, which has the Initial {or Final) Point of 
 
 * This Limit is closely analogous to a definite integral, of the ordinary kind ; or 
 rather, we may say that it is a Definite Integral, but one of a new kind, which could 
 not easily have been introduced without Quaternions. In fact, if we did not employ 
 the non-commutative property (168) of quaternion multiplication, the Products here 
 considered would evidently become each equal to imity : so that they would fur- 
 nish no expressions for spherical or other areas, and in short, it would be useless to 
 speak of them. On the contrary, when that property or principle of multiplication 
 is introduced, these expressions of product-form are found, as above, to have ex- 
 tremely useful significations in spherical geometry ; and it will be seen that they sug- 
 gest and embody a remarkable <Aeorem, respecting ihQ resultant of rotations of a sys- 
 tem, round any number of successive axes, all passing through one fixed point, but in 
 other respects succeeding each other with any gradual or sudden changes. 
 
CHAP. I.] AREA OF POLYGON OR CURVE ON SPHERE. 341 
 
 the Polygon for its Pole^ and represents the Semi-area of the Figure;'' it being un- 
 derstood tliat this resultant arc is reversed in direction, when the half-sides are (ar- 
 cually) added in an opposite order. 
 
 (56.) As regards the order thus referred to, it may be observed that in the arcual 
 addition, which corresponds to the quaternion multiplication in LXXVI., we con- 
 ceive a point to move, first, from b to F, through half i\iQ arc df ; which half-side 
 of the triangle def answers to the right-hand factor, or square-root, (^5~0^- ^^ 
 tlien conceive the same point to move next from f to A, through half the arc fe, 
 which answers to the factor placed immediately to the left of the former ; having 
 thus moved, on the whole, so far, through the resultant arc ba (as a transvec- 
 tor, 180, (3.))j or through any equal arc (163), such as ml in Fig. 68. And 
 finally, we conceive a motion through half the arc ed, or through any arc equal to 
 that half, such as the arc ln in the same Figure, to correspond to the extreme left- 
 hand factor in the formula ; the final resultant (or total transvector arc), which 
 answers to ihQ product of the three square roots, as arranged in the formula, being 
 thus represented by i\\Q final arc mn, which has the point d for its positive pole, and 
 the half-area, ^S, for the angle (51.) of the quaternion (or versor) product which 
 it represents. 
 
 (57.) Now the direction o^ positive rotation on the sphere has been supposed to 
 be that round d, from f to e; and therefore along the perimeter, in the order dfe, 
 as seen* from any point of the surface within the triangle : that is, in the order in 
 which the successive sides df, fe, ed have been taken, before adding (or compound- 
 ing) their halves. And accordingly, in the conjugate (or reciprocal) formula 
 LXXIX., we took the opposite order, def, in proceeding as usual from right-hand 
 to left-hand factors, whereof the former are supposed to be multiplied hgf the latter; 
 while the result was, as we saw in (52.), a new versor^ in the expression for which, 
 the area S of the triangle was simply changed to its own negative. 
 
 (58.) To give an example of the reduction of the area to zero, we have only to 
 conceive that the three points D, e, f are co-arcwaZ (165), or situated on one great 
 circle ; or that the three lines d, e, K are complanar. For this case, by the laws+ 
 of complanar quaternions, we have the formula, 
 
 LXXXII. . . (^ri)i {sK-^)i (?^')* = h if S^£?= ; 
 
 thus cos iS = l, and 2 = 0. 
 
 * In this and other cases of the sort, the spectator is imagined to stand on the 
 point of the sphere, round which the rotation on the surface is conceived to be per- 
 formed ; his body being outside the sphere. And similarly when we say, for exam- 
 ple, that the rotation round the line, or radius, OA, from the line OB to the line oc, 
 is negative (or left-handed), as in the recent Figures, we mean that such would ap- 
 pear to be the direction of that rotation, to a person standing thus with h\s feet on 
 A, and with his body in the direction of OA prolonged : or else standing on the centre 
 (or origin) o, with his head at the point A. Compare 174, II. ; 177; and the Note 
 to page 153. 
 
 t Compare the Notes to pages 146, 159. 
 
 X Compare the Second Chapter of the Second Book. 
 
342 ELEMENTS OF QUATERNIONS. [bOOK HI. 
 
 (59.) Again, in (_53.) let the point H be co-arcual with d and f, or let Sd^rj = ; 
 then, because 
 
 LXXXir. . . {KT^)i (»j^-i> = (^^i>, if S^^7/ = 0, 
 
 the product of four factors LXXXI. reduces itself to the product of three factors 
 
 'LXXVI. ; the geometrical reason being evidently that in this case the added area 
 
 2' vanishes ; so that the quadrilateral defh has only the same area as the triangle 
 
 DEF. 
 
 (60.) But this added area (53.) may even have a negative* effect^ as for exam- 
 ple when the new point H falls on the old side de. Accordingly, if we write 
 
 LXXXIII. . . Qi=:(t^J)^ {W)' (»?£-')*. 
 and denote the product LXXXI. of four square-roots by Qi, we shall have the trans- 
 formation, 
 
 LXXXIV. . . Q2 = (^£-' )i Q) (£5-» )i, if ^^tr, = ; 
 
 which shows (comp. (15.) ) that in this case the angle of the quaternary/ product Qz 
 is that of the ternary product Qi, or the half-area of the triangle efh (= def — dhf), 
 although the axis of Qz is transferred from the position of the axis of Qi, by a ro- 
 tation round the pole of the arc ed, which brings it from oe to od. 
 
 (Gl.) From this example, it may be considered to be sufficiently evident, how the 
 formula LXXXL may be applied and extended, so as to represent (comp. (54.) ) the 
 area of any closed figure on the sphere, with any assumed point D on the surface as 
 a sort of spherical origin ; even when this auxiliary point is not situated on the pe- 
 rimeter, but is either external or internal thereto. 
 
 (62 ) A new quaternion Qo, with the same axis od as the quaternion Q of (50,), 
 but with a double angle, and with a tensor equal to unity, may be formed by simply 
 squaring the versor UQ ; and although this squaring cannot be effected by removing 
 the fractional exponents,^ in the formula LXXVI., yet it can easily be accomplished 
 in other ways. For example we have, by LXXIII. LXXIV., and by VII. IX. X., 
 the transformations :| 
 
 LXXXV. . . Qo = UQ2 = r-2(5yo/3)2 = - ^^ ya/3^.^yo/3 
 = - (y«/5)2 = - (e - (5)2 = r2 - e« + 2ed ; 
 
 and in fact, because S — r. od, by XII., the trigonometric values LIV. for r and e 
 enable us to write this last result under the form, 
 
 LXXXVI. . . Qo = - (7a/3)2 = cos S + od . sin 2. 
 
 (63.) To show its geometrical signification, let us conceive that abc and lmn 
 
 * In some investigations respecting areas on a sphere, it may be convenient to 
 distinguish (comp. 28, 63) between the two symbols def and dfe, and to consider 
 them as denoting two opposite triangles, of which the sum is zero. But for the pre- 
 sent, we are content to express this distinction, by means of the two conjugate qua- 
 ternion products, (51.) and (52.). 
 
 t Compare the Note to (54.). 
 
 X The equation 5ya/3 = ya/?^ is no< valid generally ; butwehave/jere d=~y-/aj3; 
 and in general, qp ■= pq, if p || Yq. 
 
CHAP. I.] CASE OF SIDES GREATER THAN QUADRANTS. 343 
 
 have the same meanings in tlie new Fig. 70, as in Fig. 68 ; and that AiBiMi are 
 three new points, determined by the three arcual equations (163), 
 
 LXXXVII. OAC = '>CAi, <^BC='^CBi, 
 
 r> MN = n NMi ; 
 
 which easily conduct to this fourth equation of 
 the same kind, 
 
 LXXXVir. . . n LMi = " BiAi. 
 This new arc LMi represents thus (comp. 167, and 
 Fig. 43) the product aiy-*.y/3rJ = ya-i./3y-i ; 
 
 while the old arc ml, or its equal ba (31.), represents afl-^ ; whence the arc mmi, 
 which has its pole at d, and is numerically equal to the whole area S of def (be- 
 cause MN was seen to be equal (50.) to half that area), represents the product 
 ya-i]3y-i. a(3-\ or - (ya/3)2, or Qq. The formula LXXXVI. has therefore been 
 interpreted^ and may be said to have been proved anew, by these simple geometri- 
 cal considerations. 
 
 (64.) We see, at the same time, how to interpret the symbol^ 
 
 LXXXVIII. . . Qo=--^; 
 a y /3 
 
 namely as denoting a versor, of which the axis is directed to, or from, the corner d 
 of a certain auxiliary spherical triangle def, whereof the sides, respectively o/>/)osj7e 
 to D, E, F, are bisected (5.) by the given points A, b, o, according as the rotation round 
 a from /3 to y is negative or positive; and of which the angle represents, or is numeri- 
 cally equal to, the area S of that auxiliary triangle : at least if we still suppose, as 
 we have hitherto for simplicity done (1.), that the sides of the^'it'ew triangle abc are 
 each less than a quadrant. 
 
 298. The case when the sides of the given triangle are all greater, 
 instead of being all less, than quadrants, may deserve next to be 
 (although more briefly) considered; the case when they are all 
 equal to quadrants, being reserved for a short subsequent Article: 
 and other cases being easily referred to these, by limits, or by passing 
 from a given line to its opposite, 
 
 (1.) Supposing now that 
 
 I. . , / < 0, m<0, n < 0, 
 
 or that II. ..a>-, o>—, c>— , 
 
 we may still retain the recent equations lY. to XI. ; XIII. ; and XV. to XXVI., of 
 297 ; but we must change the sign of the radical, r, in the equations XII. and XIV., 
 and also the signs of the versors JJd, Ue, U^ in XII., if we desire that the sides of 
 the auxiliary triangle, def, may still be bisected (as in Figures 67, 68) by the cor- 
 ners of the given triangle ABC, of which the sides a, 6, c are now each greater than 
 a quadrant. Thus, r being still the common tensor of d, i, ^, and therefore being still 
 supposed to be itself >0, we must write now, under these new conditions I. or II., 
 the new equations. 
 
344 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 III. . . OD = -m = -r-i5; OE=-U£=-r-'6; OF = -U^ = -r-»^; 
 I V. . . cos a = — r cos a', cos b =—r cos 6', cos c = — r cos c'. 
 (2.) The equations IV. and VIII. of 297 still holding good, we may now write, 
 V. . . + 2r cos a cos b' cos c = cos a'2 + cos 6'2 + cos c'^—l, 
 according as we adopt positive values (297), or negative values (298), for the co- 
 sines I, m, n of the sides of the bisecting triangle ; the value of r being still supposed 
 to be positive. 
 
 (3.) It is not difficult to prove (comp. 297, LIV., LXIX.), that 
 
 VI. . . r=4:C0S |S, according as ^>0, &c., or l<0,&c.; 
 the recent formula V. may therefore be written unambiguously as follows : 
 
 VII. . . 2 cos a cos b' cos c' cos ^2 = cos a'2 -|- cos 6'2 + cos c'^ — 1 ; 
 and the formula 297, LV. continues to hold good. 
 
 (4.) In like manner, we may write, without an ambiguous sign (comp. 297, LI.), 
 the following expression for the fourth proportional /3a"iy to three unit-vectors a, /3, 
 y, the rotation round the first from the second to the third being negative : 
 
 VIII. . . j3a-Jy = sin AS + 0D. cos IS; 
 
 where the scalar part changes sign, when the rotation is reversed. 
 
 (6.) It is, however, to be observed, that although this ^rmwZa VIII. holds good, 
 not only in the cases of the last article and of the present, but also in that which has 
 been reserved for the next, namely when Z= 0, &c. ; yet because, in the present case 
 (298) we have the area S> tt, the radius on is no longer the (positive) axis XJd of 
 the fourth proportional jSa-^y ; nor is Att — iS any longer, as in 297, L., the (posi- 
 tive) angle of that versor. On the contrary we have noWy for this axis and angle, 
 the expressions : 
 
 IX. . . Ax. /3rt-Jy = DO=-OD; X. . . Z./3a-iy = i(2-7r). 
 
 (6.) To illustrate these results by a construction, we may remark that if, in Fig. 
 67, the bisecting arcs bc, ca, ab be supposed each greater than a quadrant, and if 
 we proceed to form from it a new Figure, analogous to 68, the perpendicular CQ will 
 also exceed a quadrant, and the poles p and k will fall between the points c and Q ; 
 also M and k will fall on the arcs lq and ql' prolonged: and although the arc km, 
 or the angle kdm, or l'dr, or edp, may still be considered, as in 297, (34.), to re- 
 present the versor /3a"' y, yet the corresponding rotation round the point d is now o' 
 a negative character. 
 
 (7.) And as regards the quantity of this rotation, or the magnitude of the angle 
 at D, it is again, as in Fig. 68, a base-angle of one p 
 
 of three isosceles triangles, with p for their common , /-'^l^^v^ ; / 
 
 vertex ; but we have now, as in Fig. 71, a new ar- \>^, y^ \ ^s^/' 
 range7nent, in virtue of which this angle is to be B^>^~ C \ ~^^ 
 
 found by halving what remains, when the sum of """^^^TTrrrr^^^^' 
 
 the supplements of the angles at d and e, in tlietri- Yig. 71. 
 
 angle def, is subtracted /ro?» the angle at f, instead 
 
 of our subtracting (as in 297, (35.) ) the latter angle from the former sum ; it i^i 
 therefore now, in agreement with the recent expression X., 
 XL . . Z. /3a-«y = ^(d f e 1 f) - tt. 
 
CHAP. I.] MODIFICATIONS OF THE CONSTRUCTION. 345 
 
 (8.) The negative of the conjugate of the formula VIII. gives, 
 XII. . . ya-^j3 = - sin IS + OD . cos iS ; 
 and by taking the negative of the square of this equation, we are conducted to the 
 following : 
 
 XIII. . . ^ 5 ^ = _ (y a-i/3)3 = cos S + OD . sin S ; 
 
 ay 13 
 
 a result which had only been proved before (comp. 297, (62.), (64.)) for the case 
 2 < TT ; and in which it is still supposed that the rotation round a from /3 to y is 
 negative. 
 
 (9.) With the same direction of rotation, we have also the conjugate or recipro- 
 cal formula, 
 
 XIV. . . ^^- = -(/3a-»y)2 = cos2-OD.sin2. 
 
 a(3y 
 
 (10.) If it happened that only one side, as ab, of the given triangle abc, was 
 greater, while each of the two others was less than a quadrant, or that we had Z > 0, 
 tn > 0, but n < ; and if we wished to represent the fourth proportional to a, /?, y by 
 means of the foregoing constructions ; we should only have to introduce the point c' 
 opposite to c, or to change y to y' = — y ; for thus the new triangle abc' Avould have 
 each side greater than a quadrant, and so would fall under the case of the present 
 Article; after employing the construction for which, we should only have to change 
 the resulting versor to its negative. 
 
 (11.) And in like manner, if we had I and m negative, but n positive, we might 
 again substitute for c its opposite point c', and so fall back on the construction of 
 Art. 297: and similarly in other cases. 
 
 (12.) In general, if we begin with the equations 297, XII., attributing any arbi- 
 trary (but positive) value to the common tensor, r, of the three co-initial vectors 
 ^, f, ^, of which the versorsy or the unit-vectors Vd, &c., terminate at the corners of 
 a given or assumed triangle def, with sides = 2a', 26', 2c', we may then suppose 
 (comp. Fig. 67) that another triangle abc, with sides denoted by a, 6, c, and with 
 their cosines denoted by /, m, n, is derived from this one, by the condition of bisect- 
 ing its sides ; and therefore by the equations (comp. 297, LVIII.), 
 
 XV. ..OA=a = U(€ + 0, OB = ^=U(^+5), oc = y = U(5 + e), 
 with the relations 297, IV. V. VI., as before; or by these other equations (comp. 
 297, XIII. XIV.), 
 
 XVI. . . 6 + ^ = 2mco3a', <^ + S=2rl3 cos b', d+€=2ry cose'. 
 
 (13.) When this simple construction is adopted, we have at once (comp. 297, 
 LX.), by merely taking scalars of products of vectors, and without any reference to 
 areas (compare however 297, LXIX., and 298, VII.), the equations, 
 
 XVII. . . 4 cos a cos 6' cos c' = 4 cos b cos c cos a' = 4 cos c cos a' cos b' 
 = - r-2S (? + 6) (5 + f) = &c. = 1 + cos 2a' + cos 26' + cos 2c' ; 
 or 
 
 cos a _ cos6 _ cose _ cos a'^ + cos b'^ + cos c'^ ~ 1 
 cos a' cos b' cos c 2 cos a' cos b' cos c' ' 
 
 which can indeed be otherwise deduced, by the known formulae of spherical trigo- 
 nometry. 
 
 2 Y 
 
346 ELEMENTS OF QUATERNIONS. [BOOK III. 
 
 (14.) We see, then, that according as the sum of the squares of the cosines of 
 the half-sides, of a given or assumed spherical triangle, def, is greater than unity, 
 or equal to unity, or less than unity, the sides of the inscribed and bisecting triangle^ 
 ABC, are together less than quadrants, or together equal to quadrants, or together 
 greater than quadrants. 
 
 (15.) Conversely, t/the sides of a given spherical triangle abc be thus all less, 
 or all greater than quadrants, a triangle def, but only one* such triangle, can be 
 exscrihed to it, so as to have its sides bisected, as above : the simplest process being 
 to let fall a perpendicular, such as CQ in Fig. 68, from c on ab, &c. ; and then to draw 
 new arcs, through c, &c., perpendicular to these perpendiculars, and therefore coin- 
 ciding in position with the sought sides de, &c., of def. 
 
 (16.) The trigonometrical results of recent sub-articles, especially as regards the 
 area\ of a spherical triangle, are probably all well known, as certainly some of them 
 are ; but they are here brought forward only in connexion with quaternion formulcB ; 
 and as one of that class, which is not irrelevant to the present subject, and includes 
 the formula 294, LIIL, the following may be mentioned, wherein a, (3, y denote any 
 three vectors, but the order of the factors is important : 
 
 XIX . . (a/3y)2 = 2a2^2y9 + a2 (/3y)^ + /32 (ay)2 + y2 (a/3 ^a _ Any Sa/3 S/3y . 
 
 (17.) And if, as in 297, (1.), &c., we suppose that a, (3, y are three unit-vec- 
 tors, OA, OB, oc, and denote, as in 297, (47.), by a the area of the triangle abc, 
 the principle expressed by the recent formula XIII. may be stated under this appa- 
 rently different, but essentially equivalent form : 
 
 ^v n + /3y-I-aj3+y 
 
 XX. . . . . - — - = cos 0- + a sin (T ; 
 
 /8 + 7 a + /3 y + a 
 
 which admits of several verifications. 
 
 (18.) We may, for instance, transform it as follows (comp. 297, LXVII.) : 
 
 XXI -(« + ig)(<3+y)(y + «) ^ -2e+2a(l + ^+m + n) 
 ' • • K(a-l-/3)(i3f y) (y + a) + 2^+ 2a(l + ^+ w + n) 
 
 . , - , , 1 + a tan - cos - + a sm - 
 
 _l4-/ + w + n-fca_ 2 2 2 
 
 l + / + m + » — ea _ a a , a 
 
 \ ~ a tan ~ cos - — a sm - 
 
 2 2 2 
 
 -[ 
 
 i- + a sm - = cos (T + a sm <T, as above. 
 
 * In the next Article, we shall consider a case of indeterminateness, or of the ex- 
 istence of indefinitely many exscribed triangles def : namely, when the sides of abc 
 are all equal to quadrants. 
 
 t This opportunity may be taken of referring to an interesting Note, to pages 
 96, 97 of Luby's Trigonometry (Dublin, 1852); in which an elegant construction, 
 connected with the area of a spherical triangle, is acknowledged as having been men- 
 tioned to Dr. Luby, by a since deceased and lamented friend, the Rev. William Digby 
 Sadleir, F.T.C.D. A construction nearly the same, described in the sub-articles to 
 297, was suggested to tlie present writer by quaternions, several years ago. 
 
CHAP. I.] CASE OF SIDES EQUAL TO QUADRANTS. 347 
 
 (19.) This seems to be a natural place for observing (comp. (16.) ), that if a, j8, 
 y, d be any four vectors^ the lately cited equation 294, LIII., and the square of the 
 equation 294, XV., with S written in it instead of p, conduct easily to the following 
 very general and symmetric formula : 
 
 XXII. . . a2/32y252 + (S,8ySa^)2+ (SyaS/3^)H (Sa(3Sydy 
 
 + 2a^SPyS(35Sy5 + 2/3^SyaSy^Sa5 + 2y^Sa[5SaSS(5d + 2S^Sa(5S^ySya 
 
 = 2SyaSal3Sl3SSyd + 28a(3SI3ySydSa8 + 2S[3ySyaSadS(3d 
 
 + ^2y2(Sa^)3 + y2a2 (8/3^)2+ a2/32(Sy^)2 
 
 + a2^2(S/3y)2 + /32^2(Sya)2 + y2^2(Sa/3)2. 
 
 (20.) If then we take anr/ spherical quadrilateral abcd, and write 
 
 XXIII. . . r = cos AD = — SVad, m' = cos bd = - SU/3^, «' = cos cd = &c., 
 
 treating a, (3, y as the unit-vectors of the points A, b, c, and /, m, n as the cosines 
 
 of the arcs bc, ca, ab, as in 297, (1.), we have the equation, 
 
 XXIV. . . 1 + M'2 + m^ra"- + n^n'a + 2Zm'»'+ 2mnl' + 2nl'm\ 2lmn 
 
 = 2mnm'n' + 2nln'l' + 2lml'm 
 
 + Z34 m2+ n2 + Z'2 + m'2 + n'2 ; 
 
 which can be confirmed by elementary considerations,* but is here given merely as 
 an interpretation of the quaternion formula XXII. 
 
 (21.) In squaring the lately cited equation 294, XV., we have used the two 
 following formulae of transformation (comp. 204, XXIL, and 210, XVIII.), in 
 which a, /3, y may be any thr^e vectors^ and which are often found to be useful : 
 XXV. . . (Va/3)2 = (Sa/3)2 - a2/32 ; XXVI. .. S (V/3y . Vy a) = y2Sa/3 - S/3ySya. 
 
 299- The two cases, for which the three sides «, b^ c, of the given 
 triangle abc, are all less, or all greater, than quadrants, having been 
 considered in the two foregoing Articles, with a reduction, in 298, 
 (10.) and (11.), of certain other cases to these, it only remains to 
 consider that third principal case, for which the sides of that given 
 triangle are all equal to quadrants : or to inquire what is, on our 
 general principles, the Fourth Proportional to Three Rectangular 
 Vectors. And we shall find, not only that tJiis fourth proportional 
 is not itself a Vector, but that it does not even contain any vector 
 part (292) different from zero : although, as being found to be equal 
 to a Scalar, it is still included (131, 276) in the general conception 
 of a Quaternion. 
 
 (1.) In fact, if we suppose, in 297, (1.), that 
 
 I. . . Z = 0, TO = 0, n = 0, or that II. . . a = 5 = « 
 
 * A formula equivalent to this last equation of seventeen terms, connecting the 
 six cosines of the arcs which join, two by two, the corners of a spherical quadrilateral 
 abcd, is given at page 407 of Carnot's Geometric de Position (Paris, 1803). 
 
348 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 or III. . . S/37 = Sya = Sa/3 = 0, while IV. . . Ta = T/3=:Ty = 1, 
 
 the formulae 297, VII. give, 
 
 V. ..a = 0, £ = 0, ^=0; 
 but these are the vector parts of the three pairs of fourth proportionals to the three 
 rectangular unit-lines, a, (3, y, taken in all possible orders ; and the same evane- 
 scence of vector parts must evidently take place, if the three given lines be only at 
 right angles to each other, without being equally long. 
 
 (2.) Continuing, however, for simplicity, to suppose that they are unit lines, and 
 that the rotation round a from /3 to y is negative, as before, we see that we have now 
 r=0, and e=l, in 297, (3.); and that thus the six fourth proportionals reduce 
 themselves to their scalar parts, namely (here) to positive or negative unit?/. In 
 this manner we find, under the supposed conditions, the values : 
 
 VI. . . j3a-'y = y/3-Ja = ay-i/3 = +l; VI'. . . ya-^(3 =al3-^y = l3y-^a = -l. 
 
 (3.) For example (comp. 295) we have, by the laws (182) of i, j, A, the values, 
 
 VII. . . ij-^k =jk-H = ki-^j = + 1 ; VII'. . . Jcj-H = ik-^j =ji-^k = - 1. 
 
 In fact, the two fourth proportionals, ij'^k and kj-^i, are respectively equal to the 
 two ternary products, — ijk and - kji, and therefore to + 1 and - 1, by the laws in- 
 cluded in the Fundamental Formula A (183). 
 
 (4.) To connect this important result with the constructions of the two last Ar- 
 ticles, we may observe that when we seek, on the general plan of 298, (15.), to 
 exscrihe a spherical triangle, def, to a given tri-quadrantal (or tri-rectangular) 
 triangle, ABC, as for instance to the triangle ijk (or jik) of 181, in such a manner 
 that the sides of the new triangle shall be bisected by the corners of the old, the 
 problem is found to admit of indefinitely many solutions. Any point p may be as- 
 sumed, in the interior of the given triangle abc ; and then, if its reflexions D, E, f 
 be taken, with respect to the three sides a, b, c, so that (comp. Fig. 72) the arcs 
 PD, PE, PF are perpendicularly bisected by those 
 three sides, the three other arcs ef, fd, de will be 
 bisected by the points A, b, c, as required : because 
 the arcs ae, ap have each the same length as ap, 
 and the angles subtended at a by pe and pf are to- 
 gether equal to two right angles, &c. 
 
 (5.) The positions of the auxiliary points, d, e, 
 F, are therefore, in the present case, indeterminate, 
 or variable ; but the sum of the angles at those three 
 points is constant, and equal to four right angles ; 
 because, by the six isosceles triangles on pd, pe, pf as bases, that sum of the 
 three angles d, e, f is equal to the sum of the angles subtended by the sides of the 
 given triangle abc, at the assumed interior point p. The spherical excess of the 
 triangle def is therefore equal to two right angles, and its area 2 = tt ; as may be 
 otherwise seen from the same Figure 72, and might have been inferred from the for- 
 mula 297, LV., or LVI. 
 
 (6.) The radius od, in the formula 297, XLVII., for the fourth proportional 
 /3a-'y, becomes therefore, in the present case, indeterminate ; but because the angle 
 i/dp., or ^ (tt - S), in the same equation, vanishes, the formula becomes simply 
 
CHAP. I.] OTHER VIEW OF A FOURTH PROPORTIONAL. 349 
 
 /3a-iy = 1, as in the recent equations VI. ; and similarly in other examples, of the 
 class here considered. 
 
 (7.) The conclusion, that the Fourth Proportional to Three Rectangular Lines 
 is a Scalar, may in several other ways be deduced, from the principles of the present 
 Book. For example, with the recent suppositions, we may write, 
 
 Vlir. ..^a-i = -y, y(3-^ = -a, ay-i=-/3; 
 Vlir. . . ya-J= + /3, a/3-»= + y, /3y-» = + a; 
 
 the three fourth proportionals VI. are therefore equal, respectively, to — y^, -a^, 
 - (3^, and consequently to + 1 ; while the corresponding expressions VI'. are equal 
 to + /3*, + y2, + a2, and therefore to - 1. 
 
 (8.) Or (comp. (3.) ) we may write generally the transformation (comp. 282, 
 
 XXI.*") 
 
 IX. . . /3a-iy = a-2./3ay, if a-3=l: aS 
 
 in which the factor a'^ is always a scalar, whatever vector a may be ; while the 
 vector part of the ternary product j3ay vanishes, by 294, III., when the recent con- 
 ditions of rectangularity III. are satisfied. 
 
 (9.) Conversely, this terna?-y product jSay, and this fourth proportional (3a-^y, 
 can never reduce themselves to scalars, unless the three vectors a, (3, y (supposed to 
 be all actual (Art. 1)) are perpendicular each to each. 
 
 Section 8. — On an equivalent Interpretation of the Fourth 
 Proportional to Three Diplanar Vectors^ deduced from the 
 Principles of the Second Book. 
 
 300. In the foregoing Section, we naturally employed the results 
 of preceding Sections of the present Book, to assist ourselves in at- 
 taching a definite signification to the Fourth Proportional (297) 
 to Three Diplanar Vectors ; and thus, in order to interpret the sym- 
 bol /3a"^7, we availed ourselves of the interpretations previously ob- 
 tained, in this Third Book, of a"' as a line, and of a/3, u^^ as quater- 
 nions. But it may be interesting, and not uninstructive, to inquire 
 how the equivalent symbol, 
 
 I. . . {^\a),^i, or -7, with 7 not \\\ a, /3, 
 
 might have been interpreted, on the principles of the Second Booh, with- 
 out at first assuming as known, or even seeking to discover, any in- 
 terpretation of the three lately mentioned symbols, 
 II. . . a-', up, 0^7. 
 
 It will be found that the inquiry conducts to an expression of the 
 form, 
 
 * The formula here referred to should have been printed as Ra = 1 : a = a-*. 
 
350 ELEMENTS OF QUATERNIONS. [bOOK HI. 
 
 III. . . (i3:a).7=^^-ew; 
 where S is the same vector, and e is the same scalar, as in the recent 
 sub-articles to 297; while u is employed as a temporary symbol, to 
 denote a certain Fourth Proportional to Three Rectangular Unit 
 Lines, namely, to the three lines oq, ol', and op in Fig. 68; so 
 that, with reference to the construction represented by that Figure, 
 we should be led, by the principles of the Second Book, to write the 
 equation : 
 
 IV. . . (oB : oa) . oc = CD . cos JS + (ol' : oq) . op. sin ^2. 
 And when we proceed to consider what signification should be at- 
 tached, on the principles of the same Second Book, to that particular 
 fourth proportional, which is here the coefficient of sin 12, and has 
 been provisionally denoted by u, we find that although it may be 
 regarded as being in one sense a Line, or at least homogeneous with a 
 line, yet it must not he equated to any Vector: being v2ii\mT analogo^is, 
 in Geometry, to the Scalar Unit of Algebra, so that it may be naturally 
 and conveniently denoted by the usual symbol 1, or + 1, or be equated 
 to Positive Unity. But when we thus write u=\, the last term 
 of the formula III. or IV., of the present Article, becomes simply 
 e, or sin -^2 ; and while this term (or part) of the result comes to be 
 considered as a species of Geometrical Scalar, the complete Expres- 
 sion for the General Fourth Proportional to Three Diplanar Vectors 
 takes the Form of a Geometrical Quaternion: and thus the fortnula 
 297, XL VII., or 298, VIIL, is reproduced, at least if we substitute 
 in it, for the present, (/3: a).7 for ^a^r^, to avoid the necessity of 
 interpreting here the recent symbols II. 
 
 (1.) The construction of Fig. 68 being retained, but no principles peculiar to the 
 Third Book being employed, we may write, with the same significations of c, jo, &c., 
 as before, 
 
 V. . . OB : OA = OR : OQ = cos c + (ol' : oq) sin c ; 
 VI. . . oc = OQ . cos/> + OP . sin/) . 
 
 (2.) Admitting then, as is natural, for the purposes of the sought interpretation, 
 that distributive property which has been proved (212) to hold good for the multi- 
 plication of quaternions (as it does for multiplication in algebra); and writing for 
 abridgment, 
 
 VII. . . M = (ol' : oq) . op; 
 
 we have the quadrinomial expression : 
 
 VIII. . . (oB : oa). oc = ol'. sin c cos/)i- OQ . cos ccos/> 
 + OP . cos c sin j3 + « . sin c sin j9 ; 
 
 in which it may be observed that the sum of the squares of the four coefficients of the 
 
CHAP. I.] SCALAR UNIT IN SPACE. 351 
 
 three rectangular unit-vectors, oq, ol', op, and of their fourth proportional, m, is 
 equal to unity, 
 
 (3.) But the coefficient of this fourth proportional, which may be regarded as a 
 species oi fourth unit, is 
 
 IX, . . sin c sin p = sin mn = sin 12 = e ; 
 we must therefore expect to find that the three other coefficients in VIII., when di- 
 vided by cos 12, or by r, give quotients which are the cosines of the arcual distances 
 of some point x upon the unit-sphere, from the three points i!, Q, p ; or that a point 
 X can be assigned, for which 
 
 X. . . sin c cosp = /' cos l'x ; cos c cos /? = r cos Qx ; cos c sin p = r cos px. 
 
 (4.) Accordingly it is found that these three last equations are satisfied, when we 
 substitute d for x ; and therefore that we have the transformation, 
 
 XI. . . OL,'. sinccos/j + OQ .cose cos/? + OP.coscsin/j = OD . cosiS = ^, 
 whence follow the equations IV. and III. ; and it only remains to study and interpret 
 t\\Q fourth unit, u, which enters as a factor into the remaining part of the quadrlno- 
 mial expression VIII., without employing any principles except those of the Second 
 Booh : and therefore without using the Interpretations 278, 284, of (3a, &c. 
 
 301. In general, when two sets of three vectors, a, /3, 7, and 
 «'» ^\ 7'* are connected by the relation, 
 
 1. ..--—= 1, or 11... ,= -7^ 
 
 a Y P a ^ a' 
 
 it is natural to write this other equation, 
 
 III. . .-7 = - 7 ; 
 a ft 
 
 and to say that these two fourth proportionals (297), to a, /3, 7, and 
 to a', [i\ 7^ are equal to each other: whatever the /wZ^ signification oi 
 each of these two last symbols III., supposed for the moment to be 
 not yet fully known, may be afterwards found to be. In short, we 
 may propose to make it a condition of the sought Interpretation, on 
 the principles of the Second Book, of the phrase, 
 
 ^''Fourth Proportional to Three Vectors,'^ 
 
 and of either of the two equivalent Symbols 300, I., that the recent 
 Equation III. sha.\\ follow from I. or II.; just as, at the commence- 
 ment of that Second Book, and before concluding (112) that the ge- 
 neral Geometric Quotient /3: a of any two lines in space is a Quaternion, 
 we made it a condition (103) of the interpretation of such a quotient, 
 that the equation {fi:a).a = jS should be satisfied. 
 
 302. There are however two tests (comp. 287), to which the re- 
 cent equation III. must be submitted, before its final adoption; in 
 
352 ELEMENTS OF QUATERNIONS. [bOOK 111. 
 
 order that we may be sure of its consistency^ 1st, with the previous 
 interpretation (226) of a Fourth Proportional to Three Complanar 
 Vectors, as a Line in their common plane; and Ilnd, with the gene^ 
 ral principle of all mathematical language (105), that things equal to 
 the same thing, are to be considered as equal to each other. And it 
 is found, on trial, that both these tests are home : so that they form 
 no objection to our adopting the equation 301, III., as true hy defini- 
 tion^ whenever the preceding equation II., or I., is satisfied. 
 
 (1.) It may happen that the first member of that equation III. is equal to a.line 
 df as in 226 ; namely, when a, j3, y are complanar. In this case, we have by II. 
 the equation, 
 
 y y Y a a a ' 
 
 so that a', /3', y are also complanar (among themselves), and the line B is their 
 fourth proportional likewise : and the equation III, is satisfied, both members being 
 symbols for one common line, ^, which is in general situated in the intersection of 
 the two planes, ajSy and a'jS'y' ; although those planes may happen to coincide, 
 without disturbing the truth of the equation. 
 
 (2.) Again, for the more general case oi diplanarity of a, /3, y, we may con- 
 ceive that the equation* II. co-exists with this other of the same form, 
 
 V. . . ^ 1- = ^ ; which gives VI. . . ^ y =Cr", 
 a y a a a 
 
 if the definition 301 be adopted. If then that definition be consistent with general 
 principles of equality, we ought to find, by III. and VI., that this third equation be- 
 tween two fourth proportionals holds good : 
 
 VII. . . ^'y' = ^'y" ; or that VIII. . . ^L = ^ 
 a a a y a 
 
 when the equations II. and V. are satisfied. And accordingly, those two equations 
 give, by the general principles of the Second Book, respecting quaternions considered 
 as quotients of vectors, the transformation, 
 
 (B'y' /3 y y' /3 y jS" 
 
 — -i^ = C: ± . _L_ = c: _i- = '--^ as required. 
 
 a' y ay y' ay" a 
 
 303. It is then permitted to interpret the equation 301, III., on 
 the principles of the Second Book, as being simply a transformation 
 (as it is in algebra) of the immediately preceding equation II., or I.; 
 and therefore to write, generally, 
 
 I. . . 57 = 2V> if II. . . 5(7:7')=?'; 
 
 * In this and other cases of reference, the numeral cited is always supposed to be 
 the one which (with the same number) has last occurred before, although perhaps 
 it may have been in connexion with a shortly preceding Article. Compare 217, (1.). 
 
CHA1\ T.] FOURTH PROPORTIONAL RESUMED. 353 
 
 where 7, 7' are any two vectors^ and q, q' are any tivo quaternions^ 
 which satisfy this last condition. Now, if v and v be any two right 
 quaternions^ we have (by 193, comp. 283) the equation, 
 
 III, . . Iv'.lv' - v'.v' = vv'^ ; 
 or 
 
 IV. . . v~^{l.v: Iv') - «j'"^ ; whence V. . . v^ . Iv = v'-^. lv\ 
 
 by the principle which has just been enunciated. It follows, then, 
 that '■''if a right Line (Iv) he multiplied hy the Reciprocal (v") of the 
 Right Quaternion (v), of which it is the Index, the Product {v^lv) is 
 independent of the Lengthy and of the Direction, of the Line thus ope- 
 rated on ;" or, in other words, that this Product has one common Va- 
 lue., for all possible Lines (a) in Space: which common or constant 
 value may be regarded as a kind of new Geometrical Unit, and is equal 
 to what we have lately denoted, in 300, III., and VII., by the tem- 
 porary symbol u; because, in the last cited formula, the line op is 
 the index of the right quotient oq,: ol'. Retaining, then, for the 
 moment, this symbol, u, we have, for every line a in space, considered 
 as the index of a right quaternion, v, the four equations : 
 
 VI. . . v'^a = u ; VII. . . a = Vlt ; VIII. . . V- a:u; 
 
 IX. . . V"' = w: «; 
 in which it is understood that a = Jv, and the three last are here re- 
 garded as being merely transfoi^mations o{ the fivst, which is deduced 
 and interpreted as above. And hence it is easy to infer, that for 
 any given system of three rectangular lines a, /3, 7, we have the general 
 expression : 
 
 X. . . (/3 : a) . 7 = XU, if aJ-^,^JL^,<^_i,a\ 
 
 where the scalar co-efficient, x, of the new unit, u, is determined by 
 
 the equation, 
 
 XI. . .a; = ±(Ty3:Ta).T7, according as XII. . . U7 = + Ax. (a: /3). 
 
 This coefficient x is therefore always equal, in magnitude (or absolute 
 quantity), to the fourth proportional to the lengths of the three given 
 lines 0^7 ; but it is positively or negatively taken, according as the 
 rotation round the third line 7, from the second line /3, to the first line 
 a, is itseM positive or negative: or in other words, according as the 
 rotation round the first line, from the second to the third, is on the 
 contrary negative ox positive (compare 294, (3.) ). 
 
 (I.) In illustration of the constancy of that fourth proportional whicli has been, 
 for the present, denoted by u, while the system of the three rectangular unit-lines 
 
 2 z 
 
354 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 from \vhi<'h it is conceived to be derived is in any manner turned about, we may ob- 
 serve that the three equations, or proportions, 
 
 XIII. . . u : y =j3: a ; y:a = a:-y; i8:-y = y:/3, 
 
 conduct immediately to this fourth equation of the same kmd, 
 
 XIV. . . M:a = y:/3, or* « = (y:j3).a; 
 
 if we admit that this new quantity, or symbol, u, is to be operated on at all, or com- 
 bined with other symbols, according to the general rules of vectors and quaternions. 
 
 (2.) It is, then, permitted to change the three letters a, /3, y, by a cyclical per- 
 mutation, to the three other letters /3, y, a (considered again as representing unit- 
 lines), without altering the value of the fourth proportional, w, or in other woi'ds, it 
 is allowed to make the system of the three rectangular lines revolve, through the third 
 part of four right angles^ round the interior and co-initial diagonal of the unit-cube, 
 of which they are three co-initial edges. 
 
 (3.) And it is still more evident, that no such change of value will take place, if 
 we merely cause the system of the two first lines to revolve, through any angle, in 
 its own plane, round the third line as an axis ; since thus we shall merely substitute, 
 for the factor (i : «, another factor equal thereto. But by combining these two last 
 modes of rotation, we can represent ang rotation whatever, round an origin supposed 
 to be fixed. 
 
 ('{.) And as regards the scalar ratio of any one fourth proportional, such as 
 (3' : a' . y', to any other, of the kind here considered, such as j3 : a . y, or «, it is suffi- 
 cient to suggest that, mthout any real change in the former, we are allowed to sup- 
 pose it to be so prepared, that we shall have 
 
 XV. ..a' = a; /3' = /3; y' = xy; 
 
 X being some scalar coefficient, and representing the ratio required. 
 
 304. In the more general case, when the three given lines are 
 not rectangular, nor unit-lines, we may on similar principles de- 
 termine their fourth proportional, without referring to Fig. 68, as 
 follows. Without any real loss of generality, we may suppose that 
 the planes of a, /3 and a, 7 are perpendicular to each other; since 
 this comes merely to substituting, if necessary, for the quotient 
 )3 : a, another quotient equal thereto. Having thus 
 I. . . Ax.(/3:a) JL Ax.(7:a), let II. . . /3 = /3' + )3'^ ry = y + y', 
 where /3' and 7' are parallel to a, but ^" and 7'' are perpendicular 
 to it, and to each other; so that, by 203, I. and II., we shall have 
 the expressions, 
 
 III. ..^' = S^.a, y=S^.a, 
 a tt 
 
 * In equations of this form, the parentheses may be omitted, though for greater 
 clearness they are here retained. 
 
CHAP. I.] SPHEUICAL PARALLELOGRAM. 355 
 
 and W... ^" = Y^.a, y/ = V^.o. 
 
 a a 
 
 We may then deduce, by the distributive principle (300, (2.) ), the 
 t ran s formations, 
 
 
 a a a a 
 
 where 
 VI.. 
 
 . ^ = )3S^+7''S^=7S- + /3''S^, and VII. . . o^w = ^' 7' 
 a a a a a 
 
 The latter part, xu, is what we have called (300) the (geometrically) 
 scalar part, of the sought fourth proportional ; while the former part 
 B may (still) be called its vector part : and we see that this part is 
 represented by a line^ which is at once m thetwo planes^ of /3, 7'', and 
 of 7, ^" ; or in two planes which may be generally constructed as fol- 
 lows, without now assuming that the planes ajS and ar^ are rectangu- 
 lar, as in I. Let 7' be the projection of the line 7 on the plane of 
 a, j3, and operate on this projection by the quotient yS: a as a multi- 
 plier ; the plane which is drawn through the line /3 : a . 7' so obtained, 
 at right angles to the plane a^, is one locus for the sought line d : 
 and the plane through 7, which is perpendicular to the plane 77^ 
 is another locus for that line. And as regards the length of this line, 
 or vector part ^, and the magnitude (or quantity) of the scalar part 
 xu, it is easy to prove that 
 
 VIII. . . T^ = / cos 5, and IX. . . a; = + ^sin 5, 
 where 
 
 X... . ^ = T/3:Ta.T7, and XI. . . sin 5 = sine sin p, 
 
 if c denote the angle between the two given lines a, )3, and jo the 
 inclination of the third given line 7 to their plane: the sign of the 
 scalar coefficient, x, being positive or negative, according as the rota- 
 tion round a from yS to 7 is negative or positive. 
 
 (L) Comparing the recent construction with Fig. 68, we see that when the con- 
 dition L is satisfied, the four unit-lines Uy, Ua, U/3, Vd take the directions of the 
 four radii oc, oq, or, od, which terminate at the four comers of what may be called 
 a tri -rectangular quadrilateral CQRD on the sphere. 
 
 (2.) It may be remarked that the area of this quadrilateral is exactly equal to 
 h(dfthe area 2 of the triangle def ; which may be inferred, either from the circum- 
 
356 ELEMENTS OF QUATERNIONS. [boOK 111. 
 
 stance that its spherical excess (over four right angles) is constructed by the angle 
 MDN ; or from the triangles dbr and eas being together equal to the triangle abf, 
 60 that the area of desk is 2, and therefore that of cqrd is ^S, as before. 
 
 (3.) The two sides CQ, qr of this quadrilateral, which are remote from the obtuse 
 angle at d, being still called p and c, and the side cd which is opposite to c being 
 still denoted by c', let the side dr which is opposite to p be now called p' ; also let 
 the diagonals CR, qd be denoted by d and d' ; and let s denote the spherical excess 
 (ODR - ^tt), or the area of the quadrilateral. "We shall then have the relations, 
 
 !cos d = cosp cos c ; cos d' = cosp cos c' ; 
 tanc'= cosp tan c ; tan p' = cos c tan j» ; 
 cos s = cos p sec/>' = cos c sec c' = cos d sec d' ; 
 
 of which some have virtually occurred before, and all are easily proved by right-an- 
 gled triangles, arcs being when necessary prolonged. 
 
 (4.) If we take now two points, A and b, on the side qr, which satisfy the arcual 
 equation (comp. 297, XL., and Fig. 68), 
 
 XIII. . . (^ AB= nQB; 
 
 and if we then join AC, and let fall on this new arc the perpendiculars bb', dd' ; it 
 is easy to prove that the projection b'd' of the side bd on the arc AC is equal to that 
 arc, and that the angle dbb' is right : so that we have the two new equations, 
 
 XIV. . . n b'd' = o AC ; XV. . . dbb' = |7r ; 
 
 and the neiv quadrilateral bb'd'd is also tri-rectangular. 
 
 (5.) Hence the point d may be derived from the three points A'BC, by any two of 
 the four following conditions: 1st, the equality XIII. of the arcs ab, qr ; Ilud, the 
 cori'espondiug equality XIV. of the arcs AC, b'd'; Ilird, the tri-rectangular charac- 
 ter of the quadrilateral CQRD ; IVth, the corresponding character of bb'd'd. 
 
 (6.) In other words, this derived point D is the common intersection of the four 
 perpendiculars, to the four arcs ab, ac, cq, bb', erected at the four points R, d', C, b ; 
 CQ, bb' being still the perpendiculars from c and b, on ab and AC; and r and d' 
 bohig deduced from Q and b', by equal arcs, as above. 
 
 305. These consequences of the construction employed in 297, 
 &c., are here mentioned merely in connexion with that theory of 
 fourth proportionals to vectors, which they have thus served to illus- 
 trate; but they are perhaps numerous and interesting enough, to 
 justify us in suggesting the name^ ''^ Sp>herical Parallelogram,''^* for 
 the quadrilateral cabd, or bacd, in Fig. 68 (or 67) ; and in proposing 
 to say that d is the Fourth Point, which completes such d^ parallelogram, 
 when the three points c, A, B, or B, a, c, are given upon the sphere, 
 {kS first, second, and third. It must however be carefully observed, 
 that the analogy to the plane is here thus far imperfect, that in the 
 
 * By the same analogy, the quadrilateral cqrd, in Fig. 68, may be called a 
 Sjiherical Rectangle. 
 
CHAP. 1.] SERIES OF SniERICAL PARALLELOGRAMS. 357 
 
 gefieral case, when the three given points are not co-arcucd, but on the 
 contrary are corners of a spherical triangle abc, then if we take c, d, b, 
 or B, D, c, for the three first points of a new spherical parallelogram^ of 
 the kind here considered, the new fourth pointy say a„ will not coin- 
 cide with the old second point a; although it will very nearly do so, 
 if the sides of the triangle abc be small: the deviation aAj being in 
 fact found to be small of the third order, if those sides of the given 
 triangle be supposed to be small of the first order; and being always 
 directed towards the foot of the perpendicular, let fall from a on bc. 
 
 (L) To investigate the Zaw of this deviation, let /3, y be still any two given 
 unit-vectors, ob, oc, making with each other an angle equal to a, of which the co- 
 sine is I ; and let p or op be any third vector. Then, if we write, 
 
 I. . . pi = ^(p) = ANp. -y+-/3 , OQ=Up, OQi = Upi, 
 \9 P I 
 
 the new or derived vector, <pp or pi, or OPi, will be the common vector pai't of the 
 two fourth proportionals, to p, /3, y, and to p, y, (3, multiplied hy the square of the 
 length of Q ; and BQCQi will be what we have lately called a spherical parallelogram. 
 We shall also have the transformation (compare 297, (2.)), 
 
 IL..pi = 0p=^S^+yS|-pS|; 
 
 and the distributive symbol of operation <p will be such that 
 
 III...^p|||Ay, and >V = P, if Plll/^,y; 
 but IV. ..^p = -Zp, if p II Ax. (y : /3). 
 
 (2.) This being understood, let 
 
 V. ..p = p' + p"; ^p' = p'i; p'lli/3, y, p"|| Ax.(y:/3); 
 so that p', or op', is the projection of p on the plane of (3y ; and p", or op", is the 
 part (or component) of p, which is perpendicular to that plane. Then we shall have 
 an indefinite series of derived vectors, pi, pg) P3» • • or rather two such series, suc- 
 ceeding each other alternately, as follows : 
 
 VI. . . fP^'^'^P"^ f'^ ~ '^ " ' P2 = <P^9 = p' + l^p" ; 
 
 lp3 = <p^p = p'l - i^p"'-, p4 = 0V = p' + i^p" ; &c- ; 
 
 the two series of derived points, Pi, P2, P3, P4, . . . being thus ranged, alternately, 
 on the two perpendicular SfW' and PiP'i, which are let fall from the points p and Pi, 
 on the given plane BOO ; and the intervals, PP2, P1P3, P2P4, • • . forming a geometri- 
 cal progression, in which each is equal to the one before it, multiplied by the con- 
 stant factor - I, or by the negative of the cosine o£ the given angle boc. 
 
 (3.) If then this angle be still supposed to be distinct from and tt, and also 
 in general from the intermediate value ^tt, we shall have the two limiting values, 
 
 VII. . . p2n = p', p2rt+l = p'l, if n = 00 ; 
 or in words, the derived points r2, P4, . . of even orders, tend to the point p', and the 
 other derived points, Pi, 1% . . oi odd orders, tend to the other point p'l, as limiting 
 
358 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 positions: these two limit points being the feet of the two (rectilinear) perpendicu- 
 lars, let fall (as above) from p and p' on the plane boc. 
 
 (4.) But even \h.Q first deviation ppg, is small of the third order, if the length Tp 
 of the line op be considered as neither large nor small, and if the sides of the spheri- 
 cal triangle BQC be small otth.Q first order. For we have by VI. the following ex- 
 pressions for that deviation, 
 
 VIII. . . pp2 = p2-p = (^^ -l)p"=-sina2.sinpa.Tp .Up"; 
 
 where pa denotes the inclination of the line p to the plane (3y ; or the arcual perpen- 
 dicular from the point Q on the side bc, or a, of the triangle. The statements lately 
 made (305) are therefore proved to have been correct. 
 
 (5.) And if we now resume and extend the spherical construction, and conceive 
 that Di is deduced from baiC, as Ai was from bdc, or d from bac ; while A2 may 
 be supposed to be deduced by the same rule from bd^c, and D2 from BA2C, &c., 
 through an indefinite series of spherical parallelograms, in which t\i.Q fourth point 
 of any one is treated as the second point of the next, while the first and third points 
 remain constant : we see that the points Ai, A2, . . are all situated on the arcual 
 perpendicular let fall from A on bc ; and that in like manner the points Dj, D2, . . 
 are all situated on that other arcual perpendicular, which is let fall from d on bc. 
 We see also that the ultimate positions, a<x. and Dw, coincide precisely with the feet 
 of those two perpendiculars : a remarkable theorem, which it would perhaps be diffi- 
 cult to prove, by any other method than that of the Quaternions, at least with calcu- 
 lations so simple as those wliich have been employed above. 
 
 (6.) It may be remarked that the construction of Fig. 68 might have been other- 
 wise suggested (comp. 223, IV.), by the principles of the Second Book, if we had 
 sought to assign i]ie fourth proportional (297) to three right quaternions; for ex- 
 ample, to three right versors, v, v', v", whereof the unit lines a, (5, y should be sup- 
 posed to be the axes. For the result would be in general a quaternion v'v^v", with 
 e for its scalar part, and with d for the itidex of its right part : e and d denoting 
 the same scalar, and the same vector, as in the sub-articles to 297. 
 
 306. Quaternions may also be employed to furnish a new con- 
 struction, which shall complete (comp. 305, (5.)) the ^mj^/izW deter- 
 mination of the two series of derived points, 
 
 I. . . D, Ai, D„ A2, D2, &C., 
 
 when the three points a, b, c are given upon the unit-sphere ; and 
 thus shall render visible (so to speak), with the help of anew Figure, 
 the tendencies of those derived points to approach, alternately and 
 indefinitely, to the/ee^, say D'and a', of the two arcual perpendiculars 
 let fall from the two opposite corners, d and a, of the first spherical 
 parallelogram, baod, on its given diagonal bc ; which diagonal (as we 
 have seen) is common to all the successive parallelograms. 
 
 (1.) The given triangle abc being supposed for simplicity to have its sides ahc 
 less than quadrants, as in 297, so that their cosines Imn are positive, let a', b', c' be 
 
CHAP. 
 
 ••] 
 
 CONSTRUCTION OF THK SERIES. 
 
 359 
 
 the feet of the perpendiculars let fall on these three sides from the points A, b, c ; 
 also let M and n be two auxiliary points, determined by the equations, 
 
 II. . . r> BM = r> MC, ^ AM = r\ mn ; 
 so that the arcs an and bc bisect each other in m. Let fall from n a perpendicular 
 nd' on BC, so that 
 
 III. . . «-> bd'= n a'c ; 
 
 and let b", o" be two other auxiliary points, on the sides b and c, or on those sides 
 prolonged, which satisfy these two other equations, 
 
 IV. . . o b'b" = r^ AC, f^ C'C" = n AB, 
 
 (2.) Then the perpendiculars to these last sides, CA and AB, erected at these last 
 points, b" and c", will intersect each other in the point D, which completes (ZQb^ the 
 spherical parallelogram bacd ; and the foot of the perpendicular from this point d, 
 on the third side bc of the given triangle, will coincide (comp. 305, (2.) ) with the 
 foot d' of the perpendicular on the same side from n ; so that this last perpendicular 
 nd' is one locus of the point D. 
 
 (3.) To obtain another locus for that point, adapted to our present purpose, let 
 E denote now* that new point in which the two diagonals, ad and bc, intersect each 
 other ; then because (comp. 297, (2.) ) we have the expression, 
 
 V. . . OD = u(mj3 + ny - ?a), 
 we may write (comp. 297, (25.), and (30.)), 
 
 VI. . . OE = u (m/3 + ny), whence VII. . . sin be : sin ec = w : m = cos ba' : cos a'c ; 
 the diagonal ad thus dividing the arc bc into segments, of which the sines are pro- 
 portional to the cosines of the adjacent sides of the given triangle, or to the cosines 
 of their projections ba' and a'c on bc ; so that the greater segment is adjacent to the 
 lesser side, and the middle point M of bc (1.) lies between the points a' and E. 
 
 (4.) The intersection e is therefore a known point, and the great circle through 
 A and e is a second known locus for 
 D ; which point may therefore be 
 found, as the intersection of the arc 
 AE prolonged, with the perpendicular 
 nd' from N (1.). And because e lies 
 (3.) beyond the middle point m of bc, 
 with respect to the foot a' of the per- 
 pendicular on bc from a, but (as it 
 is easy to prove) not so far beyond 
 M as the point d', or in other words 
 falls between M and d' (when the arc 
 BC is, as above supposed, less than a 
 quadrant), the prolonged arc ae cuts 
 nd' between N and d'; or in other 
 words, the perpendicular distance of 
 the sought fourth point D, from the 
 given diagonal BC of the parallelo- 
 gram, is less than the distance of the 
 given second point A, from the same given diagonal, (Compare the annexed Fig. 73.) 
 
 Fig. 73. 
 
 It will be observed that m, n, e have not here the same significations as in 
 
360 ELKMENTS OF QUATERNIONS. [bOOK III. 
 
 (6.) Proceeding next (305) to derive a new point Ai from b, i>, c, as d has been 
 derived from b, a, c, we see that we have only to determine a new* auxiliary point 
 F, by the equation, 
 
 VIII. . . --> EM = r. MF ; 
 
 and then to draw df, and prolong it till it meets a a' in the required point Ai, which 
 will thus complete the second parallelogram, bdcai, with bc (as before) for a given 
 diagonal. 
 
 (6.) In like manner, to complete (comp. 305, (5.) ), the third parallelogram, 
 BAiCDi, with the same given diagonal bc, we have only to draw the arc AiE, and 
 prolong it till it cuts nd' in Di ; after which we should find the point A2 of a fourth 
 successive parallelogram BD1CA2, by drawing DiF, and so on for ever. 
 
 (7.) The constant and indefinite tendency, of the derived points d, Di, . . to the 
 limit-point d', and of the other (or alternate^ derived points Ai, Ag, . • to the other 
 limit-point a', becomes therefore evident from this new construction ; the final (or 
 limiting') results of which, we may express by these two equations (comp. again 
 
 305,(5.)), 
 
 IX. . . Dd) = d' ; A<p = a'. 
 
 (8.) But the smallness (305) of the first deviation AAi, when the sides of the 
 given triangle abc are small, becomes at the same time evident, by means of the 
 same construction, with the help of the formula VII. ; which shows that the intervalf 
 EM, or the equal interval mf (5.), is small of the third order, when the sides of the 
 given triangle are supposed to be small of ihe first order: agreeing thus with the 
 equation 305, VIII. 
 
 (9.) The theory of such spherical parallelograms admits of some interesting ap- 
 plications, especially in connexion with spherical conies ; on which however we can- 
 not enter here, beyond the mere enunciation of a Theorem, % of which (comp. 271) 
 the proof by quaternions is easy : — 
 
 Fig. 68 ; and that the present letters c' and c" correspond to q and r in that Fi- 
 gure. 
 
 * This new point, and the intersection of the perpendiculars of the given trian- 
 gle, are evidently not the same in the new Figure 73, as the points denoted by the 
 same letters, f and p, in the former Figure 68 ; although the four points A, b, c, d 
 are conceived to bear to each other the same relations in the two Figures, and indeed 
 in Fig. 67 also ; bacd being, in that Figure also, what we have proposed to call a 
 spherical parallelogram. Compare the Note to (3.). 
 
 t The formula VII. gives easily the relation, 
 
 VII'. . . tan EM = tan ma' ( tan - T ; 
 
 hence the interval em is small of the third order, in the case (8.) here supposed ; and 
 
 generally, if o < -, as in (1,), while 6 and c are unequal, the formula shows that this 
 
 interval em is less than ma', or than d'm, so that e falls between m and d', as in (4.), 
 X This Theorem was communicated to the Royal Irish Academy in June, 1845, 
 as a consequence of the principles of Quaternions. See the Proceedings of that date 
 (Vol. III., page 109). 
 
CHAP. I.J THIRD INTERPRETATION OF A PRODUCT. 3G I 
 
 " T/'klmn be any spherical quadrilateral, and.l any point on the sphere ; if also 
 we complete the spherical parallelograms, 
 
 X. . . KILA, LIMB, MINC, NIKD, 
 
 and determine the poles E and F of the diagonals km and ln of the quadrilateral : 
 then these two poles are the foci* of a spherical conic, inscribed in the derived quadri- 
 lateral ABCD, or touching its four sides." 
 
 (10.) Hence, in a notationf elsewhere proposed, we shall have, under these con« 
 ditions of construction, the formula : 
 
 XL . . EF (. .) ABCD ; or XI'. . . EF (. .) BCDA ; &C. 
 
 (11.) Before closing this Article and Section, it seems not irrelevant to remark, 
 that the projection y' of the unit-vector y, on the plane of a and /3, is given by the 
 formula, 
 
 _,__ , a sin a cos B + /3 sin i cos A 
 
 XII. . . y = . ; 
 
 smc 
 
 and that therefore the point p, in which (see again Fig. 73) the three arcual perpen- 
 diculars of the triangle abc intersect, is on the vector, 
 
 XIII. . . p = a tan a + /3 tan B + y tan c. 
 
 (12.) It may be added, as regards the construction in 305, (2.), that the right 
 lines, 
 
 XIV. . . PPi, P1P2, P2P3, P3P4, . . • 
 
 however far their series may be continued, intersect the given plane boc, alternately, 
 in two points s and T, of which the vectors are, 
 
 VTT 9 1 + Ip' P'+Ip'l 
 
 XV...03=-j^, OT=-^-; 
 
 and which thus become two fixed points in the plane, when the position of the point 
 p in space is given, or assumed. 
 
 Section 9. — On a Third Method of interpreting a Product or 
 Function of Vectors as a Quaternion ; and on the Consis- 
 tency of the Results of the Interpretation so obtained^ with 
 those which have been deduced from the two preceding Me- 
 thods of the present Book. 
 
 307. The Conception of the Fourth Proportional to Three 
 Rectangular Unit-Lines^ as being itselfa species of i^6W?*^^ Uyiit 
 in Geometry^ is eminently characteristic of the present Calcu- 
 lus ; and offers a Third Method of interpreting a Product of 
 two Vectors as a Quaternion : which is however found to be 
 
 * In the language of modem geometry, the conic in question may be said to 
 touch eight given arcs ; four real, namely the sides ab, bc, CD, da ; and/owr ima- 
 ginary, namely two from each of the focal points, B and F. 
 
 t Compare the Second Note to page 295. 
 
 3 A 
 
362 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 consistent^ in all its results^ with the two former methods (278, 
 284) of the present Book ; and admits of being easily extended 
 to products of three or more lines in space ^ and generally to 
 Functions of Vectors (289). In fact we have only to conceive* 
 
 * It was in a somewhat aaalogous way that Des Cartes showed, in his Geome- 
 <na (Schooten's Edition, Amsterdam, 1659), that all products and powers of lines, 
 considered relatively to their lengths alone, and without any reference to their direc- 
 tions, could be interpreted as lines, by the suitable introduction of a line taken for 
 unity, however high the dimension of the product or power might be. Thus (at 
 page 3 of the cited work) the following remark occurs: — 
 
 " Ubi notandum est, quod per a2 vel 6^, similesve, communiter, non nisi lineas 
 omnino simplices concipiam, licet illas, ut nominibus in Algebra usitatis utar, Qua- 
 drata aut Cubos, &c. appellem." 
 
 But it was much more difficult to accomplish the corresponding multiplication of 
 directed lines in space ; on account of the non-existence of any such line, which is 
 symmetrically related to all other lines, or common to all possible planes (comp. the 
 Note to page 248). The Unit of Vector -Multiplication cannot properly be itself a. 
 Vector, if the conception of the Symmetry of Space is to be retained, and duly com- 
 bined with the other elements of the question. This difficulty however disappears, 
 at least in theory, when we come to consider that new Unit, of a scalar kind (300), 
 which has been above denoted by the temporary symbol u, and has been obtained, 
 in the foregoing Section, as a certain Fourth Proportional to Three Rectangular 
 Unit-Lines, such as the three co-initial edges, AB, AC, ad of what we have called an 
 Unit- Cube : for this fourth proportional, by the proposed conception of it, undergoes 
 no change, when the cube abcd is in any manner moved, or turned ; and therefore 
 may be considered to be symmetrically related to all directions of lines in space, or to 
 all possible vections (or translations) of a pointy or body. In fact, we conceive its de- 
 termination, and the distinction of it (as + u) from the opposite unit of the same kind 
 (— «), to depend only on the tisual assumption of an uiiit of length, combined with 
 the selection of a hand (as, for example, the right hand), rotation towards which 
 hand shall be considered to he positive, and contrasted (^as such) with rotation to- 
 wards the other hand, round the same arbitrary axis. Now in whatever manner the 
 supposed cube may be thrown about in space, the conceived rotation round the edge 
 AB, from AC to AD, will have the same character, as right-handed or left-handed, at 
 the end as at the beginnhig of the motion. If then the fourth proportional to these 
 three edges, taken in this order, be denoted by + «, or simply by + 1, at one stage of 
 that arbitrary motion, it may (on the plan here considered) be denoted by the same 
 symbol, at er^ery other stage: while the opposite character of the (conceived) rota- 
 tion, round the same edge ab, from AD to AC, leads us to regard the fourth propor- 
 tional to AB, AD, AC as being on the contrary equal to — «, or to — 1, It is true that 
 this conception of a new unit for space, symmetrically related (as above) to all linear 
 directions therein, may appear somewhat abstract and metaphysical ; but readers 
 who think it such can of course confine their attention to the rules of calculation , 
 which have been above derived from it, and from other connected considerations : and 
 which have (it is hoped) been stated and exemplified, in this and in a fonner Vo- 
 lume, with sufficient clearness and fullness. 
 
CHAP. I.] CONCEPTION OF THE FOURTH UNIT. 363 
 
 that each proposed vector, a, is divided by the neio or fourth 
 unit, u, above alluded to ; and that the quotient so obtained, 
 which is always (by 303, VIII.) the ripht quaternion T^a, 
 whereof the vector a is the index^ is substituted for that vec- 
 tor ; the resulting quaternion being finally, if we think it con- 
 venient, multiplied into the same fourth unit. F6r in this way 
 we shall merely reproduce the process of 284, or 289, although 
 now as a consequence of a different train of thought ^ or of a dis- 
 tinct but Consistent Interpretation : which thus conducts, by a 
 new Method, to the same Rules of Calculation as before. 
 
 (1.) The equation of the unit-sphere, p2 + i = Q (282, XIV.), may thus be con- 
 ceived to be an abridgment of the following fuller equation : 
 
 i...(ey=-i; 
 
 \uj 
 the quotient p : u being considered as equal (by 303) to the rigfit quaternion, I'/o, 
 which must here be a right versor (154), because its square is negative unity. 
 (2.) The equation of the ellipsoid, 
 
 T(tp + pfc) = fc2 - t2 (282, XIX.), 
 may be supposed, in like manner, to be abridged from this other equation : 
 
 \u u uu j \u I \tt/ 
 and similarly in other cases. 
 
 (3.) We might also write these equations, of the sphere and ellipsoid, under these 
 other, but connected forms : 
 
 III...^p = -«; IV...Tf-p+- 
 
 u 
 
 with intepretations which easily offer themselves, on the principles of the foregoing 
 Section. 
 
 (4.) It is, however, to be distinctly understood, that we do not propose to adopt 
 this Form of Notation, in the practice of the present Calculus : and that we merely 
 suggest it, in passing, as one which may serve to throw some additional light on the 
 Conception, introduced in this Third Book, of a Product of two Vectors as a Qua- 
 ternion. 
 
 (5.) In general, the Notation of Products, which has been employed throughout 
 the greater part of the present Book and Chapter, appears to be much more conve- 
 nient, for actual use in calculation, than any Notation of Quotients : either such as 
 has been just now suggested for the sake of illustration, or such as was employed in 
 the Second Book, in connexion with that First Conception of a Quaternion (112), 
 to which that Book mainly related, as the Quotient of two Vectors (or of two di- 
 rected lines in space). The notations of the two Books are, however, intimately con- 
 nected, and the former was judged to be an useful preparation for the latter, even as 
 
3G4 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 regarded the quotient-forms of many of the expressions used : while the Characteris- 
 tics of Operation, such as 
 
 S, V, T, U, K, N, 
 
 are employed according to exactly the same laws in both. In short, a reader of the 
 Second Book has nothing to unlearn in the Third; although he may be supposed to 
 have become prepared for the use of somewhat shorter and more convenient pro- 
 cesses, than those before employed. 
 
 Section 10 — On the Interpretation of a Power of a Vector 
 as a Quaternion. 
 
 308. The only symbols, of the kinds mentioned in 277, 
 which we have not yet interpreted, are the cube a% and the 
 general power a\ of an arbitrary vector base^ a, with an arbi- 
 trary scalar exponent, t ; for we have already assigned inter- 
 pretations (282, (1.), (14.), and 299, (8.)) for th^ particular 
 symbols a^, a'S a"'^, which are included in this last^rm. And 
 we shall preserve those particular interpretations if we now 
 define, in fall consistency with the principles of the present and 
 preceding Books, that this Power a* is generally a Quaternion, 
 which may be decomposed into two factors, of the tensor and 
 vers or kinds, as follows : 
 
 I. ..a^=Ta^Ua'; 
 
 IV denoting the arithmetical value of the t^^"- power of the po- 
 sitive number Ta, which represents (as usual) the length of the 
 base-line a ; and Ua^ denoting a versor, which causes any line 
 p, perpendicular to that line a, to revolve round it as an axis, 
 through t right angles^ or quadrants, and in a positive or nega- 
 tive direction, according as the scalar exponent, t, is itself a 
 positive or negative number (comp. 234, (5.) ). 
 
 (1.) As regards the omission of parentheses in the formula I., we may observe 
 that the receut definition, or interpretation, of the symbol a*, enables us to write 
 (comp. 237, 11. III.), 
 
 II. . . T(aO = (Tay = Ta«; III. . . U (a*) = (Ua)< = Ua*. 
 
 (2.) The ascis and angle of the power a*, considered as a quaternion, are generally 
 determined by the two following formulae : 
 
 IV. . . Ax. a< = ± Ua ; V. . . ^ . a' = 2n7r ± ^tn ; 
 the signs acctmipanying each other, and the (positive or negative or null) integer, «, 
 being so chosen as to bring the angle within the usual limits, and it. 
 
CHAP. 1.] POWER OF A VECTOR A QUATERNION. 365 
 
 (3.) In general (comp. 235), we may speak of the (positive or negative) product 
 i<7r, as being the amplitude of the same power, with reference to the line a as an 
 axis of rotation ; and may write accordingly, 
 
 VI. , . am. a* = ^tir. 
 
 (4.) We may write also (comp. 234, VII. VIII.), 
 
 VII. . . Ua< = cos Y + Ua . sin — ; or briefly, VIII. . . Ua« = cas —. 
 
 (5.) In particular, 
 
 IX. . . Ua^"^ cas «7r = ± 1 ; IX'. . . Ua2«+i= ± Ua ; 
 upper or lower signs being taken, according as the number n (supposed to be whole) 
 is even or odd. For example, we have thus the cubes, 
 
 X. . . Ua3 = -Ua; X'. . . a3 = -aNa. 
 
 (6.) The coiijugate and norm of the power a' may be thus expressed (it being 
 remembered that to turn a line -^ a through - |f7r round + a, is equivalent to turn- 
 ing that line through + Itir round -a): 
 
 XI. . . Ka< = Ta« . Ua' = (- a)« ; XII. . . Na* = Ta*« ; 
 parentheses being unnecessary, because (by 295, VIII.) Ka = — a. 
 
 (7.) The scalar, vector, and reciprocal of the same power are given by the for- 
 mulae : 
 
 XIII. . . S.a« = Ta<.cos~; XIV. . . V. a« = Ta^.Ua. sin ^; 
 
 2 A 
 
 XV. . . 1 : a^= Ta-«.Ua-' = a-«= Ka« : Na< (comp. 190, (3.)). 
 (8.) If we decompose any vector p into parts p' and p", which are respectively 
 parallel and perpendicular to a, we have the general transformation :* 
 
 XVI. . . atpa-t=^at{p + p") a-«= p' + Va^K p", 
 = the new vector obtained by causing p to revolve conically through an angular quan- 
 tity expressed by tir, round the line a as an axis (comp. 297, (15.)), 
 (9.) More generally (comp. 191, (5.) ), if q be any quaternion, and if 
 XVII. ..a*qa-*=q, 
 the new quaternion q is formed from q by such a conical rotation of its own axis 
 Ax. 5, through tir, round a, without any change of its angle L q, or of its tensor Tq. 
 (10.) Treating ijk as three rectangular unit-lines (295), the symbol, or expres- 
 sion, 
 
 XVIII. . .p = rktjskj-^kt, or XIX. . . p = r¥j^^k^^, 
 in which 
 
 XX. ..r>0, s>0, s^l, t^O, *<2, 
 
 may represent any vector ; the length or tensor of this line p being r ; its inclina- 
 tion\ to k being sir ; and the angle through which the variable />Zane kp may be 
 
 * Compare the shortly following sub-article (11.). 
 
 t If we conceive (compare tlie first Note to page 322) that the two hnes i andy 
 are directed respectively towards the south and west points of the horizon, while the 
 third line k is directed towards the zenith^ then sir is the zenith-distance of p; and 
 tTT is the azimuth of the same line, measured /rom south to west, and thence (if ne- 
 cessary) through north and east, to south again. 
 
366 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 conceived to have revolved, frem the initial position ki, with an initial direction to- 
 wards the position kj, being t-jr. 
 
 (11.) In accomplishing the transformation XVI., and in passing from the ex- 
 pression XVIII. to the less symmetric but equivalent expression XIX., we employ 
 the principle that 
 
 XXI. . . */-* = S-i = - K (kj-o) =j^k ; 
 
 which easily admits of extension, and may be confirmed by such transformations as 
 VII. or VIII. 
 
 (12.) It is scarcely necessary to remark, that the definition or interpretation I., 
 of the power a* of an?/ vector a, gives (as in algebra) the exponential property, 
 
 XXII. ..a*a< = a«+«, 
 
 whatever scalars may be denoted by s and t ; and similarly when there are more than 
 two factors of this form. 
 
 (13.) As verifications of the expression XVIII., considered as representing a vec- 
 tor, we may observe that it gives, 
 
 XXIII... p = -Kp; and XXI V. . . p2 = _ r2. 
 
 (14.) More generally, it will be found that if m* be any scalar, we have the 
 eminently simple transformation : 
 
 XXV. . . |0« = (rk^j^kj-^k-^y = r^'ktfktfj-«k-*. 
 
 In fact, the two last expressions denote generally two equal quaternions, because 
 they have, 1st, equal tensors, each = r" ; Ilnd, equal angles, each = L (^'0 ; and 
 Ilird, equal (or coincident) axes, each formed from + A by one common system of 
 two successive rotations, one through stt round j, and the other through tn round k. 
 
 309. Ani/ quaternion, q, which is not simply a scalar^ may 
 be brought to i\\Qform a\ by a suitable choice of the base, a, 
 and of the exponent, t ; which latter may moreover be supposed 
 to fall between the limits and 2 ; since for this purpose we 
 have only to write, 
 
 1...^=^^; II. . .Ta = T^^ III. . .Ua = Ax.^; 
 
 TT 
 
 and thus the general dependence of a Quaternion, on a Scalar 
 and a Vector Element, presents itself in a new ivay (comp. 17, 
 207, 292). When the proposed quaternion is a versor, T^- = 1, 
 
 * The emplojonent of this letter u, to denote what we called, in the two preced- 
 ing Sections, a. fourth unit, &c., was stated to be a merely temporary' one. In gene- 
 ral, we shall henceforth simply equate that scalar unit to the number one ; and die- 
 note it (when necessary to be denoted at all) by the usual symbol, 1, for that num- 
 ber. 
 
CHAP. I.] EXPRESSIONS FOR VERSORS AS POWERS. 367 
 
 we have thus Ta = 1 ; or in other words, the base a, of the 
 equivalent jooi^er a', is an unit-line. Conversely, every versor 
 may be considered as a power of an unit-line^ with a scalar ex- 
 ponent^ t^ which may be supposed to be m. general positive, ^-rA 
 less than two ; so that we may write generally^ 
 
 lY...Vq^a\ with V. . .a = Ax.y = T-U, 
 and VI. . . ^ > 0, t<2\ 
 
 although if this versor degenerate into 1 or - 1, the exponent 
 t becomes or 2, and the base a has an indeterminate or ar- 
 bitrary direction. And from such transformations ofversors 
 new methods may be deduced, for treating questions of sphe- 
 rical trigonometry, and generally of spherical geometry. 
 
 (1.) Conceive that p, q, k, in Fig. 46, are replaced by a, b, c, with unit-vec- 
 tors a, j3, y as usual ; and let a;, y, z be three scalars between and 2, determined 
 by the three equations, 
 
 VII. . . x7r = 2A, ^7r = 2B, 27r = 2c', 
 
 where a, b, c denote the angles of the spherical triangle. The three versors, indi- 
 cated by the three arrows in the upper part of the Figure, come then to be thus de- 
 noted : 
 
 VIII. . . 9 = a^ ; 9' = /32/ ; q'q = y2-z . 
 
 so that we have the equation, 
 
 IX. . . /3J/a*= 72-a ; or X. . . y^^va^^- 1 ; 
 
 from which last, by easy divisions and multiplications, these two others immediately 
 
 follow : 
 
 X'. . . a^y^i^v = - 1 ; X", . . ^va^'y^ = - 1 ; 
 
 the rotation round a from /3 to y being again supposed to be negative. 
 (2.) In X. we may write (by 308, VIII.), 
 
 XI. . . a»^ = casA ; /3J' = c/3sb; y« = cySC; 
 and then the formula becomes, for any spherical triangle, in which the order of ro- 
 tation is as above : 
 
 XII. . . cysc . c/3sB . caSA = — 1; 
 or (com p. IX.), 
 
 XIII. . . - COS c + y sin c = (cos b + jS sin b) (cos a + a sin a). 
 (3.) Taking the scalars on both sides of this last equation, and remembering that 
 S/3a= - cos c, we thus immediately derive one form of ihQ fundamental equation of 
 spherical trigonometry ; namely, the equation, 
 
 XIV. . . cos c + cos a cos b = cos c sin a sin b, 
 (4.) Taking the vectors, we have this other formula : 
 
 XV. . . y sin c = a sin a cos B + jS sin b cos a + V/3« sin a sin E ; 
 which is easily seen to agree with 306, XII., and may also be usefully compared 
 with the equation 210, XXXVII. 
 
368 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (5.) The result XV. may be euunciated in the form of a Theorem^ as follows : — 
 " If there be any spherical triangle abc, and three lines he drawn from the 
 centre O of the sphere, one towards the point a, with a length = sin A cos B ; another 
 towards the point b, with a length = sin b cos A ; and the third perpendicular to the 
 plane aob, and towards the same side of it as the point C, with a length = sin c sin A 
 sin B ; and if with these three lines as edges, we construct a parallelepiped : the 
 intermediate diagonal from o will he directed towards c, and will have a length 
 = sinc." 
 
 (6.) Dividing both members of the same equation XV. by p, and taking scalars, 
 we find that if p be any fourth point on the sphere, and q ih.^ foot of the perpendi- 
 cular let fall from this point on the arc ab, this perpendicular pq being considered as 
 positive when c and p are situated at one common side of that arc (or in one common 
 hemisphere, of the two into which the great circle through a and b divides the sphe- 
 ric surface), we have then, 
 
 XVI. . . sin c cos pc = sin a cos b cos pa + sin b cos a cos pb + sin a sin b sin c sin pq ; 
 
 a formula which might have been derived from the equation 210, XXXVIIL, by first 
 cyclically changing aftcABC to 6caBCA, and then passing from the former triangle to 
 its polar, or supplementary : and from which many less general equations may be 
 deduced, by assigning particular positions to p. 
 
 (7.) For example, if we conceive the point p to be the centre of the circumscribed 
 small circle abc, and denote by R the arcual radius of that circle, and by s the 
 se7nisum of the three angles, so that 2s=A + B4-c=7r + <T, if<7 again denote, as in 
 297, (47.), the area^ of the triangle abc, whence 
 
 XVII. . . PA = PB = po = iE, and sin pq = sin R sin (s — c), 
 the formula XVI. gives easily, 
 
 XVIII. . . 2 cot ^ sin — = sin a sin b sin c : 
 2 
 
 a relation between radius and area, which agrees with kno\^n results, and from which 
 we may, by 297, LXX., &c., deduce the known equation : 
 
 abc 
 XIX. . . e tan i? = 4 sin - sin - sin - ; 
 2 2 2' 
 
 in which we have still, as in 297, (47.), &c., 
 
 XX. . . e = (Sa/3y =) sin a sin & sin c = &c. 
 (8.) In like manner we might have supposed, in the corresponding general equa- 
 tion 210, XXXVIII., that p was placed at the centre of the inscribed small circle, 
 and that the arcual radius of that circle was r, the semisum of the sides being s ; 
 and thus should have with ease deduced this other known relation, which is a sort 
 of polar reciprocal of XVI II., 
 
 XXI . . . 2 tan r . sin s = e. 
 But these results are mentioned here, only to exemplify the fertility of the formulae, 
 to which the present calculus conducts, and from which the theorem in (5.) was 
 early seen to be a consequence. 
 
 Compare the Note to the cited sub-article. 
 
CHAP. I.] EXPONENTIAL FORMULA FOR THE SPHERE. 369 
 
 (9.) We might devefope the ternary product in the equation XII., as we deve- 
 loped the 6^nary/Jrorfttc< XIII. ; compare scalar and vector parts; and operate on 
 the latter, by the symbol S . p-\ New general theorems, or at least new general 
 forms, wonld thus arise, of which it may be sufficient in this place to have merely 
 suggested the investigation. 
 
 (10.) As regards the order of rotation (1.) (2.), it is clear, from a mere inspec- 
 tion of the formula XV., that the rotation round y from /3 to a, or that round c from 
 B to A, must be positive, when that equation XV. liolds good; at least if the angle* 
 A, B, c, of the triangle ABC, be (as usual) treated as positive : because the rotation 
 round the line Yj3a from /3 to a is always positive (by 281, (3.) ). 
 
 (11.) If, then, for any given spherical triangle, ABC, with angles still supposed 
 to be positive, the rotation round c from b to a should happen to be (on the con- 
 trary) negative, we should be obhged to modify the formula XV. ; which could be 
 done, for example, so as to restore its correctness, by interchanging a with j8, and at 
 the same time A with b. 
 
 (12.) There is, however, a sense in which the formula might be considered as 
 still remaining true, without any change in the mode of writing it ; namely, if we were 
 to interpret the symbols A, b, c as denoting negative angles, for the case last sup- 
 posed (11.)- Accordingh', if we take the reciprocal of the equation X., we get this 
 other equation, 
 
 XXII. . . a-^/3-yy-^=-l; 
 
 where x, y, z are positive, as before, and therefore the new exponents, —x, —y, — z, 
 are negative, if the rotation round a from j8 to y be iV^e//" negative, as in (1). 
 
 (13.) On the whole, then, if a, j3, y be any given system of three co-initial and 
 diplanar unit-lines, OA, OB, oc, we can always assign a system of three scalars, 
 X, y, z, which shall satisfy the exponential equation X., and shall have relations of 
 the form VII. to the spherical angles A, b, c; but these three scalars, if determined 
 so as to fall between the limits + 2, will be all positive, or all negative, according as 
 the rotation round a from /3 to y is negative, as in (1.), or positive, as in (11.). 
 
 (14.) As regards the limits just mentioned, or the inequalities, 
 
 XXIII. .. a; < 2, y<2, z<2; x>-2, y>-2, z>-2, 
 
 they are introduced with a view to render the problem of finding the exponents xyz 
 in the formula X. determinate ; for since we have, by 308, 
 
 XXIV. . .a4 = ^4 = y4=+l, if Ta = T/3 = Ty = l, 
 
 we might otherwise add any multiple (positive or negative) of the number four, to 
 the value of the exponent of any unit-line, and the value of the resulting /jower would 
 not be altered. 
 
 (15.) If we admitted exponents = + 2, we might render the problem of satisfy- 
 ing the equation X. indeterminate in another way ; for it would then be sufficient to 
 suppose that any one of the three exponents was thus equal to + 2, or —2, and that 
 the two others were each = ; or else that all three were of the form + 2. 
 
 (16) When it was lately said (13.), that the exponents, x, y, z, in the formula 
 X., if limited as above, would have one common sign, the case was tacitly excluded, 
 for which those exponents, or some of them, when multiplied each by a quadrant, 
 give angles not equal to those of the spherical triangle abc, whether positively or 
 
 3 B 
 
370 ELEMENTS OF QUATERKIONS. [bOOK III. 
 
 negatively taken ; but equal to the supplements of those angles, or to the negatives 
 of those supplements. 
 
 (17.) In fact, it is evident (because a^ = /32 = 72 = _ 1), that the equation X., or 
 the reciprocal equation XXII., if it be satisfied by any one system of values of xj/z, 
 will still be satisfied, when we divide or multiply any two of the three exponential 
 factors, by the squares of the two unit-vectors, of which those factors are supposed to 
 he powers: or in other words, if we subtract or add the number two, in each of two 
 exponents, 
 
 (18.) We may, for example, derive from XXII. this other equation : 
 XXV. . . a2 ^/3^-3/y-s' = - 1 ; or XXVI. . . a^-^(3^-y= y'-^ ; 
 which, when the rotation is as supposed in (1.), so that xyz are positive, maybe in- 
 terpreted as follows. 
 
 (19.) Conceive a lune cc', with points A and b on its two bounding semicircles, 
 and with a negative rotation round A from b to c ; or, what comes to the same thing, 
 with a positive rotation round A from b to c'. Then, on the plan illustrated by Fi- 
 gures 45 and 46, the supplements tt - A, 7r — B, of the angles A and b in the triangle 
 ABC, or the angles at the sa7ne points A and b in the co-lunar triangle abc', will 
 represent two versors, a multiplier, and a multiplicand, which are precisely those 
 denoted, in XXVI., by the two factors, a^"^ and (S^-v ; and the product of these two 
 factors, taken in this order, is that third versor, which has its axis directed to o', 
 and is represented, on the same general plan (177), by the external angle of the lune, 
 at that point c' ; which, in quantity, is equal to the external angle of the same lune 
 at c, or to the angle rr-c. This product is therefore equal to that power of the 
 
 2 
 unit-line oc', or - y, which has its exponent = - (tt — c) = 2 — z ; we have there- 
 fore, by this construction, the equation, 
 
 XXVII. . . a2-*/3«-y = (-y)2-«; 
 which (by 308, (6.) ) agrees with the recent formula XXVI. 
 
 310. The equation, 
 
 2c 2b 2a 
 
 I. . . 7'^P'^d^ = -l, 
 which results from 309, (1.), and in which a, j3, 7 are the 
 unit-vectors oa, ob, oc of any three points on the unit-sphere ; 
 while the three scalars a, b, c, in the exponents of the three 
 factors, represent generally the angular quantities of rotation, 
 round those three unit-lines, or radii, a, j3, 7, from the plane 
 Aoc to the plane aob, from boa to bog, and from cob to coa, 
 and are positive or negative according as these rotations of 
 planes are themselves positive or negative : must be regarded 
 as an important formula, in the applications of the present 
 Calculus. It includes^ for example, the whole doctrine of 
 Spherical Triangles; not merely because it conducts, as we 
 
CHAP. I.] SPHERICAL SUM OF ANGLES. 371 
 
 have seen (309, (3.) ), to one form of the fundamental scalar 
 equation of spherical trigonometry^ namely to the equation, 
 
 II. . . cos c + cos A cos B = COS c sin A sm b ; 
 but also because it gives a vector equation (309, (4.) ), which 
 serves to connect the angles^ or the rotations^ a, b, c, with the 
 directions* of the radii, a, j3, 7, or OA, ob, go, for any system 
 of three diverging right lines from one origin. It may, there- 
 fore, be not improper to make here a few additional remarks, 
 respecting the nature, evidence and extension of the recent 
 formula I. 
 
 (1.) Multiplying both members of the equation I., by the inverse exponential 
 20 
 y "" , vfe have the transformation (comp. 309, (1.) ) : 
 
 2b 2a 2c 2(7r — c) 
 
 IIL . . j3^ a^ =-y ^ =y '^ . 
 
 2a 
 
 (2.) Again, multiplying both members of I. intof a t, we obtain this other for- 
 mula: 
 
 2c 2b 2a 2(ff — a) 
 
 IV. . . y'' (3^ =-a~^ =a ^ . 
 
 2a 20 
 
 (3.) Multiplying this last equation IV. by a'^, and the equation III. into y"^, 
 we derive these other forms : 
 
 * This may be considered to be another instance of that habitual reference to 
 direction, as distinguished from mere quantity (or magnitude), although combined 
 therewith, which pervades the present Calculus, and is eminently characteristic of 
 it ; whereas Des Cartes, on the contrary, had aimed to reduce all problems of geo- 
 metry to the determination of the lengths of right lines : although (as all who use 
 his co-ordinates are of course well aware) a certain reference to direction is even in 
 his theory inevitable, in connexion with the interpretation of negative roots (by him 
 called inverse or false roots) of equations. Thus in the first sentence of Schooten's 
 recently cited translation (1659) of the Geometry of Des Cartes, we find it said: 
 
 " Omian Geometriae Problemata facile ad hujusmoditerminosreduci possunt, ut 
 deinde ad illorum constructionem, opus tantum sit rectarum quarundam longitudinem 
 cognoscere." 
 
 The very different view of geometry, to which the present writer has been led, 
 makes it the more proper to express here the profound admiration with which he re- 
 gards the cited Treatise of Des Cartes : containing as it does the germs of so large a 
 portion of all that has since been done in mathematical science, even as concerns 
 imaginary roots of equations, considered as marks of geometrical impossibility. 
 
 t For the distinction between multiplying a quaternion into and by a factor, see 
 the Notes to pages 146, 159. 
 
372 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 2a 2c 2b 2b 2a 2c 
 
 V. . . a'T y'r /S" =-1; VI. . . /3^ o*^ y'^ = - 1 ; 
 
 so that cyclical permutation of the letters, a, /3, y, and A, B, c, is allowed in the 
 equation I. ; as indeed was to be expected, from the nature of the theorem which 
 that equation expresses. 
 
 (4.) From either V. or VI. we can deduce the formula: 
 
 2a 2c 2b 2 (tt — b) 
 
 VII. . . a'T y?r = _^ 7r = ^ n ; 
 
 by comparing which with III. and IV , we see that cyclical permutation of letters 
 is permitted, in these equations also. 
 
 (5.) Taking the recijaroca/ (or conjugate) of the equation I., we obtain (com- 
 pare 309, XXII.) this other equation : 
 
 2a 2b 2c 
 
 VIII. . . a~»r j3~7r y T=_l; 
 
 2 (tt — A) 2(7r — B) 2(7r-c ) 
 
 or IX. . . a If (3 If y •" = + 1; 
 
 in which cyclical permutation of letters is again allowed, and from which (or from 
 III.) we can at once derive the formula, 
 
 2a 2b 2c 
 
 X. . . a «• ^" TT = _ y »r. 
 
 (6.) The equation X. may also be thus written (comp. 309, XXVII.) : 
 
 2(7r — A) 2 (TT — B) 2(7r — c) 2 (tt — c) 
 
 XI. . . a '^ TT =.j,~ TT =(-y) T . 
 
 (7.) And all the foregoing equations may be interpreted {cqvc\'^. 309, (19.) ), and 
 at the same time/jrorerf, by a reference to that general construction (177) for the 
 multiplication ofversors, which the Figures 45 and 46 were designed to illustrate; if 
 we bear in mind that a power a*, of an unit-line a, with a scalar exponent, t, is (by 
 308, 309) a versor, which has the effect of turning a line -^ a, through t right an- 
 gles, round a as an axis of rotation. 
 
 (8.) The principle expressed by the equation I , from which all the subsequent 
 equations have been deduced, may be stated in the following manner, if we adopt the 
 definition proposed in an earlier part of this work (180, (4.) ), for the spherical sum 
 of two angles on a spheric surface : 
 
 " For any spherical triangle, the Spherical Sum of the three angles, if taken in a 
 suitable Order, is equal to Two Right Angles." 
 
 (9.) In fact, when the rotation round A from B to c is negative, i{ we spherically 
 add the angle b to the angle a, the spherical sum so obtained is (by the definition 
 referred to) equal to the external angle at c; if then we add to this sum, or supple- 
 ment of c, the angle c itself, we get di final or total sum, which is exactly equal to 
 7r ; addition of spherical angles at one vertex, and therefore in one plane, being ac- 
 complished in the usual manner; but the spherical summation of angles with diffe- 
 rent vertices being performed according to those new rules, which were deduced in the 
 Ninth Section of Book II., Chapter I. ; and were connected (180, (6.) ) with the 
 conception of angular transvection, or of the composition of angular motions, in dif- 
 ferent and successive planes. 
 
CHAP. I.] ADDITION OF ARCS ON A SPHERE. 373 
 
 (10.) "Without pretending to attach importance to the following notation, we may- 
 just propose it in passing, as one which may serve to recall and represent the con- 
 ception here referred to. Using a plus in parentheses, as a symbol or characteristic 
 of such spherical addition of angles, the formula I. may be abridged as follows: 
 
 XII. . . c(+)B(+)A=7r; 
 
 the symbol of an added angle being written to the left of the symbol of the angle to 
 which it is added (comp. 264, (4.) ) ; because such addition corresponds (siS above) 
 to a multiplication ofversors, and we have agreed to write the symbol of the multi- 
 plier to the left* of the symbol of the multiplicand, in every multiplication of qua- 
 ternions. 
 
 311. There is, however, another view of the important equation 
 310, I., according to which it is connected rather with addition of 
 arcs (180, (3.) ), than with addition of angles (180, (4.) ); and may 
 be interpreted) and proved anew^ with the help of the supplementary 
 or polar triangle^ a'b'c', as follows. 
 
 (1.) The rotation round a from b to o being still supposed to be negative, let 
 a', b', c' be (as in 175) the positive poles of the sides bc, ca, ab ; and let a', (5', y' 
 be their unit- vectors. Then, because the rotation round a from y' to /3' is positive 
 (by 180, (2.) ), and is in quantity the supplement of the spherical angle a, the pro- 
 duct y'j3' will be (by 281, (2.), (3.)) a versor, of which a is the axis, and a the 
 angle; with similar results for the two other products, a'y', (5' a'. 
 
 (2.) If then we write (comp. 291), 
 
 I. . . a' = UV/3y, /3' = UVya, y' = UVa|3, 
 supposing that 
 
 II. ..Ta = T/3 = Ty = l, and III. . . Sa/3y > 0, 
 
 we shall have (comp. again 180, (2.) ), 
 
 IV. . . a = UVy'/3', (3 = Way', y = UV)3V, 
 and V. .. A=z.y'/3', B = z.a'y', c = lfS'a'', 
 
 whence (by 308 or 309) we have the following exponential expressions for these 
 three last products of unit-lines, 
 
 2a 2b 2c 
 
 VI. . . y '^' = a~' ; a'y ' = j3^ ; (i'a = y^. 
 
 (3.) Multiplying these three expressions, in an inverted order, we have, there- 
 fore, the new product : 
 
 2c 2b 2a 
 
 VII. . .y-^ ^ a"^ = j3'a'. a'y'. y'jS' = y'2/3'2<i'2 = - 1 ; 
 
 and the equation 310, I. is in this way proved anew. 
 
 (4.) And because, instead of VI., we might have written, 
 
 Compare the Note to page 146. 
 
374 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 - ' — a' — 3' 
 
 VIII. . . a'r=- L; (5^ ■■=--,; y" = --„ 
 P y « 
 
 we see that the equation to be proved may be reduced to the form of the identity 
 
 « 7 /5' 
 aud may be interpreted as expressing, what is evident, that if a point be supposed to 
 move first along the side b'c', of the polar triangle a'b'c', from b' to c' ; then along 
 the successive side c'a', from c' to a' ; and finally along the remaining side a'b', 
 from a' to b', it will thus have returned to the position from which it set out, or will 
 on the whole have not changed place at all. 
 
 (5.) In this view, then, we perform what we have elsewhere called an addition of 
 arcs (instead of angles as in 310) ; and in a notation already used (264, (4.) ), we 
 may express the result by the formula, 
 
 X. . . '^ a'b' + r> c'a' + o b'c' = ; 
 each of the the two left-hand symbols denoting an arc, which is conceived to be added 
 (as a successive vector-arc, 180, {d.) ), to the arc whose symbol immediately /o//ow« 
 it, or is written next it, but towards the right-hand. 
 
 (6.) The expressions VI. or VIII., for the exponential factors in 310, I., show 
 in a new way the necessity of attending to the order of those factors, in that formula : 
 for if we should invert that order, without altering (as in 310, VIII.) the exponents, 
 we may now see that we should obtain this new product : 
 
 2a 2b ic , , 
 
 XI. . . a^ /S"^ y^ =- ^ -, ^ = + (/i8'a')2 ; 
 (5 y a 
 
 which, on account of the diplanarity of the lines a', (3', y', is not equal to negative 
 unity, but to a certain other versor ; the properties of which may be inferred from 
 what was shown in 297, (64.), and in 298, (8.), but upon which we cannot here 
 delay. 
 
 312. In general (comp. 221), an equation^ such as 
 
 1...?'=?, 
 
 between two quaternions, includes a system o//our* scalar equa- 
 tions, such as the following : 
 
 II. . . Sq = ^q; Saq' = Saq ; Sj3^' = S(5q ; Syq = Syq ; 
 
 where a, j3, y may be ani/ three actual and diplanar vectors : 
 and conversely, if* a, /3, y be any three such vectors, then the 
 four scalar equations II. reproduce, and are sufficiently re- 
 
 * The propriety, which such results as this establish, for the use of the name, 
 Quaternions, as applied to this whole Calculus, on account of its essential connexion 
 with the number Four, does not require to be again insisted on. 
 
CHAP. I.] A QUATERNION EQUATION INCLUDES FOUR. 375 
 
 placed by, the one quaternion equation I. But an equation 
 between two vectors is equivalent only to a system of three sca- 
 lar equations^ such as the three last equations II. ; for exam- 
 ple, in 294, (12.), the one vector equationXXll. is equivalent 
 to the three scalar equations XXI., under the immediately 
 preceding condition of diplanarity XX. In like manner, an 
 equation between two versors of quaternions,* such as the equa- 
 tion 
 
 III. ..JJq'=\Jq, 
 
 includes generally a system of three, but of not more than 
 three, scalar equations ; because the versor \]q depends gene- 
 rally (comp. 157) on a system of three scalars, namely the two 
 which determine its axis Ax. q, and the one which determines 
 its angle /. q ; or because the versor equation III. requires to 
 be combined with the tetisor equation, 
 
 IV. . . Tq=Tq, compare 187 (13.), 
 
 in order to reproduce the quaternion equation I. Now the re- 
 cent equation, 310, I., is evidently of this versor-form III., if 
 a, j3, 7 be still supposed to be unit-lines. If then we met that 
 equation, or if one of its form had occurred to us, without any 
 knowledge of its geometrical signification, we might propose to 
 resolve it, with respect to the three scalars a, b, c, treated as 
 three unknown quantities. The few following remarks, on the 
 problem thus proposed, may be not out of place, nor unin- 
 structive, here. 
 
 (1.) Wiitiug for abridgment, 
 
 V, . . cot A = t, cot B = M, cot c = V, 
 and VI. . . « = — cosec a cosec b cosec c, 
 
 the equation to be resolved becomes (by 308, VII., or 309, XII.), 
 
 VII. ..(y + y) («+/S) {t + a) = s; 
 in which the tensors on both sides are already equal, because 
 
 * An equation, Up'= Up, or UV9' = UV9, between two versors of vectors (156), 
 or between the axes of two quaternions (291), is equivalent only to a system of ^ujo 
 scalar equations ; because the direction of an axis^ or of a vector^ depends on a sys- 
 tem of two angular elements (111). 
 
376 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 VIII. . . «2 = (y2 + 1) (a2 + 1) (<« + 1). 
 (2.) Multiplying the equation VII. by t + a, and into t-a, and dividing the re- 
 sult by i^ + 1, we have this new equation of the same form, but differing by cyclical 
 permutation (comp. 310, (3.) ) : 
 
 IX. ..(« + a)(«+y)(«+)8) = «; 
 
 and in like manner, 
 
 X. . . (u + p)(t+a)(:o-\-y) = 8. 
 
 (3.) Taking the half difference of the two last equations, and observing that (by 
 279, IV., and 294, II.) 
 
 XI V|(i3ar-ariS)=V./3Vay = ySa/3-aS/3y, 
 "'\i(l3a-a(i) = Y(3a, K(^y-y^) = Y(3y, 
 we arrive at this new equation, of vector form : 
 
 XII. . . = vYpa + tY(3y + ySa/3 - aS/3y ; 
 which is equivalent only to a system of two scalar equations, because it gives = 0, 
 when operated on by S./8 (comp. 294, (9.) ). 
 
 (4.) It enables us, however, to determine the twoscalars, t and v ; for if we ope- 
 rate on it by S.a, we get (comp. 298, XXVI. ), 
 
 XIII. . . fSa/3y = a2S/3y-S^aSay = S(V/3a.Vay); 
 
 and if we operate on the same equation XII. by S . y, we get in like manner, 
 
 XIV. . . rSa/3y = y2Sa/3 - SaySy(3 = S(Vay.Vy/3). 
 (5.) Processes quite similar give the analogous result, 
 
 XV. . . uSal3y = |32Sya - Sy/3 S/3a = S (Vy/3 . V/3a) : 
 
 and thus the problem is resolved, in the sense that expressions have been found for 
 the three sought scalars t, u, v, or for the cotangents V. of the three sought angles 
 A, B, c : whence the fourth scalar, s, in the quaternion equation VII., can easily be 
 deduced, as follows. 
 
 (6.) Since (by 294, (6.), changing S to a, and afterwards cyclically permuting) 
 we have, for any three vectors a, j3, y, the general transformations, 
 
 XVI. . . aSa/3y = Y(Y(3a . Vay), /3Sa)3y = V(Vy/3 .Yl3a), 
 ySa/3y = V(ay.Vy/3), 
 the expressions XIII. XV. XIV. give, 
 
 Ut +a)Sai3y = Vi3a.Vay; 
 XVII. ..)(u + (B) Sa(5y = Vy/3 .Y(3a ; 
 ((» + y)Sa/3y = Vay .Vy/3; 
 whence, by VII , 
 
 XVIII, . . «(Sa/3y)3 = (Vy/3)2(Vi8a)2 (Vay)2; 
 
 and thus the remaining scalar, s, is also entirely determined. 
 
 ( 7.) And the equation VIII. may be verified, by observing that the expressions 
 
 XVII. give, 
 
 ((«« + 1) (Sa/3y)2 = (V/3a)2 (Vay)2 ; 
 
 XIX. . . («2 + 1) (Sa/3y)2 = (Vy;8)^ (V,8a)2 ; 
 
 ( («2 + 1) (SafSyy = (Vay)2 ( Vy/3)=*. 
 
 (8.) The equations XIII. XIV. XV. XVI. give, by elimination of Saf3y, these 
 
 new expressions : 
 
CHAP. I.J SOLUTION OF THE EXPONENTIAL EQUATION. 377 
 
 XX. . . a<-» = (V : S) (Vi3a . Vay) ; /3«-' = (V : S) (Vy/3 . Y(3a) ; 
 y«-l=(V:S)(Vay.Vy/3); 
 
 by comparing which Avith the formula 281, XXVIII., after suppressing (291) the 
 characteristic I, we find that the three scalars, t, u, v, are either 1st, the cotangents 
 of the angles opposite to the sides a, b, c, of the spherical triangle in which the three 
 given unit-lines a, (3, y terminate^ or Ilnd, the negatives of those cotangents, the 
 angles themselves of that triangle being as usual supposed to hepositive (309, (10.) ), 
 according as the rotation round a from /3 to y is negative or positive : that is (294, 
 (3.) ), according as Sa/3y >or < ; or finally, by XVIIL, according as the fourth 
 scalar, s, is negative or positive, because the second member of that equation XVIII. 
 is ahvays negative, as being the product of three squares of vectors (282, 292). 
 
 (9.) In the 1st case, which is that of 309, (1.), we see then anew, by V. and VI., 
 that we are permitted to interpret the scalars A, B, c, in the exponential formula 
 310, L, as equal to the angles of the spherical triangle (8.), which are usually de- 
 noted by the same letters. But we see also, that we may add any even multiples of 
 TT to those three angles, without disturbing the exponential equation ; or any one 
 even, and two odd multiples of tt, in any order, so as to preserve o, positive product 
 of cosecants, because s is, for this case, negative in VI., by (8.). 
 
 (10.) In the Ilnd case, which is that of 309, (11.), we may, for similar reasons, 
 interpret the scalars A, B, c, in the formula 310, 1., as equal to the negatives of the 
 angles of the triangle; and as thus having, what VI. now requires, because s is now 
 positive (8.), a negative product of cosecants, while their cotangents have the values 
 required. But we may also add, as in (9.), any multiples of tc, to the scalars thus 
 found for the formula, provided that the number of the odd multiples, so added, is 
 itself even (0 or 2). 
 
 (11.) The conclusions of 309, or 310, respecting the interpretation of the expo- 
 nential formula, are therefere confirmed, and might have been anticipated, by the 
 present new analysis : in conducting which it is evident that we have been dealing 
 with real scalars, and with real vectors, only. 
 
 (12.) If this last restriction were removed, and imaginary values admitted, iu 
 the solution of the quaternion equation VII., we might have begun by operating, aa 
 in II., on that equation, by i\\Q four characteristics, 
 
 XXI. . . S, S . a, S . /3, and S . y ; 
 
 which would have given, with the significations 297, (1.), (3.), of/, m, n, and e, 
 and therefore with the following relation between those ybi/r scalar data, 
 
 XXII. . . e2 = 1-/2-^2 -n2+2Zm», 
 
 a system of four scalar equations, involving theyb«r sought scalars, s, t, u, v; from 
 which it might have been required to deduce the (real or imaginary) values of those 
 four scalars, by the ordinary processes of algebra. 
 
 (13.) The four scalar equations, so obtained, are the following: 
 
 = e + lt-\- mu 4- nr — tuv + s ; 
 = c< + tiitu + ntv + «w - Z : 
 ^ . . = - ew + ftw + *w + nuv + m - 2/n ; 
 = ew + <?f + Uv + muv — n ; 
 
 3 c 
 
378 ELEMENTS OF QUATERNIONS. [bOOK III, 
 
 = e(<2 + l) (fiu-n+Z7n)i 
 
 eliminating uv and u between the three last of which, we find, with the help of XXII., 
 the determinant, 
 
 1, mt, ntv -\-et — l 
 XXIV. . . = m, t, Itv + ev-n 
 
 n, li — e, tv-'r 171 — 2171 
 and analogous eliminations give, 
 
 XXV. . . = c(<2+l) (eu-m+nl), 
 and XXVI. . . = (t^ + 1) [e^uv - (m - nl) (»-//») + (1 - /«) (et-l + mn)}. 
 
 (14.) Rejecting then the factor i^ + 1 we find, as the only real solution of the 
 problem (12.), the following system of values : 
 
 XXVII. .. c^ = ^ — »ran; eu = m-nl; ev = n—lm\ 
 and XXVIII. .. e^s = -(l- P) (1 - m^) (1 - n^) ; 
 
 which correspond precisely to those otherwise found before, in (4.) (5.) (6.), and might 
 therefore serve to reproduce the interpretation of the exponential formula (310). 
 
 (15.) But on the purely algebraic side, it is found, by a similar analysis, that 
 the four equations XXIII. are satisfied also by a system offour imaginary solutions^ 
 represented by the following formulae : 
 
 XXIX. ..f'+ 1 = ^5 t,2+l = 0; 
 
 \s = tuv — It — mu — «?j — e = ; 
 
 which it may be sufficient to have mentioned in passing, since they do not appear to 
 have any such geometrical interest, as to deserve to be dwelt on here : though, as 
 regards the consistency of the different processes employed, it may be remembered 
 that in passing (2.) from the equation VII. to IX., after certain preliminary multi- 
 plications, we divided by f^ + 1, as we were entitled to do, when seeking only for real 
 solutions, because t was supposed to be a scalar. 
 
 (16.) This seems to be a natural occasion for remarking that the following gene- 
 ral transformation exists, whatever three vectors may be denoted by a, /3, y : 
 
 XXX. . . S(V/3y .Vya .Va/3) =- (Sa/3y)2 ; 
 
 which proves in a new way (comp. 180), that the rotation round the line Y(3y, from 
 Vya to Va/3, is always positive ; or is directed in the same sense (281, (3.) ), as the 
 rotation round Vaj3 from a to (3, &c. 
 
 (17.) In like manner we have generally, 
 
 XXXI. . . S (Va/3 .Vya .V/3y) = + (Sa/3y)2, 
 and XXXII. . . S (Vy/3 . Vay . V/3a) = + (Sa/3y )2 ; 
 
 80 that the rotation round Yy (3 from Vay to V/3a is negative, whatever arrange- 
 ment the three diplanar vectors a, /3, y may have among themselves. 
 
 (18.) If then a", b", c" be the negative poles of the three successive sides, BC, CA, 
 AB, of any spherical triangle, the rotation round a" from b" to c" is negative: which 
 is entirely consistent with the opposite result (180), respecting the system of the 
 three positive poles a', b', c'. 
 
 (19.) A quantitative interpretation^ of the equation XXX. may also be easily as- 
 signed : for we may infer from it (by 281, (4.), and 294, (3.) )'that (/"oabo be any 
 pyramid, and if normals oa', ob', oo' to the three faces BOC, COA, aob have their 
 lengths numerically equal to the areas of those faces (as bearing the same ratios to 
 
CHAP. I.] EXTENSION TO SPHERICAL POLYGONS. 379 
 
 units^ &c,), then (with a similar reference to units) the volume of the new pyramid^ 
 Oa'b'c', will he three quarters of the square of the volume of the old pyramid^ 
 
 OABC. 
 
 313. But an allusion was made, in 310, to an extension oi 
 the exponential formula which has lately been under discus- 
 sion ; and in fact, that formula admits of being easily extended, 
 from triangles to polygons upon the sphere : for we may write, 
 generally, 
 
 2A„ 2A„_i 2A3 2Ai 
 
 I. . . a„~ an.~ ... 02"^ ai~= (- 1)», 
 
 if A1A3 . . . A„.i A„ be any spherical polygon^ and if the scalars 
 Ai, A2, . . . in the exponents denote the positive or negative 
 angles of that polygon, considered as the rotations a^AiAj, 
 A1A2A3, . . . namely those from AiA„ to A1A2, &c. ; while n is any 
 positive whole number* > 2. 
 
 (1.) One mode of proving this extended formula is the following. Letoc = y 
 be the unit-vector of an arbitrary point c on the spheric surface ; and conceive that 
 arcs of great circles are drawn from this point c to the n successive corners of the 
 polygon. We shall thus have a system of « spherical triangles, and each angle of 
 the polygon will (generally) be decomposed into two (positive or negative) partial 
 angles, which may be thus denoted : 
 
 II. . . CA1A3 = Ai', CA2A3 = A2', . . . ; 
 
 III. . . AnAiC = Ai", A1A2C = A2", . . . ; 
 
 so that, with attention to signs of angles in the additions, 
 
 IV. . . Ai = Ai' + Ai", A2 = A2' + A2", &c. 
 Also let 
 
 v. . . AoCAi = Ci, A3CA2 = C3, &c. ; 
 and therefore 
 
 VI. . . Ci + C2 + . . + C,» = an even multiple of tt, 
 
 which reduces itself to 27r in the simple case of a polygon with no re-entrant angles, 
 and with the point c in its interior. 
 
 (2.) Then, for the triangle CA1A2, of which the angles Are Ci, Ai', A2", we have, 
 by 310, III., the equation, 
 
 2A2" 2Ai' 2Ci 
 
 VII. . . a2 '^ aj T = — y n- ; 
 and in like manner, for the triangle CA2A3, we have 
 
 * The formula admits of interpretation, even for the case n~2. 
 
380 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 2A:j 2A2' 2C2 
 
 VIII. . . 03 '^ a2 '^ - - y "■ » &c. 
 Bat, when we multiply VII. by VIII., we obtain, by IV., the product, 
 
 243" 8A2 2Ai' 2(Ci + C2) 
 
 IX. . . as "^ a2 '^ ai T = + y 'f ; 
 and so proceeding, we have at last, by VI., a product of the form, 
 
 2Ai" 2A;j 2A2 2Ai' 
 
 X. . . ai T a„ T . . . a2 «■ ai «■ = (- 1)'* ; 
 
 2Ai" 2Ai" 
 
 which reduces itself to I., when it is multiplied hy a "^ , and into a "^ (comp. 
 310, (3.) ). The theorem is therefore proved. 
 
 (3.) In words (comp. 310, (8.) ), " the spherical sum of the successive angles of 
 any spherical polygon, if taken in a suitable order, is equal to a multiple of two right 
 angles, which is odd or even, according as the number of the sides (or corners) of the 
 polygon is itself odd or even'''' : the definition formerly given (180, (4.) ), of a Sphe- 
 rical Sum of Angles, being of course retained. And the reasoning may be briefly 
 stated thus. When an arbitrary point c is taken on the spherical surface, as in (1.), 
 the spherical sum of the two partial angles, at the ends of any one side, is the supple- 
 ment of the angle which that side subtends, at the point c ; but the sum of all such 
 subtended angles is either four right angles, or some whole multiple thereof: there- 
 fore the sum of their supplements can differ only by some such multiple from nir, if 
 n be the number of the sides. 
 
 (4.) Whatever that number may be, if we denote by p„ the exponential product 
 in the formula I., we have for every vector p, and for every quaternion q, the equa- 
 tions : 
 
 XI. . . pnppn'^ = p', XII. . . pnqpn'^ = ? ; 
 
 whereof the former may (by 308, (8.), be thus interpreted: — 
 
 " If any line OP, drawn from the centre O of a sphere, he made to revolve coni' 
 cally round any n radii, OAi, . . OA^, as n successive axes of rotation, through an- 
 gles equal respectively to the doubles of the angles of the spherical polygon Ai . .An, 
 the line will be brought back to its initial position, by the composition of these n rota- 
 tions.^^ 
 
 (5.) Another way of proving the extended formula I., for anj' sphencal polygon, 
 is analogous to that which was employed in 311 for the case of a triangle on a sphere, 
 and may be stated as follows. Let Ai', A2', . . . A,/ be the positive poles of the arcs 
 A1A2, A2A3, . . . ArtAi ; and let ai, az, . . . an be the unit- vectors of those n poles. 
 Then the point Ai is the positive pole of the new arc Ai'a,/, and the angle Ai of the 
 polygon at that point is measured by the supplement of that arc ; with similar re- 
 sults for other corners of the polygon. Thus we have the system of expressions 
 (comp. 311, VI.): 
 
 2Ai 2A^ 
 
 XIII. . . ai w = aiUn ; . . . a,i '^ = an'a'n-i ; 
 80 that the product of powers in I. is equal to the following product of n squares of 
 unit-lines, and therefore to the n'^ poiver of negative unity, 
 
CHAP. I.] FORMULA FOR A SPHERICAL QUADRILATERAL. 381 
 
 XIV. . . a'na'n-\ . a'n-ia'„-3 . . . a'oa'i . a'ia'„ = (- 1)" ; 
 and thus the extended theorem is proved anew. 
 
 (6.) This latter process may be translated into another theorem of rotation, on 
 which it is possible that we may briefly return,* in the Second and last Chapter of 
 this Third Book, but upon which we cannot here delay. 
 
 (7.) It may be remarked however here (comp. 309, XII.), that the extended 
 exponential formula I. may be thus written : 
 
 XV. . . CanS An • Ca„-iSAn_i . . . Ca2S A2 . caiS Ai = (- I)'*. 
 
 (8.) For example, if abcd be any spherical quadrilateral, of which the angles 
 (suitably measured) are denoted by A, . . d, so that a represents the positive or ne- 
 gative rotation from ad to AB, &c., while a, j3, y, 5 are the unit vectors of its cor- 
 ners, then 
 
 XVI. . . c^sD .cysc . c/3sB.casA = + l. 
 
 (9.) Hence (comp. 309, XIII.), we may write also, 
 
 XVII. . . (cos c - y sin c) (cos d - ^ sin d) = (cos b + /3 sin b) (cos a + a sin a) ; 
 and therefore, by taking scalars on both sides, and changing signs, 
 
 XVIII. . . - cos c cos D + sin c sin d cos cd = — cos b cos A + sin b sin a cos ba ; 
 
 in fact, each member of this last formula is equal (by 309, XIV.) to the cosine of 
 the angle aeb, or ced, if the opposite sides ad, bc of the quadrilateral intersect in e. 
 (10.) Let jO = OP be the unit vector of any fifth point, p, upon the spheric sur- 
 face; then operating by S . p on XVII., we obtain this other general formula. 
 
 JO = sir 
 'I + sir 
 
 „ = sin A cos B cos ap + sm b cos a cos bp + sin a sin b sin ab sin pq 
 ■ sin c cos D cos cp + sin d cos c cos dp + sin c sin d sin cd sin pr : 
 
 in which the sines of the sides AB, CD are treated as always positive ; but the sines 
 of the perpendiculars pq and PR, on those two sides, are regarded as positive or ne- 
 gative, according as the rotations round p, from A to b and from c to d, are negative 
 or positive : and hence, by assigning particular positions to p, several other but less 
 general equations of spherical tetragonometry can be derived. 
 
 (11.) For example, if we place p at the intersection, say F, of the opposite sides 
 ab, CD, the two last perpendiculars will vanish, and two of the six terms will disap- 
 pear, from the general formula XIX. ; and a similar reduction to four terms will 
 occur, if we make the arbitrary point p the pole of a side, or of a diagonal. 
 
 314. The definition o£ the power a\ which was assigned in 308, 
 enables us to form some useful expressions, by quaternions, for cir- 
 cular^ elliptic, and spii'al loci^ in a given plane, or in space, a few of 
 which may be mentioned here. 
 
 (1.) Let a be any given unit- vector oa, and /3 any other given line ob, perpendi- 
 cular to it ; then, by the definition (308), if we write, 
 
 Compare 297, (24.). 
 
382 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 J...OF = p = a% Ta = l, Sa/3 = 0, 
 the locus of the point p will be the circumference of a circle, with o for centre, and 
 OB for radius, and in a plane perpendicular to OA. 
 
 (2.) If we retain the condition Ta = 1, but not the condition Sa/3 = 0, then the 
 product a^^ will be in general a quaternion, and not merely a vector ; but if we take 
 its vector-part (292), we can form this new vector- expression, 
 II. . . OP = jO = V. a*/3 = /3 cos a; + y sin x, 
 where III. . . 2x = tir, and IV. . . y = oc = Va/3 ; 
 
 and now the locus of p is a plane ellipse, with its centre at o, and with OB and oc 
 for its major and minor semiaxes : while the angular quantity, x, is what is often 
 called the excentric anomaly. 
 
 (3.) If we write, under the same conditions (2.), 
 
 V. . . OB'=/3' = V/3a: a = a-iy, and VI. .. op' =p' = Vpa: a = aV|Oa, 
 60 thatB' and p' are the projections (203) of b and v ona plane drawn through o, at 
 right angles to the unit-line OA, we have then, by II., the equation, 
 
 VII. . . p' = j8' cos ic + y sin aj = a*^' -, 
 so that the locus of this projected point p' is a circle, with ob' and oc for two rectan- 
 gular radii. 
 
 (4.) Under the same conditions, the elliptic locus (2.), of the point p itself, is the 
 section of the right cylinder (compare 203, (5.) ), 
 
 VIII. . . TVap = TVa/3 = Ty, 
 made by the plane, 
 
 IX. . . = Sy(3p, or IX'. . . /32Sap = Sa^S/3p (comp. 298, XXVI.) ; 
 as a confirmation of which last form we have, by II. and IV., 
 
 X. . . Sap = Sa/3 cos x, S/3p = (3^ cos x. 
 (5.) If we retain the condition Sa/3 = (1.), but no^ now the condition Ta= 1, 
 we may again write the equation I. for p ; but the locus ofv will now be a loga- 
 rithmic spiral, with o for its pole, in the plane perpendicular to OA ; because equal 
 angular motions, of the turning line OP, correspond now to equal multiplications of 
 the length of that line p. 
 
 (6.) For example, when the scalar exponent t is increased by 4, so that the re- 
 volving unit line, 
 
 XL. .Up = Ua*.U/3 
 
 returns (comp. 309, XXIV.) to the direction which it had before the increase of* 
 was made, the length Tp of the turning line p itself or of the radius vector of the 
 locus, is multiplied by Ta* ; which constant and positive scalar is not now equal to 
 unity. 
 
 (7.) If we reject both the conditions (1.), 
 
 Ta=l, and Sa/3 = 0, 
 so that the line a, or the base of the power a*, is now neither an unit-line, nor per- 
 pendicular to /3, namely to the line on which that power operates, as & factor, we 
 must again take vector parts, but we have now this new expression : 
 
 XII. . . OP = p = V. a*/3 = a'(/3 cos a; + y sin a?) 
 in which we have written, for abridgment, 
 
CHAP. I.] EXPRESSIONS FOR CERTAIN SPIRALS. 383 
 
 XIII. ..« = Ta, y = V(Ua./3). 
 
 (8.) In this more complex case, the locus of p is still a plane curve, and may be 
 said to be now an elliptic* logarithmic spiral; for if we suppress the scalar factor, 
 a', we fall back on the /orm II., and have again an ellipse as the locus: but when 
 we ifa^c accoMw* of that factor, we find (comp. (2.)) that equal increments of ex- 
 centric anomaly («), in the auxiliary ellipse so determined, correspond to equal mul- 
 tiplications of the length (Tp), of the vector of the new spiral. 
 
 (9.) We may also project b and p, as in (3.), into points b' and p', on the plane 
 through o perpendicular to OA, which plane still contains the extremity c of the 
 auxiliary vector y ; and then, since it is easily proved that y = Ua./3', the equa- 
 tion of the projected spiral becomes (with Ta > or < 1), 
 
 XIV. . . jo' = a^(/3' cos a; -I- y sin x') = a^/3' ; 
 so that we are brought back to the case (5.), and the projected curve is seen to be a 
 logarithmic spiral, of the known and ordinary kind. 
 
 (10.) Several spirals of double curvature are easily represented, on the same ge- 
 neral plan, by merely introducing a vector-term proportional to t, combined or not 
 with a constant vector-term, in each of the expressions above given, for the variable 
 vector p. For example, the equation, 
 
 XV. .. p = cta + a% with Ta = 1, and Sa/3 = 0, 
 while c is any constant scalar different from zero, represents a helix, on the right 
 circular cylinder VIII. 
 
 (11.) And if we introduce a new and variable scalar, «, as Si factor in the right- 
 band term, and so write, 
 
 XVI. . . p = cta + ua% 
 
 we shall have an expression for a variable vector p, considered as depending on two 
 variable scalars {t and ?/), which thus becomes (99) the expression for a.'vector of a 
 surface : namely of that important Screw Surface, which is the locus of the perpen- 
 diculars, let fall from the various points of a given helix, on the axis of the cylinder 
 of revolution, on which that helix, or spiral curve, is traced. 
 
 315. Without at present pursuing farther the study of these loci 
 by quaternions, it may be remarked that the definition (308) of the 
 power a\ especially for the case when Ta= 1, combined with the 
 laws (182) of i,j, k, and with the identification (295) of those three 
 important right versors with their own indices, enables us to esta- 
 blish the following among other transformations, which will be found 
 useful on several occasions. 
 
 (1.) Let a be any unit-vector, and let t be any scalar ; then, 
 
 I. . . S.a-'=S.a'; II. . . S. a-*-i = S . a«^i = - S . a<-'; 
 
 ♦ The usual logarithmic spiral might perhaps be called, by contrast to this one, 
 a circular logarithmic spiral. Compare the following sub-article (9.), respecting the 
 projection of what is here called an elliptic logarithmic spiral. 
 
384 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 III. . . a«=S.a« + aS.a'-i; IV. . . a-« = S. a'- aS. a<-i; 
 
 V. . . (S . a02 + (S . a^i)2 = aiat= 1. 
 
 (2.) Let a and t be any two unit-vectors, and let t be still any scalar ; then 
 
 VI. . . S.a' = S.i*; VII. . . V.a« = aS. a'l ; 
 
 VIII. . . aV. a« = a^S . a'l = S . a*+'. 
 (3.) Hence, by the laws of z, j, k, 
 
 IX. . . iV. it =jY.jt = kY.kt = ^. a«+». 
 (4.) We have also, by the same principles and laws, 
 
 X. . . iY.jt = V. A' ; jV. ¥ = V. it ; kV. it = Y.jt ; 
 XI. . . jV. it=-Y.kt; kY.jt = - V. it ; iV. kt = - Y.jt, 
 (5.) The expression 308, (10.), for an arbitrary vector p, may be put under the 
 following form : 
 
 XII. . . p = rV.;i2s+i + rA2«V.t2«. 
 
 (6.) And it may be expanded as follows : 
 
 XIII. . . p = r { (i cos tTT +j sin <7r) sin «7r + A cos stt } . 
 
 (7.) We shall return, briefly, in the Second Chapter of this Book, on some of 
 these last expressions, in connexion with differentials and derivatives of powers of 
 vectors ; but, for the purposes of the present Section, they may suffice. 
 
 Section 11 — On Powers and Logarithms ofJDiplanar Qua- 
 ternions; with some Additional Formulce. 
 
 316. We shall conclude the present Chapter with a short Sup- 
 plementary Section, in which the recent definition (308) of a power 
 of a vector^ with a scalar exponent, shall be extended so as to include 
 the general case, of a Power of a Quaternion, with a Quaternion Ex- 
 ponent, even when the two quaternions so combined are diplanar: 
 and a connected definition shall be given (consistent with the less 
 general one of the same kind, which was assigned in the Second 
 Chapter of the Second Book), for the Logarithm of a Quaternion in 
 an arbitrary Plane ;* together with a few additional Formulas, which 
 could not be so conveniently introduced before. 
 
 (1.) We propose, then, to write, generally, 
 
 q being any quaternion, and c being the real and known base of the natural (or Na- 
 pierian) system of logarithms, of real and positive scalars : so that (as usual), 
 
 * The quaternions considered, in the Chapter referred to, were all supposed to be 
 in the plane of the right versor i. But see the Second Note to page 265. 
 
CHAP. I.] LOGARITHMS OF DIPLANAR QUATERNIONS. 385 
 
 II. ..£ = £1=1+1+ -i-+&c. = 2-71828... 
 
 (Compare 240, (1.) and (2.).) 
 
 (2.) We shall also write, for any quaternion q^ the following expression for what 
 we shall call its principal logarithm^ or simply its Logarithm : 
 
 III. . .\q = \Tq+ Lq.UYq; 
 and thus shall have (comp. 243) the equation, 
 
 IV. . . 6i* = q. 
 (3.) When q is any actual quaternion (144), which does not degenerate (131) 
 into a negative scalar, the formula III. assigns a definite value for the logarithm, 
 \q ; which is such (comp, again 243) that 
 
 V. ..Sl^^lT^; VI. .. Vlg=^9.UVg; 
 VII. . . UVl^ = UV9 ; VIII. . .TVl5 = Z9; 
 the scalar part of the logarithm being thus the (natural) logarithm of the tensor ; 
 and the vector part of the same logarithm \q being constructed by a line in the direc- 
 tion of the axis Ax. q, of which the length bears, to the assumed unit of length, the 
 same ratio as that which the angle L q bears, to the usual unit of angle (comp. 241, 
 
 (2.),W)- 
 
 (4.) If it were merely required to satisfy the equation, 
 
 IX... £9' = 9, 
 
 in which q is supposed to be a given and actual quaternion, which is not equal to 
 
 any negative scalar (3.), we might do this by writing (compare again 243), 
 
 X. . . g' = (log q)n = 1? + 2n7rUVg', 
 
 where n is any whole number, positive or negative or null ; and in this view, what 
 
 we have called the logarithm, \q, of the quaternion q, is only what may be considered 
 
 as the simplest solution of the exponential equation IX., and may, as such, be thus 
 
 denoted : 
 
 XI. ..19 = (log 9)0. 
 
 (5.) The excepted case (3.), where 9 is a negative scalar, becomes on this plan 
 a case of indetermination, but not of impossibility : since we have, for example, by 
 the definition III., the following expression for the logarithm of negative unity, 
 
 XII. .. l(-l) = 7rV-l; 
 which in its form agrees 'fvith old and well-known results, but is here interpreted as 
 signifying any unit-vector, of which the length bears to the uriit of length the ratio 
 of TT to 1 (comp. 243, VII.). 
 
 (6.) We propose also to write, generally, for any two quaternions, q and q', even 
 ifdiplanar, the following expression (comp. 243, (4.) ) for what may be called the 
 principal value o^thQ power, or simjjly the Power, in which the former quaternion q 
 is the base, while the latter quaternion q is the exponent : 
 
 XIII. . . 92' = £9'13; 
 
 and thus this quaternion power receives, in general, with the help of the definitions I. 
 and III., a perfectly definite signification. 
 
 (7.) When the base, q, becomes a rector, p, its angle becomes a right angle ; the 
 definition III. gives therefore, for this case, 
 
 3d 
 
386 KLEMKNTS OF QUATERNIONS. [bOOK III. 
 
 XIV. . .lp=lTp + |Up; 
 
 and this is the quaternion which is to be multipled by 9', in the expression, 
 XV. . . p9' = 6«''P. 
 (8,) When, for the same vector-base, the exponent q' becomes a scalar, t, the 
 last formula becomes : 
 
 XVI. . . p* = £<1P = Tp^ £^Up^ if 2a; = f TT ; 
 and because, by I., the relation (Up)^ =—1 gives, 
 
 XVII. . . £'UP = cos a? + Up sin x, or briefly, XVII'. . . t^^p = cpscc, 
 we see that the former definition, 308, 1., of the power a\ is in this wsiy reproduced, 
 as one which is included in the more general definition XIII., of the power qi' ; for 
 we may write, by the last mentioned definition, 
 
 XVIII. . . (Up)« = £^UP = cps y (comp. 234, VIII.), 
 
 with the recent values XVI. and XVII., of x and t^^p. 
 
 (9.) In the present theory of diplanar quaternions, we cannot expect to find 
 that the sum of the logarithms of any two proposed yhc^ors, shall be generally equal 
 to the logarithm of the product ; but for the simpler and earlier case of complanar 
 quaternions, that algebraic property may be considered to exist, with due modifica- 
 tions for 7nultiplicity of value* 
 
 (10.) The definition III. enables us, however, to establish generally the very 
 simple formula (comp. 243, II. III.) : 
 
 XIX. . .lq=] (Tq . Vq) = \Tq + Wq ; 
 in which (comp. (3.) ), 
 XX. . . IU5 = Z 9 . TJYq = Y\q ; XXI. . . TlUg = ^ 9 ; XXII. . . UIU9 = UYq. 
 
 (11.) We have also generally, by XIII., for any scalar exponent, t, and any 
 quaternion lase, q, the power, 
 
 XXIII. . . 9« = 6^1? = (Tqy. (cos t ^q+ \JYq .sint Iq); 
 or brieflv, 
 
 XXIIV. ..q*=Tqt. CVS tig, if v = \JYq; 
 
 in which the parentheses about Tq may be omitted, because 
 
 XXIV. . . T(90 = (T9)« = T9*(comp. 237, II.). 
 
 (12 ) When the base and exponent of a power are two rectangular vectors, p and 
 p', then, whatever their lengths may be, the product p'lp is, by XIV., a vector; but 
 £1 is always a versor, 
 
 XXV. . . £« = cos Ta + Ua sin la, if a be any vector ; 
 Ave have therefore, 
 
 * In 243, (3.), it might have been observed, that every value of each member of 
 the formula IX., there given, is one of the values of the other member; and a similar 
 remarli applies to the forraulca I. and II. of 236. 
 
CHAP. I.] POWERS AND FUNCTIONS OF QUATERNIONS. 387 
 
 XXVI. . . T.pP'=l, if S.pjo'=0; 
 
 or in words, the power pp' is a versor, under this condition of rectangularity . 
 
 (13.) For example (comp. 242, (7.),* and the shortly following formula 
 
 XXVIII.), 
 
 XXVII. . . V = c^l' = -k; ji = £'U- = + A ; 
 
 and generally, if the base be an unit- line, and the exponent a line of any length, but 
 perpendicular to the base, the axis of the power is a line perpendicular to both ; un- 
 less the direction of that axis becomes indeterminate, by the power reducing itself to a 
 scalar, which in certain cases may happen. 
 
 (14.) Thus, whatever scalar c may be, we may write, 
 
 XXVIII. . . i<^= f'yi' = f-»<^*'' = cos A sin — ; 
 
 2 2 
 
 this power, then, is a versor (12.), and its axis is generally the line + k ; but in the 
 case when c is any whole and et?en number, this versor degenerates into positive or ne- 
 gative ««zVy (153), and the axis becomes indeterminate (131). 
 
 (15.) If, for any real quaternion q, we write again, 
 XXIX. . . UVg = V, and therefore XXX. . . vq^ gv, and XXXI. . . i;2 = - 1, 
 the process of 239 will hold good, when we change i to v; the series, denoted in I. 
 by eS', is therefore always at last convergent,^ however great (but finite) the tensor 
 'Iq may be ; and in like manner the two following other series, derived from it, which 
 represent (comp. 242, (3.) ) what we shall call, generally, by analogy to known ex- 
 pressions, the cosine and sine of the quaternion q, are always ultimately convergent : 
 
 XXXII. . . C089 = i(s''9+ s-'9) = l - j«i+ j-^^-&C.l 
 XXXIII.. .stoj = l(a^^-r^') = f-j^+j-^^.-&c. 
 
 (16.) We shall also define that the secant, cosecant, tangent, and cotangent of 
 a quaternion, supposed still to be real, are the functions : 
 
 2 2v 
 XXXIV. . . sec o = ; cosec q = ; 
 
 XXXV. . . tan o = —^ ^ ; cot q =-^ ^ ; 
 
 ^ £"9 4- £-"9 c"? _ C-W9 
 
 and thus shall have the usual relations, sec g = 1 : cos q, &c. 
 (17.) We shall also have, 
 
 XXXVI. . . 6"' = cos 9 + V sin q, £-"* = cos 5 - w sin g ; 
 
 * In the theory of complanar quaternions, it was found convenient to admit a 
 certain multiplicity of value for & power, when the exponent was not a whole num- 
 ber; and therefore a notation for the principal value of a power was employed, with 
 which the conventions of the present Section enable us now to dispense. 
 
 t In fact, it can be proved that this final convergence exists, even when the qua- 
 ternion is imaginary, or when it is replaced by a biquaternion (214, (8.) ) ; but we 
 have no occasion here to consider any but real quaternions. 
 
388 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 and therefore, as in trigonometry (comp. 315, (!•))> 
 
 XXXVII. . . (cos 9)2 + (sin 9)2 = e''^ . e-""^ =e° = 1, 
 whatever quaternion q may be. 
 
 (18.) And all the formulce of trigonometry, for cosines and sines of sums oUwo or 
 more arcs, &c., will thus hold good for quaternions also, provided that the quater- 
 nions to be combined are in any common plane ; for example, 
 
 XXXVIII. . . cos {q' + g) = cos q cos q - sin q' sin q, if q' \\\ q- 
 (19.) This condition of eomplanarity is here a necessary one; because (comp. 
 (9. ) ) it is necessary for the establishment of the exponential relation between swns and 
 powers. 
 
 (20.) Thus, we may indeed write, 
 
 XXXIX. . . £2'+? = £«'. £9, if q\\\q; 
 but, in general, the developments of these two expressions give the difference, 
 
 XL. . . £9'-^ 9 - £9' £9 = — — i-^ -f terms of third and higher dimensions ; 
 
 and XLI. . . ^ {qq' - q'q) =Y {Yq .Yq'), 
 
 an expression which does not vanish, when the quaternions q and q' are diplanar. 
 
 (21.) A few supplementary formulae, connected with the present Chapter, maybe 
 appended here, as was mentioned at the commencement of this Article (316). And 
 first it may be remarked, as connected with the theory of powers of vectors, that if 
 a, (3, y be any three unit-lines, OA, OB, oc, and if a denote the area of the spherical 
 triangle abc, then the formula 298, XX. may be thus written : 
 
 xLii...t±i:.liS.ll±y = X 
 
 /S+ya + jS y + a 
 
 the exponent being here a scalar. 
 
 (22.) The immediately preceding formula, 298, XIX., gives for any three vec- 
 tors, the relation : 
 
 XLIIL . . (Ua/3y)2 + (U/3y)2 + (Uay)2+(Ua/3)2+4Uay.SUa/3.SU/3y = -2; 
 for example, if a, j8, y be made equal to i, j, k, the first member of this equation be- 
 comes, l-l-l-H-0 = -2. 
 
 (23.) The following is a much more complex identity, involving as it does not 
 only three arbitrary vectors a, j3, y, but also/o«r arbitrary scalar s, a, b, c, and r ; 
 but it has some geometrical applications, and a student would find it a good exercise 
 in transformations, to investigate a proof of it for himself. To abridge notation, the 
 three vectors a, (3, y, and the three scalars a, b, c, are considered as each composing 
 a cycle, with respect to which are formed sums S, and products U, on a plan which 
 may be thus exemplified : 
 
 XLIV. . . SaV/3y = aV/3y + 6Vya + cVa/3 ; Ua^ = a^b^c^. 
 This being understood, the formula to be proved is the following : 
 
 XLV. . . (Sa/3y)2 + (2aVi3y)^- + r2(SV/3y)2-r2(2a(/3-y))2 
 + 2n (r2 + SjSy + be) = 20 (r2 + a2) + 2 ria2 
 + 2(r2 + a2 + a«){(Vi3y)« + 26c(r2 + S/3y)-r2 03-y)«}; 
 the sign of summation in the last line governing all that follows it. 
 
CHAP. I.] ADDITIONAL FORMULA, CONTACTS ON A SPHERE. 389 
 
 (24.) For example, by making the four scalars a, 6, c, r each =0, this formula 
 gives, for any three vectors a, j3, y, the relation, 
 
 XLVI. . . (Sa/3y)2 + 2nS/3y = "lUa^ -f 2 . a2(V/3y)2 ; 
 
 which agrees with the very useful equation 294, LIII., because 
 
 XLVII. . . a2(V/3yy = a^ {(Si3y)2 - /32y2 } = (aS/3y)2 - UaK 
 
 (25.) Let a, /3, y be the vectors of three points A, b, c, which are exterior to a 
 given sphere^ of which the radius is r, and the equation is, 
 
 XLVIII. . . p2+ r2 = (comp. 282, XIII.) ; 
 
 and let a, 6, c denote the lengths of the tangents to that sphere, which are drawn 
 from those three points respectively. We shall then have the relations : 
 
 XLIX. .. a2 + a2 = /32+fc2 = ^2-^c2 = -r2; 
 
 thus r2+ a2= - a2, &c., and the second member of the formula XLV. vanishes ; the 
 first member of that formula is therefore also equal to zero, for these significations of 
 the letters : and thus a theorem is obtained, which is found to be extremely useful, 
 in the investigation by quaternions of tbe system of the eight (real or imaginary) 
 small circles^ which touch a given set of three small circles on a sphere. 
 
 (26.) We cannot enter upon that investigation here; but may remark that be- 
 cause the vector p of the foot p, of the perpendicular op let fall the origin o on the 
 right line ab, is given by the expression, 
 
 L. . . p = aS -i— + i3S = -^, 
 
 as may be proved in various ways, the condition of contact of that right line ab with 
 the sphere XLVIII. is expressed by the equation, 
 
 LL . . TVjSa = rT {a-^); or LII. . . (Vj3a)2 = r2 (a - /3)2 ; 
 or by another easy transformation, with the help of XLIX., 
 
 LIII. . . (r2 + Sa/3)2 = (r2 + a2) (r2 + pi) = a2i2. 
 
 (27.) This last equation evidently admits of decomposition into two factors, re- 
 presenting two alternative conditions, namely, 
 
 LIV. ..r2 + Sai3-a6 = 0; LV. . . r2 + Sa/3 + o6 = ; 
 and if we still consider the tangents a and h (25.) &s positive, it is easy to prove, in 
 several different ways, that the frst or the second factor is to be selected, according 
 as the point p, at which the line ab touches the sphere, does or does not fall between 
 the points A and b ; or in other words, according as the length of that line is equal 
 to the sum, or to the difference, of those two tangents. 
 (28.) In fact we have, for the first case, 
 LVL . .T(j3-a) = 6 + a, or = (/S - a)2 + (6 + a)2 = -2 (r2+ Sa/3 -a6), 
 in virtue of the relations XLIX. ; but, for the second case, 
 
 LVIL . .T(/3-a)=±(6-a), or = (/3 -a)2 + (6-a)2 = -2(r2 + Sa/3 + a6) ; 
 and it may be remarked, that we might in this way have been led to find the system 
 of ihe two conditions (27.), and thence the equation LIII., or its transformations, 
 LII. and LI. 
 
390 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (29.) We may conceive a cone oftangetits from A, circumscribing i\\Q sphere 
 XLVIII., and touching it along a small circle, of which ih^ plane, or \hB polar plane 
 of the point A, is easily found to have for its equation, 
 
 LVIII. . . Sa/) + r2 = (comp. 294, (28.), and 215, (10.) ) ; 
 and in Uke manner the equation, 
 
 LIX. . . S/3.o + r2 = 0, 
 represents the polar plane of the point b, which plane cuts the sphere in a second 
 small circle : and these two circles touch each other, when either of the two con- 
 ditions (27.) is satisfied; such contact being external for the caseLIV., hwtinternal 
 for the case LV. 
 
 (30.) The condition of contact (26.), of the line and sphere, might have been 
 otherwise found, as the condition of equality of roots in the quadratic equation 
 (comp. 216, (2.)), 
 
 LX, . . = (a:a + y/3)2 + (a; + yY r^, 
 or LXI. . . = 0:2 (r2 + ^2) + 2xy (r2 + Sa/3) + y3 (r^ + /32) ; 
 
 the contact being thus considered here as a case of coincidence of intersections. 
 
 (31.) The equation of conjugation (comp. 215, (13.)), which expresses that 
 each of the two points a and b is in the polar plane of the other, is (with the present 
 notations), 
 
 LXII. . .r2 + Sa/3 = 0; 
 
 the equal but opposite roots of LXI., which then exist if the line cuts the sphere, 
 answering here to the well-known harmonic division of the secant line ab (comp. 
 215, (16.) ), which thus connects two conjugate points. 
 
 (32.) In like manner, from the quadratic equation* 216, III., we get this analo- 
 gous equation, 
 
 connecting the vectors X, ju of any two points l, m, which are conjugate relatively to 
 the ellipsoid 216, II. ; and if we place the point L on the surface, the equation LXIII. 
 will i-epresent the tangent plane at that point i., considered as the locus of the conju- 
 gate point M ; whence it is easy to deduce the normal, at any point of the elhpsoid. 
 But all researches respecting normals to surfaces can be better conducted, in con- 
 nexion with the Differential Calculus of Quaternions, to which we shall next pro- 
 ceed. 
 
 (33.) It may however be added here, as regards Powers of Quaternions with 
 scalar exponents (11.), that the symbol q^rq-* represents a quaternion formed from r, 
 by a conical rotation of its axis round that of q, through an angle — 2tLq', and that 
 both members of the equation, 
 
 LXIV. . . {qrq-^y=zqrtq'\ 
 are symbols of one common quaternion. 
 
 Corrected as in the first Note to page 298. 
 
 Lxm. ..S- s^-s^-^ 
 
CHAP. II.] DEFINITION OF DIFFERENTIALS. 391 
 
 CHAPTER II. 
 
 ON DIFFERENTIALS AND DEVELOPMENTS OF FUNCTIONS OF 
 quaternions; and on some APPLICATIONS OF QUATER- 
 NIONS, TO GEOMETRICAL AND PHYSICAL QUESTIONS. 
 
 Section 1. — On the Definition of Simultaneous Differentials. 
 
 317. In the foregoing Chapter of the present Book, and in 
 several parts of the Book preceding it, we have taken occasion 
 to exhibit, as we went along, a considerable variety of Exam- 
 pies, of the Geometrical Application of Quaternions : but these 
 have been given, chiefly as assisting to impress on the reader 
 the meanings of new notations, or oi new combinations of sym- 
 bols, when such presented themselves in turn to our notice. 
 In this concluding Chapter, we desire to offer a few additional 
 examples, of the same geometrical kind, but dealing, more 
 freely than before, with tangents and normals to curves and 
 surfaces ; and to give at least some specimens, of the applica- 
 tion of quaternions to Physical Inquiries, But it seems ne- 
 cessary that we should first establish here some Principles, and 
 some Notations, respecting Differeritials of Quaternions^ and 
 of their Functions, generally. 
 
 318. The usual definitions, o{ differential coefficients, and 
 of derived functions, are found to be inapplicable generally to 
 the present Calculus, on account of the (generally) wow-cowz- 
 Twz^^a^iue character of quaternion- multiplication (168, 191). It 
 becomes, therefore, necessary to have recourse to a new Defi- 
 nition of Differentials, which yet ought to be so framed, as to 
 be consistent with, and to include, the usual Rules of Diffe- 
 rentiation: because scalars (131), as well as vectors (292), 
 have been seen to be included, under the general Conception 
 of Quaternions. 
 
 319. In seeking? for such a new definition, it is natural to 
 
392 ELEMENTS OF QUATERNIONS. [boOK III. 
 
 go back to the first principles of the whole subject of Diffe- 
 rentials : and to consider how the great Inventor of Fluxions 
 might be supposed to have dealt with the question, if he had 
 been dq)rived of that powerful resource o^ common calculation, 
 which is supplied by the commutative property of algebraic 
 multiplication ; or by the familiar equation, 
 
 xy=^yx, 
 considered as a general one, or as subsisting for every pair of 
 factors, X and y ; while limits should still be allowed, but in- 
 finitesimals be still excluded: and indeed the fluxions them- 
 selves should be regarded as generally finite,* according to 
 what seems to have been the ultimate view of Newton. 
 
 320. The answer to this question, which a study of the 
 Principia appears to suggest, is contained in the following 
 Definition, which we believe to be a perfectly general one, as 
 regards the older Calculus, and which we propose to adopt 
 for Quaternions : — 
 
 '* Simultaneous Differentials (or Corresponding Fluxions) 
 are Limits of Equimultiples] of Simultaneous and Decreasing 
 Differences." 
 
 * Compare the remarks annexed to the Second Lemma of the Second Book of the 
 Principia (Third Edition, London, 1726) ; and especially the following passage (page 
 244): 
 
 " Neque enim spectatur in hoc Lemmate magnitude momentorum, sed prima 
 nascentium proportio, Eodem recidit si loco momentorum usurpentur vel velocitates 
 incrementorum ac decrementorum (quas etiam motus, mutationes et fluxiones quan- 
 titatum nominare licet) vel finitae quaevis quantitates velocitatibus hisce proportion- 
 ales." 
 
 f As regards the notion of multiplying such differences, or generally any quanti- 
 .ties which all diminish together, in order to render their ultimate relations more evi- 
 dent, it may be suggested by various parts of the Principia of Sir Isaac Newton ; but 
 especially by the First Section of the First Book. See for example the Seventh Lemma 
 (p. 31), under which such expressions as the following occur : " intelligantur semper 
 AB et AD ad puncta longinqua b Qi d produci," . . . . " ideoque rectae semper finitae 
 Ah, Ad, . . ." The direction, " ad puncta longinqua produci," is repeated in con- 
 nexion with the Eighth and Ninth Lemmas of the same Book and Section ; while 
 under the former of those two Lemmas we meet the expression, " triangula semper 
 finita," applied to the magnified representations of three triangles, which all diminish 
 indefinitely together : and under the latter Lemma the words occur, " manente longi- 
 tudine Ae,'" where Ae is a finite and constant line, obtained by a constantly increas- 
 ing multiplication of a constantly diminishing line AE (page 33 of the edition 
 cited). 
 
CHAP. II.] LIMITS OF EQUIMULTIPLES OF DIFFERENCES. 393 
 
 And conversely, whenever any simultaneous differences^ of 
 any system of variables, all tend to vanish together^ according 
 to any law^ or system of laws ; then, if any equimultiples of 
 those decreasing differences all tend together to any system of 
 finite limits, those Limits are said to be Simultaneous L>iffe- 
 rentials of the related Variables of the System ; and are de^ 
 noted, as such, by prefixing the letter d, as a characteristic of 
 differentiation, to the Symbol of each such variable. 
 
 321. More fully and symbolically, let 
 I. , .q,r,s,. .. 
 
 denote any system of connected variables (quaternions or others); and 
 
 let 
 
 II. . . A^', Ar, As, . . . 
 
 denote, as usual, a system of their connected (or simultaneous) diffe- 
 rences ; in such a manner that the sums, 
 
 III. , . q + Aq, r + Ar, s + As, . . . 
 
 shall be a new system of variables, satisfying the same laws of con- 
 nexion, whatever they may be, as those which are satisfied by the old 
 system I. Then, in returning gradually from the new system to the 
 old one, or in proceeding gradually from the old to the new, the 
 simultaneous differences II. can all be made (in general) to approach 
 together to zero, since it is evident that they may all vanish together. 
 But if while the differences themselves are thus supposed to decrease* 
 indefinitely together, we multiply them all by some one common but 
 increasing number, n, the system of their equimultiples, 
 
 IV. . . nAq, nAr, uAs, . . . 
 
 may tend to become equal to some determined system of finite limits. 
 And when this happens, as in all ordinary cases it may be made to do, 
 by a suitable adjustment of the increase of n to the decrease of Aq, 
 &c., the limits thus obtained are said to be simultaneous differentials 
 of the related variables, q,r,s; and are denoted, as such, by the sym- 
 bols, 
 
 V. . . d^-, dr, d5, . . . 
 
 * A quaternion may be said to decrease, when its tensor decreases ; and to de- 
 crease indefinitely, when that tensor tends to zero. 
 
 3e 
 
394 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 Section 2. — Elementary Illustrations of the Definition^ from 
 Algebra and Geometry. 
 
 322. To leave no possible doubt, or obscurity, on the im- 
 port of the foregoing Definition, we shall here apply it to de- 
 termine the differential of a square^ in algebra, and that of a 
 rectangle, in geometry; in doing which we shall show, that 
 while for such cases the old rules are reproduced, the differen- 
 tials treated of need not be small; and that it would be a vitia- 
 tion, and not a correction, of the results, if any additional terms 
 were introduced into their expressions, for the purpose of ren- 
 dering all the differentials equal to the corresponding diffe- 
 rences : though some of them may be assumed to be so, 
 namely, in the first Example, one, and in the second Exam- 
 ple, two. 
 
 (1.) In Algebra, then, let us consider the equation, 
 
 I. ..y = a:2, 
 which gives, 
 
 II. . . y + Ay = (a; -f Ax)\ 
 
 and therefore, as usual,* 
 
 III. . . Ai/ = 2xAx+ Ax"^; 
 
 or what comes to the same thing, 
 
 IV. . . nAy = 2xnAx + w"*(nAa;)2, 
 
 where n is an arbitrary multiplier^ which may be supposed, for simplicity, to be a 
 positive whole number. 
 
 (2.) Conceive now that while the differences Ax and Ay, remaining always con- 
 nected with each other and with x by the equation III., decrease, and tend together 
 to zero, the number n increases, in the transformed equation IV., and tends to infi- 
 nity, in such a manner that ^q product, or multiple, nAx, tends to ?,om& finite limit 
 a ; which may happen, for example, by our obliging Ax to satisfy always the con- 
 dition, 
 
 V. . . Aa; = n-^a, or nAx = a, 
 
 after a previous selection of some given and finite value for a. 
 
 * We write here, as is common, Aa;' to denote (Aa;)2; while A.a;2 would be 
 written, on the same known plan, for A {x^), or Ay. In like manner we shall write 
 da;2, as usual, for (da;)2 ; and shall denote d(x^) by d.x^. Compare the notations 
 S92, S.g2, and Yq^, Y.q^, in 199 and 204. 
 
CHAP. II.] ILLUSTRATION FROM ALGEBRA. 395 
 
 (3.) We shall then have, with this last condition V., the following expression 
 by IV., for the equimultiple nAi/, of the other difference, Ay ; 
 
 VI. . . »Ay = 2xa + n" 'a^ = 6 + n-ia2, if 6 = 2xa. 
 
 But because a, and therefore a2, is given and finite, (2.), while the number n in- 
 creases indefinitely, the term n-^a^, in this expression VI. for 7iAy, indefinitely tends 
 to zero, and its limit is rigorously null. Hence the two finite quantities, a and b 
 (since x is supposed to be finite), are two simultaneous limits, to which, under the 
 supposed conditions, the two equimultiples, nAxandnAy, tend;* they are, therefore, 
 by the definition (320), simultaneous differentials of x and y : and we may write ac- 
 cordingly (321), 
 
 VII. . . dx = a, ([y = b = 2xa ; 
 
 or, as usual, after elimination of a, 
 
 VIII. . . dy = d..r2 = 2a;da;. 
 
 (4.) And it would not improve, but vitiate, according to the adopted definition 
 (320), this usual expression for the differential of the square of a variable x in alge- 
 bra, if we were to add to it the term da;'^, in imitation of the formula III. for the 
 difference A.x^. For this would come to supposing that, for a given and finite 
 value, a, of da;, or oi nAx, the term n'^a^, or n'^dx^, in the expression VI. fornAy, 
 could fiiil to tend to zero, while the number, n, by which the square oidx is divided^ 
 increases without limit, or tends (as above) to infinity. 
 
 (5.) As an arithmetical example, let there be the given values, 
 IX...x = 2, y = x^=4,, daj = 1000; 
 and let it be required to compute, as a consequence of the definition (320), the arith- 
 rithmetical value of the simultaneous differential, dy. We have now the following 
 equimultiples of simultaneous differences, 
 
 X. . .nAa; = da; = 1000; ndy = 4000+ 1000000 n-i ; 
 but the limit of the n^^part of a million (or of any greater, hwt given and finite num- 
 ber') is exactly zero, if n increase without limit ; the required value of dy is, therefore, 
 
 rigorously, in this example, 
 
 XI. . .dy = 4000. 
 
 (6.) And we see that these two simultaneous differentials, 
 XII. . .da; = 1000, dy = 4000, 
 are not, in this example, even approximately equal to the two simultaneous diffe- 
 rences, 
 
 XIII. . . Aa; = da; = 1000, Ay = 10022 - 22 = 1004000, 
 
 which answer to the value n = 1 ; although, no doubt, from the very conception of 
 simultaneous differentials, as embodied in the definition (320), they must admit of 
 having such equisubmultiples of themselves taken, 
 
 XIV. . . n'^da; and n'dy, 
 
 * In this case, indeed, the multiple wAa; has by V. a constant value, namely a ; 
 but it is found convenient to extend the use of the word, limit, so as to include the 
 case of constants : or to say, generally, that a constant is its own limit. 
 
396 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 as to be nearly equal^ for large values of the number n, to some system of simulta- 
 neous and decreasing differences, 
 
 XV. . . Ax and Ay; 
 
 arid more and more nearly equal to such a system, even in the way of ratio, as they 
 all become smaller and smaller together, and tend together to vanish. 
 
 (7.) For example, while the differentials themselves retain the constant values 
 XII., their millionth parts are, respectively, 
 
 XVI. .. n-Ma; = 0-001, and n-'dy = 0004, if n=1000000; 
 
 and the same value of the number n gives, by X., the equally rigorous values of two 
 simultaneous dfferenees, as follows, 
 
 XVII. .. Ax = 0*001, and Ay = 0-004001; 
 
 so that these values of the decreasing differences XV. may already be considered to 
 be nearly equal to the two equisubmultiples, XIV. or XVI., of the two simultaneous 
 differentials, XII. And it is evident that this approximation would be improved^ 
 by taking higher values of the number, n, without the rigorous and constant values 
 XII., of diC and dy, being at all affected thereby. 
 
 (8.) It is, however, evident also, that after assuming y = x^, and a; = 2, as in IX., 
 we might have assumed any other finite value for the differential dx, instead of the 
 value 1000 ; and should then have deduced a different (but &i\l\ finite') value fox the 
 other differential, dy, and not the formerly deduced value, 4000 : but there would 
 always exist, in this example, or for this form of the function, y, and for this value 
 of the variable, x, the rigorous relation between the two simultaneous differentials, 
 dx and dy, 
 
 XVIII. ..dy = 4da;, 
 
 which is obviously a case of the equation VIII., and can be proved by similar rea- 
 sonings. 
 
 323. Proceeding to the promised Example from Geometry (322), 
 we shall again see that differences and differentials are not in gene- 
 ral to be confounded with each other, and that the latter (like the 
 former) need not he small. But we shall also see that the differentials 
 {like the differences), which enter into a statement of relation, or into 
 the enunciation of a proposition, respecting quantities which vary to- 
 gether, according to any law or laws, need not even he homogeneous 
 among themselves : it being sufficient that each separately should be 
 homogeneous with the variable to which it corresponds, and of which it 
 is the differential, as line of line, or area of area. It will also be seen 
 that the definition (320) enables us to construct the differential of a 
 rectangle, as the sum of two other {finite) rectangles, without any refe- 
 rence to units of length, or of area, and without even the thought of 
 employing any numerical calculation whatever. 
 
CHAP. II.] ILLUSTRATION FROM GEOxMKTRY. 397 
 
 (1.) Let, then, as in the annexed Figure 74, abcd be any given rectangle, and 
 let BE and dg be any arbitrary but given and finite -j^ 
 
 increments of its sides, ab and ad. Complete the 
 increased rectangle gaef, or briefly af, which will 
 thus exceed the given rectangle ac, or CA, by the sum 
 of the three partial rectangles, ce, cf, cg ; or by 
 what we may call the gnomon^ * cbefgdc. On the 
 diagonal cf take a point i, so that the line ci may 
 be any arbitrarily selected submultiple of that diago- 
 nal ; and draw through i, as in the Figure, lines hm, Fig. 74, 
 KL, parallel to the sides ad, ab ; and therefore in- 
 tercepting, on the sides ab, ad prolonged, equisubmultiples bh, dk of the two given 
 increments, be, dg, of those two given sides. 
 
 (2.) Conceive now that, in this construction, ih&point i approaches to c, or that 
 we take a series of new points i, on the given diagonal of, nearer and nearer to the 
 given point c, by taking the line ci successively a smaller and smaller part of that 
 diagonal. Then the two new linear intervals^ bh, dk, and the new gnomon, cbhikdc, 
 or the sum of the three new partial rectangles, CH, ci, ck, will all indefinitely de- 
 crease, and will tend to vanish together : remaining, however, always a system of 
 three simultaneous differences (or increments), of the two given sides, AB, AD, and 
 of the given area, or rectangle, AC. 
 
 (3.) But the given increments, be and dg, of the two given sides, are always 
 (by the construction) equimultiples of the two fir st^ of the three new and decreasing 
 differences ; they may, therefore, by the definition (320), be arbitrarily taken as two 
 simultaneous differentials of the two sides, AB and AD, provided that we then treat, 
 as the coiTCsponding or simultaneous differential of the rectangle Ac, the limit of the 
 equimultiple of the new gnomon (2.), or of the decreasing difference between the two 
 rectangles, AC and Ai, whereof i\iQ first is given. 
 
 (4.) We are then, first, to increase this new gnomon, or the difference of AC, ai, or 
 the sum (2.) of the three partial rectangles, ch, ci, ck, in the ratio of be to bh, or 
 of DG to DK ; and secondly, to seek the limit of the area so increased. For this last 
 limit will, by the definition (320), be exactly and rigorously equal to the sought dif- 
 ferential of the rectangle AC ; t/the given and finite increments, be and dg, be as- 
 sumed (as by (3.) they may) to be the differentials of the sides, ab, ad. 
 
 (5.) Now when we thus increase the two new partial rectangles, ch and ck, we 
 get precisely the two old partial rectangles, ce and cg ; which, as being given and 
 constant, must be considered to be their own limits, f But when we increase, in the 
 same ratio, the other new partial rectangle ci, we do not recover the old partial 
 rectangle CF, corresponding to it ; but obtain the new rectangle cl, or the equal 
 rectangle cm, which is not constant, but diminishes indefinitely as the point i ap- 
 proaches to c ; in such a manner that the limit of the area, of this new rectangle cl 
 or CM, is rigorously null. 
 
 * The word, gnomon, is here used with a slightly more extended signification, 
 than in the Second Book of Euclid, 
 t Compare the Note to page 395. 
 
398 ELEMENTS OF QU ATKllNIONS. [booK III. 
 
 (6.) //", then, the given increments, be, dg, be still assumed to be the differen- 
 tials of the given sides ab, ad (an assumption which has been seen to he permitted^, 
 the differential of the given area, or rectangle, AC, is proved (not assumed) to be, as 
 a necessary consequence of the definition (320), exactly and rigorously equal to the 
 sum of the two partial rectangles CE and CG ; because such is the limit (5.) of the 
 multiple of the new gnomon (2.), in the construction. 
 
 (7.) And if any one were to suppose that he could improve this known value for 
 the differential of a rectangle, by adding to it the rectangle CF, as a new term, or 
 part, so as to make it equal to the old or given gnomon (1.), he would (the definition 
 being granted) commit a geometrical error, equivalent to that of supposing that the 
 two similar rectangles ci and cr, bear to each other the simple ratio, instead of bear- 
 ing (as they do) the duplicate ratio, of their homologous sides. 
 
 Section 3. — On some general Consequences of the Definition. 
 
 324. Let there be any proposed equation of the form, 
 
 I. ..Q = i^(5r, r, ...); 
 
 and let d^', dr, . . . be any assumed (but generally finite) and 
 simultaneous differentials of the variables^ q, r, . . . whether 
 scalars, or vectors, or quaternions, on which Q is supposed to 
 depend, by the equation I. Then the corresponding (or simul- 
 taneous) differential o^ th^w function, Q, is equal (by the de- 
 finition 320, compare 321) to the following limit: 
 
 II. . . dQ = lim.w (2^(5' + 72-^5', r + w'dr, .. .)-F(q,r, ...)); 
 
 71= CO 
 
 where n is any whole number (or other positive* scalar) which, 
 as the formula expresses, is conceived to become indefinitely 
 greater and greater, and so to tend to infinity. And if, in 
 particular, we consider the function Q as involving only one 
 variable q, so that 
 
 III. ..Q =/(?)=/?, 
 then 
 
 TV. . .dQ = dfq = \im,n{f(q ^n-'dq)--fq]; 
 
 n= 00 
 
 a formula for the differential of a single explicit function of a 
 single variable, which agrees perfectly with those given, near 
 the end of the First Book, for the differentials of a vector, and 
 of a scalar, considered each as a function (100) of a single sea- 
 
 * Except in some rare cases of discontinuity, not at present under our considera- 
 tion, this scalar n may as well be conceived to tend to negative infinity. 
 
CHAP. II.] CONSEQUENCES OF THE DEFINITION. 399 
 
 lar variable^ t : but which is now extended^ as a consequence 
 of the general definition (320), to the case when the connected 
 variables, q, Q, and their differentials, Aq, dQ, are quaternions : 
 with an analogous application, of the still more general For- 
 mula of Differentiation II., to Functions of several Quater- 
 nions, 
 
 (1.) As an example of the use of the formula IV., let the function of 5 be its 
 square, so that 
 
 Then, by the formula, 
 
 VI. . .dQ = d/9 = lim. «{(5 + n-id9)2-g2} 
 
 n= 00 
 
 = lira. {q.dq+ 6q.q + n,-^ d^^)} 
 
 »= 00 
 
 where dq"^ signifies* the square of dq ; that is, 
 
 VII. . .d.q^=q.diq + dq.q; 
 or without the pointsf between q and dq, 
 
 Vir. . . d.q''- = qdq + dq q; 
 an expression for the differential of the square of a quaternion, which does not in gene- 
 ral admit of any further reduction : because q and dq are not generally commutative, 
 as, factors in multiplication. When, however, it happens, as in algebra, that q.dq 
 = dq.q, by the two quaternions q and dq being complanar, the expression Vll. then 
 evidently reproduces the usual form, 322, VIII., or becomes, 
 
 VIII. . . d.9^ = 23dg, if d5||l5(123). 
 (2.) As another example, let the function be the reciprocal, 
 
 IX... Q=fq=q-\ 
 
 Then, because 
 
 X. . ./(g + n-idgr)-/^ = (5 + »-id9)-i-g-» 
 
 = (9 + n-i dg)-l {9 - {q + n-idg)}g'-l 
 
 = —n-'^{q-\-n''^dqy^.dq.q-'^, 
 
 of which, when multiplied by n, the limit is - q'^dq.q'"^, we have the following ex- 
 pression for the differential of the reciprocal of a quaternion, 
 
 XL . .d.g-i=-g-i.dg.5-J; 
 
 * Compare the Note to page 394. 
 
 t The /Jom< between d and q"^, in the first member of VII., is indispensable, to 
 distinguish the differential of the square from the square of the differential. But 
 just as this latter square is denoted briefly by dq\ so the products, q . dg- and dq . q, 
 may be written as qdq and d^ q ; the symbol, dq, being thus treated as a whole one, 
 or as if it were a single letter. Yet, for greater clearness of expression, we shall re- 
 tain the point between q and dq, in several (though not in all) of the subsequent for- 
 mulae, leaving it to the student to omit it, at his pleasure. 
 
400 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 or without the points* in the second member, 6q being treated (as in VII'.) as a 
 whole symbol, 
 
 XI'. . . d.^-i = -5-id^ 5-1 ; 
 
 an expression which does not generally admit of being any farther reduced, but be- 
 comes, as in the ordinary calculus, 
 
 XII...d.5-l=-5-2dg, if dglll^, 
 that is, for the case of complanarity , of the quaternion and its differential. 
 
 325. Other Examples of Quaternion Differentiation will be given 
 in the following Section ; but the two foregoing may serve sufficiently 
 to exhibit the nature of the operation, and to show the analogy of 
 its results to those of the older Calculus, while exemplifying also 
 the distinction which generally exists between them. And we shall 
 here proceed to explain a notation^ which (at least in the statement of 
 the present theory of dijQferentials) appears to possess some advan^ 
 tages ; and will enable us to offer a still more brief symbolical defi- 
 nition, of the differential of a function fq, than before. 
 
 (1.) We have defined (320, 324), that if dg- be called the differential of a Cqua- 
 temion or other) variable, q, then the limit of the multiple, 
 
 L..n{/(^ + n-id(?)-/?}, 
 of an indefinitely decreasing difference of i\iQ function fq, of that (single) variable q, 
 when taken relatively to an indefinite increase of the multiplying number, n, is the 
 corresponding or simultaneous differential of that function, and is denoted, as such, 
 by the symbol d/g. 
 
 (2.) But before we thus pass to the limit, relatively to n, and while that multi- 
 plier, n, is still considered and treated a.s finite, the multiple I. is evidently a func- 
 tion of that number, n, as well as of the two independent variables, q and dq. And 
 we propose to denote (at least for the present) this new function of the three variables, 
 
 II. ., n,q, and dq, 
 of which the form depends, according to the law expressed by the formula I., on the 
 form of the given function, f, by the new symbol, 
 
 III... fn(q,dqy, 
 
 in such a manner as to write, for any two variables, q and q, and any number, n, the 
 equation, 
 
 IV. . .fnCq, q') = n{f^q + n-iq')-fq}; 
 
 which may obviously be also written thus, 
 
 V. . . fiq + n-l q') =fq + n"'/,, (q, q'), 
 and is here regarded as rigorously exact, in virtue of the definitions, and without 
 anything whatever being neglected, as small. 
 
 Compare the Note immediately preceding. 
 
CHAP. II.] DISTRIBUTIVE PROPERTY. 401 
 
 (3.) For example, it appears from the little calculation in 324, (1.), that, 
 VI. . . fn{q, q) = qq + q'q + n"! q\ ii fq = q^ ; 
 and from 324, (2.), that, 
 
 VII. . . Mq, q') =-(q + n-^ 5')"^ 9r\ ^^fl = T'- 
 (4.) And the definition of difq may now be briefly thus expressed : 
 VIII. ..d/5=/J^,d5); 
 or, if the sub-index ^ be understood^ we may write, still more simply, 
 
 IX...d/g=/(g,d5); 
 this last expression, /(g', d^), orf^q, q'), denoting thus a. function of two indepen- 
 dent variables, q and 5', of which the form is derived* or deduced (comp. (2.) ), from 
 the given or proposed form of the function _/5 of a single variable, g, according to a 
 law which it is one of the main objects of the Differential Calculus (at least as re- 
 gards Quaternions) to study. 
 
 326. One of the most im-poYt2int general properties, of the 
 Junctions of this class f(q, q'), is that they are all distributive 
 with respect to the second independent variable, q\ which is in- 
 troduced in the foregoing process of what we have called de- 
 rivation,^ from some ffiven function fq, of a single variable, q: 
 a theorem which may be proved as follows, whether the two 
 independent variables be, or be not, quaternions. 
 
 (1.) Let q" be any third independent variable, and let n be any number ; then 
 the formula 325, V. gives the three following equations, resulting from the law of de- 
 rivation offniq, q) from/g : 
 
 I. . .f(q + n-^q")=fq+n-%(iq, q"); 
 
 II. ..f(q + n'iq" + n^q') =f(q + n'^q") + nifn(ci + n'l q\ q') ; 
 
 III. . .f{q + nlq+n-'q)=^fq + n-^fn(iq, q+q"); 
 
 * It was remarked, or hinted, in 318, that the usual definition oi a, derived func- 
 tion, namely, that given by Lagrange in the Calcul des Fonctions, cannot be taken 
 as a foundation for a differential calculus of quaternions : although such derived 
 functions of scalars present themselves occasionally in the applications of that cal- 
 culus, as in 100, (3.) and (4.), and in some analogous but more general cases, which 
 will be noticed soon. The present Law of Derivation is of an entirely different 
 kind since it conducts, as we see, from a given function of one variable, to a derived 
 function of two variables, which are in general independent of each other. The 
 function /n(9, 9')> of the three variables, n, 9, q, may also be called a derived func- 
 tion, since it is deduced, hj the jftxed law IV., from the same given function fq, 
 although it has in general a less simple form than its own limit, f^ (5, y'), or 
 
 f(g. 9'). 
 
 t Compare the Note immediately preceding. 
 
 3 F 
 
402 ELEMENTS OF QUATERNIONS. [boOK III. 
 
 by comparing which we see at once that 
 
 IV. . ./„(?, ^'+ q")=Mg + n~iq", q')+Mq, q"), 
 
 the form of the original function, fq, and the values o£ the four variables, q, q', q'\ 
 and n, remaining altogether arbitrary : except that n is supposed to be a number, or 
 at least a scalar, while q, q', q" may (or may not^ be quaternions. 
 
 (2.) For example, if we take the particular function /g' = 2'2^ which gives the 
 form 325, VI. of the derived function /„ (§-, q'), we have 
 
 V. . .fn{q, q")=qq"-Vq"q-^n-^q"^; 
 VI. . . fn{q, q'+ q") = q{q' + q') + {q + q"\q + n"' (^' + q'^ \ 
 and therefore 
 
 VII. . .fn{q, q'^-q")-fn(iq, f^ ^ qq' ^ q q ^ rc^^i \ c[ c[' ^ <j[' q^ 
 = {q^n-^q")q' + q'{q + n\q")+n^q'^ 
 =fniq+n-iq", q'), 
 as required by the formula IV. 
 
 (3.) Admitting then that formula as proved, for all values of the number n, we 
 have only to conceive that number (or scalar) to tend to infinity, in order to deduce 
 this lim.iting form of the equation : 
 
 VIII. . . f^{q, q'+q") ^f^q. q) +/«((?, q") ', 
 or simply, with the abridged notation of 325, (4.), 
 
 IX. . ./(?, q' + q")=f(q, q)+f(q,q")', 
 which contains the expression of the functional property, above asserted to exist. 
 (4.) For example, by what has been already shown (comp. 325, (3.) and (4.)), 
 X. . .Ufq = q\ then /(^, c[) = qq + qq ; 
 and XL . . if /? = q-\ then /(^, j') = - q-\ qq-^ ; 
 in each of which instances we see that the derived function f(jj^ q') is distributive 
 relatively to q', although it is only in the frst of them that it happens to be distri- 
 butive with respect to q also. 
 
 (5.) It follows at once from the formula IX. that we have generally* . , 
 
 XII.../(^, 0)=0; , JL^ ^ 
 
 and it is not difficult to prove, as a result including this, that 
 
 XIII. . .f(^q, xq') = xf(q, q'), if a; be any scalar. 
 (6.) As a confirmation of this last result, we may observe that the definition of 
 f(jl, q) may be expressed by the following formula (comp, 324, IV., and 325, IX.): 
 
 XIV. . .f{q, q) = \\vc^.n{f(^q^n-^q')-fq]', 
 
 we have therefore, if x be any finite scalar, and m = a;-^ n, 
 
 XV. . ./(«?, x^^ = x.\\m,m{f{q^■m-^q')-fq} ; 
 
 a transformation which gives the recent property XIII., since it is evident that the 
 letter m may be written instead of n, in the formula of definition XIV. 
 
 * We abstract here from some exceptional cases of discontinuity, &c. 
 
chap.ii.]differentialquotients,derivedquaternions.403 
 
 327. Resuming then the general expression 325, IX., or 
 writing anew, 
 
 ■l...&fq^f{q,Aq), 
 
 we see (by 326, IX.) that this derived function^ dfq, of q and 
 dq, is always (as in the examples 324, VII. and XI.) distribu- 
 tive with respect to that differential dq, considered as an inde- 
 pendent variable, whatever the^rm of the given function fq 
 may be. We see also (by 326, XIIL), that if the differential 
 dq of the variable^ q^ be multiplied by any scalar^ x, the diffe- 
 rential dfq^ of the function fq, comes to be multiplied, at the 
 same time, by the same scalar, or that 
 
 II. . *f(q, xdq) = xf(q, dq), if x be any scalar. 
 And in fact it is evident, from the very conception and defini- 
 tion (320) of simultaneous differentials, that every system of 
 such differentials must admit of being all changed together to 
 any system of equimultiples, or equisubmultiples, of themselves, 
 without ceasing to be simultaneous differentials: or more gene- 
 rally, that it is permitted to multiply all the differentials^ of a 
 system, by any common scalar. 
 
 (1.) It follows that the quotient^ 
 
 III. ..d/^:d^=/(^,d?):d?, 
 of the two simultaneous differentials, dfq and d^', does not change when the differen- 
 tial d^- is thus multiplied hy any scalar ; and consequently that this quotient III. is 
 independent of the tensor Td^-, although it is not generally independent of the versor 
 Ud^, if q and diq be quaternions : except that it remains in general unchanged, when 
 we merely change that versor to its own opposite (or negative),or to — Ud^, because 
 this comes to multiplying d^ by — 1, which is a scalar. 
 
 (2.) For example, the quotient, 
 
 IV. . . d.j2 : diq=q^diq.qAq-^ = q^\]dq.q.\]dq-\ 
 in which d^'"! and Udg'~i denote the reciprocals of Aq and Ud^', is very far from being 
 independent of d(/, or at least of Ud^' ; since it represents, as we see, the sum of the 
 given quaternion q, and of a certain other quaternion, which latter, in its geometrical 
 interpretation (comp. 191, (5.)), may be considered as being derived from q, by a 
 conical rotation of Ax.q round Ax.dq, through an angle = 2/_dq : so that both the 
 axis and the quantity of this rotation depend on the versor Ud^, and vary with that 
 versor. 
 
 (3.) In general we may, if we please, say that the quotient III. is a Differential 
 Quotient; but we ought not to call it a Differential Coefficient (comp. 318), be- 
 cause dfq does not generally admit of decomposition into two factors, whereof one 
 shall be the differential dq, and the other a. function ofq alone. 
 
404 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (4.) And for the same reason, we ought not to call that Quotient a Derived 
 Function (comp. again 318), unless in so speaking we understand a Function of Two* 
 independent Variables, namely of q and Ud^-, as before. 
 
 (5.) "When, however, a quaternion, q, is considered as a. function of a scalar va- 
 riable, t, so that we have an equation of the form, 
 
 V. . . g=ft, where t denotes a scalar, 
 it is then permitted (comp. 100, (3.) and (4.)) to write, 
 
 =^l\m.h-i{f{t+h)-ft} 
 
 =ft = I)tft=-Dtg', 
 
 and to call this limit, as usual, a derived function oft, because it is (in fact) 9i func- 
 tion of that scalar variable, t, alone, and is independent of the scalar differential, 
 dt. 
 
 (6.) We may also write, under these circumstauces, the differential equation, 
 
 VII. . .dq=Dtq. dt, or VIII. . . dfq ^f't.dt, 
 and may call the derived quaternion, Dtq, or ft, as usual, a differential coefficient in 
 this formula, because the scalar differentia/, dt, is (in fact) multiplied by it, in the 
 expression thus found for the quaternion differential, dq or dft. 
 
 (7.) But as regards the logic of the question (comp. again 100, (3.)), it is im- 
 portant to remember that we regard this derived function, or differential coefficient^ 
 
 IX. . . ft, or Dtft, or Dtq, 
 as being an actual quotient VI., obtained by dividing an actual quaternion, 
 
 X. . . dft, or dq, 
 by an actual scalar, dt, of which the value is altogether arbitrary, and may (if we 
 choose) be supposed to be large (comp. 822); while the dividend quaternion X. de- 
 pends, for its value, on the values of the two independent scalars, t and dt, and on 
 the form of the function ft, according to the law which is expressed by the^eraemZ 
 formula 324, IV., for the differentiation of explicit functions of any single variable. 
 
 328. It is easy to conceive that similar remarks apply to 
 quaternion fund ions of more variables than one; and that when 
 the differential of such sl function is expressed (comp. 324, II.) 
 under the form, 
 
 I. . . dQ = dF(q, r, 5, . .) = F(q, r, 5, . . d^, dr, d*, . .), 
 
 the new function F is always distributive^ with respect to each 
 separately of the differentials, d^-, dr, d^, . . ; being also homo- 
 geneous of the first dimension (comp. 327), with respect to all 
 those differentials, considered as a system ; in such a manner 
 
 * Compare the Note to 325, (4.). 
 
CHAP. II.] PARTIAL AND TOTAL DIFFERENTIALS. 405 
 
 that, whatever may be the /<?rm of the given quaternion func- 
 tion, Q, or F, the derived* function F, or the third member of 
 the formula I., must possess this gQXiQvsX functional property 
 (comp. 326, XIII., and 327, II.), 
 II. , , F(q, Vf s, . . xdgi a;dr, xds . .) 
 
 = xF(q,r,s,, . d^, dr, d.9, . .), 
 
 where x may be any scalar: so that products, as well as 
 squares, of the differentials dq, dr, &c., of q, r, &c. considered 
 as so many variables on which Q depends, are excluded irom the 
 expanded expression of the differential dQ of the function Q. 
 
 (1.) For example, if the function to be differentiated be a product of two qua- 
 ternions, 
 
 III... Q = F(q,r)=qr, 
 
 then it is easily found from the general formula 324, XL, that (because the limit of 
 n-' .d^-.dr is null, when the number n increases without limit') the differential of the 
 function is, 
 
 IV. . . dQ = d.gr = 6F(q,r) = F(q, r, dq, dr) = g . dr + dg.r ; 
 with analogous results, for differentials of products of more than two quaternions. 
 (2.) Again, if we take this other function, 
 
 V... Q=F(iq,r) = q-W, 
 then, applying the same general formula 324, II., and observing that we have, for 
 all values of the number (or other scalar), n, and of the four quaternions, q, r, q\ r', 
 the identical transformation (comp. 324, (2.) ), 
 
 VL . . «{(g+n-igr')-i (r+n~h')-q-h} 
 = q'^r' — (q + n~^ q')~^ q'q~^(r + n-V), 
 
 we find, as the required limit, when n tends to infinite/, the following differential of 
 the function : 
 
 VIL . . dQ=d.q-^r = dF{q,r) = F(q,r,dq,dr) = q-Kdr-q-Kdq.q-^r; 
 which is again, like the expression IV., distributive with respect to each of the dif- 
 ferentials dq, dr, of the variables q, r, and does not involve the product of those two 
 differentials: although these two differential expressions, IV. and VII., are both en- 
 tirely rigorous, and are not in any way dependent on any supposition that the ten- 
 sors of dq and dr are small (comp. again 322). 
 
 329. In thus differentiating a function of more variables 
 than one, we are led to consider what may be called Partial 
 Differentials of Functions of two or more Quaternions; which 
 may be thus denoted, 
 
 * Compare the Note last referred to. 
 
406 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 I.. .dgQ, d,Q, d,Q,... 
 if Q be a function, as above, of q, r, s, , , . which is here sup- 
 posed to be differentiated with respect to each variable sepa- 
 rately, as //"the others were constant. And then, if dQ de- 
 note, as before, what may be called, by contrast, the Total 
 Differential of the function Q, we shall have the General For- 
 mula^ 
 
 11. . .dQ = d5Q + d,y+d,Q + ...; 
 
 or, briefly and symbolically, 
 
 III. . . d = dq + dr + ds + . . ., 
 if q, r, s, . . . denote the quaternion variables on which the 
 quaternion function depends, of which the total differential is 
 to be taken ; whether those variables be all independent, or be 
 connected with each other, by any relation or relations. 
 
 (1.) For example (comp. 328, (1.) ), 
 
 IV. . . if Q=qr, then dqQ = dq.r, and drQ = g.dr; 
 and the sum of these two partial differentials of Q makes up its total differential d Q, 
 as otherwise found above. 
 
 (2.) Again (comp. 328, (2.) ), 
 
 V. . . if Q=q-^r, then dqQ = ~q-idq.q-W; drQ = q~^dr; 
 and dg Q + dr Q = the same d Q as that which was otherwise found before, for this form 
 of the function Q. 
 
 (3.) To exemplify the possibility of a relation existing between the variables q 
 and r, let those variables be now supposed equal to each other in V. ; we shall then 
 have Q—1, d Q = ; and accordingly we have here dqQ = — q~idq = — dr Q. 
 
 (4.) Again, in IV., let 5r = c = any constant quaternion ; we shall then again 
 have 0=dQ=dqQ + drQ ; and may infer that 
 
 VI. . . dr = — g~i . dgr . r, if gr = c = const. ; 
 a result which evidently agrees with, and includes, the expression 324, XI., for the 
 differential of a reciprocal. 
 
 (5.) A quaternion, q, may happen to be expressed as a, function of two or more 
 
 scalar variables, #,«,...; and then it will have, as such, by the present Article, 
 
 its partial differentials, dtq, duq, &c. But because, by 327, VII., we may in this 
 
 case write, 
 
 VII. . . dtq = Dtq . dt, dttq = D,,?. dM, . . . 
 
 where the coefficients are independent of the differentials (as in the ordinary calcu- 
 lus), we shall have (by II.) an expression for the total differential dq, of the form, 
 
 VIII. . . dq = dtq + duq+ . . . ='Dtq .dt +Duq .du-\- . . . ; 
 and may at pleasure say, under the conditions here supposed, that the derived qua- 
 ternions, 
 
CHAP. II.] ELIMINATION OF DIFFERENTIALS. 407 
 
 IX. . . Dtq, Dug, . . . 
 are either the Partial Derivatives, or the Partial Differential Coefficients, of the 
 
 Quaternion Function, 
 
 X...q=:F(t,u,,..); 
 
 with analogous remarks for the case, when the quaternion, q, degenerates (comp. 
 289) into a vector, p. 
 
 330. In general, it may be considered as evident, from the 
 definition in 320, that the differential of a constant is zero ; 
 so that if Q be changed to any constant quaternion^ c, in the 
 equation 324, I., then dQ is to be replaced by 0, in the diffe- 
 rentiated equation^ 324, II. And if there be given any system 
 of equations^ connecting the quaternion variables^ q, r, s, , , , 
 we may treat the corresponding system of differentiated equa- 
 tions^ as holding good, for the system of simultaneous differen- 
 tials, d^', dr, d5, . . . ; and may therefore, legitimately in 
 theory, whenever in practice it shall be found to be possible, 
 eliminate any one or more of those differentials, between the 
 equations of this system. 
 
 (1.) As an example, let there be the two equations, 
 
 I. . . qr = c, and 11. . . s = r\ 
 where c denotes a constant quaternion. Then (comp. 328, (1.), and 324, (1.) ) we 
 have the two differentiated equations corresponding, 
 
 III. . . ^.dr + dg'. r = ; IV. . . ds =r. dr+ dr. r ; 
 in which t\xQ points* might be omitted. The former gives, 
 
 V. . . dr = -^-idg.r, asin 329, VL; 
 
 and when we substitute this value in the latter, we thereby eliminate the differen- 
 tial dr, and obtain this new differential equation, 
 
 VI, . . d« = — rt/'^.d^.r — g-i.d^f.r^. 
 (2.) The equation I. gives also the expression, 
 
 VII. . .r = q-^c; 
 
 the equation II. gives therefore this other expression, 
 
 VIII. . .« = (9-ic)2=g-ic5->c, 
 
 by elimination before differentiation. And if, in the formula VI., we substitute the 
 expressions VII. and VIII. for r and s, we get this other differential equation. 
 
 Compare the second Note to 324, (1.). 
 
408 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 IX. . . d.(y-Jc)2 = — q-^cq-^ .dq.g'^c — q-^ .dg.g"' cq''^ c\ 
 ■which might have been otherwise obtained (com p. again 324, (1.) and (2.) ), under 
 the form, 
 
 X. . . d.(5-»c)2 = g-ic.dC5-ic) + d(^-»c).5-ic. 
 
 33 1 . No special rules are required, for the differentiation of 
 functions of functions of quaternions ; but it may be instructive 
 to show, briefly, how the consideration of such differentiation 
 conducts (comp. 326) to 2i general property of functions of the 
 class f{q^ q')\ and how that property can be otherwise esta- 
 blished. 
 
 (1.) Let/, <p, and »// denote any functional operators, such that 
 
 l...^q = (p{fq)', 
 then writing 
 
 II. . . r =/9, and III. . .s = <pr, we have IV. . . s = \pq] 
 
 whence V. . . ds = di/zg = d^r. 
 
 That is, we may (as usual) differentiate the compound function, <pifq}, as iffq were 
 an independent variable, r ; and then^ in the expression so found, replace the diffe- 
 rential dfq by its value, obtained by differentiating the simple function, fq. For this 
 comes virtually to the elimination of the differential dr, or of the symbol dfq, in a 
 way which we have seen to be permitted (330). 
 
 (2.) But, by the definitions of d/? and/„(9, q'\ we saw (325, VIII. IX.) that 
 the differential d/5 might generally be denoted hyf^(q, dq), or briefly by f(q, dq); 
 whence d<pr and dx^/q may also, by an extension of the same notation, be represented 
 by the analogous symbols, 0„(r, dr) and i^„(<7, d^), or simply by <p(r, dr) and 
 
 (3.) We ought, therefore, to find that 
 
 VI. . . ^„ iq, dq) = <p^ (fq, /« (q, dq)), if ^Pq = <p(fq) ; 
 or briefly that 
 
 VII. ..i|/(g,9') = ^(/9,/(?,0)» if ^q='PA^ 
 for any two quaternions, q, q, and any two functions, f, <p; provided that the func- 
 tions /« (9, q), <pn(s^ q'\ ^niq, q') ^^re deduced (or derived) from the functions /g', 
 d>9, ^q, according to the law expressed by the formula 325, IV. ; and that then the 
 limits to which these derived functions fn(q, q'), &c. tend, when the number n tends 
 to infinity, are denoted by these other functional symbols, f{q, q'), &c. 
 
 (4.) To prove this otherwise, or to establish this general property VII., of func- 
 tions of this class f{q, q), without any use of differentials, we may observe that the 
 general and rigorous transformation 326, V., of the formula 325, IV. by which the 
 functions /«(g, q') are defined, gives for all values of n the equation : 
 
 VIII. . . <pf{ci-\-n-^q) = <p{fq-^n-\fn{q, q')) 
 :=<l>fq+n''(Pn(fq,fn(,q,q'))', 
 
 but also, by the same general transformation, 
 
CHxVP. II.] DIFFERENTIAL OF A PRODUCT OR POWER. 409 
 
 IX. . . i// (gr + n-l q') = i^? + «-l ^/n (?, q') ] 
 hence generally, for aW values of the number n, as well as for all values of the two 
 independent quaternions, q, q\ and for all forms of the two functions, f <p, we may- 
 write, 
 
 X. . . ^n{q, q) = (pnifq, fn {q, ?'))» ^^ ^? = 'Pfl > 
 an equation of which the limiting form, for n = co, is (with the notations used) the 
 equation VII. which was to be proved. 
 
 (5.) It is scarcely worth while to verify the general formula X., by any parti- 
 cular example : yet, merely as an exercise, it may be remarked that if we take the 
 forms, 
 
 XI...fq = q^ ^q = q^, ^j^q = q\ 
 
 of which the two first give, by 325, VI., the common derived form, 
 
 XII. . . /„ {q, q) = (Pn (?, «?') = qq' + q'q + «"' 9'^ 
 the formula X. becomes, 
 
 XIII. . . ^„(g, q') = <Pn{q% qq + q'q + n-^q') ^ 
 = f^ {fli + S'? + »*^ 3^) + (99 + 9'9 + »"^5'2) 5^ + n-l {fiq + qq + »"' ^'2)2 ; 
 which agrees with the value deduced immediately from the function -^q or q\ by the 
 definition 325, IV., namely, 
 
 XIV. . . ^n(?, q') = n{(q + n-^q'y-q^ 
 = n{(iq^ + n-Kqq + qq + n-'q'')y-iq^y}. 
 (6.) In general, the theorem, or rule, for differentiating as in (1.) a function of 
 a Junction, of a quaternion or other variable, may be briefly and symbolically ex- 
 pressed by the formula, 
 
 XV...d(^/)3 = dK/9); 
 and if we did not otherwise know it, a /jroo/ of its correctness would be supplied, by 
 the recent proof of the correctness of the equivalent formula VII. 
 
 Section 4 Examples of Quaternion Differentiation. 
 
 332. It will now be easy and useful to give a short collection of 
 Examples of Differentiation of Quaternion Functions and Equations^ 
 additional to and inclusive of those which have incidentally occurred 
 already, in treating of the principles of the subject. 
 
 (1.) If c be any constant quaternion (as in 330), then 
 
 I. ..dc = 0; Il...d(fq + c-) = dfq; 
 III. . . d.cfq = cdfq ; IV. . . d(/y . c) = dfq . c. 
 
 (2.) In general, 
 V. . . d(/g + 03 + . . ') = ^fq + ^H + • • • ; OT^ briefly, VI. . . dS = 2d, 
 if 2 be used as a mark of summation. 
 
 (3.) Also, VII. . . d{fq . (pq) = dfq .<pq+fq. dfq ', 
 
 and similarly for a product of more functions than two : the rule being simply, to 
 differentiate each factor separately, in its own place, or without disturbing the order 
 
 3 G 
 
410 ELEMENTS OF QUATERNIONS. [bOOK HI. 
 
 of the /actors (comp. 318, 319); and then to add together the partial results (comp. 
 329). 
 
 (4.) In particular, if m be any positive whole number, 
 
 VIIT. . . d . <y»» = jw* I dq + 9»»-2 d^ . ^ . . + 9d^ . 5"t-2 + d^ . J*"-! ; 
 and because we have seen (324, (2.)) that 
 
 IX. . . ei.q-\=-q-\.Aq.q-\ 
 
 we have this analogous expression for the differential oi a. power of a quaternion^ with 
 a negative but whole exponent ^ 
 
 X. . . d . 5r-»« = - q-mA . gw^. q-m 
 = - 5"! dgr . g-"*" - 5"2 ^^ ^ ^i-w _ , ^ _ \j\-m ^q . ^-2 _ ^-m (J^ , ^-i, 
 
 (5.) To differentiate a square root, we are to resolve the linear equation* 
 XI. . . ^i . d . ja + d . gi. 5* = dg ; or XI'. . . rr'-\- r'r = q\ 
 if we write, for abridgment, 
 
 XII. . . r = gri, 9' = d^, r' = d . gi = dr. 
 (6.) Writing also, for this purpose, 
 
 XIII. .. s = Kr=K.^J, 
 whence (by 190, 196) it will follow that 
 
 XIV. . . r« = Nr = Tr2=T9, and XV. . . r + * = 2Sr = 2S.9i, 
 
 the product and sum of these two conjugate quaterrdons, r and s, being thus scalars 
 (140, 145), we have, by XI'., ^ /%''/%At!s -f /k-'A'-*-^ 
 
 XVL . . r-ig'« = r'5 + sr'; ^ a '> ^a'X^^-^ 
 
 whence, by addition, = a' s ^ x/t' 
 
 and finally, 
 
 A 
 
 XVn. . . g' + r-i^'s = (r + fi) r '+ r' (r + «) = 2r' (r -y s) 
 
 an expression for the differential of the square -root of a quaternion, which will be 
 found to admit of many transformations, not needful to be considered here. 
 
 (7.) In the three last sub-articles, as in the three preceding them, it has been sup- 
 posed, for the sake of generality, that q and d^ are two diplanar quaternions ; but 
 if in any application they happen, on the contrary, to be complanar, the expressions 
 are then simplified, and take usual, or algebraic forms, as follows: 
 
 XX. . . d .^« = mg'»-i dg ; XXI. . . d . g-»» = - mq'^-^Aq ; 
 and XXII. . . d . ^^ = Iq-^Aq, if XXIII. . . dg 1 1 1 ^ (123) ; 
 
 * Although such solution of a linear equation, or equation of the^rs< degree, in 
 quaternions, is easily enough accomplished in the present instance, yet in general the 
 problem presents diflSculties, without the consideration of which the theory of diffe- 
 rentiation of implicit functions of quaternions would be entirely incomplete. But a 
 general method, for the solution of all such equations, will be sketched in a subse- 
 quent Section. , , 
 
CHAP.II.] DIFFERENTIALSOrTRANSCENDENTALFUNCTIONS. 411 
 
 because, when q is complanar with g, and therefore with gi, or with r, in the ex- 
 pression XVIIL, the numerator of that expression may be written as r"^ q (r + s). 
 
 (8.) More generally, if x be awy scalar exponent, we may write, as in the ordi- 
 nary calculus, but still under the condition of complanarity XXIII., 
 
 XXIV. . . d.9« = a;^-Jdg'; or XXV. . . ^d . 9* = arg* dar. 
 
 333. IhQ functions of quaternions, which have been lately diffe- 
 rentiated, may be said to be of algebraic form ; the following are a 
 few examples of differentials of what may be called, by contrast, 
 transcendental functions of quaternions : the condition oi complanarity 
 (dg' III 5') being however here supposed to be satisfied, in order that 
 the expressions may not become too complex. In fact, with this sim- 
 plification, they will be found to assume, for the most part, the known 
 and usual forms, of the ordinary differential calculus* 
 
 (1.) Admitting the definitions in 316, and supposing throughout that d^ ||j q, 
 we have the usual expressions for the differentials of ii and \q, namely, 
 I. . . d.£2 = £«dg ; II. . . dlgr = qr-Mg'. 
 (2.) We have also, by the same system of definitions (316), 
 
 III. . . d sin 9 = cos gdj ; IV. . . d cos 5' = - sin gd^ ; &c. 
 
 (3.) Also, if r and dr be complanar with q and dg, then, by 316, 
 IV'. . . d . g*- = d. 6»-i« = 5*-d.rl9 = g''(lgdr + g-'rdg) ; 
 or in the notation of partial differentials (329), 
 
 V. . . dg.g*"= rg»-id(7, and VI. . . dr . g*" = g^'lgdr. 
 (4.) In particular, if the base 9 be a given or constant vector, a, and if the ex- 
 ponent r be a variable scalar, t, then (by the value 316, XIV. of Ip) the recent for- 
 mula IV. becomes, 
 
 VII. . . d.a« = [ lTa + |Ua ja'dt. 
 
 (5.) If then the base a be a given unit line, so that ITa = 0, and Ua = a, we 
 may write simply, J U-^ " " * I . 
 
 VIII. ..d.a* = -a«+id^, if da = 0, and Ta=l. ^n.<.^^^^r p I 
 
 (6.) This useful formula, for the differential of a power of a constant unit line, 
 with a variable scalar exponent, may be obtained more rapidly from the equation 
 308, VII., which gives, 
 
 IX. - . a'=c03-— + asin— , if Ta = 1 ; 
 
 since it is evident that the differential of this expression is equal to the expression 
 itself multiplied by ^Tradt, because a2 = - 1. 
 
 (7.) The formula VIII. admits also of a simple geometrical interpretation, con- 
 nected with the rotation through t right angles, in a plane perpendicular to a, of 
 
 fM. 
 
412 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 which rotation, or version, the power a*, or the versor Va*, is considered (308) to be 
 the instrument* or agent, or operator (comp. 293). 
 
 334. Besides algebraical and transcendental forms, there are other 
 results of operation on a quaternion, q^ or on a function thereof, 
 which may be regarded as forming a new class (or kind) oi func- 
 tions^ arising out oi \hQ principles and rules of the Quaternion Cal- 
 culus itself: namely those which we have denoted in former Chapters 
 by the symbols, 
 
 I. . . Kg, Sg, Yq, Ng, Tq, \Jq, 
 
 or by symbols formed through combinations of the same signs of 
 operation, such as 
 
 II. . . SUg, VUg, UVg, &c. 
 
 And it is essential that we should know how to differentiate expres- 
 sions of these forms, which can be done in the following manner, 
 with the help of the principles of the present and former Chapters, 
 and without now assuming the complanarity , ^q\\\q. 
 
 (1.) In general, let /represent, for a moment, any distributive symbol, so that for 
 any two quaternions, q and q', we shall have the equation, 
 
 III. ../(g + 0=/9+/9'; 
 
 and therefore alsof (comp. 326, (5.)), 
 
 IV. . . f(xq^ = xfq, if x be any scalar. 
 (2.) Then, with the notation 325, IV., we shall have 
 
 V. . ./«(?, q)=n{f{q^n-^q')-fq} =.fq' ; 
 and therefore, by 325, VIII., for any such function /j', we shall have the differential 
 expression, 
 
 VI. . . ^fq=fAq. 
 
 (3.) But S, V, K have been seen to be distributive symbols (197, 207) ; we can 
 therefore infer at once that 
 
 VII. . . dK<7 = Kd^ ; VIII. . . dS? = Sd^ ; IX. . . dV^= Vd^ ; 
 
 or in words, that the differentials of the conjugate, the scalar, and the vector of a 
 quaternion are, respectively, the conjugate, the scalar, and the vector of the differen- 
 tial of that quaternion. 
 
 (4.) To find the differential of the norm, Ng, or to deduce an expression for 
 dN^, we have (by VII. and 145) the equation, 
 
 * Compare the second Note to page 133. 
 
 f In quaternions the equation III. is not a necessary consequence of IV., al- 
 though the latter is so of the former; for example, the equation IV., but not the 
 equation III., will be satisfied, if we assume fq = qcq^ c'q, where c and c' are any 
 two constant quaternions, which do not degenerate into scalars. 
 
CHAP. II.] DIFFERENTIALS OF TENSOR AND VERSOR. 413 
 
 X. . . d^q = d . qKq = 6q .Kq + q .Kdq ; 
 but qKq = K . q'Kq, by 145, and 192, II. ; 
 
 and (1 + K) . q'Kq = 2S . q'Kq = 2S(Kq . q'), by 196, II., and 198, I. ; 
 therefore XI. . . dNg = 2S(Kq . dq). 
 
 (5.) Or we might have deduced this expression XI. for dNg, more immediately, 
 by the general formula 324, IV., from the earlier expression 200, VII., or 210, XX., 
 for the norm of a sum, under the form, 
 
 XI'. . . dN^ = lim . n { N(5 + n^ dq) - N^} 
 
 = lim. {2S(Kq.dq) + n-^-Sdq} 
 
 *» = «> 
 
 = 2S(K^.d^), 
 as before. 
 
 (6.) The tensor, Tq, is the square-root (190) of the norm, 'Nq; and because Tq 
 and Ng are scalars, the formula 332, XXII. may be applied ; which gives, for the 
 differential of the tensor of a quaternion, the expression (comp. 158), 
 
 XII. . . dT9 = ^^ = S(KU5.d5) = S-^, 
 
 a result which is more easily remembered, under the form, 
 
 Tq q 
 
 y: 
 
 (7.) The versor \Jq is equal (by 188) to the quotient, q : T^^, of the quaternion 
 q divided by its tensor Tq ; hence the differential of the versor is, 
 
 XIV...dU, = df = (^^-S^-iU=V^.U,; ^3t«'-^^' 
 
 whence follows at once this formula, analogous to XIII., and like it easily remem- 
 bered, Q . 
 
 XV... ^^=v^. "- -^ 7^-J 
 
 ^q q 
 
 (8.) We might also have observed that because (by 188), we have generally 
 q^Tq. U?, therefore (by 332, (3.)) we have also, 
 
 XVI. . . d^ = dT^ . U^ + T^ . dU^, 
 and 
 
 q Tj ^ % • 
 if then we have in any manner established the equation XIII., we can immediately 
 deduce XV. ; and conversely, the former equation would follow at once from the 
 latter. 
 
 (9.) It may be considered as remarkable, that we should thus have generally, or 
 for any two quaternions, q and dq, the formula :* 
 
 * When the connexion of the theory of normals to surfaces, with the differential 
 calculus of quaternions, shall have been (even briefly) explained in a subsequent 
 Section, the student will perhaps be able to perceive, in this formula XVIII., a re- 
 cognition, though not a very direct one, of the geometrical principle, that the radii 
 of a sphere are its normals. -' 
 
414 ELEMENTS OF QUATERNIONS. [booK III. 
 
 XVIII. . . S (dU^ : U^) = ; or XVIII'. . . dU^ : Ug = Si ; 
 but this vector character of the quotient dJJq : JJq can easily be confirmed, as fol- 
 lows. Taking the conjugate of that quotient, we have, by VII. (comp. 192, II. ; 
 158 ; and 324, XI,), 
 
 XIX. . . K (dU^ . U^-i) = KU^-i . dKU^ = U^ . d (U^ i) = - dU^ . Ug-i ; 
 whence 
 
 XX. . . (1 + K) (dU? . U^O = ; 
 
 which agrees (by 196, II.) with XVIII. 
 
 (10.) The scalar character of the tensor, Tj, enables us always to write, as in 
 the ordinary calculus, 
 
 XXI. . . dlTg = dTq : Tq ; 
 
 but 1T(7 = SI$', by 316, v.; the recent formula XIII. may therefore by VIII. be 
 thus written, 
 
 XXII. . . Sd\q = dSlq = dTq:dq = S (dq : q) ; or XXII'. . . dig - q'^ dq = S"' 0. 
 (11.) "When dg I II g, this last difference vanishes, by 333, II. ; and the equation 
 
 XV. takes the form, 
 
 XXIII. . . dlUg = Vdlg = dVlg. 
 
 And in fact we have generally y 1U5' = V15', by 316, XX., although the differentials 
 of these two equal expressions do not separately coincide with the members of the re- 
 cent formula XV., when q and dq are diplanar. We may however write generally 
 (comp. XXII.), 
 
 XXIV. . . dlUg - dVq : Vq = V(dlg - dg : g) = d]q -dq : q. 
 
 335. We have now differentiated the six simple functions 334, I., 
 which are formed by the operation of the six characteristics^ 
 
 K, S,V, N, T, U; 
 and as regards the differentiation of the compound functions 334, II., 
 which are formed by combinations of those former operations, it is 
 easy on the same principles to determine them, as may be seen in 
 the few following examples. 
 
 (1.) The axis Ax. g of a quaternion has been seen (291) to admit of being re- 
 presented by the combination JJYq ; the differential of this axis may therefore, by 
 334, IX. and XIV., be thus expressed : 
 
 I. . . d (Ax. q) = dUV^ = V (Ydq : Yq) . VYq ; 
 whence 
 
 Tl d(Ax.g) _dUVg^ Vdg 
 Ax. g TJYq Yq ' 
 
 The differential of the axis is therefore, generally, a line perpendicular to that axis, 
 or s\tua.ted in the plane of the quaternion; but it vanishes, when the plane (and 
 therefore the axis) of that quaternion is constant ; or when the quaternion and its 
 differential are complanar. 
 
 (2.) Hence, 
 
 III. ..dUVg = 0, if IV. ..dg III 7; 
 
 and conversely this complanarity IV. may be expressed by the equation III. 
 
CHAP. II.] DIFFERENTIALS OF AXIS AND ANGLE. 415 
 
 (3.) It is easy to prove, on similar principles, that 
 
 and 
 
 VL . . dSU^ = SdUg = S f V ^. U^l 
 
 (4.) But in general, for any two quaternions, q and q, we have (comp. 223, 
 (o.) ) the transformations, 
 
 VII. . . S (V^' . ^) = S (V^' . V^) =S. ^Vj ; 
 and when we thus suppress the characteristic V before dg' : q^ and insert it before 
 Ug', under the sign S in the last expression VI., we may replace the new factor VU^' 
 by TVU^. UVU^ (188), or by TVU^ . JJYq (274, XIIL), or by - TVUg : UV^ 
 (204, v.), where the scalar factor TVUg' may be taken outside (by 196, VIII.) ; 
 also for q-^ : UV^f we may substitute 1 : (UVg' . $'), or 1 : q\^Yq, because \JYq \\\q'^ 
 the formula VI. may therefore be thus written, 
 
 VIIL..dSU^ = -S^.TVUg. 
 
 (5.) Now it may be remembered, that among the earliest connexions of quater- 
 
 ternions with trigonometry^ the following formulae occurred (196, XVI., and 204, 
 
 XIX.), / f //, 
 
 IX. . . SUg = cos L q, TVU? = sin ^ 5 ; -C -^ > 
 
 we had also, in 316, these expressions for the angle of a quaternion, 
 
 X. . . Zg=TVl2 = TlU<?; 
 
 we may therefore establish the following expression for the differential of the angle 
 
 of a quaternion, 
 
 XI. . . d Z ? = dTVl^ = dTlU^ = S ^^ 
 
 q\JYq 
 
 (6.) The following is another way of arriving at the same result, through the 
 differentiation of the sine instead of the cosine of the angle, or through the calcula- 
 tion of dTVU^-, instead of dSU^'. For this purpose, it is only necessary to remark 
 that we have, by 334, XII. XIV., and by some easy transformations of the kind 
 lately employed in (4.), the formula, 
 
 dividing which by SUg', and attending to IX. and X,, we arrive again at the ex- 
 pression XI., for the differential of the angle of a quaternion. 
 
 (7.) Eliminating S (dg' : g'UVg') between VIII. and XII., we obtain the differen- 
 tial equation, 
 
 XIII. . . SU^ . dSU(? + TVUj . dTVUg = ; 
 
 of which, on account of the scalar character of the differentiated variables, the inte- 
 gral is evidently of the /orni, 
 
 XIV. . . (SUg)2 + (TVUg)2 = const.; 
 
 and accordingly we saw, in 204, XX., that the sum in the first member of this equa- 
 tion is constantly equal to positive unity. 
 
416 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (8.) The formula XI. may also be thus written, 
 
 XV...dlq = S (V(d$? : q) : UV<?); 
 ■with the verification, that when we suppose d^|[|$', as in IV., and therefore 
 dJJYq = by III., the expression under the sign S becomes the differential of the 
 quotient, Vl^: UV^, and therefore, by 316, VI., of the angle Z g itself. 
 
 336. An important application of the foregoing principles and 
 rules consists in the differentiation of scalar functions of vectors, when 
 those functions are defined and expressed according to the laws and 
 notations of quaternions. It will be found, in fact, that such diffe- 
 rentiations play a very extensive part, in the applications of quater- 
 nions to geometry/ ; but, for the moment, we shall treat them here^ as 
 merely exercises of calculation. The following are a few exam- 
 ples. 
 
 (1.) Let p denote, in these sub-articles, a variable vector; and let the following 
 equation be proposed, 
 
 I. . . r2 + p2 = 0, in which Vr = 0, 
 so that r is a (generally variable) scalar. Differentiating, and observing that, by 
 279, III., pp' + p'p = 2Spjo', if p' be any second vector, such as we suppose dp to be, 
 we have, by 322, VIII., and 324, VII., the equation, 
 
 II. . . rdr + Spdp = 0; or III. . . dr = -r-^Qp6^^=rSp-^dp. f 
 
 In fact, if r be supposed positive, it is here, by 282, II., the tensor of p ; so that^his 
 last expression III. for dr is included in the general formula, 334, XIII. \ ^ ^jt^ 
 
 (2.) K this tensor, r, be constant, the differential equation II. becomes simply, I 
 
 IV. . . Spdp = 0, if - p2 = const., or if dTp = 0. 
 (3.) Again, let the proposed equation be (comp. 282, XIX.), 
 V. . . r2 = T(tp + pK), with dt = 0, dK = 0, 
 so that I and k are here two constant vectors. Then, squaring and differentiating, 
 we have (by 334, XL, because Ktp = pi^ &c.), - \ 
 
 VL . . 2r3dr = idN(ip + pK)=:S(pt + K;p)(td|o+dp/c) . J 
 
 VII. .. 2r-idr = Si/dp, 
 if V be an auxiliary vector, determined by the equation, 
 
 VIII. ..r^v = (i2 + ic2) p + 2VKpi ; 
 which admits of several transformations. 
 
 (4.) For example we may write, by 295, VIL, 
 
 IX. . . r^v = (i2 4- k2^ p ^ ^pt ^ tp,j 
 = t (ip + pK) + K (pn- Kp) ; 
 or, by 294, III., and 282, XIL, 
 
 X. . .r^v = (t2 + k2) p + 2 (icStp - pStK + iS<cp) 
 = (t - «:)2 p + 2 (iSfcp + kSip) ; &c. 
 
 - sft'' 
 
CHAP. II.] SCALAR FUNCTIONS OF VECTORS. 417 
 
 (5.) The equation V. gives (comp. 190, V.), when squared without differentia- 
 tion, 
 
 XI. . . r* = N (ip + p/c) = (tp + pic) (pi + fcp) 
 
 = (l* + K^)p^ + l'pKp + pTpl 
 = (l2+K2)p2 + 2St'pKp 
 
 = (t - k)2 p2 + 4Sip S/cp = &C., 
 by transformations of the same kind as before; we have therefore, by the recent ex- 
 pressions for r*v, the following remarkably simple relation between the two variable 
 vectors^ p and v, 
 
 XII. . . Si/p = 1 ; or Xir. . . Spv = 1. 
 
 (6.) When the scalar^ r, is constant, we have, by VII., the differential equa- 
 tion, 
 
 XIII. . . Svdp = ; whence also XI V. . . Spdv = 0, by XII. ; 
 
 a relation of reciprocity thus existing, between the two vectors p and v, of which the 
 geometrical signification will soon be seen. 
 
 (7.) Meanwhile, supposing r again to rary, we see that the last expression VI. 
 for 2rMr may be otherwise obtained, by taking half the differential of either of the 
 two last expanded expressions XI. for r* ; it being remembered, in all these little 
 calculations, tliat ctjcHcuI permutation of factors, under the sign S, is permitted 
 (223, (10.)), even if those factors be quaternions, and whatever their number may 
 be : and that if they be vectors, and if their number be odd, it is then permitted, 
 under the sign V, to invert their order (295, (9.)), and so to write, for instance, 
 YipK instead of Ykqi, hi the formula VIII. 
 
 (8.) As another example of a scalar function of a vector, let p denote the proxi- 
 mity (or nearness) of a variable point p to the origin o ; so that 
 
 XV. . .p = (-p2)-i = Tp->, or XV'. . . p-2 + p2 = 0. 
 Then, 
 
 XVI. . . dp = Sv6p, if XVII. . . j/=/)V=P^Up; 
 
 V being here a new auxiliary vector, distinct from the one lately considered (VIII.) 
 and having (as we see) the same versor (or the same direction) as the vector p it~ 
 self, but having its tensor equal to the square of the proximity of 'P too ; or equal 
 to the inverse square of the distance, of one of those two points from the other. 
 
 337. On the other hand, we have often occasion, in the applica- 
 tions, to consider vectors as functions ofscalars, as in 99, but now 
 with forms arising out of operations on quaternions^ and therefore 
 such as had not been considered in the First Book. And whenever 
 we have thus an expression such as either of the two following, 
 
 I. ..f.=i0(O, or 11. . .p = 0(5, 0, 
 for the variable vector of a curve^ or of a surface (comp. again 99), s 
 and t being two variable scalars, and 0(«) and '^(s^t) denoting anj/ 
 functions of vector form, whereof the latter is here supposed to be en- 
 tirely independent* of the former, we may then employ (comp. 100, 
 
 * We are therefore not employing here the temporary notation of some recent 
 Articles, according to which we should have had, d^q = f(q, d^). 
 
 3 H 
 
418 ELEMENTS OF QUATERNIONS. [bOOK 111. 
 
 (4.) and (9.)» and the more recent sub-articles, 327, (5.), (6.), and 
 329, (5.) ) the notation of derivatives, total or partial; and so may 
 write, as the differentiated equations^ resulting from the forms I. and 
 TT. respectively, the following : 
 
 III. . . d/3 = ^'^d« = />'d« = D,/>.d«; 
 IV. . . d/> = d^ + d</> = D^pAs + Dtp . d« ; 
 of which the geometrical significations have been already partially 
 seen, in the sub-articles to 100, and will soon be more fully deve- 
 loped. 
 
 (1.) Thus, for the circular locus, 314, (1.), for which 
 Y...p = a% Ta=l, Saj3 = 0, 
 we have, by 333, VIII., the following derived vector, 
 
 (2.) And for the elliptic locus, 314, (2.), for which 
 
 VII. . . p=V.a«/3, Ta = l, but «o< S«j3 = 0, 
 we have, in like manner, this other derived vector, 
 
 VIII. ..p'=D,p = -V.a'-i/3. 
 
 (3.) As an example of a vector-function of more scalars than one^ let us resume 
 the expression (308, XVIII.), 
 
 IX. . . p^rk*j»kj-»k-t; 
 in which we shall now suppose that the tensor r is given, so that p is the variable 
 vector of a point upon a given spheric surface, of which the radius is r, and the cen- 
 tre is at the origin ; while s and t are two independent scalar variables, with respect 
 to which the two partial derivatives of the vector p are to be determined. 
 
 (4.) The derivation relatively to t is easy ; for, since ijk are vector-units (295), 
 and since we have generally, by 333, VIII., 
 
 X. . . d . a* = - a^i da?, and therefore XI. . . D< . a* = — a^+' D^a?, 
 
 if Ta = 1, and if x be any scalar function of t, we may write, at once, by 279, IV., 
 
 XII. . . D,p = "^ {kp - pk) = TrYIcp ; 
 
 and we see that , 
 
 XIII. ..SpD,p = 0, 
 
 a result which was to be expected, on account of the equation, 
 
 XIV. . . p2 + ;.2=0, 
 which follows, by 308, XXIV., from the recent expression IX. for p. 
 
 (5.) To form an expression of about the same degree of simplicity, for the other 
 partial derivative of p, we may observe that/'+>^j-» is equal to its own vector part 
 (its scalar vanishing) ; hence 
 
CHAi'. II.] VECTOR FUNCTIONS, DERIVED VECTORS. 419 
 
 XV. . . Dgjo = nktjk-ip ; or XVI. . . D^p = irk'^tjp = Trjk-i^p, 
 by the transformation 308, (11.). And because the scalar of k^jk-^ is zero, we have 
 thus the equation, 
 
 XVII. . . SpD,p = 0, 
 
 which is analogous to XIIL, and might have been otherwise obtain^, by taking the 
 derivative of XIV. with respect to the variable scalar s. 
 
 (6.) The partial derivative T>sp must be a vector ; hence, by XV. or XVI., p 
 must be perpendicular to the vector k^jk'^^ or A^'j, orJA-^'; a result which, under 
 the last form, is easily confirmed by the expression 315, XII. for p. In fact that 
 expression gives, by 315, (3.) and (4.), and by the recent values XII. XVI., these 
 other forms for the two partial derivatives of p, which have been above considered : 
 
 XVIII. . . Dip = 7rr/v2«V.72»; XIX. . . D,jO = Trr(Jc^^ .i^^^ -Y.k^) ; 
 which might have been immediately obtained, by partial derivations, from the ex- 
 pression 315, XII. itself, and of which both are vector-forms. 
 
 (7.) And hence, or immediately by derivating the expanded expression 316, 
 XIII., we obtain these new forms for^the partial derivatives of p : 
 XX. . . Dtp = 7rr (j cos tTr — i sin <7r") sin stt ; 
 
 XXI. . . Dgp = 7rr { (t cos tir + j sin <7r) cos stt — A; sin «7r } . 
 
 (8.) We may add that not only is the variable vector p perpendicular to each of 
 tiie two derived vectors, Dap and Dtp, but also they are perpendicular to each other ; 
 for we may write, by XII. and XVI., 
 
 XXII. . . ^(Dsp . Dtp) = - 7r2 S . /t3'ip2 k^^2r2S.k^ti=o. 
 and the same conclusion may be drawn from the expressions XX. and XXI. 
 
 (9.) A vector may be considered as a function of three independent scalar varia- 
 bles, such as r, s, t; or rather it rmist be so considered, if it is to admit of being the 
 vector of an arbitrary point of space ; and then it will have a total differential (329) 
 of the trinomial form, 
 
 XXIII. . . d|0=drp + dsp + d<jO = D,.p.dr + D«|0.d«+D<jo.d^; 
 and will thus have three* partial derivatives. 
 
 (10.) For example, when p has the expression IX., we have this third partial 
 
 derivative, 
 
 XXIV.. . D.p = r-ip = Up, 
 
 which may also be thus more fully written (comp. again 315, XIIL), 
 
 XXV. . . Drp = k*j^kj'^k'* = (t cos tTT +j sin ttr) sin ^tt + A: cos stt ; 
 
 and we see that the three derived vectors, 
 
 XXVI. . . Drp, Dsp, Dtp, 
 
 compose here a rectangular system. 
 
 * That is to say, three of the first order ; for we shall soon have occasion to con- 
 sider successive differentials, of functions of one or more variables, and so shall be 
 conducted to the consideration of orders of differentials and derivatives, higher than 
 the first. 
 
420 ELEMENTS OF QUATERNIONS. [bOOK 111. 
 
 Section 5 On Successive Differentials^ and Developments^ 
 
 of Functions of Quaternions. 
 
 338. There will now be no difficulty in the successive dif- 
 ferentiation, total or partial, of functions of one or more qua- 
 ternions ; and such differentiation will be found to be useful^ 
 as in the ordinary calculus, in connexion with developments of 
 functions : besides that it is necessary for many of those geo- 
 metrical and physical applications of differentials of quater- 
 nions, on which we have not entered yet. A few examples of 
 successive differentiation may serve to show, more easily than 
 any general precepts, the nature and effects of the operation ; 
 and we shall begin, for simplicity, with explicit functions of one 
 quaternion variable. 
 
 (1.) Take then the square, q^, of a quaternion, as a function /7, which is to be 
 twice differentiated. We saw, in 324, VII., that a first differentiation gave the 
 equation, 
 
 I. . . d/g = d.</2 = 5.dg + d5.g; 
 
 but we are now to differentiate again, in order to form the second differential d^fq 
 of the function q"^, treating the differential of the variable q as still equal to d^, and 
 in general writing dc'.^ = d-9, where d^^ is a new arbitrary quaternion, of which the 
 tensor, Td2g, need not be small (comp. 322). And thus we get, in general, this 
 twice differentiated expression, or differential of the second order, 
 
 II. . . d2/^ = d2.g2=j.d«^ + 2dj« + d2j.5. 
 
 (2.) The second differential of the reciprocal of a quaternion is generally (comp. 
 
 324, XL), 
 
 III. . . d3.5-i = 2(9->dj)2j-i-7-id2^.g-i. 
 
 (3.) If p be a variable vector, then (comp. 336, (1.)) we have, for the first and 
 second differentials of its square, the expressions : 
 
 I V. . . d . p2 = 2Spdp ; V. . . d2. p2 = 2S/jd2p + 2dp2. 
 (4.) If/p be any o^Aer scalar function of a variable vector p, and if (comp. again 
 the sub-articles to 336) \t& first differential be put under the form, 
 
 VI. . . d/p = 2Sj/dp, when v is another variable vector, 
 then the second differential of the same function may be expressed as follows : 
 
 VII. . . d2/p = 2Svd2p + 2Sdrdp ; 
 in which we have written, briefly, Sdj^dp, instead of S(dj/.dp). 
 
 (5.) The following very simple equation will be found useful, in the theory of 
 motions, performed under the influence of central forces : 
 , Y i VIII. . . dVpdp = Vpd"-p ; because V. dp2 = 0. 
 
 (6.) As an example of the second difterential of a quaternion, considered as a 
 
CHAP. II.] SUCCESSIVE DIFFERENTIALS, DEVELOPMENTS. 421 
 
 function of a scalar variable (comp. 333, VIII., and 337, (1.))? th»following may- 
 be assigned, in which a denotes a given unit line, so that a^ =— 1, da = 0, but x 
 is a variable scalar : 
 
 IX. . . d2.a*=^df |a^»da; j = |a^^id2a;-f I ]a*da;2. 
 
 (7.) The second difierential oiihQ product oi any two functions oi& quaternion 
 q may be expressed as follows (comp. II.) : 
 
 X. . . d^(ifq.il>q) = d2fq.<pq + 2dfg.d<l>q+fq.d^q. 
 
 339. The second differential, d^q^ of the variable quaternion q, 
 enters generally (as has been seen) into the expression of the second 
 differential d^fq, of the function /9', as a new and arbitrary quater- 
 nion : but, for that very reason, it is permitted, and it is frequently- 
 found to be convcfiient, to assume that this second differential d^^ is 
 equal to zero : or, what comes to the same thing, that the first dif- 
 ferential dq is constant. And when we make this new supposition, 
 I. , . dq = constant, or P. . . d^^- = 0, 
 
 the expressions for d^q become of course more simple, as in the 
 following examples. 
 
 (1.) With this last supposition, I. or I'., we have the following second differen- 
 tials, of the square and the reciprocal of a quaternion : 
 
 II. . . d2.g2 = 2d52; IIL . . d2.9-i=2(<7-idg)2g-i = 27-»(d5. j-i)/ *. 
 
 (2.) Again, if we suppose that cq, Ci, cg are any thi-ee constant quaternions, and 
 
 take the function, 
 
 IV. . .fq=CoqCiqC2, 
 
 we find, under the same condition I. or I'., that its first and second differentials are, 
 
 V. . . d/g = codq . ciqc2 + coqcidq. c^; VI. . . d2/^ = 2codq . cidq . cz ; 
 
 in writing which, the points* may be omitted. 
 
 (3.) Theirs* differential, dq, remaining still entirely arJiVrary (comp. 322, (8.), 
 
 and 325, (2.) ), so that no supposition is made that its tensor Tdq is small, although 
 
 we now suppose this differential dq to be constant (I.) we have rigorously, 
 
 Yll...(iq + dqy=q^ + d.q'-\-^dKq^; 
 
 an equation which may be also written thus, 
 
 VIII. ..({?+ dg)2 = (1 + d + ^d2) . ^2. 
 
 (4.) And in like manner we shall have, more generally, under the same condi- 
 tion of constancy of d^, the equation, 
 
 IX. ../(?+ d9) = (l + d+ id2)/^, 
 if the function /7 be the sum of any number of nionomes, each separately of the ybrwi 
 
 * Compare the second Note to page 399. 
 
422 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 IV., and there%re each rational, integral, and homogeneous of the second dimension, 
 with respect to the variable quaternion, 9 ; or of such monomes, combined with others 
 of the Jirst dimension, and with constant terms : that is, if a©, bo, 61, 6'o, 6'i, . . and 
 Co, ci, C2, c'o, c'l, c'a, . . be ani/ constant quaternions, and 
 
 X. . .fq = ao+ ^buqbi + Sco^cigca. 
 
 340. It is easy to carry on the operation of diiFerentiating, to the 
 third and higher orders ; remembering only that if, in any former 
 stage, we have denoted the first differentials of q, dq, . . by d^', d^q, . . 
 we then continue so to denote them, in every subsequent stage of the 
 successive differentiation : and that if we find it convenient to treat 
 any one differential as constant, we must then treat all its successive 
 differentials as vanishing. A few examples may be given, chiefly 
 with a view to the extension of the recent formula 339, IX., for the 
 function f{q + d^) of a sum, of any two quaternions, q and d^', to po- 
 lyn^mial forms, of dimensions higher than the second. 
 
 (1.) The third differential of a square is generally (comp. 338, II.), 
 
 I.. . d3.g3 = ^.d3g+d35.9 + 3(d7. d^^ + da^.d^). 
 (2.) More generally, the MtVd differential of a product of two quaternion func- 
 tions (comp. 338, X.) may be thus expressed : 
 
 II. . . d^ (fq.<pq) = d^fq .<})q + 3d.^fq.d(pq + Sdfq.d^q +fq.d^(pq. 
 (3.) More generally still, the n'* differential of a product is, as in the ordinary 
 calculus, 
 
 III. . .d»(ifq.(pq') = d>*fq.<Pq-[-ndn-^fq.d(pq + n.id»-^fq.d^(l)q+ . . +fq.dyq, 
 n(n-l) n(»-l)( n-2) 
 if n2= , n3= 2~3 ' '' 
 
 the only thing peculiar to quaternions being, that we are obliged to retain (gene- 
 rally) the order of the factors, in each term of this expansion III. 
 
 (4.) Hence, in particular, denoting briefly the function /§ by r, and changing 
 
 H to q, 
 
 IV. . . d'*.rq = d»r.q + nd»-^r.dq, if d^q = 0. 
 
 (5.) Hence also, imder this condition that dg- is constant, if c be any other con- 
 stant quaternion, we have the transformation, 
 
 (l+d + |d» + ^d3 + .. •+ ^3^^^^^_^^ d"-'y-<^g + ^g)^' '^ ^"'■ = ^- 
 
 (6.) Hence, by 339, (4), it is easy to infer that if we interpret the symbol t"* by 
 the equation (comp. 316, I.), 
 
 VI. . . t*" = 1 + d + id2 + -^ d3+ &c., 
 
 that is, if we interpret this other symbol e'^fj, as concisely denoting the series which 
 
CHAP. II.] FUNCTIONS WHICH VANISH TOGETHER. 423 
 
 is formed from fq, by operating on it with this symbolic development ; and if the 
 function fq, thus operated on, be any finite polynome^ involving (like the expression 
 339, X.) no fractional nor negative exponents; we may then write, as an extension 
 of a recent equation (339, IX.), the formula : 
 
 VIL..£<l/9=/(9+d(?), if 6?q=0', 
 which is here a perfectly rigorous one, all the terms of this expansion for di. function 
 of a sum of two quaternions, q and dgr, becoming separately equal to zero, as soon as 
 the symbolic exponent of d becomes greater than the dimension of the polj'nome. 
 
 (7.) We shall soon see that there is a sense, in which this exponential transfor- 
 mationYll. may be extended, to other functional forms which are not composed as 
 above : and that thus an analogue of Taylor's Theorem can be established for Qua- 
 ternions. Meanwhile it may be observed that by changing dq to Aq, in the^ntVe 
 expansion obtained as above, we may write the formula as follows : 
 
 VII I. . . t<^fq =f{q + Aq) = (1 + A)fq, OX briefly, IX. . . tJ = 1 + A ; 
 which last symbolical equation may be operated on, or transformed, as in the usual 
 calculus of differences and differentials. For instance, it being understood that we 
 treat A'^q as well as d^*/ as vanishing, we have thus (for any positive and whole ex- 
 ponent m), the two following transformations of IX., 
 
 X. . . A"* = (fd - !)»«, and XI. . . d»« = (log ( 1 + A) )"« ; 
 the results of operating, with the symhols thus equated, on any polynomial function 
 fq, of the kind above described, being always ^neYe expansions, which are rigorously 
 equal to each other. 
 
 341. Let Fx and (px be any two functions of a scalar va- 
 riable, of which both vanish with that variable ; so that they 
 satisfy the two conditions, 
 
 I. . . FO = 0, 00 = 0. 
 Then the three simultaneous values, 
 
 II. . . rr, Fx, (px, 
 
 of the variable and the two functions, are at the same time 
 (comp. 320, 321) three simultaneous differences, as compared 
 with this other system of three simultaneous values, 
 
 III. . . 0, i^O, 00. 
 If, then, any equimultiples, 
 
 IV. . . nx, nFx, n(px, 
 of the three values II., can be made, by any suitable iyicrease 
 of the number, n, combined with a decrease of the variable, x, 
 to tend together to any system of limits, those limits must (by 
 the definition in 320, compare again 321) admit of being con- 
 sidered as a system of simultaneous differentials, 
 
424 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 V. . . d.T, ^Fx, d(pX, 
 
 answering to the system of initial values III. ; and must be 
 proportional to the ultimate values of the connected system of 
 derivatives, 
 
 VI. . . 1, F'x, (p'x, when x tends to zero. 
 
 We may therefore write, as expressions for tliose ultimate va- 
 lues of the two last derived functions, 
 
 VII. . . FO = llm. nF^, = lim. w0 -, if FO = 00 = 0. 
 
 n = =o ^ n = =«> n 
 
 And even if these last values vanish, or if the ^z<;o new condi- 
 tions 
 
 VIII. . . /^O = 0, 00 = 0, 
 
 are satisfied, so that x, F'x, and 0'ic are now (comp. II.) a w^z« 
 system of simultaneous differences, we may 5^z7/ establish the 
 following equation of limits of quotients, which is independent 
 of these last conditions VIII., 
 
 IX. . .lim(Fa;:0aj)=lim(F'a':0'a;), if FO = 0O = O; 
 
 it being understood that, in certain cases, these two quotients 
 may 6o^7e vanish with x ; or may tend together to infinity, when 
 X tends^ as before, ^o zero. 
 
 (1.) This theorem is so important, that it will not be useless to confirm it by a 
 geometrical illustration, which may at the same time serve for a geometrical proof ; 
 at least for the extensive case where both the functions fx and ^x are of scalar forms, 
 and consequently may be represented, or constructed, b}' the corresponding ordi- 
 nates, XY and XZ (or ordinates answering to one common abscissa OX), of two 
 curves OyY and OzZ, which are in one plane, and set out from (or pass through) 
 one common origin O, as in the annexed Figure 75. We shall afterwards see that 
 the result, so obtained, can be extended to quaternion functions. 
 
 (2.) Suppose then, first, that the ordinates of these two curves are proportional, 
 or that they bear to each other one fixed and constant ratio', so that the equation, 
 
 X. .. XY : liZ = xy'.xz, 
 is satisfied for every pair of abscisses, OX and Ox, however great or small the corre- 
 sponding ordinates may be. Prolonging then (if necessary) the chord Yy of the 
 first curve, to meet the axis of abscissae in some point t, and so to determine a sub- 
 secant tX, we see at once (by similar triangles) that the corresponding chord Zz of 
 thes econd curve will meet the same axis in the same point, t ; and therefore that 
 it will determine {rigorously^ the same subsecant, tX. 
 
CHAP. II.] GEOMETRICAL ILLUSTRATIONS. 425 
 
 (3.) Hence, if the point x be conceived to approach to X, so that the secant Yt/t 
 of the first curve tends to coincide with the tangent YT to that curve at the point Y, 
 the secant Zzt of the second curve must tend to 
 coincide with the line ZT, which line therefore 
 must be the tangent to that second curve : or in 
 other words, corresponding suhtangents coincide, 
 and of course are equal, under the supposed con- 
 dition X., of a constant proportionality ofordi- 
 nates. 
 
 (4.) Suppose next that corresponding ordi- 
 nates only tend to bear a given or constant ratio 
 to each other ; or that their (now) variable ratio 
 tends to a given or fixed limit, when the com- 
 mon abscissa is indefinitely diminished, or when 
 die point X tends to ; and let T be still the 
 variable point in which the tangent to the^rs^ curve at Y meets the axis, so that the 
 line TX is still the first suhtangent. Then the corresponding tangent to the second 
 curve at Z will not in general pass through the point T, but will meet the axis in 
 some different point U. But the ratio of the two corresponding suhtangents, TX and 
 UX, which had been a ratio o^ equality, when the condition oi proportionality X. 
 was satisfied rigorously, will now at least tend to such a ratio; so that we shall have, 
 under this new condition, of tendency to proportionalit}' of ordinates, the limiting 
 equation, 
 
 XL . . lim (TX : UX) = 1 ; 
 
 whence the equation IX. results, under the geometrical fi)rm, 
 
 XII. . . lim (tan XTY : tan XUZ) = lim (XY : XZ). 
 
 (5.) We might also have observed that, when the proportion X. is rigorous, cor- 
 responding areas ^ (such as xXYy and xXZz) of the two curves are then exactly in 
 the given ratio of the ordinates ; so that this other equation, or proportion, 
 
 XIIL , . OXYyO : OXZ2O = XY : XZ, 
 is then also rigorous. Hence if we only suppose, as in (4.), that the ordinates tend 
 to some fixed limiting ratio, the areas must tend to the same ; so that ey the second 
 member of the equation IX. have any definite value, as a limit, ihQ first member 
 must have the same: whereas the recent proof, hy suhtangents, served rather to 
 show that if )JaQ first (or left hand) limit in IX. existed, then the second limit in 
 that equation existed also, and was equal to the first. 
 
 (6.) U the function Fx be a quaternion, we may (by 221) express it as follows, 
 XIV. .. Fx=: Jr+ iX + jV+kZ, 
 where JV, X, Y, Z axe four scalar functions of x, of which each separately can be 
 
 * Compare the Fourth Lemma of the First Book of the Principia ; and see espe- 
 cially its Corollary, in which the reasoning of the present sub-article is virtually an- 
 ticipated. 
 
 • 3i 
 
426 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 constructed^ as the ordinate of a plane curve ; and the recent geometrical* reasonmg 
 will thus apply to each of them, and therefore to their linear combination Fx: which 
 quaternion function reduces itself to a vector function oi x, when JV=0. 
 
 (7.) And if \l/x were another quaternion or vector function, we might first sub- 
 stitute it for Fx, and then eliminate the scalar function <px ; so that a limiting equa- 
 tion of the form IX. may thus be proved to hold good, when both the functions com- 
 pared are vectors, or quaternions, supposed still to vanish witli x. 
 
 (8.) The general considerations, however, on which the equation IX. was lately 
 established, appear to be more simple and direct ; and it is evident that they give, 
 in like manner, this other but analogous equation, in which F"x and ^"x are second 
 derivatives, and the conditions VIII. are now supposed to be satisfied : 
 
 XV. . . lim {F'x : (p'x) = lim (^F"x : ^"x}, if F'O = 0, 0'O = 0. 
 
 And so we might proceed, as long as successive derivatives, of higher orders, conti- 
 nue to vanish together. 
 
 (9.) Hence, in particular, if we take this scalar form, 
 
 XVI.. . 0a; = 
 
 2.3. ..m 
 which evidently gives the values, 
 
 XVII. ..00 = 0, 0'O = O, 0"O = O, . . . 0^'»"«O = O, 0WO=1, 
 and if we suppose that the function F,t is such that 
 
 XVIII. . . FO = 0, i^'O = 0, F"0 = 0, . . . F(»»- »)0 = 0, 
 while i5'(»»)0 has any finite value, we may then establish this limiting equation : 
 XIX. . . lim {Fx : <px) = FWO ; 
 
 x = Q 
 
 in which the function Fx, and the vahie i^('»)0, are here supposed to be generally 
 quaternions ; although they may happen, in particular cases, to reduce themselves 
 (292) to vectors, or to scalars. 
 
 * Instead of the equation IX., it has become usual, in modem works on the Dif- 
 ferential Calculus, to give one of the following form (deduced from principles of La- 
 grange) : 
 
 F(x) F'(Ox) 
 
 (p(x) (i>xexy 
 
 if F(O) = 0(O) = O: 
 
 9 denoting some proper fraction, or quantity between and 1. And a geometrical 
 illustration, which is also a geometrical proof, when the functions Fx and (px can be 
 constructed (or conceived to be constructed) as the ordinates of two plane curves, is 
 sometimes derived from the axiom (or geometrical intuition), that the chord of any 
 finite and ;>/ane arc must be parallel to the tangent, draw^n at some point of that 
 finite arc. But this parallelism no longer exists, in general, when the curve is one 
 of double curvature ; and accordingly the equation in this Note is not generally true, 
 when the functions are quaternions ; or even when one of them is a quaternion, or 
 a vector. 
 
CHAP. II.] Taylor's series extended to quaternions. 427 
 
 342. It will now be easy to extend the Exponential Trans- 
 formation 340, VII.; and to show that there is a sense in 
 which that very important Formula, 
 
 which is, in fact, a known* mode of expressing the Series or 
 Theorem of Taylor, holds good for Quaternion Functions ge- 
 nerally, and not merely for those functions of finite and poly- 
 nomial form, \\\i\\ positive and lohole exponents, for which it 
 was lately deduced, in 340, (6.). For let^ and/(<7 + d^) 
 denote any two states, or values, of which neither is infinite, of 
 any function of a quaternion; and of the m first differentials, 
 
 II. . . dfq, d-fq, . . d"fq, in which dq = const., 
 
 let it be supposed that no one is infinite, and that the last of 
 them is different from zero ; while all that precede it, and the 
 functions^ and f{q + d^') themselves, may or may not happen 
 to vanish. Let the first m terms, of the exponential develop- 
 ment of the symbol (e^ - l)fq, be denoted briefly by qi, qz, . . 
 qm ; and let r^ denote what may be called the remainder of the 
 series, or the connection which must be conceived to be added 
 to the sum of these m terms, in order to produce the exact value 
 of the difference, 
 
 III. . . Afq =^f(q + A^) -fq =f{q + dq) -fq ; 
 in such a manner that we shall have rigorously, by the nota- 
 tions employed, the equation, 
 
 d"fq 
 IV. . . /(^ + d^) =fq^qi + q^-^'^ + q,n + r„„ where ^„, = ^^ ^ ; 
 
 this term q^ being different from zero, but 7io one of the terms 
 being infinite, by what has been above supposed. Then we 
 shall prove, as a Theorem, that 
 
 * Lacroix, for instance, in page 168 of the First Volume of his larger Treatise 
 
 on the Differential and Integral Calculus (Paris, 1810), presents the Theorem of 
 
 Taylor under the form, 
 
 dw d^M d^M d^M ^ 
 
 ^ 1 1.2 1.2.3 1.2.3.4 
 ■where u' denotes the value which the function u receives, when the variable x re- 
 ceives the arbitrary increment dx (I'accroissement quelconque d:r). 
 
428 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 V. . . lim {Tr,n : Tq,,) = 0, if lim. Td^ = ; 
 or in words, that the tensor of the remainder may he made to 
 bear as small a ratio as we please^ to the tensor of the last term 
 retained, by diminishing the tensor, without changing the ver- 
 sor, of the differential (or difference) ^q. And this very gene- 
 ral result, which will soon be seen to extend to functions of 
 several quaternions, is in the present Calculus that analogue 
 of Taylor's theorem to which we lately alluded (in 340, (7.) ) ; 
 and it may be called, for the sake of reference, " Taylor s 
 Theorem adapted to Quaternions'^ 
 
 (1.) Writing 
 
 we shall have the following successive derivatives with respect to a;, 
 
 ra; = d/(? + a:dg) - d/9 -. ^dy^ - . . - 2-^— -^— d'H - 1/9 ; 
 
 VII. . . \ F'x = d2/(9 + xd.q) - dV? - . . - ^ 3 "'"'^"^_3^ d-i/g ; . . . 
 
 F(»»- ^)x = d"*- ' f{q + xAq) - d»»- ' /5 ; and finally, 
 jp(»»)a; = d»'/(9 + xdiq) ; 
 
 because, by 327, VI., and 324, IV., 
 VIII, . . D/(9 + a;dg) = lim.n{/(g + a;dg' + «-idg)-/(^ + a;d5)} =d/(5'+a;dj), 
 
 and in like manner, 
 
 IX. . . D2/(? + xdiq) = d2/(9 + xdq), &C. ; 
 the mark of derivation D referring to the scalar variable a:, while d operates on q 
 alone, and not here on x, nor on Aq. 
 
 (2.) We have therefore, by VI. and VII., the values, 
 
 X. . . FO = 0, F'O = 0, F"Q = 0, . . F(»» - J)0 = 0, F(»»)0 = d'"/? ; 
 whence, by 341, XIX., we have this limiting equation. 
 
 XL . .lim.f Fx:—^ V 
 
 ^=0 V 2.3. ..mj 
 
 d-/r. 
 
 Xll...Mm(Fx:^x) = l, if ,Lx = f-^^^5^\ 
 x=Q \2.3...m) 
 
 (3.) But these two functions, Fx and v^.r, are formed by IV. from qm + r„j and 
 qm> by changing d^ tox'dg ; and instead of thus multiplying dq by a decreasing sca- 
 lar^ x, we may diminish its tensor Tdq, Avithout changing its versor Vdq. We may 
 therefore say that, when this is done, the quotient (qm + »m) : qm tends to unitt/, or 
 this other quotient r,„ : qm to zero, as its limit; or in other words, the limiting equa- 
 tion V. holds good. 
 
CHAP. II.] EXAMPLES OF QUATERNION DEVELOPMENT. 429 
 
 (4.) As an example, let the function fq be the reciprocal, q-^ ; then (comp. 339, 
 IIL) its m*^ differential is (for d5 = const.), 
 
 XIII. .. d»*/^ = d»».g-i = 2.3. ..w.^r-i (-r)"', if r = dq.q-^i 
 and it is easy to prove, without differentials, that 
 
 XIV. . . (r? + rg)-i =9-^(1 + r)-i=5-i{l - r + r2-. . + (- r)'» + (-r>»^i (1 + r)-i }; 
 we have therefore here 
 
 XV. ..g^ = 5-'(-r)-, r„. = -<7,,r(l + r)-', T(r„, : 9,„) = Tr . T(l + r)"!; 
 and this last tensor indefinitely diminishes with Td(jf, the quaternion q being sup- 
 posed to have some given value different from zero. 
 
 (5.) In general, if we establish the following equation, 
 
 XVI. ../(<? + n-id^) =fq + n-id/5 + _ dy^ + . . + ^ 3^^^^_.^^ d-i/y 
 
 2.8 
 
 as a definitional extension of the equation 325, V. ; and if we suppose that neither 
 the function /9 itself, nor any one of its differentials as far as dP'-^fq is infinite ; the 
 result contained in the limiting equation XI. may then be expressed by the formula, 
 
 XVIL ../-)(g, d<?) = d-/g; 
 which for the particular value to = 1, if we suppress the upper index, coincides with 
 the form 325, VIII. of the definition d/r, but for higher values of m contains a theo- 
 rem : namely (when (l"fq is supposed neither to vanish, nor to become infinite), 
 what we have called Taylor's Theorem adapted to Quaternions. 
 
 343. That very important theorem may be applied to cases, in 
 which a quaternion (as in 327, (o.) ), or a vector (as in 337), is ex- 
 pressed as Q. function of a scalar ; also to transcendental forms (333), 
 whenever the differentiations can be effected; and to those new 
 forms (334), which result from the peculiar operations of the present 
 Calculus itself. A few such applications may here be given. 
 
 (1.) Taking first this transcendental and quaternion function of a variable scalar, 
 I. . . q=ft = a*, with Ta = l, da = 0, d^ = const., 
 we have, by 333, VIII., the general term, 
 
 d»* .a* a^ I Tradt 
 
 "•••ff-^TT-T— r=^^— ll-ir- =^H5--' if 2x=7rd*; 
 
 \w _ a*(a;a)'' 
 
 2.3 
 
 2.3..mV 2 j 2. 3.. TO 
 
 dividing then t'^ . a' by a*, we obtain an infinite series, which is found to be correct, 
 and convergent ; namely (comp. 308, (4.)), 
 
 IIL . . a<i' = 1 + xa + ^-^ +..-f--\-^- + ..=£^« = cos-^ + asm— -. 
 2 2 . 3 . . m 2 2 
 
 (2.) Correct and finite expansions, for S(q + dq), Y(q + dg), K(q + dq}, and 
 N(j + dg), are obtained when we operate with c"^ on S^', Yq, Kq, and N5 ; for ex- 
 ample (dq being still constant), the third and higher differentials of 'Rq vanish by 
 334, XL, and we have 
 
430 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 IV. . . £'iNg = (l + d + id2)N^=N5 + 2SCK9.cl^)+Ndr? = N(<7+dg); 
 an expression for the norm of a sum, which agrees with 210, XX., and with 200, 
 VII. 
 
 (3.) To develope, on like principles, the tensor and versor of a sum, let us again 
 write r for dq : q, and denote the scalar and vector parts of this quotient by s and v ; 
 so that, by 334, XIII. and XV., 
 
 v....=s.= s5? ««; vi....=vr=v^ = :lH?. 
 q Tq * q ^q 
 
 (4.) Then writing also, for abridgment, as in a known notation of factorials, 
 
 m 
 
 VII...[-l] = (-l).(-2).(-3)....(-,n), 
 we shall have, by 342, XIII., dq being still treated as constant, the equation, 
 
 m m 
 
 VIII. . . d"^(s + u) = d"*/- = [- 1] r'w+i = [- 1] (s -I- r)'«+i, 
 of which it is easy to separate the scalar and vector parts ; for example, 
 
 IX. . . ds = -S.(s + «)2=:-(52+t,2). dv = ~Y. (s + v)^ = -2sV. 
 
 (5.) We have also, by V. and VI., 
 
 x...^^=(. + a)£^=... = (^ + ci)-i; ■■J^lT^I^(i*'i 
 XI. . /^^=(„ + d)^' = .. = (<. + d).»i;,-.r^^^y.^K.y/ 
 
 the notation being such that we have, for instance, by IX., 
 
 XII. . . (6- + d)l=s; (« + d)2l = (s + d)s = s'^ + ds = -«;2. 
 XIII. . . (w + d)l = r; (u+d)21=(y+d)r = u2 + clu=tj2-2sr. 
 (6.) The exponential formula 342, I., gives, therefore, 
 
 XIV. . . T(9+d5)=£dT^ = £''+di.T^. 
 
 XV. . . U(9 + d9) = £<JU^ = £^^n.Ug; 
 
 or, dividing and substituting, j 
 
 XVI. ..T(l + s + v)=f»+di; XVIL .. U(l + 5+t;) = £''^dl; ^^* j 
 s and V being here a scalar and a vector, wjiich are entirely independent of each 
 other ; but of which, in the applications, the tensors must not be taken too large, ia 
 order that the series may converge. 
 
 (7.) The symbolical expressions, XVI. and XVII., for those two series, may be 
 developed by (4.) and (5.) ; thus, if we only write down the terms which do not exceed 
 the second dimension, with respect to s and v, we have by XII. and XIII. the deve- 
 lopment, 
 
 XVIII. . . T(H-s + v)=l + s-ir2+..., 
 XIX. . . U(l+s + «)=! + « + (it;2-si,) + ... • 
 of which accordingly the product is 1 + s + r, to the same order of approximation. 
 
 (8.) K function of a sum of two quaternions can sometimes be developed, with- 
 out differentials, by processes of a more algebraical character ; and when this hap- 
 pens, we may compare the result with the form given by Taylor's Series, as adapted 
 to quaternions in 342, and so deduce the values of the successive differentials of the 
 function ; for example, we can infer the expression 342, XIII. for d'" . q'\ from the 
 series 342, XIV., for the reciprocal of a sum. 
 
CHAP. II.] SUCCESSIVE DIFFERENCES AND DERIVATIVES. 431 
 
 (9.) And not only may we verify the recent developments, XVIII. and XIX., by 
 comparing them with the more algebraical forms, 
 
 XX.. .T(l+s + i;)=(l+s+«)Kl + s-r>, 
 XXI. . . U(l+s+r) = (H-s + t)>(l + s-vH 
 but also, if the first of these for example (when expanded by ordinary processes, 
 which are in this case applicable) have given us, without differentials, 
 
 XXII. . .T{g + q') = (^l + s- |z?2 . .)T;7, where s ■= Sqq-^, and v = Yq'q-\ 
 
 we can then infer the values of they?7's* and second differentials of the tensor of a 
 quaternion, as follows : 
 
 XXIII. . . dT^ = S — . T7 ; d^Tq = - I V-^ YTq ; 
 
 whereof the first agrees with 334, XII. or XIII., and the second can be deduced from 
 it, under the form, 
 
 xxiv...a«T,=a(s^l.T,) = ((syy-s.(jy)T, 
 
 (10.) In general, if we can only develope a function /(g 4- 5') as far as the term 
 or terms which are of the first dimension relatively to q', we shall still obtain thus 
 an expression for theirs* differential d/j, by merely writing dq in the place of ^'. 
 But we have not chosen (comp. 100, (14.) ) to regard this property of the differen- 
 tial of a function as the fundamental one, or to adopt it as the definition of dfq ; be- 
 cause we have not chosen to postulate the general possibility of such developments of 
 functions of quaternion sums, of which in fact it is in many cases difficult to discover 
 the laws, or even to prove the existence, except in some such way as that above ex- 
 plained. 
 
 (11.) This opportunity may be taken to observe, that (with recent notations) we 
 have, by VIII., the symbolical expression, 
 
 XXV. .. ts+^'+d 1 = 1 + 5 + 0; or XXVI. ..£'-+di = i+r. 
 
 .344. Successive differentials are also connected with successive dif- 
 ferences, by laws which it is easy to investigate, and on which only 
 a few words need here be said. 
 
 (1.) We can easily prove, from the definition 324, IV, of dfq, that if d^- be con- 
 stant, 
 
 L . . d2/^ = lim . «2 [f(^q + 2n-» dg) - 2f(q 4 n'^ d^) +fq]; 
 
 n= CD 
 
 with analogous expressions for differentials of higher orders. 
 
 (2.) Hence we may say (comp. 040, X.) that the successive differentials, 
 11. ..dfq, d^fq, d^fq,.. for d''q==0, 
 are limits to which the following multiples of successive differences, 
 lll...nAfq, n^A^fq, n^A^fq,.. for A^q = 0, 
 all simultaneously tend, when the multiple nAq is either constantly equal to d^, or at 
 least tends to become equal thereto, while the number n increases indefinitely. 
 
 (3.) And hence we might prove, in a new way, that if the function f(q + d^) 
 
432 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 can be developed, in a series proceeding according to ascending and whole dimemions 
 with respect to dq, the parts of this series, which are of those successive dimensions, 
 must follow the law expressed by Taylor^s Theorem* adapted to Quaternions 
 (342). 
 
 345. It is easy to conceive that the foregoing results may be ex- 
 tended (comp. 338), to the successive differentiations of functions of 
 several quaternions; and that thus there arises, in each such case, a 
 system of successive differentials^ total and partial: as also a system of 
 partial derivatives^ of orders higher than the first, when a quaternion^ 
 or a vector^ is regarded (comp. 337) as a function of several scalars. 
 
 (1.) The general expression for the second total differential, 
 I. ..d2Q=d2F(5r,r, . .), 
 involves d^g, d2r, . . ; but it is often convenient to suppose that all these second dif- 
 ferentials vanish, or that the Jirst differentials dq, dr, . . are constant ; and then 
 d^Q, or d.»^F(q, r, . .), becomes a rational, integral, and homogeneous function of 
 the m^^ dimension, of those first differentials dq, dr, . . , which may (comp. 329, 
 III.) be thus denoted, 
 
 II. . . d'"Q = (d3 + dr + ..)'»Q; or briefly. III. . . d'» = (d2 + dr + . .)»», 
 in developing which symbolical power, the midtinomial theorem of algebra may be 
 employed : because we have generally, for quaternions as in the ordinary calculus, 
 iV. . . drdq = d,,dr. 
 (2.) For example, if we denote dq and dr by q and r', and suppose 
 
 V. . . Q = rqr, then VI. . . dqQ = rq'r; VII. . . d,Q = r'qr + rqr ; 
 and VIII. . . drdq Q = d^dr Q = r'q'r + rq'r. 
 
 And in general, each of the two equated symbols IV. gives, by its operation on 
 F(q, r), the limit of this other function, or product (comp. 344, I.), 
 
 IX. ..nn{ F(q + «-» d^, r + n''^ dr) - F((q, r + »'-» dr) - F{q + n-i d^, r) + F((j, r) } ; 
 
 in which the numbers n and n ' are supposed to tend to infinity. 
 (8.) We may also write, for functions of several quaternions, 
 
 X... Q-\-^Q = F(^q^dq,r^dr,..)=e\^\*'-F{q,r)', 
 or briefly, XI. . . 1 + A = e'^3+'^r+"= e''; 
 
 with interpretations and transformations analogous to those which have occurred 
 already, for functions of a single quaternion. 
 
 (4.) Finall}'^, as an example of successive and partial derivation, if we resume 
 the vector expression 308, XVIII. (comp. 315, XIL and XIII.), namely, 
 XII. . . p = r¥j«kj'«k-*, 
 
 * Some remarks on the adaptation and proof of this important theorem will be 
 found in the Lectures, pages 589, &c. 
 
CHAP. II.] SCALAR AND VECTOR INTEGRALS. 433 
 
 ■which has been seen to be capable of representing the vector oi any point of space, 
 we may observe that it gives, without trigonometry, by the principle mentioned in 
 308, (11.), and by the sub-articles to 315, not only the form, 
 XIII. . . p = rk*j2^k^-*, as in 308, XIX., 
 but also, if a be any vector unit, 
 
 XIV. . . p =- rk*^\j-^^k't = rki(k^ . a2« + iS. a«»-0- *"' ; 
 whence XV. . . p = rV. k^'^^ + rk^^Y. i^^, as in 315, XII. 
 
 (5.) We have therefore the following new expressions (compare the sub- articles 
 to 337), for the two partial derivatives o{t\\e first order, of this variable vector p, 
 taken with respect to s and t : 
 
 XVI. . . Dsp = Trrk^jHpk-i = -Trp¥jJc-\ 
 with the verification, that 
 
 XVII. . . p'Dsp = 7rr^.kifkj-'>k'i.k<jHj-'k-* = '7rr^¥jk'f', 
 and XVIII. . . D^p = Trrk^ty.j^^ = Trrk-^tjS . a2«-i = r'lpD^p . S. a2»-i, 
 
 whence XIX. . . pD<p = -rDsp.S.a2«-i, and XX. . . Dsp.D<p= TrVpS.a^*'; 
 while XXI. ..Drp = r-ip = ktj^kj-^k-*, as in 337, XXV. ; 
 
 so that we have the following ternary product of these derived vectors of the first 
 order, 
 
 XXII. . . Drp.Dsp.D<p= 7r2p2S.a2«-i = 7rr2D,S.a2*; 
 
 tho^ scalar character of which product depends (comp. 299, (9.)) on the circum- 
 stance, that the vectors thus multiplied compose (337, (10.) ) a rectangular system. 
 
 (6.) It is easy then to infer, for the six partial derivative!^ of p, of the second 
 order, taken with respect to the same three scalar variables, r, s, t, the expressions : 
 
 XXIII. . . DrV = ; D,.D,p = D.Drp = r-iD^p ; D^D<p = D^D^p = r-»D<p ; 
 XXIV. . . Ds2p =- 7r2p ; D,D<p = D^D.p = 7r2r^2<y.j2s+i. Di2p = - 7r«rA2<V. i2*. 
 
 (7.) The three ;?ar<«aZ differentials oiihQ first order, of the same variable vector 
 p, are the following: 
 
 XXV. . . d,.p = r-ipdr ; dsp = Djp . ds ; d<p = D^p . dt ; 
 with the products, 
 
 XXVI. . . d,p . d^p = - TrrpdS . a2» . dt ; 
 XXVII. . . d,.p . dsp . dtp = Trr^dr . dS . a^^ . dt. 
 
 (8.) These differential vectors, d,p, dsp, d^p, are (in the present theory) gene- 
 rally finite ; d,p, like Dfp, being a line in the direction of p, or of the radiMs of this 
 sphere round the origin, at least if dr, like r, be positive ; while d«p, like D«p, is 
 (comp. 100, (9.) ) a tangent to the meridian of that spheric surface, for which r 
 and t are constant ; but dtp, like D<p, is on the contrary a tangent to the small circle 
 (or parallel), on the same sphere, for which r and s are constant. 
 
 (9.) Treating only the radius r as constant, and writing p = op, if we pass from 
 the point P, or (s, t), to another point q, or (s + As, t), on the same meridian, the 
 chord PQ is represented by the^ni^e difference. Asp ; and in like manner, if we pass 
 from p to a point R, or (s, t + At), on the same parallel, the new chord pk is repre- 
 sented by the other partial and finite difference, Atp ; while the point (s + As, < + At) 
 may be denoted by s. 
 
 (10.) If now the two points Q and n be conceived to approach to p, and to come 
 to be very near it, the chords pq and PR will very nearly coincide with the two cor- 
 
 3 K 
 
434 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 responding arcs of meridian and parallel ; or with the tangents to the same two cir- 
 cles at p, so drawn as to have the lengths of those two arcs : or finally with the dif- 
 ferential and tangential vectors, dsp and d^p, if we suppose (as we may, comp. 322) 
 that the two arbitrary and scalar differentials, d« and d*, are so assumed as to be 
 constantly equal to the two differences, As and At, and consequently to diminish 
 with them. 
 
 (11.) Whether the differentials d^ and dt be large or small, the product dsO.dtp, 
 like the product D^p . Dtp, represents rigorously a normal vector (as in XXVI. and 
 XX.) ; of which the length bears to the unit of length (comp. 281) the same ratio, as 
 that which the rectangle under the two perpendicular tangents, dgO and dtp, to the 
 sphere, bears to the unit of area. Hence, with the recent suppositions (10.), we 
 may regard this /)roc?Mc^ dsp . dtp as repi*esenting, with a continually and indefinitely 
 increasing accuracy, even in the way of ratio, what we may call the directed element 
 of spheric surface, pqrs, considered as thus represented (or constructed) by a nor- 
 mal at p ; and the tensor of the same product, namely (by XXVI.), 
 
 XXVIII. . . T(dsp . dtp) = - TrrMS . a.^'. dt, 
 in which the negative sign is retained, because S . a^* decreases from -f 1 to - 1, while 
 s increases from to 1, is an expression on the same plan for what we may call by con- 
 trast the undirected element of spheric area, or that element considered with reference 
 merely to quantity, and not with reference to direction. 
 
 (12.) Integrating, then, this last differential expression XXVIII., from < = to 
 t = 2, and from s — sq to s =s\, that is, taking the limit of the sum of all the elements 
 PQRS between these bounding values, we find the following equation : 
 
 XXIX. . . Area of Spheric Zone = 27rr2S (a^^o - a^'i) ; 
 whence 
 
 XXX. . . Area of Spheric Cap (s) = 27rr2(l - S . aS*) = 47rr2 (TV. a^y ; 
 
 and finally, 
 
 XXXI. . . Ai'ea of Sphere =4.7rr^, as usual. 
 
 (13.) In like manner the expression XXVII., with its sign changed (on account 
 of the decrease of S . a^*, as in (11.) ), represents the element of volume ; and thus, 
 by integrating from r = To to r = ri, from « = to s = l, and from t = to t-2, 
 we obtain aneAV the known values : 
 
 XXXII, . . Volume of Spheric Shell = — (n^ - ro^) ; 
 
 and 
 
 4:7rr^ 
 
 XXXIII. . . Volume of Sphere (r)= — -, as usual. 
 
 o 
 
 (14.) These are however only specimens of what may be called Scalar Integra- 
 tion, although connected with quaternion forms ; audit will be more characteristic 
 of the present Calculus, if we apply it briefly to take the Vector Integral, or the limit 
 of the vector-sum oi the directed elements (11.), of a portion of a spheric surface: 
 a problem which corresponds, in hydrostatics, to calculating the reswZ^aw^ of the pres- 
 sures on that surface, each pressure having a normal direction, and a quantity pro- 
 portional to the element of area. 
 
 (15.) For this purpose, we may employ the expression XXVI. with its sign 
 changed, in order to denote an inward normal, or a. pressure acting /ro/n without ; 
 and if we then substitute for p its value XV., and observe that 
 
CHAP. II.] DIFFERENTIALS OF IMPLICIT FUNCTIONS. 435 
 
 XXXIV. . . pA2<d^=o, because k^=-l, 
 and remember that V.^^s+i —^g , „2s^ ^e easily deduce the expressions: 
 
 XXXV. . . Sum of Directed Elements of Elementary Zone = Trr^M . (S . a2*)2 ; 
 XXXVI. . . Sum of Directed Elements of Spheric Cap (s) = - 7rrU-(l - (S.a2*)2) 
 = Trr^k (V. a202 = tt'U- (D<p)2 = ttIc (Ykpy. 
 (16.) But the radius of the plane and circular base, of the spheric segment cor- 
 responding, is TVA-p, so that its area is in quantity =- tt (Ykpy ; and the common 
 direction of all its inward normals is that of + ^ ; hence if we still represent the di- 
 rected elements by normals thus drawn inwards^ we have this new expression : 
 XXXVII. . . Sum of Directed Elements of Circular Base = ~7rk(Ykpy ; 
 comparing which with XXXVI., we arrive at the formula, 
 
 XXXVIII. . . Sum of Directed Elements of Spheric Segment = Zero ; 
 a result which may be greatly extended, and which evidently answers to a known 
 case of equilibrium in hydrostatics. 
 
 (17.) These few examples may serve to show already, thsX Differentials of Qua- 
 ternions (or of Vectors') may be applied to various geometrical and physical ques- 
 tions ; and that, when so appUed, it \b permitted to treat them as small, if any con- 
 venience be gained thereby, as in cases of integration there always is. But we must 
 now pass to an important investigation of another kind, with which differentials will 
 be found to have only a sort of indirect or suggestive connexion. 
 
 Section 6. — On the Differentiation of Implicit Functions of 
 Quaternions; and on the General Inversion of a Linear 
 Function, of a Vector or a Quaternion : loith some connected 
 Investigations. 
 
 346. We saw, when differentiating the square-root of a 
 quaternion (332, (5.) and (6.) ), that it M^as necessary for that 
 purpose to resolve a linear equation,* or an equation of the 
 first degree; namely the equation, 
 
 I . . rr + rr = q\ 
 in which r and q represented two given quaternions, q^ and 
 d^, while r represented a sought quaternion, namely dr or d . qh. 
 And generally, from the linear or distributive form (327), of 
 the quaternion differential 
 
 of any given and explicit function fq, when considered as de- 
 pending on the differential dg' of the quaternion variable q, we 
 see that the return from the former differential to the latter, 
 
 * Compare the Note to page 410. 
 
436 ELEMENTS OF QUATERNIONS, [bOOK III. 
 
 that is from dQ to dg', or the differentiation of the inverse or 
 iwpZiczY function /'^Q, requires for its accomplishment the So- 
 lution of an Equation of the First Degree: or what may be 
 called the Inversion of a Linear Function of a Quatejmion. 
 We are therefore led to consider here that general Problem ; 
 to which accordingly, and to investigations connected with 
 which, we shall devote the present Section, dismissing how- 
 ever now the special consideration of the Differentials above 
 mentioned, or treating them only as Quaternions, sought or 
 given, of which the relations to each other are to be studied. 
 
 347. Whatever the particular form of the given linear or dis- 
 tributive fmiction, fq, may be, we can always decompose it as follows: 
 
 I. . ./? =/(§!? + Vj)=/S2+/V2 = Sj./l +/Vr, 
 
 taking then separately scalars and vectors, or operating with S and 
 V on the proposed linear equation, 
 
 IL.. fq^r, 
 
 where r is a given quaternion, and q a sought one, we can in general 
 eliminate Sq, and so reduce the problem to the solution of a linear 
 and vector equation, of the form, 
 
 III. . . (pp=(r; 
 
 where o- is a given vector, but p{= Yq) is a sought one, and ^ is used 
 as the characteristic of a given linear and vector function of a vector, 
 which function we shall throughout suppose to be a real one, or to 
 involve no imaginary constants in its composition. But, to every such 
 function fp, there always corresponds what may be called a conjugate 
 linear and vector function 0'/>» connected with it by the following 
 Equation of Conjugation, 
 
 IV.. . S>Mp = Spf\; 
 
 where A and p are any two vectors. Assuming then, as we may, that 
 fi and V are two auxiliary vectors, so chosen as to satisfy the equa- 
 tion, 
 
 and therefore also, 
 
 VI. . . SXff = SX/ii/, S/i(T = 0, Si/a = 0, 
 
 where X is a f^ercf auxiliary and arbitrary vector, we may (comp. 312) 
 replace the otie vector equation III. by the three scalar equations, 
 

 CHAP. II.] INVERSION OF LINEAR AND VECTOR FUNCTIONS. 437 
 
 VII. . . Sp^'A = S//ti/, S/J0> =-• 0, S/>^' v = 0. 
 And these give, by principles with which the reader is supposed to 
 be already familiar,* the expression, 
 
 VIII. . . mp = yjra, or IX. . . p^<p~^a = m'^yfra-, 
 if w be a vector-constant^ and Y^ an auxiliary linear and vector function, 
 of which the value and the form are determined by the two following 
 equations : C » i ^^C ^ > J 
 
 X. . . m^Xfiv = S 
 
 XL . . V^CV^i.) ^ 
 
 or briefly, i/ Ujl^c ' ( ^'^"^^ ^ 
 
 X'. . . m'&\fjiu=^.<p'\<p',x<p'v, ^ 
 
 and 
 
 XP. .. YrV/*i/ = V.0>0V. 
 
 And thus the proposed Problem of Inversion, of the linear and vector 
 function 0, may be considered to be, in all its generality, resolved; 
 because it is always possible so to prepare the second members of the 
 equations X. and XL, that they shall take the forms indicated in the 
 first members of those equations. 
 
 (1.) For example, if we assume any three diplanar vectors a, a\ a", and deduce 
 from them three other vectors /3o, /3'o, jS'o, by the equations, 
 
 XII. . . (3oSaa'a" = Ya'a', /3'oSaa'a" = Ya'a, /3"oSaa'a" = Yaa, 
 
 then ani/ vector p may, by 294, XV., be expressed as follows, 
 
 XIIL . . p = (3oSap + /3'oSa'p + (5"oQa"p ; 
 if then we write, 
 
 XIV. . .(3 = ^i3o, /3' = V'/3o', jS" = (PI3\ 
 
 we shall have the following General Expression, or Standard Trinomial Form, for 
 a Linear and Vector Function of a Vector, 
 
 XV. . . 0p = jSSajo + jS'Sa'p + /3"Sa"p ; 
 containing, as we see, three vector constants, (3, /3', /3", or nine scalar constants, 
 such as 
 
 XVI. . . Sa|3, Sa'(3, Sa"/3; SafS', Sa'jS', Sa"(5' ] Sa(5", Sa'jS", Sa"(i" ; 
 
 which may (and generally will) all vary, in passing from one linear and vector func- 
 tion <Pp to another such function ; but which are all supposed to be real, and given, 
 for each particular form of that function. 
 
 (2.) Passing to what we have called the conjugate linear function 0'p, the form 
 XV. gives, by IV., the expression, 
 
 * A student might find it useful, at this stage, to read again the Sixth Section of 
 the preceding Chapter; or at least the early sub-articles to Art. 294, a familiar ac- 
 quaintance with which is presumed in the present Section. 
 
438 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XVII. . . ^'p = aSfSp + a'S/3'p + a"S/3"p ; 
 but 
 
 V. (aSiSfx 4- a'S/3V) (aS(3v + a'S/3V) = Yaa'S . (3' (i/S/3/i - fiS(3v) 
 = Yaa'S. (i'y.fiVnv = Yaa'S.(3'(3Vfiv ; 
 therefore the transformation XI. succeeds, and gives, 
 
 XVIII. . . i//p = Va'a"S^"/3> + Va"aS/3i3> + Vaa'S/3'/3p, 
 
 as an expression for the auxiliary function ;// ; of which the conjugate may be thus 
 written, 
 
 XIX. . . ^/'p = V/3'/3"Sa"a'p + V/3"i3Saa"p + V/3/3'Sa'ap ; \ T -^ ^ 
 
 so that •«// is changed to ;p', when ^ is changed to 0', by interchanging each of the 
 three alphas with the corresponding 6e<a. 
 
 (3.) If we write, as in this whole investigation we propose to do, 
 XX. . .\'= V/iv, /i' = Yv\ v = V\/«, 
 the formulae XI. and X. become, 
 
 XXI. . .^\' = V. ^>0V, and XXII. . . mSW = S. 0'X^\', 
 with the same sort of abridgment of notation as in XI'. ; and because the coefficient 
 of Saa'a" in this last expression XXII. is by XVII. XVIII., 
 
 S/3XS/3"/3'\' + S(3'KS(3(3"\' + Sf5"XSl3'^X' = S/3"/3'/3SXX', 
 the division by SXX', or by SXjuv, succeeds, and we find the expression, 
 
 XXIII. ..m = Saa'a"S/3"/3'/3; 
 which may also be thus written, 
 
 XXIir. . . m = S/3/3'i3"SaVa, 
 so that m does not change when we pass from ^ to (p', on which account we may 
 write also, 
 
 XXIV. . . mSXX'= S.^X-f X', or XXIV. . . mSX/^i/ = S.^X^/i^v, 
 because, by (2.), we can deduce from XI. the conjugate expression, 
 XXV. .. ;//'X'=V.0/i0i/. 
 (4.) We ought then to find that the linear equation, 
 
 XXVI. . . (T = <pp=^Sap + (S'Sa'p + (3"Sa"p, 
 has its solution expressed (comp. VIII.) by the formula, 
 
 XXVII. . . pSaa'a"Si3"|8'/3 = Va'a"Si3"/3'(T + Va"aS/3|3"(r + Vaa'S/3'j3(r; 
 and accordingly, if we operate on the expression XXVI. for <t with the three sym- 
 bols, 
 
 XXVIII. .. S./3"/3', S.i8/3", S./3'/3, 
 
 we obtain the three scalar equations, 
 
 XXIX. . . S/3"/3'(T = S/3"/3'/3Sap, &c., 
 from which the equation XXVII. follows immediately, without any introduction of 
 the auxiliary vectors \, ft, v, although these are useful in the theory generally. 
 
 (5.) Conversely, if the equation XXVII. were given, and the value of c sought, 
 we might operate with the three symbols, 
 
 XXX. ..S. a, S.(3, S.y, 
 
 and so obtain the three scalar equations XXIX., from which the expression XXVI. 
 for <j would follow. 
 
CHAP. II.] STANDARD TRINOMIAL FORM. 439 
 
 (6.) It will be found an useful check on formulaj of this sort, to consider each beta, 
 in what we have called the Standard Form (l.)of0|O, as being oi the Jirst dimension ; 
 for then we may say that <p and 0' are also of the first dimension, but \p and ^' of 
 the second, and m of the third; and every formula, into which these symbols enter, 
 will thus be homogeneous: a, a', a", and X, ft, v, p, being not counted, in this mode 
 of estimaimg dimensions, but <t being treated as of the/r4< dimension, when it is 
 taken as representing (pp. 
 
 (7.) And although the trinomial form XV. has been seen to be sufficiently gene' 
 ral, yet if we choose to take the more expanded form, 
 
 XXXI. . . 0p = 2/3Sap, which gives XXXII. . . 0'p = 2aS/3|0, 
 any number of terms of ^p, such as jSSap, /3'Sa'p, &c , being now included in the 
 sum 2, there is no difficulty in proving that the equations "VIII. and IX. are satis- 
 fied, when we write, 
 
 XXXIII. . . i|/p = 2 Vaa'S/3'/3p, with XXXIV. . . t//> = 2V|3/3'Sa'«p, 
 and 
 
 XXXV. ..m = 2Saa'a"S/3"/3'|3= 2S|S/3'/3"Sa"a'a. 
 
 (8.) The important property (2.), that the auxiliary function y^j is changed to its 
 own conjugate i// , when ip is changed to <p\ may be proved without any reference to 
 the form S(5Sap of ^p, by means of the definitions IV. and XI., of ^' and \p, as fol- 
 lows. Whatever four vectors fi, v, fii, and vi may be, if we write 
 
 XXXVI. . . \'i = V/iivi, and XXXVII. . . ;//'V/xv = V. 0^t0v, 
 adopting here this last equation as a definition of the function •<//', we may proceed to 
 prove that it is conjugate to ^, by observing that we have the transformations, 
 XXXVIII. . . S\'i^l^X = ^(Yfiivi.Y.<PiJi<pv)=8.lxi(Y.viV.<pfi<pv) 
 = SfXicpv . Svi^n ~ SjUl^jU . Svicpv 
 = SfKp'vi . Sv0'/ii — SfKp'ni . Sj-'0Vi 
 
 = S . jw ( V. j^V. <p'ni<p'v\) = S (V;u V .V. (p'lxitp'vi) = SX'-*|/\'i ; 
 which establish the relation in question, between ;// and t//'. 
 
 (^9.) And the not less important property (3,), that m remains unchanged when 
 we pass from (^ to 0', may in like manner be proved, without reference to the/orm 
 XV. or XXXI. of 0p, by observing that we have by XXXVII., &c. the transfor- 
 mations, 
 
 XXXIX. . . S . (p\(Pix<pv = S . (/)Xt^'/V = SX'»//0X = mSX'X = mSXjui/, 
 because the equations III. and VIII. give, 
 
 XL. . . ■<\/(pp = mp, whatever vector p may be ; 
 so that the value of this scalar constant m may now be derived from the original 
 linear function (p, exactly as it was in X. or X'. from the conjugate function 0'. 
 
 348. It is found, then, that the linear and vector equation, 
 
 1. . . (pp = (ri gives II. . . mp = \P(T, 
 
 as its formula of solution; with the general method, ahoy e ex- 
 plained, of deducing m and ^ from 0. We have therefore the 
 two identitieSy 
 
440 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 III. . . ma = ^^(T, mp = ;//<^jO ; 
 or briefly and symbolically, 
 
 Iir. . . m = (p-ip = ipcj) ; 
 with which, by what has been shown, we may connect these 
 conjugate equations, 
 
 III". . . m = 0'i//' = i//'^'. 
 
 Changing then successively ju and v to ;/^'/x and ■\p'v, in the 
 equation of definition of the auxiliary function i//, or in the 
 formula, 
 
 ^v^v = v.0>^'v, 347, xr., 
 
 we get these two other equations, 
 
 IV. . . - ^V. v^'ju = mY-iK^'v ; V. . . -^Y.-^'ii-^'v = m^Vjuv ; 
 
 in the former of which i\iQ points may be omitted, while in 
 each of them accented may be exchanged with unaccented 
 symbols of operation : and we see that the law of homogeneity 
 (347, (6.) ) is preserved. And many other transformations of 
 the same sort may be made, of which the following are a few 
 examples. 
 
 (1.) Operating on V. by ^-^ or by m"'^, we get this new formula, 
 VI. . . Y,\l>'ix\p'v = m^Ynv; 
 comparing which with the lately cited definition of tp, we see that we may change 
 (f> to \p, if we at the same time change tp to mip, and therefore also m to m^ ; 0' being 
 then changed to \p', and ip' to mtp'. 
 
 (2.) For example, we may thus pass from IV. and V. to the formulis, 
 VII. . . - (l>Yv(p'fi =Yfji\p'v, and VIII. . . <pY.ffi(p'v =mYfiv ; 
 in which we see that the lately cited law of homogeneity is still observed. 
 
 (3.) The equation VII. might have been otherwise obtained, by interchanging 
 fi and V in IV., and operating with - m-'0, or with -;|/-i ; and the formula VIII. 
 may be at once deduced from the equation of definition of ■^, by operating on it 
 with 0. In fact, our rule of inversion^ of the linear function 0, may be said to be 
 contained in the formula, 
 
 IX. . . ^-Wfiv = m-^Y.(p'fi,(p'v; 
 where m is a scalar constant, as above. 
 
 (4.) By similar operations and substitutions, 
 
 X. . . (p'^Y.(p'ij.^'v — m^Yfiv = Y.\j/'iitxp'v, 
 
 XI. . . m^Y. <p'fi(p'v = mWfiv - 1// V. -.//'/ii/zV ; 
 
 XII. . . m2V. 0'ju0V =7n2-^V/i5/ = \p^Y. xp'/x-ip'r ; 
 
 XIII. . . Y.<p'^n(p'^u=\pY.<b'fi<p'v = xp^Yfiv; &c. 
 
CHAP. II.] SECOND FUNCTIONS, QUATERNION CONSTANTS. 441 
 
 (5.) But we have also, 
 
 XIV. . . S . X^3p = S . 0pf \ = S . p(p'% 
 so that the second functions 0^ and <p'^ are conjugate (compare 347, IV.) ; hence, by 
 XIII., i//2 is formed from ^2^ as \p from ^ ; and generally it will be fomid, that if 
 n be any whole number, and if we change <p to <p'\ we change at the same time ^' to 
 ^'w, ■tp to i//»*, \p' to ;//'»», and m to ni>K 
 
 (6.) It may also be remarked that the changes (1.) conduct to the equation, 
 
 XV. , . (S . <p\<pfi(pvy = SXfiv^^. ^\yp^-4jv ; 
 and to many other analogous formulae. 
 
 349. The expressions, 
 
 with the significations 347, XX. of X', fi\ v, and others of the 
 same type, are easily proved to vanish when X, jw, v are corn- 
 planar, and therefore to be divisible by SX/xv, since each such 
 expression involves each of the three auxiliary vectors X, fij v 
 in the^r^^ degree only ; the quotients of such divisions being 
 therefore certain constant quaternions, independent of X, ju, v, 
 and depending only on the particular form of 0, or on the 
 (scalar or vector, but real) constants, which enter into the 
 composition of that given function. Writing, then, 
 
 I. . . ^i = (X'0X + ix(pfx + v'0v) : SX/xv, 
 and II. . . ^2 = (X'^X + ju'T//,a + v'^v) : SX^v, 
 
 we shall find it useful to consider separately the scalar and 
 vector parts of these two quaternion constants, q^ and q^*, 
 which constants are, respectively, of thejirst and second di- 
 mensions, in a sense lately explained. 
 
 (1.) Since VX'0X=fiSi^0\- i'SX^'^, &c., it follows that the vector parts of^'i 
 and 93 change signs, when is changed to ^', and therefore i^ to 4/'. On the other 
 hand, we may change the arbitrary vectors X, /z, v to X', /*', v., if we at the same 
 time change X' to V/^V, or to - XSXfiv, &c., and SXfiv, or SXX', to - (SX/iv)^ ; di- 
 viding then by - SX/zv, we find these new expressions, 
 
 III. . . qi = (X<pX' + fifpfi' + v(pv') : SX/Ltr, 
 
 IV. ..9^3 = (X^X' + fx-ipfi' + vxpv') : SXjUJ/ ; 
 
 operating on wliich by S, we return to the scalars of the expressions I. and II., with 
 and 1// changed to <p' and i//'. 
 
 (2.) Hence the conjugate quaternion constants, Kqi and Kg-o, are obtained by 
 passing to the conjugate linear f mictions ; and thus we may write, 
 
 3 L 
 
 1 . ■) IT t2^A 
 
 — y f r^ ^t 
 
442 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 V. . . Kj, = (X'^'A + ii(p'ii + r'^V) : SX/tv ; 
 
 VI. . . K52 = (X't//'X -f fx'i/y + v->\j'v) : SX^v ; 
 
 or, interchanging \ with X', &c., in the dividends, 
 
 VII. . . Kqx = (X^'X' + /*0'|ii' + v<p'v') : SX/xv ; 
 VIII. . . Kq2= (Xt//>'+ /i^y + vxl^'v) : SXfiv ; 
 
 where X' = Yfiv, &c., as before. 
 
 (3.) Operating with Y.p on Vg'i, and observing that 
 
 V. pVX>X = 0(XSX'|t)) - X'SX0>, &c., 
 while * ^ (XSX'p + fiSfi'p + vSv'p) = 0(oSX/iv, 
 
 and X'SX^'p + fi'SfKp'p + v'Sv^'p - ^'pSX/xr, 
 
 with similar transformations for Y.pVqz, we find that 
 
 IX. . . Y.pYqi = (pp-(p'p; 
 and X. . . V. pYq2 =4^p- ^''p- 
 
 (4.) Accordingly, since 
 
 Sp ((pp - ip'p) = - Sp (0p - ip'p) = 0, 
 the vector 0p — ^'p, if it do not vanish, must be a line perpendicular to p, and there- 
 fore of the form, 
 
 XI. . . (pp-(p'p = 2Yyp, 
 
 in which y is some constant vector ; so that we may write, 
 
 XII. . . 0p = 0op + Vyp, 0'p = 0op-Vyp, 
 where the function ^oP is rts own conjugate, or is the common self- conjugate part of 
 0p and 0'p ; namely the part, 
 
 XIII. . . 0op = K^P + f |o)- 
 And we see that, with this signification of y, 
 
 XIV. . . V(X>X + fi'iPfi + v'(pv) = - 2ySX/*v, or XIV'. . . V91 = - 2y ; 
 while we have, in like manner, 
 
 XV. . . V(X'^X + fi'^ix + v'^v) = - 25SX/XV, or XV'. . . V92 = - 2^, 
 if XVI. . .^|/p-4''p = 2V^p. 
 
 As a confirmation, the part <po of has by (1.) no effect on Yqi ; and if we change 
 0X to VyX, &c., in the first member of XIV., we have thus, 
 
 (XSyX' + fi^yp-' + vSyv') - yS (XX' + fifi' + vv) = ySX/iv - 3ySX/ij/. 
 (5.) Since VX';|/'X = - 6YX(p'\\ &c., by 348, VII., while we may write, by (1.), 
 
 (2.), and C4.), 
 
 XVII. . . Y(X(p\' + fi(Pfi' + v(l)v') = - 2ySXfiv, 
 
 XVIII. . . Y(X^X'+^l^fl'-\■v^pv') = - 2dSXfiv, 
 or XIX. . . Y(X(p'X' + fKp'fi' + v(l>'v') = + 2ySX/iv, 
 and XX. . . Y^X^'X + /^/'/^ + v'»//V) = + 2^SX/iv, 
 we have this relation between the two new vector constants, 
 
 XXI. . . ^ = -^y = -fy=-0oy; 
 for 0, <p', and 0o have all the same effect, on this particular vector, y. 
 
 (6.) We may add that the vector constant y is of theirs? dimension, and that 
 5 is of the second dimension, with respect to the betas of the standard form ; in fact, 
 with that /or/w, 347, XV., of 0p, we have the expressions, 
 
CHAP. II.] SYMBOLIC AND CUBIC EQUATION. 443 
 
 XXII. . . y = lYQBa + fi'a' + p"a"), 
 and XXIII. . .d = |V(V/3')3".Va'a" + V/3"(3. Va"a + V/3/3'. Va«')- 
 
 (7.) If we denote by ^o and mo, what 4' and m become when ^ is changed to ^o, 
 we easily iSnd that , 
 
 XXIV. . ,^p = ^Qp- -ySyp + V^p ; XXV. . . i//'p = y^Qp - ySyp - V^p ; 
 
 so that the seZ/'-con/w^'afe por* of i//p contains a <erm, — ySyp, which involves the 
 vector y, but only in the second degree; and in like manner, 
 XXVI. . . m = mo + Syd = wiq — Sy^y ; 
 y again entering only in an even degree, because m remains unchanged, when we pass 
 from to 0', or from y to - y. 
 ■ (8.) It is evident that we have the relations, 
 
 XXVII. . . mo= ^o'/'o = i^o^o ; 
 and that, in a sense already explained, ^o, ^o, and mo are of the^r*^, second, and 
 third dimensions, respectively. 
 
 350. After thus considering the vector parts of the two 
 quaternion constants, q^ and ^2? we proceed to consider their 
 scalar parts ; which will introduce two new scalar constants, 
 m!' and m\ and will lead to the employment of two new conju- 
 gate auxiliary functions, ^p and \p ; whence also will result 
 the establishment of a certain Symbolic and Cubic Equation, 
 
 I. . . = m - m'(^ + m"<p^ - (j)\ 
 
 which is satisjied by the Linear Symbol of Operation, (p, and 
 is of great importance in this whole Theory of Linear Func- 
 tions. 
 
 (1.) Writing, then, 
 
 II. . . m"=Sgi, and III. . . m' = Sgz, 
 we see first that neither of these two new constants changes value, when we pass from 
 (p to 0', or from y to — y ; because, in such a passage, it has been seen that we only 
 change qi and q2 to Kqi and Kg2- Accordingly, if we denote by m'o and m"o what 
 m' and m" become, when (p is changed to ^o, we easily find the expressions, 
 IV. . . m" = m"o ; and V. . . m' = m'o— y^. 
 (2.) It may be noted that m", or m"o, is of theirs* dimension, but that m' and 
 m'o are of the second, with respect to the standard form of ; and accordingly, with 
 
 that form we have, 
 
 VI. . . m" = Sa/3 + Sa'/3' + Sa"/3" ; 
 and VII. . . m' = S (Va'a". V/3"/3' + Va"a.V/3|3" + Yaa'.Y(5'(3). 
 
 (3.) If we introduce two new linear functions, xp and x'p» such that 
 VIII. . . X Vjuv = Y^fKp'v - v(p'fi), 
 aiul IX. . . x'V/*i' = V(/i^j/ - r^^i), 
 
444 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 it is easily proved that these functions are conjugate to each other, and that each is 
 of thejirst dimension ; in fact, with the standard form of <pp, we have the expres- 
 sionSf 
 
 X. . . xP=y(iaY^p + a'Y(3'p + a"V/3», 
 XI. . . x'p = V()3Vap+/3'Va'p+/3"Va"p); 
 and S . XaV/3p = S . p(SVa\, &c. Also, if xo be formed fi'om ^o as x ftom ^, it will 
 be found that 
 
 XII. . . xp = Xop-Vyp, and XIII. . . x'p =Xop + Vyp; 
 where xo is of the first dimension. 
 (4.) Since 
 
 SXxV = S. X(/t0'i/ — v^'fi) = S(/i'^'/u + v'^'v), 
 
 the expression II. gives, by 349, V., the equation, 
 
 XIV. . . 7»"S\\' = S. \(0 + x)X', 
 
 X and X' being two arbitrary and independent vectors ; which can only be, by our 
 having i\iQ functional relation, 
 
 XV. . . ^p-\- X9~ "* P » 
 or briefly and symbolically, 
 
 XVI. . . X + (ft = m". 
 
 Accordingly it is evident that the relation XV. is verified, by the form X. of xp, 
 combined with the standard form of ^p, and with the expression VI. for the con- 
 stant m". 
 
 (5.) The formula XVI. gives, 
 
 XVII. . . x0 = »*"0 - 0^ = 0X ? 
 and accordingly the identity of x^ ai^d 0x ™^y easily be otherwise proved, by 
 changing fi and v to ^'fx and -^'v in the definition VIII. of x, and remembering that 
 
 Y.xp'lJi.yp'v = m(pYfjtv, <p'xl^' = m, and Yfji\p'v = — (pYv^'iJi.) 
 for thus we have, 
 
 XVIII. . . x^V/iv = Y(jiyp'v - vy\j'n) = <pY{fi(p'v - v((>'n) = 0x^/"''» 
 as required. 
 
 (6.) Since, then, 
 
 S . X^x^' = S . X(/Ln//V - vyp'ix) = S(/u'»/''/tt + v'tp'v), 
 the value III. of m' gives, by 349, VI., the equation, 
 
 XIX. . . m'SXX'=S.X(»p+^x)^', 
 X and X' being independent vectors ; hence, 
 
 XX. . .■^p + (pxp = m'pj 
 or briefly, 
 
 XXI. . . -tp + <px = ni'. 
 
 And in fact, with the standard form of 0p, we have 
 
 XXII. . . ^xp = X0P=V(V|3'/3".VpVa'tt"+ V/3"/3.VpVa"a + V/3/3'.VpVaa'); 
 which verifies the equation XX., when it is combined with the value VII. of w', and 
 with the expression 347, XVIII. for xj/p. 
 
 (7.) Eliminating the symbol x, between the two equations XVI. and XXL, and 
 remembering that ^\p =\ly<p = m, we find the symbolic expression. 
 
CHAP. II.] BINOMIAL FORM, FIXED LINES AND PLANES. 445 
 
 XXIII. . . m<p-^ = rp = m'- m"(p + ^2 . 
 
 and thus the symbolic and cubic equation I. is proved. 
 
 (8.^ And because the coefficients, m, m\ m'\ of that equation, have been seen to 
 remain unaltered, in the passage from to 0', we may write also this conjugate 
 
 equation, 
 
 XXIV. . . = m - m'(p' + m'>'2 - f 3. 
 
 (9.) Multiplying symbolically the equation I. by — m-'i//3^ and reducing by 
 •^(p = 7», we eliminate the symbol 0, and obtain this cubic in ?^, 
 
 XXV. . . = m2 - mm"yl^ + m';|/2 _ ^3 . 
 
 in which i//' may be substituted for i//. 
 
 (10.) In general, it may be remarked, that when we change to ^, and there- 
 fore il/ to m^, as before, we change not only m to m2, but also m' to mm", and m" to 
 m ; while x is at the same time changed to 0x> ^^ ^^ X^i ^^^ *^^ quaternion qx is 
 changed to 52- Accordingly, we may thus pass from the relation XVI. to XXI. ; 
 and conversely, from the latter to the former. 
 
 (11.) And if the two new auxiliary functions, x and x', be considered as defined 
 by the equations VIII. and IX., their conjugate relation (3.) to each other may be 
 proved, without any reference to the standard form of 0p, by reasonings similar to 
 those which were employed in 347, (8.), to establish the corresponding conjugation 
 of the functions ip and ;//'. 
 
 (12.) It may be added that the relations between 0, 0', x, %', and m" give the 
 following additional transformations, which are occasionally useful : 
 
 XXVI, . . ^'Vjuv = Y{nxv + v^/i) = - y^vxi^ + jw^v) ; 
 XXVII. . . (pYfiv = Yifix'v + vtp'n) = - V(vxV + f^fv) ; 
 
 with others on which we cannot here delay. 
 
 35 1 . The cubic in ^ may be thus written : 
 
 1, . . = mp - m'(j)p + m"(^'^p ~ (p^p ; 
 
 where p is an arbitrary vector. If then it happen that for some 
 particular but actual vector, p, the linear function (j)p vanishes, 
 so that (pp = 0, (p'^p = 0, (p^p = 0, &c., the constant m must be 
 zero ; or in symbols, 
 
 II. . . if 0p = 0, and Tp > 0, then m = 0. 
 
 Hence, by the expression 347, XXIII. for m, when the 
 standard form for 0p is adopted, we must have either 
 
 III. . . Saa'a = 0, or else IV. . . S/3' '^jS = ; 
 
 so that, in each case, that generally trinomial for m^ 347, XV., 
 must admit of being reduced to a binomial. Conversely, when 
 we have thus a function of the particular form^ 
 
446 ELEMENTS OF QUATERNIONS. [boOK III. 
 
 V. . . ^p = j3Sa/o + f5'Sap, 
 we have then, 
 
 VI. . .^Vda' = 0; 
 
 so that if a and a be actual and non-parallel lines, the real and 
 actual vector Yaa will be a value of |0, which will satisfy the 
 equation 0/> = ; but no other real and actual value of /o, ex- 
 cept Q = a; Vaa, will satisfy that equation, if j3 and j3' be actual^ 
 and non-parallel. In this case V., the operation reduces 
 every other vector to the fixed plane of j3, )3', which plane is 
 therefore the Z<^cw5 of ^p ; and since we have also, 
 
 we see that the locfus of the functionally conjugate vector., (p'p, 
 is another fixed plane, namely that of a, a . Also, the normal 
 to the latter plane is the line which is destroyed hj the former 
 operation, namely by ^ ; while the normal to the former plane 
 is in like manner the line, which is annihilated by the latter 
 operation, (p', since we have, 
 
 VIII.. .fVj30' = O, 
 but not (p'p= 0, for any actual p, in any direction except that 
 of Vj3/3', or its opposite, which may however, for the present 
 purpose, be regarded as the same.*. In this case we have 
 also monomial forms for ^p and \ilp, namely 
 
 IX. ..^Pp^ yaa'Si5'j5p, and X. . . ^'/o = Y(5(5'Sa'ap ; 
 so that the operation xp destroys every line in the first fixed 
 plane (of j3, j3'), and the conjugate operation ^p' annihilates 
 every line in the second fixed plane (of a, a). On the other 
 hand, the operation ^ reduces every line, which is out of the 
 first plane, to \hQ fixed direction of the normal to the second 
 plane; and the operation xfj reduces every line which is out of 
 the second T^\2i\i% to that other fixed direction, which is normal 
 to the^r^^ plane. And thus it comes to pass, that whether we 
 operate first with ;//, and then with ; or first with 0, and 
 then with ^ ; or first with \p' and then with 0' ; or first with 4>', 
 
 * Accordingly, in the present investigation^ whenever we shall speak of a ^^ fixed 
 direciion,^^ or the " direction of a given line,''' &c., we are always to be understood 
 as meaning, " or the opposite of that direction." 
 
CHAP. II.] EQUAL ROOTS, RECTANGULAR LINES AND PLANES. 447 
 
 and then with \p' ; in all these cases, we arrive at last at a null 
 line, in conformity with the symbolic equations, 
 
 XI. . . 0;// = ^0 = 0'^' = ^'0' = m = 0, 
 
 which belong to the case here considered. 
 
 (L) Without recurring to the standard form of <pp. the equation 348, VI,, 
 namely Y.t^'yi^'v = m(fVfiv, and the analogous equation Y.;pfi\pv = mp'Yuv, 
 might have enabled us to foresee that ^'p and ^pp, if they do not both constantly va- 
 nish, must (if m= 0) have each a. fixed direction; and therefore that each must be 
 expressible by a monome, as above : the fixed direction of xj/p being that of a line 
 which is annihilated by the operation (p, and similarly for t|/'p and <p'. 
 
 (2.) And because, by 347, XI. and XXV., we have 
 
 ^pYp.v = Y.^'[ip'v, and ip'V/iv = V. ^/i^j/, 
 
 so that the line ^'/z, if actual, is perpendicular to xpYfiv, and the line 0/i perpendicu- 
 lar to \l/'Yij,v, we see that each of the two lines, (p'p and and 0p, must have (in the 
 present case) a. plane locus ; whence the binomial forms of the two conjugate vector 
 functions, (pp and (p'p, might have been foreseen : xj/p and i//'|0 being here supposed to 
 be actual vectors. 
 
 (3.) The relations of rectangularity, of the two fixed lines (or directions^, to the 
 two fixed planes, might also have been thus deduced, through the two conjugate bi- 
 nomial forms, V. and VIL, without the previous establishment of the more general 
 trinomial (or standard) form of <pp. 
 
 (4.) The existence of a plane locus for <pp, and of another for <p'p, for the case 
 when m = 0, might also have been foreseen from the equations, 
 
 S.(pXflx<pv = S.(p'X(p'lj.^'v = mS>Xfj,v, 
 and the same equations might have enabled us to foresee, that the scalar constant 
 m must be zero, if for any one actual vector, such as X, either (pX or (p'X becomes 
 null. 
 
 (5.) And the reducihility of the trinomial to the binomial form, when this last 
 condition is satisfied, might have been anticipated, without any reference to the com- 
 position of the constant m, from the simple consideration (comp. 294, (10.)), that 
 no actual vector p can be perpendicular, at once, to three diplanar lines. 
 
 352. It may happen, that besides the recent reduction 
 
 (351) of the linear function (^tp to a binomial form, when the 
 
 relation 
 
 I. . . m = 
 
 exists between the constants of that function, in which case the 
 symbolic and cubic equation 350, I. reduces itself to the form, 
 
 thus losing its absolute term, or having one root equal to zero. 
 
448 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 this equation may undergo a further reduction^ by two of its 
 roots becoming equal to each other ; namely either by our 
 having 
 
 III. ..m' = 0, and IV. . . ^^(0 - m") = 0; 
 
 or in another way, by the existence of these other equations, 
 
 V. . . m"^ - 4/w' = 0, and VI. . . ^ (^ - ^m'y = 0. 
 
 In each of these two cases, we shall find that certain new geo- 
 metrical relations arise, which it may be interesting briefly to 
 investigate ; and of which the principal is the mutual rectan- 
 gular ity q£ two Jixed planes ^ which are the loci (comp. 351) of 
 certain derived, ^ndi functionally conjugate vectors : namely, in 
 the case III. IV., the loci of <l>p and <p'p ; and in the case V. 
 VI., the loci of ^p and O'/o, if 
 
 VII. . . O = ^ - im", and VIII. . . ^' = 0' - \m\ 
 so that, in this last case, the symbol <I> satisfies this new cubic, 
 
 IX. . . = <I>2 (O + \m") ; 
 while ^' satisfies at the same time a cubic equation with the 
 same coefficients (comp. 350, (8.)), namely 
 
 (1.) We saw in 351, (1.), (2.). that when m = the line y\/'p has generally afixed 
 
 direction, to which that of the line (pp is perpendicular ; and that in like manner the 
 
 line ;//jO has then another fixed direction, to which (/)'p is perpendicular. If then the 
 
 plane loci of 0p and 0'p be at right angles to each other, we must also have the 
 
 fixed lines i//'\ and »^/i rectangular, or 
 
 XI. . . = S.i/z'X^'/i = SXi^V, 
 
 independently of the directions of \ and n ; whence 
 
 XII. . . = i//3^, or XIII. . . x^2 = 0, 
 
 since jit is an arbitrary vector. 
 
 (2.). Now in general, by the functional relation 350, XXI. combined with 
 
 »//0 = w, we have the transformation, 
 
 XIV. . . rp^ = tp^m' — <p-)() = m'l// - m^ ; 
 
 if then w = 0, as in I., the symbol i// must satisfy the depressed qv quadratic equa- 
 
 Hon, 
 
 XV.. . 0=m';// -;//«; 
 
 which is accordingly a. factor of the cubic equation, 
 
 XVI... = m'-.//2-i//^ 
 whereto the general equation 350, XXV. is reduced, by this supposition of m va- 
 nishing. 
 
CHAP. II.] CASE OF DEPRESSED EQUATION. 449 
 
 (3.) If then we have not only m = 0, as in I., but also m' = 0, as in III., the 
 condition XIII. is satisfied, by XV. ; and the two planes, above referred to, are ge- 
 nerally rectangular. 
 
 (4.) We might indeed propose to satisfy that condition XIII., by supposing that 
 we had always, 
 
 XVII. . . i// = 0, that is, XVII'. . . ^p = 0, 
 
 for evert/ direction of p ; but in this case, the quaternion constant q^ would vanish (by 
 349, II.) ; and therefore the constant m', as being its scalar part (by 350, III.), 
 would still be equal to zero. 
 
 (5.) The particular supposition XVII. would however alter completely the geo- 
 metrical character oi ihQ (.\viQSiion '., for it would imply (comp. 351, (2.)) that the 
 directions of the lines (pp and (p'p (when not evanescent^ are Jixed, instead of those 
 lines having only certain planes for their loci, as before. 
 
 (6.) On the side of calculation, we should thus have, for the tAVO conjugate 
 functions, ^p and 0'p, monomial expressions of the forms, 
 
 XVIII. . . 0p = (3Sap, (p'p = aS/3p ; 
 whence, by 347, XVIII., and 350, VII., we should recover the equations, \pp = 
 and m' = 0. 
 
 (7.) We should have also, in this particular case, 
 
 XIX. . .<pp = 0, if pMa, and XX. . . fp = 0, if p ^ (3; 
 so that <pp now vanishes, if p be any line in the fixed plane perpendicular to a ; and 
 in like manner (p'p is a null line, if p be in that other fixed plane, which is at right 
 angles to the other given line, /3. 
 
 (8.) These two planes, or their normals a and j3, or the fixed directions of the 
 two lines <}>'p and <pp, will be rectangular (comp. (1.) ), if we have this new equa- 
 tion, 
 
 XXL . . 02 = 0, or XXI'. . . 02p = 0, 
 
 for every direction of p ; and accordingly the expression XVIII. gives 
 02p = 8a(B. (pp = Q, if /3 -i- a, and reciprocally. 
 (9.) Without expressly introducing a and /3, the equation 350, XXIII. shows 
 that when a^ = 0, and therefore also m' = 0, as in (4.), the symbol satisfies (comp« 
 (2.) ) the new quadratic or depressed equation, 
 
 XXII. . . = (^2 - m"0 ; 
 which is accordingly a, factor of the cubic IV., but to which that cubic is not redu- 
 cible, unless we have thus ^ = 0, as well as m' = 0. 
 
 (10.) The condition, then, of the existence and rectangularity of the two planes 
 (7.), for which we have respectively ^p = and tp'p = 0, without ^p generally va- 
 nishing (a case which it would be useless to consider), is that the four following 
 equations should subsist : 
 
 XXIII. . . ?« = 0, m' = 0, m" = 0, and XVII. . . ^ = ; 
 or that the cubic IV., and its quadratic factor XXII., should reduce themselves to 
 the very simple forms, 
 
 XXIV. . . 03 = 0, and XXV. . . 02 = Q; 
 the cubic in having thus its three roots equal, and null, and »//p vanishing. 
 
 3 M 
 
450 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (11.) We may also observe that as, when even one root of the general cubic 350, 
 I. is zero, that is when m = 0, the vector equation 
 
 XXVI... 0p = O 
 was seen (in 351) to be satisfied by one real direction of p, so when we have also 
 m' ~ 0, or when the cubic in <p has two null roots, or takes the form IV., then the 
 two vector equations, 
 
 XXVII. .. ^p=0, ^//p = 0, 
 are satisfied by one common direction of the rea? and actual line p ; because we have, 
 by 350, XVII. and XX., the general relation, 
 
 \pp = m'p - x<pp' 
 
 (12.) And because, by 350, XV., we have also the relation xp = m"p - ^p, it fol- 
 lows that when the three roots of the cubic all vanish, or when the three scalar 
 equations XXIII. are satisfied, then the three vector equations, 
 XXVIII. .. 0p = 0, xpp=0, xp = o, 
 have a common {real and actual) vector root ; or are all satisfied by one common 
 direction of p. 
 
 (13.) Since m" — (J) = x, the cubic IV. may be written under any one of the fol- 
 lowing forms, 
 
 XXIX. . . = 0^x — ^X'P ~ X'P ~ ^ ' 0X ~ ^*^M 
 in which accented may be substituted for unaccented symbols : and its geometrical 
 signification may be illustrated by a reference to certain ^a;cd/mes, and fixed planes, 
 as follows. 
 
 (14.) Suppose first that m and m' both vanish, but that m"is dififerent from zero, 
 so that the cubic in ^ is reducible to the form IV., but not to the form XXIV. ; and 
 that the operation i//, which is here equivalent to — ^x» or to — x0> does not annihi- 
 late every vector p, so that (comp, (4.) (5.) (6.) ) <pp and ^'p have no? the directions 
 of two fixed lines, but have only (comp. (1.) and (3.) ) two fixed and rectangular 
 planes, H and U', as their loci ; and let the normals to these two planes be denoted 
 by X and X', so that these two rectangular lines, X and X', are situated respectively 
 in the planes n' and 11. 
 
 (15.) Then it is easily shown (comp. 351) that the operation destroys the line X' 
 itself, while it reduces* every other line (that is, every line which is not of the form 
 irX', with Yx = 0) to the plane IT -J- X ; and that it reduces every line in that 
 plane to a fixed direction, p., in the same plane, which is thus the common direction 
 of all the lines ^^p, whatever the direction of p may be. And the symbolical equa- 
 tion, X • 02 = 0, expresses that this fixed direction p, of ^^p may also be denoted by 
 X~'0 ; or that we have the equation, 
 
 XXX. . . = x/i = w^V - <pp, if p = ^^pi 
 which can accordingly be otherwise proved : with similar results for the conjugate 
 symbols, <p' and x- 
 
 * We propose to include the case where an operation of this sort destroys a line, 
 or reduces it to zero, under the case when the same operation reduces a line to afixed 
 direction, or to n fixed plane. 
 
XXXII. 
 
 CHAP. II.] CASE OF MONOMIAL FORM. 451 
 
 (16.) For example, we may represent the conditions of the present case by the 
 following system of equations (comp. 351, V. VII. IX. X., and 350, VI. VII. X. 
 XI): 
 
 Up = (3Sap + |8'Sa'p, <p'p = aS/3p + a'SjS'p, 
 XXXI. ■ . 1 = m' = S (Vaa'.V/3'/3) = Sa(3 Sa'fi' - Sa/3'Sa'i3, 
 (m" = Sa|3 + Sa'/3'; 
 
 r XP = y(aYl3p + a'V/3'p) = m"p - cpp, 
 I X'P = V(/3Vap + /3'Va'p) = m"p - 0'p, 
 I - ^/'p = 0XP = X^P = V««'S/3^V» 
 \_-^'p = fx'p = xYP = V/3^'Saa'p ; 
 and may then write (not here supposing \' = V/wv, &c.), 
 
 XXXIII f^ = ^f^l^'^ \' = Yaa', S\\'=0, 
 
 ' ■ ' \/x = 0/3 II 0/3', n' = (p'a' 11 0'a, SX/x = S\>' = ; 
 after which we easily find that 
 
 XXXIV /^'^'=^' ^Vll/^, <p]x=m"ii, x/* = 0; 
 
 \<p-\ = 0, 0'2p II /, <p'ix'==m"fA,', x>'= 0. 
 (17.) Since we have thus xV = ^» where /*' is a line in the fixed direction of 
 0'2p, we have also the equation, 
 
 XXXV. . . = Spx>'= Sfi'xp, or xP "»-/*' ; 
 the locus of xp is therefore a plane perpendicular to the line ju' ; and in like manner, 
 H is the norjnal to a plane, which is the locus of the line x'p« And the symbolical 
 equations, <P'(l>X = ^i P^'X-^: °^^^y ^® interpreted as expressing, that the operation 
 reduces every line in this new plane of xp to thej^a;ec? direction of 0-^0, or of \' ; and 
 that the operation 0^ destroys every line in this plane -L ju'; with analogous results, 
 when accented are interchanged with unaccented symbols. Accordingly we see, by 
 XXXII., that (pxp has the fixed direction of Yaa', or of V ; and that . ^xP = 0, 
 because 0\' = 0. 
 
 (18.) We see also, that the operation 0x> or X0» destroys every line in the plane 
 n, to which the operation reduces every line ; and that thus the symbolical equa- 
 tions, <Px-^ = 0, x0 • ^ = 0> inay be interpreted. 
 
 (19.) As a verification, it may be remarked that the Jixed direction X', of cpxp 
 or x0p> ought to be that of the line of intersection of the two fixed planes of 0p and 
 Xp; and accordingly it is perpendicular by XXXIII. to their two normals, X and 
 fjf : with similar remarks respecting the fixed direction X, of 0'x'p or X'P'Pi which 
 is perpendicular to X' and to ju. 
 
 (20.) Let us next suppose, that besides m=0, and m' — 0, we have 4' = 0, but 
 that to" is still diff'erent from zero. In this case, it has been seen (6.) that the expres- 
 sion for 0p reduces itself to the monomial form, jGSap; and therefore that the opera- 
 tion destroys every line in a. fixed plane (-L- a), while it reduces every other line to 
 a. fixed direction (|1 (3), which is not contained in that plane, because we have not 
 nowSaj3=0. 
 
 (21.) In this case we have by (16.), equating a or ^S' to 0, the expressions, 
 
 XXXVI. . . 1^'° " ^^"l"' ^'^ " "^^^' """ " ^"^ < ^' 
 
 j XP = V. aV/3p = (to" - 0) p, x'P = V. fiYap = (to" - 0')p, 
 so that the equations XVIII. are reproduced ; and the depressed cubic, or the qua~ 
 dratic XXII. in 0, may be written under the very simple form. 
 
452 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XXXVII. . . O = 0x = X0- 
 
 (22.) Accordingly (comp. (5.) and (7.) ), the operation ^ here reduces an arbi- 
 trary line to the fixed direction of f3, while x destroys every line in that direction ; 
 and conversely, the operation x reduces an arbitrary line to the fixed plane perpen- 
 dicular to a, and (p destroys every line in that fixed plane. But because we do not 
 here suppose that m" = 0, the Jixed direction of ^p is not contained in the Jixed plane 
 of XP ) and (comp. (8.) and (10.) ) the directions of (pp and <p'p are not rectangular 
 to each other. 
 
 (23.) On the other hand, if we suppose that the three roots of the cubic in va- 
 nish, or that we have m=0, m' = 0, and m" = 0, as in XXIII., but that the equa- 
 tion xpp = is not satisfied for all directions of p, then the binomial forms XXXI. 
 of ^p and (p'p reappear, but with these two equations of condition between their wec^or 
 constants, whereof only one had occurred before : 
 
 XXXVIII. . . = Sa/3Sa'/3' - Sa/3'Sa'/3, = SajS + Sa'/3'. 
 
 (24.) We have also now the expressions, 
 
 XXXIX... xp = -0p, x'p=-'P'pi 
 and the cubic in ^ becomes simply <p^ = 0, as in XXIV. ; but it is important to ob- 
 serve that we have not here (comp. (9.) ) the depressed or quadratic equation ^2 = q, 
 since we have now on the contrary the two conjugate expressions, 
 
 XL. ..<p'^p = ^p = YaaS(3'l3p, f^p = i//'p = Y(3i3'Saap, 
 which do not generally vanish. And the equation ^^=0 is now interpreted, by ob- 
 serving that ^2 here reduces ever?/ line to the fixed direction of 0"iO ; while <p reduces 
 an arbitrary vector to that fixed plane, all lines in which are destroyed by ^2, 
 
 (25.) In this last case (23.), in which all the roots of the cubic in (p are equal, 
 and are null, the theorem (12.), of the existence of a common vector root of the three 
 equations XXVIII., may be verified by observing that we have now, 
 
 XLI. . . fYaa' = 0, •.//Vaa'= 0, x"^""' = ^ J 
 the third of which would not have here held good, unless we had supposed m"= 0. 
 (26.) This last condition allows us to write, by (16.), 
 
 XLII. . . ^/i = 0, ffi' = 0, V/i\' = 0, V/i'X = 0, S/iju' = 0, 
 
 the lines fi' and ft thus coinciding in direction with the normals \ and X', to the 
 planes 11 and 11' ; if then we write, 
 
 XLIII. . . V = VXX' II Yfifi', 60 that Sfiv = 0, SjuV = 0, 
 
 this new vector v will be a line in the intersection of those two rectangular planes^ 
 which were lately seen (14.) to be the loci of the lines <pp and ^'p, and are now 
 (comp. (17.) ) the loci of xp and x'p \ and the three lines ^, fi', v (or X', X, v) will 
 compose a rectangular system. 
 
 (27.) In general, it is easy to prove that the expressions, 
 
 XLIV. . //3 = °^i + *^'i. ^' = a'i3i + 6'i3'i, 
 \ai = aa-\- a'a, a'l =ba + Va, 
 
 in which a, /3, a', ^' may be any four vectors, and a, b, a\ b' may be any four sca- 
 lar*, conduct to the following transformations (in which p may be any vector) : 
 
CHAP. II.] VECTOR AND QUATERNION INVARIANTS. 453 
 
 XLV. . . Saij3i-f Sa'ii3'i = Sai3 + Sa'j8'; 
 XLVI. . . jSiSaip + jS'iSa'ip = /3Sap + /3'Sa> ; 
 XLVII. . . Yaia'i .Yj3'il3i = Vaa'. V/3'/3 ; 
 80 that the sea ?ar, Saj3 + Sa'/3' ; the vector, /3Sap + jS'Sa'p ; and the quaternion * 
 Yaa.Yj3'(3, remain unaltered in value, when we pass from a given system oi four 
 vectors a(Ba'(i\ to another system of four vectors ai/3ia'i/3'i, by expressions of the 
 forms XLIV. 
 
 (28.) With the help of this general principle (27.), and of the remarks in (26.), 
 it may be shown, without difficulty, that in the case (23.) the vector constants of 
 the binomial expression /3Sap + (3'Sa'p for <pp may, without any real loss of genera- 
 lity, be supposed subject to the four following conditions, 
 
 XLVIII. . . = Sa(3 = Sa'(3 = S/3/3' = Sa'/3' ; 
 which evidently conduct to these other expressions, 
 
 XLIX. . . ^2p =l3Sa(3'Sap, <p^p = 0; 
 and thus put in evidence, in a very simple manner, the general non-depression of the 
 cubic (p^ = 0, to the quadratic, <p = 0. 
 
 (29.) The case, or sub-case, when we have not only 7w = 0, m' = 0, m"= 0, but 
 also i// = 0, and therefore 02 = 0, as a depressed form of 0^ = 0, by the linear function 
 0/0 reducing itself to the monomial /3Sap, with the relation Sa/3 = between its con- 
 stants, has been already considered (in (10.)) ; and thus the consequences of the 
 supposition III., that there are (at least) two equal but null roots of the cubic in 0, 
 have been perhaps sufficiently discussed. 
 
 (30.) As regards the other principal case of equal roots, of the cubic equation in 
 0, namely that in which the vector constants are connected by the relation V., or by 
 the equation of condition, 
 
 L. . . = m"2 - Am' = (Sa/3 + ^a'^y - 4S(Vaa'.V|8'/3) 
 
 = (Sa/3 - Sa'/3')2 + 4Sa;8'Sa'/3, 
 it may suffice to remark that it conducts, by VI., or by VII. and IX., to the sym- 
 bolical equation, 
 
 LI. ..0 = 0*2^ if * = 0-|m"; 
 
 and that thus its interpretation is precisely similar to that of the analogous equation, 
 X02 = O, where x = m"-0, XXIX., 
 
 as given in (14.), and in the following sub-articles. 
 
 353. When v^^e have m = 0, but not /w'=0, nor m"^=4tm\ 
 the three roots of the cubic in are all unequal^ while one of 
 them is still null^ as before ; and the two roots of the quadratic 
 and scalar equation, with real coefficients (347), 
 
 I. . . = c2+ m"c^m\ 
 
 * We have, in these transformations, examples of what may be called Quater- 
 nion Invariants. 
 
454 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 which is formed from the cubic by changing (p to ~c, and then 
 dividing by c, are also necessarily unequal^ whether they be 
 real or imaginary. We shall find that when these two scalar 
 roots, Ci, C2, are real, there are then two real directions, pi and 
 P2, in ihsitjlxed plane 11 which is the locus (351, 352) of the 
 line (J)p, possessing the property that for each of them the ho- 
 mogeneous and vector equation of the second degree, 
 
 II. . . V|O0/o = 0, or 0|O II p, 
 is satisfied, without p vanishing; namely by our having, for the 
 Jirst of these two directions, the equation 
 
 III. . . 0/Oi = -Cipi, or ^ipi = 0, if 0i = + Ci; 
 and for the second of them the analogous equation, 
 
 IV. . . 0/)2 = -C2p25 or 02)02=0, if 02=0+^2: 
 
 but that no other direction of the real and actual vector p, sa- 
 tisfies the equation V., except that third which has already 
 been considered (351), as satisfying the linear and vector equa- 
 tion, 
 
 Y. , . (Jip = 0, with Tp > 0. 
 
 It will also be shown that these two directions, pi, pz, are not 
 only real, but rectangular, to each other and to the third 
 direction p, when the linear function (pp is self conjugate (349, 
 (4.) ), or when the condition 
 
 VI. . . 0> = ^p, or Vr. . . SX<f>p = Sp(l>X, 
 
 is satisfied by the given form of 0, or by the constants which 
 enter into the composition of that linear symbol; but that when 
 this condition of self- conjugation is wo^ satisfied, the roots of the 
 quadratic I. may happen to be imaginary : and that in this 
 case there exists no real direction of p, for which the vector 
 equation II. of the second degree is satisfied, by actual values 
 of p, except that one direction which has been seen before to 
 satisfy the linear equation V. 
 
 (1.) The most obvious mode of seeking to satisfy II,, otherwise than through V., 
 is to assume an expression of the form, p = x(3 + x'(i', and to seek thereby to satisfy 
 the equation, (^ + c) p = 0, with <pp = (3Sap + jS'Sa'p, by satisfying separately the two 
 scalar equations, 
 
 VII. . . = a; (c + Sa/3) + a'Sa/3', = a;' (c -(- Sa'/3') + a;Sa'/3, 
 
CHAP. II.] CASE OF UNEQUAL REAL ROOTS. 455 
 
 which give, by elimination o^af-.x^ the following quadratic in c, 
 VIIL . . (c + Sa/3) (c + Sa'/3') = Sa/3'Sa')3, 
 which is easily seen to be only another form of I. Denoting then, as above, by Cj 
 and C2, the roots of that quadratic I., supposed for the present to be real, we have 
 these two real directions for p, in the plane IT of /3, j8' : 
 
 IX. . . |0i = )3(ci + Sa'/3') - j3'Sa'i3 = ci/3 + Va'VjS'/B ; 
 
 X. . . p2 = j8 (C2 + Sa'iS') - )8'Sa'/3 = C3j3 + Va'V/3'/3 ; 
 which satisfy the equations III. and IV. In fact, the expression IX. gives 
 
 0pi = ci^j3 + m'/3 = -cipi, or 0i|Oi = 0, 
 because we may write it thus, 
 
 XI. . .pi = (m"+ci)/3-0/3 = -C2/3-0i3=-02/3 = -^/3-»»Vi3; 
 and in like manner, the expression X. may be thus written, 
 
 XII. . . p2 = (m" + C2)i3 - 0j8 = - ci/3 - 0i3 = - 0i/3 = - ^/3 - »i'c2-i/3, 
 and gives, 
 
 0P2 = C20/3 + m'^ = - C2P, or 02P2 = 0. 
 
 (2.) We may also write, 
 
 XIII. . . p'l = /3'(ci + Sfl/3) - /3Sa|8' = ci/3' + VaV/3/3' = - 02i8' || pi ; 
 XIV. . . p'2 = /3'(c2 + SajS) - i8Sa/3' = C2/3' + VaV/3/3' = - 0ii3' |1 p2 ; 
 
 and shall then have the equations, 
 
 XV. . . (p\p\ = 0, ^2p'2 = ; 
 but the directions of p'l and p'2 will be the same by VIIL as those of p\ and p2, and 
 so will furnish no new solution of the problem just resolved. 
 (3.) Since we have thus, 
 
 XVL..02i3'||^2/3||pi||^riO, and XVI'. . . 0i/3' |1 0i/3 || p2 11 562 '0, 
 
 it follows that the operation 02 reduces every line in the fixed plane of <pp to the 
 fixed direction of ^r'O ; and that, in like manner, the operation 0i reduces every line, 
 in the same fixed plane of 0p, to the other fixed direction of 02"'O. 
 (4.) Hence we may write the symbolic equations, 
 
 XVIL .. 01.02^ = 0, 02-?>i0 = O, 
 in which the points may be omitted ; and in fact we have the transformations, 
 XVIIL . . 0102 = 0201 = (^ + ci) (0 4- cz) = 02 - m"0 -\-m'=\p, 
 
 so that 0102.0= 0201.0 = »p0 =»i = 0. 
 
 (5.) If we propose to form t//i from 0i, by the same general rule (347, XI.) by 
 which -.// is formed from 0, we have 
 
 XIX. . . ifjiYfiv = V. 0'1^0'iv = V.(0'/i + ciju) (0V + cip), 
 and therefore, by the definition 350, VIII. of x? 
 
 XX. . . ;//ip = i//p+cixp + ci2p, or XXI. . . t//i = T^ + cix + ci2 ; 
 and in like manner, 
 
 XXII. . . ^//2 = ^ + C2X + ^22> 
 
 even if m be different from zero, and if ci, c^ be arbitrary scalars. 
 
 (6.) Accordingly, without assuming that m vanishes, if we operate on xj/ip with 
 
456 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 ^1, or symbolically multiply the expression XXI. for xpi by ^i, we get the symbolic 
 product, 
 
 XXIII. . . ^1^1 = (0 + ci) (»// + cix + Ci2) 
 
 = m + c\m' + ci^m" + c^ = mi, 
 
 where mi is what the scalar m becomes, when is changed to ^i, or is such that 
 
 XXIV. . . miSX/iV = S.0'iX0'i/i^'iv = S.(0'\+ci\) (^V + ci^i) (^V+civ); 
 
 as appears by the definitions of 0', t^, x, »»j W, m", and by the relations between 
 those symbols which have been established in recent Articles, or in the sub- articles 
 appended to them. 
 
 (7.) Supposing now again that m = 0, and that ci, c^ are the roots of the quadra- 
 tic I. in c, we have by XXIII., 
 
 XXV. . . 0i;//i = TWi = ; and in like manner XXVI. . . ^2^2 = »*2 = 0, 
 if m% be formed from mi, by changing c\ to a. 
 
 (8.) Comparing XXV. with XVII., we may be led to suspect the existence of an 
 intimate connexion existing between ipi and 020, since each reduces an arbitrary vec- 
 tor to the fixed direction of 0r'O, or of pi ; and in fact these two operations are iden- 
 tical, because, by XXI., and by the known relations between the symbols, we have 
 the transformations, 
 
 XXVII. . . t//i = ip + cix + Ci2 = (m' - m"0 + 02) ^ ci (m" - 0) + c^ 
 = ,p2-(ni"+ ci)0 = 03 + C20 = 002 ; 
 and similarly, XXVIII. . . \p2 = 0^ + ci0 = 00i ; 
 
 while \p = 0102, as before. 
 
 (9.) We have thus the new symbolic equation, 
 
 XXIX. . . 00102 = 0, 
 
 in which the three symbolic factors 0, 0i, 02 may be in any manner grouped and 
 transposed, so that it includes the two eqxiations XVII. ; and in which the subject 
 of operation is an arbitrary vector p. Its interpretation has been already partly 
 given ; but we may add, that while reduces every vector to the Jixed plane IT, 
 01 reduces every line to another fixed plane, 11 1, and 02 reduces to a third plane, 
 1X2 ; thus 0102, or 020i, while it destroys two lines pi, po, and therefore every line in 
 the plane 11, reduces an arbitrary line to the fixed direction of the intersection of the 
 two planes 111112, which intersection must thus have the direction of 0-10 ; and in 
 like manner, the fixed direction pi of 0r^O, as being that to which an arbitrary" vec- 
 tor is reduced (3.) by the compound operation 020, or 002, must be that of the inter- 
 section of the planes 11112 ; and p2, or 02"'O, has the direction of the intersection of 
 IIIIi ; while on the other hand 002 destroys every line in IIi, and 00i every line in 
 1X2: so that these three planes, with their three lines of intersection, are the chief 
 elements in the geometrical interpretation of the equation 00i02 = 0. 
 (10.) The conjugate equation, 
 
 XXX. . . 0'0 10 2 = 0, 
 may be interpreted in a similar way, and so conducts to the consideration of a con- 
 jugate system o^ planes and lines ; namely the planes 11', II'i, n'2, which are the 
 loci of 0'p, 0'ip, 0'3p, while the operations 0'i0'2. 0'20'i, and 0'0'i destroy all lines 
 
CHAP. II.] CASE OF IMAGINARY ROOTS AND DIRECTIONS. 457 
 
 in these three planes respectively, and reduce arbitrary lines to the fixed directions 
 of the intersections, Xl'ill'a, n'2n', II'II'i, which are also those of ^'-'O, ^V'O, 
 
 (11.; It is inaportant to observe that these three last lines are the normals to the 
 three first planes, IT, IT, IT"; and that, in like manner, the three ^rmer lines 
 are perpendicular to the three latter planes. To prove this, it is sufficient to ob- 
 serve that 
 
 XXXI. . . Sp>p = Sp^'p' = 0, if 0'p' = 0, or that (pp 4- 0'-'O ; 
 and similarly, ^'p -J- ^-'0, &c. 
 
 (12.) Instead of eliminating x' : cc between the two equations VII., we might 
 have eliminated c ; which would have given this other quadratic, 
 
 XXXII. . . = a;2Sa'/3 + .r.r(Sa'/3'-Sai8)-a;'2Sa|3'; 
 also, if x'\ : x\ and x^ : X2 be the two values of x' : x, then 
 
 XXXIII. . . pi II Xl(3 + X\(3', P2 II 232/3 + Xz(3', 
 
 and XXXIV. . . Xix^ : (a;ia;'2 + x^x'i) : x'lx'^ = - Sa/3' : (Saj3 - Sa'/3') : Sa'^; 
 hence the condition of rectangularity of the two lines pi, p2, or ^r^O, ^2''0, is ex- 
 pressed by the equation, 
 
 XXXV. . . = - )82Sa/3 ' + S/3/3'(Sa)3 - Sa'/^) + ^'^a'^ = S . ^|3'V(/5a + /3'a') ; 
 and consequently it is satisfied, if the given function be self-conjugate (VI.), be- 
 cause we have then the relation, 
 
 XXXVI. .. V/3a+V/3'a' = 0; 
 in fact the binomial form of ^ gives (comp. 349, XXII.), 
 
 XXXVII. . . ^'p - 0p = (aS/3p - (38ap)+ (a'S/3'p -/3'Sa'p) = V.pV(|3a +/3'a'), 
 which cannot vanish independently of p, unless the constants satisfy the condition 
 XXXVI. 
 
 (13.) With this condition then, of self-conjugation of ^, we have the relation of 
 rectangularity, 
 
 XXXVIII. .. Spip3=0, or ^1-10 -L ^2-'0 ; 
 
 at least if these directions pi and p2 be real, which they can easily be proved to be, 
 as follows. The condition XXXVI. gives, 
 
 XXXIX. . . = S . aa'Y(i(3a + /3'a') = a^Sa'13 + Saa'(Sa'|8' - Sa/3) - a'^Safi' ; 
 hence (a^ Sa'/3 - a'2Sa/3')2 = (Saa')^ {Sa(3 - Sa'fty, 
 
 a^a'Hrn"^ - 4m') = a^a"^ { (Sa^S - Sa'/S')^ + 4Sa|8'Sa'/3} 
 = {a^a'^ - (Saa')2) (a/3 ~ Sa'/3')^ + (a2Sa'/3 + a'2Sa/3')2 > 0, 
 and XL. . . (Saj3 - Sa'/3')2 + 4Sa/3'Sa'/3 = m"2 - 4m' > ; 
 
 so that each of the two quadratics, I. (or VIII.), and XXXII., has real and unequal 
 roots : a conclusion which may also be otherwise derived, from the expressions 
 /3 = aa + 6a', (i' = ba + a' a', which the condition allows us to substitute for /3 and j3'. 
 (14.) The same condition XXXVI. shows that the /owr vectors a(3a'(3' are corn- 
 planar, or that we have the relations, 
 
 XLI. . . Sa(3(i' = 0, Sa'i3/3' = 0, V(Vaa'. V^'|3) = ; 
 
 hence Yaa', or ^"'0 is now normal to the ja^ane 11 ; and therefore by (13.), when 
 the function <p is self-conjugate (VI.), the three directions, 
 
 3 N 
 
458 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XLII. . . p, pu P2, or ^^10, ^r^O, 02-10, 
 compose a real and rectangular si/stem. 
 
 (15.) In the present series of subarticles (to 353), we suppose that the three 
 roots of the cubic in <p are all unequal^ the cases of equal roots (with m = 0) having 
 been discussed in a preceding series (352) ; but it may be remarked in passing, that 
 when a self- conjugate function <pp is reducible to the monomial form (3Sap^ we must 
 have the relation V/3a = ; and that thus the line /3, to thefxed direction of which 
 (comp, 352, (5.) and (6.) ) the operation then reduces an arhitranj vector, is per- 
 pendicular to ihQ fixed plane (352, (7.) ), every line in which is destroyed by that 
 operation 0. 
 
 (16.) In general, if be thus self-conjugate, it is evident that the three planes 
 XT', n'l, II'2, which are (comp. (10.) ) the loci of ^'p, ^I'p, (p^p, coincide with the 
 planes TI, lli, 1X3, which are the loci of 0jo, 0ip, fp%p. 
 
 (17.) When is not self- conjugate, so that 0p and 0'p are not generally equal, 
 it has been remarked that the scalar quadratic I,, and therefore also the symbolical 
 cubic in 0, may have imaginary roots; and that, in this case, the vector equation IT. 
 of the second degree cannot be satisfied by any real direction of p, except that one 
 which satisfies the linear equation V., or causes 0p itself to vanish, while p remains 
 real and actual. As an example of such imaginary scalars, as roots of I., and of 
 what may be called imaginary directions^ or imaginary vectors (comp. 214, (4.) ), 
 which correspond to those scalars, and are themselves imaginary roots of II., we may 
 take the very simple expressions (comp. 349, XII.), 
 
 XLIII. . . 0p = Vyp, <p'p = -Yyp ; 
 in which y denotes some real and given vector, and which evidently do not satisfy 
 the condition VI., the function being here the negative of its own conjugate, so that 
 its self-conjugate part 0o is zero (comp. 349, XIII.). We have thus, 
 
 XLIV. ..mo=0, m'o = 0, w"o=0, 0o = O, i^o = 0, xo = 0, 
 and consequently, by the sub-articles to 349 and 350, 
 
 XLV. . . m = 0, m' = - y^, m" =0, ^p = - y^yp, XP — ~ ^7P » 
 the quadratic I., and its roots ci, C2> become therefore, 
 
 XLVI. ..c2-y2 = 0, Ci = + yri.Ty, C2 = --/rT.Ty, 
 
 where v— 1 is the imaginary of algebra (comp. 214, (3.) ) ; thus by XX. or XXI., 
 and XXII.) we have now 
 
 XLVII. . . i//i(r = - ySy<T - ciYyff 4- CiV = (y - ci)Vy(r, i//o(r = (y - C2)Vy(r ; 
 
 hence 
 
 Syi^i<T = 0, Vy;|/i(T= yi//i(r, &c., 
 and 
 
 XLVIII. . . 0n//i<T ={<p+ ci)4/i<r = (y + ci) (y - ci)Yy(T = (y2 - ci^)Yy(r = 0, 
 and in like manner XLVIII'. . . 02'A2<^ = ; 
 
 if then we take an arbitrary vector cr, and derive (or rather conceive as derived) from 
 it two (imaginary) vectors pi and pz by the (imaginary) operations i^i and \p2, we 
 shall have (comp. III. and IV.) the equations, 
 
 XLIX. . . pi = ^la, <pipi = 0, (ppi = - cipi, Vpi^pi = 0, 
 
 and L. . • P2 = 4'i'^^ 02P2 = 0> 0P2 = ~ ^^2^2) Vp20p2 = 0, 
 
CHAP. II.] VECTOR AND QUADRATIC EQUATION. 459 
 
 as ones which are at least symbolically true. We find then that the two imaginary 
 directions, p\ and jOo, satisfy (at least in a symbolical sense, or as far as calculation 
 is concerned) the vector equation II., or that p\ and p^ are two imaginary vector roots 
 of yp(pp = ; but that, because the scalar quadratic I. has here imaginary roots, 
 this vector equation II- has (as above stated) no real vector root p, except one in the 
 direction of the given and real vector y, which satisfies the linear equation V., or 
 gives <Pp = 0. 
 
 (18.) This particular example might have been more simply treated, by a lees 
 general method, as follows. We wish to satisfy the equation, 
 
 LI. . . = V. pYyp = pSyp - p^y ; 
 which gives, when we operate on it by V. y and V.p, these others, 
 
 LII. . . 0=Vyp.Syp, = p2Vyp; 
 if then we wish to avoid supposing <pp =Vyp = 0, we must seek to satisfy the two 
 scalar equations, 
 
 LIII. . . Syp = 0, p2=0; 
 
 and conversely, if we can satisfy these by any (real or imaginary) p, we shall have 
 satisfied (really or symbolically) the vector equation LI. Now the first equation 
 LIII. is satisfied, when we assume the expression, 
 
 LI V. . . p = (c + y)Vy(r = Vy(T . (c - y), 
 
 where a is an arbitrary vector, and c is any scalar, or symbol subject to the laws of 
 scalars; and this expression LIV. for p, with its transformation just assigned, gives 
 
 LV. . . p2=(c2-y2) (Vy<T)2 = 0, if c2 - y2 = Q ; 
 
 the quadratic XLVI. is therefore reproduced, and we have the same imaginary roots, 
 and imaginary directions, as before. 
 
 (19.) Geometrically, the imaginary character oi the recent problem, of satisfying 
 the equation V. juVyp = by any direction of p except that of the given line y, is 
 apparent from the circumstance that <pp, or Vyp, is here a vector perpendicular to p, 
 if both be actual lines ; and that therefore the one cannot be also parallel to the 
 other, so long as both are real* 
 
 354. In the three preceding Articles, and in the sub-arti- 
 cles annexed, we have supposed throughout that the absolute 
 term of the cubic in (^ is wantmg, or that the condition m = 
 is satisfied ; in which case we have seen (351) that it is always 
 possible to satisfy the linear equation (pp = 0, by at least one 
 real and actual value of p (with an arbitrary scalar coefficient) ; 
 or by at least one real direction. It will be easy now to show, 
 
 * Accordingly the two imaginary directions, above found for p, are easily seen to 
 be those which in modern geometry are called the directions of lines drawn in a given 
 plane (perpendicular here to the given line y), to the circular points at infinity : of 
 which supposed directio7is the imaginary character may be said to be precisely this, 
 that each is (in the given plane) its oivn perpendicular. 
 
460 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 that although conversely (comp. 351, (4.) ) the function «^^ 
 cannot vanish for any actual vector /d, unless we have thus 
 m = 0, yet there is always at least one real direction for which 
 the vector equation of the second degree^ 
 I. . . Yp^p = 0, 
 which has already been considered (353) in combination with 
 the condition m = 0, is satisfied ; and that if the function be 
 a self-conjugate one, then this equation I. is always satisfied 
 by at least three real and rectangular directions^ but not gene- 
 rally by more directions than three; although, in this case of 
 self-conjugation, namely when 
 
 II. . . (f>'p = <l>p, or ir. . . SX^/o = Sp^A, 
 
 for all values of the vectors p and X, the equation I. may hap- 
 pen to become true, for one real direction of p, and for every 
 direction perpendicular thereto : or even for all possible direc- 
 tions, according to the particular system of constants, which 
 enter into the composition oi t\iQ function (pp. We shall show 
 also that the scalar (or algebraic) and cubic equation, 
 
 III. . . =m + m'c + nfc^ + c^ 
 
 which is formed from the symbolic and cubic equation 350, I., 
 by changing to - c, enters importantly into this whole 
 theory ; and that if it have one real and two imaginary roots, 
 the quadratic and vector equation I. is satisfied by only one 
 real direction of p ; but that it may then be said (comp. 353, 
 (17.)) to be satisfied also by tivo imaginary directions, or to 
 have two imaginary and vector roots : so that this equation 
 I. may be said to represent generally a system of three right 
 lines, whereof one at least must be real. For the case IL, the 
 scalar roots of III. will be proved to be always real; so that 
 if TWo, m'o, and m"o be formed (as in sub-articles to 349 and 350) 
 from the self -conjugate part <pop of any linear and vector func- 
 tion- (pp, as m, m', and m" are formed from that function (pp it- 
 self, then the new cubic, 
 
 IV. . . = w?o f m'oC + m"oC- + c^, 
 
 which thus results, can never have imaginary roots. 
 
CHAP. II.] CASE OF SELF-CONJUGATION. 461 
 
 (1.) If we write, 
 V. . . */o = 0|O f cp, ^'p = (p'p + cp, or briefly, V. . . * = -f c, *' = 0' + c, 
 where c is an arbitrary scalar, and if we denote by ■*", ''ir', and M what i^, ;//', and 
 m become, by this change of ^ to ^ + c or $, the calculations in 353, (5.), (6.), 
 show that we have the expressions, 
 
 VI. ..■*■=<// + ex + c2, >i^' = ]//'+ ex' +c2, 
 and VII. . . M= m + m'c + mV + c^, 
 
 with VIII. . . M= ^^ = ^(p = <E»'^' = ^'*'. 
 
 (2.) Hence it may be inferred that the functions x> x'j ^^^ the constants m', 
 m" become, 
 
 IX. . . X= Dc^ = X + 2c, X'= D,^' = x' + 2c, 
 j-M' = DcM= m' + 2m"c + 3c2, 
 • " \M"=|Dc2iW = m"+3c; 
 with the verifications, 
 
 XI. . . $ + X = *' + X'=M", *X + ^ = $'X' + >t''=iW', 
 as we had, by the sub-articles to 350, 
 
 + X = 0' + x' = ^"» 0x + 'Z' = ^'x' + '/''= w'- 
 
 (3.) The new linear symbol $ must satisfy the new cubic, 
 XII. ..0 = M- M'^ + M"<E)3 - $3 . 
 which accordingly can be at once derived from the old cubic 350, I., under the form, 
 XIII. . . = m 4 rn(c - *) + m\e - $)2 + (c - *)3. 
 (4.) Now it is always possible to satisfy the condition, 
 XIV. . . M=0, 
 by substituting for c a real root of the scalar cubic III. ; and thereby to reduce the 
 new symbolical cubic XII. to the ^brm, 
 
 XV. . . = *3 _ iVf""I>2 + Af' $; 
 which is precisely similar to the form, 
 
 Q = (f>^-m'y^vi<p, 352,11., 
 
 and conducts to analogous consequences, which need not here be developed in detail, 
 since they can easily be supplied by any one who will take the trouble to read again 
 the few recent series of sub-articles. 
 
 (5.) For example, unless it happen that "^p constantly vanishes, in which case 
 M' = 0, and ^p (if not identically null) takes a monomial form, which is reduced to 
 zero (comp. 352, (7.) ) for every direction of p in a given plane, the operation ^ 
 reduces (comp. 351) an arbitrary vector to a given direction; and the operation ^ 
 destroys every line in that direction : so that, in every case, there is at least one real 
 way of satisfying the vector equation $p = 0, and therefore also (as above asserted) 
 the equation I., without causing p itself to vanish. 
 
 (6.) And since that equation I. may be thus written, 
 XVI, . . Vp*p = 0, or *pI1p, 
 we see that it can be satisfied withoid <I»p vanishing, if this new; scalar and quadratic 
 
 equation, 
 
 XVII. . . = C2 + M"C-V M\ comp. 353, I., 
 
462 ELEMENTS OF QUATERNIONS. [boOK III. 
 
 have real and unequal roots Ci, C^ ; for if we then write, 
 
 XVIII. .. *i = *+ Ci, $2=*+C2, 
 the line ^p will generally have for its locus a given plane, and there will be two real 
 and distinct directions pi and pz in that plane, for one of which 4>ipi = 0. while 
 ^2P2= for the other, so that each satisfies XVI., or I. ; and these are precisely the 
 fixed directions of ^ip and "ir-zp, if ■*■! and ^2 he formed from ^ by changing $ to 
 $1 and $2 respectively. 
 
 (7.) Cases of equal and of imaginary roots need not be dwelt on here ; but it may 
 be remarked in passing, that if the function 0p have the particular form {g being 
 any scalar constant), 
 
 XIX. . . 0p = gp, then XX. . .{g-<l>y = 0, and XXI. . . M= (^ + c)3 ; 
 the cubic XIV. or III. having thus all its roots equal, and the equation I. being sa- 
 tisfied by every direction of p, in this particular case. 
 
 (8.) The general existence of a real and rectangular system of three directions 
 satisfying I., when the condition II. is satisfied, may be proved as in 353, (14.) ; 
 and it is unnecessary to dwell on the case where, by two roots of the cubic becoming 
 equal, all lines in a given plane, and also the normal to that plane, are vector roots 
 of I., with the same condition II. 
 
 (9.) And because the quadratic, = c^ + m"c + m' (353, I.), has been proved to 
 have always real roots (353, (13.) ) when (}>'p = <l>p, the analogous quadratic XVII. 
 must likewise then have real roots, Ci, Cz ; whence it immediately follows (comp. 
 XII. and XIII.), that (under the same condition of self-conjugation) the cubic III. 
 has three real roots, c, c + Ci, c + C2 ; and therefore that (as above stated) the other 
 cubic IV., which is formed from the self-conjugate part ^q of the general linear and 
 vector function 0, and which may on that account be thus denoted, 
 XXII. . . Mo= 0, has its roots always real. 
 (10.) If we denote in like manner by <E>o the symbol ^o f c, the equation 
 w =mo - Sy0oy (349, XXVI., comp. 349, XXI.) becomes, 
 
 XXIII. . . M=Mo- Sy^oy ; 
 whence, by comparing powers of c, we recover the relations, 
 
 ni = m'o - y2, and m" = m"o, as in 350, (1.). 
 (11.) On a similar plan, the equation m^'Y^v = Y.^'fi'^l^v becomes, 
 
 XXIV. . . M^'Vfiv==Y.^n^v, comp, 348, (1.), 
 
 in which p. and v are arbitrary vectors, and c is an arbitrary scalar ; or more fully, 
 XXV. . . (w + m'c + m"c^ + c^) {^' + c)Ypv = V. (j^p + cxp + c2/i) (i//j/ -j- c^v + c^v) ; 
 whence follow these new equations, 
 
 XXVI. . . (to + m'<p')Ypv = Y(-^p . xv - ^'V. x/a), 
 
 XXVII. . . (m' + m"f)Ypv = Y(p\pv - vxpp + xfi- x**)) 
 
 XXVIII. . . (m"+<l>')Ypv = Y(px^-^XfJ^)> 
 
 which can all be otherwise proved, and from the last of which (by changing to i/^, 
 
 &c.) we can infer this other of the same kind, 
 
 XXIX. . . (m' + \ly')Ypv = Y{p(pxv - v<Pxi*)- 
 (12.) As an example of the existence of a real and rectangular system of three 
 directions (8.), represented jointly by an equation of the form I., and of a system of 
 
CHAP. II.] REAL AND RECTANGULAR SYSTEM. 463 
 
 three real roots of the scalar cubic III., when the condition II. is satisfied, let us 
 take the form, 
 
 XXX. . . (f)p=gp-irY\pfi = i>'p, 
 
 g being here any real and given scalar, and X, /i any real and non-parallel given 
 vectors ; to which/orm, indeed, we shall soon find that ewery self-conjugate function 
 ^op can be brought. We have now (after some reductions), 
 
 XXXI. . . xj/p = VXpjuSX/A - Y\iJiS\pfi-g(\Sfip + fjiSXp) + g^p, 
 XXXII. . . XP = - (>^S/tp + /iSXp) + 2gp, 
 and XXXIII, . . m = (5- - SX/*) (g^ - X^^), m' = - X^ - 2^SX/i + 3^2, 
 
 m" = - SX/i + 3g ; 
 where the part of \pp which is independent of^* may be put under several other forms, 
 such as the following, 
 XXXIV. . . V(Xp/iSX/A - XfiSXpn) = XpfiSXfi - XfiSXpfi 
 
 = X(pSX/i + SXfxp)fi = ^X(X/ip + pXfi)ii = X(XSfxp + juSXp - Xp/i)/i, &c. ; 
 and 4>, ""i^, X, M, M', M" may be formed from ^, ;//, Xi ™> "* » *^ » ^7 simply 
 changing g to c^g. The equation M= has therefore here three real and unequal 
 roots, namely the three following, 
 
 XXXV. . . c = -^ + SX/i, c+Ci = -^ + TX^, c+C2 = -<7-TX/[i; 
 and the corresponding forms of "^p are found to be, 
 
 XXXVI. . . ^p = YXfiSXfip, ^ip= - (XTju + /iTX; S . p(XT/i + /nTX), 
 
 xp2p = - (XT/i - /i*TX)S. p(XT/i - /xTX). 
 
 Thus ^p, ^ip, and '^'ap have in fact the three fixed and rectangular directions of 
 YX/x, XTix + /xTX, and XTfi - ^TX, namely of the normal to the given plane of X, 
 fi, and the bisectors of the angles made by those two given lines ; and these are ac- 
 cordingly the onlg directions which satisfy the vector equation of the second degree, 
 
 XXXVII. . . (Vp0p=V.pVXp/[t=)VpXS/ip + Vp/xSXp = O; 
 80 that this last equation represents (as was expected) a system of three right lines, 
 in these three respective directions. 
 
 (13.) In general, if ci, C2, C3 denote the three roots (real or imaginary) of the 
 cubic equation M=0, and if we write, 
 
 XXXVIIL . . $x = + Ci, $2=0+C2, *3 = 0+C3, 
 
 the corrresponding values of ■*" will be (comp. VI.), 
 
 XXXIX. . .^l = xf^+CiX+Ci^, ^2 = '/' + C2X + C22, ^3=»^ + C3X + <'32; 
 
 also we have the relations, 
 
 (ci + C2 + cz = -m" =-^ - X, 
 XL. . . < C2C3 + c^ci + C1C2 = +m'=(px+i', 
 { C1C2C3 = — m = — 0i// ; 
 
 whence it is easy to infer the expressions, 
 
 XLI. . . a»i = (C2 - C3)-i (^3 - ^2), *2= (c3 - ciyi (^1 - ^3), 
 
 *3 = {Cl - C2)-» (^2 - ^1) ; 
 
 which enable us to express the functions *ip, 4>2p, *3p as binomials (comp. 351, 
 &c.), when ^ip, *^2p, ^sp have been expressed as monomes, and to assign the 
 planes (real or imaginary), which are the loci of the lines *ip, fp2p, *3p- 
 
464 ELEMENTS OF QUATERNIONS, [bOOK III. 
 
 (14.) Accordingly, the three operations, $, <I>i, 4>2, by which lines in the three 
 lately determined directions (12.) are destroyed, or reduced to zero, and which at 
 first present themselves under the forms, 
 
 XLII. . . *p = XS//p + /iSXp, *ip = VXpjLi + pTX^, *2 = VXp/i - pTX/i, 
 are found to admit of the transformations, 
 
 ^ 2TX/i ' ^^ TX/i+SX/i' ^^ TXfi-SXix' 
 
 where *, *i, ^2 have the recent forms XXXVI., and the loci of $p, 4»ip, *2io com- 
 pose a system of three rectangular planes. 
 
 (IS.') In general, the relations (13.) give also (comp. 363, (8.)), 
 
 XLIV. . . ^1 = $2^3, ^2 = *3$i, ^3 = $1*2, 
 
 and XLV. . . *i^i = $2*2 = *3^3 = *i*2*3 = 0, 
 
 whence also, XLVI. . . ^i'*^2 = ^2^3 = ^3'*'i = 0, 
 
 the symhols (in anyone system of this sort) admitting of being transposed and grouped 
 at pleasure; if then the roots of M— be real and unequal, there arises a system 
 of three real and distinct planes, which are connected with the interpretation of the 
 symbolical equation, $1*2*3 = 0, exactly as the three planes in 353, (9.) were con- 
 nected with the analogous equation, <p(p].<P'2, = 0. 
 
 (16.) And when the cubic has two imaginary roots, it may then be said that there 
 is one real plane (such as the plane -^ y in 353, (18.), (19.) ), containing the two 
 imaginary directions which then satisfy the equation I. ; and two imaginary planes, 
 which respectively contain those two directions, and intersect each other in one real 
 line (such as the line y in the example cited), namely the one real vector root of 
 the same equation I. 
 
 355. Some additional light may be throv^rn upon that vector 
 equation of the second degree, by considering the system of the two 
 scalar equations, 
 
 I. . . Sx/3^/> = 0, and II, . . Sa/j = 0, 
 
 and investigating the condition of the reality of the two* directions, 
 pi and p2, by which they are generally satisfied, and for each of 
 which the plane of p and (pp contains generally the given line A in 
 I., or is normal to the plane locus II. of p. We shall find that these 
 two directions are always real and rectangular (except that they may 
 become indeterminate), when the linear function (p is its own conju- 
 gate ; and that then, if A be a root po of the vector equation, 
 
 III. . . Yp9p^0, 
 
 * Geometrically, the equation I. represents a cone of the second order, with X 
 for one side, and with the three lines p which satisfy III. for three other sides ; and 
 II. represents a plane through the vertex, perpendicular to the side X. The tivo direc- 
 tions sought are thus the tivo sides, in which this plane cids the cone. 
 
CHAP. II.] NEW PROOF OF EXISTENCE OF THE SYSTEM. 465 
 
 which has been already otherwise discussed, the lines px and p^ are 
 also roots of that equation ; the general existence (354) of a system of 
 three real and rectangular directions, which satisfy this equation III. 
 when (p'p = (ppy being thus proved anew : whence also will follow a 
 new proof of the reality of the scalar roots of the cubic 31= 0, for this 
 case of self -conjugation of ^ ; and therefore of the necessary reality of 
 the roots of that other cubic, Mq = 0, which is formed (354, IV. or 
 XXII.) from the self -conjugate part <po of the general linear and vec- 
 tor function ^, as if =0 was formed from 0. 
 
 (1.) Let \, fi, V he a. system of three rectangular vector units, following in all 
 respects the laws (182, 183), of the symbols i, j, k. Writing then, 
 
 IV, . . p = y/i + cv, and therefore, \p = yv— zfi, ^p = y^/i + z<pv, 
 the equation II. is satisfied, and I. becomes, 
 
 V. . . = i/'^Sv<pn + y^ {.^v(l)v - SfKpfi) - z'^Sft<pv ; 
 the roots of which quadratic will be real and unequal, if 
 
 VI. . . (Sv<J)v-Sfi(piJiy+4:Sfi(pvSv(pfi>0; 
 and the correspondmg directions of p will be rectangular, if 
 
 VII. . . = S (y i/i + ziv) (r/2fi + z%v) = - (j/iy-i -f 212:2) ; 
 that is, if 
 
 VIII. . . Sv0ft = S/i0v, 
 
 at least for this particular pair of vectors, ju and v. 
 
 (2.) Introducing now the expression, <pp = 0op + Vyp (349, XII.), the condi- 
 tions VI. and VIII. take the forms, 
 
 IX. . . {^vfpQV - Sfxcpofiy + 4S (in<pQvy > 4 (Sy/tiv)2, and X. . . Syfiv = ; 
 
 which are both satisfied generally when y = 0, or = ^' = ^q ; the only exception 
 being, that the quadratic V. may happen to become an identity, by all its coefficients 
 vanishing : but the opposite inequality (to VI. and IX.) can never hold good, that 
 is to say, the roots of that quadratic can never be imaginary, when is thus self- 
 conjugate. 
 
 (3.) On the other hand, when y Is actual, or 0'p not generally =^p, the condi- 
 tion X. of rectangularity can only accidentally be satisfied, namely by the given or 
 fixed line y happening to be in the assumed plane of //, v ; and when the two direc- 
 tions of p are thus not rectangular, or when the scalar Syixv does not vanish, we 
 have only to suppose that the square of this scalar becomes large enough, in order to 
 render (by IX.) those directions coincident, or imaginary. 
 
 (4.) When ^' = 0, or y = 0, we may take p and v for the two rectangular direc- 
 tions of p, or may reduce the quadratic to the very simple form yz = ; but, for this 
 purpose, we must establish the relations, 
 
 XL .. Sp<pv = Svfp = 0. 
 
 (5.) And if, at the same time, \ satisfies the equation III., so that <p\ || \, we 
 shall have these other scalar equations, 
 
 3 o 
 
466 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XII. . . ^ Sfi(p\ = Sv(pX = S\(l>fi = S\(pv ; 
 ■whence ^fi \\ Yv\ || ;x, and 0v || YXfi || v, 
 
 or, XIII. . . = YX(p\ = YfjL<pix = Yv(})v ; 
 
 X, )u, V thus forming (as above stated) a system, of three real and rectangular roofs 
 of that vector equation III. 
 
 (6.) But in general, if III. be satisfied by even two real and distinct directions 
 of p, the scalar and cubic equation M= can have no imaginary root; for if those 
 two directions give two unequal but real and scalar values, ci and C2, for the quo- 
 tient — 0p : p, then ci and ci are <wo reaZ roof* of the cubic, of which therefore 
 the third root is also real ; and if, on the other hand, the two directions pi and p2 
 give one common real and scalar value, such as ci, for that quotient, then ^p = — cip, 
 or *ip = (^ + ci)p = 0, for every line in the plane of pi, p^ ; so that <pp must be of 
 the form, — cip + i8Spip2P, and the cubic will have at least two equal roots, since it 
 will take the form, 
 
 XIV. . . = (c - ci)2 (c - ci + Spip3/3), 
 
 as is easily shown from piinciples and formulae already established. 
 
 (7.) It is then proved anew, that the equation M=0 has a/Z its roots real, if 
 0'p = fp ; and therefore that the equation Mo = (as above stated) can never have 
 an imaginary root. 
 
 (8.) And we see, at the same time, how the «caZar cubic M= might have been 
 deduced from the symbolical cubic 350, L, or from the equation 351, I., as the con- 
 dition for the vector equation III, being satisfied by any actual p ; namely by ob- 
 serving that if (pp = -cp, then 02p = c^p, 03p =- c^p, &c., and therefore Mp = 0, in 
 which p, by supposition, is different from zero. 
 
 (9.) Finally, as regards the case* of indetermination, above alluded to, when 
 the quadratic V. fails to assign any definite values to y : z, or any definite directions 
 in the given plane to p, this case is evidently distinguished by the condition, 
 
 XV. . . S/^^/t = Sv^j/, 
 in combination with the equations XI. 
 
 356. The existence of the Symbolic and Cubic Equation (350), 
 which is satisfied by the linear and vector symbol ^, suggests a Theo- 
 rem\ of Geometrical Deformation, which may be thus enunciated: — 
 
 * It will be found that this case corresponds to the circular sections of a surface 
 of the second order; while the less particular case in which (p'p = (pp, but not 
 Sfi^fi^Sv^v, so that the two directions of p are determined, real, and rectangular, 
 corresponds to the axes of a non-circular section of such a surface. 
 
 f This theorem was stated, nearly in the same way, in page 568 of the Lectures; 
 and the problem of inversion of a linear and vector function was treated, in the few 
 preceding pages (559, &c.), though with somewhat less of completeness and perhaps 
 of simplicity than in the present Section, and with a slightly different notation. The 
 general form of such a function which was there adopted may now be thus ex- 
 
 0p = S/3Sap + Yrp, r being a given quaternion ; 
 the resulting value of m was found to be (page 561), 
 
CHAP. II.] THEOREM OF SUCCESSIVELY DERIVED LINES. 467 
 
 "7/*, hy any given Mode, or Law^ of Linear Derivation^ of the 
 kind above denoted by the symbol ^, we pass from any assumed Vec- 
 tor p to a Series of Successively Derived Vectors, />!, /jg, pz, . , . or ^ V» 
 ^V) ^V> • • ; «^^ i/*» % constructing a Parallelepiped, we decompose any 
 Line of this Series, such as p^, into three partial or component lines^ 
 mp, - m'pi, m"pi^ in the Directions of the three which precede it, as here 
 of P) Pii Pi '•> i^^^ i^^ Three Scalar Coefficients, m, - m\ m", or the Three 
 Ratios which these three Components of the Fourth Line pz hear to the 
 Three Preceding Lines of the Series, will depend only on the given Mode 
 or Law of Derivation, and will be entirely independent of the assumed 
 Length and Direction of the Initial Vector y 
 
 (1.) As an Example of such successive Derivation, let us take the law, 
 I. . . pi = 0p = - \ftpy, p2 = ^2p = _ V^piy, &c., 
 which answers to the construction in 305, (1.), &g., when we suppose that /3 and y 
 are unit-lines. Treating them at first as any two given vectors, our general method 
 conducts to the equation, 
 
 II. . . p3 = mp - m'pi + »»"p2, 
 with the following values of the coefiicients, 
 
 III. . .m = -i82y2S/3y, m'=-(3^y^, m" = S/3y; 
 as may be seen, without any new calculation, by merely changing p, X, and /*, 
 in 354, XXXIII., to 0, (B, and - y. 
 
 (2.) Supposing next, for comparison with 306, that 
 
 IV. . . /32=y2=-l, and S/3y=-Z, 
 so that (3, y are unit lines, and I is the cosine of their inclination to each other, the 
 
 values III. become, 
 
 V. . . m = Z, w' = - 1, m" = — l; 
 
 and the equation II., connecting /owr successive lines of the series, takes the form, 
 VI. . . p3 = ^p + Pi-^p3, or VII. . .p3-pi = -?(p3-p); 
 
 m = :SSaa'a"Si8"/3'/3 + 2S (rVaa'. V^G'/S) + SrSSa^Sr - SSarS/3r + SrTr^ ; 
 and the auxiliary function which we now denote by t// was, 
 
 m<p-^<j = 4^a= SVaa'Si3'i3(T + 2 V. aV(V/3cr.r) + (VarSr - VrSirr) ; 
 where the sum of the two last terms of ;|/<t might have been written as arSr — rSar. 
 A student might find it an useful exercise, to prove the correctness of these expres- 
 sions by the principles of the present Section. One way of doing so would be, to 
 treat 2/3Sflp and r as respectively equal to ^op + Vyp and c + e; which would 
 transform m and \p(T, as above written, into the following. 
 
 Mo- S(y + £) (00 + c) (y +0, and >iro(T-iy + t) S(y + e) a + Ya(<Po + c) (y + «) i 
 that is into the new values which the M and "Va of the Section assume, when *p 
 takes the new value, *p = (0o + c) p + V(y + e) p. 
 
468 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 a result which agrees with 305, (2.), since we there found that if p = op, &c., the 
 interval P1P3 was = — ? x ppz. 
 
 (3.) And as regards the inversion of a linear and vector function (347), or the 
 return from any one line pi of such a series to the line p which precedes it, our ge- 
 neral method gives, for the example I., by 354, (12.), 
 
 VIII. ..;//pi= 1/3 (/3ypi + pi/3y)y. 
 and 
 
 a result which it is easy to verify and to interpret, on principles already explained. 
 
 357. We are now prepared to assign some new and gene- 
 ral Forms, to which the Linear and Vector Function (with real 
 constants) of a variable vector can be brought, without assum- 
 ing its self-conjugation ; one of the simplest of which forms is 
 the following, 
 
 I. . . 0|O = V^'op + VX|OjU, with T. . . qo = ff + y; 
 
 qo being here a j^eal and constant quaternion, and X, /x two real 
 and constant vectors, which can all be definitely assigned, when 
 the particular form of is given : except that X and jul may be 
 interchanged (by 295, VII.), and that either may be multiplied 
 by any scalar, if the other be divided by the same. It will 
 follow that the scalar, quadratic, and homogeneous function of 
 a vector, denoted by Sp(pp, can always be thus expressed : 
 
 II. . . S^0jO = gp^ + SXpjujO ; 
 or thus, 
 
 ir. . . 8p(pp = gp^ + 2S\pSpp, if g' = g- SX/ul ; 
 
 a general and (as above remarked) definite transformation, 
 which is found to be one of great utility in the theory of Sur- 
 faces* of the Second Order, 
 
 (1.) Attending first to the case of seZ/'-cow/M^afe functions 0op, from which we 
 can pass to the general case by merely adding the term Yyp, and supposing (in vir- 
 tue of what precedes) that aia2as are three real and rectangular vector-units, and 
 C1C2C3 three real scalars (the roots of the cubic Mo = 0), such that 
 
 * In the theory of such surfaces, the two constant and real vectors, \ and //, 
 have the directions of what are called the ci/clic normals. 
 
CHAP. II.] RECTANGULAR AND CYCLIC TRANSFORMATIONS. 469 
 
 III. . . 0iai= (^o + ci)«i=0, ^2^2= (0o + <?2)a2= 0, ^aota = (0o + C3) as = 0, 
 
 we may write 
 
 IV. . . p = - (aiSaip + aiSazp + asSasp), 
 and therefore 
 
 V. . . <Pop — CiaiSaip + C2a2Sa2p + caaaSasp ; 
 so that 
 
 ( 0ip = (<?2 - <?i) a2Sa2P + (C3 — ci) asSaap, 
 VI. . . J 02P = (<^3 - c-i) asSasp + (ci - 02) aiSaip, 
 ( ^sp = (ci - C3) aiSaip + (c2 — C3) a2Sa2p, 
 the binomial forms of ^1, 02, 03 being thus put in evidence. 
 (2.) We have thus the general but scalar expressions : 
 
 VII. . . - p2 = (Saip)2 + (Sa2p)2 + (Sa3p)2 ; 
 
 VIII. . . Sp0p = Sp56op=ci(Saip)2 + C2(Sa2p)2+C3(Sa3p)2 
 
 = - Cip2 + (C3 - ci) (Sa2p)2 + (C3 - Ci) (Sasp)^ 
 
 = - C2p2 - CC2 - Ci) (Saip)2 + (C3 - ^2) (Sa3p)2 
 
 = - C3P* - (C3 - ci) (Saip)2 - (C3 - C2) (Sa2p)2 : 
 iu which it is in general permitted to assume that 
 
 IX. . . ci<cz< cs, or that X. . . C2 - ci = 2e^, cz-c% = 2e'2, 
 c and e' being real scalars, and the numerical coefficients being introduced for a mo- 
 tive of convenience which will presently appear. 
 
 (3.) Comparing the last but one of the expressions VIII. with II'., we see that 
 we may bring Sp^p to the proposed form II., by assuming, 
 
 XL . . \ = erti + e'a3, /i = — eai + e'as, 5' = SX/x — C2 = — 5 (ci + C3), 
 
 because SX/t = e2 — e'2 = C2 — | (ci + C3). 
 
 (4.) But in general (comp. 349, (4.) ) we cannot have, for all values of p, 
 
 XII. . . Sp0p = Sp0'p, unless XIII. . . 0op=0'op> 
 that is, unless the self-conjugate parts of and <f be equal ; we can therefore infer 
 from II. that ^Qp=gp + Y\pfi, because VXp/* = V/ipX = its own conjugate; and 
 thus the transformation I. is proved to be possible, and real. 
 
 (5.) Accordingly, with the values XI. of X, fx, g, the expression, 
 
 XIV. . .<}>op=9p-\- VXp/i = p(g- SXfi) + XS/^p + /iSXp, 
 becomes, 
 
 XV. . . (pop = - C2P + (e'as + eai) S(e'a3- eai)p + {e'az - eai) S(e'a3+ eai)p 
 = - C2P - 2e2aiSaip + 2e'2aoSa3p ; 
 which agrees, by X., with VI. 
 
 (6.) Conversely if ^r, X, and p, be constants such that ^Qp-=gp-\-N\pp, then 
 0oVX/i = (/'VXyw, where g' =g - SX/i, as before ; hence — g' must be one of the three 
 roots ci, C2, C3 of the cubic Mq = 0, and the normal to the plane of X, p. must have 
 one of the three directions of ai, 02, az ; if then we assume, on trial, that this plane 
 is that of ai, 03, and write accordingly, 
 
 XVI. . . X = aai + a'as, p, = bai-\-b'azi (p2P = XSpp-i- [iSXp, 
 
 we are, by VI., to seek for scalars aa'bb' which shall satisfy the three conditions, 
 
 XVII. . .2ab = ci~ C2, 2a'6' = C3 - co, ab' + 6a' = ; 
 
 but these give 
 
 XV in. . . (2aJ')2 = (26a')2 = (cs ^ cz) (c2 - ci). 
 
470 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 so that if the transformation is to be a real one, we must suppose that C2 — cy and 
 C3 — c-z are either both positive, as iu IX., or else both negative ; or in other words? 
 we must so arrange the three real roots of the cubic, that 03 may be (algebraically) 
 intermediate in value between the other two. Adopting then the order IX., with 
 the values X., we satisfy the conditions XVII. by supposing that 
 
 XIX. . . a' = b' = e', a = — 6 = e ; 
 and are thus led back from XVI. to the expressions XI., as the onl^ real ones for \, 
 fi, and g which render possible the transformations I. and II. ; except that \ and fi 
 may be interchanged, &c., as before. 
 
 (7.) We see, however, that in an imaginary sense there exist two other solutions 
 of the problem, to transform (pp and Sp0p as above ; for if we retain the order IX., and 
 equate g' in II'. to either - ci or - C3, we may in each case conceive the corresponding 
 sum of two squares in VIII. as \)&ii\g the product of two imaginary but linear fac- 
 tors ; the planes of the two imaginary pairs of vectors which result being real, and 
 perpendicular respectively to a\ and as. 
 
 (8.) And if the real expression XIV. for 0op he given, and it be required to pass 
 from it to the expression V., with the order of inequality IX., the investigation in 
 354, (12.) enables us at once to establish the formulae : 
 
 XX. . . ci = -5r-TX/i, C2=-^+S\ju, cz = -g^T\fi; 
 XXI. . . ai = V(\Tfx - fiTX), az = UVX/*, as = V(\Tfx + fiTX) ; 
 in which however it is permitted to change the sign of any one of the three vector 
 units. Accordingly the expressions XI. give, 
 
 TX/i + SX/z = 2e2 = C2 - ci, TX)ti-SX/i = 2e'2=c3-c2, SX/i = ^ + C2; 
 TX = T/i, . X- ju = 2eai, VXj[i = — 2ee'a3ai = + 2ee'a2, \ + fi=2e'a3. 
 (9.) We have also the two identical transformations, 
 
 XXII. . . SXpup = p^TXfi + { (SX/ip)2 + (SXpTfjL + SfxpTXy } (TXfx - SX/i)-i, 
 XXIII. . . SXpfip = - p2TX/* - {(SXfipy + (SXpT/i - SupTXy] (TXh + SX/i)-«, 
 which hold good for any three vectors, X, ft, p, and may (among other ways) be de- 
 duced, through the expressions XX. and XXI., from II. and VIII. 
 
 (10.) Finally, as regards tbe expressions VI. for ^ip. Sec, if we denote the cor- 
 responding forms of »//p by ipip, &c., we have (comp. 354, (15.) ) these other ex- 
 pressions, which are as usual (comp. 351, &c.) oi monomial form : 
 I ■^ip = (p2(pzp = (C2 - ci) (ci - C3) aiSaip ; 
 XXIV. . . j ■4>2p = 4>3(p\p = (c3 - C2) (c2 - ci) a2Sa2p ; 
 ( 4'3p = 0i^2p = (ci - cs) (c3 - C2) asSasjO ; 
 
 and which verify the relations 354, XLI., and several other parts of the whole fore- 
 going theory. 
 
 358. The general linear and vector function <Pp of o. vector has been 
 seen (347, (1.)) ^° contain, at least implicitly, nii^e scalar constants ; 
 and accordingly the expression 357, 1, involves that number, namely 
 four in the term YqoP, on account of the constant quaternio7i q^^ and 
 five in the other term YXpfi^ each of the two unit-vectorSy UX and Uyt«, 
 counting as two scalarSj and the tensor TX/* as one more. But a self- 
 
CHAP. II.] FOCAL TRANSFORMATIONS. 471 
 
 conjugate linear and vector function, or the self-conjugate part 0oP of 
 the general function ^/), involves only six scalar constants; either be- 
 cause three disappear with the term Y^p of (pp ; or because the con- 
 dition of self-conjugation, 2Vy3a = 27 = (comp. 349, XXII. and 353, 
 XXXVI.), which arises when we take for (pp the form 2y3Sa/3 (347» 
 XXXI.), is equivalent to a system of thi^ee scalar equations, connect- 
 ing the nine constants. And for the same reason the general quadra- 
 tic but scalar function, Spcpp, involves in like manner only six scalar 
 constants. Accordingly there enter only six such constants into the 
 expressions 357, II-, 11^, V., YIII., XIV.; Ci, Cg, C3, for instance, 
 being three such, and the rectangular unit system ai, ag, "3 answer- 
 ing to three others. The following other general transformations of 
 Sp^p and ^Qp, although not quite so simple as 357, II. and XIV., in- 
 volve the same number (six) of scalar constants, and deserve to be 
 briefly considered : namely the forms, 
 
 I.. .Sp^p = a{Vapy + h(S^py', 
 II. . . V = - aaYap + bfiS^p ; 
 
 in which a, h are two real scalars, and a, y3 are two real unit-vec- 
 tors. We shall merely set down the leading formul£e, leaving the 
 reader to supply the analysis, which at this stage he cannot find 
 difficult. 
 
 (1.) In accomplishing the reduction of the expressions, 
 
 S|O0p = ci(Saip)2+C2(Sa2p)2+C3(Sa3p)2, 357, VIII. 
 
 and ^op = ciaiSaip + CiUzSazp 4- caasSasp, 357, V., 
 
 to these new forms I. and II., it is found that, if the result is to be a real one, —a 
 must be that root of the scalar cubic Mo=0, the reciprocal of which is algebraically 
 intermediate, between the reciprocals of the other two. It is therefore convenient 
 here to assume this new condition, respecting the order of the inequalities, 
 
 III. . . cr^ > C2"' > C3-1 ; 
 
 which will indeed coincide with the arrangement 357, IX., if the three roots ci, ci, 
 03, be all positive, but will be incompatible with it in every other case. 
 
 (2.) This being laid down (or even, if we choose, the opposzYe order being taken), 
 the (real) values of a, h; a, (3 may be thus expressed : 
 
 IV, . . a = - C2, b = ci-C2 + cz; 
 V. . . a = xai + zas, /3 = x'ai + z'a^ ; 
 
 in which 
 
 VI.. :r3 = £LiZ 
 
 ci 1 - C3 
 
 ATT 2 <'1^-C2'' „ C-Z- 
 
 V I. . x^ = , z^= — 
 
 C\T C^Z 
 
 VII. . . -p = -p = b(xx + zz') = - 6Sa/3 = (8ay) b' ; 
 
472 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 VIII. . . 6'2 = C1C2-IC3& = Ci2a;2 + C32z2 ; IX. . . x2 + y2 = a;'2 4-/2 = 1; 
 
 X. . . hx'z' = c^xz ; 
 
 XI. . . cix3 + C3z2 = ciC2-»C3 = ft-ifc'z = & (Sa/3)2, C1C3 = - a6 CSa|3)2 . 
 
 XII. . . 6'/3 = -6]8Sa/3 = cia;ai + cszas; &c. 
 
 (3.) And there result the transformations : 
 
 XIII. . . ^2p =(ci - C2)aiSaip + (03 — C2) asSasp 
 
 = - ci{xa\ + zaz) S (a;ai + zaz) p H • (a?<riai + 20303) S (arciai + zc^a-^ p ; 
 
 C1C3 
 
 XIV. . . 0op = ciaiSaip + C2a2Sa2p + C3a3Sa3p 
 
 C2 
 
 = C2(a;ai + 2a3) V(a;ai+za3)pH (ascioi + 20303) S(a;ciai + zc303)p ; 
 
 C1C3 ■ 
 
 XV. . . Sp^p =- C2(V(a;ai + 2a3)p)2+ — (SCxciai + 20303) p)2; 
 
 C1C3 
 
 which last, ifcicz he positive, gives this o^Aer real form, 
 
 XVI. . . Sp0p = N { S(a;ciai + 203^3) p + (0103)^ V(a;ai + 203) p } ; 
 
 <-l03 
 
 a;2 and z^ being determined by the expressions VI. 
 
 (4.) Those expressions allow us to change the sign of 2 : ic, and thereby to deter- 
 mine a second pair of real unit lines, a and /3', which may be substituted for a and j3 
 in the forms I. and II. ; the order of inequalities III. (or the opposite order), and the 
 values IV. of a and b, remaining unchanged. We have therefore the double trans- 
 formations : 
 
 XVII. . . Sp^p = - C2(Vap)2 + (ci - C2 + C3) (S/3p)2 = - C2(Va'p)2 
 
 + (ci-C2 + C3)CS^'p)2; 
 
 XVIII. . . (pop = C2aYap + (ci - C2 + C3)/3S/3p = Cza'Ya'p + (ci - C2 -f- C3) /3'S/3'p. 
 
 (5.) If either of the two connected /orws I. and II. had been ^-iwen, we might 
 have proposed to deduce from it the values of C1C2C3, and of 010203, by the general 
 method of this Section. We should thus have had the cubic, 
 
 XIX. .. = i[fo = (c + a){c2+(a-6)c-a6CSo/3)2}; 
 
 and because the quadratic (c+ a)'iAfo = may be thus written, 
 
 XX. . . (c-i+a-i)2 (So/3)3-(c-i+ a- 1) (o-iS.(o/3)2 + 6-i) + a-2(Vo^)2 = 0, 
 
 it gives two real values of c"i + o-i, one positive and the other negative ; if then we 
 arrange the reciprocals of the three roots of Mo=0 in the order III., we have the 
 expressions, 
 
 lc3 = K* - «) - 1«& V(a-« + 2a-li-iS. (ia(3y + 6-2) ; 
 the signs of the radical being determined by the condition that (ci — C3) : ab (Sa/3)2 
 = ci'^ — 03"^> 0. Accordingly these expressions for the roots agree evidently with 
 the former results, IV. and XI., because S . (afSf = 2 (Sa/3)2 - 1. 
 
 (6.) The roots ci, C2, C3 being thus known, the same general method gives for 
 the directions of oi, 02, 03 the versors of the following expressions (or of their nega- 
 tives) : 
 
CHAP. II.] SECOND FOCAL TRANSFORMATIONS. 473 
 
 !»^ip = ac3->(c3a + 6/3SajS) S (c^a + b(3Sa(3) p ; 
 4/2|0 = a6Va/3S/3ap; 
 ^s9 = acr'(cia + 6/3Sa/3) S(cia + 6/3Saj3)p ; 
 
 of which the monomial forms may agaiu be noted, and which give, 
 
 XXII'. .. ai = ± U(c3a + &;3Saj8), as = ± UVa/3, as = ± U(cia + 6/3Sa^). 
 
 (7.) Accordingly the expresssions in (2.), give (if we suppose a^ai = + az), 
 
 XXIII. . . cza + */3Saj3 = (ca - ci)a;ai, Va^S = (x'z - xz) ai, cia + 6/3Sa/3 
 
 = (ci - C3)za3 ; 
 
 and as an additional verification of the consistency of the various parts of this whole 
 theory, it may be observed (comp. 357, XXIV.), that 
 
 XXIV. . . - ac3-J(c3a + h^^a^y = (c2 - ci) (ci - cg), ab(Yaf3y 
 
 = (c3 - C2) (c3 - ci), - acri(cia + 6/3Sa/3)2 = (ci - C3) (03 - C2). 
 
 (8.) As regards the second transformations, XVII. and XVIII., it is easy to 
 prove that we may write, 
 
 XXV. . . (C3 - ci) a' = 5j3a/3 - aa, (cg - ci)/3' = aa/3a - b(3, 
 
 XXVI. . . - (C3 - ci)2 = (bj3a(3 - aay = [aa^a - b(3y ; 
 
 so that we have the following equation, 
 
 XXVII. . . (a(Vap)2+ 6(Si3p)«) (a2 + 2a6S.(a]3)2+62) 
 = a (V(6/3a/3 - aa)p)2 + b (S(aa/3a - fe/3)p)2, 
 
 which is true for any vector p, any two unit lines a, /3, and any two scalars a, 6. 
 
 (9.) Accordingly it is evident from (4.), that ai, as must be the bisectors of the 
 angles made by a, a', and also of those made by (3, (i' ; and the expressions XXV. 
 may be thus written (because 6 - a = ci + C3), 
 
 XXVIII. . . (C3 - ci) a' = (C3 + ci) a + 26/3Saj3, (ci - C3)/3' = (ci + C3)/3 - 2aaSa/3 ; 
 
 whence, by XXIII., we may write, 
 
 XXIX. . . a + a=^ 2xai, a-a' = 2za3 ; 
 
 so that ai bisects the internal angle, and as the external angle, of the lines a, a'. 
 
 (10.) At the same time we have these other expressions, 
 XXX. . . (ci - C3) (j3 + i3') = 2 (ci/3 - aaSa/3), (C3 - ci) (/3 - jS') = 2 (csiS - aaSa/S) . 
 which can easily be reduced to the simple forms, 
 
 XXXI. . . /3 + /3' = 2x'ai, (B-I3'= 2z'az, 
 
 with the recent meanings of the coefficients x' and z'. 
 
 (11.) And although, for the sake of obtaining real transformations, we have 
 supposed (comp. III.) that 
 
 XXXII. . . (ci-i - cg-i) (C3-^ - C3-O > 0, 
 
 because the assumed relation a = xax + zaz between the three unit vectors aaiuz, 
 whereof the two latter are rectangular, gives a;2-f 2:3= 1^ as in jx., so that each of 
 the two expressions VI. involves the other, and their comparison gives the ratio, 
 
 XXXIII. . . ar2 : 22 = (ci-i - C2-J) : {c^^ - C3-'). 
 
 3 P 
 
474 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 yet we see that, without this inequality XXXII. existing, the foregoing transforma- 
 tions hold good in an imaginary (or merely symbolical) sense : so that we may say, 
 in general, that the functions Sp^p and tpop can be brought to the forms I. and II. 
 in six distinct ways, whereof two are real, and the four others are imaginary. 
 
 (12.) It may be added that the first equation XXII. admits of being replaced by 
 the following, 
 
 XXXIV. . . ^i^ip=-bci-i(cil3-aaSaj3)S{ei(B-aaSaj3)p, 
 with a corresponding form for \pzp; and that thus, instead of XXII'., we are at 
 liberty to write the expressions, 
 
 XXXV. . . ai = U (tfii3 - aaSa/3), a^ = UVa^, 03 = U (<?3i3 - aaSa/3), 
 for the rectangular unit system, deduced from I. or II. 
 
 359. If we call, as we naturally may, the expressions 
 
 I. . . 0o/> = CiftiSai^ + Caa^Sag/o + CsU^Sa^^ 357, V., 
 
 and II. . . Sp(pp = c,{Sa,py + c,(Sa,py + c,{Sa,p)\ 357, VIII., 
 
 the Rectangular Transformations of the Functions (poP and Sp(pp, 
 then by another geometrical analogy^ which will be seen when we 
 come to speak briefly of the theory of Surfaces of the Second Order, 
 we may call the expressions, 
 
 . III. ..1>oP=gp + VV/*, 357, XIV., 
 
 and TV. . .Sp4>p = gp^±S\pfip, 357,11., 
 
 the Cyclic* Transformations of the same two functions; and may 
 say that the two other and more recent expressions, 
 
 V. . . 0o/> = - auYap + b^SjSp, 358, II., 
 
 and VI. . . Sp(f)p = a(Vapy + b(S^py, 358, L, 
 
 are Focalf Transformations of the same. We have already shown 
 
 (357) how to exchange rectangular forms with cyclic ones; and also 
 
 (358) how to pass from rectangular expressions to focal ones, and 
 reciprocally : but it may be worth while to consider briefly the mu- 
 tual relations which exist, between cyclic and /oca^ expressions, and 
 the modes of passing from either to the other. 
 
 (1.) To pass from IV. to VI., or from the cyclic to the focal form, we may first 
 accomphsh the rectangular transformation II., with the values 357, XX., and XXI., 
 of ci, C2, C3, and of ai, a%, 03, the order of inequality being assumed to be 
 
 * Compare the Note to Art. 357. 
 
 f It will be found that the two real vectors a, a', of 358, are the two real focal 
 lines of the real or imaginary cone, which Is asymptotic to the surface of the second 
 order, S/o0p = const. 
 
CHAP. II.] PASSAGE FROM CYCLIC TO FOCAL FORMS. 475 
 
 VII. . . C3 > C2> ci, as in 357, IX. ; 
 
 and then shall have (comp. 358, XV.) the following expressions : 
 
 VIII. . . 4Sp(pp = { S . p (cii(UX - U/i) + cs^UX + U/i)) } 2 
 -{Y.p (ci^(U\ + U/i) + ezi(JJ\ - U/i)) } 2 ; 
 Vlir. . . 4Sp0p = - (S . p ((- ci)i (UX ~ U/i) + (- C3)i (U\ + U/i*)) }2 
 + { V. p ((- cx)i (U\ + U/x) + (- C3)KUX - U/.))} 2 ; 
 IX. . . (C3 - C2)2 Sp^p = { V. p (csi VX/z + (- C2)i (XT/i + fiTX)) } « 
 + {S.p((- C2)J VX/t - C3KXT/i + /.TX))}2; 
 X. . . (c2-ci)2Sp^p = -{V.p((-ci)^VX/i + C2iCXT/i-;*TA))}2 
 
 - {S.p(- C2^VX/i + (- ci)i (XT/* -juTX))}2; 
 
 in which it is to be remembered that (by 357, XX.), 
 
 XL . . ci = -5r-TX/i, C2 = -^ + SX/t, cs = -g-T\fi; 
 and of which all are symbolically true, or give (as in IV.) the real value gp^ + SXp/xp 
 for Sp^p, if g, X, /t, p be real. And in #Ats symbolical sense, although they have 
 been written down as four, they only count as three distinct /ocaZ transformations, 
 of a ^it'e7i and reaZ cyclic form i because the expression VIII'. is an immediate con- 
 sequence of VIII. ; and other formulae IX'. and X'. might in like manner be at once 
 derived from IX. and X. 
 
 (2.) But if we wish to confine ourselves to real focal forms, there are then four 
 cases to be considered, in each of which some one of the four equations VIII. VIII'. 
 IX. X. is to be adopted, to the exclusion of the other three. Thus, 
 if XII. . . C3 > C3 > ci > 0, and therefore ci'^ > C2-1 > C3-' > 0, 
 
 the form VIII. is the only real one. If 
 
 XIII. . . C3 > C2 > > ci, C3-1 > C3-1 > > ci-i, then X. is the real form. 
 If XIV. . . C3 > > C2 > ci, C3-1 > > ci-i > C2-\ the only real form is IX. 
 Finally if XV. . . > C3 > cg > ci, > cr^ > cg"^ > C3"', 
 
 that is, if all the roots of the cubic ifo = be negative, then VIII'. is the form to be 
 adopted, under the same condition of reality. 
 
 (3.) When all the roots c axe positive, or in the case when VIII. is the realfo- 
 calform, the unit lines a, /3 in VI. may be thus expressed : 
 
 XVI. 
 
 [i3 = ^(^i^y(UX-U/i) + ^(^|y(UX + U/i); 
 
 with 6 = ci - ci + C3 as before (358, IV.). 
 
 (4.) In the same case VIII., the expressions for 4Sp0p may be written (comp. 
 358, XVI.) under either of these two other real forms : 
 
 XVII. . . 4Sp^p = N { (cgi + cii) p .UX 4- (csi - ci^) U/i . p } ; 
 XVII'. . . 4Sp0p = N {(cgi + cii) UX . p + (C3i - Cii)p.U/t } ; 
 so that if we write, for abridgment, 
 
 XVIII. . . *o = i (C3* + cii) UX, Ko = i (c^^ - cii) U/i, . 
 we shall have, briefly, 
 
 XIX. . . Sp^p = N(top + pK:o)=N(p(o+«-op)- 
 
476 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (5.) Or we may make 
 XX. . . t = ^ (ci-i + C3-i) UX, K = ^ (ci-i - C3"-i)U)u, whence k« - i^ = cfi C3~i ; 
 and shall then have the transformation, 
 
 which may be compared with the equation 281, XXIX. of the ellipsoid, and for the 
 reality of which form, or of its two vector constants, i, k, it is necessary that the 
 roots c of the cubic should all be positive as above. 
 
 (6.) It was lately shown (in 358, (8.), &c.), how to pass from a, given and real 
 focal form to a second of the same kind, with its new real unit lines a', /3' in the 
 same plane as the two old or given lines, a, /3 ; but we have not yet shown how to 
 pass from a focal form to a ct/clic one, although the converse passage has been re- 
 cently discussed. Let us then now suppose that the ^rm VI. is real and given, or 
 that the two scalar constants a, 6, and the two unit vectors a, (3, have real and 
 given values ; and let us seek to reduce this expression VI. to the earlier form IV. 
 
 (7.) We might, for this purpose, begin by assuming that 
 XXII. . . ci-i > C2-1 > C3-\ as in 358, IIL ; 
 which would give the expressions 358, XXI. and XXII., for C1C2C3 and rtiaaos, and 
 so would supply the rectangular transformation, from which we could pass, as be- 
 fore, to the ct/clic one. 
 
 (8.) But to vary a little the analysis, let us now suppose that the given focal 
 form is some one of the four following (comp. (1.) ) : 
 XXin. . . Sptpp = (S/3o|o)2 - (Vaop)2; XXIII'. . . Sp0p = (Vaop)^ - (S/3o|o)2 ; 
 XXIV. . . Sp^p = (S/3op)2 + (Vaop)2 ; XXIV. . . Sp^p = - (Vaop)^ - (S^op)^; 
 in each of which ao and /3o are conceived to be given and real vectors, but not gene- 
 rally unit lines; and which are in fact the four cases included under the general 
 form, a(Vap)2 ■}- 6(S/3p)2, according as the scalars a and 6 are positive or negative. 
 It will be sufficient to consider the two cases, XXIII. and XXIV., from which the 
 two others will follow at once. 
 
 (9.) For the case XXIII. we easily derive the real cyclic transformation, 
 
 XXV. . . Sp^p = (S/3op)2 - (Saop)2 + ao^p^ 
 
 = S(|3o+ ao)p.S(^o- ao) p + aoV 
 = ^p2 + SXp/xp = {g- SXju)p2 + 2SX/iS/*p, 
 where XXVI. . . X = jSo + ao, M = | (jSo - ao), 9 = IW + M ; 
 and the equations 357, (9.) enable us to pass thence to the two imaginary cyclic 
 forms. 
 
 (10.) For example, if the proposed function be (comp. XIX.), 
 
 XXVIL . . Sp^p =N(top + pKo) = (SO0+ »co)p)2 -(V(to - ko)p)2, 
 we may write 
 
 ao = «o - KG, /3o = to + Ko, X = 2to, /i = Ko, £/ = to* + Ko' ; 
 and the required transformation is (comp. 336, XL), 
 
 XXVIIL . . N(iop + p/co) = (to2+ 'co«)p2 + 2SiopKop. 
 (11.) To treat the case XXIV. by our general method, we may omit for simpli- 
 city the subindices 0, and write simply (comp. V. and VI.) the expressions, 
 
CHAP. II.] PASSAGE FROM FOCAL TO CYCLIC. 477 
 
 XXIX. . . ^p = - aVap + /3Si3p, and XXX. . . Sp^p = (Vap)2 + (Si3p)2 ; 
 in which however it is to be observed that a and j3, though real vectors, are not now 
 unit lines (8.). Hence because - aVap = aSap - a?p^ we easily form the expres- 
 sions : 
 
 XXXI. . . m = a2 (Sai3)2, m = a^ (a^ - 132) - (Sa/3)2 , m" = f5^ - 2a2 ; 
 
 XXXII. . . j = Vap/3Sa/3 + a (a^ - ^2j Sap, 
 
 ( XP = - (aSap + i8S/3p) + (/32 - ««) p ; 
 and therefore XXXIII. .. M=(c-a^) (c^ + (j32 - a«)c - (Sa/3)2), 
 and XXXIV. . . ^p = Yap(3Saf3 + (fP _ a2) (cp - aSap) -claSap + /3S/5p) + c2p 
 = (a(a2 - |82 - c) + /3Saj8) Sap + (aSa/3 - c/3) S(3p + (c2 + (/32 -a^)c- (Sa(3y)p. 
 (12.) Introducing then a real and positive scalar constant, r, such that 
 XXXV. . . r* = (a2 - /32)2 + 4(Sa/3)2 = (a^ + /32)2 -f 4 (Ya(3y 
 = a* + (a/3)2 + (/3a)2 + )8< = a* + 2S. (a/3)2 + ^S* 
 = a-2 (a3 + /3a;3)2 = /3-2 (/S^ + a)3a)2 = &c., 
 in which (by 199, &c.), 
 
 S . (a/3)2 = (Sa/3)2 + (Va/3)2 = 2 (Sa(3y - a^^-i = 2 (Va/3)2 + a2|8a, 
 the roots oi AI=0 admit of being expressed as follows : 
 
 XXXVI. . . ci = i («^ - i3^ + »•'), Co = a2, C3 = ^ (a2 - /32 - »•«) ; 
 and when they are thus arranged, we have the inequalities, 
 
 XXXVII. . . Ci > > C3 > C2, ci-i > > C2-1 > C3-1. 
 (13.) The corresponduig forms of ^p are the three monomial expressions, 
 
 T Y Y VTTT f'^'P = '^^''^'"'^ + '^^"^^ ^ ^""'^ + ^^"^^ <"' '^'^ " Va/3S/3ap, 
 js^JLViii. . . |^3p^^^-,(„^^^^Sa/3)S(aci + i8Sap)p; 
 
 which may be variously transformed and verified, and give the three following rect- 
 angular vector units, 
 
 XXXIX. .. ai = U(aC3 + /3Sa;8), a2 = UVa(3, a3 = U(aci + i8Sa/3) ; 
 in connexion with which it is easy to prove that 
 
 ( T (ac3 + (5Sa[3) = (- ca)^ {d. - c%f (<?i - ^3)^ = r (ci - ci)^ (- ^3)^ 
 
 XL. . . j TVa/3 = (ci - C2)i (<?3 - ^2)^ ; 
 
 the radicals being all real, by XXXVII. 
 
 (14.) "We have thus, for the given focal form XXX., the rectangular transfer' 
 mation, 
 
 XLL . . Sp^p = (Vap)2 + (S/3p)2 
 
 _ ci(S(ag3+i8Sa/3)p)2 C2(Sai8p)2 f3(S(flCi + /3Sa/3)p)2 
 
 C3 (<;i - C2) r2 ^ ^ (Ci - C2) (^3 - C2) ' Ci (C3 - C2) r^ 
 
 or briefly, 
 
 XLII. . . Sp0p = (Vap)« + (S/3p)2 = tfx (S . pU(ac3 + l3Sa(5)py 
 
 + a2(S.pUVa/3)« + C3(S.pU(aci + i8Sai3))2 ; 
 
 in which the first term is positive, but the two others are negative, and ci, 03 are 
 the roots of the quadratic, 
 
 • XLin. . . = c2+ (i32 - a2)(j- (Sa^y-. 
 
478 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (15.) We have also the parallelisms, 
 
 XLIV. . . aC3 + l5Sa(3 \\ jBci - aSajS, aci + /3Sa/3 || (Sc^ - aSa(3, 
 because cics = - (Sa/S)^ ; 
 
 and may therefore write, 
 
 XLV. . . Sp0p = (Yapy + (S(3p)^ =Ci{S. pV(l3ci - aSa/3))2 
 
 + a2 (S. |oUVa/3)3 + C3(S. pU(/3c3 - aSaj8))2 ; 
 while 
 
 XLVI. . . T(j8ci-aSa|3) = r(?ii(ci-C3)i, T ((3cs - aSa(3) = r (^- e^y (03-02^, 
 
 and r = (ci— cs)^, with real radicals as before. 
 
 (16.) Multiplying then by r^(TVa(3)^, or by (ci - C2) (ci-cz) (C3-C2), we ob- 
 tain this new equation, 
 
 XLVII. . . (tfi - cs) {(TVa/3)2 ({Yapy + (Si3p)2) - a2 (Sa/3p)2} 
 
 = (tf3- a2) (ciS/3p - Sa/3Sap)3 - (ci - a2) (cgS/Bp - aSa/3)2 ; 
 which is only another way of expressing the same rectangular transformation as be- 
 fore, but has the advantage of being freed from divisors. 
 
 (17.) Developing the second member of XLVII., and dividing by C1-C3, we 
 obtain this new transformation : 
 
 XLVIII. . . (TVa/3)2 SpPp = - (Va/3)2 {{Yapy + (SfSp^) 
 
 = a2 (Sa/3p)2 - (Sa/3)2 (Sap)^ + 2a2Sa/3SapS/3p + C(Sj3p)2 ; 
 in which we have written for abridgment, 
 
 XLIX. . . C= C1C3 - a2 (ci + C3). 
 (18.) The expressions XXXVI. for ci, C3 give thus, 
 L. . . C=-a*-(Va/3)2; 
 
 and accordingly, when this value is substituted for C in XLVIII., that equation 
 becomes an identity^ or holds good for all values of the three vectors^ a, /3, p ; as 
 may be proved* in various ways. 
 
 (19.) Admitting this result, we see that for the mere establishment of the equa- 
 tion XLVII. , it is not necessary that ci and cg should be roots of the particular qua- 
 dratic XLIII. It is sufficient, for this purpose, that they should be roots of any qua- 
 dratic, 
 
 LI. . . c2 + ^c + i? = 0, with the relation LII. . . Aa^ + ^ + a* + (Ya^y = 0, 
 between its coefficients. But when we combine with this the condition ofrectangu- 
 larity, 03 -*- ai, or 
 
 LIII. . . = S . QcijS - aSa/3) (cs/S - aSa/3) = A (Sa^S)? + B(3^ + a« (Saj8)2, 
 
 we obtain thus a second relation, which gives definitely, for the two coefficients, the 
 values, 
 
 LIV. . . ^ = /32 - a', B=- (Sa/3)2 ; 
 
 and so conducts, in a new way, to the equation XLIII. 
 
 * Many such proofs, or verifications, as the one here alluded to, are purposely 
 left, at this stage, as exercises, to the student. 
 
CHAP. II.] IDENTITIES, REAL AND IMAGINARY, 479 
 
 (20.) In this manner, then, we might have been led to perceive the truth of the 
 rectangular transformation XLVII., with the quadratic equation XLIII. of which 
 ci and C3 are roots, without having previously found the cubic XXXIII., of which 
 the quadratic is a factor^ and of which the other root is co, = a^. But if we had not 
 employed the general method of the present Section, which conducted us to form Jirst 
 that cubic equation, there would have been nothing to suggest the particular form 
 XLVII., which could thus have only been by some sort of chance arrived at. 
 
 (21.) The values of aia2az give also (comp. 357, VII.), 
 
 LV. . . -p2 = (S.pU(i8ci-aSa/3))H (S.joUVa/3)2 + (S.pU(|8c3- aSa/3))2 ; 
 that is, by XL. and XLVL, 
 
 LVI. . . ciC3(ci-C3) (pKVa/3)2-(Sai3p)2) = C3(c3-a2) (ciS/3p - Sa^Sa|o)2 
 
 - ci (ci - a2) (caS^p - SafSSap) ; 
 
 and accordingly the values XXXVI. of ci, C3 enable us to express each member of 
 this last equation under the common form, — ci 03(01 — C3) (aS^p — j8Sap)2. 
 
 (22.) Comparing the recent inequalities ci>C3>C3 (XXXVII.) with the ar- 
 rangement 357, IX., we see, by 357, (6.), that for the real cyclic transformation 
 (6.) at present sought, the plane of X, fi is to be perpendicular to 03 (and not to 02, 
 as in 357, (3.), &c.). We are therefore to eliminate (csSjSp — SajSSap)^ between 
 the equations XLVII. and LVI., which gives (after a few reductions) the real trans- 
 formation : 
 
 LVIL . . ((Sa/3)2- Ci/32) ((Vap)2+ (S/3p)2) - (oi - a2) (Sa^)2p2 
 
 = (ciS(3p - Saj3Sap)2 - a (Sa(3py 
 
 = S . p (ci/3 - aSa^ + Ci^VajS) S . p (ci/3 - aSa/3 - Ci^Va/S) ; 
 
 which is of the kind required. 
 
 (23.) Accordingly it will be found that the following equation, 
 
 LVIII. . . ((Sa/3)2-cj32) (Vap)2 + (c - a2) (c(S/3p)2 - p2S(a/3)2) 
 
 = (cS/3p - Sa/3Sap)2 - c(Sa(3p)\ 
 is an identity, or that it holds good for all values of the scalar c, and of the vectors 
 a, (3, p; since, by addition of c(Yal3)^p^ on both sides, it takes this obviously iden- 
 tical form, 
 
 LIX. . . ((Sa/3)2 - c^2) (Sap)2 + c(e - a2) (S/3p)2 =. (cS/3p - Saf^Sap^ 
 
 -c(aS/3p-|8Sap)^; 
 so that if ci be either root of the quadratic XLIIL, or if ci(ei - a^) = (Sa/3)2 - cij3^, 
 the transformation LVIL is at least symbolically valid : but we must take, as above, 
 the positive root of that quadratic for ci, if we wish that transformation to be a real 
 one, as regards the constants which it employs. And if we had happened (comp. 
 (20.)) to perceive this identity LIX., and to see its transformation LVIII., we 
 might have been in that way led to form the quadratic XLIIL, without having 
 previously formed the cubic XXXIII. 
 
 (24.) Already, then, we see how to obtain one of the two imaginary cyclic trans- 
 formations of the given focal form XXX., namely by changing ci to C3 in LVIL ; 
 and the other imaginary transformation is had, on principles before explained, by 
 eliminating (SafSpy between XLVII. and LVI. ; a process which easily conducts to 
 the equation, 
 
480 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 LX. . . (Yapy + (S/3p)2 + a2/)2 = (ci - ca)-' {cr'(cS/3p - SajSSap)^ 
 
 -C3-i(c3S)3/o-Sa/3Sap)2}, 
 
 where the second member is the sum of two squares (ci being > 0, but cz < 0), as the 
 second expression LVII. would also become, if ci were replaced by C3. Accordingly, 
 each member of LX. is equal to (Sap)2 + (Sj3p)3, if ci, C3 be the roots of any quadra- 
 tic LI., with only the one condition, 
 
 LXL . . ciC3 = 5 = -(Sa/3)2; 
 which however, when combined with the condition of rectangularity LIIL, suffices 
 to give also J. = j82 — a', as in LIV., and so to lead us back to the quadratic XLIII., 
 which had been deduced by the general method, as & factor of the cubic equation 
 XXXIIL 
 
 (25.) And since the values XXXVI. of ci, C3 reduce, as above, the second mem- 
 ber of LX. to the simple form (Sap)2 + (S/3jo)2, we may thus, or even without em- 
 ploying the roots ci, cz at all, deduce the following expression for the last imaginary 
 cyclic transformation : 
 
 LXII. . . Sp^p = (Vap)2 + (S)3p)2 = - a2p2 + S (a + a/^/3) p . S (a - \/^/3) p, 
 
 where ^/— 1 is the imaginary of algebra (comp. 214, (6.)) ; while the real scalar 
 r* of XXXV. may at the same time receive the connected imaginary form , 
 
 LXIII. . . r4 = (a2 - /32)2 + 4 (Sa/3)5 = (a + V^l^f (a - \/~li3)2. 
 
 (26.) Finally, as regards the passage from the given form XXX., to a second 
 real focal form (comp. 358, (4.) ), or the transformation, 
 
 LXIV. . . (Vap)2+ (S/3p)2= (Va'p)2 + (S/3'p)2, 
 in which a' and j8' are real vectors, distinct from + a and + /3, but in the same plane 
 with them, it may be sufficient (comp. 358, (8.) ), to write down the formulae : 
 
 LXV. . . r2a' = - (a' + jSa/S), r2/3' = - (i33 + a/3a), 
 
 with the same real value of r' as before ; so that (by XXXV., &c.) we have the 
 relations, 
 
 LXVI. . . Ta' = Ta, T/3' = T/3, Sa'/3' = Sa/3 ; 
 rr2(a + a') = a(r2-a2 + /32)-2/3Sa/3 = -2(ac3 + ^Saj8)[|ai, 
 \r2(a -a')^a (r^ + a^ - fi^) + 2(3Sa(3 = 2 (an + f3Sa(3) \\ az ; 
 >2(/3 + /3') = /3(r2 + a2 - j82) - 2aSa|3 = 2 (|3ei - aSa/3) II ai, 
 ! (|3 - /3') = /3 (r2 - a2 + j82) + 2aSa/3 = - 2 (/Sca - aSaj3) || as- 
 
 (27.) We have then the identity, 
 LXIX. . . (V(a3 + /3ai3)p)2 + (S(i33 + a^a)py 
 
 = (a4+2S.(a/3)^ + ^0 ((Vap)H (S^p)^); 
 with which may be combined this other of the same kind, 
 
 LXX. . . - (V(a3 - /3a/3)p)2 + (S(i33 - a^a)py 
 
 = (a4 - 2S. (a/3)2 + /30 (-(Vap)2 + (S/3p)2), 
 
 which enables us to pass from the focal form XXIIL, to a second real focal form, 
 with its two new lines in the same plane as the two old ones : and it may be noted 
 that we can pass from LXIX. to LXX,, by changing a to a\/- 1. 
 
 rr2( 
 
 Lxvm. . . 1^,; 
 
CHAP. II.] BIFOCAL AND MIXED TRANSFORMATIONS. 481 
 
 360. Besides the rectangular, cyclic, and focal transformations 
 of S/30/>, which have been already considered, there are others, al- 
 though perhaps of less importance: but we shall here mention only 
 two of them, as specimens, whereof one may be called the Bifocal^ 
 and the other the Mixed Transformation. 
 
 (1.) The two lines a, a', of 359, LXV., being called /oca? lines,* an expression 
 which shall introduce them both may be called on that account a bifocal transforma- 
 tion. 
 
 (2.) Eetaining then the value 359, XXXV. of r*, and introducing a new auxi- 
 liary constant e, which shall satisfy the equation, 
 
 I. . . /32 - a2 = r^e, and therefore II. . . 4 (Sa(3y = H (1 - e"^), 
 so that III. . . 4c2 (Sa(Sy= (1 - e^) (/32 - a2)2, 
 
 the first equation 359, LXV. gives, 
 
 IV. . . r2 (ea - a') = 2/3S«j3, V. . . r^ (eSap - Sa» = 2S«/3S/?p ; 
 and therefore, with the form 359, XXX. of Sp^p, 
 
 VI. . . (1 - e2) Spi>p = (1 - c2) ((Vap)2 + (S/3p)2) 
 
 = (1 - e2) (Vap)2 + (eSap - Sa'p)2 
 = (e2 - i; rt2p2 -1- (Sap)2 - 2eSapSa'p + (Sa'p)2 ; 
 in which ft2 = a'2, by 359, LXVL, so that a and a' may be considered to enter st/m- 
 metrically into this last transformation, which is of the bifocal kind above men- 
 tioned. 
 
 (3.) For the same reason, the expression last found for Sp0p involves again 
 (comp. 358) six scalar constants; namely, e, Ta(=Ta'), and the four involved in 
 the two unit lines, Ua, Ua'. 
 
 (4.) In all the foregoing transformations, the scalar and quadratic function Sp0p 
 has been evidently Jwmogeneous, or has been seen to involve no terms below the se- 
 cond degree in p. We may however also employ this apparently heterogeneous or 
 mixed form, 
 
 VII. ..Sp^p=y(p-£)2 + 2SX(p-OSp(p-0 + e; 
 
 in which g\ X, ^ have the same significations as in 357, but e, t, K are three new 
 constants, subject to the two conditions of homogeneity, 
 
 YIU. . . g's + \SfiK + fJiSXK = 0, 
 and IX. . . g't^ + 2SX^Sp^ + e = 0, 
 
 in order that the expression VII. may admit of reduction to the form, 
 X. . . Sp0p =/p2 + 2SXpS^p, as in 357, If. 
 (5.) Other general homogeneous transformations of Sp0p, which are themselves 
 real, although connected -with, imaginary^ cyclic forms (comp. 357, (7.)), because 
 
 * Compare the Note to Art. 359. 
 
 t Xi + ^Z— 1 pi, and X3 + ^- 1 p3, may here be said to be two pairs 0^ ima- 
 ginary cyclic normals, of that real surface of the second order, of which the equa- 
 tion is, as before, Sp^p = const. Compare the Notes to pages 4G8, 474. 
 
 3 Q 
 
482 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 a sum of two squares of linear and scalar functions is, in an imaginary sense, a pro' 
 duct of two such functions, are the two following (comp. 357, (9.) ) : 
 
 XI. . . Sp0p=5rp2 + SXp/ip= 5^1102 +(S\ip)2 + (S/iip)2; 
 
 XII. . . Spipp = ^p2 + SXpfxp = pip^ - (SX3/t))2 - (S/i3p)2 ; 
 in which (comp. 357, (2.) and (8.) ), 
 
 XIII. . .ffir=p-^TXfl = -Cl, p3=p-T\fl = -C3, 
 
 XIV. . . Xi = VX/i (TX/i - SX/x)-i, fii = (XT/x + fiTX) (TX/z - SX/4)-i, 
 and XV. . . X3 = VX/x (TXju + SX;i)-i, nz = (XT/i - /xTX) (TX/t + SX/>i)-i ; 
 so that gi, Xi, ^1, and ^3, X3, fis are reaZ, if ^r, X, fi be such. 
 
 (6.) We have therefore the two new mixed transformations following : 
 
 XVI. . . Sp0p=^i(p-£i)2+(sxi(p-^x)yH(S;ii(p-?i))2 + ei; 
 
 XVII. . . Sp0p = 5r3(p-£3)2-(SX3(p-^3))2-(S^3(p-?3))2 + e3; 
 with these two new pairs of equations, as conditions of homogeneity, 
 XVIII. . . giti + XiS^iXi + iiSlm = 0, 
 XIX. . . «7if i2 + (S?iXO» + (S:iiui)2 + ei = 0, 
 and XX. . . <73f3 - X3S^3X3 - //sS^sjwa = 0, 
 
 XXI. . . ^3*32 - (8^3X3)2 - i^flzKzy + 63 = 0. 
 
 361. We saw, in the sub-articles to 336, that the diffe- 
 rential, d/)b, of a scalar function of a vector, may in general be 
 expressed under the form, 
 
 I. . . d^ = wSvdjO, 
 where i/ is a derived vector function, of the same variable vec- 
 tor |0, and w is a scalar coefficient. And we now propose to 
 show, that if 
 
 II. . ,fp = Sp(j)p, 
 
 (pp still denoting the linear and vector function which has been 
 considered in the present Section, and of which ^op is still the 
 self-conjugate part, we shall have the equation I. with the va- 
 lues, 
 
 III. . . w = 2, v = (Pop; 
 so that the part (pop may thus be deduced from <pp by operat- 
 ing with |dS.|o, and seeking the coefficient of d/o under the 
 sign S. in the result: while there exist certain general rela- 
 tions of reciprocity (comp. 336, (6.)), between the two vectors 
 p and V, which are in this way connected, as linear functions of 
 each other. 
 
 (1.) We have here, by the supposed linear form of cpp, the differential equation 
 (comp. 334, VI.), 
 
 IV. . . d^p = ^dp ; 
 
CHAP. II.] RECIPROCITY OF FORMS. 483 
 
 also S(dp.^p)=S(0|O.dp), and S(p.^dp) = SC^'p.dp) ; 
 
 heuce, by 349, XIII., we have, as asserted, 
 
 V. . . dSp^p = S{<pp+ (p'p) dp = 2S . 0opdp. 
 
 (2.) As an example of the employment of this formula, in the deduction of ^op 
 from 0p, let us take the expression, 
 
 VI. . .<pp = S/3Sap, 347, XXXI., 
 
 which gives, 
 
 VII. . . /p = Sp0p = 2SapS(3p, 
 and therefore 
 
 VIII. . . d/p = 2(/3Sap + aS/3p)dp. 
 
 mparing this with the general formula, 
 
 IX. . . |d/p = Si/dp = S. 0opdp, 
 we find that the form VI. of 0p has for its self-conjugate parf, 
 
 X. . . v= 0o|O = |2 ((3Sap + aS/3p) ; 
 and in fact we saw (347, XXXII.) that this form gives, as its conjugate, the ex- 
 pression, 
 
 XI. . . fp = 2aS/3p. 
 
 (3.) Supposing now, for simplicity, that the function tp is given, or made, self- 
 conjugate, by taking (if necessary) the semisum of itself and its own conjugate func- 
 tion, we may write ^ instead of 0o> and shall thus have, simply, 
 
 XII. ..v = <Pp, XIII. . . /p = Svp, XIV. . . d/p = 2Svdp ; 
 whence also (comp. 348, I. II.), 
 
 XV. . . p = 0-V = m-»i//i/, and XVI. . . Sj^dp = Spdj/. 
 (4.) Writing, then, 
 
 XVII. . . Fv = Sv0-V = m-iSj/i/'V, 
 we shall have the equations, 
 
 XVIII. . . Fv =fp, XIX. . . dFv = 2Spdj/ = 2S. ^"ivdi/ ; 
 so that p may he deduced from Fv, as v was deduced from fp ', and generally, as 
 above stated, there exists a perfect reciprocity of relations, between the vectors p and 
 V, and also between their scalar functions, fp and Fv. 
 
 (5.) As regards the deduction, or derivation, of v from/p, and of p from Fv, it 
 may occasionally be convenient to denote it thus : 
 
 XX. . . 1/ = i (S . dp)-id/p ; XXI. . . p - KS . dv)-^dFv ; 
 
 in fact, these last may be considered as only symbolical transformations of the ex- 
 pressions, 
 
 XXII. . . d/p = 2S (dp. v), dFv = 2S(dv. p), 
 
 which follow immediately from XIV. and XIX. 
 
 (6.) As an example of the passage from an expression such as fp, to an equal 
 
 expression of the reciprocal form Fv, let us resume the cyclic form 357, II., writing 
 
 thus, 
 
 XXIII. . . /p = Sp0p = gp'- + SXppp, 
 
 and supposing that g., \, and p. are real. Here, by what has been already shown (in 
 sub-articles to 354 and 357), if ^p be supposed self-conjugate, as in (3.), we have, 
 
 * 
 
484 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XXIV. . . v = (pp=gp + YXpfji ; 
 
 XXV. ..m = (g- SXfi) (^2 _ X2/,2) = _ c,c2C3 ; 
 
 XXVI. . . xPv = YXvnS\n-Y\ixSXvn-g(\Sixv-^nSXv)^g^v; 
 
 and therefore 
 
 XXVII. . . mFv = Spx^v 
 = SXvfivQXfi + (S\v/Li)2 - 2gSXvSfip + g^v^ 
 = (g-i-X^fi^) v^ + X2(S/iiv)2 + /«2(sXv)2 - 2^S\x'S/iv ; 
 which last, when compared with 360, VI., is seen to be what we have called a bifo- 
 cal form : its focal lines a, a' (360, (1.)) having here the directions of \, ju, that is 
 of what may be called the cyclic lines* of the /orm XXIII, The ct/clic and bifocal 
 transformations are therefore reciprocals of each other. 
 
 (7.) As another example of this reciprocal relation between cyclic and focal lines, 
 in the passage from fp to Fp^ or conversely from the latter to the former, let us now 
 begin with the focal form, 
 
 XXVItl. . .fp= Sp<}>p = (Yapy + (S/3p)2, 359, XXX., 
 
 in which a and (5 are supposed to be given and real vectors. We have now, by 359, 
 
 fp = <l>p = -aYap + ^S(3p, m = a2(Sa/3)2, 
 
 and therefore, 
 
 XXX. . . mFp = a2(Sa^)2 Fp = Spx^p 
 
 = Sapf3pSa(3 + (a* - /32) (Sa py 
 
 = - 1.2 (Sa/3)2 -h Sai/((a2 - /32) Sa J/ -f 2Sa/?S/3j/) 
 
 = -p\Sa(3y + SavS(a^ + f3a^)p, 
 
 an expression which is of cyclic ybrm ; one cyclic line oi Fp being the given focal 
 line a of fp ; and the other cyclic line of Fp having the direction of 4; (a^ + /3«j3), 
 and consequently (by 359, LXV.) of Hha', where a' is the second real and focal line 
 of/0. 
 
 (8.) And to verify the equation XVIII., or to show by an example that the two 
 functions fp and Fp are equal in value, although they are (generally) different in 
 form, it is suflficient to substitute in XXX. the value XXIX. of p ; which, after a 
 few reductions, will exhibit the asserted equality. 
 
 362. It is often convenient to introduce a certain scalar and sym- 
 metric function of two independent vectors, p and p% which is linear 
 with respect to each of them, and is deduced from the linear and 
 self-conjugate vector function 0/3, of a single vector p, as follows: 
 
 I. . 'f(P^ P') =f(p\ P) = ^P'^P = ^P^P'- 
 With this notation, we have 
 
 * They are in fact (compare the Note to page 468) the cyclic normals, or the 
 normals to the cyclic planes, of that surface of the second order, which has for its 
 equation /p = const. ; while they are, as above, the focal lines of that other or re- 
 ciprocal surface, of which p is the variable vector, and the equation is Fp - const. 
 
CHAP. II.] DERIVED LINEAR FORMS, 485 
 
 11. ../(/> + p') =fp + 2/(/>, p') +fp^ ; 
 III. . . /(/>, / + p^') =f{p, p') ^f{p, p") ; 
 IV. . ^f{p,p)-fp; V, . . d//> = 2/(/>,d/>); 
 VI. . . f{xp, yp) = xyf{p, p% if Vx = Yy=0; 
 
 and as a verification, 
 
 YIL.. f(xp):=x^fp, 
 
 a result which might have been obtained, without introducing this 
 new function I. 
 
 (1.) It appears to be unnecessarj'-, at this stage, to write down proofs of the fore- 
 going consequences, II. to VI., of the definition I.; but it may be worth remarking, 
 that we here depart a little, in the formula V., from a notation (325) which was 
 used in some early Articles of the present Chapter, although avowedly only as a 
 temporary one, and adopted merely for convenience of exposition of the principles of 
 Quaternion Differentials. 
 
 (2.) In that provisional notation (comp. 325, IX.) we should have had, for the 
 differentiation of the recent function /p (361, II.), the formulae, 
 
 d/P=/(p>dp), /(p,p') = 2SpVp; 
 the numerical coeflScient being thus transferred from one of them to the other, as 
 compared Avith the recent equations, I. and V. But there is a convenience now in 
 adoptmg these last equations V. and I., namely, 
 
 d/P=2/(p, dp), f(p,p') = Sp'<Pp', 
 because ihla function Sp'<pp, or Sp^p', occurs frequently in the applications of qua- 
 ternions to surfaces of the second order, and not always with the coefficient 2. 
 
 (3.) Retaining then the recent notations, and treating dp as constant, or d^p as 
 null, successive differentiation of/p gives, by IV. and V., the formulae, 
 
 VIII. . . d2/p = 2/(dp) ; d3/p = ; &c ; 
 so that the theorem 342, I. is here verified, under the form, 
 IX. ..6yp = (l + d+id2)/p 
 
 =//> + 2/(p,dp)-|-/dp; 
 or briefly, X. . . e'^fp =f(p + dp), 
 
 an equation which by II. is rigorously exact (comp. 339, (4.)), without any suppo- 
 sition whatever bemg made, respecting any smallness of the tensor^ Tdp. 
 
 36.3. Linear and vector functions of vectors, such as those con- 
 sidered in the present Section, although not generally satisfying the 
 condition of self-conjugation^ present themselves generally in the dif- 
 ferentiation of non-linear but vector functions of vectors. In fact, if 
 we denote for the moment such a non-linear function by w(/>), or 
 simply by icp^ the general distributive property (326) of differential 
 expressions allows us to write, 
 
 I. . . dtt?(/>) = (/)(^p)i or brielly, 1'. . . dw/j = <56d/); 
 
486 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 where has all the properties hitherto employed, including that of 
 not hQing generally self- conjugate, as has been just observed. There 
 is, however, as we shall soon see, an extensive and important case, 
 in which the property of self-conjugation exists, for such a function 
 0; namely when the differentiated function, wp, is itself thQ result v 
 of the differentiation of a scalar function fp of the variable vector /», 
 although not necessarily a function of the second dimension, such as 
 has been recently considered (361); or more fully, when it is the 
 coefficient of d/>, under the sign S., in the differential (361, I.) of 
 that scalar function//), whether it be multiplied or not by any sca- 
 lar constant (such as n, in the formula last referred to). And gene- 
 rally (comp. 346), the inversion of the linear and vector function 
 in I. corresponds to the differentiation of the inverse (or implicit) func- 
 tion u)-^ ; in such a manner that the equation I. or F. may be writ- 
 ten under this other form, 
 
 II. . . dw-'o- = 0-'dff = mr^-^dia, if o- = ivp, 
 
 (1.) As a very simple example of a non-linear but vector function, let us take 
 the form, 
 
 III. . . (T = io{p) = pap^ where a is a constant vector. 
 
 This gives, if dp = p', 
 
 IV. . .<pp' = 0dp = dwjo = p'ag + pap' — 2Ypap' ; 
 
 V. . . 8\(Pp' = 2S\pap' = Sp<p'X ; 
 VI. . . 0'X = 2 VXpa = 2 VapX, (p'p' = 2Vapp' ; 
 
 so that 0p' and (p'p' are tmequal, and the linear function (pp' is not self- conjugate. 
 
 (2.) To find its self-conjugate part (pop', by the method of Art. 361, we are to 
 form the scalar expression, 
 
 VII...|/p' = |SpV = P''Sap; 
 of which the differential, taken with respect to p', is 
 
 VIII. . . id/p' = S . ^op'dp' = 2SapSp'dp', giving IX. . . 0op' = 2p'Sap ; 
 
 and accordingly this is equal to the semisura of the tw^o expressions, IV. and VI., for 
 (pp' and its conjugate. 
 
 (3.) On the other hand, as an example o/thc self- conjugation of the linear and 
 vector function, 
 
 X. . . dv = dwp = 0dp, when X'. . . d/p = 2Si^dp = 2S.wpdp, 
 
 even if the scalar function fp be of a higher dimension than the second, let this 
 last function have the form, 
 
 XI. • 'fp = ^9P9PQ"Pi 9i 9% 9" being three constant quaternions. 
 Here XII. . . v = cjp- h^{9P9'p9" + 9P9'p9 + 9"p9P9') ; 
 
 XIII. . . 6v = (p6p=<pp=^Y(qp'qpq"-\^q'pq"p'q) + iYCq'p'q"pq + 9"p9p'9') 
 
 + ^y(9"p'9P9' -\- 9P9P'9") i 
 
CHAP. II.] LINEAR FUNCTION OF A QUATERNION. 487 
 
 and XIV. . . SX^p' = |S . q'pq"(\qp' + p'?^) + &c- = Sp'cpX ; 
 
 so that 0' = 0, as asserted. 
 
 (4.) In general, if 8 be used as a second and independent symbol of differentia- 
 tion, we may write (comp. 345, IV.), 
 
 XV. .. ddfq = dSfq, 
 where /g' may denote any function of a quaternion ; in fact, each member is, by the 
 principles of the present Chapter (comp. 344, I., and 345, IX.), an expression for 
 the limit,* 
 
 XVI. . . lim. nn'{f(q + n-^dq + n'-^Sq) -f(^q + n-^dq) -f{q + n'-i^^) -^fq } . 
 
 (5.) As another statement of the same theorem, we may remark that a first dif- 
 ferentiation of /j, with each symbol separately taken, gives results of the forms, 
 
 XVII. . . dfq =f{q, dq\ dfq =f(q, Sq) ; 
 and then the assertion is, that if we differentiate the first of these with d, and the se- 
 cond with d, operating only on q with each, and not on dq nor on dq, we obtain 
 equal results, of these other forms, 
 
 XVIII. . . ddfq=f(q, dq, dq) =f(q, Sq, dq) = ddfq. 
 
 For example, if 
 
 XIX. . .fq = qeq, where c is a constant quaternion, 
 the common value of these last expressions is, 
 
 XX. . . 8dfq = d8fq^-^.C.dq-Vdq.C.dq. 
 (6.) Writing then, by X., 
 
 XXI. . . dfp = 2Swpdp, 8fp = 2So)p8p, 
 and XXII. . . dcjp — <p8p, with dwp = (jtdp, as before, 
 
 we have the general equation, 
 
 XXIIL . . S(dp.^^p) = S(^p.^dp), 
 in which dp and 8p may represent atiy two vectors ; the linear and vector function, 
 0, which is thus derived from a scalar function fp by differentiation, is therefore (as 
 above asserted and exemplified) always self-conjugate, 
 (7.) The equation XXIII. may be thus briefly written, 
 XXIV. . . Sdp^v=S^pdj/; 
 and it will be found to be virtually equivalent to the following system of three known 
 equations, in the calculus of partial differential coeflScients, 
 
 XXV. . . T>jyy = DyD;,, Dj/Ds = D^Dj,, DzT>x = I>xT)z. 
 
 364. At the commencement of the present Section, we 
 reduced (in 347) the problem of the inversion (346) of a linear 
 (or distributive) quaternion function of a quaternion, to the 
 
 * We may also say that each of the two symbols XV. represents the coeflicient 
 of aj'y', in the development o^f{q + xdq-\-ydq) according to ascending powers of a; 
 and y, when such development is possible. 
 
488 ELEMENTS OF QUATERNIONS. [bOOK IIT. 
 
 corresponding problem for vectors; and, under tliis reduced 
 or simplified form, have resolved it. Yet it may be interest- 
 ing, and it will now be easy, to resume the linear and quater- 
 nion equation, 
 
 I...fq = r, with lL.-f(q + q')-fq+fg', 
 and to assign a quaternion expression for the solution of that 
 equation, or for the inverse quaternion function, 
 
 111. ..q^f-^r, 
 with the aid of notations already employed, and of results al- 
 ready established. 
 
 (1.) The conjugate of the linear and quaternion function yj being defined (comp. 
 347, IV.) by the equation, 
 
 lY,.. Spfq = Sqfp, 
 
 in which p and q are arbitrary quaternions, if we set out (comp. 347, XXXI.) with 
 
 the form, 
 
 Y. . .fq =tqs + t'qs' + . . = S/js, 
 
 in which s, s\ . , . and t, /',... are arbitrary but constant quaternions^ and wliich is 
 more than sufficiently general, we shall have (comp. 347, XXXII.) the conjugate 
 
 form, 
 
 VI. . .f'p = spt + s'pt' + . . . = Ilspt ; 
 whence VII. . . /I = 2^s, and VIII. . . /'I = 2sf ; 
 
 it is then possible, for each given particular form of the linear function /^j to assign 
 one scalar constant e, and two vector constants, f, e', such that 
 
 IX. ../l = c + c, fl=e+s'', 
 and then we shall have the general transformations (comp. 347, I.) : 
 X. . .Sfq = S.qfl = eSq + St'q ; 
 XL ..Yfq = tSq + Y.fYq ^t^q + fYq] 
 and XII. . . A = (e + c) 8q + S^'^ + <l>Yq ; 
 
 in which Sis'q = S.e'Yq, and (pYq or YfYq is a linear and vector function of Yq, of 
 the kind already considered in this Section ; being also such that, with the form V. 
 
 of /<?, we have 
 
 XIII. . .<Pp = -EYtps. 
 
 (2.) As regards the number of independent and scalar constants which enter, at 
 least implicitly, into the composition of the quaternion function /5, it may in various 
 ways be shown to be sixteen; and accordingly, in the expression XII., the scalar e 
 is one; the two vectors, e and e', count each as three; and the linear and vector 
 function, <pYq, counts as nine (comp. 347, (!.))• 
 
 (3.) Since we already know (347, &c,) how to invert a function of this last kind 
 ^, we may in general write, 
 
 XIV. . . r = Sr + Vr = Sr + 0|O, where XV. . . p = <p-'^Yr = m-'i>pYr ; 
 the scalar constant, m, and the auxiliary linear and vector function, i//, being deduced 
 
CHAP. II.] PROBLEM OF QUATERNION INVERSION. 489 
 
 from the function ^ by methods already explained. It is required then to express 9, 
 or Sq and Yq, in terms of r, or of Sr and p, so as to satisfy the linear equation, 
 
 XVI. . . (e + £)Sg' + Se'9 + 0Vg = Sr + 0p; 
 the constants e, e, f ', and the form of (p, being given. 
 (4.) Assuming for this purpose the expression, 
 
 XVIl...q = Q'+p, 
 
 in which q' is a new sought quaternion, we have the new equation, 
 
 XVIII. ../g'=Sr+0p-/p = S(r-£'p); 
 whence XIX. . . q'=S(r-e'p).f-n, 
 
 and XX. . . 5' = p + S(r~e'p)./-il; 
 
 iu which p is (by supposition) a known vector, and S(r- e'p) is a known scalar; so 
 that it only remains to determine the unknown but constant quaternion, /''I, or to 
 resolve the particular equation, 
 
 XXI. . . /gro = 1, in which XXII. . . ^0 = c + y =/'! 1 , 
 c being a new and sought scalar constant, and y being a wczt; and sought vector con- 
 stant. 
 
 (5.) Taking scalar and vector parts, the quaternion equation XXI. breaks up 
 into the two following (comp. X. and XI.) : 
 
 XXIII... 1 = S/(c + y) = ec -f St'y ; XXIV.. . = V/(c + y) = £C + 0y ; 
 which give the required values of c and y, namely, 
 
 XXV. . . c = (e - S£>-'0" S and XXVI. . . y = - c^p'h ; 
 whence XXVII. ../-'!= ^ "" f '' ; 
 
 and accordingly we have, by XII., the equation, 
 
 XXVIII. . . /(I - 0-i£) = e - Ssy •£ = V-iO. 
 (6.) The problem of quaternion inversion is therefore reduced anew to that of 
 rector inversion, and solved thereby ; but we can now advance some steps further, 
 in the elimination of inverse operations, and in the suhstitution for them of direct 
 ones. Thus, if we observe, that tp~'^ =m-i\p, as before, and write for abridgment, 
 
 XXIX. . . n = me-St'\pe=f(m-\Pe), 
 so that re is a new and known scalar constant, we shall have, by XV. XX. XXVII. 
 XXIX. 
 
 XXX. . .mp = 4/Vr; XXXI. . . n/-il =m - ^6 ; 
 
 and XXXII. . . mnq = n^^Yr + (mSr - Ss'-^Yr) . (m - ^t), 
 
 an expression from which all inverse operations have disappeared, but which still ad- 
 mits of being simplified, through a division by m, as follows. 
 
 (7.) Substituting (by XXIX.), in the terra n\pYr of XXXIL, the value me 
 - Si'xps for n, and changing (by XXX.) -ipYr to mp, in the terms which are not ob- 
 viously divisible by m, such a division gives, 
 
 XXXIII. . .nq = (m- i^f) Sr + e-^Yr - Sf'i/zVr + a, 
 where XXXIV. . . a = - p^exj^s + ^j^eSe'p = Y.t'Vp^s. 
 
 But (by 348, VII., interchanging accents) we have the transformation, 
 XXXV. . . Vpi//f = - fVt(pp = - <},'YeYr, 
 3 R 
 
490 KLEMENTS OF QUATERNIONS. [bOOK III. 
 
 because 0p = Vr, by XIV. or XV. ; everything inverse therefore again disappears, 
 with this new elimination of the auxiliaiy vector p, and we have this final expres- 
 sion, 
 
 XXXVI. . . nq =nf--ir = {me-^t'->\ji).f-'^r 
 = (»i - -^f) Sr + e;//Vr - St'j//Vr - Ve V'Vt Vr, 
 in which each symbol of operation governs all that follows it, except where a point 
 indicates the contrary, and which it appears to be impossible further to reduce, as 
 the formula of solution of the linear equation I., with the /orm XII. of the quater- 
 nion function^ fq. 
 
 (8.) Such having been the analysis of the problem, the synthesis, by which an 
 a posteriori proof oi the correctness of the resulting formula is to be given, may be 
 simphfied by using the scalar value XXIX. of /(m - -.//c) ; and it is sufficient to 
 show (denoting Vr by w), that for every vec/or w the following equation holds good, 
 with the same form XII. of/: 
 
 XXXVII. . .f{e^\Ju) - Se'i/zoi) -fVe'cp'Yeu) = {me - Sf'^/c). w. 
 (9.) Accordingly, that form of/ gives, with the help of the principle employed 
 in XXXV., 
 XXXVIII Z^^*^*^ = ^ (Sf 't//w + mio), -/Se'tpw = - (e + t) Sc'i// w, 
 
 X-fYe'fYeoj = - ^Ve'f Vew = V(Vf w . i^'t') =:£Ss'^u) - uSe'^t, 
 because Swip'c' = Sg'vpw, &c. ; and thus the equation XXXVI. is proved, by actually 
 operating with/ 
 
 (10.) As an example, if we take the particular form, 
 
 XXXIX. ..r=/^=;,9 + 9P, 
 in which XL. . . jp = a + a = a given quaternion, 
 
 we have then, 
 
 XLI. . ./1=/'1=2;?, e = 2a, E = i' = 2a, (pp = 2ap; 
 whence by the theory of linear and vector functions, 
 
 XLII. ^ . 0'p = 2ap, i//p = 4a2p, m = 8a', 
 and therefore, XLIII. . .\pe = Sa'^a, m-^pe = 8a^ (a - a), n = 16a^ (a2 - a^) ; 
 so that, dividing by 8a, the formula XXXVI. becomes, 
 
 XLIV. . . 2a (a2 - a2) 9 = a (a - a) Sr + a2 Vr - aS . aYr - a V. aYr, 
 or XLV. . . 2a(a + a)^ = aSr+(a + a)Vr-Sar, 
 
 or XLVI. . .2pqSp^S.rKp + pYr = rS/) + V (Yp . Vr), 
 
 or XLVII. . . 4pqSp = 2rSp + Qpr - rp) =pr + rKp ; 
 
 or finally, 
 
 XLVIII. . . q =/ V= !1±^:^ = r + Kp.rp-i 
 Accordingly, 
 
 XLIX. . . (pr + rK/)) + {rp + Kp . r) = 2r (;? + K/)) = 4rS/J. 
 
 (11.) In so simple an example as the last, we may with advantage avail our- 
 selves of special methods; for instance (comp. 346), we may use that which was 
 employed in 332, (6.), to differentiate the square root of a quaternion, and which 
 conducted there more rapidly to a formula (332, XIX.) agreeing with the recent 
 XLVIII. 
 
 (12.) We might also have observed, in the same case XXXIX., that 
 
CHAP. II.] SYMBOLIC AND BIQUADRATIC EQUATION. 491 
 
 L. . . pr - rp =p^q - qp^ = 2V(V(p2) .y^) = 43^ .Y(Yp Nq) =2Sp.(pq-qp); 
 whence pq — qp, and therefore ^95 and qp, can be at once deduced, with the same re- 
 sultuig value for q, or for/-ir, as before : and generally it is possible to differentiate, 
 on a similar plan, the nfl^ root of a quaternion. 
 
 365. We shall conclude this Section on Linear Functions, 
 of the kinds above considered, by proving the general exist- 
 ence of a Symbolic and Biquadratic Equation, of the form, 
 
 I. . . O^n-nf-^n'p-n'f^+fS 
 which is thus satisfied hy the Symbol (/) of Linear and Qua- 
 ternion Operation on a Quaternion, as the Symbolic and Cubic 
 Equation, 
 
 r. . . = m - m'0 + ?w"02 _ ^3^ 350^ I,^ 
 
 was satisfied by the symbol (0) of linear and vector operation 
 on a vector ; the/cwr coefficients, n, n, n', ri", being^wr sca- 
 lar constants ^ deduced from the function y* in this extended or 
 quaternion theory, as the three scalar coefficients m, m, rn' 
 were constants deduced from ^, in the former or vector theory. 
 And at the same time we shall see that there exists a System, 
 of Three Auxiliary Functions, F, G, H, of the Linear and 
 Quaternion kind, analogous to the two vector functions, \p and 
 X, which have been so useful in the foregoing theory of vec- 
 tors, and like them connected with each other, and with the 
 given quaternion function^ by several simple and useful re- 
 lations. 
 
 (1.) The formula of solution, 364, XXXVL, of the linear and quaternion equa- 
 tion fq = r, being denoted briefly as follows, 
 
 11. . . nq = nf-h=Fr, 
 so that (comp. 348, III'.) we may write, briefly and symbolically, 
 III.../F=iy=n, 
 
 it may next be proposed to examine the changes which the scalar n and the function 
 Fr undergo, when/r is changed to /r + cr, or/to/+c, where c is any scalar con- 
 stant; that is, by 364, XII., when e is changed to e+ c, and (p to (p + c ; <p',^, and 
 m being at the same time changed, according to the laws of the earlier theory. 
 (2.) Writing, then, 
 
 IV. ../c=/+C, €c = e + c, 0c = 0+c, (p'c = <p' + C, 
 and V. . . ?//c = »p + ex + c2, w^c = m + m'c + m'c'i + c3, 
 
 we may represent the new form of the equation 364, XXXVI. as follows : 
 VI. . . ncfi^r = For, or VII. . . foFc = Uc ; 
 
492 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 where VIII. . . Fcr = (mc - -^cf ) Sr + ec-^cYr - Ss'^pcYr - Vc'^'cVeVr, 
 and IX, . . wc = e,Wc - Se'ipcf- 
 
 (3.) In this manner it is seen that we may write, 
 
 X. .. Fc=F+cG+c^H+c^, 
 
 and XI. . . nc = n + nc + n"c2 + n'V + c* ; 
 
 where i^, G, H, are three functional symbols, such that 
 
 r Fr = (w - 1^0 Sr + e^Vr - Sc'i/zYr - Yt'<l>'YiYr ; 
 XII. . . j (?r = (m' - xO Sr + (ex + ^)Vr - St'xVr - Vt'Vf Vr ; 
 ( Hr = (m" _ £) Sr + (e + x) Vr - Se'r ; 
 and w, »', w", n'" are four scalar constants, namely, 
 
 ^ n = em- Bt^s (as in 364, XXIX) ; 
 
 XIII... J"r^+'"^':^:,'r' 
 
 n = m + em — Sf £ ; 
 n" = m" + e. 
 
 (4.) Developing then the symbolical equation VII., with the help of X. and XI., 
 and comparing powers of c, we obtain these new symbohcal equations (comp. 350, 
 XVI. XXI. XXIII.) : 
 
 (H=n"~f', 
 XIV. . . G?=n"-/a-=n"-n'7+/2; 
 
 f F= n' -fG = n' - n'/+ n'p -^ ■ 
 and finally, 
 
 XV. . . » = Ff= n'f- np + n'p -/S 
 
 which is only another way of writing the symbolic and biquadratic equation I. 
 
 (5.) Other functional relations exist, between these various symbols of operation, 
 which we cannot here delay to develope : but we may remark that, as in the theory 
 of linear and vector functions, these usually introduce a mixture of functions with 
 their conjugates (comp. 347, XL, &c.). 
 
 (6.) This seems however to be a proper place for observing, that if we write, as 
 temporary notations, for any four quaternions, p, q, r, s, the equations, 
 
 XVI. . . [pq-]=pq-qp', XVII. . . (;>?r) = S ./) [^r] ; 
 
 XVIII. . . [pqr-] = (pqr) + [rq^] Sp + [pr] Sq + [qp^ Sr ; 
 and XIX. . . (jpqrs)= S./>[jr5], 
 
 so that \^pq'] is a vector, (pqr') and (pqrs) are scalars, and [pgr^ is a quaternion, we 
 shall have, in the first place, the relations : 
 
 XX...lpq-]=-lqpl [p/)] = 0; 
 
 XXI. ..(pqr) = - {qpr) = (qrp) = &c. , (ppr) = ; 
 
 XXII. . . [pqr] = - [qpr] = [qrp] = &c., [ppr] = ; 
 
 and XXIII. . . (pqrs) = — (qprs) = (qrps) = — (qrsp) = &c., {pprs) = 0. 
 
 (7.) In the next place, if t be any fifth quaternion, the quaternion equation, 
 
 XXIV. . . =p(qrst) + q(rstp) +r(stpq) + s(tpqr) + t(pqrs), 
 
 which may also be thus written, 
 
 XXV. . . q (prst) ~p{qrst) f r(pqst) + s (prqt) +t(prsq), 
 and which is analogous to the vector equation, 
 
 XXVI. . . 0=aS(3y5-(3Syda + ySdaP-dSal3y, 
 
CHAP. II.] GENERAL QUATERNION TRANSFORMATIONS. 493 
 
 or to the continually* occurring transformation (comp. 294, XIV.), 
 
 XXVII. . . SSaf3y =■ a8d(3y + (SSady + ySa(Bd, 
 is satisfied generally^ because it is satisfied for thenar distinct suppositions, 
 XXVIII. . . q =p, q = r, q = s, q = t. 
 (8.) In the third place, we have this other general quaternion equation, 
 XXIX. . . q(prst) = [rst] Spq - [stp] Srq + [tpr^ Ssq - [prs'] Stq, 
 which is analogous to this other f useful vector formula (comp. 294, XV.), 
 
 XXX. . . dSa(3y = Y(3ySad +YyaSf3d -]-Ya[5Syd', 
 because the equation XXIX. gives true results, when it is operated on by the four 
 distinct symbols (comp. 312), 
 
 XXXI. . . S.;?, S.r, S.s, S.<. 
 
 (9.) Assuming then any four quaternions, p, r, s, t, which are not connected by 
 the relation, 
 
 XXXII. . . (prst) = 0, 
 
 and deducing from them /owr others, p\ r', s\ t\ by the equations, 
 XXXIII J^^' ^ ^^^^^ =/[^«^]' ^' (p^^O = -/[«(p]> 
 
 "'\sXprst)=j 
 
 --f\tpr']y tXprst) = -flprsl 
 
 in which /is still supposed to be a symbol of linear and quaternion operation on a 
 quaternion, the formula XXIX. allows us to write generally, as an expression for 
 the function fq, which may here be denoted by q' (because r is now otherwise used) : 
 
 XXXIV. . . q' ^fq ==pSpq + r'Srq + s'Ssq + t'Stq ; 
 
 and its sixteen scalar constants (comp, 364, (2.)) are now those which are involved 
 in its four quaternion constants, p', r, s', t'. 
 
 (10.) Operating on this last equation with the four symbols, 
 
 XXXV. ..s.[r'sV], %.[s'ep'-\, s.p'pV], s.[pVV], 
 
 we obtain the four following results : 
 
 fCqVsr) = {p'r's't') Spq ; (q's'tY) = (r's't'p') Srq ; 
 . . \(^qrp'r) = ist'p'r')Ssq; {q'p'r's')=. {t'p'rs)Stq; 
 and when the values thus found for the four scalars, 
 
 XXXVII. . . Spq, Srq, Ssq, Stq, 
 are substituted in the formula XXIX., we have the following new formula of quater- 
 nion inversion : 
 
 XXXVIII. . . (p'r's'f) (jprst')q = {p'r's't') {prst)f''^q' 
 = b'sf] {q'r's't') + Istp'] (q's't'p') + [tpr'] iq't'p'r) + [prs'] (q'p'r's') ; 
 
 * The equations XXVII. and XXX., which had been proved under slightly diffe- 
 rent forms in the sub-articles to 294, have been in fact freely employed as trans- 
 formations in the course of the present Chapter, and are supposed to he familiar to 
 the student. Compare the Note to page 437. 
 
 t Compare the Note immediately preceding. 
 
494 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 which shows, in a new way, how to resolve a linear equation in quaternions^ when 
 put under what we may call (comp. 347, (I.)) t^^ Standard Quadrinomial Fornix 
 XXXIV. 
 
 (11.) Accordingly, if we operate on the formula XXXVIII. with,/; attending to 
 the equations XXXIII., and dividing by (jprst), we get this new equation, 
 
 XXXIX. . . ip'r's't')fq =p'{q'r's't'^ — r' {q's't'p') + a' (jq't'p'r') — t' (^qp'r's'') ; 
 whence fq = 5', by XXV. 
 
 (12.) It has been remarked (9.), that /?, r, s, <, in recent formulaj, may be any 
 four quaternions^ which do not satisfy the equation XXXII. ; we may therefore as- 
 sume, 
 
 XL. ../)=1, r = i, «=j, < = ^, 
 
 with the laws of 182, &c., for the symbols t,y, A, because those laws give here, 
 
 XLI. ..(lz;-A) = -2; 
 
 and then it will be found that the equations XXXIII. ^ve simply, 
 
 XLII. ../=/l, r' = -/i, s' = -fj, t'=-fk', 
 
 so that the standard quadrinomial form XXXIV. becomes, with this selection of 
 prst, 
 
 XLIII. . .fq = fl.Bq-fi.Siq-fj.Sjq-JJ^.Skq, 
 
 and admits of an immediate verification, because any quaternion, q, may be ex- 
 pressed (comp. 221) by the quadrinomial, 
 
 XLIV. . . q = Sq- i^iq -jSjq - IcSkq. 
 
 (13.) Conversely, if we set out with the expression, 
 
 XLV. . . q = w + ix +jy + kz, 221, III., 
 
 which gives, 
 
 XLVI. ..fq = wfl + xfi + yfj + zfh, 
 or briefly, 
 
 XLVII. . . e = aw + 6a; + cy + dz, 
 
 the letters dbcde being here used to denote five known quaternions, while wxyz are 
 four sought scalars, the problem of quaternion inversion comes to be that of the se- 
 parate determination (comp. 312) of these four scalars, so as to satisfy the one 
 equation XLVII. ; and it is resolved (comp. XXV.) by the system of the four fol- 
 lowing formulae : 
 
 XL VIII Z"' («^cd) = {ehcd) ; x (abed) = (aecd) ; 
 \y {abed) = (abed') ; z(abcd) = (abce) ; 
 the notations (6.) being retained. 
 
 (14.) Finally it may be shown, as follows, that the biquadratic equation I., for 
 linear functions oi quaternions, includes* the cubic I'., or 350, I., for vectors. Sup- 
 
 * In like manner it may be said, that the cubic equation includes a quadratic 
 one, when we confine ourselves to the consideration of vectors in one plane ; for 
 which case m = 0, and also ^'p = 0, if p be a line in the given plane : for we have 
 then ^^=m' — ^1/ = m', or 
 
 02 - m"0 + m' = 0, 
 
CHAP. III.] ADDITIONAL APPLICATIONS. 495 
 
 pose, for this purpose, that the linear and quaternion function, fq^ reduces itself to 
 the last term of the general expression 364, XII., or becomes, 
 
 XLIX. ../7 = 0V9, so that L. ..e = 0, £=£'=0, /l=/'l = Oj 
 the coefficients n, n', n", n" take then, by XIII., the values, 
 
 LI. . . n = 0, n =m^ n" = m\ n" = m" ; 
 and the biquadratic I. becomes, 
 
 LIL . . = (-m + m'/-m'72+/3)/ 
 
 But/g is now a vector^ by XLIX., and it may be any vector, p ; also the operation 
 /is now equivalent to that denoted by <p, when the subject of the operation is a vec- 
 tor ; we may therefore, in the case here considered, write this last equation LII, under 
 the form, 
 
 LIII. . . = (- m + wi'^ - wi'>2 + 03) p^ 
 
 which agrees with 351, I., and reproduces the symbolical cubic, when the symbol of 
 the operand (p) is suppressed. 
 
 CHAPTER III. 
 
 ON SOME ADDITIONAL APPLICATIONS OF QUATERNIONS, WITH 
 SOME CONCLUDING REMARKS. 
 
 Section 1. — Remarks Introductory to this Concluding 
 Chapter. 
 
 366. When the Third Book of the present Elements was 
 begun, it was hoped (277) that this Book might be made a 
 much shorter one, than either of the two preceding. That 
 purpose it was found impossible to accomplish, without injus- 
 tice to the subject; but at least an intention was expressed 
 (317), at the commencement of the Second Chapter^ of render- 
 ing that Chapter the last : while some new Examples of Geo- 
 
 with this understanding as to the operand. In fact, the cubic gives here (because 
 m = 0), 
 
 (^2 _ ni"(p + m')(pp = 0; 
 
 and therefore (02 - m"^ + m')<T = 0] 
 
 if a be already the result of an operation with <p, on any vector p : that is if it be, as 
 above supposed, a line in the given plane. 
 
496 ELEMENTS OF QUATERNIONS. [bOOK HI. 
 
 metrical Applications, and some few Specimens of Physical 
 ones, were promised. 
 
 367. The promise, thus referred to, has been perhaps al- 
 ready in part redeemed ; for instance, by the investigations 
 (315) respecting certain tangents, normals, areas, volumes, and 
 pressures, which have served to illustrate certain portions of 
 the theory o^ differentials and integrals of quaternions. But it 
 may be admitted, that the six preceding Sections have treated 
 chiefly of that Theory of Quaternion Differentials, including 
 of course its Principles and Rules; and of the connected and 
 scarcely less important theory of Linear or Distributive Func- 
 tions, of Vectors and Quaternions : Examples and Applica- 
 tions having thus played hitherto a merely subordinate or illus- 
 trative part, in the progress of the present Volume. 
 
 368. Such was, indeed, designed from the outset to be, 
 upon the whole, the result of the present undertaking : which 
 was rather to teach, than to apply, the Calculus of Quaternions. 
 Yet it still appears to be possible, without quite exceeding 
 suitable limits, and accordingly we shall now endeavour, to 
 condense into a short Third Chapter some Additional Exam- 
 ples, geometrical and physical, of the application of the princi- 
 ples and rules of that Calculus, supposed to be already known, 
 and even to have become by this time familiar* to the reader. 
 And then, with a few general remarks, the work may be 
 brought to its close. 
 
 Section 2 On Tangents and Normal Planes to Curves in 
 
 Space. 
 
 369. It was shown (100) towards the close of the First Book, 
 that if the equation of" a curve in space, whether plane or of double 
 curvature, be given under the form, 
 
 I. ../> = 0(O=0^ 
 where f is a scalar variable, and is a functional sign, then the de- 
 rived vector, 
 
 11. , .T>p = T>(t)t = (p't = p'=dp : d^, 
 
 * Accordingly, even references to former Articles will now be supplied more 
 sparingly than before. 
 
CHAP. III.] TANGENTS AND NORMAL PLANES TO CURVES. 497 
 
 represents a line which is, or is parallel to, the tangent to the curve, 
 drawn at the extremity of the variable vector p. If then we sup- 
 pose that T is a point situated upon the tangent thus drawn to a 
 curve PQ, at P and that u is a point in the corresponding normal 
 plane, so that the angle tpu is right, and if we denote the vectors 
 OP, OT, ou by p, T, V, the equations of the tangent line and normal 
 plane at p may now be thus expressed : 
 
 III. ..y(T-p)p' = 0; lY. . . S(t/-p)p' = 0; 
 the vector t being treated as the only variable in III., and in like 
 manner v as the only variable in IV., when once the curve pq is 
 given^ and the point p is selected. 
 
 (1.) It is permitted, however, to express these last equations under other forms; 
 for example, we may replace p' by dp, and thus write, for the same tangent line and 
 normal plane, 
 
 V. . . V(r-p)dp = 0; VL . . S(v- p)dp = ; 
 
 where the vector differential dp may represent any line, parallel to the tangent to 
 the curve at p, and is not necessarily small (compare again 100). 
 (2.) We may also write, as the equation of the tangent, 
 
 VII. . . T — p + xp\ where a; is a scalar variable ; 
 and as the equation of the normal plane, 
 
 VIII. . . dpT(i;-p) = 0, or Vlir. . .dT(v-,o)=0, if dy = 0; 
 
 because this partial differential of T(w -p), or of fu, is (by 334, XII., &c.), 
 
 IX. . . dT(u-p) = S(U(i;-p).dp). 
 
 (3.) For the circular locus 314, (1.), or 337, (1.), of which the equation is, 
 
 X...p = a% with Ta=l, and Sa/3 = 0, 
 
 the equation of the tangent is, by VII., and by the value 337, VI. of p', 
 
 XL . . r = p +yap, where y is a new scalar variable ; 
 the perpendicularity of the tangent to the radius being thus put in evidence. 
 (4.) For the plane but elliptic locus, 314, (2.), or 337, (2*), for which, 
 
 XII. . . p = V. a% with Ta = 1, but not SafS = 0, 
 
 the value 337, VIII. of p' shows that the tangent, at the extremity of any one semi- 
 diameter p, is parallel to the conjugate semidiameter of the curve ; that is, to the 
 one obtained by altering the excentric anomaly (314, (2.)), by a quadrant: or to 
 the value of p which results, when we change < to * + 1. 
 (5.) For the helix, 314, (10.), of which the equation is, 
 
 XIII. .. p = cta^ a</3, with Ta = 1, and Sa;3 = 0, 
 c being a scalar constant, we have the derived vector, 
 
 XIV. . . p'=ca + - a<+i/3 ; whence XV. . . Sa-'p' = c, 
 
 XVI. . . TVa-ip' = |T,(3, and XVII. . . (TV : S) a-ip' = ^ ; 
 3 S 
 
498 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 the tangent line (p') to the helix is therefore inclined to the axis (a) of the cylinder 
 whereon that curve is traced, at a constant angle (a), whereof the trigonometrical 
 tangent (tan a) is given by this formula XVII. ; and accordingly, the numerator 
 7rT/3 of that formula represents the semicircwnference of the cylindric base ; while 
 the denominator 2 c is an expression for half the interval between two successive 
 spires, measured in a direction parallel to the axis. We may then write, 
 
 XVIII. . . 7rT/3 = 2c tan a = 2c cot 6, 
 
 if a thus denote the constant inclination of the helix to the axis, while 6 denotes the 
 constant and complementary inclination of that curve to the base, or to the circles 
 which it crosses on the cylinder. 
 
 (6.) In general, the parallels p' to the tangents to a curve of double curvature, 
 which are drawn from a fixed origin o, have a certain cone for their locus; and for 
 the case of the helix, the equation of this cone is given by the formula XVII., or by 
 any legitimate transformation thereof, such as the following, 
 
 XIX. . . SUa-y = + cosa=+sin6; 
 
 it is therefore, in this case, a cone of revolution, with its semiangle =a. 
 
 (7.) As an example of the determination of a normal plane to a curve of double 
 curvature, we may observe that the equation XIII. of the helix gives, 
 
 XX. . . p2= ^2 _ c2<2j and therefore XXI. . . Spp' = -c'^t', 
 
 the equation IV. becomes therefore, for the case of this curve, 
 
 XXII. . . = Sp'v + c% with the value XIV. of p'. 
 
 (8.) If then it be required to assign the point u in which the normal plane to the 
 helix meets the axis of the cylinder, we have only to combine this equation XXII. 
 with the condition v \\ a, and we find, by XIII. and XIV., 
 
 XXIII. . . ou = u = - cHa : Sap' = eta, XXIV. . . Sa (u - p) = ; 
 
 the line pu is therefore perpendicular to the axis, being in fact a normal to the cy- 
 linder. 
 
 370. Another view of tangents and normal planes may be proposed, 
 which shall connect them in calculation with Taylor'' s /S'm 65 adapted 
 to quaternions (342), as follows. 
 
 (1.) Writing I. . . p< = po + «<^p'o, or briefly, V. . . pt = p^-utp', 
 the coffieeient ut or u will generally be a quaternion, but its limiting value will be 
 positive unity y when t tends to zero as its limit ; or in symbols, 
 
 II. . . tto = lim. u=l. 
 
 t=o 
 
 (2.) Admitting this, which follows either from Taylor's Series, or (in so simple a 
 case) from the mere definition of the derived vector p', we may conceive that vector 
 p' to be constructed by some given line pt, without yet supposing it to be known that 
 this line is tangential at p to the curve PQ, of which the variable vector is OQ = p<, 
 while OP = po = p, so that the line pq = utp' is a vector chord from p, which diminishes 
 indefinitely with the scalar variable, t, and is small, if t be small. 
 
CHAP. III.] CONNEXION WITH TAYLOR's SERIES. 499 
 
 (3.) Conceiving next that w = or = the vector of some new and arbitrary point 
 R, we may let fall a perpendicular qm on the line pr, and so decompose the chord 
 PQ into the two rectangular lines, pm and mq ; which, when divided by the same 
 chord, give rigorously the two (generally) quaternion quotients, 
 
 III . . — = ^"P'(^ ~ P) IV. . ^ = V«p'(f > -P) . 
 
 * * ' PQ vp'(u) — p) ' ' * ' PQ up'{(i} — p) ' 
 
 the variable t thus disappearing through the division, except so far as it enters into 
 tt, which tends as above to 1. 
 
 (4.) Passing then to the limits, we have these other rigorous equations, 
 
 V. . . lim. ™ = Xd^l VI. . . Ita. "^ = MllZf); 
 
 PQ p{(0-p) PQ p{<o-p) 
 
 by comparing which with 369, III. and IV., we see that those two equations repre- 
 sent respectively, as before stated, the tangent and the normal plane to the proposed 
 curve at p; because, if Yp'(u) - p) = 0, the chord pq tends, by V. or VI., to coin- 
 cide, both in length and in direction, with its projection pm on the line pr ; while 
 on the other hand, if Sp'(w — p) = 0, that projection tends to vanish, even as compared 
 with the chord pq ; which chord tends now to coincide with its other projection mq, 
 or with the perpendicular to the line PR, erected so as to reach the point Q : whence 
 PR must, in this last case, be a normal to the curve at p. 
 
 (5.) We may also investigate an equation for the normal plane, by considering it 
 as the limiting position of the plane which perpendicularly bisects the chord. If r 
 be supposed to be a point of this last plane, then, with the recent notations, the vec- 
 tor w = OR must satisfy the condition, 
 
 VII. . . T(w - pi) = T(a> - po), or VIII. . . (o) - p - utp'^ = (w - p)2, 
 or IX. . . 2Sup\o) - p) = t(upy, 
 
 in which it may be noted that up' is a vector (in the direction of the chord, pq), al- 
 though u itself is generally a quaternion, as before : such then is the equation of the 
 bisecting plane, with id for its variable vector, and its limit 4K^, 
 
 X. . . S/o'(w - p) = 0, as before. 
 
 (6.) The last process may also be presented under the form, 
 
 XI. . . = lim. ri{T(w - pO - T(a> - po)} = D(T(a> - pO, when t = 0- 
 
 and thus the equation 369, VIII. may be obtained anew. 
 
 (7.) Geometrically, if we set off on rq a portion rs equal in /T 
 
 length to RP, as in the annexed Figure 76, we shall have the y^^'TTT^ 
 
 limitmg equation, /\\/ ■' 
 
 XII. . . + SQ : PQ = (rq - Rp) : PQ = (ultimately) - cos rpt ; / X \ j 
 
 which agrees with 369, IX. // ^^^^ 
 
 (8.) If then the point r be taken out of the normal ^^^^"''''''^ 
 plane at p, this limit of the quotient, rq — rp divided by pq, Y\v 76. 
 
 has & finite value, positive or negative; and if the chord pq be 
 called smaZZ of the ^r«< order, the difference of distances of its extremities from R 
 may then be said to be small of the same (first) order. But if r be taken in the nor- 
 mal plane at p (and not coincident with that point p itself), this diflfereuce of dis- 
 
500 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 tances may then be said to be small, of an order higher than the first : which an- 
 swers to the evanescence of the first differential of the tensor, T(w -p) in XI., or 
 T(t;-p)in 369, VIII'. 
 
 371. A curve may occasionally be represented in quaternions, by 
 an equation which is not of the/orw, 369, !.> although it must 
 always be conceived capable of reduction to that form : for instance, 
 this new equation, 
 
 I. . . Yap ,Ypa' = {Yaa')\ with TY aa' > 0, 
 
 is not immediately of the form p =(pt, but it is reducible to that form 
 as follows, 
 
 II. . . p = ta + t-^a'. 
 
 An equation such as I. may therefore have its differential or its deri' 
 vative taken, with respect to the scalar variable t on which p is thus 
 conceived to depend, even if the exact law of such dependence be un- 
 known : and d/>, or />', may then be changed to the tangential vector 
 up - p to which it is parallel, in order to form an equation of the tan- 
 gent, or a condition which the vector to of a point on that sought 
 line must satisfy. 
 
 (1.) To pass from I. to II., we may first operate with the sign V, which gives, 
 III. . . pSaa'p — 0, or simply, HI'. . . Saap = ; 
 whence, t and t' being scalars, we may write, 
 
 lY. . . p = ta+ t'a', Yap = t'Yaa\ Ypa = tYaa, tt' = 1, 
 
 and the required reduction is effected : while the return from 11. to I., or the elimi- 
 nation of the scalar t, is an even easier operation. 
 
 (2.) Under the form II., it is at once seen that p is the vector of a plane hyper- 
 bola, with the origin for centre, and the lines a, a' for asymptotes; and accordingly 
 all the properties of such a curve may be deduced from the expression II., by the 
 rules of the present Calculus. 
 
 (3.) For example, since the derivative of that expression is, 
 V. . . jo' = a - t-^a, 
 the tangent may (comp. 369, VII.) have its equation thus written: 
 
 VI. . . w = (« + a;)a + «-2(e_a;)a'; 
 it intersects therefore the lines a, a' in the points of which the vectors are 2ta, 2t'^a' ; 
 so that (as is well known) the intercept, upon the tangent, between the asymptotes, 
 is bisected at the point of contact : and the intercepted area is constant, because 
 Y{ta.t-^a) = Yaa', &c. 
 
 (4.) But we may also operate immediately, as above remarked, on the farm I. ; 
 and thus arrive (by substitution ofw-pfordp, &c.) at the equation of conjuga- 
 tion, 
 
 VII. . . Vaw . Vpa' + Yap . Vw«' = 2 {Yaay, 
 
CHAIMII.] NORMALS AND TANGENT PLANES TO SURFACES. 501 
 
 which expresses (comp. 215, (13.), &c.) that if p = op, and oi = or, as before, then 
 either r is on the tangent to the curve, at the point p, or at least each of these two 
 points is situated on the polar of the other, with respect to the same hyperbola. 
 
 (5.) Again, it is frequently convenient to consider a curve as the intersection of 
 two surfaces; and, in connexion with this conception, to represent it by a system of 
 two scalar equations, not explicitly involving any scalar variable : in which case, 
 both equations are to be differentiated, or derivated, with reference to such a varia- 
 ble understood, and dp or p' deduced, or replaced by w — p as before. 
 
 (6.) Thus we may substitute, for the equation L, the system of the two follow- 
 ing (whereof the first had occurred as III'.) ; 
 
 VIII. . . Saa'p = 0, p2Saa' - SapSa'p = (Jaa'y ; 
 and the derivated equations corresponding are, 
 
 IX. . . Saa'p' = 0, 2Saa'Spp' - Sap'Sa'p - SapSa'p' = ; 
 or, with the substitution of w — p for p', &c., 
 
 X . . Saa'o) = 0, 2Saa'Spa; - SawSa'p - SapSa'w = 2 (Yaay ; 
 the last of which might also have been deduced from VII., by operating with S. 
 
 (7.) And it may be remarked that the two equations VIII. represent respectively 
 in general a plane and an hyperloloid, of which the intersection (5.) is the hyperbola 
 I. or II.; or a plane and an hyperbolic cylinder^ if Saa'= 0. 
 
 Section 3. — On Normals and Tangent Planes to Surfaces. 
 
 372- It was early shown (100, (9.))» that when a curved surface 
 is represented by an equation of the form, 
 I. . . 9 = <p{x,y), 
 in which is a functional sign, and x, y are two independent and 
 scalar variables, then either the two partial differentials, or the two 
 partial derivatives^ of the^r^^ order ^ 
 
 11. . . d^, dyP, or III. . . D^p, Dyp, 
 
 represent two tangential vectors^ or at least vectors parallel to two 
 tangents to the surface, drawn at the extremity or term p of />; so 
 that the plane of these two differential vectors, or of lines parallel 
 to them, is (or is parallel to) the tangent plane at that point: and 
 the principle has been since exemplified, in 100, (11.) and (12.), 
 and in the sub- articles to 345, &c. It follows that any vector v, 
 which \s perpendicular to both of two such non-parallel differentials, 
 or derivatives, must (comp. 345, (11.)) be a normal vector at p, or at 
 least one having the direction of the normal to the surface at that 
 point ; so that each of the two vectors, 
 
 IV. . . V.d,pd,P, V. . . V. D.,pD/, 
 
 if actual, represents such a normal. 
 
502 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (1.) As an additional example, let us take the case of the ruled paraboloid^ on 
 which a given gauche quadrilateral abcd is superscribed. The expression for the 
 vector jO of a variable point p of this surface, considered as a function of two inde- 
 pendent and scalar variables, x and y, may be thus written (comp. 99, (9.)) : 
 
 VI. ..p = a;ya + (l-x)y/3 + Cl-a;) 0--y)y +x(l-y)S] 
 where the supposition y = 1 places the point p on the line ab ; x = places it on bc ; 
 y = 0, on CD ; and a; = 1, on da. 
 
 (2.) We have here, by partial derivations, 
 VII...D.p=yCa-i3) + (l-y)(5-y); D,p = a;(a-^) + (l -:r) (/3- y); 
 these then represent the directions of two distinct tangents to the paraboloid VI., at 
 what may be called the point (x, y) ; whence it is easy to deduce the tangent plane 
 and the normal at that point, by constructions on which we cannot here delay, ex- 
 cept to remark that if (comp. Fig. 31, Art. 98) we draw two right lines, QS and rt, 
 through p, so as to cut the sides ab, bc, cd, da of the quadrilateral in points Q, R, 
 s, T, we shall have by VI. the vectors, 
 
 VIII . /OQ = a;a + (!-«;) /3, OB=y/3 + (l-y)y, 
 ' * ■ \os = a;5 + (l-a;)y, OT=ya+ (1 -y)^, 
 and therefore, by VII., 
 
 IX. . . "Dxp = RT, Byp = SQ ; 
 so that these two tangents are simply the two generating lines of the surface, which 
 pass through the proposed point p. 
 
 (3.) For example, at the point (1, 1), or a, the tangents thus found are the sides 
 ba, da, and the tangent plane is that of the angle bad, as indeed is evident from 
 geometry. 
 
 (4.) Again, the equation of the screw surface (comp. 314, XVI.), 
 X. . . p = cxa+i/a''(3, with Ta = 1, and Sa/3 = 0, 
 gives the two tangents, 
 
 XI. . . Dxp = ca + ^ya^J/3, Dyp = a% 
 
 whereof the latter is perpendicular to the former, and to the axis a of the cylinder ; 
 so that the correspondmg normal to the surface X. at the point (x, y) is represented 
 by the product, 
 
 XII. . . J/ = D^p . Dy|0 = ca^+i/3 + ^y^'^a. 
 
 373. Whenever a variable vector p is thus expressed or even 
 conceived to be expressed, as a function of two scalar variables, x and 
 y (or s and t, &c.), if we assume any three diplanar vectors, such as 
 <*> A 7 (oJ^ h i^i \ &c.), the three scalar expressions^ Sap, Sfip, 87^ 
 (or Sfp, S/cp, Sap, &c.) will then be functions of the same two scalar 
 variables; and will therefore be connected with each other by some 
 07ie scalar equation, of the form, 
 
 I. . . F{Sap, S^p, S7P) = 0, 
 or briefly, 
 
CHAP. III.] CONNEXION WITH QUATERNION DIFFERENTIALS. 503 
 
 II.../P = C; 
 where C is a scalar constant, introduced (instead of zero) for greater 
 generality of expression ; and F, f are used as functional but scalar 
 signs. If then (comp. 361, XIV.) we express th.Q first differential of 
 this scalar function fp under the form, 
 
 III. ..dfp=2Svdp, 
 in which v is a certain derived vector, and is here considered as being 
 (at least implicitly) a vector function (like p) of the two scalar varia- 
 bles above mentioned, we shall have the two equations, 
 
 IV.. . Syd,p = 0, S»^d,p=0, 
 or these two other and corresponding ones, 
 
 V. .. Si/D,p = 0, SvD,P = 0; 
 from which it follows (by 372) that v has the direction of the nor- 
 mal to the surface I. or II., at the point p in which the vector p ter- 
 minates. Hence the equation of that normal (with tv for its variable 
 vector) may, under these conditions, be thus written: 
 
 VI. . .Yp{iv-p) = 0; 
 
 and the corresponding equation of the tangent plane at the same point 
 p is 
 
 VII. ..Si/(a;-p) = 0. 
 
 (1.^ For example, if we take the expression 308, XVIII., or 345, XII., namely 
 YIIL . . p = rk^j^kj-^Jc~\ in which kj-^ =j^k, &c., 
 treating the scalar r as constant, but s and t as variable, we have then (comp. 345, 
 XIV.), the equations, a denoting any unit- vector, 
 
 IX. .. Sjp = rS.a2<S.a2s+i, S/p = rS.a2<-iS.a2«+i, Skp = rS . a^'^^ ; 
 between which s and t can be eliminated, by simply adding their squares, because 
 (a^y^ + (a'-i)2 = 1, by 315, V., if Ta = 1. In this manner then we arrive at equa- 
 tions of the forms I. and IL, namely (comp. 357, VII., and 308, (10.) and (13.)), 
 
 X. . . (Sip)2 + (S/p)2 + (S/5p)2 - r2 = 0, 
 and XI. . . /p = |o2 = - r2 = const., or XI'. . . Tjo = r ; 
 
 which last results had indeed been otherwise obtained before. 
 
 (2.) With this form XI. of/p, we have the differential expression o{ the Arst 
 order, 
 
 XII. . . d/p = 2Svdp = 2SpdjO, whence XIII. . . v = p ; 
 
 and if we still conceive that p is, as above, some vector function of two scalar varia- 
 bles, s and t, although the particular law VIII. of its dependence on them may now 
 be supposed to be unknown (or to be forgotten), we may write also, 
 
 XIV. . . |d/p = S j/dp = Sjodp = Sp (d, + d^ p = SpD.p . d« + SpD^p . dt ; 
 
 if then the /wnc^ton/p have (as above) a value, = - r2, which is constant, or is tnrfe- 
 
504 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 pendent of both the variables, s and f, while their differentials are arbitrary^ and are 
 independent of each other, we shall thus have separately (comp. V., and 337, XIII., 
 
 XVII.), 
 
 XV. . . SpDsp = 0, SpDtp = 0. 
 
 The radius p of the sphere XI. is therefore in this way seen to have the direction of 
 the normal at its own extremity, because it is perpendicular to two distinct tangents, 
 Dsp and Dtp, at that point ; which are indeed, in the present case, perpendicular to 
 each other also (337, (8.)). 
 
 (3.) Instead of treating the two scalar variables, x and y, or s and t, &c., as both 
 entirely arbitrary and independent, we may conceive that one is an arbitrary (but 
 scalar) /wnciion of the other; and then the vector v, determined by the equation 
 III., will be seen anew to be the normal at the extremity p of p, because it is per- 
 pendicular to the tangent at P to an arbitrary curve upon the surface, which passes 
 through that point : or (otherwise stated) because it is a line in an arbitrary normal 
 plane at p, if a normal plane to a curve on a surface be called (as usual) a normal 
 plane to that surface also. 
 
 (4.) For example, if we conceive that s in VIII. is thus an arbitrary function of 
 t, the last expression XIV. will take the form, 
 
 XVI. . . = id/p = S . jo (s'T>sp + V>tp) dt, if d* = s'dt ; 
 whence, dt being still arbitrary, we have the one scalar equation, 
 
 XVII. . . S . p dsDsp 4- T>tp) = 0, or XVIII. . . p -i- sDsp 4 Dtp , 
 and although, on account of the arbitrary coefficient «', this one equation XVII. is 
 equivalent to the system of the two equations XV., yet it immediately signifies, as in 
 XVIII., that the directed radius p, of the sphere XL, is perpendicular to the arbi- 
 trary tangent, s'Dsp -f Dtp ; or to the tangent to an arbitrary spherical curve through 
 p, the centre o and tensor Tp (or undirected radius, r) remaining as before. 
 
 (5.) As regards the logic of the subject, it may be worth while to read again the 
 joroo/ (331), of the validity of the rule for differentiating a function of a function; 
 because this rule is virtually employed, when after thus reducing, or conceiving as 
 reduced, the scalar function /p of a vector p, to another scalar function such as Ft of 
 a scalar t, by treating p as equal to some vector function ^t of this last scalar, we 
 infer that 
 
 XIX. . .dFt = df(pt = 2S. vd<pt, if dfy = 2Svdp, as before. 
 
 (6.) And as regards the applications of the formulae VI. and VII., or of the equa- 
 tions given by them for the normal and tangent plane to a surface generally, the 
 difficulty is only to select, out of a multitude of examples which might be given : 
 yet it may not be useless to add a few such here, the case of the sphere having of 
 course been only taken to illustrate the theory, because the normal property of its 
 radii was manifest, independently of any calculation. 
 
 (7.) Taking then the equation of the ellipsoid, under the form, 
 
 XX. . . T(tp + pK) = k2 - 1^, 282 , XIX., 
 
 of which the first differential may (see the sub-articles to 336) be thus wiitten, 
 
 XXI. . . = S.{(t-K)2p + 2(iSKp + KStp)}dp = Svdp, 
 and introducing an auxiliary vector, on or ^, such that 
 
 XXII. . .ON = ? = -2(t-K)-2(tS<cp + KS(p), 
 
CHAP. III.] NORMALS TO SURFACES OF THE SECOND ORDER. 505 
 
 we have v\\p-^, and may write, as the equation of the normal at the extremity p 
 of /o, the following, 
 
 XXIII. .. V.(^-p)(a>-p) = 0, or XXIV. ..ai = p+ar(^-p), 
 in which ic is a scalar variable (conip. 369, VII.) ; making then x=l, we see that 
 ^ is the vector of the point N in which the normal intersects the plane of the two 
 fixed lines t, k, supposed to be drawn from the origin, which is here the centre of 
 the ellipsoid. 
 
 (8.) If we look back on the sub-articles to 216 and 217, we shall see that these 
 lines t, K have the directions of the two real cyclic normals^ or of the normals to the 
 two (real) cyclic planes ; which planes are now represented by the two equations. 
 
 XXV. . . Stp = 0, S/cp = 0. 
 Accordingly the equation XX. of the ellipsoid may be put (comp. 336, 357, 359) 
 under the cyclic forms^ 
 
 XXVI. . . Sp^p = (i2 + /c2)p« + 2StpK:p 
 
 = (t - /e)2 p2 + 4StpS«:p = {k^ - i3)2 = const. ; 
 hence each of the two diametral planes XXV. cuts the surface in a circle^ the com- 
 mon radius of these two circular sections being 
 
 XXVII...Tp = ,f^_^=5, 
 
 where h denotes, as in 219, (1.), the length of the mean semiaxis of the ellipsoid; 
 and in fact, this value of Tp can be at once obtained from the equation XX., by 
 making either ip = - pi, or pjc = — /cp, in virtue of XXV. 
 
 (9.) By the sub-article last cited, the greatest and least semiaxes have for their 
 lengths, 
 
 XXVIII. . . a = Ti + Tie, c = Tt - Tfc ; 
 
 and the construction in 219, (2.) shows (by Fig. 53, annexed to 217, (4.)) that 
 these three semiaxes a, b, c have the respective directions of the lines, 
 
 XXIX. . . iTk- kTi, V(/c, iTk + kTi ; 
 all which agrees with the rectangular transformation, 
 Sp<pp /T(tp + pK:)\2 
 
 XXX. . . 1 = 
 
 _/ T((p + p,c) Y 
 
 )2 \^ K2-t2 I 
 
 (K2-t2)2 
 
 _/ S.pU(tTfc-fcTO Y /^ T(t-ic)S.pUV(K) Y [ S.pU(tTfc+/cT0 \2 
 ~V Ti+Tk j "'' V Ti«-T*:2 j "^\ Tl-Tk j' 
 
 in deducing which (comp. 359, (1.)) from 357, VIII., by means of the formulae 
 357, XX. and XXI., we employ the values (comp. XXVL), 
 XXXI. . .^ = i«4-k2, \-2t, fi = K. 
 (10.) The fixed plane (7.), of the cyclic normals i and k (8.), is therefore also 
 the plane of the extreme semiaxes, a and c (9.), or that which may be called per- 
 haps the principal plane* of the ellipsoid : namely, the plane of the generating tri- 
 
 * This plane may also be said to be the plane of the principal elliptic section 
 (219, (9.)) ; or it may be distinguished (comp. the Note to page 231) as the plane 
 of the /ocaZ hyperbola, of which important curve we shall soon assign the equation 
 in quaternions. 
 
 3 T 
 
506 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 angle (218), (1.)), in that construction of the surface (217, (6.) or (7.)) which is 
 illustrated by Fig. 53, and was deduced as an interpretation of the quaternion equa- 
 tion XX., or of the somewhat less simple form 217, XVI., with the value Ti2- Tk2 
 of <-. 
 
 (11.) Let n denote the length of that portion of the normal, which is intercepted 
 between the surface and the principal plane (10.), so that, by (7.), 
 XXXII. ..n = NP = T(p-g), «2^-(p-02^ 
 with the value XXII. of ^. Let c = os be the vector of a point s on the surface of 
 a new or auxiliary sphere^ described about the point n as centre, with a radius = », 
 and therefore tangential to the ellipsoid at v ; and let us inquire in what curve or 
 curves^ real or imaginary, does this sphere cut the ellipsoid. 
 
 (12.) The equations (comp. 371, (5.)) of the sought intersection are the two fol- 
 lowing, 
 
 XXXIII. . . ((T - ^)2 + «2 = 0, and XXXIV. . . T(i(t + ok) = k:2 - |2 ; 
 whereof the first expresses that s is a point of the sphere, and the second that it is a 
 point of the ellipsoid ; while p or op enters virtually into XXXIIL, through %, and «, 
 but is here treated as a constant^ the point p being now supposed to be a given one. 
 
 (13.) We shall remove (18) the origin to this point P of the eUipsoid, if we 
 
 write, 
 
 XXXV. . . £r=p+(r', or XXXV'. . . cr'= (x-p = ps; 
 
 and thus we obtain the new or transformed equations, 
 
 XXXVL. . = cr'2+2S(p-O'y', XXXVIL .. = N(i(t'+ a'K) + 28^(7'; 
 in vfhich (as in (7.), comp. also 210, XX.), 
 
 XXXVIIL . . J/ = (t - (c)2p + 2 (iSkp + fcStp) = (t - fc)2 (p - ?), 
 and XXXIX. . . N («(t' + g'k) = (i - k)' ^'^ + 4Sia'SK(T'. 
 
 (14.) Eliminating then o-'^, we obtain from the two equations XXXVI. and 
 XXXVIL this other, 
 
 XL. . . Stff'. S/c<7' = ; 
 
 which like them is of the second degree in <t', but breaks up, as we see, into two linear 
 and scalar factors, reTpresentiXig ttvo distinct planes, parallel by XXV. to the two 
 diametral and cyclic planes of the ellipsoid. The sought intersection consists then 
 of a pair of (real) circles, upon that given surface ; namely, two circular (but not 
 diametral) sections, which pass through the given point p. 
 
 (15.) Conversely, because the equations XXXVIL XXXVIIL XXXIX. XL. 
 give XXXVI. and XXXIIL^ with the foregoing values of ^ and », it follows that 
 these two plane sections of the ellipsoid at P are on one common sphere, namely 
 that which has n for centre, and n for radius, as above ; and thus we might have 
 found, without differentials, that the line pn is the normal at p ; or that this normal 
 crosses the principal plane (10.), in the point determined by the formula XXII. 
 
 (16.) In general, the cyclic form of the equation of any central surface of the 
 second order, namely the form (comp. 357, II.), 
 
 XLI. . . ^p<pp=g'p'i + 2S\pS/ip = C= const., 
 shows that the two circles (real or imaginary) in which that surface is cut by any 
 two planes, 
 
 XLII. . . S\p = /, Siip = m, 
 
CHAP. III.] RECIPROCAL SURFACES. 507 
 
 drawu parallel respectively to the two real cyclic planes^ which are jointly repre- 
 sented (comp. XL., and 216, (7.)) by the one equation, 
 
 XLIII. . . SX|oS/i|0 = 0, 
 are homospherical^ being both on that one sphere of which the equation is, 
 XLI V. . . /p2 + 2 (ZS//p + mSXp) = 2lm + C. 
 (17.) But the centre (say n) of this new sphere, has for its vector (say 4)) 
 XLV. . . ON = g = - g-\lii + mX) ; 
 
 it is therefore situated m the plane of the two real cyclic normals, \ and /x ; and if 
 I and m in XLV. receive the values XLII., then this new ^ is the vector of intersec- 
 tion of that joZane, with the normal to the surface at p : because it is (comp. 15.)) 
 the vector of tlie centre of a sphere which touches (though also cutting^ in the two 
 circular sections) the surface at that point. 
 
 (18.) We can therefore thus infer (comp. again (15.)), without the differential 
 calculus, that the line, 
 
 XLVI. . . gXp -t) = g'9 + XS/^p + )itSXp = 0p, 
 
 as having the direction of np, is the normal at p to the surface XLI. ; which agrees 
 with, and may be considered as confirming (if confirmation were required), the con- 
 clusion otherwise obtained through the differential expression (361), 
 
 XLVIL . . dSp^p = 2Svdp = 2S^pdp; 
 the linear function 0p being here supposed (comp. 361, (3.)) to be self-conjugate. 
 
 (19.) Hence, with the notation 362, I., the equation of the tangent plane to a 
 central surface of the second order, at the same point p, may by VII. be thus 
 written, 
 
 XLVIII. . . /(w, p) = C, if Sp0p = C= const. ; 
 
 in which it is to be remembered, that 
 
 XLIX. . . /(w, p) =/(p, w) = Sw^p = Sp0w. 
 
 (20.) And if we choose to interpret this equation XLVIIL, which is only of the 
 first degree (362) with respect to each separately of the tico vectors, p and w, or op 
 and OR, and involves them symmetrically, without requiring that p shall be a point 
 on the surface, we may then say (comp. 215, (13.), and 316, (31.)), that the for- 
 mula in question is an equation of conjugation, which expresses that each of the two 
 points p and b, is situated in the polar plane of the other. 
 
 (21.) In general, if we suppose that the length and direction of a line v are so 
 adjusted as to satisfy the two equations (comp. 336, XII. XIII. XIV.), 
 
 L. . . Svp = 1, Si^dp = 0, and therefore also LI. . . Spdv = ; 
 then, because the equation VIL of the tangent plane io any curved surface may now 
 be thus written, 
 
 LIL . . ^v{o,-v-^) = 0, 
 
 it follows that v • represents, in length and direction, ihQ perpendicular from o on 
 that tangent plane at v, so that v iVse//" represents the reciprocal of thaX, perpendi- 
 cular, or what may be called (comp. 336, (8.)) the vector of proximity, of the tan- 
 gent plane to the origin. And we see, by LI., that the two vectors, p and v, if 
 drawn from a common origin, terminate on two surfaces which are, in a known and 
 
508 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 important sense (comp. the sub-arts, to 361), reciprocals* of one another: the line 
 p-i, for instance, being the perpendicular from o on the tangent plane to the second 
 surface, at the extremity of the vector v. 
 
 374. In the two preceding Articles, we have treated the symbol 
 dp as representing (rigorously) a tangent to a curve on a given surface, 
 and therefore also to that surface itself; and thus the formula 
 Svdp = has been considered as expressing that i^ has the direction of 
 the normal to that surface^ because it is perpendicular to two tangents 
 (372), and therefore generally to every tangent (373), which can be 
 drawn at a given point p. But without at present introducing any 
 other] signification for this symbol d/>, we may interpret in another 
 way, and with a reference to chords rather than to curves^ the diffe- 
 rential equation^ 
 
 I. . . d//3=2Si^d/>, 
 
 supposed still to be a rigorous one (in virtue of our definitions of dif- 
 ferentials, which do not require that dp should be smalt) ; and may 
 still deduce from it the normal property oiVaQYeciov v, but now with 
 the help of Taylor''s Series adapted to quaternions (comp. 342, 370). 
 In fact, that series gives here a differenced equation, of the form, 
 
 II. . . Afp = 2Sv^p^R', 
 where 7? is a scalar remainder (comp. again 342), having the pro- 
 perty that 
 
 III. . . lim. {R : TAp) = 0, if lim. TAp = ; 
 whence IV. . . lim. (Afp : TAp) = 2 lim. SvUAp, 
 
 whatever the ultimate direction of Ap may be. If then we conceive that 
 
 * Compare the Note to page 484. 
 
 f It is permitted, for example, by general principles above explained, to treat the 
 differential dp as denoting a chordal vector, or to substitute it for Ap, and so to re- 
 present the differenced equation of the surface under the form (comp. 342), 
 
 = A/p = (£d - l)/p =d/p + Id2/p + &c. ; 
 
 but with this meaning of the symbol dp, the equation Afp = 0, or Svdp = 0, is no 
 longer rigorous, and must (for rigour) be replaced by such an equation as the follow- 
 ing, 
 
 = 2St/dp + Sdj/dp + R, if d/p = 2Sj/dp, as before ; 
 
 the remainder R vanishing, when the surface is only of the second order (comp. 
 362, (3.)). Accordingly this last/or?« is useful in some investigations, especially 
 in those which relate to the curvatures of normal sections: but for the present it 
 seems to be clearer to adhere to the recent signification of dp, and therefore to treat 
 it as still denoting a tangent, which may or njay not be small. 
 
CHAP. III.] NEW ENUNCIATION OF TAYLOR's THEOREM. 509 
 
 A/) represents a small and indefinitely decreasing chord pq of the sur- 
 face, drawn from the extremity p of p, so that 
 
 V. . . Afp =/ {p + M -fp = 0, and lim. TA/> = 0, 
 the equation IV. becomes simply, 
 
 VI.. . lim. Si^UA/> = 0; 
 and thus proves, in a new way, that v is normal to the surface at the 
 proposed point p, by proving that it is ultimately perpendicular to all 
 the chords votfrom that point, when those chords become indefinitely 
 small, or tend indefinitely to vanish. 
 
 (1.) For example, if 
 VII. . ./jo = p2, v = p, then VIII. . . i?= Ap^, and i? : TAp = - TAp ; 
 thus, for every point of space, we have rigorously, with this form of /p, 
 
 IX. . . A/p : TAp = 2SpUAp - TAp ; 
 and for every point q of the spheric surface, fp = const., we have with equal rigour, 
 
 X. . . 2SpUAp = TAp, or XI. . . pq = 2op.cosopq ; 
 in fact, either of these two last formulae expresses simply, that the projection of a 
 diameter of a sphere, on a conterminous chord, is equal to that chord itself and of 
 course diminishes with it. 
 
 (2.) Passing then to the limit, or conceiving the point q of the surface to ap- 
 proach indefinitely to p, we derive the limiting equations, 
 
 XII. . . lim. SpUAp = ; XIII. . . Um. cos opq = ; 
 
 either of which shows, in a new way, that the radii of a sphere are its normals ; 
 with the analogous result for other surfaces, that the vector v in I. has a normal di- 
 rection, as before : because its projection on a chord pq tends indefinitely to diminish 
 with that chord. 
 
 (3.) We may also interpret the differential equation I. as expressing, through 
 II. and III., that t\iQ plane 373, VII., which is drawn through the point p in a 
 direction perpendicular to v, is the tangent plane to the surface : because the pro- 
 jection of the chord Ap on the normal v to that plane, or the perpendicular distance, 
 
 XIV. . . - S (Ui/. Ap) = |i2. Tiz-i, 
 
 of a near point qfrom the plane thus drawn through p, is small of an order higher 
 than the first (comp. 370, (8.)), if the chord fq itself he considered as small of the 
 frst order. 
 
 375. This occasion may be taken (comp. 374, 1. II. III.), to give 
 a new Enunciation of Taylor^ s Theorem, in a form adapted to Quater- 
 nions, which has some advantages over that given (342) in the pre- 
 ceding Chapter. We shall therefore now express that important 
 Theorem as follows: — 
 
 *' If none of the m^\ functions, 
 
510 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 I. . . fq, Afq, d?fq, . . . d."'fq, in which ^^q = 0, 
 become infinite in the immediate vicinity of a given quaternion g, then the 
 quotient y 
 
 II. . . Q= {/(^ + dy)-/^-d/^-^-^- S|_&e. 
 
 ~ 2.3..w) ' 2.3..m' 
 can he made to tend indefinitely to zero, for any ultimate value of the 
 versor Udg, ly indefinitely diminishing the tensor Td^." 
 
 (1.) The jproo/ of the theorem, as thus enunciated, can easily be supplied by an 
 attentive reader of Articles 341, 342, and their sub-articles; a few hints may how- 
 ever here be given. 
 
 (2.) We do not now suppose, as in 342, that d»"/^ must be different from zero; 
 we only assume that it is not infinite: and we add, to the expression 342, VI. for 
 Fx, the term, 
 
 ■ * ■ 2.3. ..?« * 
 (3.) Hence eaoA of the expressions 342, VII , for the successive derivatives oi 
 Fx, receives an additional term; the last of them thus becoming, 
 IV. . . Jy^Fx = i^W^r = d'»/(5 + xiiix) - d"»/<?; 
 so that we have now (com p. 342, X.) the values 
 
 V. ..^0=0, F'O, -F"0=0,... F('«-i)0 = 0, F('»)0 = 0. 
 (4.) Assuming therefore now (comp. 342, XII.) the new auxiliary function, 
 
 a;'"do"* 
 VI. . . i//a; = ^ „ ^ , with Tdg > 0, 
 2. o . . .m 
 
 which gives, 
 
 VII. . . ;//0 = 0, »//'0 = 0, -^"Q = 0, . . ^/C*"- »0 = 0, i//('")0 = dj"*, 
 we find (by 341, (8.), (9.), comp. again 342, XII.) that 
 VIII. . . lim. {Fx : i//a;) = 0. 
 
 (5.) But these two new functions, Fx and i|/x, are formed from the dividend and 
 the divisor of the quotient Q in II., by changing Aq to a:dg; and (comp. 342, (3.)) 
 instead of thus 7nultiplying a given quaternion differential dg', by a small and indefi- 
 nitely decreasing scalar, x, we may indefinitely diminish the tensor, Tdg, unthout 
 changing the versor, Udg-. 
 
 (6.) And even if^^q he changed, while the differential dg is thus made to tend 
 to zero, we can always conceive that it tends to some, limit ; which limiting or ulti- 
 mate value of that versor Udg may then be treated as if'it were a constant one, with- 
 out affecting the limit of the quotient Q. 
 
 (7.) The theorem, as above enunciated, is therefore fully proved ; and we are at 
 liberty to choose, in any application, between the two forms of statement, 342 and 
 375, of which one is more convenient at one time, and the other at another. 
 
CHAP. III.] OSCULATING PLANES TO CURVES IN SPACE. * 511 
 
 Section 4 On Osculating Planes, and Absolute Normals, to 
 
 Curves of Double Curvature. 
 
 376. The variable vector pt of a curve in space may in general 
 be thus expressed, with the help of Taylor's Series (corap. 370, 
 
 (1.)): 
 
 I. . . pt = p + t/ + \fup"^ with t^o = 1 ; 
 
 /), p', p", u being here abridged symbols for />oj p'q, p"oi Wj.; and the 
 product up" being a vector, although the factor u is generally a qua- 
 ternion (comp. 370, (5.)). And the different terms of this expres- 
 sion I. may be thus constructed (compare the annexed Figure 77): 
 
 II... P 
 while III.. 
 
 OP; //)' = pt; ■|«2w/' = tq; 
 
 Pt = OQ, and tp' + ^fup" = pq ; 
 
 the line tq, or the term ^t'up'\ being thus what 
 maybe called the deflexion ofthecwrrePQR, at q, 
 from its tangent pt at p, measured in a direction 
 which depends on the law according to which pt 
 varies with t, and on the distance of q, from p. 
 The equation of the plane of the triangle ptq is 
 rigorously (by II.) the following, with no for its 
 variable vector, 
 
 IV.. .0=^up"p'{w-p)', 
 
 this plane IV. then touches the curve at p, and (generally) cuts it at 
 Q,; so that if the point q, be conceived to approach indefinitely toP, 
 the resulting formula, 
 
 V. . . = Sp"p' {iv - p\ or v. . . = Sp'p" (iv - p\ 
 is the equation of the plane ptq in that limiting position, in which it 
 is called the osculating plane, or is said to osculate to the curve pq,r, 
 at the point P. 
 
 (l.) If the variable vector p be immediately given as ti function ps of a variable 
 scalar, s, which is itself a. function of the former scalar variable t, we shall then 
 have (comp. 331) the expressions, 
 
 VI. . . p't = s'Dsps, p"t = s"DsPs + «'^D,2ps5 with s' = D^s, s"= Bt^s ; 
 
 thus the vector p" may change, even in direction, when we change the independent 
 scalar vai-iable ; but p" will always be a line, either in or parallel to the osculating 
 plane ; while p will always represent a tangent, whatever scalar variable may be 
 selected. 
 
 (2.) As an example, let us take the equation 314, XV., or 369, XIII., of the 
 
512 * ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 helix. With the independent variable t of that equation, we have (comp. 369, XIV.) 
 the derived expressions, 
 
 VIL..p' = ca-f|a^^iA p" = -^^ p)3 = (^^ ] V' 
 
 P) 
 
 p" has therefore here (comp. 369, (8.)) the direction of the normal to the cylinder ; 
 and consequently, the oseulatiiig plane to the helix is a normal plane to the cylinder 
 of revolution, on which that curve is traced : a result well known, and which will 
 soon be greatly extended. 
 
 (3.) When a curve of double curvature degenerates into a. plane curve, its oscu- 
 lating plane becomes constant, and reciprocally. The condition of planarity of a 
 curve in space may therefore be expressed by the equation, 
 
 VIII. . . UVjo'p" = + a constant unit line ; 
 or, by 335, II., and 338, VIII., 
 
 ix...o = v^iP>7=v^p'p"'- 
 
 Vp'p" YqY ' 
 
 or finally, X. . . Sp'p'p"' = 0, or XI. . . p" \\\ p', p". 
 
 (4.) Accordingly, for a plane curve, if \ be a given normal to its plane, we have 
 the three equations, 
 
 XII. . . SXp' = 0, SXp" = 0, SXp'" = ; 
 which conduct, by 294, (11.), to X. 
 
 (5.) For example, if we had not otherwise known that the equation 337, (2.) 
 represented a plane ellipse, we might have perceived that it was the equation of some 
 plane curve, because it gives the three successive derivatives, 
 
 XIII. . . p' - ^ Va«-i/3, p" = - f ^ J Ya% p'" = " f | Y Va'-'/3, 
 
 which are complanar lines, the third having a direction opposite to the first. 
 
 (6.) And generally, the formula X. enables us to assign, on any curve of double 
 curvature, for which p is expressed as a function of t, the points* at which it most 
 resembles a plane curve, or approaches most closely to its own osculating plane. 
 
 377. An important and characteristic property of the osculating 
 plane to a curve of double curvature, is that the perpendiculars let 
 fall on it, from points of the curve near to the point of osculation, 
 are small of an order higher than the second, if their distances from 
 that ^om^ be considered as small of the. first order. 
 
 (1.) To exhibit this by quaternions, let us begin by considering an arbitrary 
 plane, 
 
 * Namely, in a modern phraseology, the places o^ four-point contact with a 
 plane. The equation, Vp'p"= 0, indicates in like manner the places, if any, at which 
 a curve has three-point contact with a right line. For curves of double curvature, 
 these are also called points of simple and double inflexion. 
 
CHAP. III.] CONE OF PARALLELS TO TANGENTS. 513 
 
 I. . . S\(a»-p) = 0, with T\ = l, 
 
 drawn through a point p of the curve. Using the expression 376, I., for the vector 
 OQ, or pt, of another point q of the same curve, we have, for the perpendicular dis- 
 tance of Q from the plane I., this other rigorous expression, 
 
 II. . . SX(p« - p) = iS\p + |<2SXmp" ; 
 
 which represents, in general, a small quantity of ih& first order, \it be assumed to 
 be such. 
 
 (2.) The expression II. represents however, generally, a small quantity of the 
 second order, if the direction of \ satisfy the condition, 
 
 III. . .SXp' = 0; 
 that is, if the plane I. touch the curve. 
 
 (3.) And if the condition, 
 
 IV. . . SXp"=0, 
 
 be also satisfied by X, then, but not othertoise, the expression II. tends to bear an 
 evanescent ratio to t^, or is small of an order higher than the second. 
 
 (4.) But the combination of the two conditions. III. and IV., conducts to the 
 expression, 
 
 V. ..X = + UVp'p"; 
 
 comparing which with 376, V., we see that the property above stated is one which 
 belongs to the osculating plane, and to no other. 
 
 378. Another remarkable property* of the osculating plane to a 
 curve is, that it is the tangent plane to the cone of parallels to tangents 
 (369, (6.)), which has its vertex at the point of osculation. 
 
 (1.) In general, if p = 0a; be (comp. 369, I.) the equation of a curve in space, 
 the equation of the cone which has its vertex at the origin, and passes through this 
 curve, is of the form, 
 
 I. .. p = l/(l>x; 
 
 in which x and ?/ are two independent and scalar variables. 
 (2.) We have thus the two partial derivatives, 
 
 II. . . Da-p = r/^'x, Dyp = (px ; 
 and the tangent plane along the side (x) has for equation, 
 
 IIL . . = S(w. <px . <j)'x) ; or briefly. III'. . . = Suxpf. 
 (3.) Changing then x, <p, 0', a> to t, p', p", ot — p, we see that the equation 376, 
 v., of the osculating plane to the curve 376, I., is also that of the tangent plane to 
 the cone of parallels, &c., as asserted. 
 
 379. Among all the normals to a curvCy at any one point, there 
 are two which deserve special attention ; namely the one which is in 
 
 * The writer does not remember seeing this property in print ; but of course it 
 is an easy consequence from the doctrine of infinitesimals, which doctrine however it 
 has not been thought convenient to adopt, as the hasis of the present exposition. 
 
 3 u 
 
514 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 the osculating plane, and is called the absolute (or principal) normal; 
 and the one which is perpendicular to that plane, and which it has 
 been lately proposed to name the hinormal.* It is easy to assign ex- 
 pressions, by quaternions, for these two normals, as follows. 
 
 (1.) The absolute normal, as being perpendicular to p', but complanar with p' 
 and p", has a direction expressed by any one of the following formulae (comp. 203, 
 334) : 
 
 I. .. VpY-p'"'; or II. ..dUp'; or III. . . dUdp. 
 (2.) There is an extensive classf of cases, for which the following equations hold 
 good: 
 
 IV. . . Tp' = const. ; V. . . p'2 = const. ; VI. . . Sp'p"= ; 
 
 and in all such cases, the expression I. reduces itself to p", which is therefore then a 
 representative of the absolute normal. 
 
 (3.) For example, in the case of the helix, with the equation several times be- 
 fore employed, the conditions (2.) are satisfied ; and accordingly the absolute nor- 
 mal to that curve coincides with the normal p" to the cylinder, on which it is traced ; 
 the locus of the absolute normal being here that screw surface or Selicoid, which 
 has been already partially considered (comp. 314, (11.), and 372, (4.)). 
 
 (4..) And as regards the binormal, it may be sufficient here to remark, that be- 
 cause it is perpendicular to the osculating plane, it has the direction expressed by 
 one or other of the two symbols (comp. 377, V.), 
 
 VII. . . Y^'p", or Vir. . . Vdpd2p. 
 
 (5.) There exists, of course, a system of three rectangular planes, the osculating 
 plane being one, which are connected with the system of the three rectangular lines, 
 the tangent, the absolute normal, and the binormal, and of which any one who has 
 studied the Quaternions so far can easily form the expressions. 
 
 (6.) And a constructionX for the absolute normal may be assigned, analogous 
 to and including that lately given (378) for the osculating plane, as an interpreta- 
 tion of the expression II. or III., or of the symbol dUp'or dUdp. From any origin 
 o conceive a system of unit lines (Up' or Udp) to be drawn, in the directions of the 
 successive tangents to the given curve of double curvature ; these lines will terminate 
 
 * By M. de Saint- Venant, as being perpendicular at once to two consecutive ele- 
 ments of the curve, in the infinitesimal treatment of this subject. See page 261 of the 
 very valuable Treatise on Analytic Geometry of Three Dimensions (Hodges and Smith, 
 Dublin), by the Rev. George Salmon, D. D., which has been published in the present 
 year (1862), but not till after the printing of these Elements of Quaternions (begun iu 
 1860) had been too far advanced, to allow the writer of them to profit by the study 
 of it, so much as he would otherwise have sought to do. 
 
 t Namely, those in which the arc of the curve, or that arc multiplied by a scalar 
 constant, is taken as the independent variable. 
 
 X This construction also has not been met with by the writer in print, so far as 
 he remembers ; but it may easily have escaped his notice, even iu the books which he 
 has seen. 
 
CHAP. III.] ABSOLUTE NORMALS, GEODETIC LINES. 515 
 
 on a certain spherical curve; and the tangent, say ss'', to this new curve, at the point 
 8 which corresponds to the point P of the old one, will have the direction of the ab- 
 solute normal at that old point. 
 
 (7.) At the same time, the plane oss' of the great circle, which touches the new 
 curve upon the unit sphere, being the tangent plane to the cone of parallels (378), 
 has the direction of the osculating plane to the old curve ; and the radius drawn to 
 its pole is parallel to the hinormal. 
 
 (8.) As an example of the auxiliary (or spherical) curve, constructed as in (6.), 
 we may take again the helix (369, XIII., &c.) as the given curve of double curva- 
 ture, and observe that the expression 369, XIV., namely, 
 
 TT 7r2j3^ 
 
 VIII. . . p' = ca+~a^^% gives IX. . . p'2 = - c^ -h -^ = const, (comp. (3.)); 
 
 whence Tp' is constant (as in IV.), and we have the equation (comp. 369, XV. 
 XIX.), 
 
 / 7r2/32Vi 
 X. . . SaUp' = -cl c2 ^ ] =_ cos a = const., 
 
 a being again the inclination of the helix to the axis of its cylinder ; which shows 
 that the new curve is in this case a plane one, namely a certain small circle of 
 the unit sphere. 
 
 (9.) In general, if the given curve be conceived to be an orbit described by a 
 point, which moves with a constant velocity taken for unity, the auxiliary or sphe- 
 rical curve becomes what we have proposed (100, (5.)) to call the hodograph of that 
 motion. 
 
 (10.) And if the given curve be supposed to be described with a variable velo- 
 city, the hodograph is still some curve upon the cone of parallels to tangents. 
 
 Section 5. — On Geodetic Lines, and Families of Surfaces, 
 
 380. Adopting as the definition of a geodetic line, on any proposed 
 curved surface, the property that it is one of which the osculating 
 plane is always a normal plane to that surface, or that the absolute 
 normal to the curve is also the normal to the surface, we have two 
 principal modes of expressing by quaternions this general and ckarac^ 
 teristic property. For we may either write, 
 
 I. . . Si/pV = 0, or II. . . Si/d/)dV = 0, 
 to express that the normal v to the surface (comp. 373) is perpen- 
 dicular to the hinormal Np'p" or Vd^d^ to the curve (comp. 379^ 
 VII. VII'.) ; or else, at pleasure, 
 
 III. . . Vi^(U/)' = 0, or IV. . . Vi^dUd/> = 0, 
 to express that the same normal v has the direction of the absolute 
 normal (Up')' or dUd/a (comp. 379, H* HI.), to the same geodetic 
 line. And thus it becomes easy to deduce the known relations of 
 such lines (or curves) to some im'portant families of surfaces^ on which 
 
516 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 they can be traced. Accordingly, after beginning for simplicity 
 with the sphere, we shall proceed in the following sub-articles to de-, 
 termine the geodetic lines on cylindrical and conical surfaces, with 
 arbitrary bases; intending afterwards to show how the correspond- 
 ing lines can be investigated, upon developable surfaces, and surfaces 
 of revolution. 
 
 (1.) On a sphere^ with centre at the origin, we have v || p, and the differential 
 equation IV. admits of an immediate integration ;* for it here becomes, 
 V. . . = VpdUdp =dVjoUdp, whence VI. . . VpUdp = w, and VII. . . Sw/o = 0, 
 10 being some constant vector ; the curve is therefore in this case a great circle, as 
 being wholly contained in one diametral plane. 
 (2.) Or we may observe that the equation, 
 
 VIII. . . ^pp'p"= 0, or IX. . . SpdpdV = 0, 
 obtained by changing j/ to p in I. or II., has generally for a first integral (comp, 
 335, (1.)), whether Tp be constant or variable, 
 
 X. . . U Vpp' = UVpdp = w = const. ; 
 it expresses therefore that p is the vector of some curve (or line) in a plane through 
 the origin ; which curve must consequently be here a great circle, as before. 
 (3.) Accordingly, as a verification of X., if we write 
 
 XI. . . p = ax-\- j3y, X and y being scalar functions of t, 
 where t is still some independent scalar variable, and a, /3 are two vector constants, 
 we shall have the derivatives, 
 
 XII. . . Q' = ax' + I3y\ p" = ax" + (3y" \\\p,p'] 
 so that the equation VIII. is satisfied. 
 
 (4.) For an arbitrary cylinder, with generating lines parallel to a fixed line a, 
 we may write, 
 
 XIII. . .Sav = 0, XIV. . . SadUdp = 0, XV. . . SaUdp = const. ; 
 
 a geodetic on a cylinder crosses therefoie the generating lines at a constant angle, 
 and consequently becomes a right line when the cylinder is unfolded into a. plane : 
 both which known properties are accordingly verified (comp. 369, (5.), and 376, 
 (2.)) for the case of a cylinder of revolution, in which case the geodetic is a helix. 
 (5.) For an arbitrary cone, with vertex at the origin, we have the equations, 
 
 XVI. . . Sz/p = 0, XVII. . . SpdUdp = 0, 
 XVIII. . . dSpUdp = S(dp.Udp) = - Tdp ; 
 
 multiplying the last of which equations by 2SpUdjO, and observing that - 2Spdp 
 = - d . p', we obtain the transformations, 
 
 * We here assume as evident, that the differential of a variable cannot be con- 
 stantly zero (comp. 335, (7.)) ; and we employ the principle (comp. 338, (5.)), 
 that V. dp Udp = - VTdp = 0. 
 
CHAP. III.] GEODETICS ON SPHERES, CONES AND CYLINDERS. 517 
 
 XIX. . . = d { (SpUd|o)2 + p2 } = d . ( VpUdp)«, XX. . . T VpUdp = const. ; 
 the perpendicular from the vertex^ on a tangent to any one geodetic upon a cone, has 
 therefore a constant length; and all such tangents touch also a concentric sphere^* 
 or one which has its centre at the vertex of the cone. 
 
 (6.) Conceive then that at each point p or p' of the geodetic a tangent pt or p't' 
 is drawn, and that the angles otp, ot'p' are right ; we shall have, by what has just 
 been shown, 
 
 XXI. . . OT = or' = const. = radius of concentric sphere ; 
 and if the cone be developed (or unfolded) into a plane, this constant or common 
 length, of the perpendiculars from o on the tan- 
 gents, will remain unchanged, because the length 
 OP and the angle opt are unaltered by such de- 
 velopment ; the geodetic becomes therefore some 
 plane line, with the same property as before ; 
 and although this property would belong, not 
 only to a right line, but also to a circle with o 
 for centre (compare the second part of the an- 
 nexed Figure 78), yet we have in this result 0^ 
 merely an effect of the foreign factor SpUdp, 
 which was introduced in (5.), in order to facili- 
 tate the integration of the differential equation 
 XVIII., and which (by that very equation) cannot be constantly equal to zero. We 
 are therefore to exclude the curves in which the cone is cut by spheres concentric 
 with it : and there remain, as the sought geodetic lines, only those of which the de- 
 velopments are rectilinear, as in (4 ). 
 
 (7.) Another mode of interpreting, and at the same time of integrating, the 
 equation XVIII., is connected with the interpretation of the symbol Tdp ; which can 
 be proved, on the principles of the present Calculus, to represent rigorously the dif- 
 ferential ds of the arc (s) of that curve, whatever it may be, of which p is the varia- 
 ble vector ; so that we have the general and rigorous equation, 
 
 XXII. . . Tdp = ds, if s thus denote the arc : 
 whether that arc itself, or some other scalar, t, be taken a.s the independent variable ; 
 and whether its differential ds be small or large, provided that it be positive. 
 
 (8.) In fact if we suppose, for the sake of greater generality, that the vector p 
 and the scalar s are thus both functions, pt and st, of some one independent and sca- 
 lar variable, t, our principles direct us first to take, or to conceive as taken, a suhmul- 
 tiple, n-'d<, oi ih.Q finite differential At, considered as an assumed and arbitrary in- 
 crement of that independent variable, t ; to determine next the vector pun'^dt, and 
 the scalar st+n~^dt, which correspond to the point Ptm'^dt of the curve on which pt ter- 
 minates in P;, and of which st is the arc, ^^t^ measured to Ft from some fixed point 
 Po on the same curve ; to take the differences, 
 
 Fig. 78. 
 
 * When the cone is of the second order, this becomes a case of a known theorem 
 respecting geodetic lines on a surface of the same second order, the tangents to any 
 one of which curves touch also a confocal surface. 
 
518 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 pt+n'^dt - pt, and stHi'^dt - St, 
 •which represent respectively the directed chord, and the length, of the arc Ft^t+rTht, 
 which arc will generally be small, if the number n be large, and will indefinitely di- 
 minish when that number tends to infinity; to multiply these two decreasing diffe- 
 rences, of pt and St, by n ; and finally to seek the limits to which the products tend, 
 when n thus tends to oo : such limits being, by our definitions, the values of the two 
 sought and simultaneous differentials, dp and d«, which answer to the assumed va- 
 lues of t and dt. And because the small arc, As, and the length, TAp, of its small 
 chordf in the foregoing construction, tend indefinitely to a ratio of equality, such 
 must be the rigorous ratio of ds and Tdp, which are (comp. 320) the limits of their 
 equimultiples. 
 
 (9.) Admittmg then the exact equality XXII. of Tdp and ds, at least when the 
 latter like the former is taken positively, we have only to substitute — ds for - Tdp in 
 the equation XVIII., which then becomes immediately iutegrable, and gives, 
 
 XXIII. . . s + SpUdp = s - S (p : Udp) = const. ; 
 where S(p :Udp) denotes the projection tp, of the vector p or op, on the tangent to 
 the geodetic at p, considered as a positive scalar when p makes an acute angle 
 with dp, that is, when the distance Tp or op ffom the vertex is increasing; while s 
 denotes, as above, the length of the arc PqP of the same curve, measured from some 
 fixed point Pq thereon, and considered as a scalar which changes sign, when the va- 
 riable point p passes through the position Pq. 
 
 (10.) But the length of tp does not change (comp. (6.)), when the cone is deve- 
 loped, as before ; we have therefore the equations (comp. again Fig. 78), 
 
 /'-^ — /"^s — /^*N 
 
 XXIV. . . PoP — TP = const. = PoP' - t'p', XXV. . . pp' = t'p' — tp, 
 
 which must hold good both before and after the supposed development of the conical 
 surface ; and it is easy to see that this can only be, by the geodetic on the cone be- 
 coming a right line, as before. In fact, if ot' in the plane be supposed to intersect 
 the tangent tp in a point t , and if p' be conceived to approach to p, the second 
 member of XXV. bears a limiting ratio of equality to the first member, increased or 
 diminished by tt, ; which latter line, and therefore also the angle tot' between the 
 perpendiculars on the two near tangents, or the angle between those tangents them- 
 selves, if existing, must bear an indefinitely decreasing ratio to the arc P?' ; so that 
 the radius of curvature of the supposed curve is infinite, or t' coincides with T, and 
 the development is rectilinear as before. 
 
 (11.) The important and general equation, Tdp = ds (XXII,), conducts to many 
 other consequences, and may be put under several other forms. For example, we 
 may write generally, 
 
 XXVI. . . TD«p = 1, XXVII. . . (dsp)2 + 1 = 0; 
 also XXVIII. . . (d^p)2 + (d^s)2 = 0, or XXIX. . . p'2 + s'2 = o, 
 
 if p' and s' be the first derivatives of p and s, taken with respect to any independent 
 scalar variable, such as t ; whence, by continued derivation, 
 
 XXX. . . Sp>"+ s's" = 0, XXXI. . . Sp'p"'+ p"2 + «s"' + s"2= 0, &c. 
 
 (12.) And if the arc s be itself idken as the independent variable, then (comp. 
 379, (2.)) the equations XXIX., &c., become, 
 
 XXXII. . . p'2 + 1 = 0, Sp'p" = 0, Sp'p'" + fi = 0, &c. 
 
CHAP. III.] DIFFERENTIAL AND SCALAR EQUATION. 519 
 
 381. In general, if we conceive (comp. 372, I.) that the vector p 
 of a given surface is expressed as a given function of two scalar varia- 
 bles, X and ?/, whereof one, suppose y, is regarded at first as an un- 
 known function of the other, so that we have again, 
 
 !.../> = 0(a;, y\ but now with 11. , . y -fx, 
 where the/orm of is Tcnown^ but that of/ is sought; we may then 
 regard /> as being implicitly a function of the single (or independent) 
 scalar variable^ x, and may consider the equation, 
 
 III. . .p = (p(x,fx), 
 as being that of some curve on the given surface, to be determined by 
 assigned conditions. Denoting then the unknown total derivative 
 D<p{x,fx) by p', but the known partial derivatives of the same first 
 order by d^0 and Dy0, with analogous notations for orders higher 
 than the first, we have (comp. 376, VI.) the expressions, 
 
 IV. . p' = D^0 + y'DyCp, p" =D/(^ + 2y'D^Dy(p + y'\^(p + y^'D^cp, &c. ; 
 
 in which y' ^Ti^y =f'x, y" -T)^y-f'x^ &c. Hence, writing for the 
 normal v to the surface the expression, 
 
 V. . . 1/ = V(d^0 . Dj,0) = V. D^.0Dj,0, comp. 372, V., 
 
 or this vector multiplied by any scalar, the equation 380, I. of a 
 geodetic line takes this new form, 
 
 vi. . . o=s^/)y'=s(v.D,0D,0.V/)yO; 
 
 or, by a general transformation which has been often employed 
 already (comp. 352, XXXI., &c.), 
 
 VII. . . O=S/d,0.S/'d^0-S/d^0.S/'Dj,0; 
 
 and thus, by substituting the expressions IV. for p' and p'\ we ob- 
 tain an ordinary (or scalar) differential equation, of the second order, 
 in X and y, which is satisfied hy all the geodetics on the given surface, 
 and of which the complete integral (when found) expresses, with two 
 arbitrary and scalar constants, the form of the scalar function f in II., 
 or the law of the dependence of y on x, for the geodetic curves in 
 question. 
 
 (1.) As"an example, let us take the equation, 
 
 VIII. . .9 = ^{x,y) =y4)X, comp. 378, I., 
 
 of a cone with its vertex at the origin ; which cone becomes a known one, when the 
 form of the vector function yp is given, that is, when we know a guiding curve p = ^px, 
 through which the sides of the cone all pass. We have here the partial derivatives, 
 
520 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 IX. . . i>x^ = i/Dx^px = i/ip\ Vyf^ipx^-ij/y comp. 378, II. ; 
 and X. . . Da;2^ = yDa;2i//a;=y^", d^jDj,^ = i/'', Dj,2^=0; 
 
 the expressions IV. become, tben, 
 
 XI. . . p= i/\p' + y'«//, p" = i/\p" + 2^'i//' + 1/"^ ; 
 and since only the direction of the normal is important, we may divide V. by — y, 
 and write, 
 
 XII. . . v = Y^p^'. 
 
 (2.) The expressions XI. and XII. give (comp. VI. and VII.) for the geodetic* 
 on the cone VIII., the differential equation of the second order, 
 
 XIII. . . =S(Y^PxP'.Yp'p") = Sp'rpSp'xP' - Sp"4^'&p'^ 
 
 = CyS^^p" + 27/'SxP^' + !/"^^) (yi//'2 + y'S^/^p') 
 
 - (yS;//'-^" + 2y'»//'2 + y"S;//i//') (yS-^>//' + y'i//2), 
 
 in which i//3 and yp'^ are abridged symbols for (;^x)2 and(i|/'a;)2; but this equation in 
 
 X and 2/ may be greatly simplified, by some permitted suppositions. 
 
 (3.) Thus, we are allowed to suppose that the guiding curve (1.) is the intersec- 
 tion of the cone with the concentric unit sphere, so that 
 
 XIV. ..T;^a; = l, yp'^ = -l, S-^;//'=0, S^'i/'" + ^'2 = ; 
 and if we further assume that the arc of this spherical curve is taken as the inde- 
 pendent variable, x, we have then, by 380, (12.), combined with the last equation 
 XIV., 
 
 XV. ..T;//'a;=l, »|/'2 = _i, Sf;^" = 0, S^//i//" = -iP'2= I. 
 
 (4.) "With these simplifications, the differential equation XIII. becomes, 
 XVI. . . 0-(y-y") (_y)-(-2y') (-y')=yy'-2y'2-y2; 
 and its complete integral is found bg ordinary methods to be, 
 XVII. . . y = 6 sec (ic + c), 
 
 in which 6 and c are two arbitrary but scalar constants. 
 
 (5.) To interpret now this integrated and scalar equation in x and y, of the^fo- 
 detics on an arbitrary cone, we may observe that, by the suppositions (3.), y repre- 
 sents the distance, Tp or op, from the vertex o, and x + c represents the angle aop, 
 in the developed state of cone and curve, from some Jixed line OA in the plane, to the 
 variable line op ; the projection of this new op on thatj^a;ed line OA is therefore con- 
 stant (being = b, by XVII.), and the developed geodetic is again found to be a right 
 line, as before. 
 
 382. Let ABODE . . . (see the annexed Figure 79) be any given se- 
 ries of points in space. Draw the succes- , 
 sive right lines, ab, bc, cd, de, . . and pro- - -"'■"' "^^v-c' 
 long them to points e', c', d', e', . . . the ^^--^^^'^ c" — p^^ ^— -Ap' 
 lengths of these prolongations being ar- ^ -^ ""^'e' 
 bitrary; join also b'c', c'd', d'e', . . . We ^^^' '^^' 
 shall thus have a series of plane triangles, b^bc', c'cb', d'de', ... all ge- 
 nerally in different planes ; so that bcd'c'b', cde'd'c', . . . are generally 
 gauche pentagons, while bcde'd'c'b' is a gauche heptagon, &c. But we 
 
CHAP. III.] DEVELOPABLE SURFACE, CUSP-EDGE. 521 
 
 can conceive the jftrst triangle b'bc' to turn round its sideBCc\ till it 
 C07nes into the plane of the second triangle, c'cd'; which -will trans- 
 form the first gauche pentagon into a plane one, denoted still by 
 bcd'cV. We can then conceive this plane figure to turn round its 
 side cdd', till it comes into the plane of the third triangle, d'de'; 
 whereby the first gauche heptagon will have become a plane one, de- 
 noted as before by bcdeVc'b': and so we can proceed indefinitely. 
 Passing then to the limit, at which the points abcde . . . are conceived 
 to be each indefinitely near to the one which precedes or follows it in 
 the series, we conclude as usual (comp. 98, (12.)) that the locus of 
 the tangents to a curve of double curvature is a developable surface : or 
 that it admits of being unfolded (like a cone or cylinder) into a plane, 
 without any breach of continuity. It is now proposed to translate 
 these conceptions into the language of quaternions, and to draw from 
 them some of their consequences: especially as regards the determi- 
 nation of the geodetic lines, on such a developable surface. 
 
 (1.) Let i//ar, or simply i^, denote the variable vector of a point upon the curve^ 
 or cvsp-edge, or edge of regression of the developable, to which curve the generating 
 lines of that surface are thus tangents, considered as a. function -^ of its arc, x, mea- 
 sured from some fixed point A upon it ; so that while the equation of the surface 
 will be of the form (comp. 100, (8.)), 
 
 T. . . p = (pQc, g)=\p^-\- y-y^ = t// 4 yy^j', 
 y being a second scalar variable, we shall have the relations (comp. 381, XV.), 
 II. . . Ti//'a;=l, i//'2 = -l, Sf;p"=0, S;//'i//"' = -t//"2 = 22^ if z=Tt//". 
 (2.) Hence III. . . T)x(p=^'^yV, D^^ = »//'; 
 
 IV. . . p' = {i+y'W^yi'"i /o"=y'y + (i + V)i/'" + y '/'"'; 
 
 and V. . . J/ = Y\p'\p" = i//'ip", multiplied by any scalar. 
 
 (3.) The differential equation of the geodetics may therefore be thus written 
 (comp. 381, XIIL), 
 
 VI. . . = S(Y^P'4J".Yp'p") = Sp'^p•'Sp"^P' - Sp";|/"Sp'i//' ; 
 in which, by (1.) and (2.), 
 
 VII /^P''^" = -2'^^ Sp"4/'=-y" + y22, 
 
 • • • lSp>"=-(l + 2«/0^2-y^^', Sp';^' = -(1+/); 
 the equation becomes therefore, after division by — z, 
 
 VIII. . . = z{(l +y')2 + 0^2)2} + (1 +y') {yz)'-y"yz, 
 or simply, 
 
 IX. . . z + tj'=0, or IX'. . . TdJ/' + dw=0, if X. . . tan u=-^. = ^^. 
 
 1 + y 1+y' 
 
 (4.) To interpret now this very simple equation IX. or IX'., we may observe 
 that 2, or Tt//", or Tdi|/' : da;, expresses the limiting ratio, which the angle between 
 two near tangents i//' and i//' + A\//', to tlie cusp-edge (1.), bears to the small arc Aa5 
 
 3 X 
 
522 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 of that curve which is intercepted between their points of contact ; while v is, by IV., 
 that other angle, at which such a variable tangent, or generating line of the deve- 
 lopable, crosses the geodetic on that surface ; and therefore its derivative, v' or dr : da?, 
 represents the limiting ratio, which the change Av of this last angle, in passing from 
 one generating line to another, bears to the same small arc Ax of the curve which 
 those lines touch. 
 
 (5.) Referring then to Figure 79, in which, instead of tuo continuous curves, 
 there were two gauche polygons, or at least two systems of successive right lines, con- 
 nected by prolongations of the lines of the first system, we see that the recent formula 
 IX. or IX'. is equivalent to this limiting equation, 
 
 cd'c'-bcV 
 XI, . . lim ■ — ; = - 1 ; 
 
 CCD 
 
 but these three angles remain unaltered, in the development of the surface : the bent 
 line b'c'd' for space becomes therefore ultimately a straight line in the plane, and si- 
 milarly for all other portions of the original polygon, or twisted line, b'c'd'e' . . ., of 
 which b'c'd' was a part. 
 
 (6.) Returning then to curves and surfaces in space, the quaternion analysis (3.) 
 is found, by this simple reasoning,* to conduct to an expression for the known and 
 characteristic property of the geodetics on a developable : namely that they become 
 right lines, as those on cylinders (380, (4.)), and on cones (380, (6.) and (10.), or 
 381, (5.)), were lately seen to do, when the surface on which they are thus traced 
 is unfolded into a plane. 
 
 383. This known result, respecting geodetics on developahles, may 
 be very simply verified, by means of a new determination of the ab- 
 solute^ normal (379) to a curve in space, as follows. 
 
 (1.) The arc s of any curve being taken for the independent variable, we may 
 write (comp. 376, I.), by Taylor's Series, the following rigorous expressions, 
 
 I. . . p-s = p - sp' + ^s2m_sp", po = p, P« = P + sp' -f Is'iusp", with «o = 1, 
 for the vectors of tiiree near points, p_j, Pq, p«, on the curve, whereof the second bi- 
 sects the arc, 2s, intercepted between the first and third. 
 
 (2.) If then we conceive the parallelogram p_sPoP«Ks to be completed, we shall 
 have, for the two diagonals of this new figure these other rigorous expressions, 
 
 II. . . ■P-sPs=ps-p-s=2sp' + ^s'^(us-u_s')p"; 
 
 III. . . PoRg = ps + p_s — 2po = ls~(us + u_s) p" ; 
 
 * In the Lectures (page 581), nearly the same analysis was employed, for geo- 
 detics on a developable ; but the interpretation of the result was made to depend on 
 an equation which, with the recent significations of ;// and v, may be thus written, as 
 the integral of IX'., » + jTdi//' = const. ; where jTd;//' represents i\xQ finite angle be- 
 tween the extreme tangents to i\iQ finite arc J Td;//, or A.r, oii\\Q cusp- edge, wlien 
 that curve is developed into a plane one. 
 
 t Called also, and perhaps more usually, the principal normal. 
 
CHAP. III.] GEODETICS ON DEVELOPABLES. 523 
 
 which give the limiting equations, 
 
 IV. . . lim. s-ip-sPs = 2p' ; V. . . lim. s'^PoKs - p". 
 
 (3.) But the length P-sP» of what may be called the long diagonal, or the chord 
 of the double arc, 2s, is ultimately equal to that double arc ; we have therefore, by 
 IV., the equation, 
 
 VI. . . Tp'= 1, if p' = Dsp, and if s denote the arc, 
 
 considered as the scalar variable on which the vector p depends : a result agreeing 
 with what was otherwise found in 380, (12.). 
 
 (4.) At the same time, since the ultimate direction of the same long diagonal is 
 evidently that of the tangent at Pq, we see anew that the same first derived vector p' 
 represents what may be called the unit-tangent* to the curve at that point. 
 
 (5.) And because the lengths of the two sides P-sPo and PqPs, considered as chords 
 of the two successive and equal arcs, s and s, are ultimately equal to them and to 
 each other, it follows that the parallelogram (2.) is ultimately equilateral, and there- 
 fore that its diagonals are ultimately rectangular; but these diagonals, by IV. and 
 v., have ultimately the directions of p' and p" ; we find therefore anew the equation, 
 
 VII. . . Sp'/o" = 0, if the arc be the independent variable, 
 
 which had been otherwise deduced before, in 880, (12.). 
 
 (6.) But under the same condition, we saw (379, (2.)) that the second derived 
 vector p" has the direction of the absolute normal to the curve ; such then is by V. 
 the ultimate direction of what we may call the short diagonal PqKs, constructed as 
 in (2.) ; or, ultimately, the direction of the bisector of the (obtuse) angle p.sPqPs, be- 
 tween the two near and nearly equal chords from the point Pq ; while the plane of 
 the parallelogram becomes ultimately the osculating plane. 
 
 (7.) All this is quite independent of the consideration of any surface, on which 
 the curve may be conceived to be traced. But if we now conceive that this curve 
 is formed //o/n a right line b'c'd' . . . (comp. Fig. 79), by wrapping round a develop- 
 able surface a plane on which the line had been drawn, and if the successive por- 
 tions b'c', c'd', . . of that line be supposed to have been equal, then because the two 
 right lines c'b' and c'l>' originally made supplementary angles with any other line 
 c'c in the plane, the two chords c'b' and cV of the curve on the developable tend to 
 make supplementary angles with the generatrix c'c of that surface ; on which ac- 
 count the bisector (6.) of their angle b'c'd' tends to he perpendicrdar to that generat- 
 ing line c'c, as well as to the chord b'd', or ultimately to the tangent to the curve at 
 c', when chords and arcs diminish together. The absolute normal (6.) to the curve 
 thus formed is therefore perpendicular to two distinct tangents to the surface at c', 
 and is consequently (comp. 372) the normal to that surface at that point ; whence, 
 by the definition (380), the curve is, as before, a geodetic on the developable. 
 
 (8.) As regards the asserted rectangularity (7.), of the bisector of the angle 
 b'c'd' to the line c'c, when the angles cc'b' and cc'd' are supposed to be supple' 
 mentary, but not in one plane, a simple proof may be given by conceiving that the 
 
 * Compare the Note to i)age 152. 
 
524 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 right line b'c' is prolonged to c", in such a manner that c'c" = c'd' ; for then these 
 two equally long lines from c' maiie equal angles with the line c'c, so that the one may- 
 be formed from the other by a rotation round that line as an axis; whence c"d', 
 which is evidently parallel to the bisector of b'c'd', is also perpendicular to c'c. 
 
 (9.) In quaternions, if a and p be any two vectors, and if t be any scalar, we 
 have the equation, 
 
 VIII. . .S.a(a'pa-«-p) = 0; 
 
 which is, by 308, (8.), an expression for the geometrical principle last stated. 
 
 384. The recent analysis (382) enables us to deduce with ease, 
 by quaternions, other known and important properties of develop- 
 able surfaces: for instance, the property that each such surface may 
 be considered as the envelope of a series of planes^ involving only one 
 scalar and arbitrary constant {ox parameter) in their common equation; 
 and that each plane of this series osculates to the cusp-edge of the de- 
 velopable. 
 
 (I.) The equation of the developable surface being still, 
 
 I. . . p-=<p{x,y)-^j: + y^'^ = •«//+ yi//', as in 382, I., 
 
 its normal v is easily found to have as in 382, V., the direction of V^/'^p", whether 
 the scalar variable a; be, or be not, the arc of the cusp-edge^ of which curve the 
 equation is, 
 
 II. . . p = ^x- 
 
 (2.) Hence, by 373, VII., the equation of the tangent plane takes the form, 
 III.. . Sw;P'i//" = S;//\//';//", 
 from which the second scalar variable y thus disappears : this common equation, of 
 all the tangent planes to the developable, involves therefore, as above stated, only 
 one variable and scalar parameter, namely x ; and the envelope of all these /)Zawes is 
 the developable surface itself. 
 
 (3.) The plane III., for any given value of this parameter x, that is, for 2iny given 
 point of the cusp-edge, touches the surface along the whole extent of the generating 
 line, which is the tangent to this last curve. 
 
 (4.) And b}' comparing its equation III. with the formula 376, V., we see at 
 once that this plane osculates to the same cusp-edge, at the point of contact of that 
 curve with the same generatrix of the developable. 
 
 385. If the reciprocals of the perpendiculars, let fall from a given 
 origin, on the tangent j^lcines to a developable surface, be considered 
 as being themselves vectors from that origin, they terminate on a 
 curve, which is connected with the cusp-edge of the developable by 
 some interesting relations of reciprocity (comp. 373, (21.)): in such 
 a manner that if this new curve be made the cusp-edge of a new de- 
 velopable, we can return from it to the former surface, and to its cusp- 
 edge, by a similar process of construction. 
 
CHAP. III.] THEOREM OF RECIPROCAL CURVES IN SPACE. 525 
 
 (1.) In general, if \px and Xx, or briefly ^ and %, be two vector functions of a 
 scalar variable x, such that x ^^^7 ^^ deduced from ^ by the three scalar equa- 
 tions, 
 
 I. ..St//x=c, S;//'x = 0, S4'"x=0, 
 
 in which S^x is written briefly for S(;//a;. X^:). and c is any scalar constant, we have 
 then this reciprocal system of three such equations, 
 
 11. ..Sx^ = c, Sx'V' = 0, Sx";^ = 0; 
 an intermediate step being the equation, 
 
 III. ..Sfx' = Sx'f =0. 
 (2.) Hence, generally, ^ 
 
 IV...ifx = ^;, then V...^=^^. 
 
 ^^^^ Sxxx 
 
 (3.) But if p be the variable vector of a curve in space, and p\ p" its first and 
 second derivatives with respect to any scalar variable, then, by the equation 376, V. 
 of the osculating plane to the curve, we have the general expression, 
 
 VI. . . ■ , „ = perpendicular from origin on osculating plane; 
 Vpp 
 
 so that if i// and x be considered as the vectors of two curves, each vector is c x the 
 reciprocal of the perpendicular, thus let fall from a common point, on the osculating 
 plane to the other. 
 
 (4.) We have therefore this Theorem: — 
 
 Jf from any assumed point, o, there be drawn lines equal to the reciprocals of 
 the perpendiculars from that point, on the osculating planes to a given curve of dou- 
 ble curvature, or to those perpendiculars multiplied by any given and constant sca- 
 lar ; then the locus of the extremities of the lines so drawn will be a second* curve, 
 from which we can return to the first curve by a precisely similar process. 
 
 386. The theory of developable surfaces, considered as envelopes 
 of planes -with one scalar and variahle parameter (384), may be addi- 
 tionally illustrated by connecting it with Taylor* s Series ^ as follows. 
 
 (1.) Let at denote any vector function of a scalar variable t, so that 
 I. . . ttf = ao + tuta'o = a + tua, with wq = 1 » 
 or, by another step in the expansion, 
 
 II. . . a< = ao + <a'o + |<^f <a"o = a 4- <a' + \tHa", vq = 1; 
 where u and v are generally quaternions, but ua and va" are vectors. 
 
 * The two curves may be said to be polar reciprocals, with respect to the (real or 
 imaginary) sphere, p^ = c, and an analogous relation of reciprocity exiats generally, 
 when the points of one curve are the poles of the osculating planes of the other, with 
 respect to any surface of the second order: corresponding tangents being then reci- 
 procal polars. Compare the theory oi developables reciprocal to curves, given in 
 Salmon's Analytical Geometry of Three Dimensions, page 89; see also Chapter XI. 
 (page 224, &c.), of the same excellent work. 
 
526 ELEMENTS OF QUATERNIONS. [bOOK HI. 
 
 (2.) Then, as the rigorous equation of the variable plane, the reciprocal of the 
 perpendicular on which from the origin is — at, we have either, 
 
 III. . . — 1 = Satp = Sap -r tSua'p, 
 or 
 
 IV. . . - 1 = Sap + <Sa'|0 + ^t^Sva'p, 
 
 according as we adopt the expression I., or the equally but not more rigorous ex- 
 pression II., for the variable vector at. 
 
 (3.) Hence, by the form III., the line of intersection of the two planes, which 
 answer to the two values and t of the scalar variable, or parameter, t, is rigorously 
 represented by the system of the two scalar equations, 
 
 V. ..Sa*+1 = 0, S«a'p = 0. 
 
 (4.) And the limiting position of this right line V., which answers to the con- 
 ceived indefinite approach of the second plane to the^rs^, is given with equal rigour 
 by the equations, 
 
 VI. . . Sap + 1 = 0, Sa'p = ; 
 
 whereof it is seen that the second may be formed from the^rs^, by derivating with 
 respect to t, and treating p as constant: although no such rule of calculation had 
 been previously laid down, for the comparatively geometrical process which is here 
 supposed to be adopted. 
 
 (5.) The locus of all the lines VI. is evidently some ruled surface; to determine 
 the normal v to which, at the extremity of the vector p, we may consider that vec- 
 tor to be a function (372) of two independent and scalar variables, whereof one is t, 
 and the other may be called for the moment w ; and thus we shall have the two 
 partial derivatives, 
 
 VII. . . SaD<p = 0, SaDu,p = 0, giving v || a. 
 
 (6.) Hence the line a has the direction of the required normal v; the plane 
 Sap +1 = touches the surface (comp. 884, (3.)) along the whole extent of the li- 
 miting line VI. ; and the locus of all such lines is the envelope of all the planes, of 
 the system recently considered. 
 
 (7.) The line VI. cuts generally the plane IV., in a point which is rigorously de- 
 termined by the three equations, 
 
 VIII. . . Sap -1- 1 = 0, Sa'p = 0, St7a"p = ; 
 and the limiting position of this intersection is, with equal rigour, the point deter- 
 mined by this other system of equations, 
 
 IX. . . Sap + 1 = 0, Sa'p = 0, Sa"p = ; 
 
 in which it may be remarked (comp. (4.)), that the third is the derivative of the 
 second, if p be treated as constant, 
 
 (8.) The locus of all these points IX. is generally some curve upon tlie surface 
 (5.), which is the locus of the lines VI., and has been seen to be the envelope (6.) of 
 the planes III. or IV. ; and to find the tangent to this curve, at the point answering 
 to a given value of t, or to a given line VI., we have by IX. the derived equations, 
 
 X. ..Sap' = 0, Sa'p'=0, whence p'|| Vaa'; 
 
 comparing which with the equations VI. we see that the lines VI. touch the curve, 
 which is thus their common envelope. 
 
CHAP. III.] FAMILIES OF SURFACES. 527 
 
 (9.) We see then, in a new way, that the envelope of the planes III., which have 
 one scalar parameter (t) in their common equation, and may represent ant/ system of 
 planes subject to this condition, is a developable surface : because it is in general 
 (comp. 382) the locus of the tangents to a curve in space, although this curve mat/ 
 reduce itself to a point, as we shall shortly see. 
 
 (10.) We may add that if at in III. be considered as the vector oi a given curve^ 
 this curve is the locus of the poles* of the tangent planes to the developable, taken 
 with respect to the unit sphere; and conversely, that the developable surface is the 
 envelope of the polar planes of the points of the same given curve, with respect to 
 the same sphere. 
 
 (11.) If then it happen that this given curve, with at for vector, is a plane one, 
 so that we have this new condition, 
 
 XI. . . 8(3at +1 = 0, jS being any constant vector, 
 namely the vector of the pole of the supposed plane of the given curve, the variable 
 plane III., or Spat+ 1 = 0, of which the surface (5.) is the envelope, passes con- 
 stantly through thin fixed pole ; so that the developable becomes in this case a cone, 
 with /3 for the vector of its vertex: the equations IX. giving now p = (3. 
 
 (12.) The same degeneration, of a developable into a conical surface, may also 
 be conceived to take place in another way, by the cusp-edge (or at least some finite 
 portion thereof) tending to become indefinitely small^ while yet the direction of its 
 tangents does not tend to become constant. For example, with recent notations, the 
 developable which is the locus of the tangents to the helix may have its equation 
 written thus : 
 
 2 
 XII. .. p = (a?, y) = c (a;a + - tan a . a*U/3) ^-ya (1 + tan a. a^U/3) ; 
 
 which when the quarter-interval, c, between the spires, tends to zero, without their 
 inclination a to the axis a being changed, tends to become a cone of revolution 
 round that axis, with its semiangle = a. 
 
 387. So far, then, we may be said to have considered, in the pre- 
 sent Section, and in connexion with geodetic lines, the four following 
 families of surfaces (if the first of them may be so called). First, 
 spherical sm faces ^ of which the characteristic j(?ro/?er^z/ is expressed 
 by the equation, 
 
 I. . . Vv(/j - a) = 0, if a be vector of centre; 
 second, cylindrical surfaces, with the property, 
 
 II. . . Si^a = 0, if a be parallel to the generating lines; 
 third, conical surfaces, with the property, 
 
 III. . . Sj^ (/> - a) = 0, if a be vector of vertex ; 
 
 and fourth, developable surfaces, with the distinguishing property 
 expressed by the more general equation, 
 
 * Compare the Note to page 525. 
 
528 ELEMENTS OF QUATERNIONS. [boOK 111. 
 
 IV. . . Yudv =0, if d/> have the direction of a generatrix ; 
 
 V being in each the normal vector to the surface, so that . 
 
 V. . . 81^(3/? = 0, for all tangential directions of d/j; 
 and \\iQ fourth family including the third, which in its turn includes 
 the second, A few additional remarks on these equations may be 
 here made. 
 
 (1.) The geometrical signification of the equation I. (as regards the radii) is ob- 
 vious ; but on the side of calculation it may be useful to remark, that elimination of 
 
 V between I. and V. gives, for spheres^ 
 
 VI. . . S(|0 - a) dp = 0, or VII. . . T(p - a) = const. 
 (2.) The equations II. and V. show that dp, and therefore Ap, may have the 
 given direction of a ; for an arbitrary cylinder, then, we have the vector equation 
 (372), 
 
 VIIT. . . p = 0(a;,y) = i//a; + ytr, 
 
 where ^l^x is an arbitrary vector function of x. 
 (3.) From VIII. we can at once infer, that 
 
 IX. . . S/3p = S/3i//^, Syp = Sy;^., if a = V/3y ; 
 the scalar equation (373) of a cylindrical surface is therefore generally of the ^brm 
 (comp. 371, (6.), (7.)), 
 
 X. ..0=F(S|3p, Syp); 
 
 (B and y being two constant vectors, and the generating lines being perpendicular to 
 both. 
 
 (4.) The equation III. may be thus written, 
 
 XI. . . SvVa = Ta-»Svp ; whence XII. . . SvVa = 0, if Ta = oo ; 
 the equation for cones includes therefore that for cylinders, as was to be expected, 
 and reduces itself thereto, Avhen the vertex becomes infinitely distant. 
 
 (5.) The same equation III., when compared with V., shows that dp may have 
 the direction of p - a, and therefore that p — a may be multiplied by any scalar ; the 
 vector equation of a conical surface is therefore of the form, 
 
 XIII. . . p — a + yxpx, 4'^ being an arbitrary vector function. 
 
 (6.) The scalar equation of a cone may be said to be the result of the elimination 
 of a scalar variable t, between tvvo equations of the forms, 
 
 XIV...S(p-a)x* = 0, S(p-a)x'. = 0, 
 which express that the cone is the envelope (comp. 386, (11.)) of a variable plane, 
 which passes through a. fixed point, and involves only one scalar parameter in its 
 equation : with a new reduction to a cylinder, in a case on which we need not here 
 delay. 
 
 (7.) The equation IV. implies, that for each point of the surface there is a direc- 
 tion along which we may move, without changing the tangent plane ; and therefore 
 that the surface is an envelope of planes, &c., as in 386, and consequently that it is 
 developable, in the sense of Art. 382. 
 
 (8.) The vector equation of a general developable surface may be written under 
 the form, 
 
CHAP. III.] ELIMINATION OF ARBITRARY FUNCTIONS. 529 
 
 XV. . . p = (l>(x,y) = ->Pa:+yVxl^'^; 
 the sign of a rersor being here introduced, for the sake of facilitating the passage, 
 at a certain liynit, to a cone (comp, 386, (12.)). 
 
 (9.) And the scalar equation of the same nrhitrary developable may be repre- 
 sented as the result of the elimination of t, between the two equations, 
 
 XYI. . . ^pxt +1=0, Spx'< = ; 
 in which xt is an arbitrary vector function of t. 
 
 (10.) The envelope oi a. plane with two arbitrary and scalar parameters, t and 
 M, is generally a curved but undevelopable surface, which may be represented by the 
 system of the three scalar equations, 
 
 XVII. . . ^pxu « + 1 = 0, S|oD,x = 0, SpD„x = ; 
 where — x denotes the reciprocal of the perpendicular from the origin on the tan- 
 gent plane to the surface, at what may be called the point (t, «). 
 
 388. It remains, on the plan lately stated (380), to consider 
 briefly surfaces of revolution, and to investigate the geodetic lines, on 
 this SLdd'itional family of surfaces; of which the equation, analogous 
 to those marked I. II. III. IV. in 387, for spheres, cylinders, cones, 
 and developables, is of the form, 
 
 I. . . Sapv = 0, 
 if a be a given line in the direction of the axis of revolution, sup- 
 posed for simplicity to pass through the origin ; but which may also 
 be represented by either of these two other equations, not involving 
 the normal v, 
 
 II. . . T/> =/(Sa/>), or III. . . TYap = F(Sap), 
 
 where /and F are used as characteristics of two arbitrary but sca- 
 lar functions : between which S«/) may be conceived to be eliminated, 
 and so a third form of the same sort obtained. 
 
 (1.) In fact, the equation I. expresses that v ||| a, jO, or that the normal to the 
 surface intersects the axis ; while II. expresses that the distance from sl fixed point 
 upon that axis is a. function of its own projection on the ssLxne fixed line, or that the 
 sections made by planes perpendicular to the axis are circles ; and the same circu- 
 larity of these sections is otherwise expressed by III., since that equation signifies 
 that the distance from the axis depends on the position of the cutting plane, and is 
 constant or variable with it : while the two last forms are connected with each other 
 in calculation, by means of the general relation (comp. 204, XXI.), 
 IV. . . (Tap)3 = (Sap)2 + (TVap)3. 
 
 (2.) The equation I. is analogous, in quaternions, to a. partial differential equa- 
 tion of WiQ first order, and either of the two other equations, II. and III., is analogous 
 to the integral of that equation, in the usual differential calculus of scalars. 
 
 3 Y 
 
530 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (3.) To accomplish the corresponding integration in quaternions, or to pass from 
 the form I. to II., whence III. can be deduced by IV., we may observe that the 
 equation I. allows us to write (because Svdp = 0), 
 
 V. . . v = a;« +yp, VI. . . orSadp + ySpdp = 0, 
 so that the two scalars Sap and Tp are together constant, or together variable, and 
 must therefore he functions of each other. 
 
 (4.) Conversely, to eliminate the arbitrary function from the form II., quater- 
 nion differentiation gives, 
 
 VII. . . = S (Up . dp) +/' (Sap) . Sadp = S . (Up + a/'Sap) dp ; 
 hence VIII. . . v || Up + afSap, and IX. . . v ] ] | a, p, as before ; 
 
 so that we can return in this way to the equation I., the functional sign f disappear- 
 ing. 
 
 (5.) We have thus the germs of a Calculus of Partial Differentials in Quater- 
 nions,* analogous to that employed hy Monge, in his researches respecting _/«»w7ieff 
 of surfaces: but we cannot attempt to pursue the subject farther here. 
 
 (6.) But as regards the geodetic lines upon a surface of revolution, we have only 
 to substitute for v, in the recent formula I., by 380, IV., the expression dUdp, 
 which gives at once the differential equation, 
 
 X. . . = SapdUdp = d.SapUdp (because S(adp.Udp) = - SaTdp = 0) ; 
 
 whence, by a first integration, c being a scalar constant, 
 
 XI. . . c = SapUdp = TVap.SU(Vap.dp). 
 
 (7.) The characteristic property of the sought curves is, therefore, that for each 
 of them the perpendicular distance from the axis of revolution varies inversely as the 
 cosine f of the angle, at which the geodetic crosses a parallel, or circular section of 
 the surface : because, if Ta = 1, the line Yap has the length of the perpendicular let 
 fall from a point of the curve on the axis, and has the direction of a tangent to the 
 parallel. 
 
 * The same remark was made in page 574 of the Lectures, in which also was 
 given the elimination of the arbitrary function from an equation of the recent form 
 III. It was also observed, in page 578, that geodetics furnish a very simple example 
 of what may be called the Calculus of Variations in Quaternions ; since we may 
 write, 
 
 ^Jds = 5jTdp=J^Tdp = -jS(Udp.Mp) 
 = - J S(Udp . d^p) = - AS (Udp . 5p) + J S (dUdp . 5p), 
 
 and therefore dUdp \ v, or VvdUdp = 0, as in 380, IV., in order that the expression 
 under the last integral sign may vanish for all variations ^p consistent with the 
 equation of the surface : while the evanescence of the part which is outside that sign 
 J supplies the equations of limits, or shows that the shortest line between two curves 
 on a given surface is perpendicular to both, as usual. 
 
 t Unless it happen that this cosine is constantly zero, in which case c = 0, and 
 the geodetic is a meridian of the surface. 
 
CHAP. III.] OSCULATING CIRCLES TO CURVES IN SPACE. 531 
 
 (8.) The equation XI. may also be thus written, 
 
 XII, . . cTp' = Sap|o', where p' = i>tp ; 
 
 and if the independent variable t be supposed to denote the time, while the geodetic 
 is conceived to be a curve described by a moving point, then while Tp' evidently re- 
 presents the linear velocity of that point, as being = ds : dt, if s denote the arc (conip. 
 100, (5.), and 380, (7.), (H-))? it is easy to prove that Sapp' represents the double 
 ureal velocity, projected on a plane perpendicular to the axis; the one of these two 
 velocities varies therefore directly as the other : and in fact, it is known from mecha- 
 nics, that each velocity would be constant,* if the point were to describe the curve^ 
 subject only to the normal reaction of the surface, and undisturbed by any other 
 force. 
 
 (9.) As regards the analysis, it is to be observed that the differential equation 
 X. is satisfied, not only by the geodetics on the surface of revolution, but also by the 
 parallels on that surface : which fact of calculation is connected with the obvious 
 geometrical property, that every normal plane to such a parallel contains the axis of 
 revolution. 
 
 (10.) In fact if we draw the normal plane to any curve on the surface, at a point 
 where it crosses a parallel, this plane will intersect the axis, in the point where that 
 axis is met by the normal to the surface, drawn at the same point of crossing ; but 
 this construction ^zVs to determine that normal, if the curve coincide with, or even 
 touch a parallel, at the point where its normal plane is drawn. 
 
 Section 6 On Osculating Circles and Spheres, to Curves 
 
 in Space; with some connected Constructions. 
 
 389. Resuming the expression 376, I. for pt, and referring again 
 to Fig. 77, we see that if a circle pq,d be described, so as to touch a 
 given curve pqr, or its tangent pt, at a given point p, and to cut the 
 curve at a near point Q, and if pn be the projection of the chord pq 
 on the diameter pd, or on the radius cp, then because we have, rigo- 
 rously, 
 
 1. . ,TCi=tp^-\- \fup", with u=\ for < = 0, 
 
 we have also 
 
 II. . .m = ^fVup''p'\p\ 
 and 
 
 2 _ _2_ _ 2pn _ Yupffp' 
 
 ' * * PC PD PQ2 (^pf ^ ^tUp"Yp'' 
 
 Conceiving then that the near point Q, approaches indefinitely to the 
 given point p, in which case the ultimate state or limiting position of 
 
 * This remark is virtually made in page 443 of Professor De Morgan's Diffe- 
 rential and Integral Calculus (London, 1842), which was alluded to in page 578 
 of the Lectures on Quaternions. 
 
632 ELEMENTS OF QUATERNIONS. [bOOK. Ill 
 
 the circle pqd is said to be that of the osculating circle to the curve at 
 that point P, we see that while t\\Q plane of this last circle is the os- 
 culating plane (376), the vector jc of its centre k, or of the limiting po- 
 sition of the point c, is rigorously expressed by the formula: 
 
 which may however be in many ways transformedy by the rules of 
 the present Calculus. 
 
 (1.) Thus, we may write, as transformations of the expression IV., the follow- 
 ing : 
 
 Vp"p'-» ^ Vp>'-i.U(o' "^ (Up')" 
 
 or introducing differentials instead of derivatives, but leaving still the independent 
 variable arbitrary, 
 
 _ _ dp^ dp Tdp _ _ d« 
 
 VI. . . k: - p - y^^^^2p -P^ Vd2(>dp-i ~ ^ dUp' ~ ^ dUd^' 
 
 if s be the arc of the curve ; so that the last expression gives this very simple for- 
 mula, for the reciprocal of the radius of curvature, or for the ultimate value of 
 1 : CP, 
 
 VII. . . (p - k)"^ = D«Up', where Up' = Udp, as before. 
 
 (2.) To interpret this result, we may employ again that auxiliary and spherical 
 curve, upon the cone of parallels to tangents, which has already served us to con- 
 struct, in 379, (6.) and (7.), the osculating plane, the absolute normal, and the bi- 
 normal, to the given curve in space. And thus we see, that while the semidiameter 
 PC has ultimately the direction of dUp', and therefore that of the absolute normal 
 (379, II.) at P, the length of the same radius is ultimately equal to the arc pq (or 
 A«) of the given curve, divided by the corresponding arc of the auxiliary curve; or 
 that the radius of curvature, or radius of the osculating circle at p, is equal to the 
 ultimate quotient of the arc PQ, divided by the angle between the tangents, pt and 
 (say) QU, to that arc pq itself at p, and to Its prolongation qr at Q, although these 
 two tangents are generally in different planes, and have no common point, so long 
 as PQ remains jf?wi7e: because we suppose that the given curve is in general one of 
 double curvature, although the forinulce, and the construction, above given, are ap- 
 plicable to plane curves also. 
 
 (3.) For the helix, the formula IV. gives, by values already assigned for p, p', p", 
 and a, the expression, 
 
 VIII. . . K = cta— a*j3 cot^ a, whence IX. . . p — ic = a*(3 cosec^ a, 
 a being the inclination of the given lielix to the axis ; the locus of the centre of the 
 osculating circle is therefore in this case a second helix, on the same cylinder, if 
 
 tt = — , but otherwise on a co-axal cylinder, of which the radius — the given radius 
 4 
 
 Tj3, multiplied by the square of the cotangent of a; and the radius of curvature 
 = T(p — k) = T/3 X cosec^ a, so that this radius always exceeds the radius of the cy- 
 linder, and is cut perpendicularly (without being prolonged) by the axis. 
 
CHAP. III.] VECTOR OF CURVATURE, EXAMPLES. 533 
 
 (4.) In general, if Tp' = const,, and therefore Sp'p"=0 (comp. 379, (2.)), the 
 
 expression IV. becomes,* 
 
 p'2 
 X. . . K = p+^, ; whence, XI. , . K = p — p""i, if Tp = 1, 
 
 that is, if the arc be taken as the independent variable (380, (12.)). Under this 
 last condition, then, the formula VII. reduces itself to the following, 
 
 XII. . . (p - fc)-' = p" = 'Dpp = ultimate reciprocal of radius CP ; 
 
 so that p" (for Tp'= 1) may be called the Vector of Curvature, because its tensor 
 Tp" is a mimerical measure for what is usually called the curvature^ at the point P, 
 and its versor Up" represents the ultimate direction of the semidiameter pc, of the 
 circle constructed as above. 
 
 (5.) As an example of the application (2.) of the formula IV. for k, to a, plane 
 curve, let us take the ellipse, 
 
 XUL..p = Ya% Ta=l, Sa/3^0, 337,(2.), 
 
 considered as an oblique section (314, (4.)) of a right cylinder. The expressions 
 376, (5.) for the derivatives of p, combined with the expression XIII. for that vec- 
 tor itself, give here the relations, 
 
 XIV. . . Vpp" = 0, Vp'p"' = 0; 
 and therefore comp. (338, (5.)), 
 
 XV. . . Ypp' = const4 = ^ /3y, Yp'p" = const. = I^Y(3y^ if y = Ya(3 ; 
 
 hence for the present curve we have by IV., 
 
 XVI. . . ic = p - -^ = Yat(3 - (Va'+i/3)3 (/3y)-'. 
 Ypp 
 
 (6.) To interpret this result, we may write it as follows, 
 
 XVII. . . K = p- ^^^] , , , where XVIII. . . pi = - p'= Va^+i/3; 
 \pp.p * 7r 
 
 so that pi is the conjugate semidiameter of the ellipse (comp. 369, (4.)), and Ypp':p' 
 is the perpendicular from the centre of that curve on the tangent. We recover then, 
 by this simple analysis, the known result, that the radius of curvature of an ellipse 
 is equal to the square of the conjugate semidiameter, divided by the perpendicular. 
 (7.) "We may also write the equation XVI. under the form, 
 
 lOi^ 
 
 XIX. . . *c = p - ——, where XX. . . Vppi = /Sy = const. ; 
 Vppi 
 
 * The expressions X. XI. may also be easily deduced by limits, from the con- 
 struction in 383, (2.). 
 
 f It may be remarked that the quantity z, or T\p", in the investigation (382) 
 reapect'mg geodetics on a developable, represents thus the curvature of the cusp-edge^ 
 for any proposed value of the arc, x, of that curve. 
 
 X These values XV. might have been obtained without integi-aiions, but this 
 seemed to be the readiest way. 
 
534 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 and may interpret it as expressing, that the radius of curvature is equal to the cube 
 of the conjugate semidiameter, divided by the constant parallelogram under any two 
 such conjugates ; or by the rectangle under the major and minor semiaxes, which 
 are here the vectors /3 and y (comp. 314, (2.)). 
 
 (8.) The expression XVI. or XIX. for k is easily seen to vanish, as it ought to 
 do, at the limit where the ellipse becomes a circle, by the ci/linder being cut perpen- 
 dicularlg, or by the condition Sa(3 = being satisfied ; and accordingly if we write, 
 
 XXI. . . e = SUa/5 = excentricity of ellipse, or XXII. . . y2 = (i _ ^2)^2^ 
 we easily find the expressions, 
 
 XXIII. . . p = /3S.a<+yS.a«-i, pi = -^S.a<-i + yS.a« ; 
 
 XXIV. . . px^=/3Kl-eHS.aOO, ^ = |i = /3-^ (/3S.aH 5^^]; 
 EC that the formula XIX. becomes, 
 
 XXV. . . «: = e^f/3(S.a03- I^|^^'\ = c2(^(S.a03--(S.aM)3), 
 
 thus containing e^ as a factor. 
 
 (9.) And it may be remarked in passing, that the expression XVI., or its recent 
 transformation XXV., for k as a function of t, may be considered as being in qua- 
 ternions the vector equation (comp. 99, I., or 369, I.) of the evolute* of the ellipse, 
 or the equation of the locus of centres of curvature of that plane curve; and that the 
 last form gives, by elimination of ^ (comp.f 315, (1.), and 371, (5.)), the following 
 system of two scalar equations for the same evolute, 
 
 XXVI...(s|]^+fsp)*=e^, S/3y« = 0; 
 
 or XXVr. . . (S/3Kf + (SyK)i = (e^Y, &c. ; 
 
 which will be found to agree with known results. 
 
 (10.) As another example of application to a, plane curve, we may consider the 
 hyperbola, 
 
 XXVII. . . Q^ta^- 1-% comp. 371, II., 
 
 with a and /3 for asymptotes, and with its centre at the origin. In this case the de- 
 rived vectors are, 
 
 XXVIII. . . p' = a- 1-^(3, p" = 2t-3j3, 
 whence XXIX. . . Yp"p' = 2t-3Y(3a = t-Wpp, 
 
 and the formula IV. becomes, 
 
 XXX. ..K-p== ; ,= , 
 
 Ypp : p ov 
 where ov is the perpendicular from the centre o on the tangent to the curve at p, 
 and PT is the portion of that tangent, intercepted between the same point p and an 
 asymptote (comp. (6.) and 371, (3.)). 
 
 * That is to say, of the plane evolute; for we shall soon have occasion to consi- 
 der briefly those evolutes of double curvature, which have been shown by Monge to 
 exist, even when the given curve is plane. 
 
 t In lately referring (373, (1.)) to the formula 315, V., that formula was inad- 
 vertently printed as (a^^ -i- (a<-')2= 1, the sign S. before each power being omitted. 
 
CHAP. III.] ABRIDGED GENERAL CALCULATION. 535 
 
 (11.) We may also interpret the denominator in XXX. as denoting the projec- 
 tion of the semidiameter op on the normal, or as the line NP where N is the foot of 
 the perpendicular from the curve on that normal line ; if then k be the sought centre 
 of the osculating circle, we have the geometrical equations, 
 
 XXXI. . . NP . PK = PT2, XXXII. . . Z NTK = - ; 
 
 whereof the last furnishes evidently an extremely simple construction for the centre 
 of curvature of an hyperbola, which we shall soon find to admit of being extended, 
 with little modification, to a spherical conic* and its cyclic arcs. 
 (12.) The logarithmic spiral with iispole at the origin, 
 
 XXXIII. . . p = a% Sa/3 = 0, Ta ^ 1, comp. 314, (5.) 
 
 may be taken as a third example of a plane curve, for the application of the foregoing 
 formulae. A first derivation gives, by 333, VII., 
 
 XXXIV. . . p' = (c + y)p = p(c - y), p'p-i = c + y, if c = ITa, and y = ^ Ua; 
 
 the constant quaternion quotient, p' : p, here showing that the prolonged vector op 
 makes with the tangent pt a constant angle, n, which is given by the formula, 
 
 XXXV. . . tan n = (TV : S) (p' : p) = c" iTy, or cot « = - ITa ;+ 
 and a second derivation gives next, 
 
 XXXVI. . . p" = (C + y)2 p, Yp"p' = (C2 - y«) p2y = p'2y. 
 
 The formula IV. becomes therefore, in this case, 
 
 XXXVII. . . fc = p+ p'y-» = pcy-i = -cy-ip = --^.a<+ii(3; 
 
 TT Ja 
 
 the evolute is therefore a second spiral, of the same kind as the first, and the radius 
 of curvature kp subtends a right angle at the common pole. But we cannot longer 
 here delay on applications within the plane, and must resume the treatment by qua- 
 ternions of curves of double curvature. 
 
 390. When the logic by which the expression 389, IV. was ob- 
 tained, for the vector k of the centre of the osculating circle, has 
 once been fully understood, the process may be conveniently and safely 
 abridged, as follows. Referring still to Fig. 77, we may write briefly, 
 
 * It was in fact for the spherical curve that the geometrical construction alluded 
 to was first perceived by the writer, soon after the invention of the quaternions, and 
 as a consequence of calculation with them : but it has been thought that a sub-arti- 
 cle or two might be devoted, as above, to the plane case, or hyperbolic limit, which 
 may serve at least as a verification. 
 
 f If r be radius vector, and 9 polar angle, and if we suppose for simplicity that 
 
 2 
 
 T(3 = 1, the ordinary polar equation of the spiral becomes r = o^, with a = Ta"-, and 
 cot n = la, as usual. 
 
536 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 as equations which are all ultimately true^ or true at the limits in a 
 
 sense which is supposed to be now distinctly seen: 
 
 , Vd^dp 
 I. . . PT = d/j, TQ = IdV, PN = (part of pq JL pt =) -^ — , 
 
 by 203, &c. ; whence, ultimately, 
 
 __ PQ^ PT^ d/>* 
 
 II. . . /c-/, = PC = — - = 5— 
 
 2pn 2pn VdVd/>' 
 as before: this last expression, in which Vd^dp denotes briefly 
 V(dV.d/>), being rigorous, and permitting the choice oi any scalar^ 
 to be used as the independent variable. And then, by writing, 
 
 III. . . dp = p'd^, d?t = 0, dV = p^^dt\ 
 
 the factor dt^ disappears, and we pass at once to the expression, 
 
 which had been otherwise found before. 
 
 (1.) When the arc of the curve is taken for the independent variable, then (comp. 
 380, (12.), &c.) the expresssion II. reduces itself to the following, 
 
 dp2 
 V. . . /c - p = -p-, because Sd^pdjO = ; 
 
 and accordingly the angle ptq in Fig. 77 is then ultimately right (comp. 383, (5.)), 
 so that we may at once write, with this choice of the scalar variable, 
 
 PX2 clp^ 
 YI. . . K-p = {ult.') PC = (ult.) - — = — , as above. 
 
 (2.) Suppose then that we have thus geometrically (and very siviply^ deduced 
 the expression V. for /c — p, for this particular choice of the scalar variable ; and let 
 us consider how we might thence pass^ in calculation, to the more general formula 
 II., in which that variable is left arbitrary. For this purpose, we may write, by 
 principles already stated, 
 
 ( _ yi^ ^V = ^ d ^P ^dUdp^ Vdydp-i.Udp 
 Vii. . . {^p K) ^r^^py Tdp Tdp Tdp Tdp 
 
 Vd2pdp-J _ Vdpd2p 
 dp dp' 
 
 and the required transformation is accomplished. 
 
 (3.) And generally, if « denote the arc of any curve of which p is the variable 
 vector, we may establish the symbolical equations, 
 
 (4.) For example (comp. 389, XII.), the Vector of Curvature, Ds^p, admits of 
 being expressed generally under any one of the five last forms VII. 
 
CHAP. HI.] POLAR AXIS, POLAR DEVELOPABLE. 537 
 
 391. Instead of determining the vector k of the centre of the os- 
 culating circle by one vector expression, such as 389, IV., or any of 
 its transformations, we may determine it by a system of three scalar 
 equations, such as the following, 
 
 I. ..S(«:-rt/ = 0; II. . . S(A:-rt/>''-/>'^ = 0; 
 
 III. ..S(/c-/))/>y' = 0, 
 
 of which it may be observed that the second is the derivative of the 
 first, if /c be treated as constant (comp. 386, (4.)) ; and of which the 
 first expresses (369, IY-) that the sought centre is in the normal 
 plane to the curve, while the third expresses (376, V.) that it is in 
 the osculating plane ; and the second serves to fix its position on the 
 absolute normal (379), in which those two planes intersect. 
 
 (1.) Using differentials instead of derivatives, but leaving still the independent 
 variable arbitrary, we may establish this equivalent system of three equations, 
 
 IV. ..S(K:-p)dp = 0; Y. . . S^k- p)u^-p -6p^ = ; VI. . . S(K:-p)dpd2p= 0; 
 
 of which the second is the differential of the first, if k be again treated as constant. 
 
 (2.) It is also permitted (comp. 369, (2.), 376, (3.), and 380, (2.)), with the 
 same supposition respecting k, to write these equations under the forms, 
 
 VII. ..dT((c-p) = 0; VIII. . .d'^T(K-p)=0; IX. . . dUV(K - p)dp = ; 
 
 and to connect them with geometrical interpretations. 
 
 (3.) For instance, we may say that the centre of the osculating circle is the point, 
 in which the osculating plane, III. or VI. or IX., is intersected by the axis of that 
 circle ; namely, by the right line which is drawn through its centre, at right angles 
 to its plane : and which is represented by the two scalar equations, 
 
 I. and II., or IV. and V., or VII. and VIII. 
 
 (4.) And we may observe (comp. 370, (8.)), that whereas for a point r taken 
 arbitrarily in the normal plane to a curve at a given point p, we can only say in ge- 
 neral, that if a chord pq be called small of the first order, then the difference of dis- 
 tances, RQ — ijp, is small of an order higher than the first ; yet, if the point r be 
 taken on the axis (3.) of the osculating circle, then this difference of distances is 
 small, of an order higher than the second, in virtue of the equations VII. and VIIT. 
 
 (5.) The right line I. II., or IV. V., or VII. VIII., as being the locus oi points 
 which may be called poles of the osculating circle, on all possible spheres passing 
 through it, is also called the Polar Axis of the curve itself, corresponding to the 
 ^ven point of osculation. 
 
 (6.) And because the equation II. is (as above remarked) the derivative of I., the 
 known theorem follows (comp. 386), that the locus of all such polar axes is a deve- 
 lopable surface, namely that which is called the Polar Developable, or the envelope 
 of the normal planes to the given curve; of which surface we shall soon have oc- 
 casion to consider briefly the cusp- edge. 
 
 3 z 
 
538 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 392. The following is an entirely different method of investigat- 
 ing, by quaternions, not merely the radius or the centre of the oscu- 
 lating circle to a curve in space, but the vector equation of that circle 
 itself: and in a way which is applicable alike, io plane cmwqs, and to 
 curves of double curvature. 
 
 (1.) In general, conceive that ox = r is a given tangent to a circle, at a given 
 point which is for the moment taken as the origin ; and let pp' = p' represent a va- 
 riable tangent, drawn at the extremity of the variable chord op= p : also let u be 
 the intersection, ot-pp', of these two tangents. Then the isosceles triangle cup, 
 combined with the formula 324, XI. for the differential of a reciprocal, gives easily 
 the equations, 
 
 I. . . p 11 pr- V ; XL . . Vrp-ip'p-i = - (Yrp-^)' = ; 
 III. . . Vrp-i = const. = Vra-i, as in 296, IX.", 
 if a be the vector OA of any second given point a of the circumference. 
 
 (2.) The vector equation of the circle pqd (389) is therefore, 
 
 IV. . . V -^ = V -^ = - V. (1 + |^Mp"p'-i)-i = - v.V'p'-' (1 + ¥^p"p'V ; 
 
 w — p Qt-p t 
 
 whence, passing to the limit (< = 0, m= 1), the analogous equation of the osculating 
 ciccle is at once found to be, 
 
 V...V-^=-vi', or VI...v(?^+2^M = 0; 
 
 b) - p p \^~ P ^P J 
 
 with the verification (comp. 296, (9.)), that when we suppose, 
 
 VII. . . w - p = 2 (fc - p) j_ p', 
 the vector k of the centre is seen to satisfy the equation, 
 
 VIII... -^=-vC or IX...-^ + V^^=0; 
 
 K-p p K-p dp .^ 
 
 which agrees with recent results (389, IV., &c.). 
 
 (3.) Instead of conceiving that a circle is described (389), so as to touch a given 
 curve (Fig. 77) at p, and to cut it at one near point Q, we may conceive that a circle 
 cuts the curve in the given point p, and also in two near points, Q and k, uncon- 
 nected by any given law, but both tending together to coincidence with p : and may 
 inquire what is the limiting position (if any) of the circle pqr, which thus intersects 
 the curve in three near points, whereof one (p) is given. 
 
 (4.) In general, if a, j3, p be three co-initial chords, OA, OB, op, of any one cir- 
 cle, their reciprocals a-i, /3'^ p'\ if still co-initial, are termino-collinear (260) ; ap- 
 plying which principle, we are led to investigate the condition for the three co-ini- 
 tial vectors, 
 
 X. . . (w - p)-i, (sp + |s2«,p")-i, (tp' -f it^utp")-^ 
 
 with «o = 1, thus ultimately terminating on one right line ; or for our having ulti- 
 mately a relation of the form, 
 
 xs-\-yt _ se V 
 
 JLl. . • — — — — — p- — - — 7, + 
 
 P'^W P'^r^tp' 
 
CHAP. III.] VECTOR EQUATION OF OSCULATING CIRCLE. 539 
 
 J. J J (xs^yt)£ 
 
 lo-p 1 + |«p"p'"^ I + itp"p'~^ 
 
 = x + y-^(xs + yt) p"p'-i -I- &c. : 
 in which last equation, both members are generally quaternions. 
 
 (5.) The comparison of the scalar parts gives here no useful information, on ac- 
 count of the arbitrary character of the coeflScieuts x and y ; but these disappear, with 
 the two other scalars, s and t, in the comparison of the vector parts, whence follows 
 the determinate and limiting equation, 
 
 XIIL . . 2Vp'(w-p)-i=-Vp>'-i, 
 
 which evidently agrees with V. 
 
 (6.) It is then found, by this little quaternion calculation, as was of course to be 
 expected,* that the circle (3.), through any three near points of a curve in space, 
 coincides ultimately with the osculating circle, if the latterhQ still tfe^ned (389) with 
 reference to a given tangent, and a near point, which tends to coincide with the given 
 point of contact. 
 
 393. An osculating circle to a curve of double curvature does 
 not generally meet that curve again; but it intersects generally a 
 plane curve, of the degree n, to which it osculates, in 2w - 3 points, 
 distinct from the point p of osculation, whereof one at least must be 
 realy although it may happen to coincide with that point P : and 
 such a circle intersects also generally a spherical curve of double 
 curvature, and of the degree n, in n-S other points, namely in 
 those where the osculating />/ane to the curve meets it again. An 
 example of each of these two last cases, as treated by quaternions, 
 may be useful. 
 
 (L) In general, if we clear the recent equation, 392, V. or XIII., of fractions, it 
 
 I. . . = 2p'2Vp' (w-p) + {<o-pyYp"p' ; 
 in which p = op = the vector of the given point of osculation, and p', p" are its first 
 and second derivatives, taken with respect to any scalar variable t, and for the par- 
 ticular value (whether zero or not) of that variable, which answers to the particular 
 point p ; while w denotes generally the vector of any point upon the circle, which 
 osculates to the given curve at that point p. 
 (2.) Writmg then (comp. 389, (10.)), 
 
 II. . ,p=ta + t-% p=a- r2/3, p" = 2r3^, 
 and III. . . w = OQ = ica + a;"ij8, 
 
 to express that we are seeking for the remaining intersection q of a plane hyperbola 
 
 * This conclusion is indeed so well known, and follows so obviously from the doc- 
 trine of infinitesimals, that it is only deduced here as a verification of previous for- 
 mulae, and for the sake oi practice in the present Calculus. 
 
540 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 with its osculating circle at p, the equation I. becomes, after a few easy reductions, 
 including a division by Va/3, the following hiqvadratic in x, 
 
 IV. .. - (a; - ty (fa^x - (5^) ; 
 in which the cubic factor is to be set aside, as answering only to the point p itself. 
 
 (3.) Substituting then, in III., the remaining value IV. of a;, we find the ex- 
 presssion, 
 
 comparing which with 371, (3.), we see that if the tangent to the hyperbola at the 
 given point p intersects the asymptotes in the points a, b, then the tangent at the 
 sought point q meets the same lines OA, ob in points a', b', such that 
 
 VI. . . OA . OA' = Ob2, ob . ok' = OA ' ; 
 
 whence q is at once found, as the bisecting point of the line a'b'. 
 
 (4.) A still more simple construction, and one more obviously agreeing with 
 known results, may be derived from the following expression for the chord pq : 
 VII. . . PQ = w - p = (r-/3-2 - r2a-2) {ta^(3 - t'afS^) 
 
 = (<3/3-2 - t'^a-^yp'ft II a|o'-i/3 ; 
 whence it follows (comp. 226) that if this chord pq, both ways prolonged, meets the 
 two asymptotes ob and oa in the points r and s, we have then the inverse similitude 
 
 of triangles (118), 
 
 VIII. . . A KOS a' AOB. 
 
 (5.) As regards the equality of the intercepts^ ep and QS, it can be verified 
 without specifying the second point q on the hyperbola, or the second scalar^ x. by 
 observing that the formula III., combined with the first equation II., conducts to 
 the expressions, 
 
 „ xp — toi , , ,. - fp — xu), . 
 IX. . . or = — = (a;-i + ri) 3, os = = (» + t)a ; 
 
 X — t i - X 
 
 which give, generally, 
 
 X. . . . p.? = QS = ^a - a;-ij3. 
 
 (6.) And as regards the general reduction, of the determination of the osculating 
 circle to a spherical curve of double curvature, to the determination of the oscu- 
 lating plane, it is sufficient to observe that when we take the centre of the sphere for 
 the origin, and therefore write (comp. 881, XIV.), 
 
 XI. . . p2 = const., Spp = 0, Spp" = - p'2, 
 
 then if we operate on the vector equation I. with the symbol V. p, and divide by 
 — p'3, there results the scalar equation, 
 
 XII. . . =2Sp(w - p) + (a> - p)2 = 0)2 - p3, 
 
 which expresses that the circle is entirely contained on the same spheric* surface as 
 the cur\'e ; while the other scalar equation, 
 
 XIII = Sp"pX<o - p), 
 
 obtained by operating on I. with S . p", expresses (comp. 376, V.) that the same 
 
 * This conclusion is geometrically evident, but is here drawn as above, for the 
 sake of practice in the quaternions. 
 
CHAP. Ill] CASE OF A SPHERICAL CURVE. 541 
 
 circle is in the osculating plane :* so that its centre K is the foot of the perpendi- 
 cular let fall on that plane from the origin, and we may therefore write (comp. 
 385, VI.), 
 
 XIV. . . OK = fc = -^-^, with the relations, XV. . . S-=S-=1; 
 
 Vp p K K 
 
 and with the verification that the expression XIV. agrees with the general formula, 
 389, IV., because 
 
 XVI. . . pVp'y +p'3 = Sp'p'p, 
 
 when the conditions XI. are satisfied. 
 
 (7.) And even if the given curve be not a spherical one, yet if we retain the 
 general expression for k, 
 
 XVII. ..fc = p + -^„ 389, IV., 
 
 Vp p 
 
 and operate on I. with S . p" and S. p"p\ we find again the equation XIII. of the os- 
 culating plane, combined with a new scalar equation, which may after a few reduc- 
 tions be written thus, 
 
 XVIII. .. (o>-»c)2 = (p-fc)2; 
 
 and which reprejsents a new sphere, whereon the osculating circle to the curve is a 
 great circle. 
 
 394. To give now an example of a spherical curve of double cur- 
 vature, with its osculating circle and plane for any proposed point p, 
 and with a determination of the point Q in which these meet the 
 curve again (393), we may consider that spherical conic^ or sphero- 
 conic, of which the equations are (comp. 357, II.), 
 
 I. . ./^-l-r^=0, II. ..^/>HSXp/x/> = 0; 
 
 namely the intersection of the sphere, which has its centre at the 
 origin, and its radius =r, with a cone of the second order, which has 
 the same origin for vertex, and has the given lines A, and /u. for its 
 two (real) cyclic normals. And thus we shall be led to some suffi- 
 ciently simple spherical constructions, which include, as their plane 
 limits, the analogous constructions recently assigned for the case of 
 the common hyperbola. 
 
 (1.) Since SXpfxp = 2SXpS/ip - p^SXfi (comp. 357, 11'.), the equations I. and II. 
 allow us to write, as their first derivatives, or at least as equations consistent there- 
 with, 
 
 III. . . Spp' = 0, SXp' + SXp = 0, S/ip' - Sjup = 0, 
 
 because the independent variable is here arbitrary, so that we may conceive the first 
 derived vector p' to be multiplied by any convenient scalar ; in fact, it is only the 
 
 Compare the Note immediately preceding. 
 
542 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 direction of this tangential vector p' which is here important, although we must con- 
 tinue the derivations consisteiltly, and so must write, as consequences of III., the 
 equations, 
 
 IV. . . Spp" + p'2 = 0, S\p"+ S\p' = 0, Sfip" - Snp' = 0. 
 
 (2.) Introducing then the auxiliary vectors, 
 
 Y. . . rj = YXfi, a = XS/xp + /iSXp, T=p-\- p, v — p — p, 
 whence 
 
 VI. . . = Sj7(r=SXr = S/iv, Sp(r=28\pSixp, S/*r = 2S)up, SXi; = 2SXp, 
 
 r2=t;2= p2^p'2j 
 
 and by new derivations, 
 YIL . . (t' = Yt]p, t' = p' + p", v' = p-p", SXr' = S/iu'=0, S/ir'=S/ir, 
 SXv' = -SXu, 
 
 we see first that r and v are the vectors ox and ou of the points in which the recti- 
 linear tangent to the curve at p meets the two cr/clic planes, perpendicular respec- 
 tively to X and fi ; and because the radius op is seen to be the perpendicular bisector 
 of the linear intercept Tu between those two planes, so that 
 
 VIII. . . p' = PT = UP -I- OP, we have IX. . . uop = pot, 
 
 or X. . . o AP = r> PB, 
 
 if the tangent arc on the sphere, to the same conic at the same point p, meet the two 
 cgclic arcs CA and cb in the points A and b : the intercepted arc ab being thus bi- 
 sected at its point of contact p, which is a well-known property of such a curve. 
 
 (3.) Another known property of a sphero-conic is, that for any one such curve 
 the sum of the two spherical angles CAB and ABC, and therefore also the area of the 
 spherical triangle abc, is constant. We can only here remark, in passing, that 
 quaternions recognise this property, under the form (comp. II.), 
 
 XI. . . cos (A + b) = - SUXpjup — - g\ TX/x = const. 
 
 (4.) The scalar equations III. and IV. give immediately the vector expressions, 
 
 XII. . . p = ^~^ , XIII. . . p = p —-— ; 
 
 or by (2.), 
 
 XIV...p'=g-;, and xy...p=p-?, if XYI...?=g 
 
 = t — t' = v+ y', 
 the new auxiliary vector ^ being thus that of the point x, in which the osculating 
 plane to the conic at p meets the line rj of intersection of the cyclic planes : so that 
 we have the geometrical expressions, 
 
 XVII. . . p" = xp, r' = XT, —v' = XV, if | = ox, 
 and the lines* r' and v are the traces of the osculating plane on those two cyclic 
 
 * We may also consider the derived vectors r' and v% or the lines xt and xu, 
 as corresponding tangents, at the points T and u (2.), to the two sections, made by 
 the cgclic planes, of that developable surface which is the locus of the tangents TPU 
 to the spherical conic in question. 
 
CHAP. III.] CONSTRUCTION FOR A SPHERICAL CONIC. 543 
 
 planes, or of the latter on the former ; while a and a', as being perpendicular respec- 
 tively to p' and p, while each 4- ri, are the traces on the plane X/a of the two cyclic 
 normals, of the normal plane to the conic at the point p, and of the tangent 
 plane to the sphere at that point : or at least these lines have the directions of those 
 traces. 
 
 (5.) Already, from the expression XVI. for the portion ox of the radius oc (2.), 
 or of that radius prolonged, which is cut off by the osculating plane at p, we can 
 derive a simple construction for the position of the spherical centre, or pole, say E, 
 of the «maZZ circZe which osculates at that point p, to the proposed sphero-conic. 
 For if we take the radius r for unity, we have the trigonometric expressions, 
 
 XVIII. . . sec CE cos EP = (T| = Tr^ : SIJj^-ip =) sec^ pb sec cp ; 
 
 or letting fall (comp. Fig. 80) the perpendicular cd on the normal arc pe, 
 
 XIX. . . cos DE = cos DP cos PB . COS PB COS PE = COS DB COS BE ; 
 
 or finally, XX. . . dbe (or dae) = — . 
 
 (6.) But although it is a perfectly legitimate process to mix thus spherical tri- 
 gonometry with quaternions (since in fact the latter include the former), yet it may 
 be satisfactory to deduce this last result by a more/)Mrc/yquaternionic method, which 
 can easily be done as follows. The values (4.) of p' and p" give, 
 
 XXI. . . Vp'p"S?jp = pS(Tp"-(rSpp"=pSp(r+p'2(r 
 
 = (r - p') Scrr + (rSpV = rSor + Yrpa 1 1 ] r, Yrpa, 
 
 in which pa denotes a vector J- p' (because SpV = 0), and ||| 77, p' (because Sj/p'p'<r 
 = 0) ; this line p'cr has therefore the direction of the projection of the line t] on a 
 plane perpendicular to p', and we are thus led to draw, through the line oc of inter- 
 section of the cyclic planes, aplane cod perpendicular to the normal plane to the conic 
 at p, or to let fall (as in Fig. 80) a perpendicular arc cd on the normal arc pd ; after 
 which the normal to the sought osculating plane, or the axis ob of the osculating 
 circle sought, as ijeing || Yp'p'i will be contained in the plane through the trace r, or 
 OT, or OB, which is perpendicular to the plane ofr and pV, or to the plane dob ; 
 and therefore the spherical angle dbe (or dae) will be a right angle, as before. 
 
 (7.) We may also observe that if k be the centre of the osculating circle, consi- 
 dered in its own plane, or the/00^ of the perpendicular on that plane from o, then 
 by XXI., 
 
544 ELEMENTS 01? QUATERNIONS. [BOOK 111. 
 
 Yp'p" pSp(r+pV ^ /oSp(r + pV 
 
 and therefore 
 
 XXIII. . . — = ^^ =^^9^^, XXIV. . . tan ep = sin2 pb cot pd, 
 OK K r2 S*^ 
 
 which gives again the angular relation XX. ; the quotient XXIII. being thus a vec- 
 tor^ as it ought by 393, XV. to be ; and the trigonometric formula XXIV. being ob- 
 tained from its expression, by observing that 
 
 XXV. . . TjoVi = pT : of = sin pot = sin pb, and ( V : S) per = U/o' . cot pd, 
 because <r -i- p'ff, but \\\p, p'a, or p'a -i- tr, but 1 1 ] p, o". 
 
 (8.) The rectangularity of the planes of r, k and r, pa is also expressed by the 
 equation, 
 
 XXVI. . . = S {Ykt .Yp'ar) = SicrSpVr - r^SpVic ; 
 
 in proving which we may employ the values, 
 
 XXVII. . . SrK-i = 1, Sp'(TK-i = (- r-V^Sj^p =) SpVr-". 
 (9.) We may also interpret these equations XXVIL, as expressing the system of 
 the two relations, 
 
 XXVIII. . . K-» - r-i J- r, K-i - r-i -L p'a ; 
 from which it follows that k~\ and therefore also that k, is a line in the plane so 
 drawn through r, as to be perpendicular to the plane through r and p'a, as before. 
 (10.) And the two relations XXVIII. are both included in the following ex- 
 
 XXIX. . . K-i - r-» = Vr-'p'(T : Spcr. 
 (11.) We may also easily deduce, from the foregoing spherical constrnction, the 
 following trigonometric expressions, for the arcual radius r = ep of the osculating 
 small circle (5.), and for the angle a = pae = ebp which it subtends at A or at b : 
 
 XXX. . . tan r = sin - tan a ; XXXI. . . tan a = ^ (cot A + cot b) ; 
 
 A and b here denoting, as in XL, the hose angles of the triangle abc with c for ver- 
 tex, and c denoting as usual the base ab, namely the portion of the arcual tangent 
 (2.) to the conic, which is intercepted between the cyclic arcs. 
 
 (12.) The osculating plane and circle at p being thus fully and in various ways 
 determined, we may next inquire (393) in what point Q do they meet the conic 
 again. In symbols, denoting by a> the vector of this point, we have the three sca- 
 lar equations, 
 
 which are all evidently satisfied by the value w = p, but can in general be satisfied 
 also by one other vector value, which it is the object of the problem to assign. 
 
 (13.) We satisfy the two first of these three equations XXXIL, by assuming the 
 expression, 
 
 XXXIII. . . (^ = ^-\-l{x-W-xv'), 
 
 in which x is any scalar ; in fact we have the relations, 
 
 XXXIV. . . S/c? = S/cp, S\u' = - 2S\p, S/tr' = 2S/ip, 
 = SX^ = S/i? = SXr' = S/iv' = Sicr' = S/cu', 
 whence XXXIII. gives, XXXV. . . SXw = a;SXp, S/zw = x'S/xp, &c. 
 
CHAP. III.] INTERSECTION WITH OSCULATING CIRCLE. 545 
 
 And because 
 
 XXXVI. ..p = ?+Kr'-y')> 
 we shall satisfy also the third equation XXXII., if we adopt for x any root of that 
 new scalar equation, which is obtained by equating the square of the expression 
 XXXIII. for w, to what that square becomes when x is changed to 1. 
 
 (14.) To facilitate the formation of this new equation, we may observe that the 
 relations, 
 
 ^ = p-p", t'=p' + p", v'=p'-p", Spp' = 0, Spp" = -p\ 
 which have all occurred before, give 
 
 XXXVII. . . - 4S4r' = 3t'2 + v'^, 4S^v' = r'2 + 3v'^ ; 
 the resulting equation is therefore, after a few slight reductions, the following biqua- 
 dratic in ic, 
 
 XXXVIII. . . = (ic - 1)3 {v'^x - r'2) ; 
 
 of which the cubic factor is to be rejected (comp. 393, (2.)), as answering only to 
 the point p itself. 
 
 (15.) We have then the values, 
 
 XXXIX. . . a; = r'2v'-2, and XL. . . 0Q = a> = ? + i [ ^^ - '^ "j ; 
 
 comparing which last expression with the formulae XVII., we see that the required 
 point of intersection q, of the sphero-conic with its osculating circle, can be constructed 
 by the following rule. On the traces (4.), of the osculating plane on the two cyclic 
 planes, determine two points t^ and Ui, by the conditions, 
 
 XLI. . . XT.XTi =XU2, XU.XUi = Xt2; then XLII. . . TiQ = QUi, 
 or in words, the right line TjUi is bisected by the sought point Q. 
 
 (16.) But a still more simple or more graphic construction may be obtained, by 
 investigating (comp. 393, (4.)) the direction of the chord pq. The vector value of 
 this rectilinear chord is, by XXXVI. and XL., 
 
 XLIIL . . PQ = w - p = i (y'2 - r'2) (u'-i -h t'-i) = i (r'-2- v'-2) r'(r' + v') v' 
 
 " V ^ - i/2 ) ^'9'''^\ because p' = |(r' + v') ; 
 
 the chord pq has therefore the direction (or its opposite) of ihQ fourth proportional 
 (226) to the three vectors, p\ t', and — v\ or pt, xt, and xu; if then we conceive 
 this chord or its prolongations to meet the traces xt, xu in two new points Tg, U2, 
 we shall have (comp. 393, VIII.) the two inversely similar triangles (118), 
 XLIV. . . A T2XU2 oc' CXT. 
 (17.) To deduce hence a spherical construction for q, we may conceive four 
 planes, through the axis oke, perpendicular respectively to the four following right 
 lines in the osculating plane : 
 
 XLV. . . r', - v', p', w - /o, or xt, xu, pt, pq ; 
 which planes will cut the sphere in four great circles, whereof the four arcs, 
 
 XLVI. . . EF, EG, EP, EH, 
 
 are parts, if F, G, H (see again Fig. 80) be the feet of the three arcual perpendiculars 
 from the pole e of the osculating circle on the two cyclic arcs cb, ca, and on the 
 arcual chord pq. 
 
 4 A 
 
546 MLKMENTS OF QUATEIINIONS. [bOOK III. 
 
 (18.) These four arcs XLVI. are therefore connected by the same angular rela- 
 tion as the /oMr lines XLV. ; and we have thus the very simple formula, 
 
 XLVII. . . GEH = PEF, 
 
 expressing an equality between two spherical angles at the pole e, which serves to 
 determine the direction of the arc eh, and therefore also the positions of the points 
 H and Q, by means of the relations, 
 
 TT 
 XLVIII. . . PHE = — , r> PH = O HQ. 
 
 (19.) If the arcual chord pq, both ways prolonged, or any chord of the conic, 
 cut the cyclic arcs cb and ca in the points r and s (Fig. 80), it is well kno^vn that 
 there exists the equality of intercepts (comp. 270, (2.)), 
 
 XLIX. . . '^ BP = r. Qs ; 
 and conversely this equation, combined with the formulae (11.), or with the trigono- 
 metric expression, 
 
 c 
 L. . . tan PE = tan r = ^ sin — (cot a + cot b), 
 
 for the tangent of the arcual radius of the osculating circle, enables us to determine 
 what may be called perhaps the arcual chord of osculation pq, by determining the 
 spherical angle kpb, or simply p, from principles of spherical trigonometry alone, 
 in a way which may serve as a verification of the results above deduced from quater- 
 nions. 
 
 (20.) Denoting by t the semitransversal rh = hs, and by s the semichord ph = hq, 
 the oblique-angled triangles rpb, spa give the equations, 
 
 ^ . ^ c . 
 
 cot (t — s) sm - = cos p cos - + sm p cot b, 
 
 LI. 
 
 I , . . c c . 
 
 cot (t + s) sm - = cos p cos - - sm p cot A; 
 
 while the right angled triangle phe gives, 
 
 LII. . . tan s = sin p tan r. 
 Equating then the values of cot 2s, deduced from LI. and LII., we eliminate s and t, 
 and obtain a quadratic in tan p, of which one root is zero, when tan r has the value 
 L. ; such then might in this new way be inferred to be the tangent of the arcual ra- 
 dius of curvature of the conic, and the remaining root of the equation is then, 
 
 cos - (cot B — cot a) 
 LIIL . . tanp= 2 : 
 
 c 
 
 cot a cot b + cos2 - — tan2 r 
 
 a fonnula which ought to determine the inclination p, or rpb, or qpa, of the chord 
 PQ to the tangent pa, but which does not appear at first sight to admit of any simple 
 interpretation* 
 
 * We might however at once see from this formula, that p = a - b at the plane 
 limit; which agrees with the known construction 393, (4.), for the corresponding 
 chord PQ in the case of the plane hyperbola. 
 
CHAP. III.] HYPERBOLIC CYLIDER, ASYMPTOTIC PLANES. 547 
 
 (21.) On the other hand, the construction (17.) (18.), to which the quaternion 
 analysis led us, gives 
 
 LIV. . . HEP=GEP— GEH = GEP — PEF = FEB4- GEA, 
 
 and therefore, by the four right-angled triangles, phe, bfe, age, and bpe or epa, 
 conducts to this other formula, 
 
 LV. . . cot 1 (cos r cot p) = cot-M cos r cos - tan (b + 
 
 -M cos r cos - tan (b + o) ] 
 
 — cot-' cos r cos - tan (a + a 
 
 in which a is the same auxiliary angle as in XXXI. ; we ought therefore to find, as 
 the proposed verification (19.), that this last equation LV. expresses virtually the same 
 relation between A, b, c, and p, as the formula LIII., although there seems at first to 
 be no connexion between them ; and such agreement can accordingly be proved to 
 exist, by a chain of ordinary trigonometric transformations^ which it may be left to 
 the reader to investigate. 
 
 (22.) A geometrical proof of the validity of the construction (17.) (18.) may 
 be derived in the following way. The product of the sines of the arcual perpendi- 
 culars, from a point of a given sphero-conic on its two cyclic arcs, is well known to 
 be constant ; hence also the rectangle under the distances of the same variable point 
 from the two cyclic planes is constant, and the curve is therefore the intersection of 
 the sphere with an hyperbolic cylinder, to which those planes are asymptotic. It 
 may then be considered to be thus geometrically evident, that the circle which oscu- 
 lates to the spherical curve, at any given point p, osculates also to the hyperbola, 
 which is the section of that cylinder, made by the osculating plane at this point ; 
 and that the point q, of recent investigations, is the point in which this hyperbola is 
 met again, by its own osculating circle at p. But the determination 393, (4.) of 
 such a point of intersection, although above deduced (for practice) by quaternions, 
 is a plane problem of which the solution was known ; we may then be considered to 
 have reduced, to this known and plane problem, the corresponding spherical prob- 
 lem (12.) ; and thus the inverse similarity of the two plane triangles XLIV., 
 although found by the quaternion analysis, may be said to be geometrically ex- 
 plained, or accounted for : the traces xx and xu, or r and — v', of the osculating 
 plane to the conic on the two cyclic planes (4.), being evidently the asymptotes of 
 the hyperbola in question. 
 
 (23.) In quaternions, the constant product of sines, &c., is expressed by this 
 form of the equation II. of the cone, 
 
 LYI. . . SUXp . SU/ip = (^ - SX^) : 2TX/* = const. ; 
 
 and the scalar equation of the hyperbolic cylinder, obtamed by eliminating p^ be- 
 tween I. and II., after the first substitution (1.), is 
 
 . LVII. . . SXpS/«p = ir2 (g - SX/i) = const. ; 
 
 while the expression XXXIII. for w may be considered as the vector equation of 
 the hyperbola, of which the intersection q with the circle, or with the sphere, is de- 
 termined by combining that equation with the condition w^ -.p2^-_^2^ 
 
548 KLEMENTS OF QUATERNIONS. [bOOK III. 
 
 (24.) In the foregoing investigation, Ave have tresLtedi a. sphero-conic in connexion 
 with its ci/e!ic arcs (2.) ; but it would have been about equally easy to have treated 
 the same curve, with reference to its /oca/ /7o^«<s; or to the /ocaZ /2nes of the cone, 
 of which it is the intersection with a concentric sphere. (Compare what has been 
 called the bifocal transformation^ in 360, (2.)). 
 
 (25.) We can however only state generally here the result of such an application 
 of quaternions, as regards the construction of the osculating small circle to a spheri- 
 cal conic, considered relatively to \t& foci : which construction* can indeed be also 
 geometrically deduced, as a certain polar reciprocal of the one given above. Two 
 focal points (not mutually opposite) being called f and g, let pn be the normal arc 
 at p, which is thus equally inclined, by a well-known principle, to the two vector 
 arcs, FP, GP ; so that if the focus G be suitably distinguished from its own opposite, 
 the spl'.erical angle fpg is bisected by the arc pn, which is here supposed to termi- 
 nate on the given arc fg. At N erect an arc qnr, perpendicular to FN, and termi- 
 nating in Q and R on the two vector arcs. Perpendiculars, qe, re, to these last 
 arcs, will meet on the normal arc pn, in the sought pole (or spherical centre^ E, of 
 the sought small circle, which osculates to the conic at the given point p. 
 
 (26.) The two focal and arcual chords of curvature from p, which pass through 
 F and G, and terminate on the osculating circle, are evidently bisected at q and r, 
 in virtue of the foregoing construction, which may therefore be thus enunciated :— 
 
 The great circle qr, which is the common bisector of the two focal and arcual 
 chords of curvature from a given point p, intersects the normal arc PN on the fixed 
 are fg, connecting the two foci ; that is, on the arcual major axis of the conic. 
 
 (27.) The construction (5.) fails to determine the position of the auxiliary point 
 D in Fig. 80, for the case when the given point p is on the minor axis of the conic ; 
 and in fact the expressions (4.) for p' and p" become infinite, when the denominator 
 SX/^p is zero. But it is easy to see that the auxiliary vector tr, which represents 
 generally the trace of the normal plane to the curve on the plane of the two cyclic 
 normals, becomes at the limit here considered the required axis of the osculating 
 circle ; and accordingly, if we assume simply (comp, (1.) and (2.)), 
 
 LVIII. . . p' = Vpc, and therefore p" = VpV + Vpcr', 
 we have LIX. . . tr' = 0, and ^p'g" \\ <t, when SX/ip = 0. 
 
 (28.) In general, if we determine three points l, m, s in the plane of X/x, by the 
 formulas (comp. again (2.)), 
 
 LX. . . OL = — — , OM=~-, OS= -— =i(OL + OM), 
 
 SXp Sfip So-p ^^ 
 
 then L and m will be the intersections of the cyclic normals X, p, with the tangent 
 
 * The reader can easily draw the Figure for himself. As regards the known 
 rule, lately alluded to (in 393, (4.), and 394, (22.)), for determining the chord of 
 intersection of a plane conic with its osculating circle, it will be found (for instance) 
 in page 194 of Hamilton's Conic Sections (in Latin, London, 1758). The two sphe- 
 rical constructions, for the small circle osculating to a spherical conic, were early 
 deduced and published by the present writer, as consequences of quaternion cal- 
 culations. Compare the first Note to page 535. 
 
CHAP. III.] OSCULATING CONE, OSCULATING SPHERE. 549 
 
 plane to the sphere at p, and the normal plane to the curve at the same point will 
 bisect the right line lm in the point s ; we shall also have this proportion of sines, 
 LXI. . . sin LOS : sin som = SU\p : SU/ip 
 = cos LOP : cos POM = sin ppi : sin PP2, comp. (23.), 
 
 if ppi, PP2 be the arcual perpendiculars from the point p of the conic on the two cyclic 
 arcs ; and this general rule for determining the position of the line os, or c, applies 
 even to the limiting case (27.), when that variable line becomes the axis of the oscu- 
 lating circle, at a minor summit of the curve. 
 
 (29.) As an example^ let us suppose that the constants g^ X, p, in the equation 
 II. are connected by the relation, 
 
 LXII. . . ^ = - SX^, whence LXIII. . . S (YXp .Yfip) = ; 
 
 the cyclic normals are therefore in this case sides of the cone, and the two planes 
 which connect them with any third side are mutually rectangular ; so that the conic 
 is now the locus of the vertex of a right-angled spherical triangle, of which the 
 hypotenuse is given. And by applying either the formula LXI., or the construction 
 (28.) which it represents, we find that the trigonometric tangent of the arcual radius 
 of the osculating small circle to such a conic, at either end of the given hypotenuse, 
 is equal to half* the tangent of that hypotenuse itself. 
 
 (30.) It is obvious that eveiy determination, of an osculating circle to a spherical 
 curve, is at the same time the determination of what may be (and is) called an os- 
 culating right cone (or cone of revolution'), to the cone which rests upon that curve, 
 and has its vertex at the centre of the sphere. Applying this remark to the last ex- 
 ample (29.), we arrive at the following theorem, which can however be otherwise 
 deduced: — 
 
 If a cone be cut in a circle hy a plane perpendicular to a side, the axis of the 
 right cone which osculates to it along that side passes through the centre of the sec- 
 tion. 
 
 395. When a given curve of double curvature is 7iot a spherical 
 curve, we may propose to investigate the spheric surface which ap- 
 proaches to it most closely, at any assigned point. An osculating 
 circle has been defined (389) to be the ZzmzY of a circle, which toiiches 
 a given curve, or its tangent pt, at a given point p, and cuts the same 
 curve at a near point q ; while the tangent pt itself had been regarded 
 (100) as the limit of a rectilinear secant, or as the ultimate position 
 of the small chord pq,. It is natural then to define the osculating 
 sphere, as being the limit of a spheric surface, which passes through 
 the osculating circle, at a given point P of a curve, and also cuts that 
 curve in a point q, which is supposed to approach indefinitely to P, 
 and ultimately to coincide with it. Accordingly we shall find that 
 this definition conducts by quaternions to formulce sufficiently sim- 
 
 * This may also be inferred by limits from the formulae (11.) ; in which r and 
 a were used, provisionally, to denote a certain spherical arc and angle. 
 
550 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 pie; and that their geometrical interpretations are consistent with 
 known results: for example, the centre of spherical curvature, or the 
 centre of the osculating sphere, will thus be shown to be, as usual, the 
 point in which l\iQ polar axis (391, (5.)) touches the cusp-edge of the 
 polar developable (391, (6.)). It will also be seen, that whereas in 
 general, if r be a point in the normal plane (370, (8.)) to a given 
 curve at p, we can only say that the difference of distances, rq-rp, 
 is small of an order higher than the first, if the chord pr be small 
 of the first order ; and whereas, even if r be on the polar axis (391, 
 (4.)), we can only say generally that this difference of distances is 
 small, of an order higher than the second ; yet, if r be placed at the 
 centre s of spherical curvature, the difference sq-sp is small, of an 
 order higher than the third : so that the distance of a near point Qt,from, 
 the osculating sphere at the given point p, is generally small of the fourth 
 order, the chord being still small of the first. 
 
 (1.) Operating with S.\, where X is an arbitrary line, on the vector equation 
 392, V. of the osculating circle, we obtain the scalar equation of a sphere through 
 that circle under the form, 
 
 I...0 = 2S-^+S^'; 
 
 which may however, by 393, (7.), be brought to this other form, better suited to 
 our present purpose, 
 
 II. . . (w - k)2 = (p - fe)2 + 2cS/)"|o' (w - p) ; 
 
 c being any scalar constant, while k is still the vector of the centre K of the circle : 
 and the vector a of the centre s of the sphere is given by the formula, 
 
 III. . . (T = K + cYp"p', 
 
 which evidently expresses that this last centre is on the polar axis. 
 
 (2.) To express now that this sphere cuts the curve in a near point q, we are to 
 substitute for w the expression, 
 
 IV. . . 0) = pt=p + tp' + |<2p" + xt^utp'", with tto = 1 ; 
 but K has been seen (in 391) to satisfy the three equations, 
 
 V. ..0 = Sp'(,c-p), = Sp"(«:-p)-p'2, = Sp"p'(k:-p); 
 
 reducing then, dividing by ^t^, and passing to the limit, we find for the osculating 
 sphere the condition, 
 
 VI. . . Sp"'(p-K) + 3Sp'p" = cSp"'p''p'; 
 
 so that finally the vector <t satisfies the three scalar equations, 
 
 VII. . . O^Sp'C^-p?, = Sp"(<T-p)-p'S = Sp"'((r - p) - 3Sp'p", 
 
 by which it is completely determined, and of which the two last are seen to be the 
 successive derivatives of the first, while that first is the equation of the normal plane : 
 
CHAP. III.] CONSTRUCTION FOR GEN IRE OF SPHERE. 551 
 
 whence the centre s of this sphere is (by the sub-arts, to 386, comp. 391, (6.)) the 
 point where the polar axis ks touches the cusp-edge of the polar developable. 
 
 (3.) Differentials may be substituted for derivatives in the equations VII., 
 which may also be thus written (comp. 391, (4.)), 
 
 VIII. . . = dT(p-(r), = d2T(p-(T), = d3T(p-(7), if d(T = 0; 
 the distance of a near point Q of the given curve from the osculating sphere is there- 
 fore small (as above said), of an order higher than the third, if the chord pq be small 
 of the Jirst order. 
 
 (4.) The two first equations VII., combined with V., give also 
 
 IX. ..0 = Sp'((T-/c), = S|o"(<r-K), = S(K-p)((r-/c); 
 which express that the line ks is perpendicular to the osculating plane and absolute 
 normal at p, as it ought to be, because it is part of the polar axis. 
 
 (5.) Conceiving the three points p, k, s, or their vectors p, k, a^ to vary together^ 
 the equations V. and VII., combined with their own derivatives, give among other 
 results the following : 
 
 X. . . = Sk'p' = Sct'p' = Sff'p" = S(t'(k - p) = S(t"p' ; 
 
 of which the geometrical interpretations are easily perceived. 
 (6.) Another easy combination is the following, 
 
 XI. . . = S/c'('^ + P-2k), 
 as appears by derivating the last equation IX., with attention to other relations ; 
 but 2k - p is the vector of the extremity, say m, of the diameter of the osculating 
 circle, drawn from the given point p ; we have therefore this construction : — 
 
 On the tangent kk' to the locus of the centre of the osculating circle, let fall a 
 perpendicular from the extremity M of the diameter drawn from the given point P ; 
 this perpendicular prolonged will intersect the polar axis, in the centre s of the oscu- 
 lating sphere to the given curve at p. 
 
 (7.) In general, the three scalar equations VII. conduct to the vector expres- 
 sion, 
 
 viT , 3Vp'p'V p" + p'2Vp>' 
 XII. . . a = p + ^^y^,r, ; 
 
 or with differentials, 
 
 3Vdpd2pSdod?p + dpaVd^pdp 
 XIII. . . a = p+ Sdpd2pd3p ' 
 
 the scalar variable being still left arbitrary. 
 
 (8.) And if, as an example, we introduce the values for the helix^ 
 
 XIV. ..p = cta + a% p' = ca + ^ a<+i^, P"=-\^ ]«*/?, 
 
 -w 
 
 whereof the three first occurred before, we find after some slight reductions the ex- 
 pression, in which a denotes again the constant inclination of the curve to the axis of 
 
 the cylinder, 
 
 XV. . . a — p — a^(B cosec2 a = cta — a*l3 cot^ a ; 
 
 but this is precisely what we found for k, in 389, VIII. ; for the helix, then, the 
 two centres, K and s, of absolute and spherical curvature, coincide. 
 
552 elp:ments OF QUATERNIONS. [bookiii. 
 
 (9.) This known result is a consequence, and may serve as an illustration, of the 
 general construction (6.) ; because it is easy to infer, from what was shown in 389, 
 (3.), respecting the locus of the centre k of the osculating circle to the helix, as being 
 another helix on a co-axal cylinder^ that the tangent kk' to this locus is perpendi- 
 cular to the radius of curvature kp, while the same tangent (kk' or k') is always 
 perpendicular (X.) to the tangent (pp' or p') to the curve ; kk' is therefore here at 
 right angles to the osculating plane of the given helix, or coincides with its polar 
 axis: so that the perpendicular on it from the extremity m of the diameter of cur- 
 vature falls at the point k itself, with which consequently the point s in the present 
 case coincides^ as found by calculation in (8.). 
 
 (10.) In general, if we introduce the expressions 376, VI., or the following, 
 
 XVI. . . p' = s'Vsp, p" = s'2D«2p + s"-Dsp, p'" = s'^Ds^p + Bs's'Ds^p + s"'Ds/0, 
 
 in which s denotes the arc of the curve, but the accents still indicate derivations with 
 respect to an arbitrary scalar t ; and if we observe (comp. 380, (.12.)) that the re- 
 lations, 
 
 XVII. . . D,p2 = _ 1, S . Jy,pWp = 0, S . T>,pDs^p + D,2p2 ^ Q, 
 
 in which T>sp^ and D^^pS denote the squares of r>sp and Ds^p, and S . T>spJ>s^p denotes 
 S(Dsp.Ds^p), &c., exist independently of the form of the curve ; we find that s" and 
 s'" disappear from the numerator and denominator of the expression XII. for <t— p, 
 and that they have s'^ for a common factor: setting aside which, we have thus the 
 simpler formulag, 
 
 XVIII - = V. UspT>s^p ^ Ds . DspWp 
 
 S.DspDs2pD/|0 S.DspVs'^pDs^p' 
 
 And accordingly the three scalar equations VII., which determine the centre of the 
 osculating sphere, may now be written thus, 
 
 XIX. . . S((T-p)D,p = 0, S(»T-p)D,2p+ 1 = 0, S(flr-p)D,3p = 0. 
 
 (11.) Conversely, when we have any formula involving thus the successive deri- 
 vatives of the vector p taken, with respect to the arc, s, we can always and easily 
 generalize the expression, and introduce an arbitrary variable t, by inverting the 
 equations XVI. ; or by Avriting (comp. 390, VIII.), 
 
 XX. . . Dsp =s'~^p'', T>s-p =s''>(s'-^p')' = s'-2p"- s'-Vp', &c. 
 
 (12.) It may happen (comp. 379, (2.)) that the independent variable t is only 
 proportional to s, without being equal thereto ; but as we have the general relation, 
 
 XXI. . . Bt^p = s'"D/'p, if s' = Dts = Tp' = const., 
 it is nearly or quite as easy to effect the transformations (10.) and (11.) in the case 
 here supposed, or to pass from t to s and reciprocally, as if we had «' = 1. 
 
 (13.) If the vector a be treated as constant in the derivations, or if we consider 
 for a moment the centre s of the sphere as a fixed point, and attend only to the va- 
 riations of distance of a point on the curve from it, then (remembering that T(p — a)^ 
 = — (p — (t)-) we not only easily put (comp. VIII.) the three equations XIX. under 
 
 the forms, 
 
 XXII. . . = D,T(p - (t) = D,2T(p - (t) = D,5T(p - (t), 
 
 but also obtain by XVII. this/o«;<A equation, 
 
 XXIII. . .T(p-(T)D/T(p-(T)=S.((7-p)p/p+D,2p2. 
 
CHAP. III.] COEFFICIENT OF DEVIATION FROM SPHERE. 553 
 
 (14.) If then we write, for abridgment, 
 
 XXIV. . . r = T(k — p) = TDs-p-i = radius of osculating circle ; 
 XXV. . . iB = T(<T — p) = radius of osculating sphere ; 
 and 
 
 XXVI S = ?^fjlPl?iV = S . DsQ^T>s ^ pT>s*p 
 
 - Ds^f}^ S . TispDs-p^Ds^p^ 
 
 we see that this scalar, S, must be constantly equal to unity, for every spherical 
 curve ; but that for a curve which is non-spherical, the distance SQ of a near point 
 Q, from the centre s of the osculating sphere at p, is generally given by an expres- 
 sion of the form, 
 
 XXVII... sQ=i2 + ^^^|^*, with «o=i; 
 
 80 that, at least for near points Q, on each side of the given point p, the curve lies 
 without or ioithin the sphere which osculates at that given point, according as the sca- 
 lar, s, determined as above, is greater or less than unity. 
 
 (15.) In the case (12.), the formula XXVI. may be thus written, 
 
 XXVIII... 5= li^^i:;!; 
 
 whence, by carrying the derivations one step farther than in (8.), we find for the 
 helix, 
 
 XXIX. . . 5' = cosec2a>l, or XXIX'. . . 5- l = cot2 a>0 ; 
 
 and accordingly it is easy to prove that this curve lies wholly unthout its osculating 
 sphere, except at the point of osculation. 
 
 (16.) In general, the scalar S - 1, which vanishes (14.) for aZZ spherical curves, 
 and which enters as a coefficient into the expression XXVII. for the deviation 
 SQ — sp of a near point of any other curve from its own osculating sphere, may be 
 called the Coefficient of Non- Sphericity ; and if qt be the perpendicular from that 
 near point q on the tangent pt to the curve at the given point P, we have then this 
 limiting equation, by which the value of that coefficient may be expressed, 
 
 /sq2- 
 
 XXX. . . 5'-l = lim. 
 
 1- 
 
 \, Qf2 
 
 (17.) Besides the forms XVIII., other transformations of the expressions XII. 
 XIII. for the vector a of the centre of an osculating sphere might be assigned ; but 
 it seems sufficient here to suggest that some useful practice may be had, in proving 
 that those expressions for a reduce themselves generally to zero, when the condition, 
 
 XXXI. . . Tp = const. 
 is satisfied. 
 
 (18.) It may just be remarked, that as r-i is often called (comp. 389, (4.)) the 
 absolute curvature, or simply the curvature, of the curve in space which is consi- 
 dered, so R-^ is sometimes called the spherical curvature of that curve : while r and 
 R are called the radii* of those two curvatures respectively. 
 
 * We shall soon have occasion to consider another scalar radius, which we pro- 
 pose to denote by the small roman letter r, of what is not uncommonly called the 
 torsion, or the second curvature, of the same curve in space. 
 
 4 B 
 
554 ELKxMKNTS OF QUATERNIONS. [bOOK III. 
 
 396. When the arc («) of the curve is made the independent 
 variable, the calculations (as we have seen) become considerably sim- 
 plified, while no essential generality is lost, because the transforma- 
 tions requisite for the introduction of an arbitrary scalar variable (t) 
 follow a simple and uniform law (395, (ll.)> &c.). Adopting then 
 the expression (comp. 395, IV.), 
 
 I. . . P, = P^■s^^ Is^t' + ^sXt'', with ?^o = 1 , 
 in which 
 
 II. . . T = D,/?, t' = D,V, -r" = D,V, 
 
 and therefore 
 
 III. ..tU 1=0, Stt' = 0, Stt" + t'2 = 0, 
 we shall proceed to deduce some other affections of the curve, besides 
 its spherical curvature (395, (18.)), which do not involve the consi- 
 deration of the fourth poiver of the a7'c (or chord). In particular, we 
 shall determine expressions for that known Second Curvature (or 
 torsion), which depends on the change of the osculating plane, and is 
 measured by the ultimate ratio of that change, expressed as an angle, 
 to the. «rc of the curve itself; and shall assign the quaternion equa- 
 tions of the known Rectifying Plane, and Rectifying Line, which are 
 respectively the tangent plane, and the generating line, of that known 
 Rectifying Developable, whereon the proposed curve is o^ geodetic (382) : 
 so that it would become a right line, by the unfolding of this last sur- 
 face into Qi plane. But first it may be well to express, in this new 
 notation, the principal affections or properties of the curve, which 
 depend only on the three first terms of the expansion I., or on the 
 three initial vectors />, t, t', or rather on the two last of these ; and 
 which include, as we shall see, the rectifying plane, but not the recti- 
 fying line : nor what has been called above the second* curvature. 
 
 (1.) Using then first, instead of I., this less expanded but still rigorous expres- 
 sion (eomp. 376, I.), 
 
 IV. . . ps = p + sr + |s2Msr', with «o = 1, 
 
 * In a Note to a very able and interesting Memoir, '* Sur les lignes courhes non 
 planes''' (referred to by Dr. Salmon in the Note to page 277 of his already cited 
 Treatise, and published in Cahier XXX. of the Journal de VEcoh Folytechnique), 
 M. de Saint- Venant brings forward several objections to the use of this appellatirm, 
 and also to the phrases torsion, flexion^ &c., instead of which he proposes to intro- 
 duce the new name, " camhrure ;" but the expression " second curvature''' may 
 serve us for the present, as being at least not unusual, and appearing to be suffi- 
 ciently suggestive 
 
CHAP. III.] EMANANT LINES AND PLANES. 555 
 
 and with the relations 11. and III., we have at once the following system of three 
 rectangular lines, which are conceived to be all drawn from the given point p of the 
 curve : 
 
 V. . . r = unit tangent ; VI. . . r'= vector of curvature (389, (4.) ; 
 and VII. . . J/ = rr' = — r'r = tt~^ = binomial (comp. 379, (4.)) ; 
 
 r being a line drawn in the direction of a conceived motion along the curve, in virtue 
 of which the arc (s) increases ; while r' is directed towards the centre of curvature j 
 or of the osculating circle, of which centre K the vector is now, 
 
 VIII. . . OK = K = p - r'"i = p + r*/ = p + rUr', 
 if IX. . . r~'^ = TT' = curvature at F, or IX'. . . r = Tt''^ = radius of curvature ; 
 and the third line v (which is normal at P to the surface of tangents to the curve) 
 has the same length (Tv = r-i) as r', and is directed so that the rotation round it 
 from r to r is positive. 
 
 (2.) At the same time, we have evidently a system of three rectangular vector 
 units from the same point p, which may be called respectively the tangent unit, the 
 normal unitf and the binormal unit, namely the three lines, 
 
 X. . . Ur = r, Ur' = r/, Vv = vtt ; 
 the normal unit being thus directed (like r') toivards the centre of curvature. 
 
 (3.) The vector equation (comp. 392, (2.)) of the circle of curvature takes now 
 the form, 
 
 XI...V-^ = -., 
 w — p 
 
 with the verification that it is satisfied by the value, 
 
 XII. . . w = /i = 2/c-p = p-2r'->, 
 in which fx. (comp. 395. (6.)) is the vector om of the extremity of the diameter of 
 curvature PM. 
 
 (4.) IhQ normal plane, the rectifying plane, and the osculating plane, to the 
 curve at the given point, form aj-ectangular si/stem of planes (comp. 379, (5.)), 
 perpendicular respectively to the three lines (1.) ; so that their scalar equations are, in 
 the present notation, 
 XIII. ..Sr(a>-p) = 0; XIV. . . Sr' (w - p) = ; XV. . . Si/(a;- p) = 0-, 
 
 by pairing which we can represent the tangent, normal, and binomial to the curve, 
 regarded as indefinite right lines ; or by the three vector equations, 
 
 XVI. . . Vr(w-p) = 0; XVII. . . Vr'(w -p)= 0; XVIII. .. Vj/ (a> - p) = 0. 
 
 (5.) In general, if the two vector equations, 
 
 XIX. . . V»/ (w ~ p) = 0, and XIX'. . . Yrjs (w« - pO = 0, 
 
 represent two right lines, ph and p«Hs, which are conceived to emanate according to 
 any given law from any given curve in space, the identical formula,'* 
 
 * It is obvious that we have thus an easy quaternion solution of the problem to 
 draw a common perpendicular to any two right lines in space. 
 
556 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XX...p.-p+V(^V„..V^^J = -.^,^;^^, 
 
 8how8 that the common perpendicular to these two emanants, which as a vector is re- 
 presented by either member of this formula XX., intersects the two lines in the two 
 points of which the vectors are, 
 
 YYT «. o (P«-p)^^ VYT' ,. ^ , „ g Cp^ - P ) ^ 
 XXI. . . w = p + wb — =: ; XAl . . . wj = Pj -f Mjb — — . 
 
 (6.) In general also, the passage of a right line from any one given position in 
 space to any other may be conceived to be accomplished by a sort of screw motion, with 
 the common perpendicular for the axis of the screw, and with two proportional velo- 
 cities, of translation along, and of rotation rownrf that axis : the locus of the two given 
 and of all the intermediate positions of the line (when thus interpolated) being a 
 Screw Surface, such as that of which the vector equation was assigned in 314, (1 1.), 
 and was used in 372, (4.). 
 
 (7.) AggLin, for ani/ quaternion, q, we have (by 316, XX. and XXIII.*) the two 
 equations, 
 
 XXII. . . IVq^Lq.VVq, XXII'. . . Y\Jq = sm l.q.\jYq', 
 
 comparing which we see that 
 
 XXIII. . . YJJq : 1U</ = sm Lq: lq= (very nearly) 1, 
 
 if the angle of the quaternion be small ; so that the logarithm and the vector of the 
 versor of a small-angled quaternion are very nearly equal to each other, and we may 
 write the following general approximate formula for such a versor : 
 
 XXIV. . .Vq = (£iU3 =) £VUg^ nearlg, if Z g be small; 
 
 the error of this last formula being in fact small of the third order, if the angle be 
 small of the^rs^ 
 
 (8.) And thus or otherwise (comp. 334, XIII. and XV.), we may perceive that 
 if the quaternion q have the form (comp. (5.)), 
 
 XXY. . . q = r]sV~\ with XXYl. . . ria^r] + srj' + . ., 
 and if we write for abridgment, 
 
 XXVII. . . = V -, and XXVIII. ..h = S-y 
 
 we shall then have nearly, if s be small, the expressions, 
 
 XXIX. . .Ua = U- = 6«^, and XXX. . . Tfl = T- = 1 + «A ; 
 V V 
 
 or, neglecting s"^, 
 
 XXXI. ..»;« = (1 + sA) e'Orj = tse^j + ^hrj, 
 
 in which last binomial, the frst (or exponential) term alone influences the direction 
 of the near emanant line (5.). 
 
 * Although the expression XXII'. for VUg is bere deduced from 316, XXIII., yet 
 it might have been introduced at a much earlier stage of these Elements ; for instance, 
 in connexion with the formula 204, XIX., namely TVUg' = sin iq. 
 
CHAP. 111.] VECTOR OF ROTATION, AXIS OF DISPLACEMENT. 557 
 
 (9.) At the same time, by supposing s to tend to 0, the formula XXI. gives, as a 
 limit, 
 
 XXXII. . . oH = o)Q = p + r}S;~-, = p-r]S^, 
 Yijij Or) 
 
 for the vector of the point, say h, on the given emanant ph, in which that given 
 line is ultimately intersected by the common perpendicular (6.), or by the axis of 
 the screw rotation (6.) ; but the direction of that axis is represented by the versor 
 U6^, and the angular velocity of that rotation is represented by the tensor T0, if the 
 velocity of motion (1.) along the given curve be taken as unity : we may therefore 
 say that the vector itself, or the factor which multiplies the arc, s, in the exponential 
 term XXXI., if set oW from the point H determined by XXXII., is the Vector of 
 Rotation of the Emanant, whatever the law (5.) of the emanation may be. 
 
 (10.) And as regards the screw translation (6.), its linear velocity is in like 
 manner represented, in length and in direction, by the following expression (obtained 
 by limits from XX.), 
 
 XXXIII. . . t = 0S - (set off from h) = Vector of Translation of Emanant, 
 
 = projection of unit-tangent on screw- axis (or of r on 9). 
 And the indefinite right line through the point H, of which this line t is a part, may 
 be called the Axis of Displacement of the Emanant. 
 
 (11.) It is easy in this manner to assign what may be called the Osculating 
 Screw Surface to the (generally gauche^ Surface of Emanants, or indeed to any 
 proposed skew surface ; namely, the screw surface which has the given emanant 
 (or other) line for one of its generatrices, and touches the skew surface in the whole 
 extent of that right line. 
 
 (12.) It is however more important here to observe, that in the case when the 
 surface of emanants is developable, the vector i of translation vanishes ; and that 
 conversely this vector i cannot be constantly zero, if that surface be undevelopable. 
 The Condition of Developability of the Surface of Emanants is therefore expressed 
 by the equation, 
 
 XXXIV. . . t = 0, or Sr0 = 0, or XXXIV. . . S??r/V = ; 
 and accordingly this condition is satisfied (as was to be expected) when »; = r, that 
 is, for the surface of tangents. 
 
 (13.) In the same case, of j; = or || r, the vector Q of rotation becomes equal (by 
 XXVII. and VII.) to the binomial v ; and the expression XXXII., for the vector a>o 
 of the foot H of the axis reduces itself to p ; and thus we might be led to see (what 
 indeed is otherwise evident), that the passage from a given tangent to a near one 
 may be approximately made, by a rotation round the binormal, through the small 
 angle, sTv =sr~^ = arc divided by radius of curvature. 
 
 (14.) Instead of emanating lines, we may consider a system of emanating planes, 
 which are respectively perpendicular to those lines, and pass through the same points 
 of the given curve. It may be sufficient here to remark, that the passage from one 
 to another of two such near emanant planes, represented by the equations, 
 
 XXXV. . . S// (w - p) = 0, XXXV. . . S7is((o - p) = 0, 
 may be conceived to be made by a rotation through an angle = sT9, round the right 
 line. 
 
558 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XXXVr. . . S?7 (o> - p) = 0, 8r]'{<o - p) - S»;r= 0, 
 or XXXVr. . . V6I (a> - p) + r'^r]T = 0, 
 
 in which the plane XXXV. touches its developable envelope, and which is parallel 
 to the recent vector 9, or to the vectoi- of rotation (9.) of the emanant line ; so that 
 if an equal vector be set off on this new line XXXVI., it may be said to be the Vec- 
 tor Axis of Rotation of the Emanant Plane. 
 
 (15.) For example, if we again make 7/ = r, so that the equation XXXV. repre- 
 sents now the normal plane to the curve, we are led to combine the equation XIII. of 
 that plane with its derived equation, and so to form the system of the two scalar 
 equations, 
 
 XXXVII. . . Sr(a> - p) = 0, Sr'(w - p) + 1 = 0, 
 
 whereof the second represents a plane parallel to the rectifying plane XIV., and 
 drawn through the centre of curvature VIII. ; and which jointly represent the polar 
 axis (391, (5.)), considered as an indefnite right line, which is represented otherwise 
 by the one vector equation, 
 
 XXXVIII. . . Vv (w - k) = 0, or XXXVIII'. . . Vv (w - p) = - r. 
 
 (16.) And if, on this indefinite line, we set off a portion equal to the binormal v, 
 h\xc\\ portion (wliich may conveniently be measured /rom the centre k) may be said, 
 by (14.), to be the Vector Axis of Rotation of the Normal Plane ; or briefly, the 
 Polar Axis, considered as representing not only the direction but also the velocity of 
 that rotation, which velocity =Tj/ = 7'~i = the curvature (IX.) of the given curve : 
 while another portion = \Jv = the binormal unit (2.), set off on the same axis from 
 the same centre of curvature, may be called the Polar Unit. 
 
 (17.) This suggests a new way of representing the osculating circle by a vector 
 equation (comp. (3.), and 316), as follows: 
 
 XXXIX. . . w*=/c + £«''(p-fc) = P + (£*''-l)r'-» 
 
 = p + «r 4 hs^r' + (£«" -1-sv- ^s^v^) r'-i ; 
 
 which agrees, as we see, with the expression I. or IV., if s^ be neglected; and of 
 which, when the expansion is continued, the next term is, 
 
 XL. . . 1s^v^t-'^=^s^vt' = . 
 
 (18.) The complete expansion of the exponential form XXXIX., for the variable 
 vector of the osculating circle, may be briefly summed up in the following trigono- 
 metric (but vector) expression : 
 
 XLI. . . Ws = K+| co3- + Uv.Bin- I (p-K), 
 
 in which, XLII. . . p- k = - r^r', and Uv. (p — /c) = ri/r'~* =rr; 
 
 80 that we may also write, neglecting no power ofs, 
 
 XLIII. . . Ws = p + rr sin - + r-r vers - ; 
 r r 
 
 and if this be subtracted from the full expression for the vector p,, the remainder may 
 be called the deviation of the given curve in space, from its oivn circle of curvature : 
 which deviation, as we already see, is small of the third order, and will soon be da- 
 
CHAIMII.] VECTOR AND RADIUS OF SECOND CURVATURE. 559 
 
 composed into its two principal parts, or terms, of that order, in the directions of the 
 
 normal and the binormal respectively. 
 
 (19.) Meantime we may remark, that if we only neglect terms of the fourth 
 
 order, the expansion T. gives, by III. and IX., for the length of a small chord pp,, 
 
 the formula : 
 
 XLIV. . . PPs = T(p,-p)=T(«r + |s2/ + is3/') 
 
 4,'"-^^y'-^^''''''t' 
 
 this length then is the same (to this degree of approximation), as that of the cAord of 
 an equally long arc of the osculating circle : and although the chord of even a small 
 arc of a curve is always shorter than that arc itself yet we see that the difference is 
 generally a small quantity of the third* order, if the arc be small of the first. 
 
 397. Resuming now the expression 396, 1., but suppressing here 
 
 the coefficient Wj, of which the limit is unity, and therefore writing 
 
 simply, 
 
 I. ../., = /)+ ST -^i5Vf|5V^ 
 
 with the relations, 
 
 11. . . T*2 = - 1 , Stt' = 0, Stt'^ = - t'2 = r-\ St't'^ = r-V, 
 
 if 5 = are, and r~^ = Tr' = curvature^'f as before, or r = radius of curva- 
 ture (> 0), while r' = D^r ; and introducing the new scalar, 
 
 III. . . r'^ = S — = T-^V — = Secondl Curvature. 
 
 it' V ^ 
 
 with v=.'n' = binormal, or the new vector, 
 
 t" v' 
 
 IV. . . r''T = TS — ^, = V— = Vector of Second Curvature, 
 
 supposed to be set off tangentially from the given point p of the 
 curve, or finally this other new scalar (> or < 0), 
 
 V. . . r = f S — -\ -Radius of Second Curvature^ 
 
 * This ought to have been expressly stated in the reasoning of 383, (5 ), for 
 which it was not sufficient to observe that the arc and chord tend to bear to each other 
 a ratio of equality, without showing (or at least mentioning) that their difference 
 tends to vanish, even as compared with a line which is ultimately of the same order 
 as the square of either. 
 
 t Whenever this word curvature is thus used, without any qualifying adjective, 
 it is always to be understood as denoting the absolute (ox first) curvature of the curve 
 in space. 
 
 X Compare the Note to page 564. 
 
560 ELEMENTS OF QUATERNIONS. [bOOK IIT. 
 
 •which gives the expression, 
 
 VI. . . t" = - r^T - rVr' + t-'tt' 
 = - 7--2Ut + (r-')'UT' + (rr)-^Uv ; 
 
 we proceed to deduce some of the chief affections of a curve in space, 
 which depend on the third power of the arc or chord. In doing this, 
 although everything new can be ultimately reduced to a dependence 
 on the two new scalars^ r' and r, or on the one new vector t'\ or even on 
 v' - Vtt^', yet some auxiliary symbols will be found useful, and almost 
 necessary. Retaining then the symbols v, /c, tr, R^ as well as t, t', r, 
 and therefore writing as before (comp. 396, VIII.), 
 
 VII. . . OK = /c = /3- t'~' = /)-}- rUT'=/> + rV, 
 VIII. . • (p- f^y^ = r''U(«: - /)) = t' = D//3 = Vector of Curvature, 
 
 we may now write also, by 395, XVIIL, 
 
 v' 
 IX. . . OS = ff = /) - - — 7 = /c + rr'rv = /c + r'rUi/, 
 
 and 
 
 X. . . (/5 - o-)"' = i^'U(o• -p) = j/'"'StV = Fec^or o/ Spherical Curvature^ 
 
 = projection of vector (t') of curvature on radius {R) of osculating sphere ; 
 
 because we have now, by VI., 
 
 XI. ..v' = {tt'Y = Vtt'^ = - t-'t' - rVV, 
 or XF. . . {TJpy = {rvY = - rr'r' = - r^UT', 
 
 and XII. . . StV' = - SttV = - r'r'^ = r-^r-\ 
 
 If then we denote by p and P the linear and angular elevations, of the 
 centre S of the osculating sphere above the osculating plane, we shall 
 have these two new auxiliary scalars, which are positive or negative 
 together, according as the linear height ks has the direction of + v 
 or of - t' : 
 
 XIII. . .;? = ^^ = r'r; XIV. . . P= kps = tan-i^ = sin-i^=cos-' 4; 
 
 while XV. . . i2=T(o--/3)=^(r2+;90 = V'0^ + ^''r2); 
 
 the angle P being treated as generally acute. Another important 
 line, and an accompanying angle of elevation, are given by the for- 
 mulae, 
 
 XVI. . . \ = V ^' = r'Yr'T'' = r^T + tt' = r-'Ur + r'JJv 
 r 
 
 = Nv'v'^ + 1/ = Rectifying Vector (set off from given point p), 
 ~ Vector of Second Curvature plus Binormal; 
 
CHAP. III.] OSC. CONES, CYLINDER, HELIX, AND PARABOLA. 561 
 
 \ r 
 
 XVII. . . H =/.- = tarr'^ - = Elevation of Rectifying Line (>0, <7r), 
 
 = the angle (acute or obtuse, but here regarded as positive), 
 which that known and important line (396) makes with the tangent 
 to the curve; so that (by XIII., XIV.) these two auxiliary angles,* 
 H and P, from which (instead of deducing them from r' and r) 
 all the affections of the curve depending on s^ can be deduced, are 
 connected with each other and with r' by the relation, 
 
 XVIII. ..tan P = r' tan iy. 
 Many other combinations of the symbols offer themselves easily, by 
 the rules of the present calculus ; for instance, the vector o- may be 
 determined by the three scalar equations (comp. 395, XIX.), 
 XIX. .. ST(ff-/)) = 0, St\<t-p) = -\, ST"((r-p)=0, 
 whence, by XVI., 
 
 XX. . . rV = r*V(VT^T''. {a - p)) = V\(<r - p), 
 a result which also follows from the expressions, 
 
 XXI. . . t'^ = [v ^'+ S ^X' = (X-r-V)T', 
 
 and XXII. . . a-p=r^T' + rpv = tUt' + p\Jp, 
 
 because XXIII. . . rpYXu = - rpv^t' = - tt't' ; 
 
 we may therefore replace the formula I. for the vector of the curve 
 by the following, which is true to the same order of approximation,! 
 
 XXIV. ..^, = /> + 5Tt^(/c-p)+^VX(^-/>): 
 
 and may thus exhibit, even to the eye, the dependence of all affec- 
 tions connected with s^, on the two new lines, \ and o"- />, which were 
 not required when ^ was neglected, but can now be determined by 
 the two scalars r and p (or r and r', or H and P as before). The 
 geometrical signification of the scalar p is evident from what precedes, 
 namely, the height (ks) of the centre of the osculating sphere above 
 that of the osculating circle, divided by the binormal unit (Ui^) ; and 
 
 * The angle H appears to have been first considered by Lancret, in connexion 
 with his theory of rectifying lines, planes, and surfaces : but the angle here called P 
 was virtually included in the earlier results of Monge. 
 
 t As regards the homogeneity of such expressions, if we treat the four vectors 
 ps, p, K, and a, and the five scalars s, r, i?, p, and r, as being each of theirs* di- 
 mension, we are then to regard the dimensions of r, r', k', H, and P as being each zero ; 
 those of r', v, and X as each equal to - 1 ; and that of either r" or v as being = - 2. 
 
 4 c 
 
562 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 as regards what has been called the radius r of second curvature (V.), 
 we shall see that this is in fact the geometrical radius of a second air- 
 cle, which osculates, at the extremity of the tangential vector rr, to the 
 principal normal section of the developable Surface of Tangents ; and 
 thereby determines an osculating oblique cone to that important sur- 
 face, and«/50 an osculating right cone* thereto, of which latter cone the 
 semiangle is S^, and the rectifying line X is the axis of revolution : 
 being also a side of an osculating right cylinder, on which is traced 
 what is called the osculating helix. We shall assign the quaternion 
 equations of these two cones, and of this cylinder, and helix ; and shall 
 show that although the helix has not generally complete contact of the 
 third order with the puen curve, yet it approaches more nearly to that 
 curve (supposed to be of double curvature), than does the osculating 
 circle. But an oscidating parabola will also be assigned, namely, the 
 parabola which osculates to the projection of the curve, on its own os- 
 culating plane : and it will be shown that this parabola represents 
 or constructs one of the two principal and rectangular components (396, 
 (18.)), of the deviation of the curve from its osculating circle, in a 
 direction which is (ultimately) tangential to the osculating sphere, while 
 the helix constructs the other component. An osculatitig' right cone to the 
 cone of chords, drawn /rom a given point of the curve, will also be as- 
 signed by quaternions : and will be shown to have in general a smaller 
 acute semiangle C (or tt - G), than the acute semiangle H (or tt - H), 
 of the osculating right cone (above mentioned) to the surface of tan- 
 gents, or (as will be seen) to the cone of parallels to tangents (369, 
 (6.), &c.) : the relation between these two semiangles, oitivo osculating 
 right cones, being rigorously expressed by the formula, 
 
 XXV... tanC = f tanir. 
 A new oblique cone of the second order will be assigned, which has con- 
 tact of the same order with the cone of chords, as the second right cone 
 (C), while the latter osculates to both of them; and also an oscu- 
 culating parabolic cylinder, which rests upon the osculating parabola, 
 and is cut perpendicularly in that auxiliary curve by the osculating 
 plane to the given curve. And the intersection of these two last sur- 
 faces of the second order (oblique cone and parabolic cylinder) will 
 
 * These two osculating cones, oblique and right, to the surface of tangents, 
 appear to have been first assigned, in the Memoir already cited, by M. de Saint Te- 
 nant : the osculating (circular) helix, and the osculating (circular) cylinder, having 
 been previously considered by M. Olivier. 
 
CHAP. III.] CONTACT OF THIRD ORDER, TWISTED CUBIC. 563 
 
 be found to consist partly of the hinormal at the given point, and 
 partly of a certain twisted cubic* (or gauche curve of the third degree)^ 
 which latter curve has complete contact of the third order with the 
 given curve in space. Constructions (comp. 395, (6.)) will be assigned, 
 which will connect, more closely than before, the tangent to the 
 locus of centres of curvature, with oiAer properties or affections of that 
 given curve. And finally we shall prove, by a very simple quaternion 
 analysis, as a consequence of the formula XI'., the known theorem,f 
 that when the ratio of the two curvatures is constant, the curve is a 
 geodetic on a cylinder. 
 
 (1.) The scalar expression III., for the second curvature of a curve in space, as 
 defined m 396, may be deduced from the formulae (396, (5.), &c.) of the recent 
 theory of emanants, which give, 
 
 XXVI. . . = VvV> = r-ir, wo = /o, < = 7-, if t] = v, 
 
 while the line of contact (396, (14.)), of the ema.na.nt plane with its envelope, coin- 
 cides in position with the tangent to the curve; in passing, then, from the given 
 point p to the near point Pg, the binormal (i/) and the osculating plane (4- v) have 
 (nearly) revolved together, round that tangent (r) as a common axis, through a 
 small angle =x~^s, and therefore with a velocity =x'\ if this symbol have the value 
 assigned by III., or by the following extended expression, in which the scalar va- 
 riable (t) is arbitrary (comp. 395, (ll.)> &;c.), 
 
 XXVII. . . r-i = S -^, = S =^-4— = -S-ficond Curvature : 
 
 Vp p Vdpd2p 
 
 while the binormal has at the same time been translated (nearly), in a direction 
 perpendicular to the tangent t, through the small interval is = sr, which (in the pre- 
 sent order of approximation) represents the small chord pPj. 
 
 (2.) As an example, if we take this new form of the equation of the helix, 
 
 XXVIII. .. pt = b(jxt cot a + 6«</3), with Ta = T^ = 1, and Sa/3 = 0, 
 which gives the derived vectors, 
 
 XXIX. . . pt' = ba (cot a + £«'^), pi' = - 6£"«/3, pi" = api', 
 and this expression for the arc s (supposed to begin with t), 
 
 XXX. . . s = s't, where s' = Tp' = b cosec a = const., 
 we easily find (after a few reductions) the following values for the two curvatures : 
 
 * This convenient appellation (of twisted cubic) has been proposed by Dr. Sal- 
 mon, for a curve of the kind here considered : see pages 241, &c., of his already cited 
 Treatise. The osculating twisted cubic will be considered somewhat later. 
 
 f This theorem was established, on sufficient grounds, in the cited Memoir of M. 
 de Saint Venant (page 26); but it has also been otherwise deduced by M. Serret, 
 in the Additions to M. Liouville's Edition of Monge (Pans, 1850, page 561, &c.). 
 
564 ELEMENTS OF QUATERNIONS. [bOOK 111. 
 
 XXXI. . . r-1 = 6-1 sin2 o, r-> = fti sin a cos a; 
 
 while the common centre (395), of the osculating circle and sphere, has now for its 
 vector (comp. 389, (3.)), 
 
 XXXII. . . K = (T = pt- hi'^^jS cosec2 a = 6 cot a (a« - £«*^ cot a) ; 
 h being here the raditis of the cylinder, but a denoting still the constant inclination 
 of the tangent (p') to the axis (a). 
 
 (3.) HhjQ rectifying line (396), considered merely as to imposition, being the 
 line of contact of the rectifying plane (396, XIV.) with its own envelope, is repre- 
 sented by the equations, 
 
 XXXIII. . . = Sr'(a> - p) = Sr"(w - p), or XXXIII'. . . = VX(a> - p), 
 with the signification XVI. of X ; and accordingly, if we treat the rectifying planes 
 as emanants, or change rj to r', we find the value = Vr"r'-* =X, which shows also 
 that in the passage from p to Pj the rectifying plane turns (nearly) round the rectify- 
 ing line, through a small angle = sTX, or with a velocity of rotation represented by 
 the tensor, 
 
 XXXIV. . . TX = V (r-2 + r-2) = r-i cosec JS"= r"! sec H; 
 
 so that what we have called the rectifying vector, X, coincides in fact (by the general 
 theory of emanants) with the vector axis (396, (14.)) of this rotation of the rectify- 
 ing plane : as the vector of second curvature (r"'r) has been seen to be, in the same 
 full sense (comp. (1.)), the vector axis of rotation of the osculating plane, when velo- 
 city, direction, and position are all taken into account. 
 
 (4.) When the derivative s' of the arc is only constant, without being equal to 
 unity (comp. 395, (12.)), the expression XVI. may be put under this slightly more 
 general form, 
 
 XXXV. . . X = v4-r, = V ^ = Rectifying Vector; 
 s p dsd2p 
 
 and accordingly for the helix (2.) we have thus the values, 
 
 XXXVI. . . X = a«'-' = ab-i sin a = ar-i^ cosec a, UX = a ; 
 the rectifying line is therefore, for this curve, parallel to the axis, and coincides with 
 the generating line of the cylinder, as is otherwise evident from geometry. The 
 value, TX = 6"^ sin a, of the velocity of rotation of the rectifying plane, which is 
 here the tangent plane to the cylinder, when compared with a conceived velocity of 
 motion along the curve, is also easily interpreted; and the formulae XVII., XVIII. 
 give, for the same helix (by XXXI.), the values, 
 
 XXXVII. . . r' = 0, H==a, P = 0. 
 
 (5.) The normal (or the radius of curvature), as being perpendicular to the 
 rectifying plane, revolves with the same velocity, and round a parallel line ; to de- 
 termine the position of which new line, or the point h in which it cuts the normal, 
 we have only to change t^ to r' in the formula 396, XXXIL, which then becomes, 
 
 XXXVIII. . . OH = ojo = p - r'S -^, = p - X-2r' 
 Xr 
 
 _ r'2 (k - p) _ r-Q -f r2K 
 
 "P+ r-2 + r-2 r2 + r2 
 
 = pcos2i/+ <csin2ir; 
 
CHAP. III.] AXIS OF DISPLACEMENT OF NORMAL. 565 
 
 the vector of rotation (396, (9.)) of the normal is therefore a line || and =\, which 
 divides (internally) the radius (r) of curvature into the two segments,* 
 
 XXXIX. . . PH = r sin2 H, hk = r cos2 H; 
 
 namely, into segments which ure proportional to the squares (r-2 and r~2)of thej^rsi 
 and second curvatures. 
 
 (6.) At the same time, what we have called generally the vector of translation 
 of an emanant line becomes, for the normal (by 396, (10.), changing 9 to \), the 
 line 
 
 7* 
 
 XL. . . t = XS - = U\ cos H=: — r"'\"i, set off from the same point h ; 
 
 A 
 
 and the indefinite right line, or axis, through that point h, 
 
 XLI. . . = VX(a) - Wo), or XLI'. . . = V\ (w - p cos^ B-k sin2 H), 
 
 along which axis the normal moves, through the small line si, while it turns round 
 the same axis (as before) through the small angle sTX, may be called (comp. again 
 396, (10.)) the Axis of Displacement of the Normal (or of the radius of curvature). 
 (7.) As a verification, for the helix (2.) we have thus the values, 
 
 XLII. . . PH = 6, Wo = P« — 6£«*/3 = bat cot a, t = a cos a ; 
 
 so that the axis of displacement (6.) coincides with the axis (a) of the cylinder, as 
 was of course to be expected. 
 
 (8.) When the given curve is not a helix, the values VI., XVI., XXXVIII., 
 and XL., of r", X, wq, and t, enable us to put the expression I. for ps under the 
 form, 
 
 XLIII. . . ps = W0 + SI + €»^ (p - Wo) -— ; 
 
 or 
 
 the curve therefore generally deviates, by this last small vector of the third order, 
 namely by that part of the term ^s^t" which has the direction of the normal r', or of 
 — t', and which depends on r',from the osculating helix, 
 
 XLIV. . . w« = wo + SI + £«^(p - Wo), 
 and from the osculating right cylinder, 
 
 XLV. . . TVX(w - wo) = sin H, 
 whereon that helix is traced, and of which the rectifying line (XXXIII.) is a side, 
 while its axis of revolution (comp. (7.)) is the axis of displacement (XLI.) of the 
 normal. 
 
 (9.) Another general transformation, of the expression I. for the vector of the 
 curve, is had by the substitution, 
 
 <2r' t' 
 XLVI..., = * + -+-, 
 
 in which * is a new scalar variable ; for this gives the new form, 
 
 * This law of division of a radius of curvature into segments, by the common 
 perpendicular to that radius and to its consecutive, has been otherwise deduced by 
 M. de Saint Venant, in the Memoir already referred to. 
 
566 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XLYIl. ..pt=^p + tT+^t^ It' + '^]-\-it^r- 
 
 and therefore shows that the curve deviates, by this other small vector of the third 
 order, 
 
 XLVIIL . . i<3r-iv = is3r-irr', 
 
 that is, by the part of the term ^s^r" which has the direction of the binomial v, and 
 which depends on r, from what we propose to call the Osculating Parabola, namely 
 that new auxiliary curve of which the equation is, 
 
 XLIX. . . m = 9 + tT + 
 
 *H-S)= 
 
 or from the parabola which osculates at the given point P, to the projection of the 
 given curve on its own osculating plane. 
 
 (10.) And because the small deviation XLVIIL of the curve from the parabola 
 is also the deviation of the same curve from this last plane, if we conceive that a 
 near point q of the curve is projected into three new points Qi, Q2, Q3, on the tan- 
 gent, normal, and binormal respectively, we shall have the limiting equation, 
 
 L. . . lim. = r-i = Second Ourvature ; 
 
 PQ1.PQ2 
 
 the sign of this scalar quotient being determined by the rules of quaternions. 
 
 (11.) But we may also(comp. 396, (17.), (18.)) employ this ^Afrc^^renera/ ^raws- 
 
 formation of I., analogous to the forms XLIII. and XLVIL, 
 
 LL. .p,= »c + «»>'(p-/c)+-vV, 
 
 with the value XI. of v'; in which the sum of the two first terms gives the vector of 
 the point of the osculating circle, which is distant from the given point PPg by an arc 
 of that circle equal to the arc s of the given curve ; and the third term, 
 
 LII. . . \s^v't = Is' (r" + r-'^T^ = - ^s^r-^r'r + ^s W, 
 which represents the deviation from the same circle, measured in a direction (comp. 
 IX. or X.) tangential to the osculating sphere, is (as we see) the vector sum of two 
 rectangular components, which represent respectively the deviations of the curve, 
 from the osculating helix (8.), and from the osculating parabola (9.). 
 
 (12.) It follows, then, that although neither helix nor parabola ha.s in general 
 complete contact of the third order with a given curve in space, since the deviation 
 from each is generally a small vector of that (third) order, yet each of these two 
 auxiliary curves, one on a right cylinder XLV., and the other on the osculating 
 plane, approaches in general more closely to the given curve, than does the osculating 
 circle : while circle, helix, and parabola have, all three, complete contact of the se- 
 cond* order with the curve, and with each other. 
 
 * It appears then that we may say that the helix and parabola have each a con- 
 tact with the curve in space, which is intermediate between the second and third or- 
 ders : or that the exponent of the order of each contact is the fractional index, 2~. 
 But it must be left to mathematicians to judge, whether this phraseology can pro- 
 perly be adopted. 
 
CHAP. III.] SECTION OF SURFACE OF TANGENTS. 567 
 
 (13.) As regards the geometrical signification of the new variable scalar, t, ia 
 the equation XLIX. of the parabola, that equation gives, 
 
 ff r't\ .1 r't t^ 
 
 liii...t.', = t|(i + -).+ *,'} = i + - + -..., 
 
 and therefore (to the present order of approximation), 
 
 LIV. . . Arc of Osculating Parabola (from (oq to ojt) 
 
 C* r't^ <3 
 
 = T<o'tdt=^t+-- + — = s(hy XLVI.) 
 Jo or br' 
 
 = Arc of Curve in Space (from po to ps^ ; 
 
 if then an arc =s be thus set of£ upon the parabola, with the same initial point p, 
 and the same initial direction, and if this parabolic arc, or its chord b)t — ioq, be ob- 
 liquely projected on the initial tangent r, by drawing a diameter of the parabola 
 through its final point, the oblique tangential projection so obtained will be = tr by 
 XLIX. ; and its length, or the ordinate to that diameter, will be the scalar t. 
 
 (14.) And as regards the direction of the diameter of the osculating parabola, 
 drawn as we may suppose from p, if we denote for a moment by D its inclination 
 to the normal + r', regarded as positive when towards the tangent + r, we have (by 
 XLIX. and XVIII.) the formula, 
 
 LV. . . tan Z) = - = i tan P cot H: 
 3 ^ 
 
 which is an instance of the reducibility, above mentioned, of all affections of the curve 
 depending on s^, to a dependence on the two angles, H and P. 
 
 (15.) Some of these affections, besides the direction of the rectifying line X, can 
 be deduced from the angle H alone. As an example, we may observe that the vec- 
 tor equation of the surface of tangents is of the form, 
 
 LVI. . . U}sit = Qs-\- tp's = Ps + tTs, 
 in which s and t are two independent and scalar variables, and 
 
 s3 
 
 LVIL .. ts=t + st'+ -t", 
 
 + terms depending on «* in pj. If then we cut this developable LVI. by the plane, 
 
 LVIII. . . Sr(w — |o) = — c= any given scalar constant, 
 which is, relatively to the surface, a normal plane at the extremity of the tangen- 
 tial vector cr from p, while this tangent is also a generating line, wegetthusa/>riK- 
 cipal* normal section, of which the variable vector has for its approximate expres- 
 sion, 
 
 LIX. . .o}s = (p + ct) + {cs +. .) r' + (|cs2r* + ..')v; 
 
 the terms suppressed being of higher orders than the terms retained, and having no 
 influence on the curvature of the section. "We find then thus, that the vector of the 
 centre of the osculating circle to this normal section of the surface of tangents to the 
 given curve is, rigorously, 
 
 * Some general acquaintance with the known theory of section* of surfaces is 
 here supposed, although that subject will soon be briefly treated by quaternions. 
 
568 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (cst'^"^ 
 LX. . . p + cr + ^^ ~ = p + c(r + rv) = p + cr\ ; 
 
 so that the locus of all such centres is the rectifying line XXXIII'. And if, in parti- 
 cular, we make c = r, or cut the developable at the extremity of the tangential vec- 
 tor rr, the expression LX. becomes then p-rrT + rVv ; which expresses that the 
 radius of the circle of curvature of this normal section of the surface is precisely 
 what has been called the Radius (r) of Second Curvature, of the given curve in 
 space. But this radius (r = r tan H) depends only on the angle IT, when the radius 
 (r) of (absolute) curvature is given, or has been previously determined. 
 
 (16.) The cone of the second order, represented by the quaternion equation, 
 LXI. . . = 2rSr (u) - p) Sv (w - p) + (Vr (a> - p))2, 
 has its vertex at the given point p, and rests upon the circle last determined ; it is 
 then the locus oi all the circles lately mentioned (15.), and is therefore (in a known 
 sense) an osculating oblique cone to the developable surface. of tangents : its cyclic 
 normals (comp. 357, &c.) being r and t + 2vv, or r and rr+ 2rUv. But, by 394, 
 (30.), the osculating right cone to this cone LXI., and therefore also (in a sense 
 likewise known) to the surface of tangents itself is one which has the recent locus 
 of centres (15.), namely the rectifying line (\), for its axis of revolution, while the 
 tangent (r) to the curve is one of its sides : its semiangle is therefore = H, and a form 
 of the quaternion equation of this osculating right cone is the following (comp. XLV.), 
 
 LXIL . . T VU\ (w - p) = sin H. 
 
 (17.) The right cone LXII., which thus osculates to the developable surface of 
 tangents LVI., along the given tangent r, osculates also along that tangential line 
 to the cone of parallels to tangents, which has its vertex at the given point P ; as is 
 at once seen (comp. 394, (30.)), by changing p and p" to / and r", in the general 
 expression Yp'p" (393, (6.), or 394, (6.)), for a line in the direction of the axis of 
 the osculating circle to a curve upon a sphere. And the axis of the right cone thus 
 determined, namely (again) the rectifying line (X), intersects the plane of the great 
 circle of the osculating sphere, which is parallel to the osculating plane, in a point 
 L of which the vector is, 
 
 LXIII. . . OL = p + rp\ = p + rrr + rpv. 
 
 (18.) We have thus, in general, a gauche quadrilateral, pksl, right-angled ex- 
 cept at L, with the help of which one figure all affections of the curve, not depending 
 on **, can be geometrically represented or constructed : although it must be observed 
 that when r = 0, which happens for the helix (XXXVIL), the osculating circle is 
 then itself a gi-eat circle of the osculating sphere, and the points p and L, like the 
 points K and s, coincide. 
 
 (19.) In the general case, it may assist the conceptions to suppose lines set off, 
 from the given point p, on the tangent and binormal, as follows : 
 
 LXI V. . . PT = BL = rr'r ; pb = tl = ks = rpv ; 
 
 for thus we shall have a right triangular prism, with the two right-angled triangles, 
 TPK and LBS, in the osculating plane and in the parallel plane (17.), for two of its 
 faces, while the three others are the rectangles, pksb, pblt, kslt, whereof the two 
 first are situated respectively in the normal and rectifying planesi 
 
CHAP. III.] INCLINATIONS DETERMINED BY QUATERNIONS. 569 
 
 (20.) All scalar properties of this auxiliary jonsw may be deduced, by our ge- 
 neral methods, from the three scalars, r, r, r', or r, H^ P ; and all vector properties 
 of the same prism can in like manner be deduced from the three vectors r, r', t", or 
 from r, v, v', which (as we have seen) are not entirely arbitrary, but are subject to 
 certain conditions. 
 
 (21.) As an example of such deduction (compare the annexed Figure 81), the 
 equation of the diagonal plane SPL, which contains the radius 
 (iE) of spherical curvature and the rectifying line (\), and V 
 the equation of the trace, say PU, of that plane on the oscu- \l 
 lating plane, which trace is evidently parallel (by the con- 
 struction) to the edffes Ls, TK of the prism, are in the recent 
 notations (comp. XX.), 
 
 LXV. . . = Sr"(w - p) ; LXVI. . . = V(r-ir)' (w - p) ; 
 with the verification that rSr'r" = r'Srr"=r-2/, by II. 
 
 (22.) In general, by 204, (22.), if a and (3 be any two 
 vectors, we have the expressions. 
 
 Fig. 81. 
 
 LXVII. . . tan Z. - = tan zl 
 a 
 
 tan Z. i8a = - tan ii a/3 
 
 = TV^:S^ 
 
 (TV: S) a/3, 
 
 TV /3 
 a a S * a 
 
 the angles of quaternions here considered being supposed as usual (comp. 130) to be 
 generally > 0, but < tt ; for example, we have thus, 
 
 LXVIII. 
 
 . tanJEf =tan Z -= (TV: S) Xr-i = (TV: S) (r-» 
 
 r') 
 
 as in XVII. ; and in like manner we have generally, by principles already ex- 
 plained (comp. 196, XVI.), 
 
 LXIX. . . cos A ' 
 
 cos Z. -- = - cos Z /3a = - cos Z a/3 
 
 = S^:T^=SU^ = -SUa/3. 
 a a a 
 
 (23.) Applying these principles to investigate the inclinations of the vector r", 
 which is perpendicular to the diagonal plane LXV. of the prism, to the three 
 rectangular lines r, r, v, or the inclinations of that diagonal plane itself to the nor- 
 mal, rectifying, and osculating planes, with the help of the expressions deduced from 
 VI. for the three products,* tt", tt", vt", we arrive easily at the following results : 
 
 * A student, who should be inclined to pursue this subject, might find it useful 
 to form for himself a table of all the binary products of the nine vectors, 
 
 r, t\ r", V, v\ X, or — p, (t — [i, and k', 
 
 considered as so many quaternions, and reduced to the common quadrinomial form, 
 a + &r + cr' + ev, in which a, h, c, e are scalars, whereof some may vanish, but 
 which are generally functions of r, r, and r'. 
 
 4 D 
 
570 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 r" -r-2 t" r'^r t" r-»r-> 
 
 LXX. .. cos £- = —-:;- ; cosZ.-=-7^,; cos^- = -7— -; 
 rTr T It v Lr 
 
 with the verification, that the sum of the squares of these three cosines is unity, be- 
 cause 
 
 LXXI. . . r2Tr"= V(l + r-2/?2) = V(l + r'2 + rV^) ; 
 or LXXr. . . rTr" = V(r-V2 + T\«), Tr" = V(r-4 + Tj/'«). 
 
 (24.) Or we may write, on the same general plan, 
 
 r" -R t" -rTX r" r ,, 
 
 LXXII. . . tan Z - = -7^ ; tan^- = — ;— ; tan ^ - = - V(H- r'2); 
 rTr TV V r ' 
 
 or 
 
 LXXIII. . . tan ^ tt" = RTv^ ; tan L t't" = rr'-iTX ; tan L vt"=- rr-i V(l + r'2) ; 
 
 and may modify the expressions, by introducing the auxiliary angles H and P, 
 with which may be combined, if we think fit, the following angle of the prism, 
 
 LXXI V. . . PKT = BSL = tan-i r'. 
 
 (25.) Instead of thus comparing the plane spl with the three rectangular planes 
 (379, (5.)) of the construction, we may inquire what is the value of the angle spl, 
 which the radius (i?) of spherical curvature makes with the rectifying line (X) ; and 
 we find, on the same plan, by quaternions, the following very simple expression for 
 the cosine of this angle, which may however be deduced by spherical trigonometry 
 also, 
 
 LXX V. . . cos SPL = - SUX(<T - p) = ^-- = sin PsinH; 
 or LXXV. . . cos spl = cos spb cos bpl. 
 
 (26.) In general, it is easy to form, by methods already explained, the quater- 
 nion equation of a cone which has a given vertex, and rests on a given curve in space ; 
 and also to determine the right cone which osculates (394, (30.)) to this general 
 cone, along any given side of it. 
 
 (27.) But if we merely wish to assign the osculating right cone to the cone of 
 chords from p, or to the locus of the line ppj, we may imitate a recent process : and 
 may observe that if this new cone be cut by the normal plane LVIIL, the vector of 
 the section has the following approximate expression, analogous to LIX., and like it 
 sufficient for our purpose, 
 
 LXXVI. . . ios= p + CT + ^csT + ics^r-iv ; 
 
 from which it may be inferred (comp. (15.), (16,)), that the axis of revolution oHIiq 
 new right cone has for equation, 
 
 LXXVII. . . = V(r-ir + |i/) (w - p). 
 
 This axis is therefore situated in the rectifying plane, between the rectifying line 
 (X or r-ir+ v), and the tangential vector (IV.) of second curvature (r"ir) : while the 
 semiangle C of the same new cone (measured like H from -f r towards + v) has the 
 value already assigned by anticipation in the formula XXV., and is therefore less 
 than the semiangle H if both be acute, but greater than H if both be obtuse; so that, 
 in each case, the new right cone (C) is sharper than the old right cone (^H). 
 
 (28.) The same result may be otherwise obtained, by observing that an unit- 
 
CHAP. III.] INTERSECTION OF OSCULATING SURFACES. 571 
 
 vector in the direction of the chord pp^ has (by 396, XLIV., and 397, I.) the ap- 
 proximate expression, 
 
 whence the axis of the osculating right cone to the cone of chords (27.) has rigorously 
 the direction of the line V^'x" (for s = 0), or of the vector, 
 
 LXXIX. . . ^ = Vr'(r2r" + ^r) = X - ij; = r-V + |v, as before. 
 
 (29.) This axis % makes (if we neglect s^) the same angle C, with the chord 
 PPs, as with the tangent t ; whereas the former axis X makes unequal angles with 
 those two lines, within the same order (or degree) of approximation : for our methods 
 conduct to the expression, 
 
 LXXX...zeiZ^ = ^__il_, 
 X 24rr' 
 
 from which the relation XXV., between the two right cones, may easily be deduced 
 anew. 
 
 (30.) Neglecting only s*, and employing the substitution XLVL, the expression 
 XLVII. for the vector of the given curve becomes, 
 
 LXXXI. .. p( = p + <r + i<2y + ^i3jr-v, if LXXXII. . . t; = r'+ — ; 
 
 3r 
 
 where the variable scalar t denotes, by (13.), the ordinate of the osculating para- 
 bola, and the constant vector v has the direction, by (14.), of the diameter of that 
 parabola. 
 
 (31.) In the present order of approximation, then, the proposed cwrue jn space 
 may be considered to be the common intersection of the three following surfaces of the 
 second order, all passing through the given point P : 
 
 LXXXIII. . . 2(Sr'(ai-p))2 = 3rSx/(6;-p)Svv(a>-p); 
 
 LXXXIV. . . 2Sr'(w-p) = -r2(Svi^(a>-p))2; 
 LXXXV. . . 3rSi/ (w - p) = - r2Sr'(w - p) Svv{u) - p) ; 
 
 whereof the first represents a new osculating oblique cone, which has a contact of the 
 same (^second) order with the cone of chords, as the osculating right cone (27.) ; the 
 second represents an osculating parabolic cylinder, which is cut perpendicularly in 
 the osculating parabola (9.), by the osculating plane to the curve ; and the third 
 represents a certain osculating hyperbolic (or ruled') paraboloid, whereof the tan- 
 gent (r) is one of the generating lines, while the diameter (?;) of the osculating /)a- 
 rabola is another. 
 
 (32.) Each of these three surfaces (31.) has in fact generally a contact of the 
 third order with the given curve; or has its equation satisfied, not only (as is ob- 
 vious on inspection) by the point p itself, but also when we derivate successively 
 with respect to the scalar variable t, and then substitute the values (comp. LXXXI.), 
 LXXXVI. . . w = po = p, (1)' = po~T, io" = po" = V, w'" = po"' = r-i v ; 
 
 r, r, p, T, V, and v being treated as constants of the equation, or of the surface, in 
 each of these derivations. 
 
.'>72 ELEMENTS OF QUATERNIONS. [boOK lil. 
 
 (33.) The cone LXXXIIL, and the cylinder LXXXIV., have a common gene- 
 ratrix, namely the hinormat* (v) ; and in like manner, another generating line of the 
 same cone, namely the tangent (r) to the curve, has just been seen (^31.) to be a 
 line on the paraboloid LXXXV. : and although the cylinder and paraboloid have 
 no finitely distant right line common, yet each may be said to contain the line at in- 
 finity, in the diametral plane of the cylinder, namely in the plane of v and v, of 
 which pldne the quaternion equation is (comp. (14.)), 
 
 LXXXVII. . . = Sru (oi - p), or LXXXVII'. . . = S(rrV' - 3r) (a> - p) ; 
 or the line in which this diametral meets the parallel axial plane. 
 
 (34.) On the whole, then, it is clear, from the known theory of intersections of 
 surfaces of the second order having a common generating line, that the given curve of 
 double curvature (whatever it may be) has contact of the third order with the twisted 
 cubic,f or gauche curve of the third degree, which is represented without ambiguity 
 by the system of the two scalar equations, 
 
 LXXXVIII. ..y = x2, z = x^ 
 
 if we write for abridgment, 
 
 (x=Q =)-r2Suv(w-p), 
 LXXXIX. . . !y = (<2 =) - 2r2Sr'(a> - p), 
 (z = (<3=)_6r2rSj/(w-p). 
 
 (35.) As another geometrical connexion between the elements of the present 
 theory, it may be observed that while the osculating plane to the curve, of which 
 plane the equation is, 
 
 XC. . . Sv(oj-p) = 0, as in 396, XV., 
 touches the oblique cone LXXXIII., along the tangent t to the same curve, the diame- 
 tral plane LXXXVII. touches the same cone along the binomial v, which was lately 
 seen (33.) to be, as well as r, a side of that oblique cone; but these two sides of 
 contact, T and v, are both in the rectifying plane (396, XIV.), and the two tangent 
 planes corresponding intersect in the diameter v of the parabola (9.) ; we have 
 therefore this theorem : — 
 
 The diameter of the osculating parabola to a curve of double curvature is the 
 polar of the rectifying plane, with respect to the osculating oblique cone LXXXIII. ; 
 that is, with respect to a certain cone of the second order, which has been above de- 
 duced from the expression LXXXI. for the vector pt of the curve, as one naturally 
 suggested thereby, and as having a contact of the third order with the curve at p, 
 
 * The geometrical reason, for the osculating cone LXXXIII. to the cone of chords 
 containing the binormal (v), is that if the expression LXXXI. for pt were rigorous, 
 and if the variable t were supposed to increase indefinitely, the ultimate direction of 
 the chord VPt would be perpendicular to the osculating plane. And the same binor- 
 mal is a generating line of the parabolic cylinder also, because that cylinder passes 
 through p, and all its generating lines are perpendicular to the last mentioned plane. 
 It is sufficient however to observe, on the side of calculation, that the equations 
 LXXXIII. and LXXXIV. are satisfied, when we suppose w — p || v. 
 
 t Compare again page 241, already cited, of Dr. Salmon's Treatise ; also Art. 
 285, in page 225 of the same work. 
 
CHAP. III.] TANGENT TO LOCUS OF CENTRES. 573 
 
 and therefore also a contact of the second order with the cone of chords from that 
 point. 
 
 (_36.) Conversely, this particular cone LXXXIII. \s geometrically distinguished 
 from all other* cones of the same (second) order, which have their vertices at the 
 given point p, and have each a contact of the same second order, with the given 
 cone of chords from that point, or of the tliird order with the given curve, by the 
 condition that it is touched (as above), along the binormal (v), by the diametral 
 plane {w) of the osculating parabolic cylinder LXXXIV. 
 
 (37.) We have already considered, in 395, (5.), the simultaneous variations of 
 the points p and k, or of the vectors p and k:. With recent notations, including the 
 expression fji = 2K — p, we have the following among other transformations, for the 
 first derivative of the latter vector, and therefore for the tangent kk' to the locus of 
 centres of curvature, of a given curve in space : 
 XCI. . . kk'= BsK = k= (p - r'-i)' = r + r'-ir'V'-i 
 = (p + r^T'y = r+ r^T"+ 2rrV' 
 = rr'r' + rh-^v = rr\T' -\- p'^rv) = rr'^ (j)t' -{■ rj/) 
 rr' rr' _ rr\a - ii) 
 
 ) — K (T—K ((T — k)(k; — |C 
 
 = cot //(Ur tan P + Uj/) = y-^R(\Jt' sin P + Vv cos P) 
 = r^vv'r' = r^Tv'v = v~^v't~'^ = r'-'^v'v'^ 
 = r-iv (p - <r) (k - p) = r-i (k - p) (p - (r) V 
 
 = r-'/?U«p-(r)(K-p)) = <i 
 if then we draw the diameter of curvature pm, and let fall S 
 
 a perpendicular kn from the centre k of the osculating cir- 
 cle on the new radius sm of the osculating sphere (as in the 
 annexed Figure- 82), this perpendicular vnll touchf the lo- 
 cus of the centi-e k, a result which agrees with the construc- 
 tion in 395, (6.) ; and we see, at the same time, that the 
 length of the line kk', or the tensor T/c', may be expressed 
 (comp. LXXIII.) as follows, 
 
 XCII. . . kk' = T/c' = J?Tr I = r^Tv' = tan I tt". \ 
 
 (38.) If we project the tangent kk', into its two rect- *^-.., ,..--'' 
 
 angular components, kk, and kk , on the diameter of cur- Fig. 82. 
 
 vature and the polar axis, we shall have by XCI. the expressions : 
 
 * The cone of this system (36.), which is touched along the binormal by the 
 normal plane ^ and which therefore intersects t\iQ parabolic cylinder LXXXIV. in 
 a new twisted cubic (comp. (34.)), having also contact of the third order with the 
 curve, is easily found to have, for its quaternion equation, the following : 
 
 2r2 (Sr'(w -p))2= 3rSr(a> -p)Sj/(w - p) ; 
 and with respect to this cone (comp. (35.)), the po/ar of the rectifying plane is the 
 (^absolute) normal (r') to the curve. 
 
 f Geometrically, and by infinitesimals, if we conceive k' to be an infinitely near 
 point of the locus of k, and therefore in the normal plane at p, the angle pk's (like 
 pks) will be right, and the point k' will be on the semicircle pks ; but the radius of 
 this semicircle drawn to k (comp. Fig. 82) h parallel to the line sm, to which line 
 the tangent kk' is therefore perpendicular, as above. 
 
XCTII. . 
 
 . KK, = rr'r' = r'Ur' = 
 
 XCIV. . 
 
 . KK =r2r-V = rr-iUv = 
 
 574 ELEMENTS OF QUATERNIONS. [bOOK HI. 
 
 = &c.; 
 
 = &c.; 
 a — K 
 
 these two projections then, or the vector-tangent kk' itself, would suffice to determine 
 r and r', or H and P, and thereby all the affections of the curve which depend on 
 s^, but not on s*. 
 
 (39.) We have also the similar triangles (see again Fig. 82), 
 
 XCV. . . A k^k'k oc ic kk' <x kms ; 
 and the vector equations, 
 
 XCVI. . . kk' : SM = KK^ : sk = kk' : km = kk : pk 
 — T-W = Vector of second curvature (IV.) ; 
 whence also result the scalar expressions, 
 
 XCVII. . . tan KSK, = tan kpk = r-^ = Second* Curvature (III.) : 
 
 this last scalar being positive or negative, according as the rotation ksk^ (or kpk*) 
 appears to be positive or negative, when seen from that side of the normal plane, 
 towards which the conceived motion (396, (1.)) along the given curve, or the unit 
 tangent + t, is directed. f 
 
 (40.) Besides the seven expressions, ITT., XXVIL, L., and XCVII,, this impor- 
 tant scalar r"i admits of many others, of which the following, numbered for reference 
 as 8, 9, &c., and deduced from formulae and principles already laid doAvn, are ex- 
 amples; and may serve as exercises in transformation, according to the rules of the 
 present Calculus, while some of them may also be found useful, in future geometrical 
 applications. 
 
 (41.) We have then (among others) the transformations : 
 
 XCVIII. . . Second Curvature = r~^(= seven preceding expressions') 
 
 = p-^r' = r-i cot H= TX cos H = r''/ cot P 
 
 (8, 9, 10, 11) 
 
 = r^SvT' = - SvV'-i = - r2Srr'r" = Srr'-»r" 
 
 (12, 13, 14, 15) 
 
 = - r^SvT"=Sv-W" = -SvK' = Stk't' 
 
 (16, 17, 18, 19) 
 
 = tk' (<r - )u)-i = S\r-i = (k - p)Y\v = - r'-iVXv 
 
 (20, 21, 22, 23) 
 
 = t'^t'YXv = r^SXvT' = S\r' v-i = SXr'-'v 
 
 (24, 25, 26, 27 
 
 = r'^^v'Xr = r^SvVr = Srv-V = r2Si/V>r" 
 
 (28, 29, 30, 31) 
 
 = HSvJ/V" = r"-iVv'X = r^r'-iSz/'Xr' = r^r'-^ SvXr" 
 
 (32, 33, 34, 35) 
 
 - Sv'Xr"-i - Tr"-2SXvV" - ~ ^'"''^ - ~ '"''' 
 
 (36, 37, 38, 39) 
 
 rr a - p 
 
 
 * In illustration it maybe observed, that if d« be treated as infinitely small, and 
 if the line kk' be supposed to represent (not the derivative k, but) the differential 
 vector dfc = k^s, then the projections kk^ and kk become dr and rr-'ds (comp. XCIII. 
 and XCIV.) ; while kpk (in Fig. 82) represents the infinitesimal angle r-'d*, through 
 which the osculating plane (comp. (1.)) revolves, round the tangent r to the curve 
 during the change d« of the arc. 
 
 t This direction of + r is to be conceived (comp. Fig. 81) to be towards the hack 
 of Fig. 82, as drawn, if the scalars r and r (and therefore also p) be positive. 
 
CHAP. III.] EXPRESSIONS FOR THE SECOND CURVATURE. 575 
 
 ^ = i2-i tan L rrr" = i2-» tan L (40, 41, 42, 43) 
 
 rr'-|-/?i/- r((r-p) 
 rr'v rrr' r t(^k — p') 
 
 (44, 45, 46, 47) 
 
 <T-K ((r-K)r r <r - k ((t-k)(p-k) 
 .S^-^— , = s / + f '-\ = S-^ (18,49,60) 
 
 J' 
 
 -dcosZ. - 
 
 = S-^=4|^) = 2 (51,62,63); 
 
 ^ ' rd cos Z. «- 
 ■ a 
 
 PKSL, in the forms 50 and 51, being points of the same gauche quadrilateral as in 
 (18.) ; and a, in 52 and 63,* denoting ani/ constant vector : while several other 
 varieties of form may be deduced from the foregoing by very simple processes, such 
 as the substitution of JJv for rv, &c., which gives for instance (comp. XI'.), from the 
 form 38, these others, 
 
 xcviir. . . r. = ::<M' = zff;')' = -J^ (64, 66, 66). 
 
 tt' Ur rdr 
 
 We may also write, with the significations (10.) of qi and Q3, the following expres- 
 sion analogous to L., 
 
 XCVIII". . . r-i = 6kp. lim. — , (57), 
 
 which contains the law of the inflexion of the plane curve, into which the proposed 
 curve of double curvature is projected, on its own rectifying plane : tlie sign of the 
 scalar, to which this last expression ultimately reduces itself, being determined by 
 the rules of quaternions. 
 
 (42.) And besides the varioiis expressions for the positive scalar r'^, which are 
 immediately obtained by squaring the foregoing forms, the following are a few 
 others : 
 
 XCIX. . . Square of Second Curvature = r-2 = Tr-2 
 
 = T\2 - r-2 = r2Sr"r'\ - r-2 = r2Tj/'2 - r-2r'2 (1, 2, 3) 
 
 = r^STvr" - r-^r'i = r2Tr"2 - r-2 - r-h'^ = R-^ (r<Tr"2 - 1) (4, 5, 6) 
 = R-^^Tv'^ = i?-2TK'2 = i2 2 tan2 l tt" (7, 8, 9) ; 
 
 while the important vector r", besides its two original forms VI., admits of the fol- 
 lowing among other expressions (comp. XX. XXI.) : 
 
 C. . . t'—'Ds^p (= the two expressions VI.) 
 
 = r-2V\((T - p) = Xr' - r-'^r'T = v't - r'^T (3, 4, 5) 
 
 = xNv\ = r-2r-ir ((t - p- r) = r'^p + r-^\ (tr - p) (6, 7, 8) 
 
 = ((p-/c)-i)' = r'(K:'-r)r' = -r-2r-il^-^ (9, 10, 11). 
 
 (43.) As regards the general theory (396, (5.), &c.) of emanant lines {rj) from 
 curves, it might have been observed that if we write, 
 
 * This last form 53 corresponds to and contains a theorem of M. Senret, alluded 
 to in the second Note to page 563. 
 
576 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 CI. ..K = yL with CII. . . = V -, as in 396, XXVIL, 
 d rj 
 
 the equation 396, XXXII. takes the simplified form, 
 
 cm. . . PH = wo- P = J/Sj?'^ = projection of vector Z, on emanant rj ; 
 for example, when »/ = v, then Q = r~ir, and ^ = 0, ph = 0, or wo = p, as in (1.) ; 
 and when ?; = r, then = v, ^ = r^r' -^ ?/, so that the projection ph again vanishes, 
 as in 396, (13.). 
 
 (44.) In an extensive class of applications, the emanant lines are perpendicular 
 to the given curve (jj -1- r) ; and since we have, by (43.), 
 
 we may write, for this case of normal emanation, the formula, 
 
 p_, _ y _ projection of vector of curvature (r') on emanant line (;?) _ 
 
 square of velocity (TO) of rotation of that emanant 
 
 for example, when the emanant (?/) coincides with the absolute normal (r'), we have 
 then = X, as in (3.), and the recent formula CV. becomes, 
 
 CVI. . . ph = wo - p = ^ = 7-'T\-« = rV sin2 H={K-p) sin2 H, 
 which agrees with the expression XXXVIII. 
 
 (45.) And in the corresponding case of tangential emanant planes, by making 
 Sr7y = in the second equation 396, XXXVI., and passing to a second derived 
 equation, we find for the intercept between the point P of the curve, and the point, 
 say R, in which the line of contact of the plane with its own envelope touches the 
 cusp-edge of that developable surface, the expression, 
 
 CVII. . PR= ~-Z^^-^ = -ZllLi^ll^Ill. ; 
 ST]rj'T]" projection of ij" on 9 
 
 which accordingly vanishes, as it ought to do, when r) = v, that is, when the emanant 
 plane Sr] (w — p) = coincides with the osculating plane XC. 
 
 (46.) Some additional light may be thrown on this whole theory, of the affections 
 of a curve in space depending on the third power of the arc, and even on those afl'ec- 
 tions which depend on higher pouters of s, by that conception of an auxiliary sphe- 
 rical curve, which was employed in 379, (6 ) and (7.), to supply constructions (or 
 geometrical representations) for the directions, not only of the tangent (p') to the 
 given curve, to which indeed the unit-vector (r) of the new curve is parallel, but 
 also of the absolute normal, the binormal, and the osculating plane ; while the same 
 auxiliary curve served also, in 389, (2.), to furnish a measwe of the curvature of 
 the original curve, which is in fact the velocity^ of motion in the new or spherical 
 curve, if that in the old or given one be supposed to be constant, and be taken for 
 unity. 
 
 * Accordingly the vector of velocity t , of this conceived motion in the auxiliary 
 curve, is precisely what we have called (389, (4.), comp. 396, VI.) the vector of cur- 
 vature of the proposed curve in space : and its temor (Tr') is equal to the rec/proca/ 
 of the radius (r) of that curvature. 
 
CHAP. III.] CASE OF CONSTANT RATIO OF CURVATURES. 577 
 
 (47.) We might for instance have observed, that while the normal plane to the 
 curve in space is represented (in direction) by the tangent plane to the sphere, the 
 rectifying plane (as being perpendicular to the absolute normal) is represented simi- 
 larly by the normal plane to the spherical curve : and it is not difficult to prove 
 that the rectifying line has the direction of that new radius of the sphere, which is 
 drawn to the point (say i.) where the normal arc to the auxiliary curve touches its 
 own envelope. 
 
 (48.) The point L thus determined is the common spherical centre (comp. 394, 
 (5.)) of curvature, of the auxiliary curve itself and of that reciprocal* curve on the 
 same sphere, of which the radii have the directions (comp. 379, (7.)) of the hinor- 
 mals to the original curve; the trigonometric tangent of the arcual radius of curva- 
 ture of the auxiliary curve is therefore ultimately equal to a small arc of that curve, 
 divided by the corresponding arc of the reciprocal curve (or rather by the latter arc 
 with its direction reversed, if the point L fall between the two curves upon the 
 sphere) ; and therefore to the first curvature (r~') of the given curve, divided by the 
 second curvature (r~i) : and thus we have not only a simple geometrical interpreta- 
 tion of the quaternion equation XI'., but also a geometrical proof (which may be 
 said to require no calculation'), of the important but known relation XVII., which 
 connects the ratio (r : r) of the two curvatures, with the angle (^H) between the tan- 
 gent (r) and the rectifying line (\), for atiy curve in space. 
 
 (49.) In whatever manner this known relation (tan H=r:r) has once been es- 
 tablished, it is geometrically evident, that if the ratio of the two curvatures be con- 
 stant, then, because the curve crosses the generating lines of its own rectifying deve- 
 lopable (396) under a constant angle (//), that developable surface must be cylin- 
 drical : or in other words, the proposed curve of double curvature must, in the case 
 supposed, be a geodeticf on a cylinder (comp. 380, (4.)). Accordingly the point L, 
 in the two last sub-articles, becomes then a fixed point upon the sphere, and is the 
 common pole of two complementary small circles, to which the auxiliary spherical 
 curve (46.), and the reciprocal curve (48.), in the case here considered, reduce them- 
 selves ; so that the tangent and the binormal to the curve in space make (in the 
 
 * The reciprocity here spoken of, between these two spherical curves, is of that 
 known kind, in which each point of one is a pole of the great-circle tangent, at the 
 corresponaing point of the other : and accordingly, with our recent symbols, we have 
 not only v = Vrr', but also, Yvv' ^r^Yv'v-^ = r-^r-W \\ t. 
 
 t The writer has not happened to meet with the 5'eome«nca?/>roo/of this known 
 theorem, which is attributed to M. Bertrand by M. Liouville, in page 558 of the 
 already cited Additions to Monge ; but the deduction of it as above, from the fun- 
 damental property (396) of the rectifying line, is sufficiently obvious, and appears to 
 have suggested the method employ ed by M. de Saint- Venant, in the part (p. 26) of his 
 Memoir sur les lignes courbes non planes. Sec, before referred to, in which the result is 
 enunciated. Another, and perhaps even a simpler method, suggested by quaternions, 
 of geometrically establishing the same theorem, will be sketched in the present sub- 
 article (49.) ; and in the following sub-article (50.), a proof by the quaternion ana- 
 lysis will be given, which seems to leave nothing to be desired on the side of simpli- 
 city of calculation. 
 
 4 E 
 
578 KLEMENTS OF QUATERNIONS. [bOOK III. 
 
 same case) constant angles, with t\iQ fixed radius drawn to that point : and the curve 
 itself ia therefore (as before) a geodetic line, on some cylindrical surface. 
 
 (50.) By quaternions, when the two curvatures have thus a constant ratio, tiie 
 equations XI'. and XVI. give, 
 
 CVIII. . . (rX)' = {\Jv + rr-ir)' = (rr-i)'r = 0, 
 or CIX. . . r\ = a constant vector ; 
 
 the tangent (r) makes therefore, in this case, a constant angle {H) with a constant 
 line (rX) : and the curve is thus seen again, by this very simple analysis, to be a 
 geodetic on a cylinder. And because it is easy to prove (comp. XXXI.), that we 
 have in the same case the expression, 
 
 ex. . . r sin^ H= radius of curvature of base, 
 
 or of the section of the cylinder made by a plane perpendicular to the generating 
 lines, this other known theorem results, with which we shall conclude the present se- 
 ries of sub-articles : When both the curvatures are constant, the curve is a geodetic 
 on a right circular cylinder (or cylinder of revolution') ; or it is what has been called 
 above, for simplicity and by eminence, a helix. * 
 
 398. When the fourth power {s^) of the arc is taken into account, 
 the expansion of the vector Ps involves another term^ and takes the 
 form (comp. 397, I.), 
 
 I. . .Ps = p + ST+ IfiV^ f i-S V + ^\sh^'\ 
 
 in which 
 
 II. . . T^'^ = D,V, and III. . . Stt'^' = - 3StV" = - Sr'r' ; 
 
 so that the new affections of the curve, thus introduced, depend only 
 on two new scalars, such as r' and r'^, or r^ and B\ or H' and F\ 
 &c. We must be content to offer here a very few remarks on the 
 theory of such affections, and on the manner in which it may be ex- 
 tended by the introduction of derivatives o£ higher orders. 
 
 * In general, the expression XLIV. for the vector w, of the osculating helix, in 
 which I = — r'^X"^ = r — X-'r', and p - wq = \~^t', gives Toj's = 1 ; so that the devia- 
 tion (8.) maybe considered (comp. (13.)) to be measured from the extremity of an 
 arc of the helix, which is equal in length to the arc s of the curve, and is set off from 
 the same initial point p, with the same initial direction : while wo does not here de- 
 note the value of w, answering to s - 0, but has a special signification assigned by 
 the formula XXXVIII. It may also be noted that the conception, referred to in 
 (46.), of an auxiliary spherical curve, corresponds to the ideal substitution of the 
 motion of a point with a varying velocity upon a sphere, for a motion with an uni- 
 form velocity in space, in the investigation of the general properties o( curves of dou- 
 ble curvature: and that thus it is intimately connected (comp. 379, (9 )) with the 
 general tlieory of hodographs. 
 
CHAP. III.] FOURTH POWER OF ARC, ROT. OF RADIUS R. 579 
 
 (1.) The new vector r'", on whicli everything here depends, is easily reduced to 
 the following forms,* analogous to the expressions 397, VI. for r" : 
 
 T t' V 
 
 = 3r-Vr + (r(r-i)"+X2)r'+ (r~^r-^yr^v. 
 (2.) The first derivatives of the four vectors, v\ k\ X, (t, taken in like manner 
 with respect to the arc « of the curve, are the following : 
 V. . .v"= (Yrr")' = Vrr'" + r-2\ 
 
 = r-2r-ir + (r-2r-i)'r'-i -f (r(r-i)"- r-2) v ; 
 VI. . . k"= - r-Wr+ (rr" -r^T-^)T' + (r2r-»)'v ; 
 VII. . . X' = Cr-i)V + (r-i)Vj/, or VII'. . . (rX)' = (rr-i)V (comp. 397,CVIII.) ; 
 VIII. . . <t' = (K+prvy = (p' + rr-^)rv = JiR'p-^rv; 
 
 in which last the scalar derivatives p' and JR^ are determined, in terms of r" and r', 
 by the equations, 
 
 IX. . . p' = (r'ry = r"v + r'r', 
 and X. . . i2' = i?-i (/?p' + rr') = p' sin P+r' cos P=(/;' + cot jff) sin P. 
 We have also the derivatives, 
 
 rr' — r'r r"lr' — r'^r' 
 
 XL . . e:' = 
 
 XII. . . P' = 
 
 r'i + r2 rrX2 ' 
 
 rp' — r'p (rr" — r'2) r + jv'r' 
 
 r2 + p3 /22 
 
 and the relations, 
 
 XIII. . . SrrV" = Svr"' = -(r-2r-i)'; 
 XIV. . . SrrV"=Sj/V" = -r-3r-2(p'-rrX2); 
 XV. . . SrVV" = r-2SXr"' = - r-5 (rr"!)' ; 
 which may be proved in various ways, and by the two first (or the two last) of 
 which, the derivatives r' and p\ and therefore also H' and P', can be separately 
 calculated, as scalar functions of the four vectors r, r', r", r'", or of some three 
 of them, including the new vector r'". 
 
 (3.) We may also deduce, from either V. or VIII., the following vector expres- 
 sions, of which the geometrical signification is evident from the recent theory (396, 
 397) of emanant lines and planes : 
 
 XVI. . . Vector of Rotation of Radius (i?) of Spherical Curvature 
 = Vector of Rotation of Tangent Plane to Osculating Sphere 
 
 = (say)^ = V^ = V^^ = i2-2r(W+(T-p) (1, 2, 3) 
 
 T I rR' <t — d\ 'rr I r , \ 
 
 ^:r1,7""*'~r)= ^2(^^+7+'-^ +^»') = ^''^(^^+/''--i'0 (4,5,6); 
 
 whence follows this tensor value for the common angular velocity of these two con- 
 nected rotations, compared still with the velocity of motion along the curve, 
 
 * In these new expressions, on the plan of the second Note to page 561, the 
 scalars r', p\ R', and the vector <t', are to be regarded as of the dimension zero ; r", 
 H', P\ and k" of the dimension — 1 ; X' of the dimension — 2 ; and v" and r"', as 
 being each of the dimension — 3. 
 
580 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XVII. . . Velocity of Rotation of Radivs (R), or of Tangent Plane to Sphere, 
 
 = T^ = TV *^ = 22-1 V(l + i?'2 cot2 P) = /?-i V { 1 + (p' + cot Sy cos» P} ; 
 
 with the verifications, for the case of the helix, for which p = 0, jo' = 0, P= 0, and 
 R=r, that these expressions XVI. and XVII. become, 
 
 XVI'. . .<p = \, and XVII'. . . T0 = T\ = r'l cosec H, 
 which agree with those found before, for the vector and velocity of rotation of the 
 radius (r) of absolute curvature. 
 
 (4.) As another verification, we have R' = for every spherical curve, and the 
 general expressions take then the forms, 
 
 XVI". . . = -^^, and XVII". . . T0 = R-\ 
 a-p 
 
 of which the interpretation is easy. 
 
 (5.) In general, the formula XVII. may also be thus written, 
 XVIII. . , i?202 + 1 = _ i2'3 cot2 P^R'2 -p-'iRiR'^ = i?'2 + <r'2 = <t'2 cos2 P ; 
 or thus, XIX. . . i2T0 = V(l + T<r'2 cos2 P) = V (1+ T(r'2 - /2'2) ; 
 
 or finally, XX. . . R^T<p = V(i22 - r2(r'2) = V(i22 + r2T(T'«) ; 
 
 so that the small angle, sT<p, between the two near radii of spherical curvature, R 
 and Rs, is ultimately equal to the square root of the sum of the squares of the two 
 small angles, in two rectangular planes, sR'^ and rsR'^Ta', or psPj and SPSjj, which 
 are subtended, respectively, at the centre s of the osculating sphere by the small arc 
 s of the given curve, and at the given point p by the small corresponding arc sT<t' 
 of the locus of centres s of spherical curvature, or of the cusp-edge (395, (2.)) of the 
 polar developable ; exactly* as the small angle sT\, between two near radii (397, 
 (5.)) of absolute curvature, r and Vs, is ultimately the square root of the sum of the 
 squares of the two other small angles, sr"i and sr-^, or pkp^ and kpKj, which are 
 likewise situated in two rectangular planes, and are subtended at the centre k of the 
 osculating circle by the small arc s of the curve, and at the given point p by the 
 corresponding arc sTk of the locus of the centre k (comp. 397, XXXIV., XCIV.). 
 (6.) The point, say v, in which the radius R of the osculating sphere at p ap- 
 proaches most nearly to the near radius Rg from P^, \s ultimately determined (comp. 
 397, CV. and X.) by the formula. 
 
 Vector of Spherical Curvature 
 
 XXI. ..PV=^: 
 
 Square of Angular Velocity of Radius (jR) 
 
 = (p - <rrTr' = TT^h^ = TT^^^-R'^ ' 
 
 the vector of this point v (in its ultimate position) is therefore 
 
 r^R'ip+p2ff r^E'^p + Y^r'icT 
 XXII...OV = p + ^=-^^^— = ^,^,2+^2?^' 
 
 with the verification, that (by X., comp. XVII.) the scalar p'^riZ' or R' cot P re- 
 
 * It will soon be seen that these two results, and others connected with them, 
 depend geometrically on one common principle, which extends to all systems of 
 normal emanants (397, (44.)). 
 
CHAP, in.] CIRCUMSCRIBED DEVELOPABLE, AUX. ANGLE J. 581 
 
 duces itself to cot H, or to rx~\ for the case p = 0, p' = 0, P= (corap. (3.)) : and 
 that thus the expression 397, XXXVIII., for the vector oh of tlie point of nearest 
 approach^ of a radius (r) of absolute curvature to a consecutive^ radius of the 
 same kind, is reproduced. 
 
 (7.) In general, if we introduce a new auxiliary angle, /, determined by the 
 formula, 
 
 XXIII. . . cot J =p-^rR' = R' cot P=(p'-|-cot H) cos P= ff(r-> +P'), 
 
 the expression XXII. takes the simplified form (corap. again 397, XXXVIII.), 
 
 XXIV. . . ov = p + ^= p cos2 /+ (T sin2 J; 
 
 and the segments, into which the point v divides (internally) the radius M of the 
 sphere, have the values (comp. 397, XXXIX.), 
 
 XXV. . . pv = ^sin3 j^ yg = i2co32 J. 
 
 (8.) A geometrical signification may be assigned for this new angle J, which is 
 analogous to the known signification of the angle H (397, XVII.). In fact, the 
 tangent plane to the osculating sphere at P touches its own developable envelope 
 along a new right line, of which the scalar equations are, 
 
 XXVI. ..S((r-p)(w-p) = 0, S((r'-r)(a>-p)=0; 
 
 and because the developable locus of all such lines can be shown to be circumscribed, 
 along the given curve, to the locus of the osculating circle, which is at the same time 
 the envelope of the osculating sphere, we shall briefly call this locus of the line 
 XXVI. the Circumscribed Developable. And the inclination of the generatrix of 
 this new developable surface, to the tangent to the given curve at P, if suitably mea- 
 sured in the tangent plane to the sphere, is precisely the angle which has been 
 above denoted by J. 
 
 (9.) To render this conception more completely clear, let us suppose that a 
 finite right line pj is set off from the given point p, on the indefinite line XXVI., so 
 as to represent, by its length and direction, the velocity oi the rotation of the tangent 
 plane to the osculating sphere; and so to be, in the phraseology (396, (14.)) of the 
 general theory of emanants, the vector-axis of that rotation. We shall then have 
 the values, 
 
 XXVII. . . pj= ^(= the six expressions XVI.) 
 
 = i?-ir (cot J+ U ((T - p)) = R-^ cosec /(r cos J+ rU(<T - p) sin J) (7, 8) ; 
 
 the angle / being determined by the formula XXIII., and a new expression, 
 T0 = jR-' cosec/, being thus obtained for the velocity XVII. 
 
 (10.) Hence the new angle J, if conceived to be included (like H) between the 
 limits and ir, may be considered to be measured /rom r to ip, or from the unit-tan- 
 gent to the curve at P, to the generating line Pj of the circumscribed developable 
 (8.), in the direction from r to t((t - p) : which Zasi tangent to the osculating sphere 
 
 * This usual expression, consecutive, is obviously borrowed here from the lan- 
 .guage of infinitesimals, but is supposed to be interpreted, like those used in other 
 parts of the present series of Articles, by a reference to the conception of limits. 
 
582 ELEMENTS OF QIJATKF? NIONS. [bOOK III. 
 
 makes generally, like the tangent ^ or pj itself, an acute angle with the positive 
 binormal v, as appears from the common sign of the scalar coefficients of that vec- 
 tor, in their developed expressions. 
 
 (11.) It may also be remarked, as an additional point of analogy, and as serv- 
 ing to verify some formulae, that Avhile the older angle H becomes right, •when the 
 
 given curve is plane, so the ?ieio angle /= — , for every spherical cvrve. 
 
 (12.) As another ffeometrical illustratinn of the properties of the angle /, and of 
 some other results of recent sub-articles, which may serve to connect them, still 
 more closely, with the general theory o( normal emanants from curves (397, (44.)), 
 let us conceive that ab, bc, cd are three successive right lines, perpendicular each 
 to each ; let us denote by a and b the angles bca and cbd, and by c the inclination 
 of the line ad to bc : and let us suppose that these two lines are intersected by their 
 common perpendicular in the points G and h respectively. 
 
 (13.) Then, by completing the rectangle bcde, and letting fall the perpendicular 
 bf on the hypotenuse of the right-angled triangle abe, we obtain the projections, 
 AE and fb, of the two lines ad and gh, on the plane through b perpendicular to bc ; 
 and hence, by elementary reasonings, we can infer the relations : 
 XXVIII. . . tan2c = tan«ADE = tan2a + tan2 6; 
 
 „^,„ BH AG AF Ab2 . ^ 
 
 and XXIX. . . — = — = — = — - = sm^ aeb, 
 
 BC AD AE AE^ 
 
 or XXIX'. . . BH = BC sin^y, if tan j = tan acoth; 
 
 nothing here being supposed to be small. It may also be observed, that the tivo 
 rectilinear angles, BCA and CBD, or a and 6, represent respectively the inclinations 
 of the plane acd to the plane BCD, and of the plane abd to the plane abc. 
 
 (14.) Conceive next that pq and p^Qs are two near normal emanants, touching 
 the polar developable in the points q and q,, whereof q is thus on the given polar 
 axis KS, and Qs is on the near polar axis KsQs ; and let the second emanant be 
 cut, in the points p' and q', by planes through p and q, perpendicular to the first 
 emanant pq. The line pp' will then be very nearly tangential to the given curve at 
 p ; and the line qq' will be very nearly situated in the corresponding normal plane 
 to that curve : so that these two new lines will be very nearly perpendicular to each 
 other, and the gauche quadrilateral p'pqq' will ultimately have the properties of the 
 recently considered quadrilateral abcd. 
 
 (15.) This being perceived, if we denote by e the length of the emanant line pq, 
 the small angle a is very nearly = e-'s ; and if the small angle b be put under the 
 form b's, then the new coefficient b' is ultimately equal (by XXIX'.) to e"* cotj : 
 where _/ is an auxiliary angle, not generally small, and is such that we have ulti- 
 mately PH = PQ.sin2y, if H be the point in which the given normal emanant pq 
 approaches most closely to the consecutive emanant PsQ*. 
 
 (16.) We have then the ultimate equation, 
 XXX. . . coty = c6' =pQ X lim. («-i . qpQs) 
 
 = length of emanant line (fq) 
 
 X angular velocity of the tangential plane (p'pq) containing it ; 
 this latter plane being here conceived as turning, for a moment, round the tangent to 
 the given curve at r, and the velocity of motion along that curve being still taken 
 for unity. 
 
CHAP. HI.] GENERAL THEORY OF SUCH AUXILIARY ANGLES. 583 
 
 (17.) Accordingly, when we change e to r, 6' to r ^ audj to H, we recover in 
 this way the fundamental value cot H= rr"i (397, XVII.), for the cotangent of the 
 older angle H-^ and when, on the other hand, we treat the radius of spherical curva- 
 ture as the normal emanant, supposing q to coincide with s, and therefore changing 
 e to R, and b' to r-'+ P', we recover the last of the expressions XXIII. for the co- 
 tangent of the new but analogous angle J, namely cot /= 7? (r"i + P'), together 
 with an interpretation, which may not have at first seemed obvious : although that 
 expression itself was deducible, in the following among other ways, from equations 
 previously establibhed, 
 
 XXXL . . B'^cotJ-v'^ 
 
 _ rR r _ Rl r\ __ (cos P)' .p, 
 pR p p\RJ sin P 
 
 (18.) As regards the angular velocity^ say w, of the emanant line pq, or the ul- 
 timate quotient of the angle between two such near lines, divided by the small arc s 
 of the given curve, we see by XXVIII. (comp. (5.)) that this small angle vs is ulli- 
 mately equal to the square root of the sum of the squares of the two other small an- 
 gles, above denoted by a and b, and found to be equal, nearly, to e'^s and e'^s cotj 
 respectively : Ave may then establish the general formula, 
 
 XXXII. . . Angular Velocity of Normal Emanant = v = e-^cosecj ; 
 which reproduces the values, r"icosecH, and R-^cosecJ, already found for the an- 
 gular velocities of the two radii, r and R. 
 
 (19.) And if we observe that the projection of the vector of curcature, kp', on 
 the emanant pq, is easily proved to be =qp~i = e^^.pQ, we see by XXXII. tluit if 
 this projection be divided by the square of the angular velocity (?)) of the lino 
 PQ, i\\e quotient h the line PQ.sin^y, or ph (15.): which reproduces the general 
 result, 397, CV., iovaXl systems of normal emanants, together with a geometrical 
 interpretation. 
 
 (20.) As still another geometrical illustration of the properties of the new angle 
 Jy we may observe that in the construction (12.) and (13.) the corresponding auxi- 
 liary angley was equal to aeb, or to abf, and that the line bf (= hg) was perpen- 
 dicular to both BC and ad, although not intersecting the latter. Substituting then, 
 as in (14.), the quadrilateral p'pqq' for abcd, and passing to the limit, we may say 
 that if a new line pj be a common perpendicular, at the given point p, to two conse- 
 cutive* normal emanants, PQ and p'q', the general auxiliary angle j is simply the 
 inclination p'pj, of that common perpendicular pj, to the tangent pp' to the curve. 
 
 (21.) And if, instead o{ normally emanating lines pq, we consider a system of 
 tangential emanant planes (as in 397, (45.)), to which those lines are pej-pendicular, 
 we may then (comp. 396, (14.)) consider the recent line pj as being a generating 
 line of the developable sujface, which is the envelope of all the planes of the system ; 
 the auxiliary angle, f j, is therefore generally by (20.) the inclination of this gene- 
 
 * Compare the Note to page 581. 
 
 t In these geometrical illustrations, tlie angle _/ has been treated, for simplicil}', 
 as being both positive and acute ; although the general formulce. wiiich involve the 
 corresponding angles //and J, permit and require (hat we should occasionally attri- 
 bute to them obtuse (but still positive) values : wiiile those angles may also become 
 right, in some particular cases (comp. (!!))• 
 
584 ELEMENTS OF QUATEIINIOKS. [bOOK III. 
 
 ratrix to the tangent : a result which agrees Avith, and includes, the known and fuu- 
 
 dameutal property (397, XVII.) of the angle H, in connexion with the Rectifying 
 
 Developable (396) ; and also the analogous property of the newer angle J, connected 
 
 (8.) with what it has been above proposed to call the Circumscribed Developable. 
 
 (22.) We shall soon return briefly on the theory of that new developable surface 
 
 (8.), and of the new locus (of the osculating circle, or envelope of the osculating 
 
 sphere) to which it has been said to be circumscribed: but may here observe, 
 
 that if we write for abridgment (comp. VIII. and XXIII.), 
 
 » Jiii' 
 
 XXXIII. ..« = —= = »' + cot -ff= cot /sec P, 
 
 rv p 
 
 then what has been called the coefficient of non-sphericity (comp. 895, (14.) and 
 (16.)) is easily seen to have by XIV. the values, 
 
 XXXIV. . . S- 1 = l^'"-^^ - 1 = - 74rSj/V" - 1 (I, 2) 
 
 = ^(/>'-rrX*)- 1 = ^( p' + ^ \ = »rr-i (3, 4, 5) 
 
 = — = cot-H'cot/secP=^^ (6,7,8); 
 
 TV px 
 
 whence also the deviation of a near point p^ of the curve, from the osculating sphere 
 
 at P, is ultimately (by 395, XXVII,). 
 
 _,_,Y^ _ _ {S-\) s* ns^ Ks^ 
 
 XXXV. . . sp.- sp= -2^^- = gri;:^^ = 21;:^^ ; 
 
 and accordingly, the square of the vector pg - o- is given now (comp. I.) by the ex- 
 pression, 
 
 (p,-cr)2 = (p-a)2-^{r2S((T-p)r"'-l}, 
 
 in which r2S (o- - p) t" = /S = 1 + nrr'i = &c., as above. 
 
 (23.) The same auxiliary scalar n enters into the following expressions for the arc, 
 
 and for the scalar radii of t\iQ first and second curvatures, of the locus of the centre 
 
 s of the osculating sphere, or of the cusp-edge of the polar developable (comp. 391, 
 
 (6.), and 395, (2.)) : 
 
 XXXVI. . . +J «ds = Arc of that Cusp-Edge (or of locus ofs) ; 
 
 7? J?' 
 XXXVI'. . . ri = nr = r +p'r = -— - = (^Scalar) Radius of Ourvature of same edge ; 
 r 
 
 XXXVl". . . ri = rar = a' v~^ = (^Scalar) Radius of Second Curvature of same curve ; 
 
 these two latter being here called scalar radii, because th.Q first as well as the second 
 
 (comp. 397, V.) is conceived to have an algebraic sign. In fact, if we denote by Ki 
 
 the centre of the osculating circle to the cusp-edge in question, its vector is (by the 
 
 general formula 389, IV.), 
 
 XXXVII. . . OKi = Ki= <T + - „ ; = a -nrVT = p - p rrr' -\- prv = (t - rirr', 
 
 with the signification XXXVI'. of n; because by XXXIII. (comp. 397, XI'.), 
 
 XXXVIII. . . a' = nrv, a" — n'rv -^^ n{rv)' -n'rv -nrf^r', 
 
 and therefore 
 
 XXXIX. . . a 2 =. _ «2, VcrV" = n^x-W. 
 
CHAP. III.] CURVATURES OF CUSP-EDGE OF POLAR SURFACE. 585 
 
 We may also observe that the relation o-' || v gives (by 397, IV.), 
 
 (t" v' 
 
 XL. . . V— 7 = V— = r"V= Vector of Second Curvature of given curve ; 
 a V 
 
 and that we have the equation, 
 
 XLI. . . — = ^^^— ^ = — , with r > 0, but ri > or < 0, 
 PK K— p r 
 
 according as the cusp-edge turns its concavity or its convexity towards the given 
 curve at P. 
 
 (24.) The radius of (first) curvature of that cusp-edge, when regarded as a po- 
 sitive quantity, is therefore represented by the tensor, 
 
 XLII. . . -v/n2 = ± ri = Tri = i?T ^ = ± -^ (> 0) ; 
 
 and as regards the scalar radius XXXVI". of second curvature of the same cusp- 
 edge, its expression follows by XXXVIII. from the general formula 397, XXVII., 
 which gives here, 
 
 XLIII. . . rr' = S s^,. = — S -^, = «-i/-i, because XLI 11'. . . S -^, = 1 ; 
 Ya(T nr Ypv Yvv 
 
 the two scalar derivatives, n and n", which would have introduced the derived vec- 
 tors r'^ and t", or Dg^p and Ds^p, of the fifth and sixth orders, thus disappearing 
 from the expressions of the two curvatures of the locus of the centre s of the osculat' 
 ing sphere, as was to be expected from geometrical* considerations. 
 
 (25.) For the helix, the formula XXXVI 1. gives fci = p, or Ki = p ; we have then 
 thus, as a verification, the known result, that the given point p of /A/« curve is itself 
 the centre of curvature Ki of that other helix (comp. 389, (3.), and 395, (8.)), which 
 is in this case the common locus of the two coincident centres, K and s. It is scarcely 
 necessary to observe that for the helix we have also J= H. 
 
 (26.) In general, the rectifying plane of the locus ofs is parallel to the rectify- 
 ing plane of the given curve, because the radii of their osculating circles are parallel ; 
 the rectifying lines for these two curves are therefore not only parallel but equal ; 
 and accordingly we have here the formula, 
 
 XLIV. . . Xi = V— ,' = V-, = X (by 397, XVL), 
 
 7-1 T 
 
 which will be found to agree with this other expression (comp. 397, XVII.), 
 
 XLV. . . tan Hi = ;^ = - Uri = + cot H, 
 iri r 
 
 the upper or lower sign being taken, according as the new curve is concave (as in 
 Figs. 81, 82), or is convex at s (comp. (23.)), towards the old (or given) curve at 
 p : and the new angle Hi being measured in the new rectifying plane, from the new 
 
 * In fact, n represents here the velocity of motion of the point s along its own 
 locus, while r"' and r-i represent respectively the velocities of rotation of the tangent 
 and binormal to that curve : so that nr and nr must be, as above, the radii of its 
 two curvatures. 
 
 4 F 
 
586 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 tangent a or nrv, to the new rectifying line Xi, and in the direction from that new 
 tangent to the new binormal vi, or (comp. XL.) to a line from s which is equal to 
 the vector of second curvature x-^t of the given curve^ multiplied by a positive scalar^ 
 namely by Tn-i, or by the coefficient n-i taken positively. 
 
 (27.) The former rectifying line \ touches the cusp-edge of the rectifying deve- 
 lopable (396) of the given curve, in a new point r (comp. Fig. 81), of which by 
 397, (45.), and by XV., the vector from the given point is, generally, 
 
 ^,,,, VrV r-2\ r\ UXsin^ 
 
 XLVI...PH=-g-;-;;-;,, = ^^ = -— ,^^,= -^^; 
 
 with the verification that this expression becomes infinite (comp. 397, (4.9.), (50.)), 
 when the curve is a geodetic on a cylinder. 
 
 (28.) In general, the vector or of thepoiw^ of contact r, which vector we shall 
 here denote by v, may be thus expressed, 
 
 XLVII. .. v = OR = p + m\, if XLVIII. . . Z = ^^ = 7^ ; 
 
 M' (rr-i) 
 
 and because (rX)' = (rr-')'r, by VII'., its first derivative is, 
 
 XLIX. . . u' = rX I ^-^ J = UX cosec H (I sin H)' = UX (/' + cos H) ; 
 
 in which however the new derived scalar Z' involves //", and so depends on t'*' : 
 while the scalar coefficient I itself represents the /Jor^ion (+fR)oithe rectifying line, 
 intercepted between the given curve, and the cusp-edge (27.) of the rectifying deve- 
 lopable, and considered as positive when the direction of this intercept pr coincides 
 with that of the line + X, but as negative in the contrary case. 
 
 (29.) For abridgment of discourse, the cusp-edge last considered, namely that of 
 the rectifying developable, as being the locus of a point which we have denoted by 
 the letter R, may be called simply " <Ae curve (r) ;" while the /ormer cusp-edge 
 (23.), or that of the /joZar developable, maybe called in like manner "</je curve 
 (s) ;" the ZocMs of the centre k of (absolute) curvature maybe called "fAe curve 
 (k) :" and the given curve itself (comp. again Figs. 81, 82) may be called, on the 
 same plan, " the curve (p)." 
 
 (30.) The arc RR«, of the curve (r), is (by XLIX., comp. XXXVI.), 
 
 TTu'ds=Z, -Z+ rcos.Hds; 
 
 this arc being treated as positive, when the direction of motion along it coincides with 
 that of + \. 
 
 (31.) The expression VII. for X', combined with the former expression 397, 
 XVI. for X, gives easily by the general formula 389, IV., 
 LI. . . Vector of Centre of Curvature of the Curve (r) 
 
 v v' v' ^^ , 
 
 ^ Yv"v'-i ^ VX'X-' • H' 
 
 whence LII. . . Radius of Curvature of Curve (r) = T— , = T -— , 
 
 the scalar variable being here arbitrary. 
 
 (82.) We see, at the same time, that the angular velocity of the rectifying line 
 X, or of the tangent to this curve (r), is represented by ±H' \ or that the small 
 
CHAP. III.] CURVATURES OF EDGE OF RECTIFYING SURFACE. 587 
 
 angle* between two such near lines, \ and Xj, is nearly equal to sH', or to Hg— H: 
 while the vector axis (VX'X'i) of rotation of the rectifying line, set off from the 
 point R, has —H'Vt', or — H'rr', for its expression. 
 
 (33.) As regards the second cvrvature of the same curve (r), we may observe 
 that the expression (comp. VII. and LI.), 
 
 LIII. . . X"= (r-i)"r + (r-i)"ry + r'l (rr-i)'r' = (r-i)"r + (r-i)"rv + VXX', 
 combined with the parallelism (XLIX.) of v' to X, gives, by the general formula 
 397, XXVII., 
 
 LIV. . . Radius of Second Curvature of Curve (r) 
 
 - f q '"'" V - ^7 Q _^ V^ -i ~ I' + cosH 
 ~[ Yv'v") -\\^Y\X'I ~\ tT"' 
 
 with the verification, that while V + cos B represents, by (30.), the velocity of mo- 
 tion along this curve (r), TX represents, by 397, (3.), the velocity of rotation of 
 its osculating plane, namely the rectifying plane of the given curve (p) : and it 
 is worth observing, that although each of these two radii of curvature, LII. and 
 LIV., depends on t^ through V (28.), yet neither of them depends on t^ (comp. 
 (24.)). As another verification, it can be shown that the plane of the two lines X 
 and t' from p, namely the plane, 
 
 LIV. . . Sr'X(w-p) = 0, 
 which is the normal plane to the rectifying developable along the rectifying line, and 
 contains the absolute normal to the given curve (p), touches its own developable en- 
 velope along the line rh, if H be the point determined by the formula 397, 
 XXXVIIL, or the point of nearest approach of a radius of curvature (r) of that 
 given curve to its consecutive (comp. (6.) ; this line~RH must therefore be the recti- 
 fying line of the curve (r) : and accordingly (comp. 397, XVII.), the trigonometric 
 tangent of its inclination to the tangent rp to this last curv^e has for expression 
 (abstracting from sign), 
 
 LIV". . . tan PRH = PH : PR = + ^'r sin2 H= ± rH' sin H = TX'^H' 
 
 Radius (LIV.) of Second Curvature of Curve (r)^ 
 ~ Radius (LII.) of First Curvature of same Curve 
 (34.) Without even introducing r"^, we can assign as follows a twisted cubic 
 (comp. 397, (34.)), which shall have contact of the fourth orc?er with the given 
 curve at P ; or rather an indefinite variety of such cubics, or gauche curves of the 
 third degree. Writing, for abridgment, 
 
 LV. . . a;=-Sr(w-p), y = - Srr'(w - p), z = - S>rv(u} - p), 
 so that LVI. . . b} = p-\- XT + yrr + zrv, 
 
 the scalar equation, 
 
 Lvn...(?^f=s(l)%..(^:)V..«'. 
 
 * A result substantially equivalent to this is deduced, by an entirely different 
 analysis, in the above cited Memoir of M. de Saint- Venant, and is illustrated by 
 geometrical considerations : which also lead to expressions for the two curvatures 
 (or, as he calls them, the courbure and cambrure), of the cusp-edge of the rectifying 
 developable ; and to a determination of the rectifying line of that cusp-edge. 
 
588 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 in which e is an arbitrary but scalar constant, represents evidently, by its form, a 
 cone of the second order^ with its vertex at the given point P ; and this cone can be 
 proved to have contact of the fourth order with the curve* at that point : or of the 
 third order with the cone of chords from it (comp. 397, (31.), (32.))- In fact the 
 coefficients will be found to have been so determined, that the difference of the two 
 members of this equation LVII. contains s^ as a factor, when we change w to ps, as 
 given by the formula I., or when we substitute for xyz their approximate values for 
 the curve, as functions of the arc a ; namely, by the expressions IV. for r'", and 
 397, VI. for r", 
 
 s^ r's^ 
 
 LVIII. 
 
 
 L 6rr 24 ^ ^ 
 
 where the terms set down are more than sufficient for the purpose of the proof. It 
 
 — «< 
 may be added that the coefficient of — — in y,, which is the only one at all complex 
 
 here, may be transformed as follows : 
 
 LVIII'. . . SrrV"' = -(r-i)"-r-iX2 = r-35' + /)(r-2r-i)'; 
 S being that scalar for which (or more immediately for its excess over unity) several 
 expressionsf have lately been assigned (22.), and which had occurred in an earlier 
 investigation (395, (14.), &c.). 
 
 (35.) With the same significations LV. of the three scalars xyz, this other equa- 
 tion, 
 
 LIX. . . 18ry - (3a; - r V)2 = (9 + r'2 - 3rr" - 3r2r-2) y^, 
 
 or LIX'. ..2ry-(x- l/y)2 = (1 - |r^ (r^)" - lr2r-2) y^, 
 
 will be found to be satisfied when we substitute for x and y the values LVIII. of x, 
 and ysi and neglect or suppress s^ ; it therefore represents an elliptic (or hyperbolic') 
 cylinder, which is cut perpendicularly, by the osculating plane to the given curve at 
 p, in an ellipse (or hyperbola), having contact of the fourth order with the pro/ec- 
 tion (comp. 397, (9.)), of that given curve upon that osculating plane : and the cy- 
 linder itself has contact of the same {fourth} order with the curve in space, at the 
 
 * In the language of infinitesimals, the cone LVII. contains five consecutive 
 points of the curve, or has five-point contact therewith : but it contains only /our con- 
 secutive sides of the cone of chords from the given point, or has only four -side con- 
 tact with that cone, except for one particular value of the constant, e, which we shall 
 presently assign. It may be observed that xyz form here a (scalar) system oi three 
 rectangular co-ordinates, of the usual kind, with their origin at the point p of the 
 curve, and with their positive semiaxes in the directions of the tangent t, the vector 
 of curvature r', and the hinormal v. 
 
 f It might have been observed, in addition to the eight forms XXXIV., that 
 we have also, 
 
 XXXI v. . . 5 - 1 = Rr' cot /= n cot H (9, 10). 
 
CHAP. III.] OSCULATING ELLIPTIC CYLINDER. 589 
 
 same given point p, so that we may call it (comp. 397, (31.)) the Osculating Elliptic 
 {or Hyperholic) Cylinder, perpendicular to the osculating plane. • 
 
 (36.) As a verification, if we suppress the second member of either LIX. or LIX',, 
 we obtain, imder a new form, the equation of what has been already called the Oscu- 
 lating Parabolic Cylinder (397, LXXXIV.) ; and as another verification, the co- 
 efficient of y^ in that second member vanishes, as it ought to do, when the given 
 curve is supposed to be a parabola : that plane curve, in fact, satisfying the differen- 
 tial equation of the second order, 
 
 LX. , . Brr" - r'2 = 9, or LX'. . . r^ (r^)" = 2, 
 
 or LX". • • '•■^ ( ( ^ )^+ 1 I = const. =/>-§, 
 
 if r be still the radius of curvature, considered as a function of the arc, s, while p is 
 here the semiparameter. 
 
 (37.) The hinormal v is, by the construction, a generating line of the cylinder 
 LIX. ; and although this line is not generally a side of the cone LVIL, yet we can 
 make it such, by assigning the particular value zero to the arbitrary constant, e, in 
 its equation, or by suppressing the term, ez^. And when this is done, the cone LVII. 
 will intersect the cylinder LIX., not only in this common side v (comp. 397, (33.)), 
 but also in a certain twisted cubic, which will have contact of the fourth order with 
 the given curve at p, as stated at the commencement of (34.). 
 
 (38.) But, as was also stated there, indefinitely many such cubics can be de- 
 scribed, which shall have contact of the same (fourth) order, with the same curve, 
 at the same point. For we may assume any point E of space, or any vector (comp. 
 
 LYI.), 
 
 LXI. . . OE = e = jO + ar + brr' + crv, 
 
 in which a, 6, c are any three scalar constants ; and then the vector equation, 
 
 LXIL . . a> = ps+<(6-p), 
 
 in which < is a new scalar variable, will represent a cylindric surface, not generally 
 of the second order, but passing through the given curve, and having the line pe for 
 a generatrix. We can then cut (generally) this new cylinder by the osctdating 
 plane to the curve at p, and so obtain (generally) a new and oblique projection of 
 the czirve upon that plane ; the x and y of which neto projected curve will depend on 
 the arc s of the original curve by the relations, 
 
 LXIII. . . x = Xs-ac-^Zs, y = ys-bc-^Zs', 
 
 with the approximate expressions LVIII. for XsysZs. And if we then determine two 
 new scalar constants, B and C, by the condition that the substitution of these last 
 expressions LXIII. for x and y shall satisfy this new equation, 
 
 LXIV. . . 2ry = a;2 + 2Bxy + Cy\ 
 if only 8^ be neglected (comp. (35.)), or by equating the coefficients of s^ and s*, 
 in the result of such substitution, then, on restoring the significations LV. of xyz, 
 and writing for abridgment, 
 
 LXV. . . X=a;-ac Jz, Y^y-bc'^z, 
 the equation of the second degree, 
 
590 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 LXVI. . . 2rF= jr2 + 2i?ZF+ CF2, 
 
 will represent generally an oblique osculating elliptic (or hyperbolic) cylinder, which 
 has contact of the /o«r<A order with the given curve at p, and contains the assumed 
 line PE. If then we determine finally the constant e in LVII., by the result of the 
 substitution of ahc for xyz, or by the condition, 
 
 LXVII. . . ( ?I^ Y = 6 f - V ac + f - Yftc + ec\ 
 
 ^y-(^r-H3'' 
 
 the cone LVIL, and the cylinder LXVI., will have that line pe for a common side ; and 
 ■will intersect each other, not only in that line, but also (as before) in a twisted cubic, 
 although now a new one, which will have the required {fourth) order of contact, with 
 the given curve at the given point. 
 
 (.39.) If, after the substitution (38.) in LXIV., we equate the coefficients of the 
 three powers, s^, «*, s^, and then eliminate £ and C, we are conducted to an equa- 
 tion of condition, which is found to be of thejform, 
 
 LXVIII. . . a&3 + b6'C + c6c2 -f ec^ = ac(bg + ch) ; 
 
 in which the ratios of abc still serve to determine the direction of the generating line 
 PE, while the coefficients a, b, c, e, g, h are assignable functions of r, r, r', r', r", r", 
 and r'", depending on the vector r'^ : and when this condition LXVIII. is satisfied, 
 the cylinder LXVI. has contact of the fifth order with the given curve at p. 
 
 (40.) Again, if we improve the approximate expressions LVIII. for the three 
 
 scalars Xg, ys, ««, by taking account of s^, or by introducing the new term — — — 
 
 (comp. I.) of ps, and if we substitute the expressions so improved, instead of x, y, z, 
 in the equation of the cone LVII. and then equate to zero (comp. (34.)) the coeffi- 
 cient of s^ in the difference of the two members of that equation, we obtain a definite 
 expression for the constant, e, which had been arbitrary before, but becomes now a 
 given function of rrr'r'r" and r" (not involving r"), namely the following 
 
 ,^,^ r4f9 21 r'2 Br" 3r'r' 27r'2 9r"\ 
 
 LXIX. . . e = — + + — + — ; 
 
 5 \^r* r^i-'i H r3 rh 4.r^r'i r^-r j 
 
 and when the constant e receives this value,* the cone has contact of the fifth order 
 with the curve at the given point. 
 
 (41.) Finally, if we multiply the equation LXVII. by bg+ ch, we can at once 
 eliminate a by LXVIII., and so obtain a cubic equation in b: c, which has at least 
 one real root, answering to a real system of ratios a, b, c, and therefore to a real 
 direction of the line pe in (38.). It is therefore possible to assign at least one real 
 cylinder of the second order (39.), which shall have contact of the fifth order with 
 the curve at p, and shall at the same time have one side pe common with the cone 
 of the second order (40.), which has contact of the same (fifth) order with the curve 
 (or oi the fourth order with the cone of chords) : and consequently it is possible in 
 this way to assign, as the intersection of this cylinder with this cone, at least one real 
 
 Compare the first Note to page 588. 
 
CHAP. III.] OSCULATING TWISTED CUBIC. 591 
 
 twisted cubic, which has contact of the fifth* order with the given curve of double 
 curvature, at the given point thereof. And such a cubic curve may be called, by- 
 eminence, an Osculatingf Twisted Cubic. 
 
 (42.) Not intending to return, in these Elements, on the subject of such cubic 
 curves, we may take this occasion to remark, that the very simple vector equation,^ 
 
 LXX. . . Yap = pYPp, 
 represents a curve of this kind, if a and [3 be any two constant and non-parallel 
 vectors. In fact, if we operate on this equation by the symbol S.X, in which \ is 
 an arbitrary but constant vector, the scalar equation so obtained, namely, 
 
 LXXI. . . SXap = S\pS/3|0 - p«S/3\, 
 leipresents a. surface of the second order, on which the curve is wholly contained; 
 making then successively X = a and X = /3, we get, in particular, the two equations, 
 
 LXXII. . . S {Yap .Yl3p) = 0, and LXXIII. . . (V/3p)2 + Sa/3p = 0, 
 representing respectively a cone and cylinder of that order, with the vector [3 from 
 the origin as a common side : and the remaining part of the intersection of these 
 two surfaces, is precisely the curve LXX., which therefore is a twisted cubic, in the 
 known sense already referred to. 
 
 (43.) Other surfaces of the same order, containing the same curve, would be 
 obtained by assigning other values to X ; for example (comp. 397, (31.)), we should 
 get genersdly an hyperbolic paraboloid from the form LXXI., by taking X-i-(3. 
 But it may be more important here to observe, that without supposing any acquaint- 
 ance with the theory of curved surfaces, the vector equation LXX. can be shown, by 
 
 * Accordingly, it is known (see page 242 of Dr. Salmon's Treatise, already 
 cited), that a twisted cubic can generally be described through any six given points ; 
 and also (page 248), that three quadric cylinders (or cylinders of the second order 
 or degree) can be described, containing a given cubic curve, their edges being pa- 
 rallel to the three (real or imaginary) asymptotes. 
 
 t Compare the first Note to page 563. 
 
 X Tliis example was given in pages 679, &c., of the Lectures, with some con- 
 nected transformations, the equation having been found as a certain condition for 
 the inscription of a gauche quadrilateral, or other even-sided polygon, in a. given 
 spheric surface (comp. the sub-articles to 296) : the 2» successive sides of the figure 
 being obliged to pass through the same even number of given points of space. It 
 was shown that the curve might be said to intersect the unit-sphere (p2 = _ i) jq figQ 
 imaginary points at infinity, and also in two real and two imaginary points, situated 
 on two real right lines, which were reciprocal polars relatively to the sphere, and 
 might be called chords of solution, Avitli respect to the proposed problem of inscrip- 
 tion of the polygon ; and that analogous results existed for even-sided polygons in 
 ellipsoids, and other surfaces of the second order : wha:eas the corresponding prob- 
 lem, of the inscription of an odd-sided polygon in such a surface, conducted only to 
 the assignment of a single chord of solution, as happens in the known and analogous 
 theory of polygons in conies, whether the number of sides be (in that theory) even 
 or odd. But we cannot here pursue the subject, which has been treated at some 
 length in the Lectures, and in the Appendices to them. 
 
592 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 quaternions^ to represent a curve of the third degree^ in the sense that it is cut, by 
 an arbitrary plane, in three points (real or imaginary). In fact, we may write the 
 equation as follows, 
 
 LXXIV. . . V9P = -a, if LXXV. . .^=i/ + i3, 
 q being here a quaternion, of which the vector part /3 is given, but the scalar part g 
 is arbitrary ; and then, by resolving (comp. 347) this linear equation LXXIV., 
 we may still further transform it as follows, 
 
 LXXVI. . . g(g^-iS^)p = (iSi5a +gY^a -g^a, 
 which conducts to a cubic equation in g, when combined with the equation, 
 
 LXXVII. . . Sf|0 = e, 
 of any proposed secant plane. 
 
 (44.) The vector equation LXX., however, is not sufficiently general, to repre- 
 sent an arbitrary twisted cubic, through, an assumed point taken as origin ; for 
 which purpose, ten scalar constants ought to be disposable, in order to allow of the 
 curve being made to pass through jf?re* other arbitrary points : whereas the equa- 
 tion referred to involves only five such constants, namely the /our included in Ua 
 and U/3, and the one quotient of tensors T/3 : Ta (comp. 358). 
 
 (45.) It is easy, however, to accomplish the generalization thus required, with 
 the help of that theory of linear and vector functions {(pp) of vectors, which was as- 
 signed in the Sixth Section of the preceding Chapter (Arts. 347, &c.). We have 
 only to write, instead of the equation LXX., this other but analogous form which 
 includes it, 
 
 LXXVIII. . . Yap + Yp^p = 0, or LXXVIII'. . . (pp^cp = a, 
 and which gives, by principles and methods already explained (comp. 354, (1.)), 
 the transformation, 
 
 LXXIX. . . p = (0 + c) Ja= , , 
 
 ^ ^ 7n + ?w c + m c^ -\- c^ 
 
 a, xl^a, and x« being here fixed vectors, and w, m', m" being fixed scalars, but c 
 being an arbitrary and variable scalar, whicli may receive any value, without the 
 expression LXXIX. ceasing to satisfy the equation LXXVIII. 
 
 * Compare the first Note to page 591. In general, when a curve in space is 
 supposed to be represented (comp. 371, (5.)) by two scalar equations, each new ar- 
 bitrary point, through which it is required to pass, introduces a necessity for /wo new 
 disposable constants, of the scalar kind : and accordingly each new order, say the 
 n*^, of contact with such a curve, has been seen to introduce a new vector, Ds"p, or 
 r^"''), subject to a condition resulting from the general equation TDsp = l, or 
 r2 = - 1 (comp. 380, XXVI., and 396, III.), but involving virtually two new scalar 
 constants. Thus, besides Ihe four such constants, which enter through t and r' into 
 the determination of the directions of the rectangular system of lines, tangent, nor- 
 mal, and binormal (comp. 379, (5.), or 396, (2.)), and of the length of the radius 
 of (first) curvature, r, the three successive derivatives, r', r", r", of that radius, and 
 the radius r of second curvature, with its two first derivatives, r' and r", have been 
 seen to enter, through the three other vectors, r", t'", t^^, into the determination 
 (41.) of the osculating twisted cubic. 
 
CHAP. III.] VECTOR EQUATION OF A TWISTED CUBIC. 593 
 
 (46.) The curve LXXVIII. is therefore cut (comp. (43.)) by the plane 
 LXXVII. in three points (real or imaginary), answering to and determined by the 
 three roots of the cubic in c, which is formed by substituting the expression LXXIX. 
 for p in the equation of that secant plane; and consequently it is a curve of the third 
 degree, the three (real or imaginary) asymptotes to which have directions correspond- 
 ing to the three values of c, obtained by equating to zero the denominator of that 
 expression LXXIX., or by making M=0, in a notation formerly employed : so that 
 they have the directions of the three lines /3, which satisfy this other vector equa- 
 tion (comp. 354, I.), 
 
 LXXX. . . V/30/3 = 0. 
 
 (47.) Accordingly, if j3 be such a line, and if y be any vector in the plane of a 
 and /3, the curve LXXVIII. is a part of the intersection of the two surfaces of the 
 second order, 
 
 LXXXI. . . Sap^jO = 0, and LXXXII. . . Syap + Syp^p = 0, 
 whereof the first is a cone, and which have the line ft from the origin for a common 
 side (comp. (42.)) : the curve is therefore found anew to be a twisted cubic. 
 
 (48.) And as regards the number of the scalar constants, which are to be con- 
 ceived as entering into its vector equation LXXVIII., when we take for (pp the form 
 VgoP + VXp/i assigned in 357, I., in which 50 is an arbitrary but constant quater- 
 nion, such as ^ + y, and X, ^ are constant vectors, the term gp of 0p disappears 
 under the symbol of operation V.p, and the equation (45.) of the curve becomes, 
 
 LXXXIII. . . Yap + pVyp + YpWpii = ; 
 in which the four versors, TJa, Uy, U\, U/i, introduce each two scalar constants, 
 while the two tensor quotients, Ty : Ta and TXfi : Ta, count as two others : so that 
 the required number of ten such constants (44.) is exactly made up, the curve being 
 still supposed to pass through an assumed origin, and therefore to have one point 
 given. It is scarcely worth observing, that we can at once remove this last restric- 
 tion, by merely adding a new constant vector' to p, in the last equation, LXXXIII. 
 (49.) Although, for the determination of the osculating twisted cubic (41.), to 
 a given curve of double curvature, it was necessary (comp. (40.)) to employ the 
 vector t'"^ or Dj^p, or to take account of s^ in the vector ps, or in the connected sca- 
 lars Xgi/sZs of (34.), and therefore to improve the expressions LVIII., by carrying in 
 each of them (or at least in the two latter) the approximation one step farther, yet 
 there are many other problems relating to curves in space, besides some tliat have been 
 already considered, for which those scalar expressions LVIII. are sufficiently ap- 
 proximate : or for which the vector expression I. suffices. 
 
 (50.) Resuming, for instance, the questions considered in (22.) and (23.), we 
 may throw some additional light on the law of the deviation of a near point Pg of the 
 curve, from the osculating sphere at p, as follows. Eliminating n by XXXVI'. 
 from XXXV., we find this new expression, 
 
 LXXXIV. . .ip,-sp= "^'^ 
 
 24rr2i2 ' 
 
 the direction of this deviation from the sphere (E) depends therefore on the sign 
 of the scalar radius n (23.) o{ curvature of the cusp-edge (s) of the polar deve- 
 lopable: Siwdit is outward or inward (comp. B9 5, (14.)), according as that cusp- 
 edge turns its concavity (comp. XLI.) or its convexity, at the centre s of the oscu- 
 
 4 G 
 
594 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 lating sphere, towards the point p of the given curve, that is, towards the point of 
 osculation. 
 
 (51.) Again, if we only take account of s^^ the deviation of v^from the osculat- 
 ing circle at p has been seen to be a vector tangential to the osculating sphere, which 
 may be thus expressed (comp. 397, IX., LII.), 
 
 LXXXV. . . C.P. = ^ vV = "'^^^-Pj 
 6 6r*r 
 
 if Cs be the point on the circle, which is distant from the given point p by an arc of 
 that circle =*, with the same initial direction of motion, or of departure from p, re- 
 presented by the common unit tangent r ; the quantity of this deviation is therefore 
 
 expressed by the scalar -—-: that is, by the deviation — (comp. 397, (9.), (10.)) 
 6r^r 6rr 
 
 from the osculating plane* at P, multiplied by the secant (r'i?) of the inclination 
 (P) of the radius (Z2) oi spherical curvature, to the radius (r) of absolute curva- 
 ture, and positive when this last deviation has the direction of the hinormal v. 
 
 (52.) On the other hand (comp. (5.)) the small angle, which the small arc sss 
 of the cusp-edge (s) of the polar developable subtends at the point p, is ultimately 
 expressed by the scalar, 
 
 LXXXVI. . . sps, = (ps, - ps) . R-^ cot P = ^* = ^ (by XXXIII.), 
 
 pR R^ 
 
 this on^'Ze being treated as positive, when the corresponding ro<a^io» f round + r from 
 
 * Besides the nine expressions in 397, (42.) for the s^ware r"2 of the second 
 curvature, the following may be remarked, as containing the law of the regression of 
 the projection of a curve of double curvature on its own normal plane : 
 
 r-2= -^. lim. — ' 397, XCIX., (10) ; 
 
 2kP PQ23 ' V / » 
 
 K being still the centre of the osculating circle, and Qi, Q2, Qs being still (as in 397, 
 (10.)) the projections of a near point q (or p«), on the tangent, the absolute normal 
 (or inward radius of curvature pk), and the binormal at p. In fact, the principal 
 terms of the three vector projections corresponding, of the small chord pq (or pp«), 
 are (comp. LVIII.) : 
 
 PQi = ST ; PQ2 = (isV =) - Ur' ; pqs = {\sH-'^v =) — Ui/ ; 
 
 whence, ultimately, 
 
 ^.^ = _r-2rUr' = r-2.KP. 
 2 PQ33 
 
 f Considered as a rotation, this small angle may be represented by the small 
 vector, rp'^R'R'^ST ; and if the vector deviation LXXXV. from the osculating circle 
 be multiplied bg this, the quarter of the product is (comp. XXXV.) the vector devia- 
 tion from the osculating sphere, under the form, 
 
 s*(p-(t) ^ 
 24ie ' rvp 
 
CHAP. III.] DEVIATIONS FROM CIRCLE AND SPHERE. 595 
 
 PS to PSs is positive : and if we multiply this scalar, by that which has just been 
 assigned (51.), as an expression for the deviation CjPs from the osculating circle, we 
 get, by XXXV., the product, 
 
 LXXXVn...|^/4'=^=4(5P,-sP). 
 6r2r pR 6rrp ^ ^ 
 
 (53.) Combining then the recent results (50.), (51.), (52.), we arrive at the fol- 
 lowing Theorem : 
 
 The deviation of a near point P5 of a curve in space, from the osculating sphere 
 at the given point P, is ultimately equal to the quarter of the deviation of the same 
 near point from the osculating circle at p, multiplied hij the sine of the small an- 
 gle which the arc ss*, of the locus of centres of spherical curvature (b), or of the 
 cusp-edge of the polar developable, subtends at the same point p ; and this deviation 
 (sPg — sp) from the sphere has an outward or an inward direction, according as the 
 same arc ss* is concave or convex towards the same given point. 
 
 (54.) The vector of the centre Ss, of the near osculating sphere at p^, is (in the 
 same order of approximation, comp. I.), 
 
 LXXXVIII. . . oSs = 0-5 = (7 + s(t' + is%" + \s^(t"' + -i-^s^a^" ; 
 and although <t — p is already a function (by 397, IX., &c.) of r, r', r", so that a' 
 is (as in (2.) or (22.)) a function of r', t", t", and o", a", a^^ introduce respectively 
 the new derived vectors r'^, t^, r", or Ds^p, D/p, 'Ds''p, which we are not at pre- 
 sent employing (49.), yet we have seen, in (23.) and (24.), that some useful combi- 
 nations of a" and a" can be expressed without r'^, r^ : and the following is another 
 remarkable example of the same species of reduction, involving not only a" and a"' 
 but also ff'^, but still admitting, like the former, of a simple geometrical interpreta- 
 tion. 
 
 (55.) Remembering (comp. (22.), and 397, XV.) that 
 LXXXIX. . . ((T - p)2 + i23 = 0, and XC. . . Sr'" ((X - p) = r--^S = r-2 + nr-ir'J, 
 
 and reducing the successive derivatives of LXXXIX. with the help of the equations 
 397, XIX., and of their derivatives, we are conducted easily to the following system 
 of equations, into which the derived vectors r, r', &c. do not expressly enter, but 
 which involve a, c", a'", o-^^ and R', R", R"\ ^'^ : 
 
 XCI. . . Sa'i(T-p) + RR' = 0; XCII. . . S(T'(T"(<r-p) = 0; 
 
 XCIII. . . S(r"(<r - p) + <r'2 + {RRJ = ; 
 
 XCIV. . . S<t"'(<t - p) + 3SctV' + (RR')" = ; 
 
 77 R' n. 
 
 XCV. . . S(t'^(<t-p) + 4S(tV" + 3(7"2+ {RR'y"=- =- -; 
 
 auxiliary equations being, 
 
 XCVI. . . S(tV = 0, S<t't' = 0, S(r"r = 0, comp. 395, X. 
 
 and XCVII. . . Sct't = - Sff'V = StrV" = Srr" - S (<r - p) r'" 
 
 = -r-2(5'-l) = -«r-ir-i. 
 
 (56.) But, if Rs denote the radius of the near sphere, and if we still neglect s^, 
 we have, 
 
 XCVIII. . . ^- = - (<Ts - ps)2 =Rs^ 
 
 = i?2 + 2sRR' + s-^'iRRj + '- iRR')" + ^ {RRJ" ; 
 
596 ELEMENTS OF QUATERNIONS. ' [bOOK HI. 
 
 whence follows, by LXXXVIII., and by the recent equations, this very simple ex- 
 pression, from which (comp. (24.)) everything depending on t^^, t", t^^ has disap- 
 peared, 
 
 XCIX...(„-rt.+ R..= ^*; 
 
 and which gives (within the same order of approximation, attending to XXXV.) 
 
 the geometrical relation, 
 
 — Jl'gi ns^ 
 
 C. . . PS, - FsSs = T((rs - p) - 72, = — — = - — - = sp, - sp ; 
 
 24rrp 24rri2 
 
 or C'. . . S4P — 3Ps = SsP, -ap = Rs-R. 
 
 (57.) This result might have been foreseen, from the following very sim- 
 ple consideration. When the coefficient S-1 of non- sphericity (395, (16.)), or 
 of the deviation of a curve from a sphere, is positive, so that a near point p, of the 
 curve is exterior to (what we may call) the given sphere, which osculates to that 
 curve at P, by an amount which is ultimately proportional to the fourth power 
 of the arc, s, of the curve, then the given point p must be, for the same rea- 
 son, exterior to the near sphere, which osculates at the point P, ; and the two devia- 
 tions, PS, — PaSj, and sp, — sp, which have been found by calculation to be equal 
 (C), if «5 be neglected, must in fact bear to each other an ultimate ratio of equality, 
 because the two arcs, + s and — s, from p to p,, and from p, hack to p, are equally 
 long, aMhoxigh oppositely directed ; or because (+«)*"=("*)*• ^"d precisely the 
 same reasoning applies, when the coefficient .S" - 1 is negative, so that the deviations, 
 equated in the formula C, are both inwards. 
 
 (58.) As regards the deviation (51.) of the near point p, of the curve from the 
 osculating circle at p, we may generalize and render more exact the expression 
 LXXXV,, by considering a point c< of that circle, which is distant by a circular arc 
 = t from the given pohit p ; and of which the vector is, rigorously, by 396, (18.), 
 
 . i . t 
 
 CI. . . oCt = (ot = p + rrsin- + r^T vers - : 
 r r 
 
 or if we only neglect t% 
 
 cii...oo, = .,=p+.(*-^] + r.'(|:-iLy 
 
 (69.) In this way we shall have (comp. (34.)) the vector deviation, 
 cm. . . qp, = p, - wi = Xr + Yrr' + Zrv, 
 with the scalar coefficientsy 
 
 CTSf. . . X=Xs-rsin-, F = ys-r vers-, Z=Zs; 
 r r 
 
 or, neglecting s^ and t^, and attending to the expressions LVIII. and LVIII',, 
 
 CV. .. I ^^ ^2-<2 p^ ,4_t\ ns* 
 2r r ^24/-3 24r2r' 
 
 6rr ^ 24 ^ '' ' 
 
 in which r, r', r, p, and n have the same significations as before. 
 
CHAP. III.] RELATIONS BETWEEN THE DEVIATIONS. 597 
 
 (60.) Assuming then for the circular arc t the value, 
 
 r's^ 
 CYI...« = .+ _3, 
 
 which differs (as we see) by only a quantity of i\\Q fourth order from the arc s of the 
 curve, we shall have, to the same order of approximation, the expressions, 
 
 CVII. . . Jt= 0, F= — Z - -^, Z = zs = &c., as before, 
 r 24r2r 
 
 the deviation at Pj from the circle being here measured in a direction parallel to the 
 normal plane at P ; and if «* be neglected (although the expressions enable us to 
 take account of it), this deviation is also parallel (as before) to the tangent t{(t- p) 
 to the osculating sphere in that plane : while it is represented in quantity by Rr'^Zs, 
 which agrees with the result in (51.). 
 
 (61.) The expressions CVII. give also, without neglecting s*, 
 
 CVIII. . . — =-— — = = sp-sps; 
 
 R 24/riE 
 
 such then is the component of the deviation from the osculating circle, which is pa- 
 rallel to the normal ps to the sphere at p ; and we see that it only differs in sign 
 (because it is positive when its direction is that of the inward normal, or inward ra- 
 dius Ps), from the expression XXXV. (comp. C), for the outward deviation sPs — SP 
 of the near point p«, from the same osculating sphere at the given point p. 
 
 (62.) This latter component (61.) is small, even as compared with the former 
 small component (60.) ; and the small quotient, of the latter divided by the former, 
 is ultimately (by LXXXVI.), 
 
 nX rV+pZ _-nrs_ 
 
 where the small angle spSj is positive or negative, according to the rule stated in 
 (52.), and may be replaced by its sine, or by its tangent. 
 
 (63.) Instead of cutting the given osculating circle, as in (60.), by a plane which 
 is parallel to the given normal plane at P, we may propose to cut that circle hy the 
 near normal plane at Ps, or to satisfy this new condition, 
 
 ex. . . = Srs (ps - wO, or CX'. . . = X^tTs + FSrrVs + Z^rvTs ) 
 which is easily found to give by CV. the values (s and t being still supposed to be 
 small, and s^ being still neglected) : 
 
 CXI. . .t = s- —3, and CXII. . . X= ~, V = &c., Z= &c., as in CVII. ; 
 
 so that in passing to this new near point Ct of the circle, we only change X from 
 zero to a small quantity of the fourth order, and make no change in the values of V 
 and Z. 
 
 (64.) The new deviation CtFs from the given circle may be decomposed into two 
 partial deviations, in the near normal plane, of which one has the direction of the 
 unit-tangent i?s"'rs(<Ts- ps) to the near sphere at p^, and the other has that of the 
 unit-normal i2s-'(<r«- ps) to the same sphere at the same point (or the opposites of 
 these two directions) ; and the scalar coefficients of these two vector units, if we at- 
 tend only to principal terms, are easily found to be. 
 
598 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 CX^I../-£^=^^ and CXIV../i±if±^=^. 
 R 6r2r R SttR 
 
 (65.) We may then write : 
 
 CXV. . . Deviation of near point Tsfrom given osculating circle, 
 
 measured in the near normal plane to the curve at P«, 
 
 = new CiPs = — Ur«(ors - p,) + '^^^'^'^^ ~ P«) '•> 
 
 in which it may be observed, that the second scalar coefficient is equal to three times 
 the scalar deviation sp* — sp (XXXV. or C), of the near point Vs of the curve, from 
 the given osculating sphere (at p). 
 
 (66.) But we may also interpret the new coefficient last mentioned, as represent- 
 ing a new deviation ; namely, that of the point Ct of the given circle, from the near 
 osculating sphere at Ps, considered as positive when that new point c< is exterior to 
 that near sphere ; or as denoting the difference of distances, SsCj — s«p«. We have 
 therefore (comp. (56.)) this new geometrical relation, of an extremely simple kind : 
 
 CXVI. . . SiCit- SsPs= 3(sp$ - sp) = 3 (ssP - SsPs) ; 
 or CXVr. . . SsCj = 3ssP — 28sPs. 
 
 (67.) Supposing, then, at first, that the coefficient ofnon-sphericity S-l\9, posi- 
 tive (comp. 395, (16.)), if we conceive a point to move hackivards, upon the curve, 
 from Ps to p, and th&n forwards, upon the circle which osculates at P, to the new 
 point Ct (63.), we see that it \\\l\ first attain (at p) a position exterior to the sphere 
 which osculates at p^, or will have an amount, determined in (56.), of outward devi- 
 ation, with respect to that near osculating sphere ; and that it will afterwards attain 
 (at the new point Ct) a deviation of the same character (namely outwards, if S> 1), 
 from the same near sphere, but one of which the amount will be threefold the former : 
 this last relation holding also when S <1, or when both deviations are inwards. 
 
 (68.) It is easy also to infer from (65.), (comp. (57.)), that if we go back from 
 Ps, on the near circle which osculates at that near point, through an arc (t) of that 
 circle, which will only differ by a small quantity of the fourth order (comp. (60.)) 
 from the are (s) of the curve, so as to arrive at a point, which for the moment we 
 shall simply denote by c, and in which (as well as in another point of section, not 
 necessary here to be considered) the near osculating circle is cut by the given nor- 
 mal plane at P, the vector deviation of this new point c of the new circle, from the 
 given point p of the curve, must be, nearly : 
 
 the coefficients being formed from those of the formula CXV., by first changing s to 
 — s, and then changing the signs of the results : • while the relation CXVI. or 
 CXVI'. takes now the form, 
 
 CXVIII. . . sc - SP = 3 (sp, - sp), or CXVIII'. . . sc = 3ips - 2 sp. 
 (69.) Accordingly if, after going from p to p* along the curve, we go forward or 
 backward, through any positive or negative arc, t, of the circle which osculates at 
 that point p*, we shall arrive at a point which we may here denote by Cs, t ', and the 
 vector (comp. again 396, (18.)) of this near point (more general than any of those 
 hitherto considered) will be, rigorously, 
 
CHAP. III.] LOCUS OF OSCULATING CIRCLE. 599 
 
 CXIX. . . b)s,t = ocs, t = Ps-\- TsTs sin — + Ts^t's vers — , 
 
 rs Ts 
 
 And if we develope this new expression to the accuracy of theybwr^^ order inclusive, 
 we find that we satisfy the new condition Tcomp. (63.)), 
 
 CXX. .. Sr («;„<- p) = 0, wlien CXXI. . . ^= - s - — ; 
 
 and that then the expression CXIX. agrees with CXVII., within the order of ap- 
 proximation here considered. 
 
 (70.) A geometrical connexion can be shown to exist, between the two equiva- 
 lents which have been found above, one for the quadruple (LXXXVII,, comp. (53.)), 
 and the other for the triple (CXVIII.), of the deviation sp^ - SP of a near point Ps 
 of the curve, from the sphere which osculates at the given point p : in such a manner 
 that if either of those two expressions be regarded as known, the other can be in- 
 ferred from it. 
 
 (71.) In fact if we draw, in the normal plane, perpendiculars pd and pe to the 
 lines PS and ps^, and determine points d and e upon them by drawing a parallel to 
 PS through the point c of (68.), letting fall also a perpendicular cf on ps«, the two 
 small lines pd and DC will ultimately represent the two terms or components CXVII. 
 of PC ; and the small angle dpc will ultimately be equal to three quarters of the 
 small angle spSs, and will correspond to the same direction of rotation round r, be- 
 cause 
 
 CXXII...^ = f.^=|V-^, 
 
 or 
 
 CXXIII. . . dpc = f SPS, = |DPE ; 
 
 so that we shall have the ultimate ratios (comp. the 
 annexed Fig. 83*): 
 
 CXXIV. . . DC : DE : CE (or fp) = 3 : 4 : 1. 
 But the line cf is ultimately the trace, on the given pirr. 83. 
 
 normal plane, of the tangent plane at c to the near 
 osculating sphere; the small line fp (or ce) represents therefore the deviation 
 SsP- SsPs of the given point p from that near sphere, or the equal deviation (57.) 
 SPs — SP ; its ultimate quadruple, de, represents the product mentioned in (52.) ; 
 and the ultimate triple, dc, of the same small line ce, is a geometrical representation 
 of that other deviation "sc— sp, which has been more recently considered. 
 
 (72.) When the two scalars, s and t, are supposed capable of receiving any va- 
 lues, the point Cs,t in (69.) may be any point of the Locus (8.) of the Osculating 
 Circle to the given curve of double curvature ; and if we seek the direction of the 
 normal to this superficial locus, at this point, on the plan of Art. 372, writing first 
 the equation of the surface under the slightly simplified, but equally rigorous form, 
 
 * In Figs, 81, 82, the little arc near s is to be conceived as terminating there, 
 or as being a preceding arc of the curve which is the locus of s, if r', r, n, and there- 
 fore also p and ri, he positive (comp. the second Note to page 574). In the new Fi- 
 gure 83, the triangle pde is to be conceived as being in fact much smaller than 
 PKs, though magnified to exhibit angular and other relations.. 
 
600 ELKMENTS OF QUATKKNIONS. [bOOK III. 
 
 CXXV. . . itts, n=Ps + TsTs sin u + rs'^r's vers u, 
 with CXXVI. . . M = rs-^< = p*k«c«, t, 
 
 so that « is here a new scalar variable, representing the ariffle subtended at the cen- 
 tre Ks, of the osculating circle at Pj, 6y the arc, <, of that circle, we are led, after a 
 few reductions, to the expression, 
 
 CXXVI I. . . V(D„Ws,,t . Dsw«, m) = rsrs'i(w«, « - Cs) vers « ; 
 
 which proves, by quaternions, what was to be expected from geometrical* conside- 
 rations, that the loctis of the osculating circle is also (as stated in (8.) and (22.)) 
 the Envelope of the Osculating Sphere. 
 
 (73.) The normal to this locus, at any proposed point c,s,t of any one osculating 
 circle, is thus the radius of the sphere to which that circle belongs, or which has the 
 same point of osculation Fs with the given curve, whether the arc (s) of that curve, 
 and the arc (t) of the circle, be small or large. We must therefore consider the tan- 
 gent plane to the locus, at the given point p of the curve, as coinciding with the tan- 
 gent plane to the osculating sphere at that point ; and in fact, while this latter plane 
 (J- Ps) contains the tangent r to the curve, which is at the same time a tangent to 
 the locus, it contains also the tangent r((T- p) to the sphere, which is by CXVII. 
 another tangent to the locus, as being the tangent at P to the section of that surface, 
 which is made by the normal plane to the curve. 
 
 (74.) But when we come to examine, with the help of the same equation CXVII., 
 Vihatis the law of the deviation DC (comp. Fig. 83) of that normal section of the 
 locus, considered as a new curve (c),/rom its own tangent pd, we find that this law 
 is ultimately expressed (comp. (71.)) by the formula, 
 
 CXXVm...^! = H.»!i!^^ = const.; 
 pd4 32 R^ 
 
 hence dc varies ultimately as the power of pd, which has the fraction f for its expo- 
 nent ; the limit of pd^ : DC is therefore null, and the curvature of the section is infinite 
 atv. 
 
 (75.') It follows that this point p is a singular point of the curve (c), in which 
 thelocus (8.) is cut (73.), by the normal plane to the given curve at that point ; but 
 it is not a cusp on that section, because the tangential component pd of the vector 
 chord PC is ultimately proportional to an orfd power (namely to the cube, by CXVII., 
 comp. (71.)) of the scalar variable, s, and therefore has its direction reversed, when 
 that variable changes sign : whereas the normal component DC of the same chord PC 
 is proportional to an even power (namely the fourth, by the same equation CXVII.) 
 of the same arc, s, of the given curve, and therefore retains its direction unchanged, 
 when we pass from a near point p,, on one side of the given point p, to a near point 
 p.g on the other side of it. 
 
 (76.) To illustrate this by a contrasted case, let G be the point in which thetan- 
 gent to the given curve at p^ is cut by the normal plane at p ; or a point of the sec- 
 tion, by that plane, of the developable surface of tangents. We shall then have 
 
 * In the language of infinitesimals, two consecutive osculating spheres, to any 
 curve in space, intersect each. other in an osculating circle to that curve. 
 
CHAP. III.] ENVELOPE OF SPHERE WITH VARYING RADIUS. 601 
 
 the sufficiently approximate expressions, 
 
 * + 3;i)'"*= ""2 -3^ = - PQ2 - 2pQ3, 
 
 with the significations 397, (10.) of Q2 and Q3; hence the point p of the curve is 
 (as is well known) a cusp of the section (g) of the developable surface of tangents 
 (comp. 397, (15.)), because the tangential component (— PQ2) of the vector chord 
 (pg) has here a. fixed direction^ namely that of the outward radius (kp prolonged) 
 of the circle of curvature at p : while it is now the normal component (— 2PQ3) 
 which changes direction, when the arc s of the curve changes sign. At the same 
 time we see* that the equation of this last section (g) may ultimately be thus ex- 
 pressed : 
 
 CXXX. . . — = -— = const. 
 
 (- PQ3)^ .9r2 
 
 comparing which with the equation CXXVIII., we see that although, in each case, 
 the curvature of the section is infinite, at the point p of the curve, yet the normal 
 component (or co-ordinate) varies (ultimately) as the power -| of the tangential com- 
 ponent, for the section (g) of the Surface of Tangents : whereas the former compo- 
 nent varies by (74.) as the power f of the latter, for the con-esponding section (c) 
 of the Locus of the Osculating Circle. 
 
 (77.) It follows also that the curve (p) itself, although it is not a cusp-edge of 
 the last-mentioned locus (8.), while it is such on the surface of tangents, is yet a 
 Singular Line upon that locus likewise : the nature and origin of which line will 
 perhaps be seen more clearly, by reverting to the view (8.), (22.), (72.), accord- 
 ing to which that Locus of a Circle is at the same time the Envelope of a Sphere. 
 
 (78.) In general, if we suppose that o- and R are any two real functions, of the 
 vector and scalar kinds, of any one real and scalar variable, t, and that a', R', and 
 a", R", &c. denote their successive derivatives, taken with respect to it, then <r 
 may be conceived to be the variable vector of a point s of a curve in space, and R to 
 be the variable radius of a sphere, which has its centre at that point s, but alters ge- 
 nerally its magnitude, at the same time that it alters its position, by the motion of 
 its centre along the curve (s). 
 
 (79.) Passing from one such sphere, with centre s and radius R, considered as 
 given, and represented by the scalar equation, f 
 
 (rr - p)2 + i22 = 0, LXXXIX., 
 
 in which p is now conceived to be the vector of a variable point p upon its surface, 
 to a near sphere of the same system, for which cr, s, and R are replaced by at, St, and 
 Rt, where t is supposed to be small, we easily infer (comp. 386, (4.)) that the equa- 
 tion, 
 
 S(T'((T-p) + i?/2'=0, XCI., 
 
 which is formed from LXXXIX. by once derivating a and R with respect to t, but 
 
 * Compare the first Note to page 594. 
 
 t This equation, and a few others which we shall require, occurred before in this 
 series, but in a connexion so different, that it appears convenient to repeat them 
 here. 
 
 4 H 
 
602 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 treating p as constant, represents the real plane (comp. 282, (12.)) of the (real or 
 imaginary) circle, which is the ultimate intersection of the near sphere with the 
 given one ; the radius of this circle, which we shall call r, being found by the follow- 
 ing formula, 
 
 CXXXI. . . r2(T'2 = i?2 (i?'2 + (t'«), or CXXXl'. . . r^Tcr'^ = i?2 (T(t'2 - R'% 
 and being therefore real when 
 
 C XXXII. . . i2'2 + (r'2 < 0, or CXXXII'. . . ij'2 < T(r'2 ; 
 while the centre, say k, of the circle is always real, and its vector is^ 
 CXXXI". . . OK = K = (T + i?i2'<r'-i ; 
 
 and the plane XCI. of the same circle is parallel to the normal plane of the curve 
 (s). 
 
 (80.) With the condition CXXXII., the two scalar equations, LXXXIX. and 
 XCI., represent fimn jointly a real circle ; and the locus of all such circles (comp. 
 386, (6.)) is easily proved to be also the envelope of all the spheres, of which one is 
 represented by the equation LXXXIX. alone ; each such sphere touching this locus, 
 in the whole extent of the corresponding circle of the system. 
 
 (81.) ThQ plane XCI., considered as varying with t, has a developable surface 
 for its envelope ; and the real right line, or generatrix, along which one touches the 
 other, is represented (comp. again 386, (6.)) by the system of the two scalar equa- 
 tions, XCI. and 
 
 S(T"((r-p) + <T'2 + (/i;72')'=0, XCIII.j 
 
 where p is now the variable vector of the line of contact, although it has been freaked 
 as constant (comp. 386, (4..)), in the process by which we are here conceived to pass, 
 by a second derivation, from LXXXIX. through XCI. to XCIII. 
 
 (82.) This real right line (81.) meets generally the sphere, and also i\\& circle (as 
 being in its plane), in two (real or imaginary) points, say pi, P2 ; and the curvilinear 
 locus of all such points forms generally a species of singular line,* upon the superfi- 
 cial locus (or envelope) recently considered (80.) ; or rather it forms in general two 
 branches (real or imaginary) of suclt a line : which generally two-branched line (or 
 curve) is the (real or imaginary) envelope (comp. 386, (8.)), of all the circles of the 
 system. 
 
 (83.) The equation, 
 
 S(tV'((t-p) = 0, XCII., 
 
 which now represents (comp. 376, V.) the osculating plane to the curve (s), shows 
 
 * Called by Monge an arete de rebroussement, except in the case to which we 
 shall next proceed, when its two branches coincide. The envelope (80.) of a varying 
 sphere has been considered in two distinct Sections, § XXII. and § XXVI., of the 
 Application de V Analyse a la Geometric; but the author of that great work does 
 not appear to have perceived the intei-pretation which will soon be pointed out, of the 
 condition of such coincidence. Meantime it may be mentioned, in passing, that qua- 
 ternions are found to confirm the geometrical result, that when the two branches (pi) 
 (P2) are distinct, then each is a cusp edge of the surface ; but that Avhen they are 
 coincident, the singular line (p) in which they merge has then a different character. 
 
CHAP. III.] CONDITION OF COINCIDENCE OF CUSP-EDGES. 603 
 
 that this plane through the centre s of the sphere is perpendicular to the right line 
 (81.), and consequently contains the perpendicular let fall from that centre on that 
 line : the foot p of this last perpendicular is therefore found by combining the three 
 linear and^calar equations, XCI., XCII., XCIIL, and its vector is, 
 
 ad' + RR'o" 
 CXXXIII. . . OP = p = (T + ^ 1 , „ , 
 
 if CXXXI V. ..g = -a'^-Il'2-RR"= T(r'2 - (JSi2')'. 
 
 (84.) The condition of contact of the right line (81.) with the sphere (78.), or 
 with the circle (79.), or the condition of contact between two consecutive* circles Of 
 the system (80.), or finally the condition of coincidence of the two branches (82.) 
 of that singular line upon the surface which is touched by all those circles, is at the 
 same time the condition of coexistence of the four scalar equations, LXXXIX., XCI., 
 XCII., XCIII. ; it is therefore expressed by the equation (comp. CXXXIII.), 
 
 CXXXV. . . m(Y<T'(r'y = (i/d' + RR'ff'y ; 
 which may also be thus written, f 
 
 CXXXVI. . . (i2S(T'(T" - R'gy = (R'i + <t'2) (^H^a'^ + g% 
 or thus, CXXXVII. . . J?2(ig'2 + <j'^) (Ya'a'y = (ga'^ + Jli2'S<T'(r")2 ; 
 
 the scalar variable t (78.), with respect to which the derivations are performed, re- 
 maining still entirely arbitrary, but the point p, which is determined by the formula 
 CXXXIII., being now situated on both the sphere and the circle : and its curvilinear 
 locus, which we may call the curve (p), being now the singular line itself, in its re- 
 
 * Compare the Note to page 581. 
 
 t In page 372 of Liouville's Edition already cited, or in page 325 of the Fourth 
 Edition (Paris, 1809), of the Application de V Analyse, &c., it will be found that 
 this condition is assigned by Monge, as that of the evanescence of a certain radical, 
 under the form (an accidentally omitted exponent of tt" in the second part of the first 
 member being here restored) ; 
 
 [a(0>" + il'Vr TtV") - A2]2 + A2 [a2(^"2 + ^"2 + ^"2) _ /^i] = Q y 
 
 in which he writes, for abridgment, 
 
 and 0, •^, TT are the three rectangular co-ordinates of the centre of a moving sphere, 
 considered as functions of its radius a. Accordingly, if we change R to a, and <r to 
 ifp j^j^ + JcTT, supposing also that R' = a'=l, and R" = a" = 0, whereby g is changed 
 to -h^, and R'^+ a'^ to h^, in the condition CXXXVI., that condition takes, by the 
 rules of quaternions, the exact form of the equation cited in this Note : which, for the 
 sake of reference, we shall call, for the present, the Equation of Monge, although 
 it does not appear to have been either interpreted or integrated by that illustrious 
 author. Indeed, if Monge had not hastened over this case of coincident branches, 
 on which he seems to have designed to return in a subsequent Memoir (unhappily 
 not written, or not published), he would scarcely have chosen such a symbol as h^ 
 (instead of - A2), to denote a quantity which is essentially negative, whenever (as 
 here) the envelope of the sphere is real. 
 
604 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 duced and one-hranched state. And the last form CXXXVII. shows, what was to 
 be expected from geometry, that when this condition of coincidence is satisfied, the 
 earlier condition of reality CXXXII. is satisfied also : together with this other in- 
 equality, 
 
 CXXXVIII. . . i2V'2 + ^2 < 0, 
 
 which then results from the form CXXXVI. 
 
 (85.) The equations CXXXI., CXXXIV., and the general formula 389, IV., 
 give the expressions, 
 
 where r is still the radius of the circle of contact of the sphere with, its envelope, and 
 ri is the radius of curvature of the locus of the centre s of the same variable sphere; 
 whence it is easy to infer, that the condition CXXXV. may be reduced to the fol- 
 lowing very simple form (comp. XXXVI'. and XLII.) : 
 
 CXLI. . . (r'ny = (HEy ; or CXLI'. . . ndr = + ME ; 
 the independent variable being still arbitrary. 
 
 (86.) If the arc of the curve (s) be taken as that variable t, the form CXXXVI. 
 of the same condition is easily reduced to the following, 
 
 CXLII. ..R^ = {RR'f + ^2ri2, with CXLIII. . .g=l- (ERJ ; 
 derivating then, and dividing by 2g, we have this new differential equation, which is 
 of linear form with respect to RR', whereas the condition tVseZ/'maybe considered as 
 a differential equation of the second degree, as well as of the second order,* 
 
 CXLI V. ..RR^ = rx (gr{)' ; or CXLV. . . n^u" + nriXu' - 1) + m = 0, 
 if CXLVI. . . M = RR' = RDtR, and therefore CXLVII. . .u^=R^- r^, 
 by CXXXI. or CXXXI'., because we have now, 
 
 CXLVIII. . . a'2 = -l, or Ta'=l, or df = Td<T: 
 so that the new scalar variable, RR', or u, with respect to which the linear equation 
 CXLIV. or CXLV. is only of the second order, represents the perpendicular heightf 
 of the centre s of the sphere, above the plane of the circle, considered as a, function 
 of the arc (t) of the curve (s), and as positive when the radius R of the sphere in- 
 creases, for positive motion along that curve, or for an increasing value of its arc. 
 
 (87.) If the curve (s) be given, or even if we only know the law according to 
 which its radius of curvature (/•{) depends on its arc (t), the coefficients o{ the linear 
 equation CXLV. are known ; and if we succeed in integrating that equation, so as to 
 
 * We shall soon assign the complete integral of the differential equation in qua- 
 ternions (84.), and also that of the corresponding Equation of Monge, cited in the 
 preceding Note. 
 
 t It will be found that this new scalar u, if we abstract from sign, corresponds 
 precisely to the p of earlier sub-articles, although presenting itself in a different con- 
 nexion : for the sphere (78.), and the circle (79.), under the condition (84.), will 
 soon be shown to be the osculating sphere and circle to the recent curve (p), or to 
 the singular line (84.) upon the surface at present considered, that is, on the locus 
 or envelope (80.). 
 
CHAP. III.] DETERMINATION OF THE SINGULAR LINE. 605 
 
 find an expression for the perpendicular m as a function of that arc t, we shall then 
 be able to express also, as functions of the same arc, the radii R and r of the sphere 
 and circle^ by the formulae, 
 
 CXLIX. . .+r=i^ri = ri (!-«'), and Ch. . . R"^ = 2^udit = u'i + ri^(l-uy ; 
 the third scalar constant, which the integral 2j«d< would otherwise introduce into 
 the expression for R^, being in this manner determined, by means of the other two, 
 which arise from the integration of the equation above mentioned. 
 
 (88.) For example, it may happen that the locus of the centre s of the sphere 
 has a constant curvature, or that ri = const. ; and then the complete integral of the 
 linear equation CXLV. is at once seen to be of the form, 
 CLT. . . M = a sin (r^H + b), 
 a and 6 being two arbitrary (but scalar) constants ; after which we may write, by 
 (87.), 
 CLII. . . +r = ri-ocos(rri< + 6); CLIII. . . ija = ri2-2ari cos (rri<+ i)+ a2; 
 
 so that, in this case, both the radii., r and R, of circle and sphere, ai*e periodical 
 functions of the arc of the curve (s). 
 
 (89.) In general, if that curve (s) be completely given, so that the vector tr is a 
 known function of a scalar variable, and if an expression have been/owrarf (or given) 
 for the scalar R which satisfies any one of the forms of the condition (84.), we can 
 then determine also the vector p, by the formula CXXXIIL, as a function of the 
 same variable; and so can assign the point p of the singular line (84.), which cor- 
 responds to any given position of the centre s of the sphere. For this purpose we 
 have, when the arc of the curve (s) is taken, as in (86.), for the independent varia- 
 ble t, the formula, 
 
 CLI V. . . p = <r - M0-' - (1 — w') <t"-i = Ki — ua — r^u'a", 
 if K\ be the vector of the centre, say Ki, of the osculating circle at s to that given 
 curve, so that (comp. 389, XI.) it has the value, 
 
 CLV. . . OKI = fci = <T - (t"-i = (T + nV, with CLV. . . a"^ + r^s = 0. 
 If then we denote by v the distance of the point p from this centre Ki, and attend to 
 the linear equation CXLV., we see that 
 
 CLVI. . . w = K^ = T(p - /ci) = V(«2 + nSw'z), 
 and CLVI'. . . vv' = riri V, with T(t'=1; 
 
 or more generally, CLVII. . . vv'si =rir\u, 
 
 if CLVII'. . .u = RR'si''\ and CLVII". . . si = JTd(T, 
 
 while CLVI". . . v2 = m2 + n^u'W'^ ; 
 
 so that $1 denotes the arc of the curve (s), when the independent variable t is again 
 left arbitrary. This distance, v, is therefore constant (= a) in the case (88.), namely 
 when the radius of curvature ri of that curve is itself a. constant quantity. 
 
 (90.) When si' = Tcr'= 1, as in CXLVIII., the part <t - ua' of the first expres- 
 sion CLIV. for p becomes = k, by CXXXI". and CXLVI. ; attending then to CLV., 
 we have the scalar quotient, 
 
 CLVIIL . ,1:^ = i-u'', 
 
 (T- Kl 
 
 whence generally, 
 
606 ELKMENTS OF QUATERNIONS. [bOOK III. 
 
 the independent variable t being again arbitrary. Accordingly, if we combine the 
 general expression CXXXIII. for p, with the expression CXXXI". for k, and with 
 the following for ki (comp. 389, IV.)> 
 
 (t'3 
 CLIX. . . (ci = tr + „ , , for an arhitrary scalar variable, 
 
 we easily deduce this new form of the scalar quotient, 
 
 CLIX'. . . '^^^ = 1 + ({RWy - HR'Sa'-^a") a'-^ ; 
 
 <T — K\ 
 
 o" Si" 
 
 which agrees with CLVIII'., because — cr'2 =Si'2, and S — = -7-. 
 
 or s 1 
 
 (91.) It has then been fully shown, how to determine the vector p as a function 
 of the scalar t, when a and E are two known functions of that variable, which satisfy 
 any one of the forms of the condition (84.). It must then be possible to determine 
 also the derived vectors, p, p", «S;c., as functions of the same variable; and accord- 
 ingly this can be done, by derivating any three of ih^four scalar equations, LXXXIX. 
 XCI. XCII. XCIII., of which that condition (84.) expresses the coexistence. Now 
 if we derivate a first time the two first of these, and tben reduce by the second and 
 fourth, we get the equations, 
 
 CLX. . . Sp'(<^ - p) = 0, SpV = 0, whence CLX'. . . p' || Y<T'(ff-p) ; 
 and although this last formula only determines the direction of the tangent to the sin- 
 gular line at p, namely that of the common tangent at that point to two consecutive 
 circles (84.), yet it enables us to infer, by the remaining equation XCII., that 
 
 CLXI. . . p' -I- (t", p II VffV", and CLXI'. . . Sp'(T"= ; 
 reducing by which the derivative of XCIII., we find, 
 
 S(T"'((r-p) + 3S(7V+ (i2i2')"=0, XCIV., 
 
 the scalar variable being still arbitrary. And. conversely, the system* oi thQ four 
 equations LXXXIX. XCI. XCIII. XCIV. gives the three equations CLX. CLXI'., 
 and so conducts to the equation XCII., and thence to the condition (84.) ; unless we 
 suppose that p is a constant vector a, or that the variable sphere passes through a 
 fixed point A, a case which we do not here consider, because in it the singular line 
 (p) would reduce itself to that one point. 
 
 (92.) Derivating the two equations CLX., and reducing with the help of 
 CLXI'., we find these new equations, 
 
 CLXIL . . Sp"(^ - p) - p'2 = 0, Sp'V = ; 
 whence CLXIII. . . Sp"'((T - p) - SSp'p" = 0. 
 
 * In the language of infinitesimals, this system of equations expresses that /o«r 
 consecutive spheres intersect, in one common point p. When that point happens 
 to be a fixed one, the condition (84.) requires that we should have the relation 
 ^a'a"{a - a) = ; or geometrically, that the curve (s) should be in a plane through 
 the fixed point, which is then a singular point of the envelope. 
 
CHAP. III.] ENVELOPE OF OSCULATING SPHERE. 607 
 
 We are led then, by elimination of the derivatives of(T, to the system of the three 
 equations 395, VII. ; and we conclude, that the point s is the centre, and the radius 
 R is the radius, of the osculating sphere* to the singular line (p) : whence it is easy 
 to infer also, that th.Q plane of contact (79.) of the sphere with its envelope is the 
 osculating plane, and that the circle of contact (80.) is the osculating circle (comp. 
 (72.)), to the same curve (p), at ihQ point where two consecutive circles touch one 
 another (84.). 
 
 (93.) In general, and even without the condition (84.), the tangent to a branch 
 (82.) of the curvilinear envelope of the circles of the system, at any point Pi of that 
 branch, has the direction represented by the vector V(7'((r- pi), of the tangent to the 
 circle at that point ; but when that condition is satisfied, so that the two branches 
 of the singular line coincide, the point p of that line is in the osculating plane (83.) 
 to the curve (s) : and then the equation XCII. shows that the tangent p', or 
 V<r'(<7 — p), to the line, is perpendicular to a", or parallel to Y<t'(t" (comp. CLXI.), 
 and therefore that the singular line crosses that plane at right angles. 
 
 (94.) It follows that, with the condition (84.), the singular line (p) is an ortho- 
 gonal trajectorg to the system of osculating planes to the curve (s) ; and whereas, 
 when this last curve is given, there ought to be one such trajectory for evert/ point 
 of a given osculating plane, this circumstance is analytically represented, in our re- 
 cent calculations, by the biordinnl form of the differential equation (yXltY ., of which 
 the complete integral must be conceived (87.) to involve generally, as in the case 
 (88.), two arbitrary constants. 
 
 (95.) It follows also that, vfhh the same condition of coincidence of branches, 
 the singular line (p) must have the curve (s) for the cusp-edge of its polar develop- 
 able ; or that the sphere, with s for centre, and with R for radius, must be the oscu- 
 lating sphere to the curve (p), as otherwise found by calculation in (92.) : while the 
 circle (80.) must be, as before, the osculating circle to that curve. 
 
 (96.) Accordingly, all equations, and inequalities, which have been stated in the 
 recent sub-articles (79.), &c., respecting the envelope of a moving sphere with va- 
 riable radius, under that condition (84.), and without any special selection of the 
 independent variable, admit of being verified, by means of the earlier formulae for 
 the osculating circle and sphere to a curve (p) treated as a given one, when the arc 
 (s) of that curve is taken as such a variable. 
 
 (97.) For example, we had lately the two inequalities, R"^-\- <j"^ < 0, CXXXIL, 
 and i22(T"2 +^2<o, CXXXVIII. And accordingly the earlier sub-articles (22.), 
 (23.) give, for those two combinations, the essentially negative values, 
 
 CLXIV. . . i2'2 + (t'2 = - p-2r2iJ'2 ; CLXV. . . i22<T"2 + ^2 = - ((«r)')2 ; 
 
 * In the language of infinitesimals (comp. the preceding Note), if every four 
 consecutive spheres of a system intersect in one point of a curve, then each sphere 
 l^sissefi through four consecutive points of that curve. Simple as this geometrical 
 reasoning is, the writer is not aware that it has been anticipated ; and indeed he is 
 at present led to suppose that this whole theory, of the Locus of the Osculating 
 Circle, as the Envelope of the Osculating Sphere, is new. Monge had however 
 considered, but rejected (page 374 of Liouville's Edition), the case of a system qf 
 circles having each a simple contact with a curve in space. 
 
608 ELKMENTS OF QUATERNIONS. [bOOK III. 
 
 in obtaining which last, the following transformations have been employed : 
 CLXVI. . . ff"2 = - n'2 - n2r-2 . CLXVIL . .g = -n'p+ nrt-\ 
 (98.) As regards the verification of the equations, it may be sufiicient to give one 
 example ; and we shall take for it the last general form CLVII. of the differential 
 equation of condition (84.). For this purpose we may now write, by (22.) and 
 (23.), 
 
 CLXVIII. . . sx=±n, u = ±p, u' = ±p\ riui'si-i=p'rin " =p'r; 
 
 and have only to observe that 
 
 CLXIX. . . Kp2+p'2j.2)'=p'r(r + ;,'r)', because p = r'r. 
 
 (99.) If we denote by ci, C2, C3 the first members of the equations XCI., XCIIL, 
 XCIV., then besides the equation LXXXIX., Avhich may be regarded as a mere de- 
 finition of the radius E, we have ci = for the whole of the superficial Zooms or enve- 
 lope (80.) ; but we have not also cg = 0, except for a point on one or other of the 
 two (generally distinct) branches of the singular line (82.) upon that locus. And 
 if, at any other and ordinary point, we cut the surface by a plane perpendicular to 
 the circle at tliat point, we find, by a process of the same kind as some which have 
 been already employed, expressions for the tangential and normal components of the 
 vector chord, whereof the principal terms involve the scalar cj as a factor, while the 
 latter varies (ultimately) as the sg-ware of the former, so that the curvature of the 
 section is finite and known, but tends to become infinite when C3 tends to zero. 
 
 (100.) If the condition of coincidence (84.) be not satisfied, so that the two 
 branches of the singular line (82.) remain distinct,' and. that thus C2=0, but not 
 C3 = (comp. (91.)), for any ordinary point on one of those two branches, then if we 
 cut the surface at that point by a plane perpendicular to the branch, or to the circle 
 which touches it there, we find an ultimate expression for the vector chord which 
 involves the scalar cz as a factor, and of which the normal component varies as the 
 sesquiplicate power of the tangential one : so that we have here the case of a semi- 
 cubical cusp, and each branch of the singular line is a cusp-edge* of the surface, 
 exactly in the same known sense (comp. (76.)) as that in which a curve of double 
 curvature is generally such, on the developable locus of its tangents. 
 
 (101.) But when the condition (84.) is satisfied, so that the two branches coin- 
 cide, and that thus (comp. again (91.)) we have at once the three equations, 
 
 CLXX. ..ci = 0, co = 0, C3=0, 
 then the terms, which were lately the principal ones (100.), disappear : and a new 
 expression arises, for the vector chord of a section of the surface, made by a plane 
 perpendicular to the singular line, which (when we take t = s, as in (96.)) is found 
 to admit of being identified with the formula CXVII., and of course conducts to 
 precisely the same system of consequences ; the tangential component now varying 
 ultimately as the cube, and the normal component as the fourth power of a small 
 variable, so that the cuspidal property of the point p of the section no longer exists? 
 although the curvature at that point is still infinite, as in (74.) : and the Singular 
 Line, reduced now to a single branch, to which all the circles of the system osculate, 
 
 * Compare the Note to page 602. 
 
CHAP. Ill,] INTEGRAL OF THE EQUATION OF CONDITION. 609 
 
 (92.), (95.), is not a cttspedge of the Surface, as had been otherwise found before 
 (77.), but a line of a different character,* which may thus be regarded, with refe- 
 rence to a more general Envelope (80.), as the result of a Fusion (84.) of Two Cusp- 
 Edges. 
 
 (102.) The condition o{ such fusion (or coincidence) has been seen (84.) to be 
 expressible by the differential equation of the second order, and second degree, 
 
 ( RSa'a" - R'gy = {R" + (t'2) (^2 ^"2 + g^), CXXXVI. 
 
 with gr=-(fi- (RR'Y, CXXXI V. 
 
 and with the independent variable arbitrary. And we are now prepared to assign 
 the complete general integral f of this differential equation ; namely the system of 
 the two following equations (comp. 395, (7.) and (14.)), of the vector and scalar 
 kinds, 
 
 CLXXI.... = p+?Ve:Spi;+I?V\ and CLXXII. . . iJ = T(.-p), 
 
 Spp p 
 
 in which p is an arbitrary vector function of any scalar variable, f, and which ex- 
 press, when geometrically interpreted, that (T is the variable vector of the centre s, 
 and that R is the variable radius, of the osculating sphere to 2ir\. arbitrary curve (p), 
 of which the variable vector of a point p is p. 
 
 (103.) In fact, if we met the cited equation of condition CXXXVI., g represent- 
 ing therein the expression CXXXIV., without any previous knowledge of its mean- 
 ing or origin, we might first, by the rules of quaternions, and as a mere affair of 
 calculation, transform it to the equation CXXXV. ; which would evidently allow 
 the assumption of the formula CXXXIII., p being treated as an auxiliary vector, 
 which satisfies (in virtue of the supposed condition') the system of the four scalar 
 equations, LXXXIX., XCI., XCII., XCIII. ; whence derivating and combining, as 
 in (91.) and (92.), we are led to a new systemX of /owr scalar equations, whereof one 
 
 * Compare the Note to page 602. Monge (in page 372 of Liouville's Edition) has 
 the remark, that (when a certain radical vanishes) " les deux branches de la courbe 
 touchee par toutes les cara^teristiques se confondent en une seule : et cette courbe, 
 sans cesser d'etre une ligne singuliere de la surface, n'est plus une arete de rebrous- 
 sement, elle est une ligne de striction." The propriety of this last name, " line of 
 striction," appears to the present writer questionable : although he has confirmed, as 
 above, by calculations with quaternions, the result that, in the case referred to, the 
 singular line is not a cusp-edge. Monge does not seem to have perceived that, in 
 the same case of fusion, the curved line in question is not merely touched, but oscu- 
 lated, by all the circles of the system. 
 
 t Compare the first Note to page 604. We say here, general integral, because a 
 less general one, although involving one arbitrary function (of the scalar kind), 
 will soon be pointed out. 
 
 J The Equation of Monge (comp. the second Note to page 603) may be consi- 
 dered as the condition of coexistence of the four following equations, in wliich ^, %(/, 
 TT are supposed to be functions of a, and to be differentiated or derivatcd as such : 
 
 4 I 
 
610 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 is again the equation LXXXIX., and may be written under the form CLXXII. ; 
 while the three others are those formerly numbered as 395, VII., and conduct (ex- 
 cept in a particular case which we shall presently consider) to the vector expression 
 CLXXI., which conversely is sufficient to represent them, all derivatives of <j and of 
 R being thus eliminated. 
 
 (104.) The case just now alluded to, in which the general integral (102.) is re- 
 placed by a less general form, is the case (91.) when the variable sphere passes 
 through a. fixed point A, to which point, in that case, the singular line reduces itself. 
 Andi i\\Q integral equations,* which then replace CLXXI. and CLXXII., maybe 
 thus written : 
 CLXXIIL . . <r = a + ;/3 + My, with u = F(t), and CLXXI V. .. 72 = T(</3 + «y) ; 
 
 (1). . . (a;-0)2 + (y-4/)2 + (z-7r)2 = a2; 
 
 (2). . . (a;-0)V>' + (y-^)'/''+ (2-7r)7r'+a = 0; 
 
 (3). ..(x- 0)0"+ (y -!//)•.//"+ (z - 7r)7r"+ 1 -f 2 _^'2_ ^2= ; 
 
 (4). . . (a; - 0) (i|/'7r" - 7r'-^") + (y - r//) (ttV" - ^V") +(iz-7r) {<p'i>"- ^'<p") = ; 
 
 whereof the first three have been employed by Monge himself, but the fourth does 
 not seem to have been perceived by him, the condition of evanescence of a radical 
 having been used in its stead. And by a translation of quaternion results, above 
 deduced, into the usual language of analysis, it is found that the complete and gene^ 
 ral integral, of the non-linear differential equation of the second order, which is ob- 
 tained by the elimination of x, y, z between these four, is expressed by a new system 
 of four equations, the equation (1) beuig one of them ; and the three others, in which 
 X, y, z are now tr&ated as arbitrary functions of a, and are derivated as such, being 
 the following : 
 
 (5). . . (a; - 0) a;' -f (y - ^//)y' + (z - 7r) 2' = ; 
 
 (6). . . (a; - fjx" + (y - >//)y"+ (z - tt) z" + x'i + y'^ + z'a = ; 
 
 (7). . . ix-ip)x"' ■\-{y-^)y"' + iz-'7r)z"'^ 3;(a; V + //' + z'z") = 0. 
 
 By treating a as a function of some other independent variable, t^ the terms + a and 
 + 1, in (2) and (3), come to be replaced by + aa' and + aa" + a'^ ; and the slightly 
 more general form, which Monge's Equation thus assumes, has still its complete 
 general integral assigned by the system (1) (5) (6) (7), if x, y, z (as well as a) be 
 now regarded as arbitrary functions of the new variable t, in the place of which it is 
 permitted (for instance) to take x, and so to write x' = 1, or" = : only two arbitrary 
 functions thus entering, in the last analysis, into the general solution, as was to be 
 expected from the form of the equation. 
 
 * ThQ particular ijitegral corresponding, of the Equation of Monge, is expressed 
 by the following system : 
 
 ^ = a + et+lu, \p=:b+ft + mu, '7r = e+gt + nu, 
 (et + luy + (/if + muy + {gt + nu^ = a'^ ; 
 
 abcdefglmn being nine arbitrary constants, while t and u are two functions of a, 
 whereof one is arbitrary, but the other is algebraically deduced from it, by means of 
 the fourth equation. The writer is not aware that either of these integrals has been 
 assigned before. 
 
CHAP. III.] VECTOR EQUATION OF LOCUS OR ENVELOPE. 611 
 
 the second scalar coefficient, u, being here an arbitrary function of the first 
 scalar coefficient, or of the independent variable #, and a, |8, y being three arbi- 
 trary but constant vectors : so that the curve (s) is now obliged to lie in some one 
 plane* through the fixed point A, but remains in other respects arbitrary. Accord- 
 ingly it will be found that this last integral system, although less general than the 
 former system (102.), and not properly included in it, satisfies the differential equa- 
 tion CXXXVI. ; whereof the two members acquire, by the substitutions indicated, 
 this common value, 
 
 CLXXV. . . (i2S(T'(T" - R'gy = &c. = RH"- (tu' - uj u"^ (Vj3y)4. 
 (105.) Other problems might be proposed and resolved, with the help of formulsef 
 already given, respecting the properties or affections of curves in space which depend 
 on \h& fourth power (s**) of the arc, or on ^e fourth derivative Ds^p or r"' of the vec- 
 tor ps', but it is time to conclude this series of sub -articles, which has extended to a 
 much greater length than was designed, by observing that, in virtue of the vector 
 form 396, XL for the equation of a circle of curvature, the Locus (8.) of the Oscu- 
 lating Circle may be concisely but sufficiently represented by the Vector Equation, 
 
 CLXXVL.. V-^+i/,= 0, 
 
 lO- Ps 
 
 * Compare the Note to page 606. 
 t "We might for example employ the formula VI. for k", in conjunction with 
 one of the expressions 397, XCL for »c', to determine, by the general formula 389, 
 IV., the vector (say ^) of the centre of curvature of the curve (k), and therefore also 
 the radius of curvature of that curve, which is the locus of the centres of curvature of 
 the given curve (p), supposed to be in general one of double curvature. After a few 
 reductions, with the help of XII., we should thus find the equations, 
 
 CLXXVII. . . V -, = ^^ + (r-i - P') T, 
 
 K rK 
 
 CLXXVIIL..^ = K:+-A7=fc+ ''~^'' + '* 
 
 k' ds r6.K 
 
 in which last the denominator is a quaternion, and the scalar variable is arbitrary 
 
 whence also, 
 
 CLXXIX. . . Radius of curvature of curve (k), 
 or of locus of centres of osculating circles to a given curve (p) in space, 
 
 ■Rdr a 1 dpy 
 
 pds \\x ds 
 with the verification, that for the case of a plane curve (p), for which therefore 
 
 — = 1, and - = = — , we have thus the elementary expression, 
 p T ds 
 
 rdr 
 CLXXX. . . Radius of Curvature of Plane Evolute = + — -, 
 
 ds 
 
 r l)eing still the radius of curvature, and s the arc, of the given curve. 
 
 *-— THi)r 
 
612 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 which apparently involves only one scalar variable, s, namely, the arc of the curne 
 (f), the other scalar variable, such as t, which corresponds (69.) to the arc of the cir- 
 cle, disappearing under the sign V : and that the surface, which was called in (8.) 
 the Circumscribed Developable, is now seen to be in fact circumscribed to that Lo- 
 cus, or Envelope, in a certain singular (or eminent) sense, as touching it along its 
 Singular Line. 
 
 399. When we take account of the fifth power {s^) of the are, 
 the expression for p^ receives a new term^ and becomes (comp. 
 398, L), 
 
 I. ../>. = P + ST + is^T' + |s3t'' + -,\5*t''' + rio^V- ; 
 
 and although some of the consequences of such an expression have 
 been already considered, especially as regards the general determi- 
 nation of what has been above called the Osculating Twisted Cubic 
 to a curve of double curvature, or the gauche curve of the third de- 
 gree which has contact of the fifth order with a given curve in space, 
 yet, without repeating any calculations already made, some addi- 
 tional light may be thrown on the subject as follows. 
 
 (1.) As regards the successive deduction of the derived vectors in the formula I., 
 it may be remarked that if we write (comp. 398, LVI., LXI.), 
 
 II. . . DV'p = r^") = OnT + bfJ-T + CnrVy 
 
 we shall have, generally, 
 
 III. . . a„vi = a'n-r-i6„, 6n+i = 6n + r-ia„-r-'c„, tf«+i =c'„ + ri6„, 
 with the initial values, 
 
 IV. ..ao=l, 6o = 0, co=0, or IV'. ..ai = 0, 6i = r-i, ei = 0; 
 whence V /"2 = -'--^ &2 = ('-0', H=r-'x-\ 
 
 \a3 = 3r-3r', 63 = (r-i)"" »-"' -'-^^"^ C3 = r(r-2r-0', 
 as in the expressions 397, VI. for t", and 398, IV. for t" \ the corresponding co- 
 efficients of t'^ being in like manner found to be, 
 
 /ai = - 2 (r-2)" + ((r-i)')2 + r-2(r-2 + r-2) ; 
 VI. . . )54 = (r-i)"'-2(r-»)'-3(r-Vi)'r-i; 
 
 ( C4 = r-i (r-i)" + 3 ((r-i)'r-O' - »'"*r"' (^-^ + ^2) ; 
 and being sufficient for the investigation of all affections or properties of a curve in 
 space, which depend only on thej^/M power of the arc s. 
 
 (2.) For the helix the two curvatures are constant, so that all the derivatives of 
 the two radii r and r vanish ; the expressions become therefore greatly simplified, 
 and a law is easily perceived, allowing us to sum the infinite series for ps, and so 
 to obtain the following rigorous expressions for the co-ordinates* Xs, y*, Zs of this 
 
 * We have here, and in this whole investigation, an instance of the facility with 
 which quaternions can be combined with co-ordinates, whenever the geometrical na- 
 
CHAP. III.] OSCULATING TWISTED CUBIC TO HELIX. 613 
 
 particular curve, instead of those which were developed generally in 398, LVIII., 
 but only as far as s* inclusive : 
 
 VII. , .Xs = P (x-H + r-2 sin t); y, = IH'^ vers t; Zs= Pr-^x-^ (t - sin t) ; 
 where I and t are an auxiliary constant and variable, namely, 
 
 VIII. . . ? = (r-2 + r-2)-J = rsmH, t = Z's, 
 I being thus what was denoted in earlier formulae by T\-i, and t being the angle be- 
 tween two axial planes ; while the origin is still placed at the point p of the curve, 
 and the tangent, normal, and binorraal are still made the axes of xyz. 
 
 (3.) The cone of the second order, 398, (40.), which has generally a contact of 
 the fifth order with a proposed curve in space, at a point p taken for vertex, has in 
 this case of the AeZta; the equation (comp. 398, LVII.* and LXIX.), 
 
 ,„ 3rr /3r T r \ ^ 
 
 IX. ..y^ = --<x-v\ —-\z}z. 
 
 ^ 2r\ VlOr lOr j J 
 
 Accordingly it can be shown, by elementary methods, that if we write, for a mo- 
 ment, 
 
 X. . . f(f) = 3 (« - sin (3< + 7 sin - 20 vers2 t, 
 
 we have the eight evanescent values, 
 
 XL . . /O =/'0 =/"0 =/"'0 =/'^0 =/^0 =p'0 =p"0 = ; 
 
 whence it is easy to infer that this cone IX. has (in the present example, although 
 not generally) a contact as high as the slvth orderf with the curve, of which the 
 co-ordinates have here the expressions VII. ; and consequently that the cone in ques- 
 tion must wholly contain the osculating twisted cubic to that curve. 
 
 ture of a question may render it convenient so to combine them, by offering to our 
 notice any obvious planes of reference. If it be thought useful to pass to a system 
 connected more immediately with the right cylinder than with the helix, we may 
 write, 
 
 !Xs = l(r'^Xa - r-'zs) = ^2r-i sin t, 
 y^ = pir-i -ys = ZV-i cos t, 
 
 Zs = l(T-^Xs + r-izs) = l^r-^t, 
 
 where l^r~^ = r sin2 H is the radius of the cylinder, with converse formnlse easily as- 
 signed. 
 
 * In the corresponding equation 398, LXVII., the coefficient of 6ac ought to 
 
 have been printed as I - | , like the coefficient of Qxz in the equation LVII. 
 
 f Or in modern language, seven-point contact, in the sense that the cone passes, 
 in this case, through seven consecutive points of the curve. It may be remarked 
 that the gauche curve of the fourth degree, or the quartic curve, in which this cone 
 cuts the cylinder of revolution whereon the helix is traced (cutting also in it a cer- 
 tain other cylinder of the second order), and which has the point p for a double point, 
 crosses the helix by one of its two branches at that point, while it has seven-point 
 contact with the same helix by its other branch ; and that thus the fact of calcula- 
 tion, expressed by the formula XI., is geometrically accounted for. 
 
614 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (4.) In general, to find a second locus for such a cubic curve, the method of 
 recent sub-articles (398, (38.) &c.) leads us to form the equation (398, LXVI.) 
 of a. cylinder of the second order, or briefly of a quadric* cylinder, which like 
 the quadric cone (3.) shall have contact of the fifth order with the proposed 
 curve in space, at the given point p; the ratios of ahc, which determine the 
 direction of a generating line pe, being obliged for^this purpose to satisfy a certain 
 equation of condition (398, LXVIIL), of which the /or?n indicates that the locus of 
 this line pe is generally a certain cubic cone, having the tangent (say pt) to the 
 curve for a nodal side : along which side it is touched, not only (like the quadric 
 cone) by the osculating plane (z = 0) to that given curve, but also by a second 
 plane, whereof the equation (gy + hz = 0, or after reductions y — |r'z = 0) shows 
 that the second branch of the cubic cone crosses the first branch, or the quadric cone, 
 or the osculating plane to the curve, at an angle of which the trigonometric cotan- 
 gent is equal to half the dijff'ere7itial of the radius (r) of second curvature, divided 
 by the differential of the arc (s) ; so that this second tangent plane to the cone coin- 
 cides with the rectifying plane to the curve, when the second curvature happens to be 
 constant. The tangent pt therefore counts as three of the six common sides of the 
 two cones with p for vertex : and the three other common sides, for the assigning of 
 which it has been shown (in 398, (41.)) how to form a cubic equation in b: c, are 
 the parallels from that point p to the three real or imaginary asymptotes^ of the 
 twisted cubic, and are generating lines pe of three quadric cylinders, whereof one at 
 least is necessarily real, and contains, as a second locus, that sought osculating gauche 
 curve of the third degree. 
 
 (5.) In applying this general method to the case of the helix, it is found that the 
 cubic cone breaks up, in this example, into a system of a new quadric cone, which 
 touches the former quadric cone IX. along the tangent pt to the curve (the two other 
 common sides of these two cones being imaginary'), and oi a. plane (y = 0), namely 
 the rectifying plane (comp. (4.)) of the helix, or the tangent plane to the cylinder of 
 revolution on which that given cm-ve is traced : and that this last plane cuts the 
 first quadric cone in two real right lines, the tangent being again one of them, and 
 the other having the sought direction of a real asymptote to the sought osculating 
 twisted cubic. Without entering here into details of calculation, the resulting equa- 
 tion of the realX quadric cylinder, on which that sought gauche curve is situated, 
 may be at once stated to be (with the present system of co-ordinates). 
 
 * So called by Dr. Salmon, in his Treatise already cited. Compare the first 
 Note to page 591 of these Elements. 
 
 f Compare again the Note last referred to. 
 
 X As regards the two imaginary quadric cylinders, their equations can be formed 
 by the same general method, employing as generating lines the two imaginary com- 
 mon sides (5.), of the cone IX., and of that other quadric cone above referred to, 
 which is here a separable part of the general cubic locus, and has for equation, 
 
 20 r / r2 \ 
 
 It seems suflScient here to remark, that by taking the sum and difference of the equa- 
 tions of those two imaginary cylinders, two new real quadric surfaces are obtained, 
 
CHAP. III.] DEVIATION OF CUBIC FROM HELIX. 615 
 
 in such a manner that if we set aside the right line, 
 
 XIII..., =0, . + (^i_l:]. = o, 
 
 which is a common side of the cone IX. and of the cylinder XIT., the curve, which is 
 the remaining part of their complete intersection, is the twisted cubic sought. As an 
 elementary verification of the fact, that this gauche curve of intersection IX. XII, 
 has contact of the fifth order with the helix at the point P, it may be observed that 
 if we change the co-ordinates xyz in XII. to the expressions VII., and write for 
 abridgment, 
 
 XIV. . . F{t) = (3< + It sin tf - 200 vers t + 60 vers2 1, 
 
 we have then (comp. X. XI.) the six evanescent values, 
 
 XV. ., FO = F'Q = F"0 = F"'0 = Fi^O = F^O = 0. 
 
 (6.) As another verification, which is at the same time a sufficient /jroo/J of the 
 a posteriori kind, that the gauche curve IX. XII. has in fact contact of the fifth or- 
 der with the helix, it can be shown that while the co-ordinates y^ and 23 of the latter 
 may (by VII., writing simply x for Xs, and neglecting x^) be thus developed, 
 
 ^'~2r'^ 24rY^ ^/"^T^VTi" r2r2 + 7^' 
 
 x5 I 9 1 
 
 XVI. . . 
 
 l~'~ 6rr ' 120rr\r^ r^ 
 
 the corresponding co-ordinates y and z of the former, that is, of the curvilinear part 
 of the intersection of the cone IX. with the cylinder XII., have (in the same order 
 of approximation) developments which may be thus abridged, 
 
 (r-2 + r-2V -^6 
 
 (7.) The deviation of the helix from the gauche curve IX. XII. is therefore of 
 the sixth order (with respect to x, or s), and it has an inward direction, or in other 
 words, the osculating twisted cubic deviates outwardly from the helix, with respect 
 to the right cylinder ; the ultimate (or initial) amount of this deviation, or the law 
 according to which it tends to vary, being represented by the formula, 
 XVir u (r-2 4»-2)2,6 _,4y. 
 
 which also contain the osculating twisted cubic, and intersect each other in that 
 gauche curve : namely two hyperbolic paraboloids, which have a common side at in- 
 finity, and of which the equations can be otherwise deduced (by way of verifica- 
 tion), without imaginarieSf through easy algebraical combinations of the two real 
 equations IX. and XII. 
 
616 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 where t denotes as in (2.) the angle, which a plane drawn through a near point p«, 
 and through the axis of the right cylinder* 
 
 XVIII.. .2r3,= (x-^.)'+[n-^y, 
 
 whereon the helix is traced, makes with the plane drawn through the same axis of 
 revolution, or through the right line, 
 
 XIX. . .x = -z, y = r-i(r;2-i-r-2)-i = i2y-i, 
 
 and through the given point p : while ys is still the (inward) distance of the same 
 near point Pj, from the tangent plane to the same cylinder at the same given point p. 
 (8.) If we cut the cone IX., and the cylinder XII., by any plane, 
 
 XX...2^=»{.+ (ll-l'-)4, 
 
 drawn through their common side XIII., we obtain two other sides, one for each of 
 these two quadric surfaces ; and these two new right lines, in this plane XX., inter- 
 sect each other in a a new point, f of which the co-ordinates xyz are given, as func- 
 tions of the new variable w, by the three fractional expressions^X 
 
 Ar2 r^l60 
 
 w + \ — 
 ^^T \r2 r^ j60 ^ Mj2 
 
 w2 ' ^ 3 m;2 ' 3 m;« ' 
 
 20/2 20/2 '■^ 20 /3 
 
 while the twisted cubic, which osculates (as above) to the helix at p, is the locus of 
 all the points of intersection thus determined. Accordingly, if we develop xyz by 
 XXI., in ascending powers oiw, neglecting w^ (or z^), we are conducted, by elimi- 
 nation of w, to expressions for y and z in terms of x, which agree with those found 
 in (6.), and thereby establish in a new way the existence of the required contact of 
 the fifth order, between the two curves of double curvature. 
 
 * With the co-ordinates VII'. of a recent Note (to page 612), the equation of 
 this cylinder would be, 
 
 XVIir. ..x2 + y2 = ZV-2. 
 
 t The plane XX., as containing the line XIII., is parallel to an asymptote, 
 and therefore meets the cubic at infinity ; it also passes through the given point p : 
 and therefore it can only cut the twisted cubic in one other point, of which the posi- 
 tion is expressed by the equations XXI. 
 
 J Quaternions suggest such fractional expressions, through the formula 398, 
 LXXIX. for the vector (0 4-c)"*a; but it is proper to state that expressions of 
 fractional form, for the co-ordinates of a curve in space of the third order (or degree) 
 were given by Mobius, who appears to have been the first to discover the existence 
 of such gauche curves, and who published several of their principal properties in his 
 Barycentric Calculus (der barycentrische Calcul, Leipzig, 1827). Compare the 
 Notes to pages 23 and 35, and Note B at the end of these Elements. 
 
CHAP. III.] REAL AND IMAGINARY ASYMPTOTES. 617 
 
 (9.) The real asymptote to the cubic curve is found by supposing the auxiliary 
 variable w to tend to infinity in the expressions XXI. ; it is therefore the right line 
 (comp. XX.), 
 
 XXII.. .y = --, .+ (^---^^_).= 0, 
 
 namely the second side in which the elliptic cylinder XII. is cut by a normal plane 
 through the side XIII. ; and by comparing the value of its y with the equation 
 XIX., we see that the least distance between the real asymptote to the osculating 
 twisted cubic, and the axis of reiwlution of the cylinder on which the helix is traced, 
 is equal to seven-thirds of the radius of that right cylinder. 
 
 (10.) As regards the two imaginary asymptotes, they correspond to the two ima- 
 ginary values of «?, which cause the common denominator of the expressions XXI. to 
 vanish ; but it may be suflBcient here to observe, that because those expressions give, 
 generally, 
 
 XXIIL . . x+l-- + l-]z = w, 
 \5r 5 r / 
 
 the two imaginary lines in question are to be considered as being contained in two 
 imaginary planes, which are both parallel to the real plane* through P, 
 
 XXIV...;.+ (f- + ^-V = 0; 
 \5r 5r / 
 
 namely to a certain common normal plane to the two real cylinders XII. and XVIII., 
 or to the elliptic and right cylinders already mentioned, 
 
 (11.) /« ^enera^, instead of seeking to determine, as above, a. cylinder of the 
 second order, which shall have contact of the fifth order with any given curve of 
 double curvature, at a given point P, we may propose to find a second cone of the 
 same (second) order, which shall have such contact with that curve at that point, 
 its vertex being at some other point of space (a6c). Writing (comp. 398, LXVI.) 
 the equation of such a cone under the form, 
 
 XXV. . . 2r(cy - bz) (c - 2) = {ex - azj- + 25(ca; - az) (cy - bz) + C{cy - bz^ ; 
 
 substituting for xyz the co-ordinates Xsy^Zg of the curve, under the forms (comp. 
 398, LVIIL), 
 
 XXVI. 
 
 in which the coefficients 036303 and 04640^ have the values assigned in (1.) ; develop- 
 ing according to powers of s, neglecting s^, and comparing coefficients of s^, s"*, s*; 
 we find first the expressions, 
 
 * The right line at infinity, in this plane XXIV., is the common side of the 
 two hyperbolic paraboloids mentioned in the third Note to page 614, as each con- 
 taining the whole twisted cubic. 
 
 4k 
 
 Xs 
 
 ^s- 
 
 «3 
 
 67^ + 
 
 24 ' 120 
 
 ys 
 
 s-i 
 ~2r~ 
 
 r's^ 
 
 63S* 64«* 
 
 24 ' 120' 
 
 Zs 
 
 *3 
 
 = — 
 6rr 
 
 C3S* 
 '*"24 
 
 C4.5 
 
 "^ 120' 
 
618 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 which are the same for cone as for cylinder : and then are led to the new equation of 
 condition, 
 
 r I h \ a 2 I h 2 2a\ 
 
 XXVIIL. -U,--C4 =a3--C3+ +5U3--C3---— ) 
 
 Q\ c j c ^ err \ c r3 cr^r } 
 
 which differs from the corresponding equation for the determination of a cylinder 
 having the same (fifth) order of contact with the curve, but only by the one term 
 
 2 
 
 — in the second member, which term vanishes when the co-ordinate c of the vertex 
 err 
 
 is infinite. 
 
 (12.) Eliminating B and C, and substituting for 036303 and 046404 their values 
 
 V. and VI., we find that the condition XXVIII. may be thus expressed (comp. 
 
 398, LXVIII.) : 
 
 XXIX. .. acf 6 - 1- c J - re2 = a63 4- b62c + c6c2 -(- ec^ ; 
 
 in which we have written, for abridgment, 
 
 _ 4 r V _ ** ^ ^ 
 
 "9r' "3~r2' 
 
 XXX. . . < o= — (6r"r- 3rr" - 2r-ir'^r - 6r'r' + 6rr-ir'2 - ISr'ir + 12rr-') ; 
 e = — (9r"'r2 - 9r-ir'r"r2 + 4r-2r'3r2 + 36r-2r'r2 + 18r' - 27rr-ir'). 
 
 yu 
 
 The locus of the vertex of the sought quadric cone XXV. is therefore that cubic sur- 
 face, or surface of the third order, which is represented by the equation XXIX. in 
 abc ; this surface, then, is a second locus (comp. (4.)) for the osculating twisted cu- 
 bic, whatever the ^fiwen curve in space may be; a. first locus for that cm6ic curve 
 being still the quadric cone (comp. (3.)), of which the equation in abc is (by 398, 
 LXVIL* and LXIX.), 
 
 XXXI...4(^).^ = 6(^)^ac + (^)V 
 
 r*/ 9 21 r'2 3r" 3r'r' 27r'2 
 5\H~r«r2''"r*~73~'''737~ 4r2r2 
 
 and which has contact of the ffth order with the curve, while its vertex is at the 
 given point p of osculation. 
 
 * After making the correction indicated in a former Note (to page 613), so as to 
 bring the cited equation into agreement with the earlier formula 398, LVII. The 
 quadric cone XXXI. may be said to have five-side contact with the cone of chords 
 of the given curve (compare the first Note to page 588). 
 
CHAP. III.] RULED CUBIC LOCUS, CUBIC CYLINDER. 619 
 
 (13.) Instead of thus introducing, as data, the derivatives of the two radii of 
 curvature, r and r, taken with respect to the arc, s, it may be more convenient in 
 many applications to treat the two co-ordinates y and z of the curve as functions of 
 the third co-ordinate x, assumed as the independent variable : and so to write 
 (comp. (6.)) these new developments, 
 
 and then the equation of the quadric cone XXXI. will be found to become (in xyz), 
 
 XXXIII. . .y^^-~xz^ 1gyz^hz\ 
 with the coeflScients, 
 
 XXXIV... 5' = rr(y"'-^rz-), h = \rx'^-[y^- - ^^xz^\^ 
 
 - r2r2 f y"'2 + ^ rz'V" - ^ r^z^^ ; 
 
 while the cubic surface XXIX. will also come to be represented by an equation of 
 the same form as before, namely (in xyz) by the following, 
 
 XXXV. . . ajz (y + hz) - rz2 = ay^ + by2z + cyz"^ + e^^, 
 
 in which the coefficients are, 
 
 r a = - ^ (as before) ; b = - g rY' + y ?'^; h = - rry '" + |rr«2'^ ; 
 
 = \ rhy'"^ - ^rhY'z" - |r2ry'^ + ^r^vH^ ; 
 
 XXXVI. . u 
 
 4 
 e = - - r4r2y"'3 + ir3r2y"y^ - ^^rh^. 
 
 (14.) Whichever set of expressions for the coefficients we may adopt, some ge- 
 neral consequences maybe drawn from the mere /or/«s of the equations, XXXI. 
 and XXIX., or XXXIII. and XXXV., of the quadric cone and cubic surface, con- 
 sidered as two loci (12.) of the osculating twisted cubic to a given curve of double 
 curvature. Thus, if we eliminate ac (comp. 398, (41.)) from XXIX. by XXXI., 
 or xz by XXXIII. from XXXV., we get an equation between 6, c, or between y, z, 
 which rises no higher than the third degree, and is of the/orm, 
 
 XXXVII. . . 2rz^ = ay3 + h^^z + cj/z^- + e^z^, 
 
 with the same value of a as before ; such then is the equation of the projection of 
 the twisted cubic, on the normal plane to the curve; and we see that, as was to be 
 expected, the plane cubic thus obtained has a cusp at the given point p, which 
 (when we neglect «' or x"^) coincides with the corresponding cusp * of the projec- 
 tion of the given curve of double curvature itself, on the same normal plane. 
 
 (15.) The equation XXXVII. may also be considered as representing a cubic 
 cylinder, which is a third locus of the twisted cubic ; and on which the tangent pt 
 
 Compare the first formula of the first Note to page 594. 
 
620 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 to the curve is a cusp-edge^ in such a manner that an arbitrary plane through this 
 
 line, suppose the plane 
 
 XXXVIII. .. BTZ = vt/, 
 
 where v is any assumed constant, cuts the cylinder in that line twice, and a third 
 time in a real and parallel right line, which intersects the quadric cone in a point at 
 infinity (because the tangent pt is a side of that cone), and in another real pointy 
 which is on the twisted cubic, and may be made to be any point of that sought curve, 
 by a suitable value of v : in fact, the plane XXXVIII. touches both curves at p, and 
 therefore intersects the cubic curve in one other real point. And thus may fractional 
 expressions (comp. (8.)) for the co-ordinates of the osculating cubic be found gene- 
 rally, which we shall not here delay to write down. 
 
 (16.) Without introducing the cubic cylinder XXXVII., it is easy to see that 
 any plane, such as XXXVIII., which is tangential to the given curve at P, 
 cuts the cubic surface XXXV. in a section which may be said to consist of the 
 tangent twice taken, and of a certain other right line, which varies with the 
 .direction of this secant plane, so that the locus XXXV. or XXIX. is a Ruled 
 Cubic Surface, with the given tangent pt for a singular* line, which is in- 
 tersected by all the other right lines on that surface, determined as above : and if we 
 set aside this line, the remaining part of the complete intersection of that cubic sur- 
 face with the quadric cone XXXIII. or XXXI, is the twisted cubic sought. We 
 may then consider ourselves to have completely and generally determined the Oscu- 
 culating Twisted Cubic to a curve of double curvature, without requiring (as in 398, 
 (41.)), the solution of any cubic or other equation.^ 
 
 (17.) As illustrations and verifications, it may be added that the general ruled 
 cubic surface, and cubic cylinder, lately considered, take for the case of the helix 
 (2.), the particular forms, + 
 
 * If the cubic surface be cut by a plane perpendicular to the tangent pt, at any 
 point T distinct from the point p itself, the section is a plane cubic, which has T for 
 a double point ; and this point counts for three of the six common points, or points of 
 intersection, of the plane cubic just mentioned with the plane conic in which the 
 quadric cone is cut by the same secant plane, because one branch, or one tangent, 
 of the plane cubic at t touches the plane conic at that point, in the osculating plane 
 to the given curve at p, while the other branch, or the other tangent, cuts that plane 
 conic there. 
 
 f It may be remarked that, by equating the second member of XXXVII. to 
 zero, and changing y, z to b, c, we obtain generally the cubic equation, referred to 
 in 398, (41.) ; and that by suppressing the term — rc2 in XXIX., or the term — rz' 
 in XXXV., we pass, in like manner generally, from the cubic surface of recent sub- 
 articles, to the earlier cubic cone (4.). 
 
 t By suppressing the term — tz^, dividing by — , and transposing, we pass for the 
 
 or 
 
 case of the helix from the equation XXXIX. of the cubic locus, to the equation IX'. 
 
 in the last Note to page 614 ; namely to the equation of that quadric cone which forms 
 
 (in this example) a separable part of the general cubic cone, the other part being here 
 
 the tangent plane (y = 0) to the right cylinder. 
 
CHAP. III.] INVOLUTES AND EVOLUTES IN SPACE. 621 
 
 and 
 
 and that accordingly these two last equations are satisfied, independently of w, when 
 the fractional expressions XXI. are substituted for xyz. 
 
 400. The general theory"^ of evolutes of curves in space may be 
 briefly treated by quaternions, as follows : a second curve (in space, 
 or in one plane) being defined to bear to 2k first curve the relation of 
 evolute to involute, when the^r^^ cuts the tangents to the second at right 
 angles. 
 
 (1.) Let p and a be corresponding vectors, op and os, of involute and evolute, 
 and let p', a\ p", tr" denote their first and second derivatives, taken with respect to 
 a scalar variable t, on which they are both conceived to depend. Then the two fun- 
 damental equations, which express the relation between the two curves, as above 
 defined, are the following : 
 
 I. . . S((T-p)p'=0; II. . . V((T-p)(7'=0; 
 which express, respectively, that the point s is in the normal plane to the involute 
 at p, and that the latter point is on the tangent to the evolute at s : so that the locus 
 ofv (the involute) is a rectangular trajectory to all such tangents to the locus ofs 
 (the evolute). 
 
 (2.) Eliminating cr - p between the two preceding equations, and taking their 
 derivatives, we find, 
 
 III. ..SpV = 0, IV. . . S((r-p)p"-p'2 = 0, V. . . V(<r- p)(r"-VpV = 0; 
 whence also, VI. . . Sp'a'(T"= 0. 
 
 (3.) Interpreting these results, we Bee first, by IV. combined with I. (comp. 391, 
 (5.)), that the point s of the evolute is on the polar axis of the involute at p, and 
 therefore that the evolute itself ia some curve on the polar developable of the invo- 
 lute ; and second, by VI. (comp. 380, I.), that this curve is a geodetic line on that 
 polar surface, because the osculating plane to the evolute at s contains the tangent 
 to the involute at p, and thei'efore also the (parallel) normal to the locus of evolutes. 
 
 (4.) The locus of centres of curvature (395, (6.)) of a curve in space is not ge- 
 nerally an evolute of that curve, because the tangentsf kk' to that locus do not gene- 
 rally intersect the curve at all ; but a given plane involute has always the locus just 
 
 * Invented by Monge. 
 
 t It might have been remarked, in connexion with a recent series of sub-arti- 
 ticles (397), that this tangent kk' or k' is inclined to the rectifying line X, at an an- 
 gle of which the cosine is, 
 
 - SU/c'\ = + i2-iT\-i = ± sin ^cos P ; 
 upper or lower signs being taken, according as the second curvature r-' is positive 
 or negative, because Sic'X = - ^"^ 
 
622 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 mentioned for one. of its evolutes ; and has, besides, indefinitely many others* which 
 are all geodetics on the cylinder which rests perpendicularly on that onQ plane evolute 
 as its base. 
 
 (5.) An easy combination of the foregoing equations gives, 
 
 VII. . . (T((T - p)y = - S (U((r - p) . ((t' - p')) = + Sj'JJct' = ± Ttr', 
 or with diflferentials, VIII. . . dT((T - p) = ± Tda ; 
 
 whence by an immediate integration (comp. 380, XXII. and 397, LIV.), 
 
 IX. . . AT((T-p) = ±jTd(r = + arc of the evolute: 
 this arc then, between two points such as s and Si of the latter curve, is equal to the 
 difference between the lengths of the two lines, PS and PiSi, intercepted between the 
 two curves themselves. 
 
 (6.) Another quaternion combination of the same equations gives, after a few 
 steps of reduction, the differential formula (comp. 335, VI.), 
 
 X. . . dcosoPS=-dSU ^-l£= _^IP-,.S^; 
 
 p ^C<^-p) p 
 
 if then the involute be a curve on a given sphere, with its centre at the origin o, so 
 that the evolute is a geodetic on a concentric cone, this differential X. vanishes, and 
 we have the integrated equation, 
 
 XI. . . cos OPS = const., or simply, XI'. . . ops = const. ; 
 the tangents ps to the evolute being thus inclined (in the case here considered) at a 
 constant angle,-]- to the radii op of the sphere. 
 
 (7.) In general, if we denote by R the interval ps between two corresponding 
 points of involute and evolute, we shall have the equation, 
 
 XII. . . ((T-p)2+i22=o, or Xir. . . T((r-p) = i?; 
 and the formula VIL may be replaced by the following, 
 
 XIII. . . i2'2 + (7'2 = 0, or Xlir. . . D«i2 = + TD<(r, 
 in which the independent variable t is still left arbitrary. 
 
 (8.) But if we take for that variable the arc SoS* of the evolute, measured from 
 some fixed point of that curve, we may then write, 
 
 XIV. . . * = JTd(r, XV. . . di2, = + d*, XVI. . . 'DtRt=-± 1 ; 
 
 * Compare the first N'ote to page 534; from the formulae of which page it now 
 appears, that if the involute be an ellipse, with /3 = ob and y = oc for its major and 
 minor semiaxes, and therefore with the scalar equations, 
 
 (S/3-V)2 + (SrV)2=l, S/3yp = 0, 
 
 the evolutes are geodetics on the cylinder of which the corresponding equation is, 
 
 (S/3^)l+(SyO^ = 032-y2)S. 
 
 t This property of the evolutes of a spherical curve was deduced by Professor 
 De Morgan, in a Paper On the Connexion of Involute and Evolute in Space (Cam- 
 bridge and Dublin Mathematical Journal for November, 1851); in which also a 
 definition of involute and evolute was proposed, substantially the same as that above 
 adopted. 
 
CHAP. III.] INVOLUTES AS LIMITS OF ENVELOPES. 623 
 
 whence 
 
 XVIL . . Bt (Ht +0 = 0, and XVIII. . . Et + t = const. = Hq, 
 the integral IX. being thus under a new form reproduced. 
 (9.) In this last mode of obtaining the result, 
 
 XIX. . . A PS = -R« — -Ro = + ' = ± «»'c SoSt of evolute, 
 no use is made of infinitesimals,* or even of small differentials. We only infer, as 
 in XVIII. (comp. 380, (9.)), that the quantity Et + t is constant,^ because its deriva- 
 tive is null: it having been previously proved (380, (8.)), as a consequence of our 
 definition of differentials (320, 324) that if « be the arc and p the vector of any 
 curve, then the equation d« = Tdp (380, XXII.) is rigorously satisfied, whatever 
 the independent variable t may be, and whether the two connected &.ndi simultaneous 
 differentials be small or large. 
 
 (10.) But when we employ the notation of integrals, and introduce, as above, 
 the symbol jTd*, we are then led to interpret that symbol as denoting the limit of a 
 sum (comp. 345, (12.)) ; or to write, generally, 
 
 XX, . . J Tdjo = lim. STAp, if lim. Ap = 0, 
 with analogous formulae for other cases of integration in quaternions. Geometri- 
 cally, the equation, 
 
 XXI. . . J Tdjo = As, or XXI'. . . J Td(r = A ^, 
 if s and t denote arcs of curves of which p and a are vectors, comes thus to be in- 
 terpreted as an expression of the well-known principle, that the perimeter of any 
 curve (or of any part thereof) is the limit of the perimeter of an inscribed polygon 
 (or of the corresponding portion of that polj'gon), when the number of the sides is 
 indefinitely increased, and when their lengths are diminished indefinitely. 
 
 (11.) The equations I. and XII. give, 
 
 XXII. . . S(t' {(t-p)^ RR' = 0, 
 the independent variable t being again arbitrary ; but these equations XII. and 
 XXII. coincide with the formulae 398, LXXXIX. and XCI. ; we may then, by 
 398, (79.) and (80.), consider the locus of the point p as the envelope of a variable 
 sphere, namely of the sphere which has s for centre and R for radius, and is repre- 
 sented by the recent equation XII., if p = op be the vector of a variable point 
 thereon. 
 
 (12.) But whereas such an envelope has been seen to \>q generally a surface, which 
 is real or imaginary (398. (79.)) according as R"^ + (t'2 < or > 0, we have here by 
 XIII. the intermediate or limiting case (comp. 398, CXXXI.), for which the circles 
 
 * In general, it may have been observed that we have hitherto abstained, at 
 least in the text of this whole Chapter of Applications, from making any use of 
 infinitesimals, although they have been often referred to in these Notes, and employed 
 therein to assist the geometrical investigation or enunciation of results. But as 
 regards the mechanism of calculation, it is at least as easy to use infinitesimals in 
 quaternions as in any other system : as will perhaps be shown by a few examples, 
 farther on. 
 
 t Compare the Note to page 516. 
 
624 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 of the system become points, and the surface itself degenerates into a curve, which is 
 here the involute (p)above considered. The involutes of a given curve (s) are there- 
 fore included, as a. limit, in that general system of envelopes which was considered in 
 the lately cited subarticles, and in others immediately following. 
 
 (13.) The equation of condition, 398, CXXXVI., is in this case satisfied by 
 XIII., both members vanishing; but we cannot now put it under the form 398, 
 CXLI., because in the passage to that form, in 398, (85.), there was tacitly efiected 
 a division hy r^, which is not now allowed, the radius r of the circle on the envelope 
 being in the present case equal to zero. For a similar reason, we cannot now divide 
 hy g, as was done in 398, (86.); and because, in virtue of IL, the two equations 
 398, CLX. reduce themselves to one, they no longer conduct to the formulae 398, 
 CLX'. CLXI. CLXr. CLXIII. XCIV. ; nor to the second equation 398, CLXII. 
 
 (14.) The general geometrical relations of the curves (p) and (s), which were 
 investigated in the sub-articles to 398 for the case when the condition* above re- 
 ferred to is satisfied, are therefore only very partially applicable to a system of invo- 
 lute and evolute in space : at least if we still consider the former curve (the involute) 
 as being a rectangular trajectory to the tangents to the latter (the evolute), instead 
 of being, like the curve (p) previously considered, a rectangular trajectory (398, (94.)) 
 to the osculating planesf of the curve (s). 
 
 * If, without thinking of evolutes, we merely suppose that the condition 398, 
 CXXXVI. is satisfied, as lately in (13.), by our having the relation i?'2 + ff'2= 0, 
 it will be fomid (comp. the symbolical expression 274, XX. for Oi, and the imagi- 
 nary solution in 353, (18.) of the system Syp = 0, p2= 0), that the envelope of the 
 sphere ((x - p)^ + i22 = 0, or the locus of the (null) circles in which such spheres are 
 (conceived to be) cut by the (tangent) planes. So*' (c — p) + RR'= 0, may be said to 
 be generally the system of all those imaginary points, of which the vectors (or the 
 bivectors, comp. 214, (6.)) are assigned by the formula, 
 
 p = (T - HH'-^a' + (Uff' + \/~l) Ya'fi ; 
 where fi is an arbitrary vector, and v — 1 is the old imaginary of algebra. By 
 making /i = we reduce this expression for p to the real vector form, 
 
 p = a-RR'-^a' = <t + RR'a'-\ 
 = the K of 398, CXXXI."; and thus the curve (p), which is here the locus of the 
 centres of the null circles of contact, and coincides with the involute in the present 
 series of sub-articles, may still be called a Singular Line upon the Envelope of the 
 Sphere (with One Variable Parameter), as being in the present case the only real part 
 of that elsewhere imaginary surface. 
 
 t The curve to the osculating planes of which another curve is thus an or- 
 thogonal trajectory, and which is therefore (398, (95.)) the cusp-edge of the polar 
 developable of the latter curve, was called by Lancret its evolute by the plane (de- 
 veloppee par le plan) ; whereas the curve (s) of the present series (400) of sub-ar- 
 ticles, to whose tangents the corresponding curve (p) is an orthogonal trajectoiy, has 
 been called by way of distinction the evolute by the thread (developpee par le fil) of 
 this last cui-ve. It would be improper to delay here on subjects so well known to 
 geometers : but the student may be invited to read again, in connexion with them, 
 the sub-articles (88.) and (89.) to Art. 398. 
 
CHAP. III.] CURVATURE OF HODOGRAPH OF EVOLUTE. 625 
 
 (15.) If the arc of the evolute be again taken for the independent variable t, and 
 if the positive direction of motion along that arc be always towards the involute, we 
 may write, 
 
 XXIII. .. p = ff + R<T', i2' = -l, <r'2=-l, &c.; 
 whence 
 
 XXIV. . . p' = Ra", p" = It(y"' - ff", Yp"p' = mYa"'a" ; 
 
 if then k = ok be the vector of the centre k of the circle which osculates to the invo- 
 lute at p, the general formula 389, IV. gives, after a few reductions,* the expression 
 (comp. 397, XVI. XXXIV., and XCVIII. (15)), 
 
 XXV. . . K = p+ -^,= (T +i2f (t' + 
 
 Ypp \ 
 
 (T + 
 
 Ya"-<T" 
 
 V(t"'<t" V(t"'(t"-> 
 
 = (T - iJri-JXr' = (T + UXi. R cos Hi, 
 if ri, JTi, and \i be what r, H, and X in 397 become, when we pass from the curve 
 (p) to the curve (s), with the present relations between those two curves ; this cen- 
 tre of curvature K is therefore the/oo< of the perpendicular let fall /rom the point p 
 of the involute, on the rectifying line Xi of the evohite : as indeed is evident from 
 geometrical considerations, because by (3.) this rectifying line of the curve (s) is the 
 polar axis of the curve (p). 
 
 (16.) If Ave conceive (comp, 389, (2.)) an auxiliary spherical curve to be de- 
 scribed, of which the variable unit- vector shall be, 
 
 XXVI.. . OT=T = a'=V(p-(T) = R-^{p-(T), 
 and suppose that v is the vector ov of the centre of curvature of this new curve, at 
 the point t which corresponds to the point s of the evolute, we shall then have by 
 XXV. the expression, 
 
 XXVII.. .T(T = v-r = -— -,= -— 7-7,= ^: =pk:ps; 
 Vr r V(7 0- R 
 
 we have therefore this theorem, that the inward radius of curvature of the hodograph 
 of the evolute (conceived to be an orbit described, as in 379, (9.), with a constant 
 velocity taken for unity) is equal to the inward radius of curvature of the involute, 
 divided by the interval It between the two curves (p) and (s) : and that these two 
 radii of curvature, TU and PK, have one common direction, at least if the direction 
 of motion on the evolute be supposed, as in (15.), to be towards the involute. 
 
 (17.) The following is perhaps a simpler enunciation of the theoremf just sta- 
 ted : — IfF, Pi, P2, . . and s, Si, Sg, . • be corresponding points of involute and evo~ 
 
 * Especially by observing that Vo-'Vd'V" = - (t"^, because StrV = 0, and Sa'cr'" 
 
 = -«t"2. 
 
 f Some additional light may be thrown on this theorem, by comparing it with 
 the construction in 397, (48.) ; and by observing that the equations 397, XVI. 
 XXXIV. give generally, in the notations of the Article referred to, for the vector of 
 the centre of curvature of the hodograph of any curve, the transformations, 
 
 r + —L = r + - = - r- iX"i = UX . cos /T. 
 
 VrV-i X 
 
 4 L 
 
626 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 lute, and if we draw lines STi || SiPi, ST2II S2r2, . • ^ith a common length = sp, the 
 ■spherical curve PT1T3 . . will then have contact of the second order with the curve 
 PP1P2 . ., that is with the involute at p. 
 
 401. The fundamental formula 389, IV., for the vector of the 
 centre of the osculating circle to a curve in space, namely the for- 
 mula, 
 
 ^•••'^^^■'Ayy' "" "•••'^ = ^^vdvd?' 
 
 which has been so extensively employed throughout the present 
 Section, has hitherto been established and used in connexion with 
 derivatives and differentials of vectors, rather than with differences, 
 great or small. We may however establish, in another way, an es- 
 sentially equivalent formula, into which differences enter by their 
 limits (or rather by their limiting relations'), namely, the following, 
 
 111. . , K-p + lim. ==r—-—-, if lim.A/>=0, and lim. — — = 0, 
 V A^pAp Ap 
 
 the denominator YA'^pAp being understood to signify the same thing 
 as V (AV . A/)) ; and then may, if we think fit, interpret the differen- 
 tial expression II. as if dp and d^/> in it denoted infinitesimals,^ of the 
 first and second orders : with similar interpretations in other but 
 analogous investigations. 
 
 (1.) If in the second expression 316, L., for the perpendicular from o on the line 
 AB, we change a and /3 to their reciprocals (comp. Figs. 58, 64) and then take the 
 reciprocal of the result, we obtain this new expression, 
 
 TV _ -_a-i-/3-i _ a(/3-cf)/3 _ OA.AB.OB 
 
 ■ ' • ^^ " ~ V/3-ia-^ Y(3a ~ V(ob.oa)' 
 
 in the denominator of which, ob may be replaced by ab, or by AG + ab, for the 
 diameter CD of the circle gab ; so that if c be the centre of this circle, its vector 
 y = oc = |oD = ^d = &c. Supposing then that p, q, r are any three points of any 
 given curve in space, while o is as usual an arbitrary origin, and writing 
 
 V. ..OP = (0, GQ = p+A|t), OR = p+2Ap+ A2|0, 
 
 and therefore 
 
 VI. ..PQ = Ap, QR = Ap+A2p, iPR = Ap + JA2p, 
 
 the centre c of the circle pqr has the following rigorous expression for its vector : 
 VII. . . GC==y = p+ ^KAp + AIPKAp +|AV) 
 
 Compare 345, (17.)> and the first Note to page 623. 
 
CHAP. III.] OSCULATING CIRCLES BY INFINITESIMALS. 627 
 
 whence passing to the limit, we obtain successively the expressions III. and II. for 
 the vector k of the centre of curvature to the curve pqr at p ; the two other points, 
 Q and R, being both supposed to approach indefinitely to the given point p, accord- 
 ing to any law (comp. 392, (6.)), which allows the two successive vector chords, pq 
 and QR, to bear to each other an ultimate ratio of equality. 
 
 *(2.) Instead of thus^rs< forming a rigorous expression, such as VII., involving 
 the differences Ap and A^p ; then simplifying the formula so found, by the rejection 
 of terms, which become indefinitely small, with respect to the terms retained ; and 
 finally changing differences to differentials (comp. 344, (2.))r, namely AptodjO, and 
 A^p to d2p, in the homogeneous expression which results, and of which the limit is to 
 be taken : we may abridge the calculation, by at once writing the differential sym- 
 bols, in place of differences, and at onee suppressing any terms, of which we foresee 
 that they must disappear from the final result. Thus, in the recent example, when 
 we have perceived, by quaternions, that if k be the centre of the circle pqr, the 
 equation 
 
 V{(qr-pq)pq} 
 is rigorous, we may at once change each of the three factors of the numerator to djO, 
 while the factor qr — pq in the denominator is to be changed to d^p ; and thus the 
 differential expression II., for the inward vector-radius of curvature k — p, is at 
 once obtained. 
 
 (3.) It is scarcely necessary to observe, that this expression for that radius, as a 
 vector, agrees with and includes the known expressions for the same radius of curva- 
 ture of a curve in space, considered as a (positive) scalar, which has been denoted in 
 the present Section by the italic letter r (because the more usual symbol p wotdd 
 have here caused confusion). Thus, while the formula II. gives immediately (be- 
 cause Tdp =ds) the equation, 
 
 IX. . . r-ids3 = TVdpd2|t), 
 
 it gives also (because dp^ =- ds3j and Sdpd^/o = - dsd^s) the transformed equation, 
 
 X. . . r-Jds2 = V(Td2p2 _ d««2) j 
 
 and it conducts (by 389, VI.) to this still simpler formula (comp. the equation r"i 
 = Tr', 396, IX.), 
 
 XI. . .r-id5 = TdUdp. 
 (4.) Accordingly, if we employ the standard trinomial form (295, 1.) for a vectory 
 XII. . . p = ix -\-jy + kz, 
 ■which gives, by the laws of the symbols ijk (182, 183), 
 
 f dp = tdx +ydy + Mz, d« = Tdp = V(d^-2 ^ ^yS ^. ^22), 
 
 I d?p = id'^x + jd2y + kd^z, Td2p = V {d^x^ + d^y^ + dH^), 
 
 XIII. . . 7 Vdpd2p = i (dyd2z - dzdV) + JCdzd^x - dxd^^z) + k {dxd^y - dyd^x), 
 1 da; ,dy , dz ,^, , , dx 
 
 LU''''='d7+^d; + *d7' iudp=.d-+.., 
 
 the recent equations IX. X. XI. take these known forms: 
 IX'. . . r-ids3 = V((d3/d2z - dzd^y^ + . .) ; 
 X'. . . r-id«2 = V(d2;r« + d2y2 -t d222 - d2«2) ; 
 
628 ELEMENTS OF QUATERNIONS. [BOOK III. 
 
 (5.) The formula IV., which lately served us to determine a diameter of a circle 
 through three given points, may be more symmetrically written as follows, /f ad 
 he a diameter of the circle ABC, then 
 
 XIV. . . AD .V(AB . BC) = AB . BC. CA ; 
 
 an equation* in which V(ab.bc) may be changed to V(ab.ac), &c., and in whj^h 
 it may be remarked that each member is an expression (comp. 296, V.) for a vector 
 AT, which touches at A the segment abc : while its length is at once a representa- 
 tion of the prorfwc^ oy/Ae Zew^</js oy^Ae sides of the triangle ABC, and also of the 
 double area of that triangle (comp. 281, XIII.), multiplied by the diameter of the 
 circumscribed circle. 
 
 (6.) In general, if pqrs be any four concircular points, they satisfy (by 260, 
 IX., comp. 296, (3.)) the condition of concircularity, 
 
 xv...v(^.?5^=o, 
 
 \ SQ RP / 
 
 which may be thus transformed rf 
 
 XVI. . . v( ?3 + '?Lt«5\=v( -i-.PQ. 2!l±^\ 
 
 \PS PR / \PS PR I 
 
 Writing then (comp. VI., and the remarks in (2.)), 
 
 XVII. . . PS = W - p, PQ = dp, PR=2dp + d2|0, QP + QR = d2p, 
 
 the second member is seen to be, on the present plan, an infinitesimal of the second 
 order, which is therefore to be suppressed, because the first member is only of the 
 first order ; and thus we obtain at once the following vector equation of the osculat- 
 ing circle to the curve pqr at p, 
 
 * A student might find it useful practice to verify, that if we write in like man- 
 ner, 
 
 XIV'. . . BE . V(bC . CA) = BC . CA . AB, 
 
 so that BE is a second diameter, then AB = ed, or abde is a parallelogram. He may 
 employ the principles, that a(5y = y/3a, if Sa(3y = 0, and that /3y - y/3 = 2V/3y ; in 
 virtue of which, after subtracting XIV'. from XIV., and dividing by V(bc.ca), or 
 by its equal V(ab.bc), the equation ad - be = 2ab is obtained, and proves the re- 
 lation mentioned. It is easy also to prove that 
 
 XIV". . . bd.V(bc.ca) = ab.S(bc.ca), 
 and therefore that abde is a rectangle. 
 
 t Without having recourse to this transformation XVI., we might treat the 
 condition XV. by infinitesimals, as follows : 
 
 r^s _ 1 , PQ 1 , ^P . , ^P 
 
 — = 1 + _ = 1 + = 1 + 
 
 xvir. . . ]^^ ^^ o-p-dp o)-p 
 
 2qr _ QP+ QR _ d2p _ d^p 
 
 PR PR ~ 2dp + d^p ~ 2dp 
 
 i- 
 
 equating then to zero the vector part of the product of these two expressions, and 
 suppressing the infinitesimal of the second order, the equation XVIII. of the osculat- 
 ing circle is obtained anew. 
 
CHAP. III.] OSCULATING SPHERES BY INFINITESIMALS. 629 
 
 XVIII... V[ -^+ 5!^^. 0; 
 
 ■which agrees with the equation 392, VI., although deduced in a quite different man- 
 ner, and conducts anew to the expression II. for k— p, under the form, 
 
 XIX. . . -^ + V ^, as in 392, VIIL 
 
 K-p dp 
 
 (7.) Again, if od = 5 be the diameter from the origin, of any sphere through that 
 point o, which passes also through any three other given points A, b, c, with OA = a, 
 «S;c., we have by 296, XXVI. the formula, 
 
 XX. . . eSa/37 = Va(|3-a) (y-fi^yy 
 writing then (comp. XVII.), 
 
 XXI. ..a = d|0, /3-a = dp + d2p, y - (i=dp + 2d^p + d% 
 and XXII. . . 5 = 2ps = 2 ((t - p), 
 
 where c is (as in 395, &c.) the vector os (from an arbitrary origin o) of the centre 
 s of the osculating sphere to a curve of double curvature at p, we have by infinitesi- 
 mals^ suppressing terms which are of the seventh and higher orders^ because the first 
 member is only of the sixth order, and reducing* by the rules of quaternions, 
 XXIII. . . ((T - p) Sdpd2pd3p = |Vdp (dp + d2p) (dp + 2d2p + d^p) (3dp + SdV + d^p) 
 = .3Vdpd2pSdpd2p + dp2 Vd3pdp ; 
 
 which agrees precisely with the formula 395, XIII., although obtained by a process 
 so different. 
 
 (8.) Finally as regards the osculating plane, and ihQ second curvature, of a curve 
 in space, infinitesimals give at once for that plane the equation, 
 
 XXIV. . . S ((u - p) dpd2p = 0, agreeing with 376, V. ; 
 and if three consecutive elements of the curve be represented (comp. XXL) by the 
 differential expressions, 
 
 XXV. . . PQ = dp, QR = dp + d2p, Rs = dp + 2d2p + d3p, 
 the second curvature r~', defined as in 396, is easily seen to be connected as follows 
 with the angle of a certain auxiliari/ quaternion q, which differs infinitely little 
 from unity : 
 
 XXVI...r-.d. = .,, if XXVII... g = I^5^=l + |^^ 
 
 V(pq.qr) Vdpd2p 
 
 * Of the eighteen terms which would follow the sign of operation ^V, if the se- 
 cond member of XXIII. were fully developed, one is of the fisurth order, but is a 
 scalar ; three are of the fifth order, but have a scalar sum ; nine are of orders higher 
 than the sixth ; and two terms of the sixth order are scalars, so that there remain 
 only three terms of that order to be considered. In this manner it is found that the 
 second member in question reduces itself to the sum of the two vector parts, 
 
 f V. (dpd2p)s = 3Vdpd2p . Sdpd2p, 
 and |dp2V(dpd3p + Sd^pdp) = dp2Vd3pdp ; 
 
 and thus the third member of XXIII. is obtained. 
 
630 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 we have then the expression, 
 
 XXVIII. . Second Curvature = r"! = -r^ = S ^r. .. ^ 
 
 dp \6pd'-p 
 
 which agrees with the formula 397, XXVIL, and has been illustrated, in the sub- 
 articles to 397 and 398, by numerous geometrical applications. 
 
 (9.) On the whole, then, it appears that although the logic of derived vectors, 
 and o{ differentials of vectors considered as finite lines, proportional to such deriva- 
 tives, is perhaps a little clearer than that of infinitesimals, because it shows more 
 evidently (especially when combined with Taylor's Series adapted to Quaternions, 
 342, 375) that nothing is neglected, yet it is perfectly />ossi6/e to combine* quater- 
 nions, in practice, with methods founded on the more usual notion of Differentials, 
 as infinitely small differences : and that when this combination is judiciously made, 
 abridgments of calculation arise, without any ultimate error. 
 
 Section 7 On Surfaces of the Second Order ; and on Cur- 
 vatures of Surfaces, 
 
 402. As early as in the First Book of these Elements, some spe- 
 cimens were given of the treatment or expression of Surfaces of the 
 Second Order by Vectors ; or by Anharmonic Equations Avhich were 
 derived from the theory of vectors, without any introduction, at that 
 stage, of Quaternions properly so called. Thus it was shown, in the 
 sub-articles to 98, that a very simple anharmonic equation {xz = yw) 
 might represent either a ruled paraboloid, or a ruled hyperholoid^ ac- 
 cording as a certain condition (ac = hd) was or was not satisfied, by 
 the constants of the surface. Again, in the sub-articles to 99, two 
 examples were given, oi vector expressions for cones of the second or- 
 der (and one such expression for a cone of the third order, with a 
 conjugate ray (99, (5.)); while an expression of the same sort, 
 namely, 
 
 I. . . p = xa+y^ + zy, with x^ f2/^ + z"^ = I, 
 
 was assigned (99, (2.)) as representing generally an ellipsoid,] with 
 a, ^, 7, or OA, OB, oc, for three conjugate semidiameters. And finally, 
 
 * Compare the first Note to page 62l It will however be of course necessary, 
 in any future applications of quaternions, to specify in which of these two senses, as a 
 finite differential, or as an infinitesimal, such a symbol as dp is employed, 
 f In like manner the expression, 
 
 U. , . p=xa+yf3+ zy, with x^ i- y^ - z^ = 1, or = - 1, 
 represents a general hyperboUid, of one sheet, or of two, with a/3y for conjugate semi- 
 diameters : while, with the scalar equation x^ + y^ — z'^=: 0, the same vector expres- 
 sion represents their common asymptotic cone (not generally of revolution). 
 
CHAP. III.] QUATERNION EQUATIONS OF SPHERE. 631 
 
 in the sub-articles (11.) and (12.) to Art. 100, an instance was fur- 
 nished of the determination of a tangential plane to a cone, bj means 
 o^ partial derived vectors. 
 
 403. In the Second Book, a much greater range of expression 
 was attained, in consequence of the introduction of the ;?ecw^ear s^m- 
 hols, or characteristics of operation, which belong to the present Cal- 
 culus; but still with that limitation which was caused, by the con- 
 ception and notation of a Quaternion being confined, in that Book, to 
 Quotients of Vectors (112, 116, comp. 307, (5.)), without yet admit- 
 ting Products or Powers of Directed Lines in Space : although ver- 
 sors, tensors, and even norms* of such vectors were already intro- 
 duced (156, 185, 273). 
 
 (1.) The Sphere,f for instance, which has its centre at the origin, and has the 
 vector oA, or a, with a length Ta = a, for one of its radii, admitted of being repre- 
 sented, not only (comp. 402, I.) by tlie vector expression, 
 
 I. . . p=:xa + !/l3 + zy, x-^ + t/^ + z^= 1, 
 with 
 
 r. . .Ta=T/3 = Ty = a, and I". . . S- = S ^=S | = 0, 
 
 a a (i 
 
 but also by any one of the following equations, in which it is permitted to change a 
 to -a : 
 
 " 1; 145, (8.), (12.) 
 
 186,(2.), 
 187,(1.) 
 
 200, (11.), 
 
 215,(10.), 
 
 P + « « « 273,(1.) 
 
 XI. ..(s^Y-(v^Y=l; XII...NS^ + NV^=1; 204, (6.), XXV., XXVI. 
 
 XIII...Nfs^ + V^^i = l; XIV...Tfs^ + V^yi; 204,(9.) 
 
 or by the system of equations, 
 
 XV. ..S^ = a:, f V^V = a:2-l(<0), 204,(4.) 
 
 representing a system of circles, with the spheric surface for their locus. 
 
 * The notation Na, for (Ta)^, although not formally introduced before Art. 273, 
 had been used by anticipation in 200, (3.), page 188. 
 
 t That is to say, the spheric surface through A, with o for centre. Compare 
 the Note to page 197. 
 
 P « 
 
 
 III., 
 
 ..eK^=l; 
 a a 
 
 IV.. 
 
 •<-> 
 
 V...Tp = «; 
 
 
 VI.. 
 
 .Tp = Ta; 
 
 VII. 
 
 ..T?=l; 
 a 
 
 viii...s^-"= 
 
 = 0; 
 
 IX.. 
 
 .Ni^ = N«: 
 
 X... 
 
 Np = Na: 
 
632 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (2.) Other forma of equation, for the same spheric surface, may on the same 
 principles be assigned ; for example we may write, 
 
 N- = l; XVIII. . .T- = l; 
 
 P P 
 
 i^^=l; XXI...S-^=1; 
 
 p^-a p+a 
 
 XVI.. 
 
 a p' 
 
 XVII. . . 
 
 XIX. 
 
 p-a ir ^ 
 ■•^p+a~2' 
 
 XX. . . s 
 
 or (comp. 
 
 186, (5.), and 200, 
 
 (3.)), 
 
 
 XXII. . . 
 
 ,T(p-ca) 
 
 = T(cp-a), c2^1; 
 
 under which last form, the sphere may be considered to be generated by the revolu' 
 Hon of the circle, which has been already spoken of as the Apollonian'^ Locus. 
 
 (3.) And from awy one to any other, of all these various /orms, it is possible? 
 and easy io pass, by general Rules of Transformation,-^' which were established in 
 the Second Book : while each of them is capable of receiving, on the principles of 
 the same Book, a Geometrical Interpretation. 
 
 (4.) But we could not, on the principles of the Second Book alone, advance to 
 such subsequent equations of the same sphere, as 
 
 XXIII. . . p2 = a2, or XXIV. . . p2 + ^2 = o, 282, VII. XIII. 
 whereof the latter includes (282, (9.)) the important equation p2 + 1 = 0, or p2 = _ i^ 
 of what we have called the Unit-Sphere (128) ; nor to such an exponential expres- 
 sion for the variable vector p of the same spheric surface, as 
 
 XXV. . . p = aktj^kj-^k't, 308, XVIII. 
 
 in which j and k belong to the fundamental system ijk of three rectangular unit- 
 lines (295), connected by the fundamental Formula A of Art. 183, namely, 
 
 ii=j2=k^ = ijk = -l, (A) 
 
 while s and / are two arbitrary and scalar variables, with simple geometricalX signi- 
 fications : because we were not then prepared to introduce any symbol, such as p', 
 or k\ which should represent a square or other /?ower of a vector. % And similar re- 
 
 * Compare the first Note to page 128. 
 
 f This richness of transformation, o^ quaternion expressions or equations, has 
 been noticed, by some friendly critics, as a characteristic of the present Calculus. In 
 the preceding parts of this work, the reader may compare pages 128, 140, 183, 573, 
 574, 575; in the two last of which, the variety of the expressions for the second 
 curvature (r-^) of a curve in space may be considered worthy of remark. On the 
 other hand, it may be thought remarkable that, in this Calculus, a single expression, 
 such as that given by the first formula (389, IV.) of page 532, adapts itself with 
 equal ease to the determination of the vector (k) of the centre of the osculating 
 circle, to a plane curve, and to a curve of double curvature, as has been sufficiently 
 exemplified in the foregoing Section. 
 
 J Compare the second Note to page 365. 
 
 § It is true that the formula A was established in the course of the Second Book 
 (page 160) ; but it is to be remembered that the symbols ijk were there treated as de- 
 noting a system oi three right versors, in three mutually rectangular planes (181) : 
 
CHAP. III.] EQUATION OF ELLIPSOID RESUMED. 633 
 
 marks apply to the representation, by quaternions, of other surfaces of the second 
 order. 
 
 404. A brief review, or recapitulation^ of some of the chief ex- 
 pressions connected with the Ellipsoid^ for example, which have been 
 already established in thesQ Elements ^ with references to a few others, 
 may not be useless here. 
 
 (1.) Besides the vector expression p = cca + yj^ + 2y, with the scalar relation 
 a;2 + y2+2?=l, and with arbitrary vector values of the constants a, /3, y, which 
 was lately cited (402) from the First Book, or the equations 403, I., without 
 the conditions 403, I'., II'. which are peculiar to the sphere, there were given in 
 the Second Book (204, (IB.), (14.)) equations which differed from those lately num- 
 bered as 403, XI. XII. XIII. XIV. XV., only by the substitution of V ^ for V ^ ; 
 for instance, there was the equation, 
 
 analogous to 403, XI., and representing generally* an ellipsoid, regarded as the 
 locus of a certain system of ellipses, which were thus substituted for the circles^ 
 (403, XV.) of the sphere, by a species of geometrical deformation, which led to the 
 establishment of certain homologies (developed in the sub-articles to 274). 
 
 although it has since been found possible and useful, in this Third Book, to identify 
 those right versors with their own indices or axes (295), and so to treat them as a 
 system of three rectangular lines, as above. 
 
 * In the case of parallelism of the two vector constants (j3 1| a), the equation I. 
 represents generally a Spheroid of revolution, with its axis in the direction of a; 
 while in the contrary case of perpendicularity (fi J- a), the same equation I. repre- 
 sents an elliptic Cylinder, with its generating lines in the direction of j3. Compare 
 204, (10.), (11.), and the Note to page 224. 
 
 t The equation I. might also have been thus written, on the principles of the Se- 
 cond Book, 
 
 whence it would have followed at once (comp. 216, (7.)), that the ellipsoid I. is 
 cut in two circles, with a common radius = T/3, by the two diametral planes, 
 
 r...se + s|=o, s£-s^=o. 
 
 In fact, this equation I', is what was called in 359 a cyclic form, while I. itself is 
 what was there called dL focal form, of the equation of the surface ; the lines a'^ ± (i'^ 
 being, by the Third Book, the two (real) cyclic normals, while ^ is one of the two 
 (^re&\) focal lines of the (imaginary) asymptotic cone. Compare the Note to page 
 474. 
 
 4 M 
 
634 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (2.) Employing still only quotients of vectors^ but introducing two other pairs 
 of vector-constants, y, 5 and i, k, instead of the pair a, (3 in the equation I., which 
 were however connected with that pair and with each other by certain assigned re- 
 lations, that equation was transformed successively to 
 
 II...T(e.+ K^^ = l, 216, X. 
 
 and to a form which may be written thus (comp. 217, (5.)), 
 
 III. . . T f t + K - . p Vp =Tt«- T/c2; 217, XVI. 
 
 (e + K^^.p^Tp 
 
 and this last form was interpreted, so as to lead to a Hule of Construction* (217, 
 (6.), (7.)), which was illustrated by a Biagram (Fig. 63), and from which many 
 geometrical properties of that surface were deduced (218, 219) in a very simple 
 manner, and were confirmed by calculation with quaternions : the equation and con- 
 struction being also modified afterwards, by the introduction (220) of a new pair of 
 vector-constants, i' and k', which were shown to admit of being substituted for t 
 and K, in the recent form III. 
 
 (3.) And although the Equation of Conjugation, 
 
 IV. ..S-S^-s(v^.r^Vl» 316, LXIII. 
 
 a a \ (3 i3 ; 
 
 which connects the vectors X, p, of any two points l, m, whereof one is on the polar 
 plane of the other, with respect to the ellipsoid I., was not assigned till near the end 
 of the First Chapter of the present Book, yet it was there deduced by principles and 
 processes of the Second Book alone : which thus were adequate, although not in 
 the most practically convenient way, to the treatment of questions respecting tangent 
 planes and normals to an ellipsoid, and similarly for other surfaces^ of the same 
 second order. 
 
 * This Construction of the Ellipsoid, by means of a Generating Triangle and a 
 Diacentric Sphere (page 227), is believed to have been new, when it was deduced 
 by the writer in 1846, and was in that year stated to the Royal Irish Academy 
 (see its Proceedings, vol. iii. pp. 288, 289), as a result of the Method of Quater- 
 nions, which had been previously communicated by him to that Academy (in the 
 year 1843). 
 
 f The following are a few other references, on this subject, to the Second Book. 
 Expressions for a Right Cone (or for a single sheet of such a cone) have been given 
 
 in pages 119, 179, 220, 221. In page 179 the equation S ^ S ^ = 1, has been as- 
 
 a p 
 
 signed, with a transformation in page 180, to represent generally a Cyclic Cone, or 
 a cone of the second order, with its vertex at the origin ; and to exhibit its cyclic 
 planes, and subcontrary sections (pp. 181, 182). Bight Cylinders hsive occurred in 
 pages 193, 196, 197, 198, 199, 218. A case of an Elliptic Cylinder has been 
 already mentioned (the case when /3 -J- a in I.) ; and a transformation of the equa- 
 tion III. of the Ellipsoid, by means of reciprocals and norms of vectors, was assigned 
 in page 298. And several expressions (comp. 403), for a Sphere of which the ori- 
 
CHAP. III.] TRANSFORMATIONS OF THE EQUATION. 635 
 
 (4.) But in this Third Book we have been able to write the equation III. under 
 the simpler form,* 
 
 V. . . T (tp + pk) = k:2 - t2, 282, XXIX. 
 
 which has again admitted of numerous transformations ; for instance, of all those 
 which are obtained by equating (k2-i2)2 to any one of the expressions 336, (5.), 
 for the square of this last tensor in V., or for the norm of the quaternion tp + p/c ; 
 cyc/ic ybrms t of equation thus arising, which are easily converted into focal forms 
 (359); while a rectangular transformation (373, XXX.) has subsequently been 
 assigned, whereby the lengths (abc), and also the directions, of the three semiaxes of 
 the surface, are expressed in terms of the two vector-constants, i, k : the results thus 
 obtained by calculation being found to agree with those previously deduced, from the 
 geometrical construction (2.) in the Second Book. 
 
 (5.) The equation V. has also been differentiated (336), and a normal vector 
 v = <Pp has thus been deduced, such that, for the ellipsoid in question, 
 
 VI. . . Svdp = 0, and VII. . . Srp = 1 ; 
 a process which has since been extended (361), and appears to furnish one of the 
 best general methods of treating surfacesX of the second order by quaternions : espe- 
 cially when combined with that theory of linear and vector functions {(pp) of vec- 
 tors, which was developed in the Sixth Section§ of the Second Chapter of the pre- 
 sent Book. 
 
 ^mwaswo^the centre, occurred in pages 164, 179, 189, and perhaps elsewhere, 
 without any employment oi products of vectors. 
 
 * Mentioned by anticipation in the Note to page 233. 
 
 t Compare the second Note to page 633. The vectors t and k are here the 
 cgcUc normals, and t — k is one of the focal lines ; the other being the line i — k of 
 page 232. 
 
 X The following are a few additional references to preceding parts of this Third 
 Book, which has extended to a much greater length than was designed (page 302). 
 In the First Chapter, the reader may consult pages 305, 306, 307, for some other 
 forms of equation of the ellipsoid and the sphere. In the Second Chapter, pages 
 416, 417 contain some useful practice, above alluded to, in the differentiation and 
 transformation of the equation r'* = T(tp + pK). As regards the Sixth Section of 
 that Chapter, which we are about to use (405), as one supposed to be faraiUar to the 
 reader, it may be suflScient here to mention Arts. 357-362, and the Notes (or some 
 of them) to pages 464, 466, 468, 474, 481, 484. In this Third Chapter, the sub- 
 articles (7.)-(21.) to 373 (pages 504, &c.) might be re-perused; and perhaps the 
 investigations respecting cones and sphero-conics, in 394 and its sub-articles (pages 
 541, &c.), including remarks on an hyperbolic cylinder, and its asymptotic planes 
 (in page 547). Finally, in a few longer and later series of sub-articles, to Arts. 
 397, &c., a certain degree oi familiarity vfith some of the chief properties of sur- 
 faces of the second order has been assumed ; as in pages 571, 688, 591, and generally 
 in the recent investigations respecting the osculating twisted cubic (pages 591, 620, 
 &C.), to a helix, or other curve in space. 
 
 § It appears that this Section may be conveniently referred to, as III. ii. 6 ; and 
 similarly in other cases. 
 
636 ELEMENTS OF QUATERNIONS. [boOK III. 
 
 405. Dismissing then, at least for the present, the special consi- 
 deration of the ellipsoid, but still confining ourselves, for the mo- 
 ment, to Central Surfaces of the Second Order^ and using freely the 
 principles of this Third Book, but especially those of the Section 
 (III. ii. 6) last referred to, we may denote any such cerdral and non-* 
 conical surface by the scalar equation (comp. 361), 
 
 . L..//> = S/>0/) = l; 
 the asymptotic cone (real or imaginary) being represented by the 
 connected equation, 
 
 II. ..//> = S/>0/> = O; 
 
 and the equation of conjugation, between the vectors />, p' of any two 
 points P, p', which are conjugate relatively to this surface I. (comp. 
 362, and 404, (3.), see also 373, (20.)), being, 
 
 III. . . f{p, p') =f{p', p) = Sp<f>p' = S/>'0/> = 1 ; 
 
 while the differential equation of the surface is of the form (361), 
 
 IV. . . = dfp = 2Si'dp, with V. . . 1/ = 0/> ; 
 
 this vector-function 0/>, which represents the normal v to the surface, 
 being at once linear and self- conjugate (361, (3.)) ; and the surface 
 itself being the locus of all the poiiits p which are conjugate to them- 
 selves, so that its equation I. may be thus written, 
 
 I'. ../(/>, rt = l, because f{p,p)=fp, by 362, IV. 
 
 (1.) Such being the /orm of ^p, it has been seen that there are always ^Aree real 
 and rectangular unit-lines, ai, a^i as, and three real scalars, ci, cg, C3, such as to 
 satisfy (comp. 357, III.) the three vector equations, 
 
 VI. . . <pai = -ciai, <f>az = -C2a2, ^as^-czas; 
 whence also these three scalar equations are satisfied, 
 
 VII./ai = ci, /a2 = C2, /a3 = C3; 
 and therefore (comp. 362, VII.)) 
 
 VIII. . ./(crJai)=/(c2-^a2)=/(03ia3) = l. 
 (2.) It follows then that the three (real or imaginary) rectangular lines, 
 IX. . . ^1 = cr'ai, (32 = C2-^a2, ^3 = cs^az, 
 are the three (real or imaginary) vector semiaxes of the surface I. ; and that the three 
 (positive or negative) sca/ars, c\, C2, C3, namely the ^/iree roofs of the scaZar and cm6ic 
 equation* M=. (comp. 367, (1.))j ^^^ t^® (always real) inverse squares of the three 
 (real or imaginary) scalar semiaxes, of the same central surface of the second order. 
 
 * It is unnecessary here to write Mo= 0, as in page 462, &c, because the func- 
 tion f is here supposed to be self- conjugate ; its constants being also real. 
 
CHAP. III.] GENERAL CENTRAL SURFACE. 637 
 
 (3.) For the reality of that surface I., it is necessary and suflScient that one at 
 least of the three scalars cy, C2, c^ should be positive ; if all be such, the surface is an 
 ellipsoid ; if two^ but not the third, it is a single- sheeted hyperholoid ; and if only 
 one, it is a double- sheeted hyperholoid : those scalars being here supposed to be 
 each j^niVe, and different from zero. 
 
 (4.) We have already seen (357, (2.)) how to obtain the rectangular transfor- 
 mation, 
 
 X. . .fp = ci {Saipf + C2(Sa2,o)2 + C3(Sa3p)2, 
 
 which may now, by IX., be thus written, 
 
 XI. . ./p = (S/3rW + (S/32-ip)2+(S^3->p)2; 
 
 but it is to be remembered that, by (2.) and (3.), one or even two of these three vec- 
 tors /3i/32i33 may become imaginary, without the surface ceasing to be real. 
 (5.) We had also the cyclic transformation (357, II. II.'), 
 
 XII. . .fp=gp^ + S\pfip = p^(g-SXp) + 2S\tjSpp, 
 
 in which the scalar g and the vector \, fi are real, and the latter have the directions 
 of the two (real) cyclic normals ; * in fact it is obvious on inspection, that the surface 
 is cut in circles, by planes perpendicular to these two last lines. 
 
 (6.) It has been proved that the four real scalars, ciczcsg, and the^ue real vec- 
 tors, aiaza^Xp, are connected by the relationsf (357, XX. and XXI.), 
 
 XIU. . . ci = -g-T'Kix, C2 = -5'+SX/i, C3 = -g + TXp; 
 
 XIV. . . ai= U (Xl> - fxTX), aa = UVX/x, 03 = U(\T/z + /iTX) ; 
 
 at least if the three roots C1C2C3 of the cubic M—0 be arranged in algebraically as- 
 cending order (357, IX.), so that ci<C2<C3. 
 
 (7.) It may happen (comp. (3.)), that one of these three roots vanishes ; and in 
 that case (comp. (2.)), one of the three semiaxes becomes infinite, and the surface I. 
 becomes a cylinder. 
 
 (8.) Thus, in particular, if ci= 0, or g-- TX/z, so that the two other roots are 
 hoih positive, the equation takes (by XII., comp. 357, XXII.) a form which may 
 be thus written, 
 
 XV. . . (SX/zp)2 + (SXpTyit + S/ipTX)2 = TX;t - SX/t > ; 
 
 and it represents an elliptic cylinder. 
 
 (9.) Again, if c^ ~ 0, or g = SXp, the equation becomes, 
 
 XVI. . . 2SXpS^p = 1, 
 
 and represents an hyperbolic cylinder ; the root ci being in this case negative, while 
 the remaining root cs is positive. 
 
 * Compare the Note to page 468 ; see also the proof by quaternions, in 373, (16.), 
 &c., of the known theorem, that any two subcontrary circular sections are homosphe- 
 rical, with the equation (373, XLIV.) of their common sphere, which is found to have 
 its centre in the diametral plane of the two cyclic normals X, p. 
 
 t These relations and a few others mentioned are so useful that, although they 
 occurred in an earlier part of the work, it seems convenient to restate them here. 
 
638 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (10.) But if we suppose that c^ = 0, or <7 = TX^, so that ci and C2 are both nega- 
 tive, the equation may (by 357, XXIII.) be reduced to the form, 
 
 XVII. . . (SXfipy + (SXpTfi - SfipTXy = - TXfi - S\/i < ; 
 
 it represents therefore, in this case, nothing real, although it may be said to be, in the 
 same case, the equation of an imaginary* elliptic cylinder. 
 
 (11.) It is scarcely worth while to remark, that we have here supposed each of 
 the two vectors X and {i to be not only real but actual {^krt. 1) ; iorii either of them 
 were to vanish, the equation of the surface would take by XII. the form, 
 
 XVIII. . . p2 =g-i, or XVIII'. ,.Tp= i-gyi, 
 and would represent a real or imaginary sphere, according as the scalar constant g 
 was negative or positive : X and fi have also distinct directions, except in the case 
 of surfaces of revolution. 
 
 (12.) In general, it results from the relations (6.), that the plane of the two (real) 
 cyclic normals, X, p,, is perpendicular to the (real) direction of that (real or imagi- 
 nary) semiaxis, of which, when considered as a scalar (2.), the inverse square ci is 
 algebraically intermediate between the inverse squares Ci, cz of the other two ; or that 
 the two diametral and cyclic planes (SXp — 0, S/ip = 0) intersect in that real line 
 (YXp) which has the direction of the real unit-vector a-i (1.), corresponding to the 
 mean root c^ of the cubic equation J/= : all which agrees with known results, re- 
 specting the circular sections of the (reaiy ellipsoid, and of the two hyperboloids. 
 
 406. Some additional light may be thrown on the theory of the 
 central surface 405, I., by the consideration of its asymptotic cone 
 405, II. ; of which cone^ by 405, XII., the equation may be thus 
 written, 
 
 I. . . //> ^9P^ t ^^Pf^P = pHg- SXytt) + 2S\pSjiip = ; 
 and which is real or imaginary, according as we have the inequa- 
 lity, 
 
 II. ..g^< \^jii\ or III. . . g^> XV ; 
 
 that is, by 405, (6.), according as the product c^c^ of the extreme 
 roofs of the cubic M= is negative or positive ; or finally, according 
 as the surface fp = \ is a (real) hyperboloids or an ellipsoid (real or 
 imaginaryf). 
 
 * In the Section (III. ii. 6) above referred to, many symbolical results have been 
 established, respecting imaginary cyclic normals, or focal lines, &c., on which it is 
 unnecessary to return. But it may be remarked that as, when the scalar function 
 fp admits of changing sign, for a change of direction of the real vector p, so as to be 
 positive for some such directions, and negative for others, although/(— jo)=/(+ p), 
 the two equations, /p =+ 1, /p = - 1, represent then two real and conjugate hyperbo- 
 loids, oi different species : so, when the function /p is either essentially positive, or 
 else essentially negative, for real values of p, the equations /p = 1 and fp = — 1 may 
 then be said to represent two conjugate ellipsoids, one real, and the other imaginary. 
 
 t Compare the Note immediately preceding ; also the second Note to page 474. 
 
CHAP. III.] CONES OF THE SECOND ORDER. 63^ 
 
 (1.) As regards the asserted reality of the cone I., when the condition II. is sa- 
 tisfied, it may suffice to observe that if we cut the cone by the plane, 
 
 IV...SX(p-,.)=-^, 
 the section is a circle of the real and diacentric sphere, 
 
 V. . . p2 = 2S/ip, or v. . . (p - /i)3 = ;^2 . 
 and a real circle, because it is on the real cylinder of revolution, 
 
 VI. . . TV(p - jti)UX = (T/i3 _^2T\-2)i, 
 so that its radius is equal to this last real radical. 
 (2.) For example, the cone 
 
 VII...S^S^ = 1, or Vir. ..2(SapS/3p-aV) = 0, 
 a p 
 
 which under the form VII. occurred as early as 196, (8.), and for which \ = a, 
 fi = (3f g = Sa/3 - 2a^, and therefore TXfi +g>0, the condition II. reduces itself to 
 TXfj,-g > ; or after division by 2Ta^, &c., to the form (comp. 199, XII.), 
 
 VIII. ..KT4S)^>1, or Vlir. ..sj^>l; 
 
 and accordingly, when either of these two last inequalities exists, it will be found 
 
 that the sphere S — = 1 is cm< by the plane S - = 1 in a real circle, the base of a real 
 
 p a 
 
 cone VII. 
 
 (3.) As an example of the variety of processes by which problems in this Calcu- 
 lus may be treated, we might propose to determine, by the general formula 389, IV., 
 the vector k of the centre of the osculating circle to the curve IV. V., considered 
 merely as an intersection of two surfaces. The first derivatives of the equations 
 would allow us to assume p' = VX(p — jw), and therefore p" = Xp'', whence, by the 
 formula, we have 
 
 TV , P'^ , P' SpX+V/iX . 
 
 the section is therefore a circle, because its centre of curvature is constant ; and its 
 radius is, 
 
 X.,.r = T(p-K) = T(p-/u4-^X-i) = (T/i2-^2TX-2>, 
 
 = the radius of the cylinder VI. 
 
 (4.) When the opposite inequality III. exists, the radius X., the cylinder VI., 
 the circle IV. V., and the cone I., become all four imaginary ; the plane IV. being 
 then wholly external to the sphere V., as happens, for instance, with the plane and 
 sphere in (2.), when the condition VIII. or VIII'. is reiiersed. 
 
 (5.) In the intermediate case, when 
 
 XI. . .g'^ = X^ii^, or XI'. . . g = T TXju, 
 the radius r in X. vanishes ; the right cylinder VI. reduces itself to its axis ; and 
 the circle IV. V. becomes a point, in which the sphere is touched by the plane. In 
 this case, then, the cone I. is reduced to a single (real*) right line, which has 
 
 * It may however be said, that in this case the cone consists of a pair of imagi- 
 nary planes, which intersect in a real right line. 
 
640 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (compare the equations of the elliptic cj/linders, 405, XV. XVII.) the direction of 
 XT;* — fiT\, if g = — TX/i, but the perpendicular direction of XT/* + ftTX, if g = 
 + TX/tt. 
 
 (6.) In general (comp. 405, X.), the equation of the cone I. admits of the rect- 
 angular transformation, 
 
 XII. . .fp = ciCSaip)2 + C2(Sa2p)2 + 03(803,0)' = ; 
 and the two sub-cases last considered (5.) correspond respectively (by 405, (6.)) to 
 the evanescence of the roots ci, cz of the cubic 71/= 0, with the resulting directions ai, 
 03 of the only real side of the cope. An analogous but intermediate case (comp. 405, 
 (9.)) is that when C2 = 0, or ^r = SXfi ; in which case, the cone I. reduces itself to the 
 pair of (real) planes, 
 
 XIII. . . SXp . S^p = 0, 
 
 namely to the asymptotic planes of the hyperbolic cylinder 405, XVI., or to those 
 which are usually the two cyclic* planes of the cone. 
 (7.) The case (comp. 394,^(29.)), 
 
 XIV. . .5'=-SX/i, or XIV. . . ci-C24-C3 = 0, 
 for which the equation I. of the cone becomes, 
 
 XV. . . =/p = 2(SXpS/ip - p2SX/i) = 2S(V\p .V/xp), 
 may deserve a moment's attention. In this case, the two planes, of Xp and /zp, 
 which connect the two cyclic normals X and p with an arbitrary side p of the cone, 
 are always rectangular to each other ; and these two normals to the cyclic planes 
 are at the same time*icfes of the cone, which thus is cut in circles, by planes perpen- 
 dicular to those two sides. And because the equation of the cone may (in the same 
 case) be thus written, 
 
 XVI. . . TV(X +p)p = TV(X - /i) p, 
 
 while the lengths of X and p may vary, if their product TXp be left unchanged, so 
 that X + p and X- p may represent any two lines from the vertex, in the plane of 
 the two cyclic normals, and harmonically conjugate with respect to them, it follows 
 that, /or this cone XV., the sines of the inclinations of an arbitrary side p, to these 
 two new lines, have a constant ratio to each other. 
 
 (8.) In general, the second form I. of /p shows (comp. 394, (23.)), that the con- 
 stant product of the sines of the inclinations, of a side p of the cone to the two cyclic 
 planes, has for expression, 
 
 XVn...cos.e.co,.e = j(^ + cos.^]; 
 
 while the first form I. of the same function /p reproduces the condition of reality II., 
 by showing that g : TX/t is (for a real cone) the cosine of a real angle, namely, that 
 of the quaternion product Xppp, since it gives the relation, 
 
 XVIII. . . -|- = SUXp/ip = cos L Xpfip = cos L ^^. 
 
 IXp A. 
 
 * The cones and surfaces which have a common centre, and common values of 
 the vectors X and fj, but different values of the scalar g, may thus be said, in a 
 known phraseology, to be biconcyclic. 
 
CHAP. III.] ARCUAL AXES AND FOCI OF A SPHERO-CONIC. 641 
 
 (9.) We may also observe that in the case of reality II., with exclusion of the 
 sub-case (6.), if 03 have the direction of the internal axis of the cone, so that 
 
 XIX. ..ci<0, C2<0, C3>0, or XIX'. . . ^>S\ju, g<T\fi, 
 
 the two sides (of one sheet) in the plane of Xjw have the directions, 
 
 XX. . . pi = ca-^aa + (- ci)-iai, pz = c^-has - (- ci)-^ai ; 
 
 if then their mutual inclination, or the angle of the cone in the plane of the cyclic 
 normals, be denoted by 2b, we have the values, 
 
 XXI. . . tan2 b = — , XXI'. . . cos 2h = ^^-^^—-^ = J— ; 
 
 the angle of the quaternion Xpfip is therefore (by XVIII.), equal to this angle 2b, 
 namely to the arcual minor axis of the sphero-conic, in which the cone is cut by the 
 concentric unit-sphere. 
 
 (10.) The same condition of reality II. may be obtained in a quite different way, 
 as that of the reality of the reciprocal cone, which is the locus of the normal vector^ 
 
 XXII. . .v = <l>p=gp + YXpix. 
 
 Inverting this linear function <f), by the method of the Section III. ii. 6, we find first 
 the expression (comp. 354, (12.), and 361, (6.)*), 
 
 XXIII. . .mp = \pv = fi^XSXv 4- X^S/iv - g (XSfxv + fiSXv) + (g^ - XV") v, 
 in which XXIV. . .m = (g- SXfi) (g^ - X^fi^) = - cicgca ; 
 
 and next the reciprocal equation (comp. 361, XXVII.), 
 
 XXV. . . = S V^ V = /X2 (S\j.)2 + \2 (S/, v)2 _ 2^SX»/S/i J/ + (^2 _ \2^2) ^2^ 
 
 which may be put under the form, 
 
 xxyi...cos(z^+.^) = ^:^, 
 
 the quotient g : TX/z thus presenting itself anew as a cosine, namely as that of the 
 supplement of the sum of the inclinations of the normal v (to the cone I.), to the two 
 cyclic normals X, fx (of that cone) ; or as the cosinef of tt - A — b, if a and b denote 
 (comp. Fig. 80) the two spherical angles, which the tangent arc to the sphero-conic 
 (9.) makes with the two cyclic arcs : so that by comparison of XXl'. and XXVI. 
 we have the relation, 
 
 XXVII. . . A + B = Z ^+Z.- = 7r-2b. 
 X p 
 
 (11.) Comparing the expression XXI'. for cos 2b, with the last expression 
 
 * In the expression 361, XXVI. for t//j/, the second term ought to have been 
 printed as - VXjuSXv/it ; or else the sign should have been changed. 
 
 f This relation was mentioned by anticipation in 394, (3.) ; and the relation in 
 XXVII. may easily be verified, by conceiving the point of contact p in Fig. 80 
 (page 543) to tend towards a minor summit of the conic, or the tangent arc apb to 
 tend to pass through the two points c, c', in which the cyclic arcs intersect. 
 
 4 N 
 
642 ELEMENTS OF QUATERNIONS. [bOOK 111. 
 
 XVIII. for g : TX/x, we derive the following construction for a sphero-conic, which 
 may easily be verified by geometry :* 
 
 Having assumed two points (l, m) on a sphere, and having described a small 
 circle round one of them (say l), bisect the arcs (mq) which are drawn to its circum- 
 ference from the other point ; the locus of the bisecting points (p) will be a sphero- 
 conic, with the two fixed points for its two cyclic poles (or for the poles of its cyclic 
 arcs), and with an arcual minor axis (2b) equal to the arcual radius of the small 
 circle. 
 
 (12.) As regards the arcual major axis (say 2a) of the same sphero-conic, it is 
 (with the conditions XIX.) the angle between the two sides (comp. XX.), 
 
 XXVIII. . . p3 = C3"ia3f (- C2)-ia2, p4 = c^-iaz - (- C'z)-^a% ; 
 whence (comp. XXI.), 
 
 XXIX. . . tan2 a = — , or XXIX'. . . cos 2a = ^-^HJl^ = (say) e, 
 -Ca' -C2+C3 ^ •'^ ' 
 
 and therefore, a few easy reductions being made, 
 
 sin b 
 
 XXX. . . 
 
 sma 
 
 = ^{<l.SU^)} = oosU^ 
 
 from which we can at once infer, that if o. focus of the conic be determined, by draw- 
 ing from a minor summit to the major axis an arc equal to the major semiaxis a, 
 the minor axis subtends at this focus (or at the other) a spherical angle equal to the 
 angle between the two cyclic arcs. 
 
 (13.) For the two real unifocal transformations of the equation of the cone, or 
 the /orms, 
 
 XXXI. . . a (Vap)2 + b (S(3py = 0, and XXXI'. . . a ( Va'p)2 + b (SjS'py = 0, 
 with one common set of real values of the scalar coefficients, a and b, but with two 
 real focal unit lines a, a', and two real directive tiormals (3, /3' corresponding, it 
 may be sufficient here to refer to the sub- articles to 358 ; except that it should be 
 noticed, that if the cone be real, and if the line ^3 have the direction of its internal 
 axis, so that the inequalities XIX. are satisfied, and therefore also (by 405, (6.)), 
 
 XXXII. . . C3-1 > > ci-J > C2-1, 
 instead of the inequalities 358, III., or 359, XXXVII., we are now to change, in 
 the earlier formulae referred to, the symbols cic^czoiazaz to cscicsasaiaa, so that we 
 have now the values, 
 
 XXXIII. .. a = -ci, & = C3-ci+C2, if T/3 = T/3' = 1. 
 (14.) And as regards the interpretation of the unifocal form XXXI., with these 
 last values, it is evidently contained in this other equation, 
 
 XXXIV. . . sin z ^- . sec 4 = n^ = ( 'iZSlllA^ = const. ; 
 
 the sines of the inclinations of an arbitrary side (p) of the cone, to a. focal line (a), 
 
 * In fact, the bisecting radii op are parallel to the supplementary chords m'q, if 
 mm' be a diameter of the sphere ; and the locus of all such chords is a cyclic cone, 
 resting on the small circle as its base. 
 
CHAP. III.] SYSTEM OF CONFOCAL SURFACES. 643 
 
 and to the corresponding rfiVec^or jo/ane (-1-/3), thus bearing to each other (as is 
 well known) a constant ratio, which remains unchanged when we pass to the other 
 (real) /ocaZ line (a'), and at the same time to the other (real) director plane (-1- /3') : 
 and the focal plane of these two lines (a, a') being perpendicular to that one of the 
 three axes, which corresponds to the root (here ci, by XXXIT.) of the cubic, of 
 which the reciprocal is algebraically intermediate between the reciprocals of the other 
 two. 
 
 (15.) It is, however, more symmetric to employ the bifocal transformation 
 (comp. 360, VI.*), 
 
 XXXV. . . = (Sap)2 - 2eSapSap + (Sa»2 + (1 - e2)p2 . 
 in which the scalar constant e has the value (comp. XXIX'.), 
 
 XXXVI. .. e = cos 2a; 
 and a, a' are the twof real a.nd focal unit lines, recently considered (13.). 
 
 (16.) The equation XXXV., for the case of a reaZ cone, may be thus written 
 (comp. XXVI. XXXVL), 
 
 XXXVII. .. L^+ L ^ = cos-i6=2a; 
 a a 
 
 the sumX of the inclinations of the side p to the two focal lines a, a' being thus con- 
 stant, and equal (as is well known) to the major axis of the spherical conic : and 
 although, when e> 1, the cone becomes imaginary, yet it is then asymptotic to a 
 real ellipsoid, as we shall shortly see. 
 
 407. The bifocal form (406, XXXV.) of the equation of a cone 
 may suggest the corresponding /orm, 
 
 I. . . C-=Cfp = (Sapy-2eSapSa'p + (Sa^py + {l -e^)p\ 
 
 in which a and a' are given and generally non-parallel unit-lines, 
 while e and C are scalar constants, as capable of representing gene- 
 rally (comp. 360, (2.), (3.)) a central but non-conical surface (fp = 1) 
 of the second order. And we shall find that if, in passing from one 
 such surface to another, we suppose a and a' to remain unchanged^ 
 but e and C to vary together, so as to be always connected by the 
 
 relation, 
 
 II... C={e'-l)(e + Saa^)P, 
 
 in which I is some real, positive, and y^iven scalar, then all the sur- 
 
 * It is to be remembered that, in the formula here cited, the symbols a, a' did 
 not denote unit- vectors. 
 
 t When these two vectors a, a' remain constant, but the scalar e changes, there 
 arises a system of biconfocal cones : or, by their intersections with a concentric 
 sphere, a system of biconfocal spheh-conics. Compare the Note to page 640. 
 
 + Or the difference, according to the choice between two opposite directions, for 
 one of the two focal lines. The angular transformation XXXVII. may be accom- 
 plished, by resolving the equation XXXV. as a quadratic in e, and then interpreting 
 the result. 
 
644 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 faces I. so deduced, or in other words the surfaces represented by 
 the common equation, 
 
 III. . . I -Pfp (7:rT)(e + SaaO ' 
 
 with e for the only variable parameter, compose a Confocal System. 
 
 (1.) The scalar form III. of//3 gives the connected vector form, 
 
 IV. . . Pv = P^p = °" (-ea> H^a-S(a:- «.)p + (1 - e»)p _ 
 (e2-l) (e + Saa) 
 
 which may also be thus written, with the value II. of C, 
 
 V. . . Cv= C<pp = (a - ea) Sap + (a' - ea) Sa'p + (1 - e^) p, 
 
 so that the function tp is self-conjugate, as it ought to be. 
 (2.) And because we have thus, 
 
 VI. . . Ce2 _ 1) iz^a = a' - eq, (c^ - 1) l^-i^a' = a - ea', 
 if we write, for abridgment, 
 
 VII. . .a^ = {e+l) 1-2, 63 = (e + Saa') IS c^ = (e - 1) Z^, 
 we shall have the values, 
 
 (0(a + a')=-a-2(« + «')j 
 VIII. . . I^Vaa' =-6-2Vaa', 
 
 ( ^ (a - a') = - c-^{a - a') ; 
 
 comparing which with 405, (1.), (2.), we see that the three (real or imaginary) 
 lines, 
 
 IX. . . aU(a + a'), iUVaa', cU(a - a'), 
 
 of any one of which the direction may be reversed, are the three vector semiaxes of 
 the surface fp — 1 ; and therefore, by VII., that the system III. is one of confocals, 
 as asserted. 
 
 (3.) The rectangular transformations, scalar and vector, are noAV (comp. 405, 
 X., and 357, V. VIII.) : 
 
 X...Z =Z/p_ -_^ +_____+, __ . 
 
 XI. . . Z^v = /2W,p = ^ .- . 
 
 ^^ e+1 c + Saa' 
 
 U(a-a' ).SpU Ca-a O 
 "^ e-1 ' 
 
 which can both be established, by the rules of the present Calculus, in several other 
 ways, and from the first of which it follows that (as is well known) through any pro- 
 posed point p of space there can in general be drawn thiee confocal surfaces, of a 
 given system III. ; one an ellipsoid, for which e> 1, and therefore a^ > 52 > c2 > o ; 
 another a single-sheeted hyperboloid, for which e < 1, e> — Saa', d^ > b^> 0> c^ ; 
 and the third a double-sheeted hyperboloid, for which e < - Saa', e>- 1, a2>0 
 > 62 > c2. 
 
CHAP. III.] RECTANGULAR SYSTEM OF THREE NORMALS. 645 
 
 (4.) From the other rectangular transformation XI. it follows, that if we denote 
 by vi = ^\p what the normal vector v = (pp becomes, when p remains the same, but 
 e is changed to a second root ei of the equation III. or X. of the surface, considered 
 as a cubic in e, then 
 
 XII. . . ^^^—^=P(pvi=P-<piv = l^(l>i(l>p = P({>(l)^p- 
 e\ e 
 
 but XIII. . . S|OVi = Sp V =fip =fp = 1, 
 
 fip being formed from//?, by the substitution of ei for e ; therefore, 
 
 XIV. . . 0-Sp<l>vi = Svi<l)p = Sviv, 
 and the known theorem results, that confocal surfaces cut each other orthogonally * 
 (5.) It follows, from V. and VI., that the inverse function 0"ip can be expressed 
 as follows : 
 
 XV. . . 0-Jp = Z^CaSa'p + a'Sap) - h'^p ; 
 
 or that p may be deduced from v by the formula, 
 
 XVI. . . p = ^-ii/ = l"^ (aSaV + a'Sav) - h^v, 
 which can easily be otherwise established. Hence (comp. 361, (4.)), the equation 
 of the surface reciprocal to the surface I. or III., or of that new surface which has v 
 (instead of p) for its variable vector, is 
 
 XVII. . . 1 = JV = Sv^-iv = 2Z2SavSaV - 62^2 . 
 ihQ fixed focal lines a, a' of the confocal system III., or of the corresponding system 
 of the asymptotic cones, becoming thus (in agreement with known results) thQ fixed 
 cyclic normals (or cyclic lines, comp. 361, (6.)) of the reciprocal system XVII. 
 
 (6.) In thus deducing the equation XVII. from III., no use has been made of 
 the rectangular transformations X. XI., of the functions /p and ^p. Without the 
 transformations last referred to, we could therefore have inferred, by a slight modifi- 
 cation of the form XVII., that the reciprocal surface (Fv= 1) with v for its vari- 
 able vector, which has the same rectangular system of directions for its three semi- 
 axes as the original surface (/p = 1), but with inverse squares (the roots of its 
 cubic) equal to the direct squares of the original semiaxes, has for equation (comp. 
 405, XII.), 
 
 XVIII. . . 1 = Fp = P (Sava'v - ep^) = SXvfxv + gv^, 
 if XIX. ..X = Za, fi = la', g = - eP = -eTXfi; 
 
 the values VII. of a"^, h"^, c^ being thus deduced anew, but by a process quite diffe- 
 rent from that employed in (2.), under the forms (comp. 405, XIII.), 
 
 XX. . . a2 = c3 = -^ + TX/i; h'^ = co = -g+ ^Xfx; c^ = Ci=- g -TXp; 
 while the directions IX. of the corresponding semiaxes may be deduced as those of 
 az, a2, ai, from the formulae 405, XIV. 
 
 (7.) If the symbol w (v), or simply ojv, be used to denote a new linear and self- 
 conjugate vector function of v, defined by the equation, 
 
 XXI. . . ojv = pSpv - P (^aSa'v + a'Sav), 
 
 * We shall soon see that the same formula XII,, by expressing that v, v\, and 
 ^vi or 0iv arc complanar, contains this other part of the knoAvn theorem referred to, 
 that the intersection is a line of curvature, on each of the two confocals. 
 
646 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 with p here treated as a vector constant, then (because ^pv=l^ the equation XVI. 
 may be thus written (comp. 354, &c.), 
 
 XXII. . . (w + 62)v=0; 
 the three rectangular directions^ of the three normals v, Vi, V2 to the tliree confo- 
 cals through p, are therefore those which satisfy (comp. again 354) the vector qua- 
 dratic equation, 
 
 XXIII. . . Vi/a>j/ = 0; 
 
 and they are the directions of the axes of this new surface of the second order (comp. 
 357, &c.), 
 
 XXIV. . . Sj/wv = (Spi/)2 - 2Z2SavSaV = 1, 
 
 in which p is still treated as a constant vector, but v as. a variable one. 
 
 (8.) The inverse squares of the scalar semiaxes of this new surface (Svwv = 1), 
 are the direct squares b^, 6i2, b^^ of what may be called the mean semiaxes of the 
 three confocals ; these latter squares must therefore be the roots of this new cubic, 
 
 XXV. . . = wH- m'b^ + m" (62)2 + (62)3^ 
 in which the coefficients m, m', m", deduced here from the new function w, as they 
 were deduced from (p in the Section III. ii. 6, have the values, 
 fm =Z4(Saa'p)2; 
 XXVI. . . )m' = li (Yaay + 2^23 (Yap .Va'p) ; 
 (m"=p2- 222 Saa'. 
 
 Accordingly, if we observe that (because Ta = Ta'= 1) we have among others the 
 transformation, 
 
 XXVII. . . (Saa'p)2 = p2 (Yaa'y - (Sap)2 - 2Saa'SapSa'p - (Sa'p^, 
 
 we can express this last cubic equation XXV., with these values XXVI. of its co- 
 efficients, under the form, 
 
 XXVIII. . . = (62+p2) {(62_;2Saa')2-Z4} 
 
 + 2Z2 (62 _ Z2Saa') SapSa'p - 1^ ((Sap)2 + (Sa'p)2) ; 
 
 which, when we change 6^ by VII. to its value Z2(e + Saa'), and divide by I*, be- 
 comes the cubic in e, or the equation III. under the form, 
 
 XXIX. . . = («2 - 1) {Z2(e + Saa') + p2} + 2eSapSa'p - (Sap)2- (Sa'p)2. 
 
 (9.) As an additional test of the consistency of this whole theory and method, 
 the directions of the three axes of the new surface XXIV., or those of the three 
 normals (7.) to the confocals, or the three vector roots (354) of the equation 
 XXIII., ought to admit of being assigned by three expressions of the forms, 
 
 {nv =\I^(T + b^x^ + ^*<^> 
 XXX. . . |nivi = v^(ri+6i2;^<ri + 6i*(ri, 
 
 ( niVz = \l/ff2 + bz^x^^z + &2*o-3 ; 
 in which b^, 6i2, 622 are the three scalar roots of the cubic XXV. or XXVIII., while 
 <r, (Ti, (Ti are three arbitrary vectors ; n, ni, nz are three scalar coefficients, which 
 can be determined by the conditions Spv= Spvi = Spvz= 1 (comp. XIII.); and ^, 
 X are two new auxiliary linear and vector functions, to be deduced here from the 
 function w, in the same manner as they were deduced from ^ in the Section lately 
 referred to. 
 
CHAP. III.] CONE RESTING ON FOCAL HYPERBOLA. 647 
 
 (10.) Accordingly, by the method of that Section, taking for convenience the 
 given* vector p (instead of the arbitrary vectors c, ci, tra) as the subject of the ope- 
 rations »// and X, we find the expressions, 
 
 XXXI. . . -^p = l*Yaa'Saap, XP = l^CaSa'p + a Sap - 2pSaa') ; 
 whence, after a few reductions, with elimination of n by the relation Sp v = 1, and by 
 the cubic in IP-^ the first equation XXX. becomes : 
 XXXII. . . = (6V + p) {(62 _ /2Saa')2 - Z^} 
 
 + Z2(68 - Z2Saa') (aSa'p + a'Sap) - 1^ (aSap + a'Sa'p) ; 
 which is in fact a form of the relation between v and p, for any one of the confocals, 
 as appears (for instance) by again changing 63 to P- (e + Saa'), and comparing with 
 the equation IV. 
 
 (11.) Another and a more interesting auxiliary surface^ of which the axes have 
 stiU the directions of the normals v, is found by inverting the new linear function o^, 
 or by forming from XXII. the inverse equation, 
 
 XXXIII... ((.>-i + 6-2)v = 0; 
 in which, 
 
 XXXIV. . . a>-ii/.(Saa'p)2=Vaa'SaaV + Z-2(VopSa'pi/ + Va'pSapv); 
 
 and from which it follows that the normals v to the confocals through p have the 
 directions of the axes of this new cone, 
 
 XXXV. . . Svw-' i; = 0, or XXXVI. . . = Z^ (Saa'»^)3 + 2SapvSa'p v, 
 with p treated as a constant, as before. 
 
 (12.) The vertex of this auxiliary cone being placed at the given point p, of in- 
 tersection of the three confocals, we may inquire in what curve is the cone cut, by 
 the plane of the given focal lines, a, a', drawn through the common ceritre o of all 
 the surface^ III. Denoting by (T = ta + t'a' the vector of a point s of this sought 
 section, and writing 
 
 XXXVII. . . v = (T-p = ta-\^ t'a - P, 
 
 the equation XXXVI. gives the relation, 
 
 XXXVIII. ..«' = - = — - — = const. ; 
 2 4 
 
 the section is therefore an hyperbola, which is independent of the point p, and has 
 the focal lines of the system for its asymptotes. And because its vector equation may 
 be thus written (comp. 371, II.), 
 
 XXXIX. . . a = ta^-lin-^a, 
 
 or what may be called its quaternion equation as follows (comp. 371, I.), 
 
 XL. . . 2Vatr.V(7a' = Z2(Vaa')2, 
 it satisfies the two scalar equations, 
 
 XLI. . . TO = 0, w' = 0, 
 with the significations XXVL of m and m' ; it is therefore that important curve, 
 which is known by the name of the Focal Hyperbola :\ namely the limit to which 
 
 * The general expressions for ^pcr and xcr include terms, which vanish when 
 
 (T = p. 
 
 t Compare the Notes to pages 231, 505. 
 
648 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 the section of the confocal surface by the plane of its extreme* axes tends, when the 
 mean axis (26) tends to vanish. We are then led thus to the known theorem, that 
 t/J with any assumed point v for vertex, and with the focal hyperbolaf for base, a 
 cone be constructed, the axes of this focal cone have the directions of the normals 
 to the confocals through P. 
 
 (13.) As regards the Focal Ellipse, its two scalar equations may be deduced 
 from the rectangular form X., by equating to zero both the numerator and the de- 
 nominator of its last term ; they are therefore, 
 
 XLII. . . S(«-a')p = 0, 2/2 = (SpU(a + a'))3 + f ?^^^y ; 
 
 the curve being thus given as a perpendicidar section of an elliptic cylinder, with 
 IV2 and /V(l + Saa'), or {a^ - e^)^ and (b^ — c-^-, for the semiaxes of its base, 
 or of the ellipse itself. 
 
 (14.) The same curve may also be represented by the equations, 
 XLIII. . . Sap = Sa'p, TVap = (b^ - c2)i, 
 or XLIII'. . . Sa'|0 = Sap, TVa'p = (6^ _ c2)i ; 
 
 which express that it is the common intersection of its own plane (-^ a — a') with two 
 right cylinders,X which have the two focal lines a, a' of the system for their axes of 
 revolution, and have equal radii, denoted each by the radical last written. 
 
 (15.) In general, the unifocal form (comp. 406, (13.)) of the equation III., 
 namely, 
 
 XLIV. . . = (1 - e2) (fYapy + 6^) + (S(a' - ea) p)3, 
 
 in which a and a' may be interchanged, shows that the two equal right cylinders, 
 
 XLV. . . (Vap)2 + 62 = 0, XLV. . . ( Va'p)2 + 62 = 0, 
 or XLYI. . . TVap = 6, XLVI'. . . TVa'p = 6, 
 
 which are real if their common radius 6 be such, that is, if the confocal (e) be either 
 an ellipsoid (supposed to be real^, or else a single-sheeted hyperboloid, and which 
 have ih.Q focal lines a, a' of the system for their axes of revolution, envelope^ that 
 confocal surface ; the planes of the two ellipses of contact (which again are real 
 curves, if 6 be real) being given by the equations, 
 
 XLVII. . . S(a' -ea)p = 0, XLVII'. . . S (a - ea) p = ; 
 
 so that they pass through the centre o of the surface (or of the system), and are the 
 (real) director planes (comp. 406, (14.)) of the osyw/)fo<ic cone (real or imaginary), 
 to the particular confocal (e). 
 
 * Namely, those two of which the squares algebraically include between them 
 that of the third ; this latter being, for the same reason, considered here as the mean. 
 
 t We shall soon see that quaternions give, with equal ease, a more general known 
 theorem, in which this is included as a limit. 
 
 + The reader may consult page 513 of the Lectures, for the case of this theorem 
 which answers to a given ellipsoid. The focal ellipse may also be represented gene- 
 rally by the expression (comp. page 382 of these Elements), 
 
 p = (a^ - c2)i V. a<U (a + a') ; 
 or by the same expression, with a and a' interchanged. 
 
 § Compare pages 199, 228, 233, 299. 
 
CHAP. III.] CENTRO-FOCAL ELLIPSES. 649 
 
 (16.) Whether the mean semiaxis (b) be real or imaginary, the surface III. 
 (supposed to be itself real) is always, by the form XLIV. of its equation, the loctts 
 of a system of real ellipses (comp. 404, (l.))j in V^a.nes parallel to the director plane 
 XLVII., which have their centres on the focal line a, and are orthogonally projected 
 into circles on a plane perpendicular to that line. 
 
 (17.) The same surface is also the locus of a second system of such ellipses, re- 
 lated similarly to the second focal line a', and to the second director plane XLVII'. ; 
 and it appears that these two systems of elliptic sections of a surface of the second 
 order, which from some points of view are nearly as interesting as the circular sec- 
 tions, may conveniently be called its Centro-Focal Ellipses. 
 
 (18.) For example, when the frst quaternion form (204, (14.), or 404, I.) of 
 the equation of the ellipsoid is employed, one system of such ellipses coincides with 
 the system (204, (13.)) of which, in the Jirst generation* of the surface, the ellipsoid 
 
 * Besides that first generation (I) of the Ellipsoid, which was a double one, in 
 the sense that a second system (17.) o{ generating ellipses might be employed, and 
 which served to connect the surface with a concentric sphere, by certain relations of 
 homology (274) ; and the second double generation or construction (II), by means 
 of either of two diacentric spheres (217, (4.), (6.), (7.), and 220, (3.)), which was 
 illustrated by Fig. 53 (page 226) : several other generations of the same important 
 surface were deduced from quaternions in the Lectures, to which it is only possible 
 here to refer. A reader, then, who happens to have a copy of that earlier work, may 
 consult page 499 for ageneration (III) of a system of two reciprocal ellipsoids, with 
 a common mean axis (2b), by means of a moving sphere, of which the radius (= b) 
 is given, but of which the centre has the original ellipsoid for its locus ; while the 
 corresponding point on the reciprocal surface, and also the normals at the two points, 
 are easily deduced from the construction. In page 502, he will find another and per- 
 haps a simpler generation (IV), of the same pair of reciprocal ellipsoids, by means of 
 quadrilaterals inscribed in a fixed sphere (the common mean sphere, comp. 216, 
 (10.)) ; the directions of the/o«r sides of such a quadrilateral being given, and one 
 pair of opposite sides intersecting in a point of one surface, while the other pair have 
 for their intersection the corresponding point of the other (or reciprocal) ellipsoid. 
 In the page last cited, and in the following page, there is given a new double genera- 
 tion (V) of any one ellipsoid ; its circular sections (of either system) being con- 
 structed as intersections of two equal spheres (or spheric surfaces), of which the line 
 of centres retains a fi^ed direction, while the spheres slide within two equal and 
 right cylinders, whose axes intersect each other (in the centre of the generated sur- 
 face), and of which the common raditis is the mean semiaxis (6). Finally, in page 699 
 of the same volume, there will be found a new generation (VI) of the original ellip- 
 soid (abc), analogous to the generation (IV) by the fixed {mean) sphere, but with 
 new directions of the sides of the quadrilaterals, which are also (in this last genera- 
 tion) inscribed in the circles of a certain mean ellipsoid (or prolate spheroid) of 
 revolution, which has the mean axis (2b) for its major axis, and has two medial 
 foci on that axis, whose common distance from the centre is represented by the ex- 
 pression, 
 
 V(a2-6^)V(6^-y) 
 
 V(a2-6«-fc2) ' 
 
 4o 
 
650 ELEMRNTS OF QUATERNIONS. [bOOK III. 
 
 was treated as the locus ; and an analogous generation of the two hyperholoids, by 
 geometrical deformation of two corresponding surfaces of revolution^ with certain 
 resulting homologies (comp. sub-arts, to 274), through substitution o^ (centro-focaT) 
 ellipses for circles^ conducts to equations of those hyperboloids of the same unifocal 
 form ; namely, if a and /3 have significations analogous to those in the cited equa- 
 tion of the ellipsoid (so that /3 and not a is here a. focal line), 
 
 XLVin...(sey+(v|y=H:i 
 
 the upper or the lower sign being taken, according as the surface consists of one 
 sheet or of two. 
 
 (19.) It may also be remarked that as, by changing (3 to a in the corresponding 
 equation of the ellipsoid, we could return (comp. 404, (1.)) to a form (403, XI.) of 
 the equation of the sphere, so the same change in XLVIII. conducts to equations 
 of the equilateral hyperboloids of revolution, of one sheet and of two, under the very 
 simple forms* (comp. 210. XI.), 
 
 XLIX. .. S, ^ 1 =-1, and L . . Sl ^ j =+ 1 ; 
 
 {'.)•- 
 
 in which it seems unnecessary to insert points after the signs S, and of which the 
 geometrical interpretations become obvious when then they are written thus (comp. 
 199, v.), 
 
 LI. . . T^ = Vsec2f ?- ^^\ LII. . . T^=Vsec2^'' 
 
 2 
 
 where T - = op : ol, while I - is the inclination AOP of the semidiameler op to the 
 a a 
 
 axis of revolution oa, and-- — z - is the inclination of the same semidiameter to a 
 Z CI 
 
 plane perpendicular to that axis. 
 
 (20.) The real cyclic forms of the equation of the surface III. might be deduced 
 
 from the unifocal form XLIV., by the general method of the subarticles to 359 ; but 
 
 since we have ready the rectangular form X., it is simpler to obtain them from that 
 
 form, with the help of the identity, 
 
 LIII. . . - p« = (SpU (a + a'))2 + (SpU Vaa')^ + (SpU (a - a'))\ 
 
 by eliminatuig ihQ first of these three terms for the case of a single-sheeted hyperbo- 
 
 the common tangent planes, to this mean (or medial) ellipsoid, and to the given (or 
 generated) ellipsoid (abc), which are parallel to their common axis (26), being pa- 
 rallel also to the two umbilicar diameters of the latter surface. 
 
 * The same /orms, but with a for p, and /3 for a, may be deduced from XLVIII. 
 on the plan of 274, (2.), (4.), by assuming an auxiliary vector a such that 
 
 (TO (TO 
 
 S-^ = + S -, and V ^ = V — ; the homologies, above alluded to, between the general 
 pa P P 
 
 hyperboloid of either species, and the equilateral hyperboloid of revolution of the 
 
 same species, admitting also thus of being easily exhibited. 
 
CHAP. III.] EXPONENTIAL FORMS, EQU. OF CONFOCALS. 651 
 
 laid (for which 6"2 > a-2 > > c-2) ; the second for an ellipsoid (c 2 > b'2 > a-2 > 0) ; 
 and the third for a double-sheeted hyperboloid (a-2 > > c-2 > fe-s^, 
 
 (21.) Whatever the species of the surface III. maybe, we can always derive from 
 the unifocal form XLIV. of its equation what may be called an Exponential Trans- 
 formation ; namely the vector expression, 
 
 LIV. . . p = a;a + yNa% with LV. . . x^fa + y2yuVaa'= 1 ; 
 
 the scalar exponent, t, remaining arbitrary, but the two scalar coefficients, x and y, 
 being connected by this last equation of the second degree : provided that the new 
 constant vector /3 be derived from a, a', and e, by the formula, 
 
 ^ e + 8aa' 
 
 which gives after a few reductions (comp. the expression 315, III. for a', when 
 Ta = 1), 
 
 LVII. . . Va/3=UVaa', S (a' - ca) /3 = 0, Saa'i3=0; 
 LVIII. . . Va</3 = |3S.a'+ UVaa'.S.a'-i ; LIX. . . Y.aVa^(3 = a*\]Yaa =T-n ; 
 
 LX. . . S{a' — ea)p = x(e + Saa'), Yap-ya*JJVaa' ; 
 while LXI. . . /a = a-262c-*, and LXII. . . f^ =fVYaa' = 42. 
 
 (22.) If we treat the exponent, t, as the only variable in the expression LIV. 
 for p, then (comp. 314, (2.)) that exponential expression represents what we have 
 called (17.) a centro-focal ellipse ; the distance of its centre (or of its plane) from the 
 centre of the surface, measured along the focal line a, being represented by the co- 
 efficient sc ; and the radius of the right cylinder, of which the ellipse is a section, or 
 the radius of the circle (16.) into which that ellipse is projected, on a plane -L a, 
 being represented by the other coefficient, y : while ^tir is the exeentric anomaly. 
 
 (23.) If, on the contrary, we treat the exponent t as given, but the coefficients 
 X and y as varying together, so as to satisfy the equation LV. of the second degree, 
 the expression LIV. then represents a different section of the surface III., made by 
 a plane through the line a, which makes with the focal plane (of a, a') an angle 
 
 = — ; this latter section (like the former) being always real, if the surface itself 
 
 be such : but being an ellipse for an ellipsoid, and an hyperbola for either hyperbo- 
 loid, because 
 
 LXIIL . ./a./UVaa'=a-2c-2 by LXI. and LXIL 
 
 (24.) And it is scarcely necessary to remark, that by interchanging a and a' we 
 obtain a Second Exponential Transformation, connected with the second system (17.) 
 of centro-focal ellipses, as ihQ first exponential transformation LIV. is connected with 
 the/r«< system (16.). 
 
 (25.) The asymptotic conefp = has likewise its two systems of centro-focal 
 ellipses, and its equation admits in liiie manner of two exponential transformations, 
 of the form LIV. ; the only difference being, that the equation LV. is replaced by 
 the following, 
 
 LXIV. . . x^fa + yYUYaa'=0, 
 
 in which, for a real cone, the coefficients of x^ and y^ have opposite signs by (23.). 
 (26.) Finally, as regards the confocal relation of the surfaces III., which may 
 represent any confocal system of surfaces of the second order, it may be perceived 
 
652 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 from (4.) that an essential character of such a relation is expressed by the equa- 
 tion, 
 
 LXV. . . . Vr,0v, = Vj/0,v ; 
 
 ■which may perhaps be called, on that account, the Equation of ConfocaU. 
 
 (27.) It is understood that the two confocal surfaces here considered, are repre- 
 presented by the two scalar equations, 
 
 LXVI Sp^p = 1, Sp^^p = 1, or LXVI'. . . /p = 1, /p = 1 ; 
 
 and that the two linear and vector functions, v and v , of an arbitrary vector p, 
 which represent normals to the two concentric and similar and similarly posited sut' 
 faces, 
 
 LXVII. . . fp= const., fp = const., 
 
 passing through any proposed point p, are expressed as follows, 
 LXVIII. . . V = ^p, v^ = <p,p. 
 
 (28.) It is understood also, that the two surfaces LXVI. or LXVI'. are not only 
 concentric, as their equations show, but also coaxal, so far as the directions of their 
 axes are concerned : or that the two vector quadratics (comp. 354), 
 LXIX. . . \p(pp = 0, and LXX. . . Yptp^ = 0, 
 are satisfied by one common system of three rectangular unit lines. And with these 
 understandings, it will be found that the equation LXV., which has been called 
 above the Equation of Confocals, is not only necessary but sufficient, for the BStab- 
 lishment of the relation required. 
 
 (29.) It is worth while however to observe, before closing the present series of 
 subarticles, that the equations XIL, and those formed from them by introducing 
 C2 and V2, give the following among other relations : 
 
 LXXL . . /Uvi = (b'i - &i2)-i = -fiUv ; f{Uv2 = {h^ - fta^)"! = -/aUvi ; &c. ; 
 and LXXII. . .f{vi, V2) =/i(v2, v) =fi{v, vi) = ; 
 
 and therefore, 
 
 LXXIIL . . /i { (i22 - 6i2)iUv2 ± (&i2 - 62)iUv } = ; 
 whence it is easy to see that the two vectors under the functional sign/i in this last 
 expression have the directions of the generating lines of the single-sheeted hyperho- 
 loid (ei) through p, if we suppose that b-i^ > 6i2 > > 62, so that the confocal (cg) is 
 here an ellipsoid, and (e) a double-sheeted hyperboloid. 
 
 (30.) But if <T be taken to denote the variable vector of the auxiliary surface 
 XXIV., the equation of that surface may by (7.) and (8.) be brought to the follow- 
 ing rectangular form, with the meaning XXI. of w, 
 
 LXXIV. . . 1 = S<TW(T = CSp<T)2 _ 2Z2Sa(TSa'(T = b^ (S(rUi/)2 
 
 + 6i2(S(rUi/02+ b2^(S<TVv2y ; 
 hence, with the inequalities (29.), its cyclic normals, or those of its asymptotic cone 
 S(Tw<T = 0, or the focal lines of the reciprocal cone S<Tft*"'<T= 0, that is of the cone 
 XXXVI. , or finally the /oca/ lines of the focal* cone (12.), which rests on the focal 
 hyperbola, have the directions of the lines LXXIII. ; those focal lines are therefore 
 
 * A more general known theorem, including this, will soon be proved by quater- 
 nions. 
 
CHAP. III.] CONDITION OF CONTACT WITH RIGHT LINE. 653 
 
 (by what has just been seen) the generating lines of the hyperholoid (ci), which 
 passes through the given point p. 
 
 (31.) And for an arbitrary tt we have the transformation, 
 LXXV. . . /-2(Sp<T)2 - ^aaa'a = e (S(rUi/)2 + ci (S(TUvi)a + Ca(S«TUi/8)*. 
 
 408. The general equation* of conjugation, 
 
 l...(fp,p')=h 405,111. 
 
 connecting the vectors /a, p^ of any two points p, p' which are con- 
 jugate with respect to the central but non-conical surface fp = 1, may 
 be called for that reason the Equation of Conjugate Points ; while 
 the analogous equation, 
 
 n. ../(/>, />0 = o, 
 
 which replaces the former for the case of the asymptotic conefp = 0, 
 may be called by contrast the Equation of Conjugate Directions : in 
 fact, it is satisfied by any two conjugate semidiameters, as may be at 
 once inferred from the differential equation f{p, d/)) = of the surface 
 fp = const, (comp. 362). Each of these two formulge admits of nu- 
 merous applications, among which we shall here consider the 
 deduction, and some of the transformations, of the Equation of a 
 Circumscribed Cone, 
 
 III. . . {f(p, p')-lf={fp-l)(fp'-\); 
 which may also be considered as the Condition ofContacty of the right 
 line pp' with the surface fp= 1. 
 
 (1.) In this last view, the equation III. may be at once deduced, as the condi- 
 tion of equal roots in the scalar and quadratic equation (comp. 216, (2.), and 316, 
 
 (30.)), 
 
 I V. . . =f(xp + afp') - (a; + xy, 
 or V. . . 0=ar2(/p-l)-l-2a;x'(/((0, p')- 1) + a:'2(/p'- 1); 
 
 which gives in general the two vectors of intersection, as the two values of the ex- 
 
 xp + x'p' 
 
 pression — — . 
 
 *^ x + x' 
 
 (2.) If we treat the point p' as given^ and denote the two secants drawn from it 
 
 in any given direction t by ti^T and t-f^T, then t\ and t% are the roots of this other 
 
 quadratic, f{p'+t'h) = 1, or 
 
 VI. . . =f(tp' + r) - <a = <a(/p' - 1) + 2tf(p% r) +fr ; 
 
 denoting then by tQ~W the harmonic mean of these two secants, so that 2tQ = ti + ^2, 
 
 and writing p = p' + to'^T, we have 
 
 VII. . . *o (1 -fp) =f{p\ r), /(p, pO = 1 ; 
 
 * For the notation used, Art. 362 may be again referred to. 
 
654 ELEMENTS OF QUATEllNIONS. [bOOK III. 
 
 we are then led in this way to the formula I., as the Equation of the Polar Plane 
 of the point p', if that plane be here supposed to be defined by its well-known har- 
 monic property (comp. 215, (16.), and 316, (31.), (32.)). 
 
 (3.) At the same time we obtain this other form of the condition of contact III., 
 as that of equal roots in VI., 
 
 VIIL../(p',r)2=/r.(/p'-l), 
 the first member being an abridgment of (/(p', r))' : and because this last equation 
 VIII. is homogeneous with respect to r, it represents a cone, namely the Cone of Tan- 
 gents (r) to the given surface /p = 1, from the given point p'. Accordingly it is easy 
 to prove that the equation III. may be thus written, 
 
 IX. . . /(p', p - p'Y =f(p - pO . (/p- - 1), 
 
 under which last form it is seen to be homogeneous with respect to p - p'. 
 
 (4.) Without expressly introducing r, the transformation IX. shows that the 
 equation III. represents some cone, with the given point p' for its vertex ; and be- 
 cause the intersection of this cone with the given surface is expressed by the square 
 of the equation I. of the polar plane of that point, the cone must be (as above stated) 
 circumscribed to the surface /p= 1, touching it along the curve (real or imaginary) 
 in which that surface is cut by that plane I. 
 
 (5.) Another important transformation, or set of transformations, of the equation 
 III. may be obtained as follows. In general, for any two vectors p and p', if the 
 scalar constant m, the vector function t^, and the scalar function F, be derived from 
 the linear and vector function ^, which is here self- conjugate (405), by the method 
 of the Section III. ii. 6, we have successively, 
 
 X. ../(p, p'y-fp.fp=Sp<l>p.Spipp-Sp<pp.Sp<pp' = S(^Vpp'.Y,pp,pp') 
 
 = S.pp'i//Vpp' = mS. pp'^-Wpp' = mFVpp' ; 
 and thus the equation III. of the circumscribed cone becomes, 
 
 XI. . . mFYpp' + f(p - p') = 0, or XII. . . mFVrp' +/r = 0, 
 if r = p — p' be a tangent from p'. Or because ^\p = m, and m = — C1C2C3 = - a~^b 2c"2, 
 by 406, XXIV., we may write (with r — p — p') either 
 
 XIII. . . = Sri//-»r + Su^-«t;, if v=Vrp' = Vpp', 
 or XIV. . . FYpp' = a-ib^c'if{p - p'), 
 
 as the condition of contact of the line pp' with the surface fp = 1. 
 
 (6.) A geometrical interpretation, of this \a,&t form XIV. of that condition, can 
 easily be assigned as follows. Supposing at first for simplicity that the surface is an 
 ellipsoid, let p be the point of contact, so that fp = 1, /(p, r) = ; and let the tangent 
 pp' be taken equal to the parallel semidiaraeter ot, so that/r =/(p — p') = 1. Then, 
 with the signification XIII. of v, the equation XIV. becomes, 
 
 XV. . . VFv = Tu.VFUv = aic; 
 in which the factor Tw represents the area of the parallelogram under the conjugate 
 semidiameters op, ot of the given surface /p = 1 ; while the other factor i F\Jv re- 
 presents the reciprocal of the semidiameter of the reciprocal surface Fv = 1, which is 
 perpendicular to their plane pot ; or the perpendicular distance between that plane, 
 and a parallel plane which touches the given ellipsoid : so that their product ^f Fv is 
 equal, by elementary principles, to the product of the three semiaxes, as stated in the 
 formula XV. And the result may easily be extended by squaring, to other central 
 surfaces. 
 
CHAP. III.] AXES OF CIRCUMSCRIBED CONE. 655 
 
 (7.) It may be remarked in passing, that if p, c, t be any three conjugate semi- 
 diameters of any central surface /p = 1, so that 
 
 XVI. . . /p =/(r -/r = 1, and XVII. . . /(p, a) =fi<r, r) =/(r, p) = 0, 
 and if xp + i/(T + zr be ant/ other semidiameter of the same surface, we have then the 
 scalar equation, 
 
 XVIII. . . f(xp + y<T + zt) = x^ + ya+ 22 = 1 ; 
 
 a relation between the coeflScients, x, y, z, which has been already noticed for the 
 ellipsoid in 99, (2.), and in 402, I., and is indeed deducible for that surface, from 
 principles of real scalars and real vectors alone : but in extending which to the Ay- 
 perboloids, one at least of those three coefficients becomes imaginary, as well as one 
 at least of the three vectors p, tr, r. 
 
 (8.) Under the same conditions XVI. XVII., we have also, 
 XIX. . . Ypa =±abc<l)T =±(- myiipT ', 
 XX. . . r = ± (- m^^-^Ypff = + (- myiV<pp<p<T ; 
 XXI. . . Sp<TT = + aJc = f (- »ra)-» ; 
 together with this very simple relation, 
 
 XXII. . . Spcrr.S0p0(7^r = -l. 
 (9.) Under the same conditions, if xp + ya + zt and x'p 4 y'ff + z't have only 
 conjugate directions, that is, if they have the directions of any two conjugate semi- 
 diameters, the six scalar coefficients must satisfy (com p. II.) the equation, 
 XXIII. .. xx' + yy' + zz' = 0. 
 (10.) The equation VIII., with p for p', may be written under the form, 
 XXIV. . . = S(Tr = Srwr, if XXV. . . d = wr = ^pSp^r + 0r(l -/p), 
 ~ a new linear and vector function, which represents a normal to the cone of tan- 
 gents from p, to the surface yp = 1. Inverting this last function, we find 
 
 XXVI...r=:o.-. = ^"^l^^ 
 
 the equation in <t of the reciprocal cone, or of the cone of normals to the circum- 
 scribed cone from p, is therefore, 
 
 XXVII. . . Sffw-'cr = 0, or XXVIII. . . F<t = (Sp(T)2, or finally 
 XXVII r. . .F(i<T: Spcr) = 1 ; 
 a remarkably simple form, which admits also of a simple interpretation. In fact, 
 the line <t : Spcr is the reciprocal of the perpendicular, from the centre o, on a tan- 
 gent plane to the cone, which is also a tangent plane to the surface ; it is therefore one 
 of the values of the vector v (comp. (Jo.), and 373, (21.)), and consequently it is a 
 semidiameter of the reciprocal surface Fv= 1. 
 
 (11.) As an application of the equation XXVIII., let the surface be the confo- 
 cal (e), represented by the equation 407, III. or X., of which the reciprocal is re- 
 presented by 407, XVII. or XVIII. Substituting for Fa its value thus deduced, 
 the equation of the reciprocal cone (10.), with a for a side, becomes,* 
 XXIX. . . 2l^Sa(TSa(T - (Sp«T)2 = b^a^, or XXIX'. . . Saaa'a- ^(Spffy = C(r« ; 
 if then the vertex p he^ed, but the confocal vary, by a change of e, or of 6' which 
 
 * It may be observed that, when 6 = 0, this equation XXIX. represents the 
 asymptotic cone to the auxiliary surface 407, XXIV. ; and at the same time the re- 
 ciprocal of that /oca/ cone, 407, XXXVI., which rests on t\iQ focal hyperbola. 
 
656 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 varies with it, the cone XXIX. will also vary, but will belong to a hiconcyclic sys- 
 tem ; whence it follows that the (direct or) circumscribed cones from a given point are 
 all biconfocal: and also, by 407, (30.), that their common focal lines are the gene- 
 rating lines of the confocal hyperholoid* of one sheet, which passes through their 
 common vertex. 
 
 (12.) Changing e to e^ in XXIX'., and using the transformation 407, LXXV., 
 with the identity (comp. 407, LIII.), 
 
 - (t2 = (S(rUi/)2 + (SffUi/i) + (S(rUv2)2, 
 we find that if (r be a normal to the cone of tangents from p to (cj, it satisfies the 
 equation, 
 
 XXX. . . = (e- e,) (S(rUj/)2 + (ci -O (SffUvi)^ + (c2- O (SfTUi/s)^ ; 
 and therefore that if r be a tangent from the same point p, to the same confocal (e,), 
 it satisfies this other condition, 
 
 XXXI. . . = (c - e,)-> (SrUi.)2 + (ei - e,)"! (SrUvi)s + (ea - c,)-i (SrUj/2)2, 
 which thus is a form of the equation of the circumscribed cone to (e^), with its ver- 
 tex at a given point p : the confocal character (11) of all such cones being hereby 
 exhibited anew. 
 
 (13.) It follows also from XXXI., that the axes of every cone thus circumscribed 
 have the directions of the normals v, v\, v^ to the three confocals through p ; and 
 this known theoremf may be otherwise deduced, from the Equation of Confocals 
 (407, LXV.), by our general method, as follows. That equation gives 
 V,— V II (^v (because <f>v, = <^v'), and therefore, 
 
 XXXII. . . {y^~v)^vv=ip,v{fQ-l\ Vvj/,Si^v,+ Vi/0,v(l-/p) = O; 
 changing then V to S, and v to r, we see that v, vi, 1/2, as being the roots (3.54) of 
 this last vector quadratic XXXII., have the directions of the axes of the cone, with 
 
 r for side, 
 
 XXXIII. . ./Xp, r)2+/r.(l-/p) = 0; 
 
 that is, by VIIL, the directions of the axes of the cone of tangents, from p to (ej. 
 
 (14.) As an application of the formula XIV., with the abridged symbols r and i; 
 of (5.) for p — p' and Vpp', the condition of contact of the line pp' with the confo- 
 cal (e) becomes, by the expressions 407, III., XVIII., and VII. for the functions 
 /, F, and the squares a^, b% c\ the following quadratic in e : 
 
 XXXIV. . . (Sar)« - 2eSarSaV + (Sa'r)2 + (1 - e«) t2 = l'^ (Sava'v - eu2) ; 
 
 there are therefore in general (as is known) two confocals, say (c) and (e J, of a given 
 system, which touch a given right line ; and their parameters,"^ e and e^, are the two 
 roots of the last equation : for instance, their sum is given by the formula, 
 XXXV. . . (e + e,)r2 = l-^v^ - 2SarSaV. 
 
 * This theorem (which includes that of 407, (30.)) is cited from Jacobi, and is 
 proved, in page 143 of Dr. Salmon's Treatise, referred to in several former Notes, 
 t Compare the second Note to page 648. 
 X This name of parameter is here given, as in 407, to the arbitrary constant 
 
 e = — -, of which the value distinguishes one confocal (e) of a system from another. 
 
CHAP. III.] CIRCUMSCRIBED RIGHT CONES. 651 
 
 (15.) Conceive then that p is a given semidiameter of a given confocal (e), and 
 that djO is a tangent, given in direction, at its extremity ; the equation XXXIV. will 
 then of course be satisfied,* if we change r to dp, and v to Vpdp, retaining the given 
 value of e ; but it will also be satisfied, for the same p and dp ("or for the same r and 
 v), when we change e to this new parameter, 
 
 XXXVI. ..e=-e-\- 2SaUdp . Sa'Udp - ^2 (VpUdp)2 ; 
 that is to say, the new confocal (e ), with a parameter determined by this last for- 
 mula, will touch the given tangent to the given confocal (e). 
 
 (16.) If we at once make /2 = Q in the equation 407, III. of a Confocal System 
 of Central Surfaces, leaving the parameter e finite, we fall back on the system 406, 
 XXXV. of Biconfoeal Cones ; but if we conceive that V^ only tends to zero, and 
 that e at the same time tends to positive infinity, in such a manner that ihQic pro- 
 duct tends to a. finite limit, r^, or that 
 
 XXXVII. ..lim.; = 0, lira. 6=00, lim.eZ2 = r2, 
 then the equation of the surface (e) tends to this limiting form, 
 
 XXXVIII. . . p3 + r8= 0, or XXXVIII'. . . Tp = r ; 
 a system of biconfoeal cones is therefore to be combined with a system of concentric 
 spheres, in order to make up a complete confocal system. 
 
 (17.) Accordingly, any given right line pp' is in general touched by only one 
 cone of the system just referred to, namely by that particular cone (e), for which 
 (comp. XXXIV.) we have the value, 
 
 XXXIX. . . e= SawaV, or XXXIX'. . . c + Sua' = 2SauSa'u-i, 
 with V = ypp', as before, so that v is perpendicular to the given plane opp', which 
 contains the vertex and the line ; in fact, the reciprocals of the biconfoeal cones 
 406, XXXV., when a, a' are treated as given unit lines, but e as a variable para- 
 meter^ compose the biconcyclief system (comp. 407, XVIII.), 
 
 XL. . . Sava'v = ev^. 
 But, besides the tangent cone thus found, there is a tangent sphere with the same 
 centre o ; of which, by passing to the limits XXXVIL, the radius r may be found 
 from the same formula XXXIV. to be, 
 
 r p-p 
 and such is in fact an expression (comp. 316, L.) for the length of the perpendicular 
 from the origin on the given line pp'. 
 
 (18.) In general, the equation XXXIV. is a form of the equation of the cone, 
 with p for its variable vector, which has a given vertex p', and is circumscribed to a 
 given confocal (e). Accordingly, by making e = -Saa' in that formula, we are 
 
 * In fact it follows easily from the transformations (6.), that 
 fp ./dp - a-26-2c-2FVpdp =/(p, dp)2. 
 
 t The bifocal form of the equation of this reciprocal system of cones XL. was 
 given in 406, XXV., but with other constants (\, p,, g), connected with the cyclic 
 form (406, I.) of the equation of the given system. 
 
 4 p 
 
658 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 led (after a few reductions, comp. 407, XXVII.) to an equation which may be thu3 
 written, 
 
 XLII. . . 0=Z2(SaaV)H2Sap'rSayr, 
 
 with the variable side T = p-p', as before ; and which differs only by the substitution 
 of p' and r for p and v, from the equation 407, XXXVI. for that focal cone, which 
 rests on the focal hyperbola. The other (real) focal cone which has the same arbi- 
 trary vertex p', but rests on the focal ellipse, has for equation, 
 XLIII. . . Z2(S(a-a')r)2 = Saua'u-i/2, 
 
 as is found by changing e to 1 in the same formula XXXIV. 
 
 (19.) It is however simpler, or at least it gives more symmetric results, to change 
 e^ in XXXI. to — Saa' for the focal hyperbola, and to + 1 for the focal elUpse, in 
 order to obtain the two real focal cones with p for vertex, which rest on those two 
 curves; while that third and wholly imaginary focal cone, which has the same ver- 
 tex, but rests on the known imaginary focal curve, in the plane of h and c, is found 
 by changing e^ to — 1. This imaginary focal cone, and the two real ones which rest 
 as above on the hyperbola and ellipse respectively, may thus be represented by the 
 three equations, 
 
 XLIV. . . = a-2(SrUi;)2 + «r2(SrUi/i)2 + «2-2(SrUv2)2 ; 
 
 XLV. . . = 6-2(SrU»/)2 + 6i-2(SrUvi)2 + 62-2(SrUr2)2; 
 
 XLVI. . . = c-2 (SrUj/)2 + cr^ (SrUj/i)2 + C2-2 (SrUj/a)^ ; 
 
 r being in each case a side of the cone, and v, v\, v^ having the same significations 
 
 as before. 
 
 (20.) On the other hand, if we place the vertex of a circumscribed cone at a point 
 
 p of Bifocal curve, real or imaginary, the enveloped surface being the confocal (c,), 
 
 we find first, by XXX., for the reciprocal cones, or cones of normals a, with the 
 
 same order of succession as in (19.), the three equations, 
 
 XLVII...a2(sUv(r)2 = a^2. 
 
 XLVIII...62(SUi/<t)2=6,2. 
 
 XLIX. ..c2(SUj/(t)2=c^2. 
 
 and next, for the circumscribed cones themselves, or cones of tangents r, the con- 
 nected equations : 
 
 L. .. a2(vUvr)2+«,2 = 0; 
 LI. .. 62 (VUj/r)2+ 6,2 = 0; 
 LIL. . c2(VUvr)2 + c,2 = 0; 
 all which have the forms of equations of cone* of revolution, but on the geometri- 
 cal meanings of the three last of which it may be worth while to say a few words. 
 
 (21.) The cone L, has an imaginary vertex, and is always iVse^f imaginary ; but 
 the two other cones, LI. and LII., have each a real vertex p, with b^ >0 for the 
 first, and c2 < for the second ; 6 being the mean semiaxis of the ellipsoid, which 
 passes through a given point of the focal hyperbola, and c2 being the negative and 
 algebraically least square of a scalar semiaxis of the double-sheeted hyperboloid, 
 •which passes through a given point of the /oca/ ellipse: while, in each case, v 
 has the direction of the normal to the surface, which is also the tangent to the curve 
 at that point, and is at the same time the axis of revolution of the cone. 
 
 (22.) The semiangles <ii\X\e two last cones, LI. and LIL, have for their respec- 
 tive sines the (wo quotients, 
 
CHAP. III.] TWELVE UMBILICAIl VECTORS. 659 
 
 LIII. . . 6, : 6, and LIV. . . (- c^2)i : (- c2)i ; 
 each of those two cones is therefore real, if circumscribed to a single-sheeted hyper- 
 boloid, because, for such an enveloped surface (ej, b^ is real, and less than the b of 
 c«y confocal ellipsoid, while c^ is imaginary, and its square is algebraically greater 
 (or nearer to zero) than the square of the imaginary semiaxis c of everg double' 
 sheeted hyperboloid, of the same confocal system ; but the cone LT. is imaginary, if 
 the enveloped surface (e,) be either an hyperboloid of two sheets (b^ imaginary), or 
 an exterior ellipsoid (b^> b) ; and the other cone LII. is imaginary, if the surface 
 (ej be either any ellipsoid (c, real), or else an exterior and dowfiZe-sheeted hyperbo- 
 loid (a,2< a"^, c^<c^, - c^2 >_c2). Accordingly it is known that the focal hyper- 
 bola, which is the locus of the vertex of the cone LI., lies entirely inside every double- 
 sheeted hyperboloid of the system ; while the focal ellipse, which is in like manner 
 the locus of the vertex of the cone LII., is interior to every ellipsoid: and real tan- 
 gents to a sin^'Zc- sheeted hyperboloid can be drawn, from every real point of space. 
 
 (2-3.) The twelve points (whereof only four at most can be real), in which a 
 surface (e) or (abc') is cut by the three focal curves, are called the Umbilics of that 
 surface ; the vectors, say w, w,, w^^, of three such umbilics, in the respective planes 
 of ca, ab, be, are : 
 
 LV. ..(u =^(a + a') + ^(a-a'); 
 
 ^,,^ aCa + a) \/^bYaa' 
 
 1 - baa 1 — Saa 
 
 LVlL...=i^^Z^^-^^I^I^; 
 " l + Saa' 1+Saa' ' 
 
 and the others can be formed from these, by changing the signs of the terms, or of 
 some of them. The four real umbilics of an ellipsoid are given by the formula LV., 
 &ud those oi a. double-sheeted hyperboloid by LVI., with the changes of sign just 
 mentioned. 
 
 (24.) In transforming expressions of this sort, it is useful to observe that the ex- 
 pressions for the squares of the semiaxes, 
 
 a2 = Z3(e + l), 62 = Z2(e + Saa'), e^ = P(ie-l), 407, VII. 
 
 combined with Ta = Ta'= 1, give not only a^ — c^ = 2P, but also, 
 
 and LX. . . TVaa' = V(l - (Saa')2) = sin Z - = l'^ (cfi - &2)J (62 _ c-i)h, 
 
 with the verification, that because 
 
 LXI. . . (a - a) (a + a') = 2Vaa', 
 therefore LXI'. . . T(a - a').T(a + a') = 2T Vaa'. 
 
 We have also the relations, 
 
 LXII. . . T(a-i-a')-2 + T(a-a)-2 = (TVaa')-'; 
 LXIII. . . T (a + a')-^ - T (a - a')"^ = Saa'. (T Vaa')"' ; 
 with others easily deduced. 
 
660 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (25.) The expression LV. conducts to the following among other consequences, 
 which all admit of elementary verifications,* and may be illustrated by the annexed 
 Fig. 84. Let u, u' be the two real points 
 in which an ellipsoid (ahc) is cut by one 
 branch of the focal hyperbola, with h for 
 summit, and with r for its interior focus ; 
 the adjacent major summit of the surface 
 being e, and r, r' being (as in the Figure) 
 the adjacent points of intersection of the 
 same surface with the focal lines a, a', that 
 is, with the asymptotes to the hyperbola. 
 Let also v, t be the points in which the 
 
 same asymptotes a, a' meet the tangent to ^ Y\s.. 84. 
 
 the hyperbola at u, or the normal to the 
 ellipsoid at that real umbilic, of which we may suppose that the vector ou is the w 
 of the formula LV. ; and let s be the foot of the perpendicular on this normal to the 
 surface, or tangent TV to the curve, let fall from the centre o. Then, besides the 
 obvious values, 
 
 LXIV. ..oE = «, 0F = (a'-c2)J, 3H = (a2-5^>\ 
 and the obvious relations, that the intercept tv is bisected at u, and that the point 
 F is at once a summit of the focal ellipse, and a focus of that other ellipse in which 
 the surface is cut by the plane (ac) of the tigure, we shall have these vector expres- 
 sions (comp. 371, (3.), and 407, VIII. LXI.) : 
 
 LXV. . . ov = (a + c) a, OT = («-c)a', TV = a(a-a')+ c(a + a'); 
 
 LXVI. . . su-i = 0w = - — (a + a') - — ' (a - a'), su = - ac : tu ; 
 
 LXVII. . . OR = — — = ab-^cUf ok' = - — ; = ab'ha' ; 
 V/a V/a 
 
 whence follow by (24.) these other values, 
 
 LXVIII. . . ov = « + c, oT = a-c, TV = 26; 
 
 LXIX. . . TU = uv =^ &, sij = or = or' = «6~^c; 
 
 LXX. . . 6xj = Ta» = (a2_62 + e2)i; 
 
 LXXI. . . OS = (a^ - 62 4. c2 _ «2J-2c2)J = 6-1 (<i2 _ 52)J (52 _ c2)i. 
 
 (26.) It follows that the lengths of the strfes ov, ox, tv of the umhilicar triangle 
 TOV are equal to the sum and difference (a + c) of the extreme semiaxes, and to the 
 mean axis (26) of the ellipsoid ; while the area of that triangle = OS- TU = («^ — 62)i 
 (6- - c2)i = the rectangle under the tivo semiaxes of the hyperbola, if both be treated 
 as real. The length (T^w)-', or su, of th.^ perpendicular from the centre o, on the 
 tangent plane at an umbilic u, is ab'^c ; and the sphere concentric with the ellipsoid, 
 which touches the four umbilicar tangent planes, passes through the points R, r' of 
 intersection of that ellipsoid with the focal lines a, a', that is, as before, with the 
 
 * Some such verifications were given in the Lectures, pages 691, 692, in con- 
 nexion with Fig. 102 of that former volume, which answered in several respects 
 to the present Fig. 84. 
 
CHAP. III.] EIGHT UMBILICAR GENERATRICES. 661 
 
 asymptotes to the hyperbola ; or, by (21.)(22.), with the axes of the two circum- 
 gcribed right cylinders. * And finally the length, say u, of the umhilicar semidia- 
 vieter ou, is given by the formula, 
 
 LXXII. . . «2 = «2 _ J2 4. c2 ; 
 
 all which agrees (25.) with known results. 
 
 (27.) An umbilie of a surface of the second order may be otherwise defined 
 (comp. (23.)), as a real or imaginary point at which the tangent plane is parallel to 
 a cyclic plane ; and accordingly it is easy to prove (comp. 407, (20.)) that the um- 
 hilicar normal ^w in LXVI. has the direction of a cyclic normal. To employ this 
 known property in verification of the recent expressions (25.), (26.), for the lengths 
 of ou and su, it is only necessary to observe that the common radius of the diame- 
 tral and circular sections of the ellipsoid is tlie mean semiaxis b (comp. 216, (7.) 
 (9.), &c.) ; and that, by a slight extension of the analysis in (7.), (8.), (9.), it can be 
 shown that if p, cr, r and p', c', r' be any two systems of three conjugate semidiame- 
 ters of any central surface, fp = 1, then 
 LXXIII. . . p'2 + (t'2 + r'2 = p2 + <y2 + ^2, and LXXIV. . . (Sp'a'r'y = (Sp(rr)2. 
 
 (28.) A less elementary verification of the value LXXII. of «2^ but one which is 
 useful for other purposes, may be obtained from either the cubic in 62^ or that in c, 
 assigned in 407, (8.). For if bo^, bi^, bo^ be the roots of the former cubic, and eo, 
 ci, €2 the roots of the latter, inspection of those equations shows at once that we 
 have generally, 
 
 LXXV. . . -p'' = bo^ + bi^ -^ b^"' -2l^Saa =P (eo+ ei + e2+Saa); 
 or LXXVI. . . OP 2 = Tp2 = ao2 + Ji3 + C22 = bo^ + ci^ + «2^ = &c., 
 
 where the semiaxes uq, 61, C2 belong to the three confocals through any proposed 
 point p. Making then, 
 
 LXXVII. . .ao^ = a^, b^^ = 0, C22 = c2 - b^, 
 we recover the expression assigned above, for the square of the length u of an um- 
 bilicar semidiameter of an ellipsoid. 
 
 (29.) For any central surface, the principle (27.) shows that if X, p. be, as in 
 405, (5.), &c., the two real cyclic normals, and if ^ be the reaZsca/ar associated with 
 them as before, then the vectors of the four real umbilics (if such exist) must admit of 
 being thus expressed : 
 
 LXXVIII. . . + 0-iX : MFX = ± ahc (pU\ + ^T\) ; 
 LXXIX. . . +^- V : Vi^/i =+ «6c {g\]p + Wp) ; 
 
 and thus we see anew, that an hyperboloid with one sheet has (as is well known) no 
 
 * Compare 218, (5.), and 220, (4.); in which the points b, b' (comp. also 
 Fig. 53, page 226) may now be conceived to coincide with the points k, k' of the 
 new Figure 84. It is obvious that the theory of circumscribed cylinders is included 
 in that of circumscribed cones; so that the cylinder circumscribed to the confocal (c), 
 with its generating lines parallel to a given (real or imaginary) semidiameter y of 
 that surface (/y = 1), may be represented (comp. III. XIV.) by the equation, 
 
 III'. . . f(p, y)2 =/p - 1 ; or XIV'. . . FYyp = a^b'^-c^ ; 
 with interpretations easily deduced, from principles already established. 
 
662 ELEMENTS OF QUATEIINIONS. [boOK III. 
 
 real nmbilic, because for that surface the product abc of the semiaxes is imaginary ; 
 or because it has no real tangeyit plane parallel to eitlier of its two real planes of 
 circular section. 
 
 (30.) Of whatever species the surface may be, the three umhilicar vectors (23.), 
 of which only one at most can be real^ with the particular signs there given, but 
 which have t\\& forms of lines in the three principal planes, must be conceived, in 
 virtue of their expressions LV. LVI. LVII., to terminate on an imaginary right 
 line, of which the vector equation is, 
 
 Lxxx. . . p=-^''+/)_v— /C^' + s--') ^±:zl), 
 
 a^-a Yaa' a- a 
 
 e' being a scalar variable, which receives the three values, — Saa', + 1, and — 1, when 
 p comes to coincide with w, w^, and a>,, respectively. And such an imaginary right 
 line, which is easily proved to satisfy, for all values of the variable e', both the rect- 
 angular and the bifocal forms of the equation of the surface (e), or to be (in an 
 imaginary sense) wholly contained upon that surface, may be called an Umbilicar 
 Generatrix. 
 
 (31.) There are in general eight such generatrices of any central surface of the 
 second order, whereof each connects three umbilics, in the three principal planes^ 
 two passing through each of the twelve umbilicar points (28.) ; and because e'^ dis- 
 appears from the square of the expression LXXX. for p, which square reduces itself 
 to the following, 
 
 LXXXI. . . p2 = - Z2 (2e' + e + Saa) = - 62 _ 2Pe, 
 
 they may be said to be the eight generating lines through the four imaginary points, 
 
 in which the surface meets the circle at infinity. 
 
 (32.) In general, from the cubics in e and in b'^, or from either of them, it may 
 
 be without diflSculty inferred (comp. (28.)), that the eight intersections (v&dX ox imsL- 
 
 ginary) of any three confocals (cq) (ei) (^2) have their vectors p represented by the 
 
 formula : 
 
 y ±aQa\a2 V— l&o6i&2 coC\C2 
 
 LXXXII. . . p— —^ jT + 
 
 l\a ^ a') - /3 Vaa' " ^2 (a - a) ' 
 
 comparing which with the vector expression LXXX., we see that the three confo- 
 cals, through the point determined by that former expression, for any given value of 
 c', are (e), (e'), and (e') again ; and therefore that two of the three confocal surfaces 
 ihxo-agh. any point oiaxi umbilicar generatrix {ZQ.') coincide : a result which gives 
 in a new way (comp.LXXV.) the expression LXXXI. for p2. 
 
 (33.) The locus of all such generatrices, for all the confocals (e) of the sy.stem, 
 is a certain ruled surface, of which the doubly variable vector may be thus expressed, 
 as a function of the two scalar variables, e and e' : 
 
 TYYYTTT ' +/rg + !>(«'+ 1) , M~U{ e+Saay{e'+Saa) 
 
 LiAAAiii. . . Pee = ; r — ; 
 
 a-\-a \aa 
 
 . l(e-l)He-l) 
 
 and because we have thus, for any one set of signs, the differential relation, 
 LXXXIV. . . Drp„e=aD./,p„/, 
 
LXXXV. 
 
 CHAP. III.] DEVELOPABLE LOCUS AND ENVELOPE. G63 
 
 it follows that this ruled locus is a Developable Surface : its edge of regression being 
 that wholly imaginary curve, of which the vector is pc,e, and which is therefore by 
 (32.) the locus of all the imaginary points, through each of which pass three coinci- 
 dent confocals. 
 
 (34.) The only real part of this imaginary developable consists of the two real 
 focal curves, which are double lines upon it, as are also the imaginary focal, and the 
 circle at infinity (31.) ; and the scalar equation of the same imaginary surface, ob- 
 tained by elimination of the two arbitrary scalars e and e, is found to be of the 
 eighth degree, namely the following : 
 
 = 2m2a^ + 22ni (m - n)arV + 2(/>2 - Qmn)x*y*^ 
 
 + 22 (3m2 - np)xY'z'- + 22m2(n -p)x^ + 2^m(mp - 3n2) x*yi 
 
 + 2(m-n) (n-p)(p- m) x^y^z^ + 2m2 (m^ - 6np)x^ 
 
 + 2'2mn {mn - 3p-) x'^y^ + 2 S»i2«/> (p -n)x^-\- ni^n^p^ ; 
 
 in which we have written, for abridgment, 
 
 LXXXVI. . .x = - S,oU(a + a'), y = - SpUVaa', z = - S|oU(a - a'), 
 and LXXXVIL . . m = 62 _ c2, n = «2_a2^ p = a^-b^, 
 
 so that LXXXVIIL . . »i + n+j!) = 0; 
 
 while each sign S indicates a sum of three or of six terms, obtained by cyclical or 
 binary* interchanges. 
 
 (35.) From the manner in which the equation of this imaginary surface (33.) or 
 (34.) has been deduced, we easily see by (32.) that it has the double property : 
 Lst of being (comp. (20.)) the locus of the vertices of all the (real or imaginary) 
 right cones, which can be circumscribed to the confocals of the system ; and II. nd of 
 being at the same time the common envelope of all those confocals : which envelope 
 accordingly is known to be a developable^ surface. 
 
 (36.) The eight imaginary lines (31.) will come to be mentioned again, in con- 
 nexion with the lines of curvature of a surface of the second order ; and before closing 
 the present series of subarticles, it may be remarked that the equation in (15.), for the 
 determination of the second confocal (e^ which ^owc^es a given tangent, dp or pp', to 
 a given surface (e) of the same system, will soon appear under a new form, in con- 
 nexion with that theory of geodetic lines, on surfaces of the second order, to which 
 we next proceed. 
 
 * When xyz and abc are cyclically changed to yzx and bca, then mnp are 
 similarly changed to npm ; but when, for instance, retaining x and a unchanged, we 
 make only binary interchanges of y, z, and of b, c, we then change m, n, and p, to 
 - m, -p, and - n respectively. 
 
 f This theorem is given, for instance, in page 157 of the several times already 
 cited Treatise by Dr. Salmon, who also mentions the double lines &c. upon the sur- 
 face ; but the present writer does not yet know whether the theory above given, of 
 the eight umbilicar generatrices, has been anticipated: the locus (33.) of which ima- 
 ginary right lines TSO.) is here represented by the vector equation LXXXIIL, from 
 which the scalar equation LXXXV. has been above deduced (34.), and ought to be 
 found to agree (notation excepted) with the known co-ordinate equation of the 
 developable envelope (35.) of a confocal system. 
 
664 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 409. A general theory of geodetic lines^ as treated by quater- 
 nions, was given in the Fifth Section (III. iii. 5) of the present 
 Chapter ; and was illustrated by applications to several different 
 families of surfaces. We can only here spare room for applying the 
 same theory to the deduction, in a new way, of a few known but 
 principal properties of geodetics on cetitral surfaces of the second or- 
 der ; the differential equation employed being one of those formerly 
 used, namely (comp. 380, IV.), 
 
 I. . . Yvd?p = 0, if II. . . Td/> = const. ; 
 
 that is, if the arc of the geodetic be made the independent variable. 
 
 (1.) In general, for any surface^ of which r is a normal vector, so that the first 
 differential equation of the surface is Srdp = 0, the second differential equation 
 dSi^d/>= gires, by I., for a geodetic on that surface, the expression, 
 
 III. . . d2ja = -v-»Sdvdp. 
 
 (2.) Again, the surface yjb = const, being still quite general, if we write (comp. 
 363, X'., 373, III., &c.), 
 
 IV. . . d/p = 2Si/dp = 2S<ppdp, we shall have V. . . d/dp = 2S(^dp . d2p) ; 
 
 and therefore, by III., for a geodetic, 
 
 YL.. J^ + 2S*^=0. 
 Sdpd^p 0p 
 
 (3.) For a central surface of the second order, ^p is a linear function, and we 
 may write (comp. 361, IV.), 
 
 VII. . . ^dp = d^p = dv, Sdpd^p = Sdp^dp =/dp ; 
 
 the general differential equation VI. becomes therefore here, 
 
 VIII... 3^^ + 28^ = 0; 
 Jdp V 
 
 and gives, by a first integration, with the condition II., 
 
 IX. . . j/2/dp = ^dp2, or IX'. . . Tv2/Udp = ^ = const. ; 
 
 or X. . . P-W-^=h, or X'. . . P. Z) = A"! = const. ; 
 
 where P = Tv'i = perpendicular from centre on tangent plane, 
 
 and D = (/Udp)* = semidiameter parallel to tangent ; 
 
 these two last quantities being treated as scalars, whereof the latter may be real or 
 
 imaginary,* together with the last scalar constant h~K 
 
 * For the case of the ellipsoid, for which the product P. D is necessarily real, the 
 foregoing deduction, by quaternions, of Joachimstal's celebrated first integral, 
 P.D = const., was given (in substance) in page 680 of the Lectures. 
 
CHAP. III.] GEODETICS ON CENTRAL SURFACES. 665 
 
 (4.) Tiie following is a quite different way of accoraplishing a first integration, 
 which conducts to another known result of not less interest, although rather of a 
 graphic than of a metric kind. Operating on the equation 407, XVI. by S-dp, and 
 remembering that SjOv= 1, and Sj/dp = 0, we obtain the differential equation^ 
 
 XI. . . ^pv^pAp = P- (SaVSadp + SavSa'd,o) ; 
 that is, by I. and XL, 
 
 XII. . . Spdp.Spd2jo-p?SdpdV = ^d(Sadp.Sa'dp), 
 in which the first member, like the second, is an exact differential, because 
 
 XIII. . . S(Vpdp .VpdV) = ^d(Vpdp)« ; 
 hence, for the geodetic, 
 
 XIV. . . Z-2(Vpdp)2-2SadpSa'dp=/j'dp«, 
 or XV. . . 2SaUdp . Sa'Udp - ^2 ( VpUdp)2 = h\ 
 
 h' being a new scalar constant. 
 
 (5.) Comparing this last equation with the formula 408, XXXVI., we find that 
 the new constant h' is the sttm^ e + e, of what have been above called the parame- 
 ters,* of the given surface (e) on which the geodetic is traced, and of the con/beaZ (e J 
 which touches a given tangent to that curve : whence follows the knownf theorem, 
 that the tangents to a geodetic, on ani/ central surface of the second order, all touch 
 one common confocal.^ 
 
 (6.) The new constant eX=h'-e) may, by 407, LXXV. and 408, LXXV. 
 (with c for eo)> t^e thus transformed : 
 
 XVI. . . e, = ei(TVUi/idp)2 + €2(TVUi'2dp)8 
 
 = ei(SU»'2dp)2 + eo(SUjvidp)2 = const. ; 
 
 where «i, 63 are the parameters of the two confocals through the point p of the geo- 
 detic on (e), and vi, vi are as before the normals at that point, to those two surfaces 
 
 (7.) In fact, the two equations last cited give the general transformation, 
 
 XVII. . . Z-2(Vp(r)2-2Sa(TSa'(r 
 = e (V<tUv)2 + ei (V(tUvi)2 + et (V<tUj/2)2 ; 
 
 <T being an arbitrary vector, which may for instance be replaced by dp. Equating 
 then this last expression to (e + fi,)(T2, or to e(Y<j\]vY - e^a"^, since Si/er = 0, we 
 obtain the first and therefore also the second transformation XVI., because the three 
 normals vv\v% compose a rectangular system (comp. 407, (4.), &c.). 
 
 (8.) It is, however, simpler to deduce the second expression XVI. from the equa- 
 tion 408, XXXI. of the cone of tangents from p to (e,), by changing t to Udp ; and 
 then if we write 
 
 XVIII... «i = ^ 5^, 
 
 * Compare the last Note to page 656. 
 f Discovered by M. Chasles. 
 
 X This touched confocal becomes a sphere, when the given confocal is a cone. 
 Compare 380, (5.), and 408, (16.), (17.) ; also the Note to page 517. 
 
 4 Q 
 
666 ELEMENTS OF QUATERNIONS. [bOOK IIT. 
 
 80 that vi denotes the angle at which the geodetic crosses the normal vi to (ei), 
 considered as a tangent to the given surface (e), the first integral XVI. takes the 
 form,* 
 
 XIX. . . e^ - d sin^ vt, + ez cos^ tjj, 
 or XX. . . a,2 = «^2 sin* vi + a^^ cos2 t?i, &c. ; 
 
 in which the constant a^ is the primary semiaxis of the touched confocal (5.). 
 
 (9.) Without supposing that Tdp is constant, we may investigate as follows the 
 differential of the real scalar h in IX. or X., or of the product P'^. D'-, for any curve 
 on a central surface of the second order. Leaving at first the surface arbitrary, as in 
 (L) and (2.), and resolving 6.-p in the three rectangular directions of v, djo, and vdp, 
 we get the general expression, 
 
 XXL . . d2p = - i/-iSdi^dp + dp-iSd/)d> + (vdp)-iSvdpd2p ; 
 of which, under the conditions I. and II., the two last terms vanish, as in III. 
 Without assuming those conditions, if we now introduce the relations VII. which 
 belong to a central surface of the second order, we have by V. and IX. the expres- 
 sion, f 
 
 XXII. . .\Ah. dp2 = v2Sdrd2p + Srdi/Sdrdp - ASdpdSp = Srdvdp-i.Sj/dpd^p, 
 or XXIIL . . dA = d . v^Sd vdp-i = d . p-2Z)-2 = 2S»/dvdp-iSj/dp-id2p ; 
 
 or finally, XXIV. , . dA . dp* = 2Si/di^dp . SrdpdZp, 
 
 the scalar variable with respect to which the differentiations are performed being here 
 entirely arbitrary. 
 
 (10.) For a geodetic line on any surface, referred thus to any scalar variable, 
 we have by 380, 11. the differential equation, 
 
 XXV.. . Sj^dpd2p = 0; 
 and therefore by XXIV., for such a line on a central surface of the second order, we 
 have again, as in (3.), 
 
 XXVI. . . d^ = 0, or XXVI'. . . h = const., 
 with h = p-«i}-2 as in X. 
 
 (11.) But we now see, by XXIV., that for such a surface the condition XXVI. 
 is satisfied, not only by this differential equation of the second order XXV. but also 
 by this other differential equation, 
 
 XXVII. . . Sivdj^dp = ; 
 the product P^D-^ (or PD itself) is therefore constant, not only as in (3.) for every 
 
 * Under this form XX., the integral is easily seen to coincide with that of M. 
 Liouville, 
 
 fi^ cos^ i + v2 sin2 i = n'i = const., 
 
 cited in page 290 of Dr. Salmon's Treatise. 
 
 t In deducing this expression, it is to be remembered that 
 
 dSdvdp = d/dp = 2Sdi;d2p ; 
 in fact, the linear and self -conjugate form of v = fp gives, 
 Sdp'd2j/=/(dp, d2p)= Sd»'d2p. 
 
CHAP. III.] LINES OF CURVATURE. 667 
 
 geodetic on the surface, but also for every curve of another set* represented by this 
 last equation XXVII., -which is only of the^r«^ order, and the geometrical meaning 
 of which we next propose to consider. 
 
 410. In general, if v and v ■{■ Jlv have the directions of the nor- 
 mals to any surface, at the extremities of the vectors p and p + A/5, 
 the condition of intersection (or parallelism) of these two normals 
 is, rigorously, 
 
 I. . . Si^Ai/A/) = 0; 
 
 the differential equation] of what are called the Lines of Curvature^ 
 on an arbitrary surface, is therefore (comp. 409, XXVII.), 
 
 II. . . Sj^di/d/) = 0; 
 
 from which we are now to deduce a few general consequences, toge- 
 ther with some that are peculiar to surfaces of the second order. 
 
 (1.) The differential equation of the surface being, as usual, 
 
 III. . . Sj/dp = 0, 
 
 the normal vector v is generally some function of p, although not generally linear, 
 because the surface is as yet arbitrary : its differential dv is therefore generally some 
 function of p and dp, which is linear relatively to the latter. And if, attending only 
 to the dependence of dv on dp, we write 
 
 IV. . . dv = <pdp, 
 
 it results from what has been already proved (363), that this linear and vector func- 
 tion is at the same time self-conjugate. 
 
 (2.) Denoting then by t a tangentX pt to a line of curvature, drawn at the 
 given extremity p of p, we see that the vector r must satisfy the two following sca- 
 lar equations, in which v is supposed to be given, 
 
 * Namely, the lines of curvature, as is known, and as will presently be proved 
 by quaternions. 
 
 t In this equation II., dp and dv are two simultaneous differentials, which may 
 (according to the theory of the present Chapter, and of the one preceding it) be at 
 pleasure regarded, either as two finite right lines, whereof dp is (rigorously) tangen- 
 tial to the surface, and to the line of curvature ; or else as two infinitely small vec- 
 tors, dp being, on this latter plan, an infinitesimal chord Ap. (Compare pages 99, 
 392, 497, 626, and the first Notes to pages 623, 630.) The treatment of the equa- 
 tions is the same, in these two views, whereof one may appear clearer to some readers, 
 and the other view to others. 
 
 X This symbol r is used here partly for abridgment, and partly that the reader 
 may not be obliged to interpret dp as denoting ^finite tangent, allbough the princi- 
 I)le3 of this work allow him so to interpret it. 
 
668 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 V. . . Si/r=0, and VI. . . Si/r0r=O ; 
 this tangent r admits therefore (355) oitwo real and rectangular directions, but not 
 in general of more : opposite directions being not here counted as distinct. Hence, as 
 is indeed well known, through each point of any surface there pass generally two lines 
 of curvature : and these two curves intersect each other at right angles. 
 
 (3.) A construction for the two rectangular directions of t can easily be assigned 
 as follows. Assuming, as we may, that the length of the tangent r varies with its 
 direction^ according to the law, 
 
 VII. ..Sr0r=l, 
 ■which gives 
 
 VIII. . . S(0r.dr) = O, or briefly VIII'. . . S^rdr = 0, 
 by the properties above mentioned of ; and remembering that v is treated as a con- 
 stant in v., so that we may write, 
 
 IX. . . Si/dr = 0, and therefore (by VI.), X. . . Srdr= ; 
 
 we see that, under the conditions of the question, the above mentioned length Tr, of 
 this tangential vector r, is a maximum or minimum : and therefore that the two 
 directions sought are those of the tivo axes of the platie conic V. VII., which has its 
 centre at the given point p of the surface, and is in the tangent plane at that point. 
 
 (4.) This plane conic V. VII. may be called the Index Curve, for the given sur- 
 face at the given point p ; in fact it is easily proved to coincide, if we abstract from 
 mere dimensions, with the known indicatrix (la courbe indicatrice) of Dupin,* who 
 first pointed out the coincidence (3.) of the directions of its axes, with those of the 
 lines of curvature ; and also established a more general relation of conjugation be- 
 tween two tangents to a surface at one point, which exists when they have the direc- 
 tions of any two conjugate semidiameters of that cui*ve : so that the lines of curvature 
 are distinguished by this characteristic property, that the tangent to each is per- 
 pendicular to its conjugate. 
 
 (5.) In our notations, this relation of conjugation between two tangents r, r', 
 which satisfy as such the equations, 
 
 V. . . Sj/r= 0, and V. . . Svr' = 0, 
 is expressed by the formula, 
 
 XI. . . Sr^r' = 0, or XI'. . . Sr'^r = ; 
 we have therefore the parallelisms,! 
 
 XII. . . r 11 Vi/^r', XII'. . . r' 11 Vv^r ; 
 so that the equation VI. may be written under the very simple form, 
 
 XIII.. . Srr'=0, 
 which gives at once the rectangularity lately mentioned. 
 
 * Developpements de Geome'trie (Paris, 1813), pages 48, 145, &c. 
 
 t The conjugate character of these two parallelisms, or the relation, 
 V. v(l>Yv(pT II r, if Si/r = 0, 
 may easily be deduced from the self-conjugate property of (p, with the help of the 
 formula 348, VJI., in page 440. 
 
CHAP. III.] INDEX CURVE AND SURFACE, CONJ. TANGENTS. 669 
 
 (6.) The parallelism XII'. may be otherwise expi-essed by saying (comp. (4.)) 
 that 
 
 XIV. . . dp and Vvdj/ 
 
 have the directions of conjugate tangents ; or that the two vectors, 
 
 XV. . . ^p and Yv^v, 
 have ultimately such directions, when TAp diminishes indefinitely. But whatever 
 may be this length of the chord Ap, the vector Yv^v has the direction of the line 
 of intersection of the two tangent planes to the surface, drawn at its two extremi- 
 ties : another theorem of Diipin* is therefore reproduced, namely, that if a develop- 
 able be circumscribed to any surface, along any proposed curve thereon, the generat- 
 ing lines of this developable are everywhere conjugate, as tangents to the surface, to 
 the corresponding tangeiits to the curve, with the recent definition (4.) of such con- 
 jugation. 
 
 (7.) The following is a very simple mode of proving by quaternions, that if A 
 tangent r satisfies the equation VI., then the rectangular tangent, 
 
 XVI. . . t'^vt, 
 satisfies the same equation. For this purpose we have only to observe, that the self- 
 conjugate property of ^ gives, by VI. and XVI., 
 
 XVII. . . = Sr'(/)r = Sr^r'=v-2Syr>r'. 
 
 (8.) Another way of exhibiting, by quaternions, the mutual rectangularity of 
 the lines of curvature, is by employing (comp. 357, I.) the self-conjugate /orm, 
 
 XVIII. .. 0r = <7r+VXr/i; 
 
 in which the vectors X, p,, and the scalar g, depend only on the surface and the point, 
 and are independent of the direction of the tangent. The equation VI. then be- 
 comes by v., 
 
 XIX. . . = SrrXr/A = Si^rXS/ir + Svr/iSXr; 
 
 assuming then the expression, 
 
 XX, . . r=^xYv\-^yYvp, 
 we easily find that 
 
 XXI. . . y2(Y^^)2 = a-2(VvX)2, or XXI'. . . y1Yvp-=±x1Yv\ ; 
 
 the two directions of r are therefore those of the two lines, 
 
 XXII. . .UVj/X+UVvju, 
 
 which are evidently perpendicularf to each other. 
 
 * Dupin proved ^rs< (^Dev. de Geometric, pp. 43, 44, &c.), that tw» such tangents 
 as are described in the text have a relation of reciprocity to each other, on which 
 account he called them " tangentes conjuguees :" and afterwards he gave a sort of 
 image, or construction, of this relation and of others connected with it, by means of 
 the curve which he named " I'indicatrice" (in his already cited page 48, &c.). 
 
 t This mode, however, of determining generally the directions of the lines of 
 curvature, gives only an illusory result, when the normal v has the direction of 
 either X or p, which happens at an umbilic of the surface. Compare 408, (27.), (29.), 
 and the first Note to page 466. 
 
670 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (9.) An interpretation, of some interest, may be given to this last expression 
 XXII., by the introduction of a certain auxiliary surface of i\\Q second order, which 
 may be called the Index Surface, because the index curve (4.) is the diametral sec- 
 tion of this new surface, made by the tangent plane to the given one. With the re- 
 cent signification of ^, this index surface is represented by the equation VII., if r 
 be now supposed (comp. (2.)) to represent a line pt drawn in any direction fronj 
 the given point p, and therefore not now obliged to satisfy the condition V. of tan- 
 gency. Or if, for greater clearness, we denote by p + p' the vector from the origin 
 o to a point of the index surface, the equation to be satisfied is, by the form XVIII. 
 of (comp. 357, II.), 
 
 XXIII. . . l=Sp'(pp=gp'^+S\p'np'] 
 
 the centre of this auxiliary surface being thus at p, and its two (real) cyclic normals 
 being the lines X and fi : so that YvX and Yvfi have the directions of the traces of 
 its two cyclic planes, on that diametral plane (Svp' = 0) which touches the given 
 surface. "We have therefore, by XXII., this general theorem, that the bisectors of 
 the angle formed by these two traces are the tangents to the two lines of curvature, 
 whatever the form of the given surface may be. 
 
 (10.) Supposing now that the given surface is itself one of the second order, and 
 that its centre is at the origin o, so that it may be represented (comp. 405, XII.) 
 by the equation, 
 
 XXIV. . . l = Sp(l>p=gp^+S>\pfip, 
 
 with constant values of A, ju, and g, which will reproduce with those values the form 
 XVIII. of ^, we see that the index surface (9.) becomes in this case simply that 
 given one, with its centre transported from o to p ; and therefore with a tangent 
 plane at the origin, which is parallel to the given tangent plane. And thus the 
 traces (9.), of the cyclic planes on the diametral plane of the inde.r surface, become 
 here the tangents to the circular sections of the given surface. We recover then, 
 as a case of the general theorem in (9.), this known but less general theorem : that 
 the angles formed by the two circular sections, at any point of a surface of the se- 
 cond order, are bisected by the lines of curvature, which pass through the same 
 point. 
 
 (11.) And because the tangents to these latter lines coincide generally, by (3.) 
 (4.) (9.), with the axes of the diametral section of the index surface, made by the 
 tangent plane to the ^fi yen surface, they are parallel, in the case (10.), as indeed is 
 well known, to the axes of the parallel section of a given surface of the second 
 order.' 
 
 (12.) And if we now look back to the Equation of Confocals in 407, (26.), and 
 to the earlier formulae of 40 7, (4.), we shall see tliat because the vector vi, in the 
 last cited sub-article, represents a tangent to the given surface Sp^p = 1, complanar* 
 with the normal v and the derived vector ^vi, so that it satisfies (comp. 407, XII. 
 XIV., and the recent formulai V. VI.) the two scalar equations, 
 
 XXV. . . Svvi = 0, and XXVI. . . Sj/vi^j/i = 0, 
 which are likewise satisfied (comp. (7.)) when we change vi to the rectangular tan- 
 
 * Compai'e the Note to page 645. 
 
CHAP. III.] LINES OF CURVATURE ON CENTRAL SURFACES. 671 
 
 gent vo, it follows that these two vectors, vi and V2, which are the normals to the 
 two confocals to (e) through p, are also the tangents to the two lines of curvature on 
 that given surface of the second order at that point : whence follows this other theo- 
 rem* of Dupin, that the curve of orthogonal intersection (407, (4.)), of two confocal 
 surfaces, is a line of curvature on each. 
 
 (13.) And by combining this known theorem, with what was lately shown re- 
 specting the umhilicar generatrices (in 408, (30.), (32.), comp. also (36.), (36.)), 
 we may see that while, on the one hand, the lines of curvature on a central surface 
 of the second order have no real envelope, yet on the other hand, in an imaginary 
 sense, they have for their common envelope^ the system of the eight imaginary right 
 lines (408, (31.)), which connect the twelve (real or imaginary) umbilics of the sur- 
 face, three by three, and are at once generating lines of the surface itself, and also of 
 the known developable envelope of the confocal system. 
 
 (14.) It may be added, as another curious property of these eight imaginary 
 right lines, that each is, in an imaginary sense, itself a line of curvature upon the 
 surface : or rather, each represents two coincident lines of that kind. In fact, if we 
 denote tlie variable vector 408, LXXX. of such a generatrix by the expression, 
 
 XXVII. .. p = e'(r+(r', 
 
 in which e' is a variable scalar, but (T, a' are two given or constant but imaginary 
 vectors, such that 
 
 XXVIII. . . (72 = 0, S(t<t' = - Z2, (r'2 = - 62, 
 
 and XXIX. . . /<r = S<T<p<T = 0, /((r, <t') = Sd'^ff = 0, fa' = 1, 
 
 we have the imaginary normal v, with (for the case of a real umbilic') a real tensor, 
 
 XXX. . . v = e'd,ir + (j>(T A. a, XXXI. . . Tv = +^ r^-; 
 
 * Dtv. de Ge'ome'trie, page 271, &c. 
 
 t The writer is not aware that this theorem, to which he was conducted by qua- 
 ternions, has been enunciated before ; but it has evidently an intimate connexion 
 with a result of I^rofessor Michael Roberts, cited in page 290 of Dr. Salmon's Trea- 
 tise, respecting the imaginary geodetic tangents to a line of curvature, drAwn from an 
 umbilicar point, which are analogous to the imaginary tangents to a plane conic, 
 drawn from a focus of that curve. An illustration, which is almost a visible repre- 
 sentation, of the theorem (13.) is supplied by Plate II. to Liouville's Monge (and by 
 the corresponding plate in an earlier edition), in which the prolonged and dotted 
 parts of certain ellipses, answering to the real projections of imagijiary portions of 
 the lines of curvature of the ellipsoid, are seen to touch a system of four real right 
 lines, namely the projections (on the same plane of the greatest and least axes), of 
 the four real umbilicar tangent planes, and therefore also of what have been above 
 called (408, (30.), (31.)) the eight (imaginary) umbilicar generatrices of the surface. 
 Accordingly Monge observes (page 150 of Liouville's edition), that "toutes les 
 ellipses, projections des lignes de courbure, seront inscrites dans ce parallelogramme 
 dont chacune d'elles touchera les quatre cotes :" with a similar remark in his expla- 
 nation of the corresponding Figure (page 160). 
 
672 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 and we find, after reductions, the imaginary expression, 
 
 XXXir. . . v(T = + V-1 (tTv, whence XXXIII. . .Sva = 0, Sv(70(t = 0. 
 The differential equations V. VI. of a line of curvature are therefore symholicnlli/ 
 satisfied, when we substitute, for the tangential vector r, either the imaginary line 
 (T itselfi or the apparently /7erjoenc?/cM/ar but in an imaginary sense coi«ci<fe«i* vec- 
 tor va ; and the recent assertions are justified. 
 
 (15.) As regards the real lines of curvature, on a central surface of the second 
 order, we see by comparing the general differential equation II. with the expres- 
 sion 409, XXIII. for the differential of A, or of P-2Z)-«, that this latter product, or 
 the product P.D itself, is constant^ for a line of curvature, as well as for a geo- 
 detic line, on such a surface, as indeed it is well known to be : although this last 
 constant (P. D) may become imaginary, for the case of a single- sheetedX hyperhO' 
 loid, and must be such for a line of curvature on an hyperboloid of two sheets. 
 
 (16.) And as regards the general theory of the index surface (9.), it is to be ob- 
 served that this auxiliary surface depends /)re?nan7y on the scalar function f, in the 
 equation /p = 1, or generally //3 = const., of the given surface ; and that it is not en- 
 tirely determined by means of that surface alone. For if we write, for instance, 
 
 XXXIV. .. f/p = fl, with d/p = 2Srdp as before, 
 we shall have, as the new first differential equation of the same given surface, instead 
 of III., 
 
 XXXV. . . = df/p = 2Swi/dp, with XXXVI. . . n = Vfp; 
 
 and if we then write, by analogy to IV., 
 
 XXXVII. . . d.nx^ = (6dp = n^dp + nVSvdp, with XXXVIII. . . «' = 2f"/p, 
 the new index surface, constructed on the plan (9.), will have for its equation, 
 analogous to XXIII., the following : 
 
 XXXIX. . . ^p'6p' = nSp'^jo' -f n' (Svp')^ = const. 
 
 * As regards the paradox, of the imaginary vector er being thus apparently per- 
 pendicular to itself, a similar one had occurred before, in the investigation 353, (17.), 
 (18.), (19.) ; and it is explained, on the principles of modern geometry, by observ- 
 ing that this imaginary vector is directed to the circle at infinity. Compare 408, 
 (81.), and the Note to page 459. 
 
 f Compare the first Note to page 667. 
 
 J Although the writer has been content to employ, in the present work, some of 
 these usual but rather long appellations, he feels the elegance of Dupin's phraseology, 
 adopted also by Mobius, and by some other authors, according to which the two cen- 
 tral hyperboloids are distinguished, as elliptic (for the case of two sheets), and hy- 
 perbolic (for the case of owe). The phrase " quadrie," for the general surface of the 
 second order (or second degree^, employed by Dr. Salmon and Mr. Cayley, is also 
 very convenient. It may be here remarked, that Dupin was perfectly aware of, or 
 rather appears to have first discovered, the existence of what have since his time come 
 to be called the /oca/ conies ; which important curves were considered by him, as 
 being at once limits of confocal surfaces, and also loci of umbilics. Comp. Dev. de 
 Geometric, pages 270, 277, 278, 279 ; see also page 390 of the Aperpu Historique, 
 &c., by M. Chasles (Brussels, 1837). 
 
CHAP. III.] FORMS OF DIFFERENTIAL EQUATION. 673 
 
 (17.) But if we take this last constant = n, the two index surfaces, XXIII, and 
 XXXIX., will have a common diametral section, made by the given tangent plane, 
 namely the index curve (4.) ; and they will touch each other, in the whole extent of 
 that curve. And it will be found that the construction (9.), for the directions of the 
 lines of curvature, applies equally well to the one as to the other, of these two auxi- 
 liary surfaces: in fact, it is evident that the differential equation II., namely 
 Svdvdjo = 0, receives no real alteration, when v is multiplied by any scalar, n, even 
 if that scalar should be variable. 
 
 (18.) And instead of supposing that the variable vector p is thus obliged, as in 
 373, to satisfy a given scalar equation, of the form* 
 
 fp = const.. 
 
 * U p = ix +jy + kz, and v =fp = F (x, y, z), and if we write, 
 do = joda; + q^y + rdz, dp =/)'da; + r"dy + 5"dz, 
 dg = qAy + p"dz + r"da;, dr = r'dz + q"^x 4-/>"dy, 
 we may then write also, on the present plan, which gives d/p = 2SvdjO, 
 dp = tda; \j^y -f ^dr, v = -\(ip +jq + kr), 
 
 dv = -^ (idp +jdq + Mr), SdpAv = ^ (dxdp + dydq + dzdr) ; 
 and the index surface, constructed as in (9.), and with p' changed to Ap = iAx +jAy 
 + kAz, will thus have the equation, 
 
 (a). . . ip'Ax"^ + l^'AyS + Ir'Az^ +p"AyAz + q'AzAx + r"AxAy = 1, 
 or more generally = const. ; so that it may be made in this way to depend upon, and 
 be entirely determined by, the six partial differential coefficients of the second order, 
 p' . .p" ' ., of the function v or fp, taken with respect to the three rectangular co- 
 ordinates, xyz. And by comparing this equation (a) with the following equation 
 of the same auxiliary surface, which results more directly from the principles em- 
 ployed in the text (comp. XVIII. XXIII.), 
 
 (b). . . SAp<pAp=gAp^+S\ApfiAp = l, 
 we can easily deduce expressions for those six partial coefficients, in terms o{g, X, /i. 
 Thus, for example, 
 
 |Da;2t? = lp=-g+ SXifii = SX/i -g+ 2S£XSj> ; 
 but StXSe> + SjX^Jn + SkXSkfi = - SXfi ; therefore, 
 
 (c). . . ^ (Dx2w + Dy^v + D^Sw) = SXfi-3g = ci + C2 + cs = - m", 
 if ci, C2, C3 be the roots and m" a coeificient of a certain cubic (354, III.), deduced 
 from the linear and vector function dv = ^dp, on a plan already explained. If 
 then the function v satisfy, as in several physical questions, the partial differential 
 
 equation, 
 
 (d). . .D.,2o + Dj,2» + Ds2o = 0, 
 
 the sum of these three roofs, ci, cz, cz, will vanish : and consequently, the asympto- 
 tic cone to the index -surface, found by changing 1 to in the second member of (a), 
 is real, and has (comp. 406, XXL, XXIX.) the property that 
 
 (e). . . cot2a + cot2b = l, 
 if a, b denote its two extreme semiangles. An entirely different method of trana- 
 
 4 R 
 
674 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 we may suppose, as in 372, thsit p is a, given vector function of two scalar varia- 
 bles^ X and y, between which there will then arise, by the same fundamental formula 
 II., a differential equation of the first order and second degree^ to be integrated 
 (when possible) by known methods. For example, if we write, 
 XL. . . p = ix +J7/ + kz, dz =:pdx + ^dy, 
 we shall satisfy the equation III. by assuming (with a constant factor understood), 
 
 XLL . . v = ip +jq - k, whence XLII. . . d v = i6p +Jdq ; 
 and thus the general equation II., for the lines of curvature on an arbitrary surface, 
 receives (by the laws of ijk) the form, 
 
 XLIII. . . dp (dy + qdz) = dg (da; + pdz) ; 
 which last form has accordingly been assigned, and in several important questions 
 employed, by Monge* : but which is now seen to be included in the still more con- 
 cise (and more easily deduced and interpreted) quaternion equation^ 
 
 Svdvdp = 0. 
 
 411. For a central surface of the second order, we have as usual 
 p = (pp, Aj^ = 0A/3, and therefore (by 347, 348, and by the self-con- 
 jugate form of 0), 
 
 I. . . VvAv = Y(pp4>^p = ylrYpAp = m(/)'^Yp^p; 
 the general condition of intersection 410, I. of two normals, at the 
 extremities of a. finite chord Ap, and the general differential equation 
 410, II. of the lines of curvature, may therefore for such a surface 
 receive these new and special forms : 
 
 forming, by quaternions, the well known equation (d), occurred early to the present 
 writer, and will be briefly mentioned somewhat farther on. In the mean time it 
 may be remarked, that because tn' = by (c), when the equation (d) is satisfied, we 
 have then, by the general theory III. ii. 6 of linear and vector functions, and espe- 
 cially by the subarticles to 360, remembering that fp is here self-conjugate, the for- 
 mulae, 
 
 (f). . . dv + xdjo = 0, and (g). . . tpa - <p^<T = ma, 
 
 X, i// being auxiliary functions, and m' another coeflScient of the cubic, while c is an 
 arbitrary vector. For the same reason, and under the same condition (d), the 
 function <p itself has the properties expressed by the equations, 
 
 (h). . . (t>YiK = jc^t - t^<c, and (i). . . ^2 Vck = V^i^/c - m'YiK ; 
 in which the two vectors t, k are arbitrary, and m' is the same scalar coefficient as 
 before. 
 
 * See the enunciation of the formula here numbered as XLIII., in page 133 of 
 Liouville's Monge: compare also the applications of it, in pages 274, 303, 305, 357. 
 (The corresponding pages of the Fourth Edition are, 115, 240, 265, 267, 312.) 
 The quaternion equation, Svdrdp = 0, was published by the present writer, in a 
 communication to the Philosophical Magazine, for the month of October 1847 
 (page 289). See also the Supplement to the same Volume xxxi. (Third Series); 
 and the Proceedings of the Royal Irish Academy for July, 1846. 
 
CHAP. III.] CONDITION OF INTERSECTION OF NORMALS. 675 
 
 II. . . SA/>0-> V/)A/> = 0, or ir. . . Sp^p(p''Ap = 0', 
 
 III. . . Sd/>0-i V/>d/3 = 0, or III'. . . Spdp(/>-'dp = ; 
 which admit of geometrical interpretations, and conduct to some 
 new theorems, especially when they are transformed as follows : 
 
 IV. . . SX A/9 . BpAp(p-y + SfiAp . Sp A/)0-i\ = 0, 
 
 V. . . S\dp. Spdp(p-^/ii + Sfidp .Spdp^-^X =0, 
 
 X and fi being (as in 405, (5.), &c.) the two real cyclic normals of 
 the surface: while the same equations may also be written under 
 the still more simple forms, 
 
 VI. . . SaAp . SaVA/3 + Sa'Ap . SapAp = 0, 
 
 VII. .. Sadp. Sapdp + Sa'dp . Sapdp = 0, 
 
 a, a being, as in several recent investigations, the two real focal 
 unit lines, which are common to a whole confocal system. 
 
 (1.) The vector <}}~^YpAp in II. has by I. the direction of VvAv ; whence, by 
 410, (6.), the interpretation of the recent equation II., or (for the present purpose) 
 of the more general equation 410, L, is that the chord pp' is perpendicular to its 
 own polar, if the normals at its extremities intersect. Accordingly, if their point of 
 intersection be called n, the polar of pp' is perpendicular at once to pn and p'n, and 
 therefore to pp' itself. 
 
 (2.) The equation 11'. may be interpreted as expressing, that when the normals 
 at P and p' thus intersect in a point n, there exists a point p" in the diametral plane 
 opp', at which the normal p''n" is parallel to the chord pp' : a result which may be 
 otherwise deduced, from elementary principles of the geometry of surfaces of the 
 second order. 
 
 (3.) It is unnecessary to dwell on the converse propositions, that when either of 
 these conditions is satisfied, there is intersection (or parallelism) of the two normals 
 at p and p' : or on the corresponding but limiting results, expressed by the equations 
 III. and III'. 
 
 (4.) In order, however, to make any use in calculation of these new forms II., III., 
 we must select some suitable expression for the self-conjugate function 0, and deduce 
 a corresponding expression for the inverse function 0"i. The ^brm,* 
 VIII. . . <l>p=ffp + Y\pii, 
 
 which has already several times occurred, has also been more than once inverted : 
 but the following new inverse f form, 
 
 * The vector form VIII. occurred, for instance, in pages 463, 469, 474, 484, 
 641, 669 ; and the connected scalar form, 
 
 fp = gp^ -[■ SXpfip, 357,11. 
 
 has likewise been frequently employed. 
 
 t Inverse forms, for <p'^p or m"'i^p, have occurred in pages 463, 484, 641 (the 
 
676 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 IX.. . {ff-S\ti).<p'^p = p-XSp(l>-^ix-fiSp(}>-% 
 has an advantage, for our present purpose, over those assigned before. In fact, this 
 form IX. gives at once the equation, 
 
 X. . . {g- S\fi).<l>-^Yp/^p =YpAp -XSpAp(p-^fi -fiSpAp^-^X; 
 
 and so conducts immediately from II. to IV., or from III. to V. as a limit. 
 
 (5.) The equation IV. expresses generally/, that the chord Ap, or pp', is a side of 
 a certain cone of the second order, which has its vertex at the point P of the given 
 surface, and passes through all the points p' for which the normals to that surface in- 
 tersect the ffiven normal at p ; and the equation V. expresses generally^ that the two 
 sides of this last cone, in which it is cut by the given tangent plane at the same point 
 p, are the tangents to the lines of curvature. 
 
 (6.) But if the surface be an ellipsoid, or a c?ott&/e-sheeted hyperboloid, then 
 (comp. 408, (29.)) the always real vectors* ^'^\ and 0"V» '^^^'^ *^^ directions of 
 semidiameters drawn to two of the four real umhilics ; supposing then that p is such 
 a semidiameter, and that it has the direction of + ^''X, the second term of the first 
 member of the equation IV. vanishes, and the cone IV. breaks up into a pair of 
 planes^ of which the equations in p' are, 
 
 XI. . . S\ (p' - p) = 0, and XII. . . Sp'^-iX^y = ; 
 
 whereof the ybrmer represents the tangent plane at the umhilic p, and the latter re- 
 presents the plane of the four real umhilics. 
 
 (7.) It follows, then, that the normal at the real umhilic p is not intersected by 
 any real normal to the surface, except those which are drawn at points p' of that 
 principal section, on which all the real umhilics are situated: but that the same real 
 umbilicar normal PN is, in an imaginary sense, intersected by all the imaginary nor- 
 mals, which are drawn from the imaginary points p' of either of the two imaginary 
 generatrices through P. 
 
 (8.) In fact, the locus of the point p', under the condition of intersection of its 
 normal p'n' with a given normal pn, is generally a quartic curve, namely the inter- 
 section of the given surface with the cone IV. ; but when this cone breaks up, as in 
 (6.), into two planes, whereof one is normal, and the other tangential to the surface, 
 the general quartic is likewise decomposed, and becomes a system of a real conic, 
 namely the principal section (7.), and a pair of imaginary right lines, namely the 
 two umbilicar generatrices at P. 
 
 (9.) We see, at the same time, in a new way (comp. 410, (14.)), that each such 
 generatrix is (in an imaginary sense) a line of curvature : because the (imaginaiy) 
 normals to the surface, at all the points of that generatrix, are situated by (7.) in 
 one common (imaginary) normal plane. 
 
 (10.) Hence through a real umhilic, on a surface of the second order, there pass 
 
 correction in a Note to which last page should be attended to). In comparing these 
 with the form IX., it will easily be seen (comp. page 661) that 
 
 * Compare the Note immediui.ely preceding. 
 
CHAP. III.] THREE LINES THROUGH AN UMBILIC. 677 
 
 three lines of curvature : whereof one is a 7'eal conic (8.), and the two others are 
 imaginary right lines, namely, the umhilicar generatrices as before. 
 
 (11.) If we prefer differentials to differences, and therefore use the equation V. 
 of the lines of curvature, we find that this equation takes the form = 0, if the 
 point p be an umbilic ; and that if the normal at that point be parallel to X, the 
 differential of the equation V. breaks up into two factors, namely, 
 
 XIII. . . SXd2p = 0, and XIV. . . Sdp0-'X^-i/i = ; 
 whereof the former gives two imaginary directions, and the latter gives one real di- 
 rection, coinciding precisely with the three directions (10.). 
 
 (12.) And if p, instead of being the vector of an umbilic, be only the vector of a 
 point on a generatrix corresponding, we shall still satisfy the differential equation 
 v., by supposing that dp belongs to the same imaginary right line : because we 
 shall then have, as at the umbilic itself, 
 
 XV. . . SXdp = 0, Spdp0-iX = 0. 
 
 An umhilicar generatrix is HdQTQiorQ proved anew (comT^. (9.)) to be, in its whole 
 extent, a line of curvature. 
 
 (13.) The recent reasonings and calculations apply (6.), not only to an ellipsoid, 
 but also to a double-sheeted hyperboloid, four umbilics for each of these two sur- 
 facesbeing real. But if for a moment we now consider specially the case of an ellip- 
 soid, and if Ave denote for abridgment the real quotient by h, we may then 
 
 * a + c 
 
 substitute in IV. and V. for X, fi, ^-% 0-y the expressions, 
 
 , 26UX , , 2bJJfi 
 XYI. ..a— ha = ; ha ~ a = --, 
 
 a-\- c a-\- c 
 
 , , -2i0-iUX , -2&0- iU^ 
 XVll. . . a + ha = ; -ha — a = -^ ^ : 
 
 ac{a-^c) ac{^a-{-c) 
 
 and then, after division by Az — 1, there remain only the two vector constants a a', 
 the equation IV. reducing itself to VI., and V. to VII. 
 
 (14.) The simplified equations thus obtained are not however peculiar to ellip- 
 soids, but extend to a whole confocal system. To prove this, we have only to com- 
 bine the equations II. and III. with the inverse form, 
 
 XVIII. . . Z-2^-'p = aSa'p + a Sap - p (e + Saa'), 
 which follows from 407, XV., and gives at once the equations VI. and VII., what- 
 ever the species of the surface may be. 
 
 (15.) The difi^erential equation VII. must then be satisfied by the three rectan- 
 gular directions of dp, or of a tangent to a line of curvature, which answer to the 
 orthogonal intersections (410, (12.)) of the three confocals through a given point P ; 
 it ought therefore, as a verification, to be satisfied also, when we substitute v for dp, 
 V being a normal to a confocal through that point : that is, we ought to have the 
 equation, 
 
 XIX. . . SavSa'pv + Sa'vSapv = 0. 
 
 And according!}' this is at once obtained from 407, XVI., by operating with S.pj/ ; 
 so that the three normals v are all sides of this cone XIX., or of the cone VII. with 
 dp for a side, with which the cone V. is found to coincide (13.). 
 
 (16.) And because this last equation XIX., like VI. and VII., involves owZy the 
 two /oca/ lines a, a' as its constants, we may infer from it this theorem : '' Jfinde- 
 
678 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 finitely many surfaces of the second order have only their asymptotic cones hiconfo- 
 caly* and pass through a given point, their normals at that point have a cone of the 
 second order for their locus ;" which latter cone is also the locus of the tangents, at 
 the same point, to all the lines of curvature which pass through it, when different 
 values are successively assigned to the scalar constant a'^ — c^ (or 2Z2) : that is, when 
 the asymptotes a, a' to the focal hyperbola remain unchanged in position, but the 
 semiaxes (cfl — b'^)i^, (b^ — c2)^ of that curve (here treated as both real) vary together. 
 (17.) The equation VI. of the cone of chords (5.) introduces t)xQ fixed focal 
 lines a, a' by their directions only. But if we suppose that the lengths of those 
 two lines are equal, without being here obliged to assume that each of those lengths 
 is unity, we shall then have (comp. 407, (2.), (3.)), the following rectangular sys- 
 tem of unit lines, in the directions of the axes of the system, 
 
 XX. . . U(a + a'), JJVaa', U(a-a'), * 
 
 which obey in all respects the laws of ijk, and may often be conveniently denoted by 
 those symbols, in investigations such as the present. And then, by decomposing the 
 semidiameter p, and the chord Ap, in these three directions XX., we easily find the 
 following rectangular transformation^ of the foregoing equation VL, 
 
 XXI S (« + «')"'P , S(a-aO-ip ^ S.(Vaa')-'p _ 
 '**S(a + a')Ap S(a-a')A|0 S.Uaa'Ap' 
 
 in which it is permitted to change Ap to dp, in order to obtain a new form of the 
 differential equation of the lines of curvature ; or else at pleasure to v, and so to 
 find, in a new way, a condition satisfied by the three normals, to the three confocals 
 through p. 
 
 (18.) The cone, VI. or XXI., is generally the locus of a system of three rectan- 
 gular lines ; each plane through the vertex, which is perpendicular to any real side', 
 cutting it in a real pair of mutually rectangular sides : while, for the same reason, 
 the section of the same cone, by any plane which does not pass through its vertex p, 
 but cuts any side perpendicularly, is generally an equilateral hyperbola. 
 
 (19.) If, however, the point v be situated in any one of the three principal 
 planes, perpendicular to the three lines XX., then the cone XXI. (as its equation 
 shows) breaks up (comp. (6.)) into Si pair of planes, of which one is that principal 
 
 * That is, if the surfaces (supposed to have a common centre) be cut by the 
 plane at infinity in biconfocal conies, real or imaginary. 
 
 t The corresponding form, in rectangular co-ordinates, oi t\xQ condition of in- 
 tersection, of normals at two points {xyz) and {x'y'z), to the surface, 
 
 Sfl y% zZ 
 
 is the equation (probably a known one, although the writer has not happened to 
 
 meet with it), 
 
 (62 _ c-J) x' (c2 -a^~)y' {cfl - 62) z' 
 
 J 1 1 — = ; 
 
 x-x y-y z-z 
 
 in which it is evident that xyz and x'y'z' may be interchanged. 
 
CHAP. III.] CENTRES OF CURVATURE. 679 
 
 plane itself, while the other is perpendicular thereto. And while the former plane 
 cuts the surface in a principal section, which is always a line of curvature through 
 p, the latter plane usually cuts the surface in another conic, which crosses the for- 
 mer section at right angles, and gives the direction of the second line of curvature. 
 
 (20.) But if we further suppose, as in (6.), that the point p is an umhilic, then 
 (as has been seen) the second plane is a tangent plane ; and the second conic (19.) 
 is itself decomposed, into a pair of imaginary right lines : namely, as before, the two 
 umhilicar generatrices through the point, which have been shown to be, in an ima- 
 ginary sense, both lines of curvature themselves, and also a portion of the envelope 
 of all the others. 
 
 (21.) We shall only here add, as another transformation of the general equation 
 VI. of the cone of chords, which does not even assume Ta = Ta', the following : 
 
 XXII. . . S(a+a')Ap.S(a+a')pAp = S(a-a')A|O.S(a-a')pAp; 
 where the directions of the two new lines, a + a and a — a, are only obliged to bo 
 harmonically conjugate with respect to the directions of i\xQ fixed focal lines of the 
 system : or in other words, are those of any two conjugate semidiameters of the focal 
 hyperbola. 
 
 412. The subject of Lines of Curvature receives of course an 
 additional illustration, when it is combined with the known concep- 
 tion of the corresponding Centres of Curvature. Without yet en- 
 tering on the general theory of the curvatures of sections of an arbi- 
 trary surface, we may at least consider here the curvatures of those 
 normal sections^ which touch at any given point the lines of curva- 
 ture. Denoting then by a the vector of the centre s of curvature of 
 such a section, and by R the radius PS, considered as a scalar which 
 is positive when it has the direction of + v, it is easy to see that 
 we have the two fundamental equations : 
 
 I. . . cr = p + RUv', II... i2-'d^ + dUi/ = 0; 
 whence follows this new form of the general differential equation 
 410, II. of the lines of curvature, 
 
 III. . . Vd/>dUi^ = 0; 
 with several other combinations or transformations, among which 
 the following may be noticed here : 
 
 (1.) The equation I. requires no proof; and from it the equation II. is obtained 
 by merely differentiating* as if <r and R were constant : after which the formula III. 
 follows at once, and IV. is easily deduced. 
 
 * To students who are accustomed to infinitesimals, the easiest way is here to 
 
680 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (2.) To obtain from this last equation a more developed expression for J2, we 
 may assume for dv, considered as a linear and self-conjugate function of dp (410, 
 (1.)), the general form (comp. 410, XVIII.), 
 
 Y. . . dv = gdp + VXdpjw, 
 in which 5-, \, fx are independent of dp ; and then, while the tangent dp has (by 410, 
 XXII.) one or other of the two directions^ 
 
 VI. . . dp II UVvX + UVj//i, 
 the curvature R'"^ receives one or other of the two values corresponding, 
 
 VII. . . /?-J = -Tv-'(<7 + SXUv.S/iUvi:TVXUr.TV/iUi/). 
 
 (3.) One mode of arriving at this last transformation, or of showing that if 
 (comp. again 410, XXII.) we assume, 
 
 VIII. . . r = (or ID UVXi/ ± UV/zv, 
 then IX. . . SXr/i7-i = SXUv . S/zUv + T VXUj/ . T V/iUa/, 
 
 or X. . . 2SXr.S/ir-i =S(VXUv.V/iUj/) + TVXUj/.TV/iUj/, 
 
 or finally, XI. . . 2SUXr . SU^r'^ = S (VUXv.VU/av) ± TVUXv. TVU/iv, 
 
 is to introduce the auxiliary quaternion, 
 
 XII. ..q = VUXv.VU/iV ; 
 
 and to prove that, with the value (or direction) VIII. of r, we have thus the equa- 
 tion (in which Vg^^ as usual, represents the square of V5), 
 
 XIII. . . 2SUXr.SUur-i = Sg4-To=— ^. 
 
 (4.) And this may be done, by simply observing that we have thus (with the 
 value VIII.) the expressions, 
 
 + SUX/U1. -SUXpv 
 
 ^^^•••^'^^=TVl>7' ^^^'^^-TVUX^' 
 + (SUX/ti;)2 +Vg2 
 
 XV. . . SrUX.SrU;it = 
 XVI. ..Yq: 
 and XVII. .. T2 = -2 + 2SUg = + 
 
 TVUXv.TVU/iv T^ ' 
 because XVI. . . Vj = - Ur . SUX/iv ; 
 
 2(Sg + T9) 
 Tg * 
 
 (5.) Admitting then the expression VII., for the curvature R'^y we easily see 
 that it may be thus transformed : 
 
 XVIII. . . i?-'=-T,/->L+TX//.cosf Z^ + Z-^"]; 
 
 and that the difference of the two (principal) curvatures, of normal sections of an 
 arbitrary surface, answering generally to the two (rectangular) directions of the 
 
 conceive the differentials to be such. But it has already been abundantly shown, that 
 this view of the latter is by no means necessary, in the treatment of them by quater- 
 nions. (Compare the second Note to page 667.) 
 
CHAP. III.] DIFFERENCE OF CURVATURES. 681 
 
 lines of curvature through the particular point considered, vanishes when the normal 
 V has the direction of either of the two cyclic normals, X, jm, of the index surface 
 (410, (9.)); that is, when the index curve (410, (4.)), considered as a section of 
 that index surface, is a circle : or finally, when the point in question is, in a received 
 sense, an nmhilic* of the given surface. 
 
 (6.) That surface, although considered to be a given one, has hitherto (in these 
 last sub-articles) been treated as quite general. But if we now suppose it to be a 
 central surface of the second order ^ and to be represented by the equation, 
 
 XIX. ../p = 5rp2+SXp/tp = l, 
 
 which has already several times occurred, we see at once, from the formula VII. or 
 XVIII. (comp. 410, (10.)), that the difference of curvatures^ of the two principal 
 normal sections of any such surface, varies proportionally to the />er/?enrftcttZar (Tv~^ 
 or P) from the centre on the tangent plane, multiplied by the product of the sines of 
 the inclinations of that plane, to the two cyclic planes of the surface. 
 (7.) In general (comp. 409, (3.)), it is easy to see that 
 
 XX. .. s5^ = Sr-idr = -i)-2, 
 dp 
 
 if D denote the (scalar) semidiameter of the index surface, in the direction of dp or 
 of T ; but for the two directions of the lines of curvature, these semidiameters become 
 (410, (3.), (4.)) the semiaxes of the index curve. Denoting then by ai and 82 these 
 last semiaxes, the two principal radii of curvature of any surface come by IV. to 
 be thus expressed : 
 
 XXI. . . iJi = ai^Tj/ ; E^ = as^Tv. 
 
 And if the surface be a central one, of the second order, then ai, &% are the semiaxes 
 of the diametral section, parallel to the tangent plane ; while Tv is (comp. again 409, 
 (3.)) the reciprocal P-i of the perpendicular, let fall on that plane from the centre. 
 Accordingly (comp. (6.), and 219, (4.)), it is known that the difference of the in- 
 verse squares of those semiaxes varies proportionally to the product of the sines of 
 the inclinations, of the plane of the section to the two cyclic planes. 
 
 (8.) And as regards the squares themselves, it follows from 407, LXXI., that 
 they may be thus expressed, in terms of the principal semiaxes of the confocal sur- 
 faces, and in agreement with known results : 
 
 XXII. . . ai* = a2 _ ai2 ; aa^ = a2 - «2* ; 
 
 being thus both positive for the case of an ellipsoid ; both negative, for that of a 
 double- sheeted hyperholoid ; and one positive, but the other negative, for the case of 
 an hyperboloid of one sheet (comp. 410, (15.)). 
 
 (9.) In all these cases, the normal + v is drawn towards the same side of the 
 tangent plane, as that on which the centre o of the surface is situated (because 
 Svp= 1); hence (by I. and XXI.) both the radii of curvature Ei, i?2 are drawn 
 in this direction, or towards this side, for the ellipsoid; but one such radius for the 
 6in^Z?-sheeted hyperboloid, and both radii for the hyperboloid of two sheets, are di- 
 rected towards the opposite side, as indeed is evident from the forms of these surfaces. 
 
 * Compare the second Note to page 669, 
 
 4s 
 
682 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (100 The following is another method of deducing generally the two principal 
 
 curvatures of a surface, from the self -conjugate function, 
 
 XXIII. . . dv = 0d/>, 410, IV. 
 
 which affords some good practice in the processes of the present Calculus. "Writing, 
 
 for abridgment, 
 
 XXIV. ..r = -^ = R-^Tv = - S ^ = - Sr'^r, 
 a — p djO 
 
 where r is still a tangent to a line of curvature, the equation II. is easily brought to 
 the form, 
 
 XXV. . . — rr = v~'^YvipT =(^7 — v'Sr^v = $r, 
 
 where ^ denotes a new linear and vector function, which however is not in general 
 self-conjugate, because we have not generally 0r || v. Treating then this new func- 
 tion on the plan of the Section III. ii. 6, we derive from it a new cubic equation, of 
 the form, 
 
 XXVI. . . = M + M'r + M"r-i -}- r^, 
 
 and with the coefficients, 
 
 XXVII. . . 3/=0, M' = Sv-^y\jv, M" = m" -^v-^v; 
 }p being a certain auxiliary function (= »n^">), and to" being the coefficient* analo- 
 gous to M", in the cubic derived from the function ^ itself The root r = is foreign 
 to the present inquiry; but the two curvatures, i?i"', R%~^, are the two roots of the 
 following quadratic in J?"', obtained from the equation XXVI. by the rejection of 
 that foreign root : 
 
 XXVIII. . . = {R-YYvy + M"R-nv + M. 
 (11.) As a first application of this general equation XXVIII., let ^r have again, 
 as in v., the form gr + VXr/« ; we shall then have the values, 
 XXIX. . . M"=2(iy+SXUj;.S^Uj.), 
 and XXX. . . M' = {g->r S\Ui/ . ^n^vf - ( VXUj/)2 (V/iUv)2, 
 
 = a great variety of transformed expressions ; and the two resulting curvatures agree 
 with those assigned by VII. 
 
 (12.) As a second application, let the surface be central of the second order, with 
 ahc for its scalar semiaxes (real or imaginary) ; then the symbolical cubic (350) in 
 becomes, 
 
 XXXI. . . = 03 - 7w"02 + m'0 - m = (0 + a-2) Qp ^ 6-2) (^^ + c-2) ; 
 
 and the coefficients of the quadratic XXVIII. in R'^ take the values, in which N 
 denotes the semidiameter of the surface in the direction of the normal : 
 XXXII. . . 2?ri + i?2-i = - M"Tj/-i = - (m" +/Uj/) P= (^-2 + 62 j^ ^-2 - m) P; 
 
 * Compare the Note to page 673, continued in page 674. The reason of the 
 evanescence of the coefficient M, or of the occurrence of a null root of the cubic, is 
 that we have here fp(p~^v = 0, so that the symbol $-iO may represent an actual vec- 
 tor (comp. 851). Geometrically, this corresponds to the circumstance that when we 
 pass, along a semidiameter prolonged, from a surface of the second order to another 
 surface of the same kind, concentric, similar, and similarly placed, the direction of 
 the normal does not change. 
 
CHAP. HI.] PRODUCT OF CURVATURES. 683 
 
 XXXIII. . . RfiMi-^ = M'Tv-^ = - mv-' = a^h-^c'^Pi ; 
 both of -which agree with known results, and admit of elementary verifications. * 
 
 (13.) In general, if we observe that m"-(p = x (360, XVI.), we shall see that 
 the quadratic XXVIIl. in r (or in R^'Iv) may be thus written: 
 
 XXXIV. . . = Sz;-i(r2j,+ rxv + ;//»'); 
 or thus more briefly (comp. 398, LXXIX.), 
 
 XXXV.. . = Sj/-i(^ + r)-iv. 
 (14.) Accordingly, the formula XXV. gives the expression, 
 XXXVI. . . vV = (^ + r)-ij/. Sr^v ; 
 from which, under the condition Svr = 0, the equation XXXV. follows at once. 
 
 (15.) We have therefore ^reneraZ/y, for the prorfwc^ of the two principal curva- 
 tures of sections oi any surface at any point, the expression: 
 
 XXXVII. . . Ei-^R^-'^ =rir2Tv-2 = -i/-''Sv»^v = -S ^ yh- \ 
 
 V V 
 
 which contains an important theorem of Gauss, whereto we shall presently proceed. 
 (16.) Meanwhile we may remark that the recent analysis shows, that the squares 
 ai^ a22 (7.) of the semiaxes of the index-curve are generally the roots of the follow- 
 ing equation, 
 
 XXXVIII. . . = Sj/(^ + a-2)-V, 
 
 when developed as a quadratic in a^. 
 
 (17.) And that the same quadratic assigns the squares of the semiaxes of a dia- 
 metral section, made by a plane -J- v, of the central surface of the second order which 
 has Sppp= 1 for its equation. 
 
 (18.) Accordingly, Vp0p has the direction of a tangent to this surface, which is 
 perpendicular to p at its extremity ; and therefore the vector, 
 
 XXXIX. . . <T = p-^Yp(pp = <pp- p-i = {'p- p-2) p, 
 is perpendicular to the plane of the diametral section, which has the semidiameter p 
 for a semiaxis : so that it is perpendicular also to p itself. The equation, 
 
 XL. . . S(T(^-p-2)-i(r = 0, 
 assigns therefore the values of the squares (- p^) of the scalar semiaxes of the cen- 
 tral section -J- cr ; which agrees with the formula XXXVIII. 
 
 (19.) If then a surface be derived from a given central surface of the second or- 
 der , as the locus of the extremities of normals (erected at the centre) to the diame- 
 tral sections of the given surface, each such normal (when real) having the length of 
 one of the semiaxes of that section, the equation of this new surface f (or locus) will 
 admit of being written thus : 
 
 XLI. . . Sp(^-p-2)-i|O=0. 
 
 * As an easy verification by quaternions of the expression XXXII., it may be 
 
 remarked (comp. 408, (27.)), that if a, (3, y be atiy three rectangular unit lines, 
 
 then 
 
 fa +f(3 +fy = const. = cj + Ca + C3 = a-2 + b'^ + c-2. 
 
 t When the given surface is an ellipsoid, this derived surface XLI. is therefore 
 the celebrated ffave Surface of Fresnel, which will be briefly mentioned somewhat 
 farther on. 
 
684 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (20.) The first of the values XXIV., for the auxiliary scalar r, gives the expres- 
 sion (if V = ^(0, as it is for a central surface of the second order), 
 
 XLII. . . (T = p + r-^r = (l+r-i^)p = r-J(^ + r)|0; 
 whence, by inversion, and operation with ^, 
 
 XLIII. . . p = r(^+r)-i(r; XLIV. . . v= r(0 + r)-i0(r; 
 and therefore, because SjOV= 1, 
 
 XLV. . . r-2 = S((0 + /•)'• <y-(^ + ''^<^) = S.ff (0 + O'2 ^<^- 
 (21.) The following is a quite different way of arriving at/this result, which is also 
 useful for other purposes. Considering <r as the vector os of a point s on the Surface 
 of Centres, that is, on the locus of all the centres of curvature of principal normal sec- 
 tions, the vector (say v) of the Reciprocal Surface is connected with a (comp. 373, 
 (21.)) by the equations of reciprocity * 
 
 XLVI. . . ScTV = Si;(r = 1 ; XLVII. . . S^der = ; XLVIII. . . Strdu = ; 
 which are all satisfied by the vector expression, 
 
 XLIX. . . V = -^, 
 
 Spr 
 
 where r is, as before, a tangent to the line of curvature : so that, if lo denote the va- 
 riable vector of the normal plane to this last curve, the equation of that plane (comp. 
 369, IV.) may be thus written, 
 
 L. . . Sy(w-p) = 0. 
 This normal plane, to the line of curvature at p, is therefore at the same time the 
 tangent plane to the surface of centres at s, as indeed it is known to be, from simple 
 geometrical considerations, independently of ih.Q form of the given surface, which re- 
 mains here entirely arbitrary. 
 
 (22.) The expression XLIX. for v gives generally the relation, 
 
 LI. . . Spy=l; 
 giving also, by 410, V. and VI., these two other equations. 
 
 * It is understood that d(r and dy, in the differential equations XLVIL, 
 XLVIII., are in general only obliged to have directions tangential to the surface 
 of centres, and to its reciprocal, at corresponding points: so that the equations 
 might be in some respects more clearly written thus, Su^o" = 0, So-^y = 0, the mark 
 d being reserved to indicate changes which arise from motion along a given line of 
 curvature, while 5 should have a more general signification. Accordingly if, in par- 
 ticular, we write ^p = vdp, for a variation answering to motion along the other line, 
 and denote the two radii of curvature for the two directions dp and Sp by Ri and 
 ^2, we shall have by II., iSr'dp + dUv = 0, Rz'^^p 4- SUv = 0, and therefore by I., 
 
 d(7 = dRi . Vu, 5(7 = ^p + ^ (RlVv) = (1 - RiR2r^)v6p + dRi . Vv ; 
 so that we have both Sdpd(T = 0, and Sdpd<T= 0, and therefore the tangent dp or r 
 to the given line of curvature has the direction of the normal v to the corresponding 
 sheet of the surface of centres, as is otherwise visible from geometry. And when 
 we have thus found an equation of the form <y= r, operation with S.t gives by 
 XLVI. the value t -^ Spr, as in XLIX., because <t - p |1 v -<- r. 
 
CHAP. III.] RECIPROCAL OF SURFACE OF CENTRES. 685 
 
 LII. . . Svi; = 0, and LIIL . . Svv(pv=0, 
 ■which are still independent of the form of the given surface. 
 
 (23.) But if that surface be a central quadric,* then the equation LI. may be 
 thus written, 
 
 LIV. . . 1 = Su0-V = Sj/0-1u ; 
 
 combining which with LII. and LIII., we derive the expressions : 
 
 ^„ v^v — vfv , ,^, , v^—(h~^vfv 
 
 v^-fv.Fv ^ ^ v^-Jv.Fv 
 
 wherein /y = Sv^v, and Fv = Sv^-^v, as usual. 
 
 (24.) Operating wiih S.v on this last expression for p, and attending to LII. 
 and LIV., we find the following quaternion fonns of the Equation of the Reciprocal 
 of the Surface of Centres : 
 
 LVIL . . 1 = rSrp =) — '^ ; or LXIII. . . vi=^(Fv - ])fv: 
 
 or 
 
 LIX. . . l = (Fi;-l)/i; or LX. . . Fy - -^ = 1 ; &c., 
 
 V 
 
 whereof the second, when translated into co-ordinates, is found to agree perfectly 
 with a knownf equation of the same reciprocal surface. 
 (25.) Differentiating the form LX., and observing that 
 
 LXL . . ( /- =— , d.u4 = 4Si>3dy, d/y=2S0i;du, dFy = 2S^-ii;dv, 
 
 we find, by comparison with XLVI. and XLVIIL, the expression: 
 
 2w^ v*(bv 2v (bv 
 
 or finally by XLIX., with the recent signification XXIV. of r, 
 
 LXIV. . . <r = r-2(0 +r)2^-iu, because LXV. . . r =/Ur=/Ut;: 
 and, for the same reason, the equation LX. of the reciprocal surface may be thus 
 briefly written, 
 
 LXVI. . . Fu + r-iu2 = 1, while LXVI'. . .fv + rv'^ = 0. 
 (26.) Inverting the last form for (T, and using again the relation XLVI., we first 
 find for V the expression, 
 
 LXVII. . . w=r2(^ + r)-2 0(T; 
 and then are conducted anew to the equation XLV., or to the following, 
 LXVIII. . . 1 = S. <r(i + r-i^) 2^(7. 
 
 * Compare the last note to page 672 ; see also the use made of this known name 
 "quadric," for a surface of the second order (or degree), in the sub-articles to 399 
 (pages 614, &c.). 
 
 t The equation alluded to, wliicli is one of the fourth degree, appears to have been 
 first assigned by Dr. Booth, in a Tiact on Tangential Co-ordinates (1840), cited in 
 page 163 of Dr. Salmon's Treatise. See also the Abstract of a Paper by Dr. Booth, 
 in the Proceedings of the Royal Society for April, 1858. 
 
686 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (27.) This last equation may also be thus written, 
 
 LXIX. . . l = S.(r(l + r-'0)-3 (0 + r->^^)(7; 
 but by combining XLIII. LI. LXVII. we have, 
 
 LXX. . . l=(Spw=)S.(r(l+r-V)-'^<T; 
 hence LXXI. . . = S . (r(l + r-^(f)-^^'^(j, 
 
 a result which may be otherwise and more directly deduced, under the form Svu = 
 (LII.), from the expressions XLIV. LXVII. for v and v. 
 
 (28.) If we write, 
 
 LXXn. . . r = Udjo, t' = V(vdp), and therefore LXXIII. . . tt' = Uv, 
 r and r' being thus unit-tangents to the lines of curvature, the equation III. gives, 
 generally, 
 
 LXXIV. . . = Vrd(rr') = - dr' + rSr'dr, whence LXXIV. . . dr' |l r ; 
 of which general parallelism of dr'to r, the geometrical reason is (comp. again III.) 
 that a line of curvature on an arbitrary surface is, at the same time, a line of cur- 
 vature on the developable normal surface which rests upon that line, and to which 
 the vectors r' or vdp are normals. 
 
 (29.) The same substitution LXXIII. for Uv gives by XL, if we denote by « the 
 arc of a line of curvature, measured from any fixed point thereof, so that (by 380, 
 (7.), &c.), 
 
 LXXV. . . Tdp = ds, dp = rds, Dsp = r, 
 
 the following general expression for the curvature of the given surface, in the direc- 
 tion r of the given line, which by LXXIV. is also that of dr' : 
 
 LXXVL . . i2-» = S.rDs(rr')=-S.rr'D.r = S(Uv-'.DsV); 
 
 but 0,2^ is (by 389, (4.)) what we have called the vector of curvature of the line of 
 curvature, considered as a curve in space, and B'^Vi^ is the corresponding vector of 
 curvature of the normal section of the given surface, which has the same tangent r at 
 the given point : hence the latter vector of curvature is (generally) the projection of 
 the former, on the normal v to the given surface. 
 
 (30.) In like manner, if we denote for a moment by R-^ the curvature of the de- 
 velopable normal surface (28.), for the same direction r, the general formula II. 
 gives, by LXXIV., 
 
 LXXVII. . . i?/i = rD,r' =- Sr'D.r = S-r'-^D^V 5 
 
 the vector i2/V' of this new curvature is therefore the projection on the new normal 
 t', of the vector of curvature Ds'^p o( the given line of curvature. But we shall soon 
 see that these two last results are included in one more general,* respecting all plane 
 sections of an arbitrary surface. 
 
 (31.) The general parallelism LXXIV. conducts easily, for the case of a central 
 quadric, to a known and important theorem, which may be thus investigated. Writ- 
 ing, for such a surface, 
 
 LXXVIIL . . r=/r, r'=/r'. 
 
 * Namely in Meusnier's Theorem, which can be Tproved generally by quaternions 
 with about the same ease as the two foregoing cases of it. 
 
CHAP. III.] SURFACE OF CENTRES AS AN ENVELOPE. 687 
 
 so that r retains here its recent signification LXV., and r is the analogous scalar for 
 the other direction of curvature, we have by LXXIV. the differential, 
 
 LXXIX. . . dr'= 2S^r'dr' = 2Sr^r'Sr'dr = 0, 
 because Sr0r'= 0, by 410, XI. 
 (32.) We have then the relation, 
 
 LXXX. . . /U (vdp) =/r' = r' = const. ; 
 that is to say, the square (r'-i) of the scalar semidiameter (Z)') of the surface, which 
 is parallel to the second tangent (r'), is constant for any one line of curvature (r) ; 
 and accordingly (comp. XXII., and the expression 407, LXXI. for/Uj^i), the value 
 of this square is, 
 
 LXXXI. . . (/Ui/dp)-i = r- 1 = a2 - «'2 = 62 _ h'l = c^- c% 
 
 if a\ b', c' be the scalar semiaxes of the confocal, which cuts the given quadric {abc) 
 along the line of curvature, whereof the variable tangent is r. 
 
 (33.) This constancy of/Uvdp may be proved in other ways ; for instance, the 
 general equation Svdvdp = gives, for a line of curvature on an arhitraru surface, 
 
 dv 
 LXXXII. . . dv = v^v-^dv + djoS — ; LXXXIII. . . Vdrdp = vdp^v'^dv ; 
 
 and LXXXI V. . . S . d|O0 (j^dp) = 0, because dv = ^dp; 
 
 while for a central quadric {fp = 1, 0p = j^) it is easy to show that we have also, 
 
 LXXXV. . . 0(j/dp) = Vpdp/(i/Udp) ; 
 
 hence, for such a surface, if we suppose for simplicity that ds or Tdp is constant, 
 which gives Yvd^p \\ dp, we have, 
 
 LXXXVI. . . d/(j/dp) = 2S(0(i/dp).d(vdp)) = 2Sv-idx;./(j.dp), 
 
 a differential equation of the second order, of which a. first integral is evidently, 
 
 LXXXVII. . . f(vdp) = Ci^adp^, or LXXX VII'. . . /U(vdp) = C = const. 
 
 (34.) But we see that the lines of curvature on a central quadric are thus ?'«•■ 
 eluded in a more general si/stem of curves on the same surface, represented by the 
 differential equation LXXXVI., of which the complete integral would involve two 
 constants : and which expresses that the semidiameters parallel to those tangents to 
 the surface, which cross any one such curve at right angles, have a common square, 
 and therefore (if real) a common length, so that (in this case) they terminate on a 
 sphero-conic* 
 
 (35.) Admitting however, as a case of this property, the constancy LXXX. of 
 the scalar lately called r', namely the second root of the quadratic XXXIV. or 
 XXXV., of which the coefficients and the first root r vary, in passing from one point 
 to another of what we may call for the moment a line of first curvature, we have only 
 to conceive r and v to be accented in the equations LXVI. LXVl'., in order to per- 
 ceive this theorem, which perhaps is new : 
 
 Compare the su^^-articles (6.) (7.) (8.) to 219, in page 231. 
 
688 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 The Curve* on the Reciprocal (24.) of the Surface of Centres of curvature of a 
 central qnadric, which answers to the second curvature of that given surface for all 
 the points of a given line oi first curvature, or which is itself in a known sense the 
 reciprocal (with respect to the given centre) of the developable normal surface (28.) 
 which rests upon that line, is the intersection of two quadrics ; whereof one (LXVI'.) 
 is a cone, concyclic with the given surface (fp = 1) ; while the other (LXVI.) is a 
 surface concyclic with the reciprocal of that given quadric {Fv = 1). 
 
 (36.) Again, the scalar Equation of the Surface of Centres (21.) may be said 
 to be the result of the elimination of r-i between the equations LXVIII. and LXXI., 
 whereof the latter is the derivative^ of the former with respect to that scalar ; we 
 have therefore this theorem : 
 
 An Auxiliary Quadric (LXVIII. or XLV.) touches the Second Sheet of the 
 Surface of Centres of a given quadric, along a Quartic Curve, which is the locus of 
 the centres of Second Curvature for all the points of sl Line of First Curvature (35.) ; 
 and (for the same reason) the same auxiliary quadric is circumscribed, along the 
 same quartic, by the Developable Normal Surface (28.), which rests on that^ri-^ 
 line: Avith permission, of course, to interchani^e the y\or As, first and second, in this 
 enunciation. 
 
 (37.) When the arbitrary constant r is thus allowed to take successively all va- 
 lues, corresponding to both systems of lines of curvature, the Surface of Centres ia 
 therefore at once the Envelope^ of the Auxiliary Quadric LXVIII., and the Locus 
 of the Quartic Curve (36.), in which one or other of its two sheets istouched, by that 
 auxiliary quadric in one of its successive states, and also by one of the developable 
 surfaces of normals to the given surface. 
 
 (38.) To obtain the vector equation of that envelope or locus, we may proceed 
 
 * The variable vector of this curve is easily seen (comp. XLIX.) to be, 
 
 , r' VT 
 
 Sr'p Svrp ' 
 
 and the reciprocal surface (21.) or (24.) is by (25.) the locus of this quartic (35.). 
 
 f The analogous relation, between the co-ordinate forms of the equations, was 
 perhaps thought too obvious to be mentioned, in page 161 of t)r. Salmon's Treatise ; 
 or possibly it may have escaped notice, since the quartic curve (36.) is only mentioned 
 there as an intersection of two quadrics, which is on the surface of centres, and 
 answers to points of a line of curvature upon the given surface. But as regards 
 the possible novelty, even in part, of any such geometrical deductions as those given 
 in the text from the quaternion analysis emploj^ed, the writer wishes to be under- 
 stood as expressing himself with the utmost diffidence, and as most willing to be 
 corrected, if necessary. The power of derivating (or differentiating) any symbolical 
 expression of the form LXVIII., or of any analogous /orm, with respect to any sca- 
 lar which it involves explicitly, as if the expression were a/pe6ra?ca/, is an important 
 but an easy consequence from the principles of the Section III. ii. 6, which has been 
 so often referred to. 
 
 X Compare the Note immediately preceding. , 
 
CHAP. III.] VECTOR EQUATION OF SURFACE OF CENTRES. 689 
 
 as follows, using a new expression for (T, in terms of v or of p, which may theu be 
 transformed into a function of two independent and scalar variables. Denoting 
 (comp. (32.)) by ai, 61, c\ the semiaxes of the confocal which cuts the given sur- 
 face in the given line of curvature, and by a-z, 62, C2 those of the other confocal, so 
 that the normals vi, Vi to these two confocals have the dii'ections of the tangents r', 
 T lately considered, we have not only the expressions LXXXI. for r'-i, with a'b'c' 
 changed to ai, fei, c\, but also the analogous expressions (comp. 407, LXXL), 
 LXXXVIII. . . r-i = a2 - ao2 = 62 - h.^ = c2 - c^^. 
 
 We have therefore by XLIL, combined with 407, XVI., this very simple expression 
 for a : 
 
 LXXXIX. . . (r = (0-' + r-i)j/ = 02-'*' = 02"'0(O; 
 
 containing, in the present notation, and as a result of the present analysis, a known 
 and interesting theorem,* on which however we cannot here delay. 
 
 (39.) It follows from this last value of cr, combined with the expression 408, 
 LXXXII. for p, that we may write, 
 
 \ a + a' 
 
 , w w,^2^ V- 16-161623 , c-'cica^ 
 
 JLU. . . <T = r^\ r + — ; 1 
 
 Yaa a — a 
 
 as the sought Vector Equation of the Surface of Centres of curvature of a given 
 quadric (a6c) ; ambiguous signs being virtually included in these three terms, be- 
 cause in the subsequent eliminationsf the semiaxes enter only by their squares : 
 while /, a, a are constants, as in 407, &c., for the whole confocal system, and ahc 
 are also constant here, but a^ — a^ and aP- — a-^, or r'^ and ;-i (38.), are variable, 
 and may be considered to be i\\Q two independent scalar 3 of which cr is a vector func- 
 tion. 
 
 413. Some brief remarks may here be made, on the connexion 
 of the general formula, 
 
 I. . . Si^-' (0 + r)-'v = 0, 412, XXXV. 
 
 in which r~R'^Tv (412, XXIV.), and which when developed by the 
 rules qf the Section III. ii. 6 takes (comp. 398, LXXIX.) the form 
 of the quadratic, 
 
 * Namely Dr. Salmon's theorem (page 161 of his Treatise), that the centres of 
 curvature oi A given, quadric at a given point are the poles of the tangent plane, 
 with respect to the two confocals. The connected theorem (page 136), respecting 
 the rectilinear locus of the poles of a given plane, with respect to the surfaces of a 
 confocal system, is at once deducible from the quaternion expression 407, XVI. for 
 0'V, although the theorem did not happen to be known to the present writer, or at 
 least remembered by him, when he investigated i\mt formula of inversion for other 
 applications, of which some have been already given. 
 
 f The corresponding elimination in co-ordinates was first effected by Dr. Salmon, 
 who thus determined the equation of the surface of centres of curvature of a quadric 
 to be one of the twelfth degree. (Compare pages 161, 162 of his already cited Trea- 
 tise.) 
 
 4 T 
 
690 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 II. . . ?-2 + rSv-\v + Su-'yjri^ = 0, 412, XXXIV. 
 
 with Gauss's* theory oHhe Measure of Curvature of a Surface; and 
 especially with his fundamental result, that this measure is equal to 
 the product of the two principal curvatures of sections of that surface: 
 a relation which, in our notations, may be thus expressed, 
 
 III. . . YA\^v^\5v = R^-'R,-'Ydiphp, 
 
 (1.) As regards the deduction, by quaternions, of the equation III., in which d 
 and S may be regarded as twof distinct symbols of diflferentiation, performed with re- 
 spect to two independent scalar variables, we may observe that, by principles and 
 rules already established, 
 
 IV. . . dUj/ = V-.Uj^, ^UJ/ = V — .Uj/ = -Uj/.V — ; 
 
 V V V 
 
 and that therefore the first member of III. may be thus transformed : 
 
 V. . . V. dUj^ 5Uj/ = V ( V — .V — ^ = - v-lSv-ldv^v. 
 
 (2.) Again, since we have dv = ^dp (410, IV., &c.), and in like manner ^v = 
 ipdp, the relations Svdp = 0, SvSp = 0, and the self-conjugate property of 0, allow 
 us to write, 
 
 VI. . . Vdi/5v = ^Ydpdp, and VII. . . Vdp^p = v-^SvdpSp ; 
 
 whence follows at once by V. the formula III., if we remember the general expres- 
 sion, deduced from the quadratic II., 
 
 VIII. . . i?rii?3-i = - i/-2rir2= - S - i// -. 412, XXXVII. 
 
 V V 
 
 (3.) If then we suppose that p, Pi, P2 are any three near points on an arbi- 
 trary surface, and that B, Ri, R2 are three near and corresponding points on the 
 unit sphere, determined by the condition of parallelism of the radii or, ori, oro to 
 the normals pn, PiNi, P2N2, the two small triangles thus formed will bear to each 
 other the ultimate ratio, 
 
 IX...lim.^^^^ = i?rii?.-; 
 APP1P2 
 
 a result which justifies Calthough by an entirely new analysis) the adoption by Gauss 
 
 * The.reader is referred to the Additions to Liouville's Monge (pages 505, &c.), 
 in which the beautiful Memoir by Gauss, entitled : Disquisitiones generales circa 
 superficies curvas, is with great good taste reprinted in the Latin, from the Commen- 
 tationes recentiores of the Royal Society of Gottingen. He is also supposed to look 
 back, if necessary, to the Section III. ii. 6 of these Elements (pages 435, &c.), and 
 especially to the deduction in page 437 of \p from 0, remembering that the latter 
 function (and therefore also the former) is here self- conjugate. 
 
 t Compare page 487, and the Note to page 684. 
 
CHAP. III.] MEASURE OF CURVATURE OF A SURFACE. 691 
 
 of this product* of curvatures of sections, as the measure of the curvature of the 
 surface^ with his signification of the phrase. 
 
 (4.) As another form of this important product or measure, if we conceive that 
 the vector p of the surface is expressed as a function (372) of two independent sca- 
 lars, t and «, and if we write for abridgment, 
 
 X. . . D<p = p', D„p = p,, Dt^p = p", DtDup = |0/, D„2p = p^^, 
 
 which will allow us (comp. 372, V.) to assume for the normal vector v the expres- 
 sion, 
 
 XL\.v = Yp'p, 
 
 it is easy to provef that we have generally, 
 
 XII. . . i2rii?3-» = S^S^-(s^ 
 
 V V \ V 
 
 which takes as a verification the well-known form, 
 
 XIII. . . Ef'Ri 
 
 when we write (comp. 410, (18.)), 
 
 XIV. . . p = ix +J2/ + kz, p = T)xp = i + kp, p^ - Byp =j + kq ; 
 'X.Y. . . v = Yp'p^= k — ip~jq, p" = kr, p^' = ks, p,^=kt. 
 
 (5.) In general, the equation XII. may be thus transformed, 
 
 XVI. , . v4i?i-ii?2-» = S (Yvp".Yvp,;) - (Yvp:y + 1/2 (Sp"p^^_ p;2) ; 
 also XVII. . . Td|o2 =r em + 2/dM?« + gdiu^, 
 
 if XVIII. . . e = - p'2, /= - Sp'p,, g = -p,% whence XIX. . . v^ =/2 - eg 
 
 and if we still denote, as in X., derivations relatively to t and u by upper and lower 
 accents, we may substitute in the quadruple of the equation XVI. the values, 
 
 XX. . . 2Yvp" = (e, - 2/)p' + e'p,, 2Vvp; = -g'p + c,p,, 2Vj^p„= - g/ 
 
 + (2/:-/)p,, 
 and XXI. . . 2 (Sp"p,, - p/O = e„ - 2// + g" ; 
 
 hence the measure of curvature is an explicit function of the ten scalar s, 
 
 XXU...e,fg; e\f',g'- e^,f,g^; and e„-2f'+g": 
 
 and therefore, as was otherwise proved by Gauss, this measure depends onlyX on the 
 
 * If it be supposed to be in any manner known that a limit such as IX. exists, 
 or that the quotient of the two vector areas in III. is a scalar independent of the di- 
 rections of PPi, PP2, or of dp, dp, we have only to assume that these are the direc- 
 tions of the lines of curvature, in order to obtain at once, by 412, II., the product 
 Br^Bz'^ as the value of this quotient or limit. 
 
 t The quadratic in R-^ may be formed by operating on 412, II. with S.p' and 
 S.p„ and then eliminating dt : dw. 
 
 X The proof by quaternions, above given, of this exclusive dependence, is per- 
 haps as simple as the subject will allow, and is somewhat shorter than the correspond- 
 ing proof in the Lectures : in page 605 of which is given however the equation, 
 
692 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 expression (XVII.) of the square of a linear element, in terras of two independent 
 scalars (^ u), and of their differentials (dt, du). 
 
 (6.) Hence follow also these two other theorems* of Gauss : — 
 If a surface be considered as an infinitely thin solid, and supposed to he flexible 
 but inextensible, then every deformation of it, as such, will leave unaltered^ 1st, the 
 Measure of Curvature at any Point, and Ilnd, the Total Curvature of any Area; 
 that is, the area of the corresponding portion of the unit sphere, determined as in (3.) 
 by radii parallel to normals. 
 
 (7.) Supposing now that t and u are geodetic co-ordinates, whereof the former re- 
 presents the length of a geodetic APfrora 3i fixed point A of the surface, and the latter 
 represents the angle bap which this variable geodetic makes at A with Sl fixed geo- 
 detic AB, it is easy to see that the general expression XVII. takes the shorter form, 
 
 XXIII. . , Tdp2 = d<2 + „2dM2^ in which XXIV. . . n = Tp, = Tr ; 
 so that we have now the values, 
 
 XXV. ..c=l, /=0, ^ = n2, ^' = 2w»', ^" = 2w»"+2n'2, 
 
 and the derivatives of e and /all vanish. And thus the general expression XII. for 
 the measure of curvature reduces itself by (5.) to the very simple form, 
 
 XXVI. . . i?ri/?2-i = - n-'n" = - «-iD<2„ ; 
 
 in which n is generally a function of both t and u, although here twice derivated 
 with respect to the former only. 
 
 (8.) The point r being denoted by the symbol (t, u), and any other point p' of 
 the surface by (t + A*, « + Am), we may consider the two connected points Pj, P2, of 
 which the corresponding symbols are (t + At, «) and (f, u -f Aw) ; and then the 
 quadrilateral pPiP'Pa, bounded by two portions PPi, Tzv' of geodetic lines from A, 
 and (as we may suppose) by two ai'cs PP2, Pip' of geodetic circles round the same 
 fixed point, will have its area ultimately =»A<Am (by XXIII.), and therefore (by 
 XXVI., comp. (3.), (6.)) its total ctirvature ultimately = — 11" At Au, ox =— Atn'.Au, 
 when At and A« diminish together, by an approach of p' to p. 
 
 (9.) Again, in the immediate neighbourhood of A, we have n = t, »' = 1 ; chang- 
 ing then — Atn' to — d^re', and integrating with respect to t from ^ = 0, we obtain 
 1 - n as the coefficient of Am in the result, and are thus conducted to the expres- 
 sion: 
 
 XXVII. . . Total Curvature of Triangle APp'= (1 - n') Am, ultimately, 
 
 if AP, AP' be any two geodetic lines, making with each other a small angle — Am, 
 and if pp' be any small arc (geodetic or not) on the same surface. 
 
 4 {eg _/2)2i?i-li22-l = e (g'^- 2gJ' + ge;) 
 
 +/(^V. - e,g' - 2e,f - 2g'f' + 4/'/) 
 
 + g (6,2 - le'f + cy ) - 2 {eg -/2) (e^^ _ 2// + g"), 
 
 which may now be deduced at sight from XVI., by the substitutions XIX. XX. 
 XXL, and differs only in notation from the equation of Gauss (Liouville's Monge, 
 page 523, or Salmon, page 309). 
 
 * See page 524 of Liouville's Monge. 
 
CHAP. III.] TOTAL CURVATURE, SPHEROIDAL EXCESS. 693 
 
 (10.) Conceive then that pq is a, finite arc oi any curve upon the surface, foi" 
 which therefore f, and consequently n', may be conceived to be a function of m ; we 
 shall have this other expression of the same kind, 
 
 XXVIII. . . Total Curvature of Area apq = J(1 -re')dM= Am — Jn'dw ; 
 the area here considered being bounded by the two geodetic lines Ap, aq, which 
 make with each other the finite angle Am, and by the arc pq of the arbitrary curve. 
 
 (IL) If this curve be itself a. geodetic, and if we treat its co-ordinates t, «, and 
 its vector p, as functions of its arc, », then the second differential of p, namely, 
 
 XXIX. . . d2p = p'm + p,d2M + p"d^2 + 2p;dkiw + p, dM2, 
 must be normal to the surface at p, and consequently perpendicular to p' and p^. 
 Operating* therefore with S.p', and attending to the relations XVIII. and XXV., 
 which give 
 
 XXX. . . p'2 = - 1, Sp'p, = Sp'p" = Sp'p', = 0, Sp'p,, = - sp,p; = ««', 
 
 we obtain the differential equation, 
 
 XXXI. . . d2/ = nn'du^, or XXXII. . . dw = - nWu, 
 if we observe that we may write, 
 
 XXXIII. . . dit- cos yds, ndw = sin yds, because XXXIV. . . d<^ + ^^Mm^ = ds« ; 
 V being here the variable angle, which the geodetic pq makes at p with ap pro- 
 longed. 
 
 (12.) Substituting then for -n'd«, in XXVIII., its value dw given by XXXII., 
 the integration becomes possible, and the result is Am + Ar ; where Am is still the 
 angle at A, and tt + Aw = (tt — u) + (w + Av) is the sum of the angles at P and q, in 
 the geodetic triangle Apq. 
 
 (13.) Writing then b and c instead of p and Q, we thus arrive at another most 
 remarkable Theorem f of Gauss, which may be expressed by the formula : 
 
 XXXV. . . Total Curvature of a Geodetic Triangle abc = A + b+c— tt, 
 
 = what may be called the Spheroidal Excess ; A, b, c, in the second member, being 
 used to denote the three angles of the triangle : and the total surface of the unit 
 sphere (= 47r) being represented by 720°, when the part corresponding to the geodetic 
 triangle is thus represented by the angular excess, A + b + c — 1 80°. 
 
 (14.) And it is easy to perceive, on the one hand, how this theorem admits (Jf 
 being extended, as it was by Gauss, to all geodetic polygons : and on the other hand, 
 how it may require to be modified, as it was by the same eminent geometer, so as to 
 give what would on the same plan be called a spheroidal defect, when the measure 
 of curvature is negative, as it is for surfaces (or parts of surfaces) of which^the prin- 
 cipal sections have their curvatures oppositely directed. 
 
 * To operate with S.p, would give a result not quite so simple, but reducible to 
 the form XXXI., with the help of di^s = 0. 
 
 t The enunciation of this theorem, respecting which its illustrious discoverer 
 justly says, " Hoc theorema, quod, ni fallimur, ad elegantissima in theoria superficie- 
 rum curvarum referendum esse videtur," ... is given in page 533 of the Additions to 
 
694 ELEMENTS OF QUATERNIONS. [bOOK IH. 
 
 414. The only sections of a surface, of which the curvatures 
 have been above determined, are the two principal normal sections at 
 any proposed point; but the general expressions of III. iii. 6 may 
 be applied to find the curvature o^ any plane section, normal or ob- 
 lique, and therefore also of any curve on a given surface, when only 
 its osculating plane is known. Denoting (as in 389, &c.) by p and k 
 the vectors of the given point p, and of the centre k of the osculating 
 circle at that point, and by s the arc of the curve, we have generally 
 (by 389, XII. and VI.), 
 
 I. . . Vector of Curvature of Curve = kp"' = (/> - /c)'' = D//) = t- ^ t^ i 
 
 d/> d/> 
 
 the independent variable in the last expression being arbitrary. And 
 if we denote by <r and ^ the vectors of the points s and x, in which 
 the axis of the osculating circle meets respectively the normal and 
 the tangent plane to the given surface, we shall have also, by the 
 right-angled triangles, the general decomposition, kp"' = sp"^ + xp"' 
 (as vectors), or 
 
 11. . . T)s'p = {p-'cr={p-<r)-^-^{p-^y- 
 
 where the two components admit of being transformed as follows: 
 III. . . Normal Component of Vector of Curvature of Curve (or 
 
 Section) = {p- o-)"' = /^"'S — = (^ - o-i)'^ cos^ v + {p - o-j)"' sin^ v 
 
 = Vector of Normal Curvature of Surface for the direction of 
 the given tangent ; 
 o-j, o-g being the vectors of the centres Si, s^ (comp. 412) of the ttuo 
 principal curvatures, and v being the angle at which the curve (or 
 its tangent d/?) crosses the first line of curvature (or its tangent tj), 
 while o- is the vector of the centre s of the sphere which is said to 
 osculate to the surface, in the given direction (of d/a) ; and 
 
 IV. . . Tangential Component of Vector of Curvature 
 
 = (p- |:)-i = v-'dp-'Si^dp-'d^p 
 = Vector of Geodetic Curvature of Curve (or Section) ; 
 
 this latter vector being here so called, because in fact its tensor re- 
 
 Liouville's Monge. A proof by quaternions was published in the Lectures (pages 
 606-609, see also the few preceding pages), but the writer conceives that the one 
 given above will be found to be not only shorter, but more clear. 
 
CHAP. III.] CURVATURES OF SECTIONS, OSC. SPHERES. 695 
 
 presents what is known by the name of the geodetic* curvature of a 
 curve upon a surface : the independent variable being still arbi- 
 trary. 
 
 (1.) As regards the decomposition II., if a, (3 be any two rectangular vectors 
 OA, OB, and if y = oo = the perpendicular from o on ab, then (comp. 316, L., and 
 408, XLL), 
 
 V. . . y-i = -^ + -^ = a-J + i3-i. 
 ^ Va/3 ^ V/3a " + ^ ' 
 
 (2.) To prove the first transformation III., we have, by I. and II., observing 
 that dSvdp = 0, 
 
 VI. ^ ^ s ^ ^ S. ^ V ^^^ = ~ ^^^^^ = ^^^^'^ = S ^^ 
 
 ' ' p-(T p — K dp djo dp2 dp2 dp* 
 
 (3.) Hence, by 412, (7.), if we denote the vector III. of normal curvature by 
 E'^JJv, we have the general expressions (comp. 412, I. XXI.), 
 
 VII. . . (T=p-{RVv, R = D^.Tv, with VIII. . .Tv = P-\ 
 
 for the case of a central quadric ; D being generally the semidiameter of the index 
 surface (410, (9.), &c.), or for a quadric the semidiameter of that surface itself 
 which has the direction of the tangent (or of dp) : and P being, for the latter sur- 
 face, the perpendicular from the centre on the tangent plane, as in some earlier for- 
 mulae. 
 
 (4.) To deduce the second transformation III., which contains a theoi'em of 
 Euler, let r, n, T2 denote unit tangents to the section and the two lines of curvature, 
 so that 
 
 IX. . . r^ncosu + rasinw, and r^ = ri2 = r2'^ = - 1 ; 
 
 we may then write generally (comp. 412, IV.), 
 
 X. .. i2-»Ti/ = -^ = -S^ = -Sr-i(6r = Sr0r, 
 a-p dp 
 
 and shall have the values (comp. 410, XI.), 
 
 XI. . . Sn^Ti = i?r'Tr, Sr20r2 = i?2"iTv, Sri^r2 = Sr2^ri = O5 
 
 whence XII. . . i?'' = i?r» cos2 1> + i?2"' sin2 v, 
 
 and the required transformation is accomplished. 
 
 (5.) The theorem of Meusnier may be considered to be a result of the elimination 
 
 (2.) of d^p from the expressions for the normal component III. of what we may call 
 
 the Vector D^'^p of Oblique Curvature ; and it may be expressed by the equation, 
 
 XIII. . . S ^-^ =1, or Xlir. . . S ^^ = 0, which gives XIII". . . pks = ^, 
 
 p - K p -K 2 
 
 if it be now understood that the point s, of which a is the vector, is the centre of the 
 
 * The name, '■'' courhure geodesique" was introduced by M. Liouville, and has 
 been adopted by several other mathematical writers. Compare pages 568, 575, &c. 
 of his Additions to Monge. 
 
696 ELEMKNTS OF QUATERNIONS. [bOOK III. 
 
 circle which osculates to the normal section ; or of the sphere which osculates in 
 the same direction to the surface, as will be more clearly seen by what follows. 
 
 (6.) In general, if p + Ap be the vector of any second point p' of the given sur- 
 face, the equation 
 
 V V 
 
 XIV. . . S = S — , with w for a variable vector, 
 
 <t» — p Ap 
 
 represents rigorously the sphere which touches the surface at the given point P, and 
 passes through the second point p' ; conceiving then that the latter point approaches 
 to the former, and observing that the development* by Taylor's Series of the equa- 
 tion ^p = const, gives (if <\fp = 2Si'dp, and dv = 0dp), 
 
 V (bAo 
 
 XV. . . = Ap"2A/b = 2S \- S^— ^ + terms which vanish generally with Ap, 
 
 Ap Ap 
 
 even if they be not always null, we are conducted in a new way, by the known con- 
 ception of the Osculating Sphere for a given direction to a surface^ to the same cen- 
 tre s, and radius i2, as before : the equation of this sphere being, 
 
 XVI. . . S 
 
 = lim. S — = - lim. S ^-— = - S — . 
 
 0) - p \ Ap Ap I dp 
 
 (7.) Conversely, if we assume a radius H, such that R'^ is algebraically inter- 
 mediate between Hi'^ and Jti'^, the tangent sphere, 
 
 2v Tv 2lJv 
 
 XVII. . . S = -— , or xvir. . . S = /?-!, 
 
 w — p Ji (t) — p 
 
 will cut the surface in two directions of osculation, assigned by the formula XII. ; 
 but if R~^ be outside those limits, there will be only contact, and not any (real) in- 
 tersection, at least in the vicinity of p. 
 
 (8.) If p' be again, as in (6.), any second point of the surface, and if we denote 
 for a moment by (IT) and (2) the normal jo/a»e pnp' and the normal section cor- 
 responding, we may suppose that n is the point in which the normals to the plane 
 curve (S) at p and p' intersect ; and if we then erect a perpendicular at N to the 
 plane (11), it will be crossed by every perpendicular at p' to the tangent p't' to the 
 section, and therefore in particular by the normal at p' to the surface, in a point 
 which we may call n' : so that the line p'n is the projection, on the plane pp'n, of 
 this second normal pV to the surface. Conceiving then the plane (11) to he fixed, 
 but the point p' to approach indefinitely to p, we see that the centre s of curvature 
 of the normal section (2), which is also by (6.) the centre of the osculating sphere 
 to the surface for the same direction, is the limiting position of the point x, in which 
 
 * Compare Art. 374, and the Second Note to page 508. The occasional use, 
 there mentioned, of the differential symbol dp as signifying a finite and chordal vec- 
 tor, in the development of /(p + dp), has appeared obscure, in the Lectures, to some 
 friends of the writer ; and he has therefore aimed, for the sake of clearness, in at least 
 the text of these Elements, and especially in the geometrical applications, to confine 
 that symbol to its^r*^ signification (100, 369, 373, &c.\ as denoting a tangential 
 vector (finite or infinitely small, and to a curve or surface) : p itself being generally 
 regarded as a vector function, and not as an independent variable (comp. 362, (3.^). 
 
CHAP. III.] VECTOR OF GEODETIC CURVATURE, DIDONIAS. 697 
 
 the given normal at p is intersected hy the projection* of the near normal pV, on 
 the given normal plane. 
 
 (9.) The two components III. and IV. are included in the binomial expression, 
 
 XVIII. . . Vector of Oblique Curvature (or of Curvature of Oblique Section) 
 
 = (p - k)-! = j/-iSdi/dp-i + v-idp-iSa/dp- id2p, 
 
 which is obtained by substituting in I. the general equivalent 409, XXI. for d^p, 
 
 and in which (as before) the independent variable is arbitrary ; and the tangential 
 
 component IV. may be otherwise found by observing that, by I. and II., 
 
 XIX. . . -^ =S-^ = S^^ = -S.dp-MV, 
 P-? p-K dp 
 
 and that — (vdp)-' = v"idp"i, because S*'dp = 0. 
 
 (10.) Another way of deducing the same component IV., is to resolve the follow- 
 ing system of three scalar equations, which by the geometrical definition of the point 
 X the vector ^ must satisfy : 
 
 XX. ..S(?-p)i/ = 0; S(?-p)dp = 0; S(?-p)d«p = dp2; 
 and which give, 
 
 XXI V v^p' _ v^P 
 
 " ^ Sj/dpd2p Si/dp-id2p' 
 
 or (p — ?)-! = &c., as before. We have also the transformations, 
 
 XXII. . . Vector of Geodetic Curvature = (p - ?)» 
 
 = (r^dp)-i S(vUdp . dUdp) = - i/dp S ?i^l^ = &c. 
 
 j'dp 
 
 (11.) The definition of the point x shows also easily, that if a developable sur- 
 face (d) be circumscribed to a given surface (s), along a given curve (c), and if in 
 the unfolding of the former surface^ the point x be carried with the tangent plane, 
 originally drawn to the latter surface at p, it will become the centre of curvature, at 
 the new point (p), to the new or plane curve (c') obtained by this development : so 
 that the radius (px) of geodetic curvature is equal, as indeed it is knownf to be, to 
 the radius of plane curvature of the developed curve. 
 
 (12.) This plane curve (c') is therefore a circle^ (or part of one) if the condi- 
 tion, 
 
 XXIII. . . PX = T (^ - p) = const., 
 
 * The reader may compare the calculations and constructions, in pages 600, 601 
 of the Lectures. In the language of infinitesimals, an infinitely near normal p'n' 
 intersects the axis of the osculating circle, to the given normal section. 
 
 f Compare page 576 of the Additions to Liouville's Monge. 
 
 t The curves on any given surface, which thus become circles by development, 
 have also the isoperimetrical property expressed in quaternions (comp. the first Note 
 to page 530) by the formula, 
 
 XXVI. . . ^JS(Uv.dp^p) + cajTdp = 0, 
 which conducts to the differential equation, 
 
 XXVII. . . c-idp = V.Uv dUdp (comp. 380, IV.), 
 
 4 u 
 
698 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 be satisfied ; but it degenerates into a right line^ if this radius of geodetic curvature 
 be infinite, that is, if 
 
 XXIV. . . T (p - ^)-i = 0, or XXV. . . SvdpdSp = 0, 
 or finally (by 380, II., comp. 409, XXV.), if the original curve (c) be a geodetic line 
 on the given surface (s), and therefore also on the developable (d) : which agrees 
 with the fimdamental property (382, 383) of geodetics on a developable surface. 
 
 (13.) Accordingly it may be here observed that the general formula IV., com- 
 bined with the notations and calculations of 382, conducts to the expression 
 
 (« + »') Tp'-', or — , for the geodetic curvature of any curve on a developable 
 
 surface, whereof the element ds crosses a generating line at the variable angle », while 
 zdx is the angle between two such consecutive lines : a result easily confirmed by geo- 
 iiietrical considerations, and agreeing with the diflferential equation z + v' = 0^82, 
 IX.) oi geodetics on a developable. 
 
 415. We shall conclude the present Section with a few supple- 
 mentary remarks, including a new and simplified proof of an im- 
 portant theorem (354), which we have had frequent occasion to 
 employ for purposes of geometry, and which presents itself often 
 in jp Az/5ecaZ applications of quaternions also : namely, that if the linear 
 and vector function be self-conjugate, then the Vector Quadratic, 
 I. . . V/>0/9 = 0, 354, I. 
 
 represents generally a System of Three Real and Rectangular Direc- 
 tions ; and that these (comp. 405, (1.), (2.), &c.) are the directions 
 of the Axes of the Central Surfaces of the Second Order, which are 
 represented by the scalar equation, 
 
 II. . . S/>0/> = const. ; 
 or more generally, 
 
 III. . . S/>0/> = C(i^-\- C\ where C and C are any two scalar constants. 
 
 (1.) It is an easy consequence of the theory (350) of the symbolic and cubic 
 equation in <p, that if c be a root of the derived algebraical cubic M=0 (354), and 
 if we write <E> = ^ + c (as in that Article), the neiv linear and vector function 4>p must 
 be reducible to the binomial form (351), 
 
 and in which the scalar constant c can be shown to have the value, 
 
 XXVIII. . . c = (^ - (o) V.vdp = ± T (? - p) = Eadius of Geodetic Curvature, 
 = radius of developed circle ; and each such curve includes, by XXVI., on the given 
 surface, a maximum area with a given perimeter : on which account, and in allusion 
 to a well-known classical story, the writer ventured to propose, in page 682 of the 
 Lectures, the name " Didonia" for a curve of this kind, while acknowledging that 
 the curves themselves had been discovered and discussed by M. Delaunay. 
 
CHAP. III.] NEW PROOF OF RECTANGULAR SYSTEM. 699 
 
 IV. . . ^p = ^p + cp = /3Sap + /3'Sa'p, with V. . . V/3a + V/3'a' = 0, 
 as the condition (353, XXXVI.) of self- conjugation. With this condition we may 
 then write, 
 
 YI, . . p = Aa + Ba', (5' = A'a+Ba; 
 
 and it is easy to see that no essential generality is lost, by supposing that a and a 
 are two rectangular vector units, which may be turned about in their own plane, if 
 j3 and /3' be suitably modified : so that we may assume, 
 
 VII. . . a2 = a'2 = - 1, Saa = ; whence VIII. . . $a = - /3, $a' = - j3', 
 and IX. . . V/3'a' = Baa = - V/3a, Y(3a = Aaa, V/3'a = - A'aa'. 
 (2.) The equation I., under the form, 
 
 X. . . Vp$p = 0, is satisfied by XI. . . $p = 0, or XII. . . Yaa'p - ; 
 and it cannot be satisfied otherwise, unless we suppose, 
 
 XIII. . . p = aja + x'a', and XIV. . . V (a;/3 + a:'/3') (xa + x'a) = ; 
 
 that is, by IX., 
 
 XV. . . B(x^-x^) + (A -A')xx'=0: 
 
 while conversely the expression XIII. will satisfy I., under this condition XV. But 
 this quadratic in x' : x, of which the coefiicients B and A— A' do not generally va- 
 nish, has necessarily two real roots, with a product = — 1 ; hence there always ex~ 
 ists, as asserted, a system of three real and rectangular directions, such as the fol- 
 lowing, 
 
 XVI. . . xa + x'a', x'a — xa', and aa' (or Va a'), 
 
 which satisfy the equation I. ; and this system is generally definite : which proves 
 thfi first part of the Theorem. 
 
 (3.) The lines a, a' may be made by (1.) to turn in their own plane, till they 
 coincide with the two first directions XVI. ; which will give, 
 
 XVII. ..5 = 0, ^=Aa, (3' = A'a', 
 and therefore, 
 
 XVIII. . . <l>p = -cp + AaSap + A'a'Sa'p 
 
 = (c + A^ aSap + (c + A') a'Sa'p + caa'Saa'p ; 
 
 and thus the scalar equation II. will take the form, 
 
 XIX. . . Sp(pp = (c + A) (Sap)2 + (c + A') (Sa»2 + c(Saa'p)2= const., 
 
 which represents generally a central surface of the second order, with its three 
 axes in the three directions a, a', aa' of p ; and does not cease to represent such a 
 surface, and with such axes, when for Sp^p we substitute, as in III., this new ex- 
 pression : 
 XX. . . Sp(pp - Cp2 = Sp0p + C ((Sap)2 + (Sa'p)3 + (Saa'p)2) = C'= const. ; 
 
 the second surface heing in fact concyclic (or having the same cyclic planes) with the 
 first, and the new term, - Cp, in <pp, disappearing under the sign V.p : so that the 
 second part of the Theorem is proved anew. 
 
 (4.) It would be useless to dwell here on the cases, in which the surfaces XIX., 
 XX. come to be of revolution, or even to be spheres, and when consequently the 
 directions of their axes, or of p in I., become partially or even vihoMy indeterminate. 
 But as an example of the reduction of an equation in quaternions to the form I., 
 
700 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 without its at first presenting itself under that form, we may take the very simple 
 equation, 
 
 XXI. . . pipK=ipKp, with K not \\ i, 
 
 which may be reduced (comp. 354, (12.)) to 
 
 XXII. ..Y.pYipK = 0; 
 and which is accordingly satisfied (comp. 373, XXIX.) by the three rectangular di- 
 rections, 
 
 XXIII. . . Ut - XJk, YiK, Ut + U/c, 
 
 of the axes {abc) of the ellipsoid, 
 
 XXIV. . .T(t|0 + pic) =K:2-t% 282, XIX. 
 
 which is one of the surfaces of the concyclic system (comp. III.), 
 
 XXV. . . StpKp = Cp"^ + C, 
 as appears from the transformations 836, XL, &c. 
 
 (5.) In applying the theorem thus recently proved anew, we have on several 
 occasions used the expression, 
 
 XXVI. . . dv = 0dp, 410, IV. 
 
 in which j/ is a vector normal to a surface whereof p is the variable vector, and the 
 function is treated as self-conjugate (363). 
 
 (6.) It is, however, important to remark that, in order to justify the assertion 
 of this last property, the following expression of integral form, 
 
 XXVII. . . J Sj/dp, 
 must admit of being equated to some scalar function of p, such as ^/p + const., 
 without its being assumed that p itself is a function, of any determinate form, of a 
 scalar variable, t. The self-conjugation of the linear and vector function <p in 
 XXVL, is the condition of the existence of the integral XXVII., considered as re- 
 presenting such a scalar function (comp. again 363). 
 
 (7.) There are indeed several investigations, in which it is sufiicient to regard 
 V as denoting some normal vector, of which only the direction is important, and 
 which may therefore be multiplied by an arbitrary scalar coefficient, constant or 
 variable, without any change in the results (comp. the calculations respecting ^'eode- 
 tic lines, in the Section III. iii. 5, and many others which have already occurred). 
 
 (8.) And there have been other general investigations, such as those regarding 
 the lines of curvature on an arbitrary surface, in which dv was treated as a self- 
 conjugate function of dp, while yet (comp. 410, (17.)) the fundamental differential 
 equation Svdvdp = was not affected by any such multiplication of v by n. 
 
 (9.) But there are questions in which a factor of this sort may be introduced, 
 with advantage for some purposes, while yet it is inconsistent with the self- conjuga- 
 tion above mentioned, unless the multiplier n be such as to render the new expres- 
 sion S«j/dp (comp. XXVII.) an exact differential of some scalar function of p. 
 
 (10.) For example, in the theory oi Reciprocal Surfaces (comp. 412, (21.)), it 
 is convenient to employ the system of the three connected equations, 
 
 XXVIII. . . Si/p = 1, Svdp = 0, Spdv = ; 373, L. LI. 
 
 but when the length of v is determined so as to satisfy the first of these equations, 
 v"i being then the vector perpendicular from the origin on the tangent plane to the 
 
CHAP. III.] CONDITION OF INTEGRABILITY, FACTOR. 701 
 
 ffiven but arbitrary/ surface of which p la the vector, while p"i is the corresponding 
 perpendicular for the reciprocal surface with v for vector, the differential Av loses 
 generally its self- conjugate character, as a linear and vector function of dp : although 
 it retains that character if the scalar function fp be homogeneous, in the equation 
 fp = const, of the original surface, as it is for the case of a central quadric* for 
 which V = (pp, div = 0dp, &c., as in former Articles. 
 
 (1 1.) In fact, the introduction of the first equation XXVIII. is equivalent to the 
 multiplication of v by the factor n = (Svp)-i ; and if we write (comp. 410, (16.)), 
 
 XXIX. . . d/p = 2Svdp, di/ = 0dp, dn = Sffdp, 
 we shall have this new pair of conjugate linear and vector functions, 
 
 XXX. . .a.nv- <idp = w0dp + x^Sadp, XXXI. . . (4'dp = n^dp + trSvdp ; 
 and these will not be equal generally, because we shall not in general have a |1 v. 
 But this last parallelism exists in the case of homogeneity (10.), because we have 
 then the relations, 
 
 XXXII. . . 2Sj/p = r/p, d . n-J = dSvp = rSi^dp, 
 
 if r be the number which represents the dimension offp (supposed to be whole). 
 
 (12.) On the other hand it may happen, that the differential equation Sx/dp= 
 represents a surface, or rather a set of surfaces, without the e^cpression 8v6p being 
 an exact differential, as in (6.) ; and then there necessarily exists a scalar /ac^or, 
 or multiplier, n, which renders it such a differential. 
 
 (13.) For example the differential equation, 
 XXXIII. . . Sypdp = Svdp = 0, with XXXIV. . . v = Vyp, dv = Vydp = <p6p, 
 represents an arbitrary plane (or a set of planes'), drawn through a given line y ; 
 but the expression Sypdp itself is not an exact differential, and the integral XXVII. 
 represents no scalar function of p, with the present form of v, of which the differen- 
 tial dv is accordingly a linear function 0dp, which is not conjugate to itself, but to 
 its opposite (comp. 349, (4.)), so that we have here 0'dp = — ^dp. 
 
 (14.) But if we multiply v by the ^ac^or, 
 XXXV. . . w = v-2 =(Vyp)-2, which gives XXXVI. . . dn = S(Tdp, (r = 2«2yVyp, 
 and therefore Sy(T = 0, Spc = - 2n, then the new normal vector nv, or v-^, is found 
 to have the self-conjugate differential, 
 
 XXXVII. . . d . nv = d . v-i = - v-i Vydp . v-i = <5dp = 6'dp ; 
 and accordingly the new expression, 
 
 do 
 XXXVIII. . . Snvdp = Sj;-'dp = S — ^, with y constant, 
 
 Yyp 
 
 is easily seen to be an exact differential, namely (if Ty = 1), that of the angle which 
 the plane of y and p makes with a fixed plane through y : so that, when v is thus 
 
 * It was for this reason that the symbol Tv was not interpreted generally as 
 denoting the reciprocal, P"i, of the length of the perpendicular from the origin on the 
 tangent plane, in the formulae of 410, 412, 414 : although, in several of those for- 
 mulae, as in an equation of 409, (3.), that symbol was so interpreted, for the case of 
 a central surface of the second order. 
 
702 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 changed to nv, the integral in XXVII. SiC<\mx&i a. geometrical signification^ which is 
 often useful in physical applications, since it then represents the change of this angle, 
 in passing from one position of p to another ; or the angle through which the variable 
 plane of yp has revolved. 
 
 (15.) In fact, the general formula 336, XV. for the differential of the angle of 
 a quaternion gives, if we write 
 
 Vyp 
 XXXIX. . . q = — ^, y = const., pQ = const, Ty = 1, 
 
 the two connected expressions : 
 
 XL...dZ^ = + S^; XLI...J-S^=±A^(Vyp:Vypo); 
 
 which contain the above-stated result, and can easily be otherwise established. 
 
 (16.) In general, if the linear and vector function dv = ^dp be not self-conju- 
 gate, and if the function d.nv = (bdp be formed from it as in (11.), it results from 
 that sub-article, and from 349, (4.), that we may write, 
 
 XLII. . .(<p- f) dp = 2 Vydp, (6 - <5') dp = 2 Vy,dp, 
 
 with the relation, 
 
 XLIII. . . 2y, = 2ny + Va/(r; 
 
 where y, y, are independent of dp, although they mag depend on p itself. If then 
 the new linear function 6dp is to be self-conjugate, so that y<= 0, we must have 
 
 XLIV. . . 2»y + Vvff = 0, and therefore XLV. . . Syv = ; 
 
 which latter very simple equation, not involving either n or er, is thus a form, in 
 quaternions, of the Condition of Integr ability^ of the differential equation Svdp = 0, 
 if the vector y be deduced from v as above. 
 
 (17.) The Bifocal Transformation of Sp^p, in 360, (2,), has been sufficiently 
 considered in the present Section (III. iii. 7) ; but it may be useful to remark here, 
 that the Three Mixed Transformations of the same scalar function fp, in the same 
 series of sub-articles, include virtually the whole known theory of the Modular and 
 Umbilicar Generations of Surfaces of the Second Order. 
 
 (18.) Thus, in the formulas of 360, (4.), if we make e = 1, c is the vector of an 
 Umbilicar Focus of the surface fp = 1, and Z is the vector of a point on the Umbili- 
 car Directrix corresponding; whence the umbilicar focal conic and dirigent cylin- 
 der (real or imaginary) can be deduced, as the loci of this point and li7ie. 
 
 (19.) Again, by making ei and e^ each = 1, in the formulae of 360, (6.), we ob- 
 tain Two Modular Transformations of the equation of the same surface; ci, 63 being 
 
 * If the proposed equation be 
 
 Svdp =pdx + qdy + rdz = 0, so that v — -(ip +jq + kr), 
 we easily find that 2y = iP +j Q + hR, where 
 
 P=I>^q-T>yr, Q^Bicr-DzP, R=Dyp-T>j,qi 
 the condition of integrability XLV. becomes therefore here, 
 
 pP+ qQ-\ ri? = 0, which agrees with knoAvn results. 
 
CHAP. III.] MODULAR AND UMBILICAR GENERATIONS. 703 
 
 vectors of Modular Foci, in two distinct planes, and ^i, ^3 being vectors of points 
 upon the Modular Directrices corresponding : whence the modular focal conies, and 
 dirigent cylinders (real or imaginary), are found by easy eliminations. 
 
 (20.) Thus, by assuming that either 
 
 XLVI. ..S\(p-^O = 0, S\(p-?3) = 0, 
 or XLVII. . . S/i (p - ?i) = 0, S/i (p - ^3) = 0, 
 
 the equations 360, XVI., XVII. may be brought to the forms, 
 
 XLVIII. . . (p - f 1)2 = mi2 (p - ^1)2, XLIX. . . (p - f 3)2 = »«32 (p - ^3)2, 
 with the values, 
 
 L. ..mi2 = l-^, and LI. . . m32= 1- - ; 
 
 Ci C3 
 
 in which ci, C2, C3 are the three roots of a certain cubic (M= 0), or the inverse 
 squares of the three scalar semiaxes (real or imaginary) of the surface, arranged in 
 algebraically ascending order (357, IX., XX. ; 405, (6.), &c.): and mi, m^ are the 
 two (real or imaginary) Moduli, or represent the modidar ratios, in the two modes 
 of Modular Generation* corresponding. 
 
 (21.) It is obvious that an equation of the form, 
 
 LII. . . T^p= C= const., 
 represents a central quadric, if 0p be any linear f and vector function of p, of the 
 
 * Mac Cullagh's rule of modular generation, which includes both those modes, 
 was expressed in page 437 of the Lectures by an equation of the form, 
 
 T(p-a)=TV.yV/3p; 
 
 in which the origin is on a directrix, (3 is the vector of another point of that right 
 line, a is the vector of the corresponding focus, y is perpendicular to a directive 
 (that is, generally, to a cyclic) plane, p is the vector of any point p of the surface, 
 and ± Sj3y is the constant modular ratio, of the distance ap of p from the focus, to 
 the distance of the same point p from the directrix ob, measured parallel to the di- 
 rective plane. The new forms (360), above referred to, are however much better 
 adapted to the working out of the various consequences of the construction ; but it 
 cannot be necessary, at this stage, to enter into any details of the quaternion trans- 
 formations : still less need we here pause to give references on a subject so interest- 
 ing, but by this time so well known to geometers, as that of the modular and um- 
 bilicar generations of surfaces of the second order. But it may just be noted, in order 
 to facilitate the applications of the formulae L. and LL, that if we Avrite, as usual, 
 for all the central quadrics, a^ >b^ > c^, whether 6^ and c^ be positive or negative, 
 then the roots ci, c^, C3 coincide, for the ellipsoid, with a"2, 6-2^ c-2 ; for the single- 
 sheeted, hyperboloid, with c-2, a'^, 6 2 • and for the double -sheeted hyperboloid with 
 6-2, c"2, a'^, (comp. page 651). 
 t In page 664 the notation, 
 
 dp = 2Svdp = 2S0pdp, 409, IV. 
 
 was employed for an arbitrary surface ; but with the understanding that this func- 
 tion tpp (comp. 363) was generally non-linear. It may be better, however, as a 
 
704 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 kind considered in the Section III. ii. 6, whether self-conjugate or not; but it re- 
 quires a little more attention to perceive, that an equation of this other form ^ 
 
 LIII. . . T(p-V./3Vya) = T(a-V.yV/3p), 
 
 represents such a surface, whatever the three vector constants a, (3, y may be. The 
 discussion of this lust form would present some circumstances of interest, and might 
 be considered to supply a new mode of generation^ on which however we cannot 
 enter here. 
 
 (22.) The surfaces of the second order, considered hitherto in the present Section, 
 have all had the origin for centre. But if, retaining the significations of ^, /, and F, 
 we compare the two equations, 
 
 LIV. . .f{p-K) = C, and LV. . .fp- 2S£p = C, 
 we shall see (by 362, &c,) that the constants are connected by the two relations, 
 
 LVI. ..6 = ^»c, C'=C-//c= C-S£K=C-F«; 
 so that the equation, 
 
 LVII. ..fp- 2S6P =/(p - 0-i£) - F,, 
 is an identity. 
 
 (23.) If then we meet an equation of the form LV., in which (as has been usual) 
 we have still yp= Sp0p = a scalar and homogeneous function of p, of the second 
 dimension, we shall know that it represents generally a surface of that order, with 
 the expression (comp. 347, IX., &c.), 
 
 LVIII. . . K = 0-'£ = m-^^e = rector of Centre. 
 (24.) It may happen, however, that the two relations, 
 LIX. ..m=0, ^^Ps>0, 
 exist together ; and then the centre may be said to be at an infinite distance, but in a 
 definite direction : and the surface becomes a Paraboloid, elliptic or hyperbolic, accord- 
 ing to conditions which are easy consequences from what has been already shown. 
 (25.) On the other hand it may happen that the two equations, 
 
 LX. . . m = 0, ;//£ = 0, 
 
 are satisfied together; and then the vector k of the centre acquires, by LVIII., an 
 indeterminate value, and the surface becomes a Cylinder, as has been already suffi- 
 ciently exemplified. 
 
 (26.) It would be tedious to dwell here on such details; but it may be worth 
 
 general rule, to avoid writing v = 0p, except for central quadrics ; and to confine 
 ourselves to the notation dv=(pdp, as in some recent and several earlier sub -articles, 
 when we wish, for the sake of association with other investigations and results, to 
 treat the function ^ as linear (or distributive) ; because we shall thus be at liberty 
 to treat the surface as general, notwithstanding this property of ^. As regards 
 the methods of generating a qnadric, it may be worth while to look back at the Note 
 to page 649, respecting the Six Generations of the Ellipsoid, which were given by 
 the writer in the Lectures, with suggestions of a few others, as interpretations of 
 quaternion equations. 
 
CHAP. III.] CUBIC CONE, SCREW SURFACE, SKEW CENTRE. 705 
 
 while to observe, that the general equation of a Surface of the Third Degree may 
 be thus written : 
 
 LXI. . . Sqpq'pq'p + Sp0p + Syp + C= ; 
 
 C and y being any scalar and vector constants ; <pp any linear, vector, and self-con- 
 jugate function ; and q, q, q" any three constant quaternions : while p is, as usual, 
 the variable vector of the surface. 
 
 (27.) In fact, besides the one scalar constant, C, three are included in the vector 
 y, and six others in the function (comp. 358) ; and of the ten which remain to be 
 introduced, for the expression of a scalar and homogeneous function of p, of the third 
 degree, the three versors JJq, Vq', Vq" supply nine (comp. 312), and the tensor 
 T . qq'q" is the tenth. 
 
 (28.) And for the same reason the monomial equation, 
 
 LXII. . . Sqpq'pq"p = 0, 
 
 with the same significations of 9, q', q", rei)Ye86nts the general Cone of the Third 
 Degree, or Cubic Cone, which has its vertex at the origin of vectors. 
 
 (29.) If then we combine this last equation with that of a secant plane, such as 
 Sep +1 = 0, we shall get a quaternion expression for a Plane Cubic, or plane curve 
 of the third degree : and if we combine it with the equation p2 + 1 = of the unit- 
 sphere, we shall obtain a corresponding expression for a Spherical Cubic,* or for a 
 curve upon a spheric surface, which is cut by an arbitrary great circle in three pairs 
 of opposite points, real or imaginary. 
 
 C30.) Finally, as an example of sections of surfaces, represented by transcen- 
 dental equations, let us consider the Screw Surface, or IIelicoid,-f of which the vec- 
 tor equation may be thus written (comp. the sub-arts, to 314) : 
 
 LXIIl. .. p = c{x + a) a -vya^y, with Ta = l, y = Ya(3, and yX); 
 
 a being the unit axis, while [3, y are two other constant vectors, a, c two scalar 
 constants, and x, y two variable scalara. 
 
 (31.) Cutting this surface by the plane of /3y, or supposing that 
 
 LXIV. . . = Syj3p = jS^Sap - Sa(3S[3p, and writing LXV. . . c = bSa(3, 
 
 we easily find that the scalar and vector equations of what we may call the Screw 
 Section may be thus written : 
 
 LXVI. . . 6(a; + a)=yS.a^->; LXVII. . . p=y(yS.a*- /3S.a*-i). 
 
 (32.) Derivating these with respect to x, and eliminating (3 and y', we arrive 
 at the equation, 
 
 LXVIII. . .p = (a; + a)p'-fzy, if LXIX. . . 26z = Try^ 
 
 * Compare the Note to page 43 ; see also the theorem in that page, which con- 
 tains perhaps a new mode of generation of cubic curves in a given plane : or, by 
 an easy modification, of the corresponding curves upon a sphere. 
 
 t Already mentioned in pages 383, 502, 514, 557. The condition y>0 an- 
 swers to the supposition that, in the generation of the surface, the perpendiculars 
 from a given helix on the axis of the cylinder are not prolonged beyond that axis. 
 
 4 X 
 
706 
 
 ELEMENTS OF QUATERNIONS, 
 
 [book III, 
 
 but zy in LXVIII. is the vector of the point, say g, in which the tangent to the sec- 
 tion at the point (a?, y), or p, intersects the given line y, namely the line in the plane 
 of that section which ii perpendicular to the axis a : we see then, by LXIX., that 
 this point of intersection depends only on the constant, h, and on the variable^ y, 
 being independent of the constant, a, and of the variable, x. 
 
 (33.) To interpret this result of calculation, which might have been otherwise 
 found with the help of the expression 872, XII. (with ]3 changed to y) for the nor- 
 mal V to a screw- surface, we may observe, first, that the equation LXVIL, which 
 may be written as follows, 
 
 LXX. . . p =yV. a^+i/3, and gives LXXI. . . TYap = yTy, 
 
 would represent an ellipse, if the coefiicient y were treated as constant ; namely, the 
 section of the right cylinder LXXI. by ihQ plane LXIV. ; the vector semiaxes (ma- 
 jor and minor) of this ellipse being y(3 and yy (comp. 314, (2.)). 
 
 (34.) By assigning a new value to the constant a, we pass to a new screw sur- 
 face (30.), which differs only in position from the former, and may be conceived to 
 be formed from it by sliding along the axis a ; while the value of x, corresponding 
 to a given y, will vary by LXVI., and thus we shall have a new screw section (31.), 
 which will cross the ellipse (33.) in a new point Q : but the tangent to the section at 
 this point will intersect by (32.) the minor axis of the ellipse in the same point G as 
 before. 
 
 (35.) We shall thus have a Figure* such as the following (Fig. 85) ; in which 
 if F be a, focus of the ellipse bc, and g (as 
 above) the point of convergence of the tan- 
 gents to the screw sections at the points p, Q, 
 &c., of that ellipse, it is easy to prove, by 
 pursuing the same analysis a little farther, 
 1st, that the angle (^), subtended at this 
 focus F by the minor semiaxis oc, which is 
 also a radius (r) of the cylinder LXXI., is 
 equal to the inclination of the axis (a) of 
 that cylinder to the plane of the ellipse, as may indeed be inferred from elementary 
 principles ; and Ilnd, what is less obvious, that the other angle (h), subtended at the 
 same focus (f) by the interval og, or by what may be called (with reference to the 
 present construction, in which it is supposed that 6 < 0, or that the angles made by 
 Dxp and j8 with a are either both acute, or both obtuse) the Depression (s) of the Skew 
 Centre (g), is equal to the inclination of the same axis (a) to the helix on the same 
 cylinder, which is obtained (comp. 314, (10.)) by treating y as constant, in the 
 equation LXIII. of the Screw Surface. 
 
 * Those who are acquainted, even slightly, with the theory of Oblique Arches (or 
 skew bridges), will at once see that this Figure 85 may be taken as representing rudely 
 such an arch : and it will be found that the construction above deduced agrees with 
 the celebrated Rule of the Focal Excentricity, discovered practically by the late Mr. 
 Buck. This application of Quaternions was alluded to, in page 620 of the Lec- 
 tures. 
 
CHAP. III.] STATICS OF A RIGID BODY. 707 
 
 Section 8. — On a few Specimens of Physical Application of 
 Quaternions, with some Concluding Remarks, 
 
 416. It remains to give, according to promise (368), before con- 
 cluding this work, some examples* oi physical applications of the 
 present Calculus: and as a first specimeti, we shall take the Statics 
 of a Rigid Body. 
 
 (1.) Let ai, . . an be n Vectors of Application, and let /3i, . . /3w be n correspond- 
 ing Vectors of Force, in the sense that n forces are applied at the points Ai, , . A» of 
 a. free but 7-igid system, and are represented as usual by so many right lines from 
 those points, to which lines the vectors OBi, . . ob„ are equal, though drawn from a 
 common origin ; and let y (= oc) be the vector of an arbitrary point c of space. 
 Then the Equation^ of Equilibrium of the system or body, under the action of these 
 n applied forces, may be thus written : 
 
 I. . . 2V(a--y)/3=0; or thus, I'. . . Vy2/3 = 2Va/3. 
 
 (2.) The supposed arbitrariness (I.) of y enables us to break up the formula T. 
 or I'., into the two vector equations : 
 
 II.. . 2/3 = 0; III. . . SVai8 = 0; 
 
 oi each of which it is easy to assign, as follows, the physical signification. 
 
 (3.) The equation II. expresses that if the forces, which are applied at the points 
 Ai . . of the body, were all transported to the origin o, their statical resultant, or 
 vector sum, would be zero. 
 
 (4.) The equation III. expresses that the resultant of all the couples, produced 
 in the usual way by such a transference of the applied forces to the assumed origin, 
 is null. 
 
 (5.) And the equation I., which as above includes both II. and III., expresses 
 that if all the given forces be transported to any common point c, the couples hence 
 arising will balance each other : which is a sufficient condition of equilibrium of the 
 system. 
 
 (6.) When we have only the relation, 
 
 IV. . . S(S/3.2Va/3) = 0, 
 
 without S/3 vanishing, the applied forces have then an Unique Resultant — 2/3, 
 acting along the line of which I. or I', is the equation, with y for its variable vec- 
 tor. 
 
 * The reader may compare the remarks on hydrostatic pressure, in pages 434, 
 435. 
 
 t We say here, ^^ equation :" because the single quaternion formula, I. or I'., 
 contains virtually the six usual scalar equations, or conditions, of the equilibrium at 
 present considered. 
 
708 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (7.) And the physical interpretation of this condition IV. is, that when the 
 forces are transported to o, as in (3.) and (4.) the resultant force is in the plane of 
 the resultant couple. 
 
 (8.) When the equation II., but not III., is satisfied, the applied forces compound 
 themselves into One Couple, of which the ^xis = SVa/3, whatever may be the posi- 
 tion of the origin. 
 
 (9.) When neither II. nor III. is satisfied, we may still propose so to place the 
 auxiliary/ point c, that when the fiven forces are transferred to it, as in (5.), the 
 resultant force 2j3 may have the direction of the axis 2V(a-y)/3 of the resultant 
 couple, or else the opposite of that direction ; so that, in each case, the condition,* 
 
 shall be satisfied by a suitable limitation of the auxiliary vector y. 
 
 (10.) This last equation V. represents therefore the Central Axis of the given 
 system of applied forces, with y for the variable vector of that right line : or the axis 
 of the screw-motion which those forces tend to produce, when they are not in balance, 
 as in (1.), and neither tend to produce translation alone, as in (6.), nor rotation 
 alone, as in (8.). 
 
 (11.) In general, if 5 be an auxiliary quaternion, such that 
 VI. . . 92/3 = SVa,3, 
 its vector part, Yq, is equal by (V.) to the Vector-Perpendicular, let fall from the 
 origin on the central axis ; while its scalar part, Sg, is easily proved to be the quo- 
 tient, of what may be called the Central Moment, divided by the Total Force : so 
 that Yq = () when the central axis passes through the origin, and 83 = when there 
 exists an unique resultant. 
 
 (12.) When the total force S;3 does not vanish, let Q be a new auxiliary qua- 
 ternion, such that 
 
 VIL..Q = ^ = ,+ ^ 
 ^ S/3 ^ 2)3' 
 
 with VIII. . . c = SQ = Sg, and IX. . . y = oc = VQ, 
 
 for its scalar and vector parts ; then c2j8 represents, both in quantity and in direction, 
 the Axis of the Central Couple (9.), and y is the vector of a point c which is on the 
 central axis (10.), considered as a right line having situation in space: while the 
 position of this point on this line depends only on the given system of applied forces, 
 and does not vary with the assumed origin o. 
 
 (13.) Under the same conditions, we have the transformations, 
 
 X. . . 2a/3 = (c + y) 2)8 ; XI. . . T2a/3 = (c2 - y2)JT2/? ; 
 
 XII. . . 2Va/3 = c2/3 + Vy2/3 ; XIII. . . (2Vai8)2 = c^ (2/3)2 + (Vy 2/3)2 ; 
 
 * The equation V. may also be obtained from the condition, 
 
 v. . . T2V(a — y)/3 = a minimum, 
 
 when y is treated as the only variable vector ; which answers to a known property 
 of the Central Moment. 
 
CHAP. III.] GENERAL EQUATION OF DYNAMICS. 709 
 
 ■whereof XII. contains the known law, according to which theaais of the couple (4.), 
 obtained by transferring all the forces to an assumed point o, varies generally in 
 quantity and in direction with the position of that point : while XIII. expresses the 
 known corollary from that law, in virtue of which the qtiantity alone, or the energy 
 (TSVa/S) of the couple here considered, is the same for all the points o of any one 
 right cylinder, which has the central axis of the system for its axis of revolution. 
 
 (14.) If Ave agree to call the quaternion product pa. aa' the quaternion moment, 
 or simply the Moment, of the applied force aa' at A, with respect to the Point p, the 
 quaternion sum 2a/3 in X. may then be said to be the Total Moment of the given 
 system of forces, with respect to the assumed origin o ; and the formula XI. ex- 
 presses that the tensor of this sum, or what may be called the quantity of this total 
 moment, is constant for all points o which are situated on any one spheric surface, 
 with the point c determined in (12.) for its centre : being also a minimum when o is 
 placed at that point c itself, and being then equal to what has been already called 
 the central moment, or the energy of the central couple. 
 
 (15.) For these and other reasons, it appears not improper to call generally the 
 point c, above determined, the Central Point, or simply the Centre, of the given 
 system of applied forces, when the total force does not vanish ; and accordingly in 
 the particular but important case, when all those forces are parallel, without their 
 sum being zero, so that we may write, 
 
 XIV. . . |3i = hl3, .. (3n-^ bn(3, TS^ > 0, 
 
 the scalar c in (12.) vanishes, and the vector y becomes (comp. Art. 97 on bary- 
 
 centres), 
 
 biai+ . . + bnUn 26a 
 
 XV. . . oc = y = = — — ; 
 
 ' bi+..-\-bn 26 
 
 so that the point c, thus determined, is independent of the common direction (3, and 
 coincides with what is usually called the Centre of Parallel Forces. 
 
 (16.) The conditions of equilibrium (1.), which have been already expressed by 
 the formula I., may also be included in this other quaternion equation, 
 
 XVI. . . Total Moment = 2a/3 = a scalar constant, 
 
 of which the value is independent of the origin ; and which, with its sign changed 
 represents what may perhaps be called the Total Tension of the system. 
 
 (17.) Any infiiiitely small change, in the position of a rigid body, is equivalent to 
 the alteration of each of its vectors a to another of the form, 
 
 XVII. . . a + Sa = a+E + Yia, 
 
 e and i being two arbitrary but infinitesimal vectors, which do not vary in the pas- 
 sage from one point a of the body to another : and thus the conditions of equilibrium 
 (1.) may be expressed by this other formula, 
 
 XVIII. . . 2S/3^a=0, 
 
 which contains, for the case here considered, the Principle of Virtual Velocities, and 
 admits of being extended easily to other cases of Statics. 
 
 417. The general Equation of Dynamics may be thus written, 
 I.. . 27wS(D/«-^)r^« = 0, 
 
710 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 with significations of the symbols which will soon be stated ; but as 
 we only propose (416) to give here some specimens of physical appli- 
 cation, we shall aim chiefly, in the following sub-articles, at the de- 
 duction of a few formulae and theorems, respecting Axes and Mo- 
 ments oi Inertia^ and subjects therewith connected. 
 
 (1.) In the formula L, a is the vector of position, at the time <, of an element 
 m of the system ; da is any variation of that vector, geometrically compatible with 
 the mutual connexions between the parts of that system; the vector m% represents 
 a moving force, or % an accelerating force, which acts on the element m of mass ; D 
 and S are marks, as usual, of derivating and taking the scalar ; and the summation 
 denoted by S extends to all the elements, and is generally equivalent to a triple in- 
 tegration, or to an addition of triple integrals in space. And the formula is ob- 
 tained (comp. 416, (17.)), by a combination of D'Alembert's principle with the prin- 
 ciple of virtual velocities, which is analogous to that employed in the Me'canique 
 Analytique by Lagrange. 
 
 (2.) For the case of a. free but rigid body, we may substitute for da the expres- 
 sion e-vYia, assigned by 416, XVII.; and then, on account of the arbitrariness 
 of the two infinitesimal vectors c and i, the formula I. breaks up into the two follow- 
 ing, 
 
 IL .. 2jn(D,2a-O = 0; HI. . . SmVa(D^2„-^) = 0; 
 
 which correspond to the two statical equations 416, II. and III., and contain re- 
 spectively the law of motion of the centre of gravity, and the law of description of 
 areas. 
 
 (3.) If the body have a, fixed point, which we may take for the origin o, we 
 eliminate the reaction at that point, by attending only to the equation III. ; and 
 may then express the connexions between the elements m by the formula, 
 
 IV. . . D<a = Yia, whence V. . . Ht^a = tVia - VaD^t ; 
 I being the Vector-Axis of instantaneous Rotation of the body, in the sense that its 
 versor Ut represents the direction of the axis, and that its tensor Ti represents the 
 angular velocity, of such rotation at the time t. 
 (4.) By v., the equation III. becomes, 
 
 VI. . . SmaVaDit = Sm (Vta Sta - YaK) ; 
 and other easy combinations give the laws of areas and living force, under the forms, 
 
 VII. . . 'EmaDta- 2mVJa|d<=y =a constant vector; 
 VIII. . . ^2»i(D<a)2 — 2mSJta?d« = c = a constant scalar. 
 
 (5.) When the applied forces vanish, or balance each other, or more generally 
 when they compound themselves into a single force acting at the fixed point, so that 
 in each case the condition 
 
 IX... 27»Va^=0 
 
 is satisfied, the equations (4.) are simplified ; and if we introduce a linear, Vector, 
 and self-conjugate function 0, such that 
 
 X. . , 0t= 2maVat = i2ma2 - 2maSat, 
 
CHAP. III.] MOMENTS AND AXES OF INERTIA. 711 
 
 and write h^ for - 2c, they take the forms, 
 
 XL ..>Dee + Vt0i = O; XII. . . 0i + y = 0; XIII. . . St^t= A2; 
 y and h being two real constants, of the vector and scalar kinds, connected with each 
 other and with t by the relation, 
 
 XIV. . . Sty + A« = ; also XV. . . <fDti = Vty. 
 It may be added that y is now the vector sum of the doubled areal velocities of all the 
 elements of the body, multiplied each by the mass m of that element, and each re- 
 presented by a right line oDta perpendicular to the plane of the area described 
 round the fixed point o in the time At ; and that 7*2 is the living force, or vis viva of 
 the body, namely the positive sum of all the products obtained by multiplying each 
 element m by the square of its linear velocity, regarded as a scalar (TD^a). 
 
 (6.") "When t is regarded as a variable vector, the equation XIII. represents an 
 ellipsoid, wWch \& fixed in the body, but moveable with it ; and the equation XIV. 
 represents a tangent plane to this eUipsoid, which plane is fixed in space, but changes 
 in general its position relatively to the body. And thus the motion of that body may 
 generally be conceived, as was shown by Poinsot, to be performed by the rolling 
 {without gliding') of an ellipsoid upon a plane ; the former carrying the body with it, 
 while its centre o remains j^icec? ; and the semidiameter (t) of contact being the vec- 
 tor-axis (3.) oi instantaneous rotation. 
 
 (7.) The ellipsoid XIII. maybe called, perhaps, the Ellipsoid of Living Force, 
 on account of the signification (5.) of the constant h^ in its equation ; and the fixed 
 plane XIV., on which it rolls, is parallel to what may be called the Plane of 
 Areas (Siy = 0) : no use whatever having hitherto been made, in this investigation, 
 of any axes or moments of inertia. But if we here admit the usual definition of such 
 a moment, we may say that the Moment of Inertia of the body, with respect to any 
 axis I through the fixed point, is equal to the living force h'^ divided by the square* 
 of the semidiameter Ti of the ellipsoid XIII. ; because this moment is, 
 
 XVI. . . S;n(TVaUO2=t-2Sm(Vta)8=-Si-i0t = 7*2X1-2. 
 
 (8.) The equations XII. and XIII. give, 
 
 XVII. . .• = y 2St0i - 7i2(^i)2 = Stv, if XVIII. . . v= y«0i - h-^fi ; 
 
 and this equation XVII. represents a cone of the second degree, fixed in the body 
 
 (comp. (6.)), but moveable with it, of which the axis i is always a side, and to which 
 
 the normal, at any point of that side, has the direction of the line v. But it follows 
 
 * Hence it may easily be inferred, with the help of the general construction of an 
 ellipsoid (217, (6.)), illustrated by Figure 53 in page 226, that for any solid body, 
 and any given point A thereof, there can always be found (indeed in more ways than 
 one) two other points, b and c, which are likewise jf?ice(7 in the body, and are such that 
 the square-root of the moment of inertia, round any axis ad, is geometrically con- 
 structed by the line BD, if the point D be determined on the axis, by the condition that 
 A and D shall be equally distant from c. This theorem, with some others here re- 
 produced, was given in the Abstract of a Paper read before the Koyal Irish Academy 
 on the 10th of January, 1848, and was published in the Proceedings of that date. 
 
712 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 from XI., or from XII. XV., and from the properties of the function 0, that D<i is 
 perpendicular to both 0i and (f^i, and therefore also by XVIII, to v ; the cone XVII. 
 is therefore touched, along the side i, by that other cone, which is the locus in space 
 of the instantaneous axis of rotation. We are then led, by this simple quaternion 
 analysis, to a second representation of the motion of the body, which also was pro- 
 posed by Poinsot : namely, as the rolling of one cone on another. 
 
 (9.) To treat briefly by quaternions some of Mac Cullagh's results on this sub- 
 ject, it may be noted that the line y, though ^a;ed in space, describes in the body a 
 cone of the second degree, of which the equation is, by what precedes, 
 
 XIX. . . g'^Sy^-^y + h^y^ = 0, if XX. . . ^ = Ty, or XXL . . y2 + ^2 = q ; 
 
 while, if we write y = oc, the point c is indeed fixed in space, but describes a 
 sphero-conic in the body, which is part of the common intersection of the cone 
 XIX., the sphere XXL, and the reciprocal ellipsoid (comp. XIIL), 
 
 XXII. . . Sy0-»y = /i2. 
 
 (10.) Also, the normal to the new cone (9.), at any point of the side y, has the 
 direction oi ff^<p-^y + h^y, or of t + A^y-i (comp. XIV.) ; and if a line in this direc- 
 tion be drawn through the fixed point o, it will be the side of contact oi the plane 
 of areas (7.), with the cone of normals at o to the cone XIX. ; which last (or reci- 
 procal) cone rolls on that plane of areas. 
 
 (11.) As regards the Axes of Inertia, it may be sufficient here to observe that 
 if the body revolve round a permanent axis, and with a constant velocity, the vec- 
 tor axis I is constant ; and must therefore satisfy the equation, 
 
 XXIIL . . Vi^i = 0, because XXIV. . . D^t = ; 
 
 it has therefore in general (comp. 415) one or other of Three Real and Rectangular 
 Directions, determined by the condition XXIIL: namely, those of the Axes of 
 Figure oi either of the two Reciprocal Ellipsoids, XIIL XXII. 
 
 (12.) And the Three Principal Moments, say A, B, C, corresponding to those 
 three principal axes, are by XVI. the three scalar values of— r^^i ; so that the 
 symbolical cubic (350) in ^ may be thus written, 
 
 XXV. . . (0 + yl) (0 + 5) (0 + C) = 0. 
 
 (13.) Forming then this symbolical cubic by the general method of the Section 
 III. ii. 6, we find that the three moments A, B, C, are the three roots (always real, 
 by this analysis) of the algebraic and cubic equation, 
 
 XXVI. .. A^- 2«2^2 + („4 + n'2) A - (n2«'2 _ n"2) = q ; • 
 
 in which, n^, »'2, n''2 are three positive scalars, namely, 
 
 XXVIL . . n2 = - 2jna2; w'2 = - Smm'(Vaa')2 ; »"2=2m»i'm"(Saa'a")2; 
 
 and the combination n^n'^ — »"2 is another positive scalar, of which the value may 
 be thus expressed, 
 
 XXVIII. . . ABC=nhz2 - n"2 = ^m^m'tt"^ (Yaa'y 
 
 + 2SjHm'm" (Taa'Ta'a'Ta'a + Saa'Sa'a"Sa"a), 
 
 if a, a', a", &c, be the vectors of the mass-elements m, m', m", &c. 
 
CHAP. III.] SYSTEM OF ATTRACTING BODIES. 713 
 
 (14.) And because the equation XXV. gives this other symbolical result, 
 XXIX. . . -ABC(p-^=f^ + {A + B+C)(p+ BC+ CA + AB, 
 it follows that XXX. . . ^-IQ = ; 
 
 and therefore, by XV., &c., that if a body, with a fixed point, &c., begin to rcA^olve 
 round one of its three principal axes of inertia, it will continue to revolve round that 
 axis, with an unchanged velocity of rotation. 
 
 (15.) It has hitherto been supposed, that all the moments of inertia are referred 
 to axes passing through one point o of the body ; but it is easy to remove this re- 
 striction. For example, if we denote the moment XVI. by /o, and if I^ be the cor- 
 responding moment for an axis parallel to i, but drawn through a new point O, of 
 which the vector is o>, then 
 
 XXXI. . . 7^ = r2Sjn(V*(a-a;))2 
 
 = 7o + SSm. S (wt-iVtK) + /)2s,„^ 
 if XXXII. . . /cSm = 2ma, and XXXIII. . . jo = TVwUt, 
 
 so that K is the vector of the centre of inertia (or of gravity) of the body, and p is 
 the distance between the two parallel axes. 
 (16.) If then we suppose that the condition 
 
 XXXIV. . . VtK = 
 
 is satisfied, that is, if the axis i pass through the centre of inertia, we shall have the 
 very simple relation, 
 
 XXXV. ..7^ = /o-|-/)22m; 
 
 which agrees with known results. 
 
 418. As a third specimen of physical applications of quaternions, 
 we propose to consider briefly the motions of a System of Bodies^ 
 m, m\ m', . . . regarded as free material points, of which the variable 
 vectors are a, a', a"^ . . . and which are supposed to attract each other 
 according to the law of the inverse square: the fundamental for- 
 mula employed being the following, 
 
 iTim/ 
 
 I. . . 2mSD/«aa + ^P = 0, if II. ..P = 2:— -: -: 
 
 J (a - a ) 
 
 P thus denoting the Potential {pi force-function) of the system, and 
 the variations ca, ta'^ . . . being infinitesimal, but otherwise arbi- 
 trary. 
 
 (1.) To deduce the formula I., with the signification 11. of P, from the general 
 equation 417, I. of dynamics, we have first, for the case of two bodies, the following 
 expressions for the accelerating forces, 
 
 111...^=—^, r=r-7^, if r = T(a-a'); 
 
 (a -a')r (a - a)r ^ 
 
 4 Y 
 
714 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 whence follows the transformation,* 
 
 TTT r,y vr^ .^.v. »x —mm „S(a — a') ^mm' 
 
 IV. . .-S(m^5a + m'K'Sa)= S -^ -=S ; 
 
 r a- a r 
 
 a result easily extended, as above. If the law of attraction were supposed different, 
 there would be no difficulty in modifying the expression for the potential accordingly. 
 (2.) In general, when a scalar, f (^as here P), is a, function of one or more vec^ 
 tors, a, a', . . . its variation (or differential) can be expressed as a linear and scalar 
 function of their variations (or differentials), of the form S(3da + S/3'^a' + . . (or 
 2Sj3da) ; in which [3, /3' . . . are certain new and fnite vectors, and are them- 
 selves generally /?inc^tows of o, a', . . ., derived from the given scalar function/. And 
 we shall find it convenient to extend the Notation^ of Derivatives, so as to denote 
 these derived vectors (3, (3', &c., by the symbols, Dn/, ^a'f &c. In this manner we 
 shall be able to write, 
 
 V. . . ^P=2S(DaP.oa); 
 and the differential equations of motion of the bodies m, m', m", . . will take by 
 I. the forms : 
 
 VI. . . mDt^a + D„P= 0, m"Dfia' + Da' P = 0, &c. ; 
 or more fully, 
 
 VII. . . Wa = . -^^- + ^ j-%- -^ + . . ; &c. 
 
 (a-a)T(a-a) (o-a)T(a-o/ 
 
 (3.) The laws of the centre of gravity, of areas, and of living force, result imme- 
 diately from these equations, under the forms, 
 
 VIII. . . SmD^a = /? ; IX. . . SmVaD^a = y ; 
 and X. . . r=-J-2m(D<a)2=P+JEf; 
 
 in which (5, y are constant vectors, ^ is a constant scalar, and 2 P is the living 
 force of the system (comp. 417, (5.)). 
 
 (4.) One mode (comp. 417, (2.)) of deducing the three equations, of which these 
 are the first integrals, is the following. To obtain VIII., change every variation 
 ^a in I. to one common but arbitrary infinitesimal vector, £. For IX., change c>a 
 to Vta, ^a' to Via', &c. ; i being ano^Aer arbitrary and infinitesimal vector. Finally, 
 to arrive at X., change variations to differentials (^a to da, &c.), and integrate 
 once, as for the two former equations, with respect to the time t. 
 
 (5.) The formula I. admits of being integrated by parts, without any restric- 
 tion on the variations 8a, by means of the general transformation, 
 
 XI. . . S(Pt^a.8a) = DtS(Pta.6a)-:^8.(J)iay, 
 
 combined with the introduction of the following definite integral (comp. X.), 
 
 XII. . . F= fVP+ T)dt. 
 
 * It may not be useless here to compare the expression in page 417, for the dif- 
 ferential of a proximity. 
 
 t In this extended notation, such a formula as d/p = 2Svdp would give, 
 v^-iD.fp. 
 
CHAP. III.] INTERMEDIATE AND FINAL INTEGRALS. 7l5 
 
 (6.) In fact, if we denote by ao, a'o, . . the initial values of the vectors a, a', . . 
 or their values when ^ = 0, and by Doa, Doa', . . the corresponding values of D^a, 
 Via', . , , we shall thus have, as a first integral of the equation I., the formula, 
 
 XIII. . . 2mS (Pta . 6a - Bqu . Sao) + dF=0; 
 in'which no variation dt is assigned to #, and which conducts to important conse- 
 quences. 
 
 (7.) To draw from it some of these, we may observe that if the masses m, m', . . 
 be treated as constant and known, the complete* integrals of the equations VI. or 
 VII. must be conceived to give what may be called the final vectors of position a, 
 a', . . and of velocity D<a, D^a', . . .in terms of the initial vectors ao, a'o, . . Doa, 
 Doa', . . and of the time, t : whence, conversely, we may conceive the initial vectors 
 of velocity to be expressible as functions of the initial and final vectors of position, and 
 of the time. In this way, then, we are led to consider P, T, and F as being scalar 
 functions (whether we are or are not prepared to express them as such), of a, a', . . 
 ao, a'o, . . and t; and thus, by (2.), the recent formula XIII. breaks up into the two 
 following systems of equations : 
 
 XIV. . . mDta + DaF= 0, m'Dta + Da'F= 0, &c. ; 
 and XV. . . - wDoa + Da^F =0, - m'Doa' + Da'^F = 0, &c. ; 
 
 whereof the former may be said to be intermediate integrals, and the latter to be 
 Jinal integrals, of the differential equations of motion of the system, which are in- 
 cluded in the formula I. 
 
 (8.) In fact, the equations XIV. do not involve the final vectors of acceleration 
 Dt^a, , . as the difi'erential equations VI. or VII. ha^ done; and the equations XV. 
 express, at least theoretically, the dependence of the ^na? vectors opposition a, . . on 
 the time, t, and on the initial vectors of position ao, . . and of velocity Doa, . . as by 
 (7.) the complete integrals ought to do. And on account of these and other impor- 
 tant properties, the function here denoted by F may be called the Principal* Func- 
 tion of Motion of the System. 
 
 (9.) If the initial vectors ao, . . and Doa, . . be given, that is, if we consider the 
 actual progress m space of the mutually attracting system of masses m, . . from one 
 set of positions to another, then the function F depends upon the time alone ; and 
 by its definition XII., its rate or ue^ociY^ of increase, or its total derivative with re- 
 spect to tf is thus expressed, 
 
 XVI. . . DiF=P+ T. 
 
 (10.) But we may inquire what is the partial derivative, say (jytF), of the 
 same definite integral^, when regarded (7.) as a function of the final and initial vectors 
 of position a, . . ao, . . which involves also the time explicitly, and is now to be deri- 
 vated with respect only to that variable t, as ift\\& final vectors a, . . were constant : 
 whereas in fact those vectors alter with the time, in the course of any actual mo- 
 tions of the system. 
 
 * This function was in fact so called, in two Essays by the present writer, " On 
 a General Method in Dynamics," published in the Philosophical Transactions (Lon- 
 don), for the years 1834 and 1835 ; although of course coordinates, and not qua- 
 ternions, were then employed, the latter not having been discovered until 1843 : 
 and the notation S, since adopted for scalar, was then used instead of/'. 
 
716 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (11.) For this purpose, it is sufficient to observe that the part of the total deri- 
 vative BtF, which arises, from the last mentioned changes of a, . . is (by XIV. 
 
 and X.), 
 
 XYIL..-SS(PaF.Dta) = 2T; 
 
 and therefore (by XVI. and X.), that the remaining part must be, 
 
 XVIII. . . (PtF) = P-T=-H. 
 
 (12.) The complete variation of the function Fis therefore (comp. XIII.), when 
 
 t as well as a, . . and ao, • . is treated as varying, 
 
 XIX. . . ^i?'=-A'^^-S»2SD^a^a + S»iSDoaW 
 
 (13.) And hence, with the help of the equations X. XIV. XV., it is easy to infer 
 
 that .the principal function JP must satisfy the two following Partial Differential 
 
 Equations in Quaternions : 
 
 XX. . . (D,i5')-iSm-i(DaF)2 = P; 
 
 XXI. . . (D,F)-^S»i-i(D„„F)2 = Po; 
 
 in which Pq denotes the initial value of the potential P. 
 (14.) If we write 
 
 XXII. . . r=f'2rd/, 
 
 so that F represents what is called the Action, or the accumulated living force, of 
 the system during the time f, then by X. and XII. the two definite integrals F and 
 V are connected by the very simple relation, 
 
 X-XIIL . . V=F + tH', 
 whence by XIX. the complete variation of F, considered as a function of the final 
 and initial vectors of position, and of the constant H of living force, which does not 
 explicitly involve the time, may be thus expressed, 
 
 XXIV. . . 5V = tdH- •2mSDtada+ -EmSBoaSao. 
 (15.) The partial derivatives of this new function V, which is for some purposes 
 more useful than F, and may be called, by way of distinction from it, the Charac- 
 teristic* Function of the motion of the system, are therefore, 
 
 XXV. . . Da F= - mDta, &c. ; 
 
 XXVI. . . D„oF= + 7nDoa, &c. ; 
 
 and XXVII. . .'D^V=t. 
 
 (16.) The intermediate integrals (7.) of the differential equations of motion, 
 which were before expressed by the formulae XIV., may now, somewhat less simply, 
 be regarded as the result of the elimination of £f between the formulae XXV. XXVII. ; 
 and the^waZ ijitegrals of those equations VI. or VII., which were expressed by XV., 
 are now to be obtained by eliminating the same constant -ff between the recent equa- 
 tions XXVI. XXVII. 
 
 * The Action, F, was in fact so called, in the two Essays mentioned in (he pre- 
 ceding Note. The properties of this Characteristic Function had been perceived by 
 the writer, before those of that which he came afterwards to call the Principal 
 Function, as above. 
 
C«AP. III.] PRINCIPAL AND CHARACTERISTIC FUNCTIONS. 717 
 
 (17.) The Characteristic Function, V, is obliged (comp. (13.)) to satisfy the two 
 following partial differential equations, 
 
 XXVIII. . . iSm-i (Da vy + P + H=0; 
 
 XXIX. . . i2m-i(D„^F)2+Po+^=0; 
 
 it vanishes, like F, when t = 0, at which epoch a = OQ, a' = a'o, &c. ; each of these 
 two functions, Fand F, depends symmetrically on the initial and final vectors of po- 
 sition : and each does so, only by depending on the mutual configuration of all those 
 initial and final positions. 
 
 (18.) It follows (comp. (4.), see also 416, (17.), and 417, (2.)), that the func- 
 tion F must satisfy the two conditions, 
 
 XXX. . . S(DaF+D„,F) = 0; XXXI. . . ^Y (aDaF + uoD^.F) = ; 
 
 which accordingly are forms, by XIV. XV., of the equations VIII. and IX., and 
 therefore are expressions for the law of motion of the centre of gravity, and the law of 
 description of areas. And, in like manner, the function V is obliged to satisfy these 
 two analogous conditions, 
 
 XXXII. . . S(DaF + D„^F) = 0; XXXIII. . . SV(aDaF+ aoDaoO = 0; 
 
 which accordingly, by XXV. XXVL, are new forms of the same equations VIII. IX., 
 and consequently are new expressions of the same two laws. 
 
 (19.) All the foregoing conditions are satisfied when t is small, that is, when the 
 time of motion of the system is short, by the following approximate expressions for the 
 functions Fand F, with the respectively derived and mutually connected expressions 
 for H and t : 
 
 XXXIV. ..F=^(P+Po) + |-^; 
 
 XXXV. . . F=r s(P + Po + 2Iiy ; 
 
 XXXVI. . . H = -(D,F) =-l(P+i'o) + ^,; 
 XXXVII. . .t = J)MV=s(^P+Fo+2H)i; 
 n which s denotes a real and positive scalar, such that 
 
 XXXVIII. . . s2 = _ 2wi (a - ao)2, or XXXIX. . . s = VSmT (a - ao)2. 
 
 419. As Q. fourth specimen, we shall take the case of a free point 
 or particle, attracted to a fixed centre* o, from which its variable 
 vector is a, with an accelerating force = J/r^, if r = Ta = the distance 
 
 * When two free masses, m and m', with variable vectors a and a', attract each 
 other according to the law of the inverse square, the differential equation of the re/a- 
 tive motion of m about m' is, by 418, VII., 
 
 r. . . Dn« - a') = (w« + W) (a - a')-ir ', if r = T(a - a) ; 
 
 and this equation I', reduces itself to I., Avhen we write a for a -n, and J/ for 
 
 m -\-iii'. 
 
718 ELEMENTS OF QUATERNIONS. [bOOK 111. 
 
 of the point from the centre, while J/ is the attracting mass: the 
 differential equation of the motion being, 
 
 I. . . Wa:=Ma-'r-\ 
 
 if D (abridged from D^) be the sign of derivation/ with respect to 
 the time i. 
 
 (1.) Operating on I. with V.a, and integrating, we obtain immediately the 
 equation (comp. 338, (5.)), 
 
 IT. . . VaDa =/3= const. ; 
 which expresses at once that tlie orbit is plane, and also that the area described in 
 it is proportional to the time ; U/3 being the fixed unit-normal to the pZarae, romid 
 which the point, in its angular motion, revolves positively ; and T/3 representing in 
 quantity the double areal velocity^ which is often denoted by c. 
 
 (2.) And it is important to remark, that these conclusions (1.) would have been 
 obtained by the same analysis, if r-i in I. had been replaced by any other scalar 
 function, f(r), of the distance ; that is, for any other law of central force, instead of 
 the law of the inverse square. 
 
 (3.) In general, we have the transformation, 
 
 III. .. a-'Ta-i=dUa:Vada, 
 because, by 334, XV., &c., we have, 
 
 IV. . . dUa = V(da.a-i).Ua = a-2Ua.Vada = a-iTa-i.Vada; 
 the equation I. may therefore by II. be transformed as follows, 
 
 V. . . D2a = -yDUa, if VI. . . y = - y¥/3-i ; 
 and thus it gives, by an immediate integration, 
 
 VII. . . Da = y(Ua-£), or VII'. . . Da = (c - Ua) y, 
 
 f being a new constant vector, but one situated in the plane of the orbit, to which 
 plane /3 and y are perpendicular. 
 
 (4.) But a, Da, D^a are here (comp. 100, (5.) (6.) (7.)) the vectors of joostVzon, 
 velocity, and acceleration of the moving point; and it has been defined (100, (5.)) 
 tliat if, for any motion of a point, the vectors of velocity be set off from any common 
 origin, the curve on which they terminate is the Hodograph^' of that motion. 
 
 (5.) Hence a and Da, if the latter like the former be drawn from the fixed point 
 o, are the yectors.of corresponding points of orbit and hodograph ; and because the 
 formula VII. gives, 
 
 VIII. . . SyDa = 0, and IX. . . (Da + yt)^ = y^, 
 it follows that the hodograph is, in the present question, a Circle, in the plane of tlie 
 
 * Compare Fig. 32, p. 98; see also pages 100, 515, 578, from the two latter 
 of which it may be perceived, that the conception of the hodograph admits of some 
 purely geometrical applications. 
 
CHAP. III.] LAW OF THE CIRCULAR HODOGUAPH. 719 
 
 orbit, with — ye (or + cy) for the vector of its centre^ and with Ty = iWT/3-' for its 
 radius, which radius we shall also denote by h. 
 
 (6.) The Law of the Circular* Hodograph is therefore a- mathematical conse- 
 quence of the Law of the Inverse Square ; and conversely it will soon be proved, that 
 no other law of central force would allow generally the hodograph to be a circle. 
 
 (7.) For the law of nature, the Eadius (h) of the Hodograph is equal, by (L) and 
 (5.), to the quotient of the attracting mass (M), divided by the double ureal velocity 
 (T/3 or c) in the orbit ; and if we write 
 
 X... e=Tf, 
 
 this positive scalar e may be called the Excentricity of the hodograph, regarded as a 
 circle excentrically situated, with respect to ihQ fixed centre of force, o. 
 
 (8.) Thus, if e < 1, the fixed point o is interior to the hodographic circle ; if e = 1, 
 the point o is on the circumference ; and if e> 1, the centre o of force is then exte- 
 rior to the hodograph, being however, in all these cases, situated in its plane. 
 
 (9.) The equation VII. gives, 
 
 XI. . . £-Ua=-y-iDa = Da.y-J; 
 
 operating then on this with S. a, and writing for abridgment, 
 
 XII. . .;7=/3y-i = iW-»T/32 = c2Jl[f-i, and XIII. .. SUa£ = cos r, 
 
 so that /) is a constant and positive scalar, while v is the inclination of a to — e, we 
 find, 
 
 XIY.. .r + Sae = p; or XV.. 
 
 1 + g cos w 
 
 the orbit is therefore a plane conic, with the centre of force o for a. focus, having e 
 for its excentricity, and p for its semiparameter. 
 
 (10.) And we see, by XII., that if this semiparameter p be multiplied by the 
 attracting mass M, the product is the square of the double areal velocity c ; so 
 that this constant c may be denoted by {Mp)\ which agrees with known results. 
 
 (11.) If, on the other hand, we divide the mass (iW) by the semiparameter (/j), 
 the quotient is by XII. the square of the radius (J/T/3"^ or h) of the hodograph. 
 
 (12.) And if we multiply the same semiparameter p by this radius 3/T/3"' of 
 the hodograph, the product is then, by the same formula XII., the constant T/3 or 
 c of double areal velocity in the orbit, so that h = Jic"i = cp'K 
 
 (13.) If we had operated with V. a on VII'., we should have found, 
 
 XVI. . .i3 = V.a(f-Ua)y = (Sa£ + r)y; 
 which would have conducted to the same equations XIV. XV. as before. 
 
 * This law of the circular hodograph was deduced geometrically, in a paper read 
 before the Royal Irish Academy, by the present author, on the 14 th of December, 
 1846 ; but it was virtually contained in a quaternion formula, equivalent to the re- 
 cent equation VII., which had formed part of an earlier communication, in July, 1845. 
 (See the Proceedings for those dates ; and especially pages 845, 347, and xxxix., 
 xlix., of Vol. III.) 
 
720 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (14.) If we operate on VII. with S.a, we find this other equation, 
 XVII. . . - rDr = SaDa = yYae ; 
 
 M 
 but XVIII. . . - y2 = /i2 = _ (by VI. and XII., comp. (1 1.)), 
 
 and XIX. . . -(Vae)2=:eV2-(;j-r)2=;j(2r-p-r5a-i), 
 
 if we write XX. . . a = : 
 
 1 - e^' 
 
 hence squaring XVII., and dividing by ?-3, we obtain the equation, 
 
 xxi...,|r=«'^ 
 
 r a r2 J* 
 
 (15.) It is obvious that this last equation, XXI., connects the distance, r, with 
 tlie time, t, as the formula XV. connects the same distance r with the true anomaly, 
 V ; that is, with the angular elongation in the orbit, from the position of least dis- 
 tance. But it would be improper here to delay on any of the elementary conse- 
 quences of these two known equations : although it seemed useful to show, as above, 
 how the equations themselves might easily be deduced by quaternions, and be con- 
 nected with the theory of the hodograph. 
 
 (16.) The equation II. may be interpreted as expressing, that the parallelogram 
 (comp. Fig. 32) under the vectors a and Da of position and velocity, or under any 
 two corresponding vectors (5.) of the orbit and hodograph, has a constant plane and 
 area, represented by the constant vector (3, which is perpendicular (1.) to that plane. 
 But it is to be observed that, by (2.), these constancies, and this representation, are 
 not peculiar to the law of the inverse square, but exist for all other laws of central 
 force. 
 
 (17.) In general, if any scalar function R (instead of M;-"2) represent the acce- 
 lerating force of attraction, at the distance r from the fixed centre o, the differential 
 equation of motion will be (instead of I.), 
 
 XXII. . . D2a = iJm-i = - i?Ua ; 
 
 and if we still write VaDa = /3, as in II., the formula IV. will give, 
 
 D3« 
 XXIII. . . D3a = - DR. Ua - Rr-^^\Ja, and XXIV. . . V — - = r'2/3 ; 
 
 D'-a 
 in which /3 = cU/3, if c = T/3, as before. 
 
 (18.) Applying then the general formula 414, I., we have, for any law* of force, 
 
 the expressions, 
 
 1 D3a e 
 XXV. . . Vector of Curvature of Hodograph = — — V — — = Ua/3 ; 
 
 D^a D2a Rr^ 
 
 XXVI. . . Radius (h) of Curvature of Hodograph = Rr^c'^ 
 
 Force x Square of Distance 
 
 ~ Double Areal Velocity in Orbit ^ 
 
 * The general value XXVI., of the radius of curvature of the hodograph, was 
 geometrically deduced in the Paper of 1846, referred to in a recent Note. 
 
CHAP. III.] PRODUCT OF OPP. VELOCITIES, POTENTIAL. 721 
 
 of which the hist not only conducts, in a new way, for the law of nature, to the con- 
 stant value (7.), h = A/c~', but also proves, as stated in (6.)? thfit for any other law 
 of central force the hodograph cannot he a circle, unless indeed the orbit happens to 
 be such, and to have moreover the centre of force at its centre. 
 
 (19.) Confining ourselves however at present to the law of the inverse square, 
 and writing for abridgment (comp. (5.)), 
 
 XXVII. . . K = OH = fy = Vector of Centre h of Hodograph, 
 
 which gives, by (5.) and (7.), 
 
 XXVIII. . . Tk = eh, 
 
 the origin o of vectors being still the centre o^ force, we see by the properties of the 
 circle, that the product of any two opposite velocities in the orbit is constant ; and 
 that this constant product* may be expressed as follows, 
 
 XXIX. . . (e-l)AUK.(e+l)/tUK = AHl-e2)=Ma-i, 
 
 by XVIII. and XX. 
 
 (20.) The expression XXIX. may be otherwise written as k^ - y2; and if v be 
 the vector of any point u external to the circle, but in its plane, and u the length 
 of a tangent ut from that point, we have the analogous formula, 
 
 XXX. . . m2= y2 _ (y _ k)2 = T (W - K^ - h^. 
 
 (21.) Let T and r' be the vectors ox, ot' of the two points of contact of tan- 
 gents thus draw n to the hodograph, from an external point u in its plane ; then 
 each must satisfy the system of the three following scalar equations, 
 XXXI. . . Syr = ; XXXII. . . (r - *c)2 = y2 ; XXXIII. . . S (r - fc) (v - k) = y^ ; 
 whereof the first alone represents the plane ; the two first jointly represent (comp. 
 (5,)) the circle ; and the third expresses the condition of conjugation of the points 
 T and u, and may be regarded as the scalar equation of the polar of the latter point. 
 It is understood that Syy = 0, as well as Sy/c = 0, &c., because y is perpendicular 
 (3.) to the plane. 
 
 (22.) Solving this system of equations (21.), we find the two expressions, 
 
 XXXIV.. . r = K+y(y + «)(v-K)-»; XXXIV. . . r' = K + y (y-«) (v- fc)-i ; 
 in which the scalar u has the same value as in (20.). As a verification, these ex- 
 pressions give, by what precedes. 
 
 * In strictness, it is only for a closed orbit, that is, for the case (8.) of the centre 
 of force being interior to the hodograph (e < 1), that two velocities can be opposite ; 
 their vectors having then, by the fundamental rules of quaternions, a scalar and posi- 
 tive product, which is here found to be= Ma~^, by XXIX., in consistency with the 
 known theory of elliptic motion. The result however admits of an interpretation, in 
 other cases also. It is obvious that when the centre o of force is exterior to the hodo- 
 graph, the polar of that point divides the circle into two parts, whereof one is con- 
 cuve, and the other convex, towards o ; and there is no difficulty in seeing, that the 
 former part corresponds to the branch of an hyperbolic orbit, which can be described 
 under the influence of an attracting force : while the latter part answers to that 
 other branch of the same complete hyperbola, whereof the description would require 
 the force to be repulsive. 
 
 4 z 
 
722 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 XXXV. ..S(r-K)(r-v) = 0; XXXV'.. . S(r'-fc) (r'- i;)= 0; 
 and XXXVI. . . (r - 1>)2 = (r' -vf = - u\ 
 
 In fact it is found that 
 
 XXXVII. . . 7- - V = u (« + y) (u - k)-i ; XXXVIII. . . T(« + y) = T (v - k) ; 
 and XXXIX. . . (r - v) (r - k) = «y ; 
 
 « + y being here a quaternion. 
 
 (23.) If v be the vector ou' of any point u', on the polar of the point u with 
 respect to the circle, then changing r to v', and u to 2, in XXXIV., we find this vector 
 form (comp. (21.)) of tlie equation of that polar, 
 
 XL. . . v' = K + y (y + 2) (y- k:)-', 
 or, by an easy transformation, 
 
 XLI. . . (/t2 4- m2) v' = h^y 4. „2k + zy (k - u), 
 
 in which 2 is an arbitrary scalar. 
 
 (24.) If then we suppose that u' is the intersection of the chord tt' with the 
 right line ou, the condition 
 
 XLII. . . Yv'v = will give XLI II. . . zy = f^^" ; 
 
 v^ — Skv 
 
 but XLIV. . . Vku . (k - v) = kS (kv - v^) + vS (kv - k^) ; 
 
 the coefficient then of k, in the expanded expression for v', disappears as it ought to 
 
 do : and we find, after a few reductions, 
 
 XLV. . . v = V 1 + —- = , 
 
 \ i;2 - Skv I V — v-^Skv 
 
 a result which might have been otherwise obtained, by eliminating a new scalar y 
 
 between the two equations, 
 
 XLVI. . . v'=yi;, S (yu - k) (?; - k) = y2. 
 
 (25.) Introducing then two auxiliary vectors, X, //, such that 
 
 XLVII. . . \ = v-'^Skv, or Skv^v\ = \v, 
 
 and therefore XLVII'. . . \ - k = u" Wkv, SkX = \2, (\ _ «)« = fc2 - \2, 
 
 y2-K2U 
 
 and XLVIII. . . /^ = \ I 1 +1 1 + '— — ^ , whence fi 
 
 we have the very simple relation, 
 
 XLIX. . . (v - X) (v' - X) = (/i - \)2, or L. . . Lu . lu' = lm2, 
 if X = OL, and )ti = om. Accordingly, the point l is the foot of the perpendicular let 
 fall from the centre 11 on the right line ou, while m is one of the two points m, m' of 
 intersection of that line with the circle ; so that the equation L. expresses, that the 
 points u, u' are harmonically conjugate, with respect to the chord mm', of which l is 
 the middle point, as is otherwise evident from geometry. 
 
 (26.) The vector a of the orlit (or of position), which corresponds to the vector 
 r (= Da) of the hodograph (or of velocity'), and of which the length is Ta = r = the 
 distance, may be deduced from r by the equations, 
 
 LI. . . a = r(K-r)y-i, and LIL . . Vra = -/3= Myi; 
 whence follow the expressions, 
 
 LIII. . . Potential = Mr"! = (say) P= Sr (k - r) = Sv (k - r) ; 
 
CHAP. III.] CONSTRUCTIONS FOR THE POTENTIAL. 723 
 
 the second expression for P being deduced from the first, by means of the relation 
 XXXV. 
 
 (27.) The first expression LIII. for P shows that the potential is equal, 1st, to 
 the rectangle under the radius of the hodograph, and the perpendicular from the 
 centre o of force, on the tangent at t to that circle ; and Ilnd, to the square of the 
 tangent from the same point T of the hodograph, to what may be called the Circle of 
 Uxcentricitg, namely to that new circle which has oh for a diameter. And the first 
 of these values of the potential may be otherwise deduced from the equality (7.) of the 
 mass M, to the product he of the radius h of the hodograph, multiplied by the constant 
 c of double areal velocity, or by the constant parallelogram (16.) under any two cor- 
 responding vectors. 
 
 (28.) The second expression LIII. for the potential P, corresponding to the 
 point T of the hodograph, may (by XXXIV., &c.) be thus transformed, with the 
 help of a few reductions of the same kind as those recently employed : 
 
 LIV. . . P= — = — ^— -| — , if hY. . . q = v(K-v), 
 r h^ + u' 
 
 q being thus an auxiliary quaternion ; and in like manner, for the other point t' 
 
 lately considered, we have the analogous value, 
 
 whence 
 
 and therefore, 
 
 and finally, 
 
 Lvi p>_^_ ^^Sg-"y^ g. 
 
 r' h'i+u'^ ' 
 
 LVn...P.P'=*-^i?|^^; 
 
 h^ + u^ 
 
 Lviii... ^=p-i5?i+^"!I?, 
 
 T TY ^' - P'-i _ Sg-«y-^Vg r _ 
 
 2M 2PP' „ ^«2y3 
 
 (29.) In fact, the same second expression LIII. shows, that if v and v' be the 
 feet of perpendiculars from x and x' on hl, then the potentials are, 
 LXI. . . P= ou . XV, and P' = ou . x'v' ; 
 and it is easy to prove, geometrically, that the segment u'l is the harmonic mean be- 
 tween what may be called the ordinates, xv, x'v', to the hodographic axis hl. 
 
 (30.) If we suppose the point u to take any new but near position u, in the plane, 
 the polar chord xx', and (in general) the length u of the tangent ux, will change ; and 
 we shall have the differential relations : 
 
 LXII. . . dr = (r - v)-iS (r - k) dv ; 
 LXir. . . dr' = (r' - v)->S (r' - k) dw ; 
 and » LXIIL . . d« = ?riS (fc - v) dv. 
 
 (3L) Conceiving next that u moves along the line ou, or lu, so that we may 
 write, 
 
 LU LM , , LO 
 
 LXIV.... = Gr-e')(^-X), if ^=- = ^, and e =-~, 
 we shall have, 
 
724 
 
 ELEMENTS OF QUATERNIONS. 
 
 [book 111 
 
 LXV. . . dv = {fx -\)dx = v {x- e')''da;, with ic > 1 > e', 
 if u be on lm prolonged, and if o be on the concave side of the arc tmt' ; and thus, 
 by LIII., the diflPerential expressions (30.) become, 
 
 LXVI. . . dr = (i;-r)->P(^-e')-ida:; dr' = (v- r')-iP'(^-e')->dr ; 
 and LXVII. . . d« = u-^Sg.{z - e'y^dx, with S7 = v(\ - v) ; 
 
 Pdx „. . P'dx 
 
 so that LXVIII. . . Tdr = 
 
 Tdr' = 
 
 if d;^; > 0. 
 
 « (iC — e'y ~ ' u(x — e) 
 Such then are the lengths of the two elementary arcs TT, and t't/ of the hodograph, 
 intercepted between two near secants ntt' and nt^t/ drawn from the pole n of the 
 chord mm', and having u and u, for their own poles ; and we see t^at these arcs are 
 proportional to the potentials^ F and P', or by LXI. to the ordinates, tv, t'v', or 
 finally to the lines nt, nt' : and accordingly we have the inverse similarity (comp. 
 118), of the two small triangles with n for vertex, 
 
 LXIX. . . A NTT, oc'ntV, 
 as appears on inspection of the annexed Figure 86. 
 
 N 
 
 
 4 
 
 A 
 
 \ 
 
 \ 
 
 
 ---.^ 
 
 f' 
 
 ^ 
 
 V \ 
 
 \^ 
 
 w^ 
 
 
 ii^ 
 
 
 "~\\\ 
 
 ■¥^ 
 
 It 
 
 ^ 
 
 \ "^ 
 
 \ / \ / 1 
 
 \ 1 Y/ 
 
 \\i //\ 
 
 \\\ ^ 
 
 V' 
 
 ^0 ^ 
 
 \IJ^^. 
 
 Ki 
 
 / 
 
 TV^,--'-' 
 
 \ 
 
 "^ 
 
 — 
 
 -^ 
 
 Fig. 8G. 
 
 (32.) For any motion of a point, however complex, the element dt of time which 
 corresponds to a given element dDa of the hodograph, is found by dividing the latter 
 element by the vector D-a of accelerating force ; if then we denote by dt and dt' tlie 
 times corresponding to the elements dr and dr' (31.), we have the expressions, 
 
 Mdx rdx 
 
 LXX. 
 
 . d< = M.P-«.Tdr= -- 
 
 Pu(x - e) 
 
 LXX'. . , d<'=iW.P'-2.Tdr' 
 
 Mdx 
 
 u (x - e'y 
 r'dx 
 
 r'u (x - e') u (x - e') ' 
 
CHAP. III.] THEOREM OF HODOGllAPHIC ISOCHRONISM. 725 
 
 because, for tlie motion here considered, the measure or quantity of the force is, by I. 
 and LIII., 
 
 LXXT. . . TD2a = Mr-2 = M-ip2. 
 
 (33.) The times of hodographically describing the two small circular arcs, t,t 
 and t't ', are therefore inversely proportional to the potentials, or directly propor- 
 tional to the distances in the orbit ; and their sum is, 
 
 T^^TT ■■ ■, , [ ^ M\u-'idx (r+r')dx 
 \F F jx-e u(jc-e) 
 that is, by LX. and LXIV., 
 
 LXXIII. ..d# + d*' = — — — , if LXXIV. -.p = T(u-\)=i:M. 
 
 u(x — eygi- .7 xr- y 
 
 (34.) We have also the relations, 
 
 M 
 LXXV. . . M = (a;2 - V)\ g, and LXXVI. ..— = (!_ e'2) ^2 ; 
 
 a 
 
 so that the sum of the two small times may be thus expressed, 
 
 LXXVII. . . d, + d<-= ^-^pt ■ (' -'"^X^, 
 M\ X (a;2 - 1)1 ' 
 
 or finally, 
 
 LXXVIII. . . de + d<' = 2 ^ ^ Y . 
 
 (1 - e' cos wy 
 
 if LXXIX. . . a; = sec w, or i^ = Z.MLW in Fig. 86, 
 
 in -which Figure u'w is an ordinate of a semicircle, with the chord mm' of the hodo- 
 graph for its diameter. 
 
 (35.) The two near secants (31.), from the pole n of that chord, have been here 
 supposed to cut the half chord lm itself, as in the cited Figure 86 ; but if they were 
 to cut the other half chord lm', it is easy to prove that the formulae LXXVIII. 
 LXXIX. would still hold good, the only difference being that the angle w, or mlw, 
 would be now obtuse, and its secant x<—l. 
 
 (36.) A circle, with u for centre, and m for radius, cuts the hodograph orthogo- 
 nally in the points T and t'; and in like manner a near circle, with 11, for centre, 
 and u-\- dw for radius, is another orthogonal, cutting the same hodograph in the near 
 points T and t/ (31.). And by conceiving a series of such orthogonals, and observ- 
 ing that the differential expression LXXVIII. depends only on i\\Q four scalars, 
 M'^a^, e', w, and dw, which are all known when the mass Mand the five points o, 
 I., m, u, u^ are given, so that they do not change when we retain that mass and those 
 points, but alter the radius h of the hodograph, or the perpendicular hl let fall from 
 its centre h on the fixed chord mm', we see that the sum of the times (comp. (33.), 
 of hodographically describing any two circular arcs, such as t T and t't/, even if 
 they be not small, but intercepted between any two secants from the pole n of the 
 fxed chord, is independent of the radius (A), or of the height hl of the centre h of 
 the hodograph. 
 
 (37.) If then two circular hodographs, such as the two in Fig. 86, having a com- 
 mon chord mm', which passes through, or tends towards, a common centre of force o, 
 Avith a common ma^s M there situated, be. cut by any ttvo common orthogonals, the 
 sum of the two times oi hodographically describing (33.) the two intercepted arc* 
 (small or large) will be the same for those two hodographs. 
 
726 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 (38.) And as a case of this general result, we have the following Theorem* of 
 Hodographic Isochronism (or Synchronism) : 
 
 '■'^ If two circular hodographs^ having a common chord, which passes throvgh, or 
 tends towards, a common centre of force, be cut perpendicularly by a third circle, 
 the times of hodographically describing the intercepted arcs will be equal." 
 For example, in Fig. 86, we have the equation, 
 
 LXXX. . . Time of Tim:' = time o/wsiw'. 
 (39.) The time of thus describing the arc tmt' (Fig. 86), if this arc be through- 
 out concave] towards o (so that x>l>e', as in LXV.), is expressed (comp. 
 LXXVIII.) by the definite integral, 
 
 / fl3(l-e'2)3Urw Aw 
 
 LXXXI. . . Time of tmt' = 2 ^ ,^ ^ V r. ' ^ 5 
 
 •^ V M y Jo (1 - e cos wy 
 
 and the time of describing the remainder of the hodographic circle, if this remaining 
 arc t'm't be throughout concave towards the centre o of force, is expressed by this 
 other integral, 
 
 dw 
 
 LXXXII. . . Time of t'm't = 2 ( ^ ^ Y 
 
 , (1 — e' cos wy 
 
 (40.) Hence, for the case of a closed orbit (e'^ < 1, e < 1, a > 0), if n denote the 
 mean angular velocity, we have the formula. 
 
 LXXXI 1 1. . . Periodic Time = — = 2 ( — T (1 -e -) J 
 
 (1 — e'cos w)' 
 
 LXXXI V. . . M=«%2, as usual. 
 
 'Hil 
 
 The same result, for the same case of elliptic motion, may be more rapidly obtained, 
 by conceiving the chord mm' through o to be perpendicular to oh ; for, in this posi- 
 tion of that chord, its middle point l. coincides with o, and e'- by LXIV. 
 (41.) In general, by LXXVI., we are at liberty to make the substitution, 
 
 LXXXV. . . ( ^ ^ r = -J, with g = half chord of the hodograph ; 
 
 supposing then that e' = - 1, or placing o at the extremity m' of the chord, we have 
 by LXXXI., 
 
 LXXXVI. . . Parabolic time of tmt' = — -— -^ ; 
 
 5^3 J^(l+C0St«)2' 
 
 for, when the centre of force is thus situated on the circumference of the hodographic 
 circle, we have by (8.) the excentricity e = 1, and the orbit becomes by XV. a para- 
 
 * This Theorem, in which it is understood that the common centre of force (o) 
 is occupied by a common mass (M), was communicated to the Royal Irish Aca- 
 demy on the 16th of March, 1847. (Seethe Proceedings of that date. Vol. III., page 
 417.) It has since been treated as a subject of investigation by several able writers, 
 to whom the author cannot hope to do justice on this subject, within the very short 
 f^pace which now remains at his disposal. 
 
 t Compare the Note to page 721. 
 
CHAP. III.] CHORD OF ORBIT, SUM OF DISTANCES. 727 
 
 hola. For lujperbolic motion (e'2> 1, e> 1, a<0), the formula LXXXI. (with or 
 without the substitution LXXXV.) is to be employed if e' < — 1, that is, if o be on 
 lm' prolonged ; and the formula LXXXIL, if e'>l, e'<seci^, that is, ifo be si- 
 tuated between m and u, 
 
 (42.) For any law of central force, if p, p' be the points of the orbit which corre- 
 spond to the points x, t' of the hodograph^ and if q be the point of meeting of the 
 tangents to the orbit at p, p', as in the annexed Figure 87, while the tangents to the 
 hodograph at x, x' meet as before in u, we shall have the parallelisms, 
 
 Fig. 87. 
 
 LXXXVII. . . OP II ux, op' II x'u, PQ II ox, qp' || ox' ; 
 writing then, 
 
 LXXXVIII. . . OP = a, OP' = a', ox = Da = r, ox' = Da' = r', ou = v^ oq = w, 
 most of which notations have occurred before, we have the equations, 
 
 LXXXIX. . . = Va(r-v)=Va'(v-r') = Vr((u-a)=Vr'(a'-w); 
 thus XC. . . Vav = Var = /3 = VaV' = Va'y, a'-a||v, pp' || ou, 
 
 and XCI. . . Vrw = Vra = - /3 = Vr'a' = Vr'w', r - r' || w, x'x |I OQ. 
 Geometrically, the constant parallelogram (16.) under op, ox, or under op', ox', is 
 equal, by LXXXVII., to each of the four following parallelograms : I. under op, ou ; 
 II. under op', ou; III. under oq, ox ; and IV. under OQ, ox'; whence pp'||ou, and 
 x'x II OQ, as before. 
 
 (43.) The parallelism XC. may be otherwise deduced for the law of the inverse 
 square, with recent notations, from the quaternion formulae, 
 
 xcn...!^ = ,-^ = "-^', in which, xcir....'=i:i±Ii; 
 
 r + r X- V u r + r 
 
 and which may be obtained in various ways; whence it may also be inferred, that 
 if s denote the length T (a' - a) of the chord pp' of the orbit, then (comp. Fig. 86), 
 
 s u 
 
 XCIII. . . , = ;=-7^^ . = UT : UL = &c. = sui tv ; 
 
 r + r T(X-v) 
 
 w being the same auxiliary angle as in (34.), &c. 
 
728 KLEMENTS OF QUATERNIONS. [bOOK III. 
 
 (44.) It is easy to prove that 
 
 whence 
 
 XCIV. . .\-r = (l+-^-, 
 
 XCV. . 
 
 •■^:-x=p=? -^ ^<^"- 
 
 =(-;)?■ 
 
 P'-l(r'-X)v = K.P-i(r-X)i;; 
 
 the lines lt, lt' are therefore in length proportional to the potentials^ F, P' ; and 
 their directions are equally inclined to that of ou, but at opposite sides of it, so that 
 the line lu bisects the angle tlt'. Accordingly (see Fig. 86), the three points t, l, t' 
 are on the circle (not drawn in the Figure) which has hu for diameter ; so that the 
 angles ult', tlu are equal to each other, as being respectively equal to the angles 
 utt', tt'u, which the chord tt' of the hodograph makes with the tangents at its ex- 
 tremities : the triangles tlv, t'lv' are therefore similar, and lt is to lt' as tv to 
 t'v', that is, by LXI., as P to P', or as r to r. 
 (45.) Again, calculation with quaternions gives, 
 
 xcvii. . . ^^LillS^zl) = (ii:iilC\_iiO = („_,)(„_ X) („ _ ,)-,, 
 
 V — r V — T 
 
 whence 
 
 y —r V ~ T T — V 
 
 XCVIII...T^ =:T- < = T-— =UT:i7L = sin 
 
 \ - T \—T K — V 
 
 such then is the common ratio, of the segments tu', u't' of the base tt' of the tri- 
 angle tlt', to the adjacent sides lt, lt', which are to each other as r'to r (44.) ; 
 and because this ratio is also that of s to r + /, by (43.), we have the proportion, 
 
 XCIX. . . OP : op' : PP' = »' : r' : « = lt' : LT : tt', 
 and the formula of inverse similarity (118), 
 
 C. . . A lt't a ' opp'. 
 
 Accordingly (comp. the two last Figures), the base angles opp', op'p of the second 
 triangle are respectively equal, by the parallelisms (42.), to the angles tul, t'ul, 
 and therefore, by the circle (44.), to the base angles tt'l, t'tl, of the first triangle : 
 but the two rotations, round o from p to p', and round l from t' to t, are oppo- 
 sitely directed. 
 
 (46.) The investigations of the three last subarticles have not assumed any know- 
 ledge of the form of the orbit (as elliptic, &c.), but only the law of attraction ac- 
 cording to the inverse square, or by (6.) the Law of the Circular Hodograph. And 
 the same general principles give not only the expression LXXVI. for the constant 
 Ma-\ but also (by LX. LXIV. LXXIV. LXXIX.) this other expression, 
 
 'iM r-\-r 1— e'* 
 
 CI. . . ^ = (1 — e' coa w) g"^ \ whence CII. . 
 
 r + r' 2« 1 - e cos w' 
 
 which last may be considered as a quadratic in e', assigning two values (real or 
 imaginary) for that scalar, when the first member of CII. and the angle w are given ; 
 the sine of this latter angle being already expressed by XCIII. 
 
 (47.) Abstracting, then, from any ambiguity* of solution, we see, by the definite 
 
 * That there ought to be some such ambiguity is evident from the consideration, 
 that when Sl focus o, and two points p, p' of an elliptic orbit are given j it is still 
 
CHAP. III.] LAMBERX'^iTHEOREMjPAllTIALDIFF. EQUATIONS. 729 
 
 integrals in (39.), that the time of describing an arc pr' of an orbit, with the law 
 of the inverse square, is a function (comp. (36.)) of the three ratios, 
 
 cm...!', LLI", -i_,; 
 
 which is a form of Lambert's Theorem, but presents itself here as deduced from the 
 recently stated Theorem of Hodographic Isochronism (38.), without the employment 
 of any property of conic sections. 
 
 (48.) The differential equation I. of the present relative motion may be thus 
 written (comp. 418, I., and generally the preceding Series 418) : 
 
 CIV. . . S.D2a^a + ^P = 0, whence CY... T=P±ir, 
 as in 418, X., if we now write, 
 
 CVI. . . r= - iDa2 = - |r2, and CVII. ..H= ~ ; 
 
 2a 
 
 in fact (by LIII., comp. (20.) (21.)), 
 
 CVIII. . . - 2fl-=2(P-T)=2P+ r? = «;^-y2=^. 
 
 (^49.) Integrating CIV. by parts, &c., and writing (as in 418, XII. XXII.), 
 
 CIX. . . P= P ( r+ P) df, and ex. . . r= P2 Pdf, 
 
 so that F may again be called the Principal Function and V the Characteristic 
 Function of the motion, we have the variations, 
 
 CXI. . . ^P= Sr^a - Sr'^a' - HU ; CXII. . . dV= Sr^a - Sr'^a' + tdH; 
 
 in which a, a' (instead of ao, a) denote now what may be called the itiitial and 
 final vectors (op, op') of the orbit ; whence follow the partial derivatives, 
 
 CXIII. . . D„P = DaF=T; CXIir. . . D„/P= D,/F=- r'; 
 CXIV. . . (DiF) =-H; and CXV. . . T>hV= t ; 
 
 F being here a scalar function of a, a', t, while T is a scalar function of a, a', If, 
 if M be treated as given. 
 
 (50.) The two vectors a, a' can enter into these two scalar functions, only 
 through their dependent scalars r, r', s (comp. 418, (17.)) ; but 
 
 CXVI. . . dr = - r-^8aSa, ^/•' = - r'-iSa'oa', Ss = - s'lS (a - a) (5a' - 5a) ; 
 confining ourselves then, for the moment, to the function V, and observing that we 
 have by CXII. the formula, 
 
 CXVII. . . S (jda - T'da') = Dr V. Sr + D^' V. dr + D, V. ds, 
 in which the variations da, da are arbitrary', we find the expressions, 
 CXVIII. . . r=-ar-iD,.F + (a'-a)s-iDsF; 
 CXVIir. . . r' = + aV'-'D/ r+ (a' - a) s-iD,F; 
 
 permitted to conceive the motion to be performed along either of the two elliptic arcs, 
 pp', p'p, which together make up the whole periphery. But into details of this kind 
 we cannot enter here. 
 
 5 A 
 
730 ELEMENTS OF QUATERNIONS. [boOK III. 
 
 which give these others, 
 
 CXIX. . .DrV= rY(a' -a)r: Yaa' ; 
 CXIX'. . . D/ F= r'Y(a - a) t : Yaa' ; 
 and CXX. . . D, F = s/3 : Yaa\ 
 
 because Var = Va V = |8. 
 
 (51.) But, by XCir., 
 
 CXXI. . . rr -f r V = (r + r) v' \\ v\\a'- a, 
 
 the chord tt of the hodograph, in Figures 86, 87, being divided at u' into segments 
 Tu', u't', which are inversely as the distances r, r', or as the lines op, op' in the 
 orbit ; we have therefore the partial differential equation, 
 
 CXXII. . . D,.F= D/ F, and similarly, CXXIII. . . D,F= Dr'F; 
 
 so that each of the two functions, F and F, depends on the distances r, r', only by 
 depending on their siim, r + r'. 
 
 (52.) Hence, if for greater generality we now treat M as variable, the Principal 
 Function F, and therefore by CXIV. its partial derivative H=- (D^F), are func- 
 tions of the/o?<r scalars, 
 
 CXXIV. . . r + r', s, t, and M. 
 
 (53.) And in like manner, the Characteristic Function (or Action-Function) F, 
 and its partial derivative (by CXV.) the Time, < = D//F, may be considered as 
 functions of this other system of four scalars (comp. (47.)), 
 CXXV. . . r + r', s, H, and M ; 
 
 no knowledge whatever being here assumed, of the form or properties of the orbit, 
 but only of the law of attraction. 
 
 (54.) But this dependence of the time, t, on the four scalars CXXV., is a new 
 form of LamberVs Theorem (47.) ; which celebrated theorem is thus obtained in a 
 new way, by the foregoing quaternion analysis. 
 
 (55.) Squaring the equations CXVIII. CXVIII'., attending to the relation 
 CXXII., and changing signs, we get these new partial differential equations, 
 
 CXXVI. . . 2P+ 2H = (D,F)2 + (D,F)2 + ^Iz^^jtf D.F. D.F; 
 
 CXXVr. ..2P'+2fl-=(D,.F)2 + (D,F)2+^l^:i^JLLD,.F.DsF; 
 
 r s 
 
 because CXXVII. . . a2 = - r2, a'2 = - r'2, («' - a)2 = - 52. 
 
 Hence, by merely algebraical combinations (because P = ilfr-«, and P' = i!fr'->), we 
 find: 
 
 CXXVIII. . . i ((D, F)2 + (D, F)2) = fi + ^ + ^^ 
 
 CXXIX. . . T)rV.DsV= 
 
 r-fr+s r + r —s 
 M M 
 
 r-i-r' + « r + r—s 
 
 4:M / 4 IN 
 CXXX. . .(D,F+D,F)2 = 2//4- r" =M — "; 1 
 
 ^ r+ r'+s \r + r'+ s a } 
 
 4tM I 4 1 
 
 CXXX'. ..(DrF-D.F)2 = 2iJ+ r— = M[ -. 
 
 ^ r+r -s \r+r-s a 
 
CHAP. III.] DEFINITE INTEGRALS FOR ACTION AND TIME. 731 
 (56) But, by CXII. CXVII. CXXII., we have the variation, 
 
 and the function V vanishes with t, and therefore with s, at least at the commence- 
 ment of the motion ; whence it is easy to infer the expressions,* 
 
 cxxxii. . . r= r ( _^_ + f y d.= r f -^ - f )*d, , 
 
 As a verification, t when t and s are small, and therefore / nearly = r, we have 
 thus the approximate values, 
 
 CXXXIV. .. V={2P+2H)l8 = {2T)h = 2Tt; 
 CXXXV. . . t=(2F+ 2H)-ha = (2 Tyh ; 
 
 in which « may be considered to be a small arc of the orhit, and (2 T')^ the velocity 
 with which that arc is described. 
 
 (57.) Some not inelegant constructions, deduced from the theory of the hodo- 
 graph, might be assigned for the case of a closed orbit, to represent the excentric and 
 mean anomalies ; but whether the orbit be closed or not^ the arc tmt' of the hodo- 
 graphic circle, in Fig. 86, represents the arc of true anomaly described : for it sub- 
 tends at the hodographic centre ii an angle tht', which is equal to the angular mo- 
 tion pop' in the orbit. 
 
 (58.) We may add that, whatever ihQ special form of the orbit may be, the equa- 
 tions CXVIII. CXVIII'. give, by CXXIL, 
 
 CXXXVI. ..T'-r = (Ua' + Ua) D,. F; 
 
 from which it follows that the chord tt' of the hodograph is parallel to the bisector 
 of the angle pop' in the orbit : and therefore, by XCI., that this angle is bisected by 
 OQ in Fig. 87, so that the segments PR, rp', in that Figure, of the chord pp' of the 
 orbit, are inversely proportional to the segments tu', uV of the chord tt' of the ho- 
 dograph. 
 
 (59.) We arrive then thus, in a new way, and as a new verification, at this 
 known theorem : that ^y<wo tangents (qp, qp') to a conic section be drawn from 
 
 * Expressions by definite integrals equivalent to these, for the action and time 
 in the relative motion of a binary system, were deduced by the present writer, but by 
 an entirely different analysis, in the First Essay, &c., already cited, and will be 
 found in the Phil. Trans, for 1834, Part II., pages 285, 286. It is supposed that 
 the radical in CXXXIII. does not become infinite within the extent of the integra- 
 tion ; if it did so become, transformations would be required, on which we cannot 
 enter here. 
 
 t An analogous verification may be applied to the definite integral LXXXI. ; in 
 which however it is to be observed that both r-\-r and s vary, along with the va- 
 riable w : whereas, in the recent integrals CXXXII. CXXXIII., r + / is treated as 
 constant. 
 
732 ELEMENTS OF QUATERNIONS. [UOOK III. 
 
 any common point (q), they subtend equal angles at a focus (o), whatever the spe- 
 cial form of the conio may be. 
 
 (60.) And although, in several of the preceding sub-articles, geometrical con- 
 structions have been used only to illustrate (and so to confirm, if confirmation were 
 needed) results derived through calculation with quaternions ; yet the eminently 
 suggestive nature of the present Calculus enables us, in this as in many other ques- 
 tions, to dispense with its own processes, when once they have indicated a definite 
 train of geometrical investigation, to serve as their substitute. 
 
 (61.) Thus, after having in any manner been led to perceive that, for the motion 
 above considered, the hodograph is a circle* (5.), of which the radius ht is equal 
 (7.) to the attracting mass M, divided by the constant /jara^^e/o^ra/n (16.) under 
 the vectors op, ot of position and velocity, in the recent Figures 86 and 87, which 
 parallelogram is equal to the rectangle under the distance op in the orbit, and the 
 perpendicular oz let fall from the centre o of force on the tangent ux to the hodo- 
 graph, we see geometrically that the potential P, or the mass divided by the dis- 
 tance, for the point p of the orbit corresponding to the point T of the hodograph, is 
 equal (as in (27.)) to the rectangle under ht and oz, and therefore, by the similar 
 triangles htv, uoz, to the rectangle under ou and tv (as in (29.)). 
 
 (62.) In like manner, the three potentials corresponding to the second point t' of 
 the first hodograph, and to the points w and w' of the second hodograph, in Fig. 86, 
 are respectively equal to the rectangles under the same line ou, and the three other 
 perpendiculars tV, wx, w'x', on what we have called (29.) the hodographic axis, 
 HL ; so that, for these two pairs of points, in which the two circidar hodographs, with 
 a common chord mm', are cut by a common orthogonal with u for centre, the four 
 potentials are directly proportional to the four hodographic ordinates (29.). 
 
 (63.) And because the force (Jfr"2) is equal to the square of the potential 
 (Jl/r-'), divided by the mass (_M), the four forces are directly as the squares of the 
 four ordinates corresponding ; each force, when divided by the square of the corre- 
 sponding hodographic ordinate, giving the constant or common quotient, 
 
 CXXXVII. . . ou2 : M. 
 
 (64.) It has been already seen (31.) to be a geometrical consequence of the two 
 pairs of similar triangles, ntt , nt' t', and ntv, nt V, that the two small arcs of the 
 first hodograph, near T and t', intercepted between two near secants from the pole N 
 of the fixed chord mm', or between two near orthogonal circles, with u aud u, for 
 centres, are proportional to the two ordinates, tv, t'v'. 
 
 (65.) Accordingly, if we draw, as in Fig. 86, the near radius (represented by a 
 
 * This follows, among other ways, from the general value XXVI. for the radius 
 of curvature of the hodograph, with any law of central force ; which value was geo- 
 metrically deduced, as stated in the Note to page 720, compare the Note to page 
 719, by the present writer, in a Paper read before the Royal Irish Academy in 1846, 
 and published in their Proceedings. In fact, that general expression for the radius 
 of hodographic curvature may be obtained with great facility, by dividing the ele- 
 ment /d< of the hodograph (in which /denotes the force), by the corresponding 
 element cr ^di of angular motion in the orbit. 
 
CHAT. HI.] DISTANCE OF COMET OR PLANET FROM EARTH. 733 
 
 dotted line from h) of the first hodograpb, and also the smaU perpendicular uv, 
 erected at the centre u of the first orthogonal to the tangent ut, and terminated in 
 Y by the tangent from the near centre u„ the two new pairs of similar triangles, tht,, 
 UTY, and THV, uu^y, give the proportion, 
 
 CXXXVIII. . . TT, : TV = UU, : UT ; 
 which not merely confirms what has just been stated (64.), for the case of the Jirst 
 hodograpb, but proves that the four small arcs, of the tivo circular hodographs in 
 Fig. 86, intercepted between the two near orthogonals, are directly pi'oportional to the 
 four ordinates already mentioned. 
 
 {QQ.^ But the time of describing any small hodographic arc is the quotient (32.) 
 of that arc divided by the /orce; and therefore, by (63.), (65.), the four small times 
 are inversely proportional to the four ordinates. And the harmonic mean u'l be- 
 tween the two ordinates tv, t'v' of the first hodograpb, does not vary when we pass 
 to the second, or to any other hodograph, with the same fixed chord mm', and the 
 same orthogonal circles ; it follows then, geometrically, that the sum (33.) of the 
 two small times is the same, in any one hodograph as in any other, under the condi- 
 tions above supposed : and that this sum is equal to the expression, 
 
 2M.xJu' 2M.tnj'.ui: 
 
 CXXXIX. . 
 
 ou'.UT.u'L ou'.i^ar.UT 
 
 which agrees with the formula LXXIIT. 
 
 (67.) On the whole, then, it is found that the Theorem of Hodographic Isochro- 
 nism (38.) admits of being ^eome^ncaZ/y* proved, although by processes suggested 
 (60.) by quaternions : and sufficient hints have been already given, in connexion 
 with Figure 87, as regards the geometrical passage from that theorem to the well- 
 known Theorem of Lambert, without necessarily employing any property of conic 
 sections. 
 • 
 
 420. As ajifth specimen, we shall deduce by quaternions an equa- 
 tion, which is adapted to assist in the determination of the distance 
 of a comet, or new planet, from the earth. 
 
 (1.) Let M be the mass of the sun, or (somewhat more exactly) the sum of the 
 masses of sun and earth ; and let a and a> be the heliocentric vectors of earth and 
 comet. Write also, 
 
 L..Ta=r, Tm = w, T(o)-a) = z, \J(io-a)^p, 
 
 so that r and w are the distances of earth and comet from the sun, while z is their 
 distance from each other, and p is the unit-vector, directed from earth to comet. 
 Then (comp. 419, I.), 
 
 * It appears from an unprinted memorandum, to have been nearly thus that the 
 author orally deduced the theorem, in his communication of March, 1847, to the 
 Royal Irish Academy ; although, as usually happens in cases of invention, his OAvn 
 previous processes of investigation had involved principles and methods, of a much 
 less simple character. 
 
734 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 11. . . D3a =- Mr-^a, B^io = - Mw-^w, 
 and III. . . D2.2P = D2(w - a) = A/(r-3_ m,-3) „ _ Mzw^p, 
 
 with IV. . . w^=-(a + zpy = r2 + z^-2zSap. 
 
 (2.) The vector a, with its tensor r, and the mass M, are given by the theory of 
 the earth (or sun) ; and p, Dp, D^p are deduced from three (or more) near obser- 
 vations of the comet ; operating then on III. with S.pD/o, we arrive at the formula, 
 
 which becomes by IV., when cleared of fractions and radicals, and divided by z, an 
 algebraical equation of the seventh degree, whereof one root is the sought distance* z, 
 of the comet (or planet) from the earth. 
 
 421. As a sixth specimen, we shall indicate a method, suggested 
 by quaternions, of developing and geometrically decomposing the 
 disturbing force of the sun on the moon, or of a relatively superior 
 on a relatively inferior planet. 
 
 (1.) Let a, <r be the geocentric vectors of moon and sun ; r, s their geocentric 
 distances (r = Ta, s = To-) ; M the sum of the masses of earth and moon ; and S the 
 mass of the sun ; then the differential equation of motion of the moon about the 
 earth may be thus written (comp. 418, 419), 
 
 I. . . D2a = M . 0a + .S. (0(T - ((T - a)), 
 if D be still the mark of derivation relatively to the time, and 
 
 II. . . 0a = 0(a) = a-iTa->; 
 so that (pa is here a vector-function of a, but not a linear one. 
 
 (2.) If we confine ourselves to the term M(pa, in the second member of >., we 
 fall back on the equation 419, 1., and so are conducted anew to the laws o( undisturbed 
 relative elliptic tnotion. 
 
 (3.) If we denote the remainder of that second member by ?/, then ij may be 
 called the Vector of Distwhing Force ; and we propose now to develope this vector, 
 according to descending powers of T (<r : a), or according to ascending powers of 
 the quotient r:s, of the distances of moon and sun from the earth. 
 
 (4.) The expression for that vector may be thua transformed : 
 
 III. . Vector of Disturbing Force = r} = D^a — M(pa 
 = Ss-^a-'^ { 1 - (1 - a(T-')-i T(l - aa'^)-^} 
 = /Ss-'cr-i { 1 - (1 - aa-"^)-^ (1 - (T'la)-*} 
 
 * Compare the equation in the Mecanique Celeste (Tom. I., p. 241, new edi- 
 tion, Paris, 1843). Laplace's rule for determining, by inspection of a globe, which 
 of the two bodies is the nearer to the sun, results at once from the formula V. 
 
CHAP. III.] DEVELOPMENT OF DISTURBING FORCE. 735 
 
 that is, IV. . . »; = »;i+ J73+»73+&c., 
 
 if y. . . j;i = - &-i(7-i (i(T'^a + f ao-'i) =— (a + 3(ra(T-') = J?i, i + m, 2 ; 
 
 3)S'r2 
 VI. . . »73= (aora-i+ 2(r + Sffaaa-iff'O = '^a, 1 + »72,2+ >72,3 ; &c. 
 
 the general term* of this development being easily assigned. 
 
 (5.) We have thus a first group of two component and disturbing forces, which 
 
 Sr 
 are of the same order as -5- ; a second group of three such forces, of the same order 
 
 as — ; a third group of four forces, and so on. 
 
 (6.) The^rs^ component of i\iQ first group has the following tensor and versor, 
 
 vn...T„„=^, 
 
 it is therefore a purely ablati- 
 tious force mn, acting along the 
 moon's geocentric vector em pro- 
 longed, as in the annexed Fi- 
 gure 88. 
 
 (7.) The second component 
 mn', of the same first group, has an exactly triple intensiti/, mn'= 3mn ; and its di- 
 rection is such that the angle nmn', between these two forces of the first group, is 
 bisected by a line ms' from the moon, which is parallel to the sun^s geocentric vector 
 
 ES. 
 
 (8.) If then we conceive a line em' from the earth, having the same direction as 
 the last force mn', this new line will meet the heavens in what may be called for the 
 moment a. fictitious moon Di, such that the arc J)])i of a, great circle, connecting it 
 with the true moon J) in the heavens, shall be bisected by the sun , as represented 
 in Fig. 88. 
 
 (9.) Proceeding to the second group (5.), we have by VI. for ih>i first compoltent 
 of this group, 
 
 VIII...T,.„=^', U,„, = Ua.a- = "-Hf; 
 
 a line from the earth, parallel to this new force, meets therefore the heavens in what 
 may be called & first fictitious sun, 1, such that the arc of a great circle, ©01, con- 
 necting it with the true sun, is bisected by the moon ]), as in the same Fig. 88. 
 
 * Such a general term was in fact assigned and interpreted in a communication 
 of June 14, 1847, to the Pioyal Irish Academy {Proceedings, Vol. III., p. 614) ; 
 and in the Lectures, page 616. The development may also be obtained, although 
 less easily, by Taylor's Series adapted to quaternions. Compare pp. 427, 428, 430, 
 431 of the present work ; and see page 332, &c,, for the interpretation of such sym- 
 bols as (Taof"', aaa~^. 
 
736 ELEMENTS OF QUATERNIONS. [iJOOK III. 
 
 (10.) The second component force, of the same second group, has an intensity ex- 
 actly double that of the Jirst (Tr]2,2 = 2Tr/2, i) ; and in direction it is parallel to the 
 sun's geocentric vector es, so that a line drawn in its direction from the earth would 
 meet the heavens in the place of the sun 0. 
 
 (11.) The third component of the present group has an intensity which is ipre- 
 clsely Jive-fold that of the^rs^ component (TjjsjS = STj/o, i) ; and a line drawn in its 
 direction from the earth meets the heavens in a second fictitious sun 00, such that 
 the arc 0i 0^, connecting these two fictitious suns, is bisected by the true sun 0. 
 
 (12.) There is no difficulty in extending this analysis, and this interpretation, to 
 subsequent groups of component disturbing forces, which forces increase in number ^ 
 and diminish in intensity^ in passing from any one group to the next ; their intensi- 
 ties, for each separate group, bearing numerical ratios to each other, and their direc- 
 tions being connected by simple angular relations. 
 
 (13.) For example, the third group consists (5.) of four small forces, tj3,i .. »j3, i, 
 
 Sr"^ 
 of which the intensities are represented by , multiplied respectively by the four 
 
 whole numbers, 6, 9, 15, and 35 ; and which have rfirec^ions respectively parallel to 
 lines drawn from the earth, towards a second fictitious moon ])2, the true moon, the 
 first fictitious moon J)i (8.), and a third fictitious moon ])3 ; these three fictitious 
 moons, like the two fictitious suns lately considered, being all situated in the momen- 
 tary plane of the three bodies E, m, s : and the three celestial arcs, ])2]), DDi, Di])3, 
 being each equal to double the arc JQ of apparent e/ow^afion of sun from moon 
 in the heavens, as indicated in the above cited Fig. 88. 
 
 (14.) An exactly similar method may be employed to develope or decompose the 
 disturbing force of one planet on another, which is nearer than it to the sun ; and it 
 is important to observe that no supposition is here made, r.'specting any smallness 
 of excentricities or inclinations. 
 
 422. As a seventh specimen of the physical application of quater- 
 nions, we shall investigate briefly the construction and some of the 
 properties of FresneVs Wave Surface, as deductions from his princi- 
 ples or hypotheses* respecting light. 
 
 (1.) Let jO be a Vector of Ray- Velocity, and fi the corresponding Vector of 
 Wave-Slowness (or Index-Vector), for propagation of light from an origin o, within 
 a biaxal crystal ; so that 
 
 I. . . S/i/> = - 1 ; II. . . Sfi^p = ; and therefore III. . . Sp^/i = 0, 
 
 * The present writer desires to be understood as not expressing any opinion of 
 his own, respecting these or any rival hypotheses. In the next Series (423), as an 
 eighth specimen of application, he proposes to deduce, from a quite different set of 
 physical principles respecting light, expressed however still in the language of the 
 present Calculus, Mac Cullagh's Theorem of the Polar Plane ; intending then, as a 
 ninth anCi final specimen, to give briefly a quaternion transformation of a celebrated 
 equation in partial differential coefficients, of the fiz'st order and second degree, which 
 occurs in the theory of heat, and in that of the attraction of spheroids. 
 
CHAP. III.] CONSTRUCTION OF FRESNEL's WAVE SURFACE. 737 
 
 if 5p and dfx be any infinitesimal variations of the vectors p and /;, consistent with 
 the scalar equations (supposed to be as yet unknown), of the Wave-Surface and its 
 Reciprocal (with respect to the unit-sphere round o), namely the Surface of Wave- 
 Slowness^ or (as it has been otherwise called) the Index* -Surface : the velocity of 
 light in a vacuum being here represented by unity. 
 
 (2.) The variation ^p being next conceived to represent a small displacement, 
 tangential to the wave, of a particle of ether in the crystal, it was supposed by Fres- 
 nel that such a displacement tp gave rise to an elastic force, say dt, not generally in 
 a direction exactly opposite to that displacement, but still a. function thereof, which 
 function is of the kind called by us (in the Sections III. ii. 6, and III. iii. 7) linear^ 
 vector, and self-conjugate ; and which there will be a convenience (on account of its 
 connexion with certain optical constants, a, h, c) in denoting here by 0'i^p (instead 
 of ^^p) : so that we shall have the two converse formulae, 
 
 IV. . . ^p = (pde ; V. . . 5e = ^-'^p. 
 
 (3.) The ether being treated as incompressible, in the theory here considered, 60 
 that the normal component fx-^Sfide of the elastic force may be neglected, or rather 
 suppressed, there remains only the tangential component, 
 
 VI. . . fi-^YfiSe = de-ix-^Sfide, 
 
 as regulating the motion, tangential to the wave, of a disturbed and vibrating par- 
 ticle. 
 
 (4.) If then it be admitted that, for the fro])agatior\ of a, rectilinear vibration, 
 tangential to a wave of which the velocity is T/i"i, the tangential force (3.) must be 
 exactly opposite in direction to the displacement dp, and equal in quantity to that 
 displacement multiplied by the square (T/i-^) of the wave-velocity, we have, by V. 
 and VI., the equation, 
 
 VII. . . ip-^Sp-fi-^SnS£=ix'^dp, or VIII. . . ^p = (^-i-/i-2)-i/t-iS/i^t ; 
 
 combining which with II., we obtain at once this Symbolical Form of the scalar 
 equation of the Index Surface, 
 
 IX. . . O = S/x-i(0-i-/i-2)- 1/4-1; 
 or by an easy transformation, 
 
 X.. . i=s/irKr'-/*-^)''Af'; 
 
 or finally, XI. . . 1 = S/i (jw2 - 0)-i ft ; 
 
 * This brief and expressive name was proposed by the late Prof. Mac Cullagh 
 (Trans. R. I. A., Vol. XVIII., Parti., page 38), for that rfciprocaZof the wave-sur- 
 face which the present writer had previously called the Surface of Components of 
 Wave- Slowness, and had employed for various purposes : for instance, to pass from 
 the conical cusps to the circular ridges of the Wave, and so to establish a geometri- 
 cal connexion between the theories of the two conical refractions, internal and exter- 
 nal, to which his own methods had conducted him (Trans. K. I. A., Vol. XVII , 
 Part I., pages 125-144). He afterwards found that the same Surface had been 
 otherwise employed by BI. Cauchy {Exercises de Mathematiques, 1830 p. 36), who 
 did not seem however to have perceived its reciprocal relation to the Wave. 
 
 5 B 
 
738 ELKMENTS OF QUATERNIONS. [bOOK III. 
 
 while the direction of the vibration Sp, for any given tangent plane to the wave, is 
 determined generally by the formula VIII. 
 
 (5.) That formula for the displacement, combined with the expression V. for the 
 elastic force resulting, gives 
 
 XII. . . dp=- cpvSfiSf, and XIII. . . Ss = - vSfiSs, 
 if XIV. . . {(p-fi^)v = fi, or XV. . . u = (^ - fi-y^fi, 
 
 V being thus an auxiliary vector; and because the equation XI. of the index surface 
 gives, 
 
 XVI. . . S/iu = - 1, while XVII. . . Yvde = 0, by XIII., 
 
 it follows that the vector v, if drawn like p and /x from o, terminates on the tangent 
 plane to the wave, and is parallel to the direction of the elastic force. 
 
 (6.) The equations XIV. XVI. give, 
 
 XVIII. . . fi^vz- Sv(}>v = 1, whence XIX. . . v^Sfi^fi = Sfidv = - Svdfx, 
 because ^S^t; = 0, by XVI., and dSv^v =:2S{<pv.Sv), by the self-conjugate pro- 
 perty of ; comparing then XIX. with III., we see that + p (as being -L. every d/i) 
 has the direction of fx + v^, and therefore, by I. and XVI., that we may write, 
 
 XX. . .p-> = -/i-v-i; XXI. . . p-2 = ;x2-u-2; XXII. .. Spy = ; 
 which last equation shows, by (5.), that the ray is perpendicular (on Fresnel's prin- 
 ciples) to the elastic force Se, produced by the displacement dp. 
 
 (7.) The equations XX. and XXI. show by XIV. that 
 
 XXIII. . . (p-2 - 0) u = p-\ whence XXIV. . . v = (p-2 - 0)-i p"' ; 
 we have therefore, by XXII., the following Symbolical Form (comp. (4.)) of the 
 Equation of the Wave Surface, 
 
 XXV. . . = Sp-» (^ - p"'*)"'p"M 
 or, by transformations analogous to X. and XI., 
 
 XXVI... l = Sp0(0-p-2)-ip-i; XXVII... l = Sp(p2-0-i)-ip; 
 
 and we see that we can return from each equation of the wave, to the' corresponding 
 equation of the index surface, by merely changing p to /*, and to 0"i : but this 
 result will soon be seen to be included in one more general, which may be called the 
 Rule of the Interchanges. 
 
 (8.) The equation XXV. may also be thus written, 
 
 XXVIII. ..Sp(0-p-2)-ip = O; 
 but under this last form it coincides with the equation 412, XLI. ; hence, by 412, 
 (19.), the Wave Surface may be derived from the auxiliary or Generating Ellipsoid^ 
 
 XXIX. ..Sp0p = l, 
 by the following Construction, which was in fact assigned by Fresnel* himself, but 
 as the result of far more complex calculations: — Cut the ellipsoid (abc) by an arbi- 
 trary plane through its centre, and at that centre erect perpendiculars to that plane, 
 which shall have the lengths of the seniiaxes of the section ; the locus of the extremi- 
 ties of the perpendiculars so erected will be the sought wave surface. 
 
 * See Sir John F. W. Herschel's Treatise on Light, in the EncyclopOEdia Me- 
 tropolitana, page 545, Art. 1017. 
 
CHAP. III.] CONNEXION OF RAY WITH INDEX-VECTOR. 739 
 
 (9.) And we see, by IX., that the Index Surface may be derived, by an exactly 
 similar construction, from that Reciprocal Ellipsoid, of Avhich the equation is, on 
 the same plan, 
 
 XXX. ..Sp0-V = l. 
 
 (10.) If the scalar equations, XXVII. and XI., of the wave and index surface, be 
 denoted by the abridged forms, 
 
 XXXI. . . fp = 1, and XXXII. . . F/i = 1, 
 then the relations I. II. III. enable us to infer the expressions (comp. the notation in 
 418, (2.)), 
 
 XXXIII... ,.= ^1^5 XXXIV... p = -=J?^; 
 
 in which (comp. 412, (36.), and the Note to that sub-article), 
 
 XXXV. . . iDpfp = (p2_0-i)-ip_pSp(p2-0-»)-V = -w- w2p^ 
 and XXXVI. . . ^D^F/x = (/xS - ^)-y - p^fi (^2 - 0)-^ = - v - 1/ V ; 
 
 V being the same auxiliary vector XV. as before, and w being a new auxiliary vec- 
 tor, such that (by XXIV. XXVII. and IX. XV.), 
 
 XXXVII. . . w=(0-i-p2)-ip = 0v; XXXVIII. . . 8pa> = -l; 
 XXXIX. . . S/xw = ; 
 
 whence also w || ^p by XII., so that (comp. (5.)) if w be drawn (like p, /u, and y) 
 from the point o, this new vector terminates on the tangent plane to the index sur- 
 face, and is parallel to the displacement on the wave ; also Sp : de = (o : v. 
 (11.) Hence, by XXXIII. XXXV. XXXVIII., 
 
 XL.../. = -I-^^ = -— ^=-(o;-i + p)-«. or XLI...-;ti-i = p + (.-i; 
 1 — urp* w ^ — p2 
 
 and similarly, by XXXIV. XXXVI. and XVI., 
 
 80 that, with the help of the expressions XV. and XXXVII. for v and w, the ray-vec- 
 tor p and the index-vector p. are expressed as functions of each other : which func- 
 tions are generally definite, although we shall soon see cases, in which one or other 
 becomes partially indeterminate. 
 
 (12.) It is easy now to enunciate the rule of the interchanges, alluded to in (7.), 
 as follows: — In any formula involving the vectors p, p, v, oj, and i\iQ functional 
 symbol (p, or some of them, it is permitted to exchange p with p, v with w, and 
 with ^'i; provided that we at the same time interchange dp with de (but no<* gene- 
 rally with dp)f when either dp or di occurs. 
 
 * It is true that, in passing from II. to III. (instead of passing to XLIII.), we 
 may be said to have exchanged not only p with p, but also dp with dfx. But usu- 
 ally, in the present investigation, dp represents a small displacement (2.), which 
 is conceived to have a definite direction, tangential to the wave ; whei'eas dp 
 
740 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 For example, we pass thus from XX. to XLI., and conversely from the latter 
 to the former ; from II. we pass by the same rule, to the formula, 
 
 XLIII. . . Spde = 0, which agrees by XVII. with XXII. ; 
 and, as other verifications, the following equations may be noticed, 
 XLIV. ..Sp = [xY,xdB ; XLV. ..di = pYpdp ; XLVI. . . Sfidi = Spdp. 
 (13.) The relations between the vectors may be illustrated by the annexed Fi- 
 gure 89 ; in which, 
 
 XLVII. . . OP = p, OQ = /*, 
 ou = v, ow = (1), 
 and XLVIir. . .op' = -p-i, 
 
 OQ' = -/i-l, 0U'= - V'l, OW'=-W~l 
 
 in fact it is evident on inspection, 
 that 
 
 XLIX. . . OP . op' = OQ . oq' y\s:. 89. 
 
 = ou . ou' = ow . ow' ; 
 
 and the common value of these four scalar products is here taken as negative unitj'. 
 
 (14.) As examples of such illustration, the equation XX. becomes p'o = qu'-, 
 XLI. becomes oq' = w'p; XXIII. may be written as w + p"i = p-^v, or as 
 p'w : ou = p'o : OP ; &c. And because the lines pq'u and qp'w are sections of the 
 tangent planes, to the wave at the extremity p of the ray, and to the index surface 
 at the extremity q of the index vector, made by the plane of those two vectors p and 
 fi, while dp and Se (as being parallel to w and v) have the directions of PQ'andQp'; 
 we see that the displacement (or vibration) has generally, in Fresnel's theory, the 
 direction of the projection of the ray on the tangent plane to the wave ; and that the 
 elastic force resulting has the direction of the projection of the index vector on the 
 tangent plane to the index surface : results which might however have been other- 
 wise deduced, from the formulae alone. 
 
 (15.) It may be added, as regards the reciprocal deduction of the two vectors [x 
 and p from each other, that (by XLL XXXVIII., and XX. XVI.) we have the 
 expressions, 
 
 L. . . - n'"^ =a>-iVw|0, and LI. . . — p-' = i»"iVi;/x ; 
 
 which answer in Fig. 89 to the relations, that oq' is the part (or component) of op, 
 perpendicular to ow ; and that op' is, in like manner, the part of oq -J- ou. 
 (16.) We have also the expressions, 
 
 LII. . . - fx'^ =u)-^Y(jt)Vf and LIII. . . — p'^^v'^Vvio, 
 which may be similarly interpreted ; and which conduct to the relations, 
 
 LIV. . . -(Vvw)2 = v2p-2=(^2^-2=Svw. 
 
 Hence, the Locus of each of the two Auxiliary Points u and w, in Fig. 89, is a Sur- 
 face of the Fourth Degree ; the scalar equations of these two loci being, 
 
 LV. . . (Yvtpvy + Sy^u = 0, and LVI. . . (Vw^-J w)2 + Sw0->w = ; 
 
 continues, as in (1.) to represent any infinitesimal tangent to the index siirfacCf 
 while dt still denotes the elastic-force (2.), resulting from the displacement dp. 
 
CHAP. III.] EQUATIONS OF WAVE, LINES OF VIBRATION. 741 
 
 from which it would be easy to deduce constructions for those surfaces, with the help 
 of the two reciprocal ellipsoids, XXIX. and XXX. 
 
 (17.) The equations XII. XXII., combined with the self-conjugate property of 
 0, give 
 
 LVII. . . = S (0-ip . ^p), or LVIII. . . = ^Sp0- V ; 
 
 hence (between suitable limits of the constant), every ellipsoid of the form, 
 
 LIX. . . Sp0"V = ^* = const., 
 which is thus concentric and coaxal with the reciprocal ellipsoid XXX., being also 
 similar to it, and similarly placed, contains upon its surface what may be called a 
 Line of Vibration* on the Wave ; the intersection of this new ellipsoid LIX. with 
 the wave surface being generally such, that the tangent at each point of that line (or 
 curve) has the direction of Fresnel's vibration. 
 
 (18.) The fundamental connexion (2.) of the /wnc^ion (p with the optical con- 
 stants, a, b, c, of the crystal, is expressed by the symbolical cubic (comp. 350, L, 
 and 417, XXV.), 
 
 LX. . . (^ + a-2) (<1> -F 6-2) (0 ^ c-2) = ; 
 
 from which it is easy to infer, by methods already explained, that if e be any scalar, 
 
 and if Ave write, 
 
 LXL . . E=(e-a-2) (e_6-2) (e - c'^), 
 
 we have then this formula of inversion, 
 
 LXII. . . E((p + e)-i = e2 - e (0 + <z-2 + b^ + c-2) - a-26-2c-2^-i. 
 
 (19.) Changing then e to — p"% the equation XXVIII. of the wave becomes, 
 
 LXIIL . . O = p-2 + a-2+6-2 + c-2+Sp-i0p-«-26-2c-2Sp0-ip. 
 
 the Wave is therefore (as is otherwise known) a Surface of the Fourth Degree : and 
 (as is likewise well known), the Index Surface is of the same degree, its equation 
 (found by changing p, (p, a, b, c ttj p,, ^-i, a-^, b~\ c-i) being, on the same plan, 
 LXIV. . . = /i-2+ a3 [. 62 + c2 + S/i-i ^- > - a^b^c^Spfp. 
 (20.) These equations may be variously transformed, with the help of the cubic 
 LX. in (p, which gives the analogous cubic in ^-i, 
 
 LXV. . . (0-1+ a^) (0-1 + 62) (^-1 + c2) = ; 
 for instance, another form of the equation of the wave is, 
 
 LXVI. . . = Sp0-2p + (p2 + a2 + 62 4. <,2) Sp^-ip - a262c2 ; 
 in which it may be remarked that Sp0-2p = (0""ip)2 < 0, whereas Sp0-ip > 0. 
 
 (21.) Substituting then, for Sp^'V in LXVI., its value h^ from (17.), we find 
 that this second variable ellipsoid, with h for an arbitrary constant or parameter, 
 
 LXVII. . . O=<0-'p)2 + A4(p2 + a2 + 62+c2)-a262c2, 
 contains upon its surface the same line of vibration as the first variable ellipsoid 
 LIX., which involves the same arbitrary constant h ; and therefore that the line in 
 
 * Such lines of vibration were discussed by the present writer, but by means of 
 a quite different analysis, in his Memoir of 1832 (Third Supplement on Systems of 
 Rays), which was published in the following year, in the Transactions of the Royal 
 Irish Academy. See reference in the Note to page 737. 
 
742 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 question is a quartic curve, or Curve of the Fourth Degree, as being the intersection 
 of these two variable but connected ellipsoids : and that the wave itself is the locus 
 of all such quartic curves. 
 
 (22.) The Generating Ellipsoid (Sp^p =1) has a, b, c for its semiaxes {a> b> c 
 > 0) ; and for any vector p, in the plane of be, we have the symbolical quadratic 
 (comp. 353, (9.)), 
 
 LXVIII. . . (^ + 6-2) (^ + c-2) = 0, or LXIX. . . - Mc-z^-i = ,^ + 6-2 + c-2 ; 
 
 making then this last substitution for 0+6-24. c-2 in LXIII., we find, for the sec- 
 tion of the wave by this principal plane of the ellipsoid XXIX., an equation which 
 breaks up into the two factors, 
 
 LXX. . . p-2 + a-2 = 0, and LXXI. . . 1 - b-^c-^Sp<p-ip = ; 
 
 whereof the ^rs< represents (the plane being understood) a circle, with radius = a, 
 which we may call briefly the circle (a) ; while the seconcf represents (with the same 
 understanding) an ellipse, which may by analogy be called here the ellipse («) : its 
 two semiaxes having the lengths of c and 6, but in the directions of b and c, for 
 which directions 0+6-2=0 and -f c-2 = 0, respectively, so that this ellipse (a) is 
 merely the elliptic section^ (6c) of the ellipsoid (^abc), turned through a right angle 
 in its own plane, as by the construction (8.) it evidently ought to be. And an ex- 
 actly similar analysis shows, what indeed is otherwise known, that the plane of ca 
 cuts the wave in the system of a circle (6), and an ellipse (6) ; and that the plane 
 of ab cuts the same wave surface, in a circle (c), and an ellipse (c). 
 
 (23.) The circle («) is entirely exterior to the ellipse (a) ; and the circle (c) is 
 wholly interior to the ellipse (c) ; but the circle (6) cuts the ellipse (6), in four 
 real points, which are therefore (in a sense to be soon more fully examined) cusps 
 (or nodal points) on the wave surface, or briefly Wave- Cusps : and the vectors p,» 
 say + po and + pi, which are drawn from the centre o to these four cusps, may be 
 called Lines of Single Rag- Velocity, or briefly Cusp-Bays. 
 
 (24.) It is clear, from the construction (8.), that these lines or rays must have 
 the directions of the cyclic normals of the ellipsoid (abc) ; which suggests our using 
 here the cyclic forms, 
 
 LXXII. . .^p=gp+ VXpX', and LXXIIl. . . Sp^p = gp^ + SXpX'p = 1, 
 for the function 0, and the generating ellipsoid (8.) ; X' being written, to avoid con- 
 fusion, instead of the fi of 357, &c., to represent the second cyclic normal. 
 
 (25.) Changing then p, to X', v to p, and g to g — p'^, in the expression 361, 
 
 XXVII. for Fv or Sv(p-^v ; equating the result to zero, and resolving the equation so 
 
 obtained, as a quadratic in ^ ; we find this new form of the Equation XXVIII. of the 
 
 Wave, 
 
 LXXIV. . . ^p2 = 1 + SXpSX'p + TVXpTVX'p ; 
 
 the upper sign belonging to one sheet, and the lower sign to the other sheet, of that 
 wave surface. The new equation may also be thus written, as an expression for the 
 itiverse square of the ray-velocity Tp, or of the radius-vector, say r, of the wave, 
 
 «-2+c-2 a-2 -C-2 / p_ p\ 
 
 LXXV...r- = Tp-=-^- + -— c«3(^.e+, £ j, 
 
 because, by 405, (2.), (6.), &c., 
 
 LXXVI. . .fl-2 = -^-TXX', 6 2 = _^ + SXX', c-^-^-g + TW; 
 
CHAP. III.] CONICAL CUSPS ON WAVE AND INDEX SURFACE. 743 
 
 and we have the verification, for a cusp-ray (23.), that 
 
 LXXVII. . . r-2 = 6-2, or r = Tp = 6, if p\\\ or \'. 
 (26.) If we write (comp. XXXI.), 
 
 LXXVIII. . . fp = - p-3 (1 + S|O0p) + rt-26-2c-2Sp0-ip, 
 the equation LXIII. of the wave takes the form, 
 
 LXXIX. . , fp = a-2 + J-3 4 c-2 = const. ; 
 and we have the partial derivative (comp. XXXV.), 
 
 LXXX. . . iDpfp = p-3(l + Sp^p) - p-^(l)p + a-26-2c-20-ip 
 = p-3 (1 - Yp<l>p) + a-^b-^c-^-^p ; 
 which gives by XXXIII. the expression, 
 
 p-3(Vp0p - 1) - a-26-2c-20-i 
 
 LXXXI. 
 
 /* = 
 
 and therefore a generally definite value (comp. (11-)) ^o^* the index vector fi, when 
 the ray p is given. 
 
 (27.) If the ray be in the plane of ac, then (comp. LXIX.), 
 LXXXII. . . <Pp+ (a-2 + c-2) p + a-^c-^~^p = 0, 
 whence LXXXIII. . . Yp(pp = - a-^c-^Ypcp- ip = a-2c-3 (Sp^-^p - p^'V) 5 
 and therefore by LXXXL, 
 
 1.JS.AJL1V. . • /*- J-2(Sp^-lp_^-2c2)^(p-2 + 6-2)^2c2? 
 
 an expression which gives, definitely, 
 
 LXXXV. .. fi = - p-i, if LXXXVI. . . p-2 + &-2 = 0, 
 but not LXXXVII. . . Sp0-ip = «2c2, 
 
 that is (comp. (22.)), if the ray terminate'on the circle (6), at any point which is 
 not also on the ellipse (6); and with equal definiteness, 
 
 LXXXVIII. . . ft =-a-^c-'^<p-^p, if LXXXVII. but not LXXXVI. hold good, 
 that is, if the ray terminate on the ellipse (6), at any point which is not also on the 
 circle. 
 
 (28.) The normal then to the wave, in each of the two cases last mentioned, co- 
 incides with the normal to the section, made by the plane of ac ; and if we abstract 
 for a moment from the cusps (23.), we see that the wave is touched, along the circle 
 (6), by the concentric sphere LXXXVI. with radius = h, which we may call the 
 sphere (6) ; and along the ellipse (h) by the concentric ellipsoid LXXXVII. which 
 may on the same plan be called the ellipsoid (b). 
 
 (29.) An exactly similar analysis shows that the wave is touched along the cir- 
 cles (a) and (c), by two other concentric spheres, with radii a and c, which may be 
 briefly called the spheres («) and (c) ; and along the ellipses (a) and (c) by two other 
 concentric and similar ellipsoids, which may by analogy be called the ellipsoids («) 
 and (c). And by comparing the equation LXXXVII. of the ellipsoid (6) with the 
 form LIX., we see that the three elliptic sections (a) (6) (c) of the wave, made by 
 the three principal planes of the generating ellipsoid (a6c), are lines of vibration 
 (17.) ; the constant /i* receiving the three values, b-c^, c^a^, a^\ for these three 
 ellipses respectively. 
 
744 ELEMENTS OF QUATERNIONS. [bOOK III, 
 
 (30.) But at a cusp the two equations LXXXVI. and LXXXVII. coexist, and 
 the expression LXXXIV. for fj, takes the indeterminate form - ; in fact, there is in 
 
 this case no reason for preferring either to the other of the two values, within the 
 plane of ac, 
 LXXXIX. . . fi = - po-\ XC. . . ju = fio, if XCI. . . /xo = - a^c^^-ipo ; 
 
 in which po is the cusp-ray (23.), and the first value of ju corresponds to the circle, 
 but the second to the ellipse (6). 
 
 (31.) The indetermination of fx, at a. wave-cusp, is however even ^reafer than 
 this. For, if we observe that the equations LXXIX. and LXXX. give, for this case, 
 by LXXXin. LXXXVI. LXXXVII., 
 
 XCIL . . fpo= 0-2 + 6-2+ c-2, and XCIIL . .T>pfp = 0, for p = po, 
 
 we shall see that if p be changed to po+ p' in the expression LXXVIII. for fp, and 
 only terms which are of the second dimension in p' retained, the result equated to zero 
 will represent a cone of tangents p', or a Tangent Cone to the Wave at the Cusp : 
 which cone is of the second degree, and everg normal p to Avhich, if limited by the con- 
 dition I., is here to be considered as one value of the vector p, corresponding to the 
 value po of p. 
 
 (32.) And it is evident, by the law (12.) of transition from the wave to the in- 
 dex surface, that if + vo, + vi be the Lines of Single Normal Slowness, or the four 
 values of p which are analogous* to the four cusp-rags + po, + pi (23.), then, at the 
 end of each such new line, there must be a Conical Cusp on the Index Surface, ana- 
 logous to the Conical Cusp (31.) on the Wave, which is in like manner one of four 
 such cusps. 
 
 (33.) In forming and applying the equation above indicated (31.), of the tan- 
 gent cone to the wave at a cusp, the following transformations are useful : 
 
 XCIV. . . - (p + p')-2 = - p-2 ( 1 + p-ip')-i (1 + p'p-i)-i 
 
 = - p-2 + 2p-2Sp'p-i + p-^p'^ - 4:p-e(Spp')^ + &c., 
 
 the terms not written being of the third and higher dimensions in p', and p, p' being 
 ang two vectors such that Tp'< Tp (comp. 421, (4.)) ; also, without neglecting ang 
 terms, the self-conjugate property of ^ gives (comp. 362), 
 
 XCV. . . S(p + p') (p + p') = Sp^p + 2Sp>p + Sp'^p', 
 
 with an analogous transformation for the corresponding expression in 0-' ; while the 
 cubic LX. in f , or LXV. in ^-', gives for an arhitrarg p, 
 
 XCVI. . . ^(^ + a-2) (^ + c-2)p = - 6-2 (^ ^ „-2) (^ + c-2)p^ 
 XCVIL . . 0-«(^ + a-2) (0 + c-2)p =- 62(^ + a-2) (^ + c-a) p ; 
 
 and therefore, among other transformations of the same kind, 
 XCVIII. . . (0 +a-2)2 (0 + c-2)2p = (a-2 - 6-2) (c 2 - 6-2) (0 + a-2) (* + 6-2) p. 
 
 * This word " analogous" is here more proper than " corresponding" ; in fact, 
 the cusps on each of the two surfaces will soon be seen to correspond to circles on 
 the other, in virtue of the law ofreciprocitg. 
 
CHAP. 111.] CIRCULAR RIDGES ON WAVE AKD INDEX SURF. 745 
 
 We have also for a cusp, the values, 
 
 XCIX. . .<ppo = fio- (a-2 + c-2) po ; XCIX'. . . 1 + Spo^po = {(i-^ + c-2) 52 
 
 C. . . juo^ = ff-<c-4Spo0"Vo = a-H^c-^ - (a-2 ^. ^-2). 
 (34.) In this "way the eqvation of the tangent cone is easily found to take the 
 form, 
 
 CI. . . O=6*Sp'(0 + O (^+c-2)p'-4SpVoSpVo 
 and to give, by operating with Dp^ (comp. (10.) (26.) (31.)), 
 
 CII. . . xfi = U{(p+a-^) (^ + c-2)p'- 2poSp>o - 2/[ioSp'po, 
 the scalar coefficient x being determined, for each direction of the tangent p' to the 
 wave at the cusp, by the condition I., which here becomes (31.), 
 
 cm.. . S/ipo=S/«opo = -l; 
 also, by CII., &c,, we have after some slight reductions, 
 
 CIV. . . xS/ipo= 2(62Sp>o + Sp'po) -, 
 
 CV. . . a;S/xjMo = 2(Sp'/io-fto^Sp'po); 
 
 CVI. . . 0:2^2 = 4(^2^^24. i-)Sp'poSp>o + 4 (poSp>o + i"oSp'po) ■ 
 
 = - 462 (Sp>o)2 + 4 (&2//o2 - 1) Sp'poSpVo + 4/io2 (Sp'po)2 ; 
 
 but this last expression is equal, by CIV. CV., to - x^SfxpoSfifio \ the equation of 
 the cone of perpendiculars, let fall from the wave-centre o on the tangent planea at 
 the cusp, takes then this very simple form, 
 
 CVII. . . /i2 + S/fpoS/i/io = 0; 
 
 60 that this cone of the second degree has the two vectors po and /iq at once for sides 
 and cgclie norinals (comp. 406, (7.)); and it is cut, by ihQ plane CIIL, in a circle, 
 of which the diameter is, 
 
 CVIII. . . T(/io + po-0 = (T^o^ - 6-«)i = h (6-2 » e?-2)5 (c-2 - 6-2)». 
 and therefore subtends, at the centre o, and in the plane of ac, the angle, 
 
 CIX. ., l'^ = tan-i . 62 (6-2 _ «-2)i (c-s _ 6-2)i. 
 po ^ 
 
 (35.) And by combining the equations CIIL CVII., we see that this circle (34.) 
 
 is a small circle of the sphere, 
 
 ex. . . /i2 = SnnQ, or CX'. . . S/t- Vo = 1 
 
 which passes through the wave-centre, and has the vector hq for a diameter, passing 
 also through the extremity of the vector - po-^. 
 
 (36.) This circle is, by III., a curve of contact of ihQ plane CIII. with the sur- 
 face of which p, is the vector, because every vector p of the curve corresponds, by 
 (31.), to the one vector po of the wave ; it is therefore one of Four Circular Ridges on 
 the Index Surface, the three others having equal diameters, and corresponding to 
 the three remaining cusp-rays, — po, pi, — pi (23.); and there are, in like manner, 
 Four Circular Ridges on the Wave, along which it is touched hy the four planes, 
 
 CXI. . . Spj/o = -l, Spvo = + l, Spvi = -1, Spn = + 1, 
 ± vo, ± J^i being the four lines introduced in (32.) ; also the common length of the 
 diameters, of these four circles on the wave, is (comp. CVIII.), 
 
 CXII. . . T((To + vq-^) = (T(To2 -62)i = 6-i(a2_ 62)i (ja _ c2)l, 
 where CXIII. . . cto = - a'^c^<pvQ, CXIV. . . TvQ=h-\ and CXV. . . Svo<^o = - 1 ; 
 
 5 c 
 
746 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 finally, - vq'^ and (Tq are the two values* of p, in tlie plane of ac, for the first of the 
 four new circles : and the angle between these two vectorf?, or the angle which the 
 diameter of the circle, in the same plane, subtends at the wave-centre, is (comp. 
 
 cixo, 
 
 CXVI. . . Z ^=tan-i.6-2(«2-62)i (b^-c^)i. 
 
 (37.) In the recent calculations (33.) (34.), the circle of contact (36.) on the 
 index surface was deduced /row the tangent cone at a wave-cusp, as a section of a 
 certain cone of normals CVII. to that tangent cone CI., made by the plane CIII. ; 
 but the following is a simpler, and perhaps more elegant method, of deducing and 
 representing the same circle hy means of its vector equation (comp. 392, IX. &c.), and 
 without assuming any previous knowledge of the character, or even the existence, of 
 that conical wave-cusp. 
 
 (38.) Ir^ general, by eliminating the auxiliary vector v between XX. and XXIII., 
 we arrive at the following equation, 
 
 CXVII...(0-,o-2)(;x + p-i)-i = p-i; 
 
 which holds good for every pair of corresponding vectors p and p, of the wave and 
 index surface. And in general, this relation is sufficient, to determine the index- 
 rector p, when the ray- vector p is given : because {<p + e)'iO is generally = 0. 
 
 (39.) But when e is a root of the equation E=0, with the signification LXI. of 
 E, then, by the formula of inversion LXIL, the symbol (<p 4 e)"'0 takes the indetermi- 
 nate form ^ ; and therefore, for every point of each of the three circles (a) (b) (c) of 
 
 the wave, the fo7'mula CXY 11. fails to determine p : although it is only at a cusp 
 (23.), that the value of /i becomes in fact indeterminate (comp. (27.) (28.) (29.) 
 (30.) (31.)). 
 
 (40.) At such a cusp (p = po), the equation CXVI I. takes the symbolical form, 
 
 CXVIII. . . (/x + po-i)-* ={<!> + 6-2)-Vo"' = Ca^o 4- po"')-^ + (^ + b-^y ; 
 
 /io retaining its recent signification XCI., and the symbol (0 +6 2)-io denoting any 
 vector oi the form y (3, if /3 be the mean vector semiaxis of the generating ellipsoid 
 XXIX., so that 
 
 CXIX. . . Si30i3 = 1, (0 + h-'i-) (5 = 0, T/3 = 6. 
 
 (41.) "Writing then for abridgment (comp. XX.), 
 
 CXX...Vo=-'{fio + Po-'y\ 
 the Vector Equation of the Index Ridge (36.) is obtained under the sufficiently 
 simple form, 
 
 CXXI. . . V/3 (/i + po-^-' + V/3vo = ; 
 and this equation does in fact represent a Circle (comp. 296, (7.)), which is easily 
 
 * It is not difficult to show that these are the vectors of two points, in which the 
 circle and ellipse (6), wherein the wave is cut by the plane of ac, are touched by a 
 common tangent. 
 
CHAP. III.] DIAM. OF CIRCLES, DIRECTIONS OF VIBRATION. 747 
 
 proved to be the same as the circular section (34.), of the cone CVII. by the plant 
 cm. ; its diameter CVIII. being thus found anew under the form, 
 
 CXXII. . . Tyo-i = *TV\\' = h (b'^ - a 2)1 (c-2 _ h-2y_^ 
 with the significations (24.) (25.) of \, X'; in fact we have now the expressions, 
 
 CXXII r. . . po= *UX, vo = po-'(V\\')-', 
 with the verification, that 
 
 CXXIV. . . (^ + 6-2)vo = XS\'uo + X'S\uo = i~^U\=-|Oo"'- 
 (42.) And by a precisely similar analysis, we have first the new general rela- 
 tion (comp. CXVII.), for any two correspondijig vectors, p and ;«, 
 
 cxxv. . . (r' -/^-') (p + /i-O' =/*-^ ; 
 
 aoid then in particular {comp. CXVIII.), for j« = vq, 
 
 CXXVI. . . (/o + i'o-0"^ = (r^ + *0"^»^o-' = ('yo + vo-0"*+Cr' + ^^)'^0; 
 so that finally, if we write for abridgment (comp. XLI. CXX.), 
 
 CXXVII. ..wo=-(<ro + i'o-0-^ 
 the Vector Equation of a Wave-Ridge is found (comp. CXXI.) to be, 
 CXXVIIL . . V/3(p+ J/0-0-' +"V/3wo = 0, 
 
 ^ being still (as in CXIX.) the mean vector semiaxis of the generating ellipsoid 
 i8p(}>p = 1) : and the diameter CXII., of this circle of contact of the wave with the 
 first plane CXI., is thus found anew (comp. CXXII.), without any reference to cusps 
 (37.), as the value of Two"*- 
 
 (43.) Several of the foregoing results may be illustrated, by a new use of the 
 last diagram (13.). Thus if we suppose, in that Fig. 89, that we have the values, 
 
 CXXIX. . . op = )Ooj «Q = iWo> ou = Vo, whence CXXX. . . op' = -po"S &c., 
 
 then the index-ridge (36.), corresponding to the wave-cusp v (23.), will be the cir- 
 cle which has p'q for diameter, in a plane perpendicular to the plane of the Figure, 
 which is here the plane of ac ; the cone of normals fi (34.), to the tangent cone to the 
 tvave at p, has the wave-centre o for its vertex, and rests on the last-mentioned circle, 
 having also for a subcontrary section that second circle which has pq' for diameter, 
 and has its plane in like manner at right angles to the plane of poq ; also if k and s 
 be any two points on the second and first circles, such thatOKS is a right line, namely 
 a side p, of the cone here considered, then the chord pr of the second circle is per- 
 pendicular to this last line, and has the direction of the vibration dp, -which answers 
 here to the two vectors p (= Pq) and p : because (comp. (14.)) this chord is perpen- 
 dicular to /u, but complanar with p and p. 
 
 (44.) Again, to illustrate the theory of the wave-ridge (36.), which corresponds to 
 a cusp (32.) on the index-surface, we may suppose that this cusp is at the point Q 
 in Fig. 89, writing now (instead of CXXIX. CXXX.), 
 
 CXXXI. . . OQ = Vq, op = o-Q, ow = Wo, oq' = - Vo"'» &c. ; 
 
 for then the ridge (or circle of contact) on the wave will coincide with the second circle 
 (43.), and the cone of rays p from o, which rests upon this circle, will have the fr^t 
 circle (43.) for a sub-contrary section : also the vibration, at any point r of the wave- 
 
748 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 ridge, will have the direction of the chord rq', for reasons of the same kind as be- 
 fore. 
 
 (4.5.) Let K and k' denote the bisecting points of the lines pq' and qp', in the 
 same Fig. 89 ; then k' is the centre of the index ridge, in the case (43.) ; while, in 
 the case (44.), k is the centre of the wave-ridge. 
 
 (46.) In the^rs^ of these two cases, the point k is not the centre of ant/ ridge, 
 on either wave or index-surface ; but it is the centre of a certain suhcontrary and 
 circular section (43.), of the cone with o for vertex which rests upon an index-ridge ; 
 and each of its chords PR has the direction (43.) of a vibration dpQ, at the wave-cusp 
 P corresponding : so that this cusp-vibration revolves, in the plane of the circle last 
 mentioned, with exactly half the angular velocity of the revolving radius kr. 
 
 (47.) And every one of those cusp-vibrations dpo, which (as we have seen) are 
 all situated in one plane, namely in the tangent plane at the cusp v to the ellipsoid 
 (6) of (28.), has (as by (14.) it ought to have) the direction of the projection of the 
 cusp-ray po, on some tangent plane to the tangent cone to the wave, at that point P : 
 to the determination of which last cone, by some new methods, we purpose shortly 
 to return. 
 
 (48.) In the second of the two cases (45.), namely in the case (44.), pq' is a 
 diameter of a wave-ridge, with K for the centre of that circle, and with a plane (per- 
 pendicular to that of the Figure) which touches the wave at every point of the same 
 circular ridge ; and the vibration, at any such point k, has been seen to have the 
 direction of the chord rq', which is in fact the projection (14.) of the ray or upon 
 the tangent plane at R to the wave. 
 
 (49.) And we see that, in passing from one point to another of this wave-ridge, 
 the vibration rq' revolves (comp. (46.)) round the Jixed point q' of that circle, 
 namely round the^o< of the perpendicular from o on the ridge-plane, with (again) 
 half the angular velocity of the revolving radius KR. 
 
 (50.) These laws of the two sets of vibrations, at a cusp and at a ridge upon the 
 wave, are intimately connected with the two conical polarizations, which accompany 
 the two conical refractions,* external and internal, in a biaxal crystal ; because, on 
 the one hand, the theoretical deduction of those two refractions is associated with, 
 and was in fact accomplished by, the consideration of those cusps and ridges : while, 
 on the other hand, in the theory of Fresnel, the vibration is always perpendicular 
 
 * The writer's anticipation, from theory, of the two Conical Refractions, was 
 announced at a general meeting of the Royal Irish Academy, on the 22nd of Octo- 
 ber, 1832, in the course of a final reading of that Third Supplement on Systems of 
 Rays, which has been cited in a former Note (p. 737). The verj' elegant experi- 
 ments, by which his friend, the Rev. Humphrey Lloyd, succeeded shortly afterwards 
 in exhibiting the expected results, are detailed in a Paper On the Phenomena pre- 
 sented by Light, in its passage along the Axes of Biaxal Crystals, which was read 
 before the same Academy on the 28th of January, 1833, and is published in the 
 same First Part of Volume XVIL of their Transactions. Dr. Lloyd has also given 
 an account of the same phenomena, in a separate work since published, under the 
 title of an Elementary Treatise on the Wave Theory of Light (London, Longman 
 and Co., 1857, Chapter XL). 
 
CHAP. III.] FORMS OF EQUATION OF CUSP-CONE. 749 
 
 to the plane of polarization. But into the details of such investigations, we cannot 
 enter here. 
 
 (51.) It is not difficult to show, by decomposing p' into two other vectors, pi 
 and p2, perpendicular and parallel to the plane of «<?, that we have the general trans- 
 formation, for ant/ vector p', 
 
 CXXXII. . . b^Sp' (0 + rt-2) (0 + c-2)p'= (S/ioPop> ; 
 
 the equation CI. of the tangent cone at a ivave-cusp may therefore be thus more 
 briefly written, 
 
 CXXXIII. . . (S/io|Oop')'=4Spon>'S)«op'; 
 
 and under this form, the cone in question is easily proved to be the locus of the nor- 
 mals from the cusp, to that olher cone CVII., which has /i for a side, and the wave- 
 centre o for its vertex : while the same cone CVII. is now seen, more easily than in 
 (34,), to be reciprocally the locus of the perpendiculars from o on the tangent planes 
 to the wave at the cusp, in virtue of the new equation CXXXIII., of the tangent 
 cone at that point. 
 
 (52.) Another form of the equation of the cusp-cone may be obtained as fol- 
 lows. The equation LXXIV. of the wave may be thus modified (comp. LXXVI.), 
 by the introduction of the two non- opposite cusp-rays, po = 6U\ (CXXIII.), and 
 pi = 6U\': 
 
 CXXXIV. . . 2«262c2 + (fl3 + c2) 62p2 + (^2 _ g?) Spop . Spip 
 
 = + («2_c2)TVpop.TVpip; 
 where it will be found that the first member vanishes, as well as the second, at the 
 cusp for which p = po. 
 
 (53.) Changing then p to po + p', and retaining only terms o? first dimension in 
 p' (comp. (31.)), we find an equation oi unifocal form (comp. 369, &c.), 
 
 CXXXV. . . Si3op' = + TVaop', or CXXXV. . . {Ya^p'y + (S|3op')2 = ; 
 
 with the two constant vectors, 
 
 CXXXVI. . . ao= (6-2-a-2)j (c-2-6-2)^po ; CXXXVI'. . . (3o = fio - Po'^ i 
 
 and this equation CXXXV. or CXXXV. represents the tangent cone, with p' for 
 side, S/3op' being positive for one sheet, but negative for the other. 
 
 (54.) As regards the calculations which conduct to the recent expressions for 
 oo, /3o, it may be sufficient here to observe that those expressions are found to give 
 the equations, 
 
 CXXXVII. . . 2fl262c2a^= (a2_c2) poTVpoPi; 
 
 CXXXVir. . . 2rt262c2/3o = 2 («2 + c*) 62po + («2 - c«) (poSpopi - 6 Vi) ; 
 
 and that, in deducing these, we employ the values, 
 
 62S\X' „„ 52TVX\" 
 
 CXXXVIII. ..Spopi=^;^, TVpopi = -^;^^'; 
 
 together with the formula XCIX., and the following, . 
 
 CXXXIX. . . ^ (po - pi) = - «"^ (Po - pO ; ^ (Po + pi) = - c"2 (po + pi). 
 (55.) It is not difficult to show that the equation CXXXV. or CXXXV'., of the 
 tangent cone at a cusp, can be transformed into the equation CXXXIII.; but it 
 
750 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 may be more interesting to assign here a geometrical interpretation^ or construction, 
 of the unifocal form last found (63.). 
 
 (56.) Retaining then, for a moment, the use made in (43.) of Fig. 89, as serv- 
 ing to illustrate the case of a wave-cusp at p, with the signification (45.) of the new 
 point k' as bisecting the line p'q, or as being the centre of the index-ridge ; and 
 conceiving a parallel cone, with o instead of p for vertex, and with a variable side 
 OT = p' ; then the cusp-ray op (= po II «o) is a. focal line of the new cone, and the 
 line ok' (= i {/Xq - po'O = il^o) ^^ ^^^ directive normal, or the normal to the director 
 plane corresponding ; and the formula CXXXV. is found to conduct to the follow- 
 ing) 
 
 CXL. . . cos k'ot = sin pok' sin pot, 
 
 which may be called a Geometrical Equation of the Cusp- Cone: or (more im- 
 mediately) of that Parallel Cone, which has (as above) its vertex removed to the 
 wave- centre o. 
 
 (57.) Verifications of CXL. may be obtained, by supposing the side ot to be 
 one of the two right lines, pi', pa', in which the cone is cut by the plane of the figure 
 (or of ac) ; that is, by assuming either 
 
 CXLI. . . OT = pi' = juo + po"' II ou, or CXLI'. . . ot = pa' = po + Ho'^ \\ ow ; 
 
 and it is easy to show, not only that these two sides, ov, ow, make (as in Fig. 89) 
 an obtuse angle with each other, but also that they belong to one common sheet, of 
 the cone here considered, because each makes an acute angle with the directive nor- 
 mal ok'. 
 
 (58.) Another way of arriving at this result, is to observe that the equation 
 CXXXIII. takes easily the rectangular form, 
 
 CXLII. . . (Sp'(U^o+Upo))2 = (Sp'(Ui«o-Upo))2-f T;uoPo(Sp'U/ioPo)2; 
 the internal axis of the cusp-cone has therefore the direction of U/io+ Upo, that ia, 
 of the internal bisector of the angle poq, while the external bisector of the same 
 angle is one of the two external axes, and the <AtrJaxisis perpendicular to the />/awe 
 ofpo)f*o> l^ut Sp'(U//o + Upo)<0, whether p' = pi', or=p2': and therefore these 
 two sides, pi and pa', belong (as above) to one sheet, because each is inclined at an 
 acute angle to the internal axis VfiQ + Upo- 
 
 (59.) It is easy to see that the second focal line of the parallel cone (5G.) is /Mq, 
 or OQ ; and that the second directive normal corresponding is the line ok (45.), in 
 the same Fig. 89 ; whence may be derived (comp. CXL.) this second geometrical 
 equation of the cone at o, 
 
 CXLIIL . . cosKOT = sinKOQ sin qot; with koq = pok'. 
 
 (60.) And finally, as a bifocal but still geometrical form of the equation of the 
 cusp-cone, with its vertex thus transferred to o, we maj"- write, 
 
 CXLIV. . . z POT -h z. QOT = const. = I wou. 
 
 (61.) Ant/ legitimate form of anyone of the four functions, ^p, 0-ip, Sp^p, 
 Sp0"ip, when treated by rules of the present Calculus which have been already 
 stated and exemplified, not only conducts to the connected forms of the three other 
 functions of the group, but also gives the corresponding forma of equation, of the 
 JFave and the Index- Surface. 
 
CHAP. III.1 RIDGES RESUMED, ORTHOG. TO LINES OF VIB. 751 
 
 (62.) For instance, -with the significations (32.) of vq and vi, the scalar func- 
 tion SjO0-'p, which is = 1 in the equation XXX. of the Reciprocal Ellipsoid (9.), 
 may be expressed by the following cyclic form^ with vo, v\ for the cyclic normals of 
 that ellipsoid, 
 
 CXLV. . . S/)0-ip = - 62p^ + (^^ - c2)62Sro|oSvip ; 
 
 reciprocating which (comp. 361), we are led to a bifocal form of the function 
 Sp0p, which function was made = 1 in the equation XXIX. of the Generating Ellip- 
 soid (8.), and is now expressed by this other equation (comp. 360, 407), 
 
 CXLVI. . . ___^(Sp^p + 6>2) = (Sx.op)2 + (S.'ip)2-2 -^SvopSrip; 
 
 vo, v\ being here the two (real) /ocaZ lines of the same ellipsoid (8.), or of its (ima- 
 ginary) asymptotic cone. 
 
 (63.) Substituting then these forms (62.), of Sp^p and Sp^'^p, in the equation 
 LXIII., we find (after a few reductions) this new form of the Equation of the 
 Wave : 
 
 CXLVII. . . (2p2 - («3 _ c2) SvopSj/ip + a2 -f c2)2 = (a2 _ c2)2 { 1 - (Svop)2} 
 
 {i-(Svip)M; 
 
 whence it follows at once, that each of the four planes CXI. touches the wave, along 
 the circle in which it cuts the quadric, with vo, vi for cyclic normals, which is found 
 by equating to zero the expression squared in the first member of CXLVII. For 
 example, the j^rs^ plane CXI. touches the wave along that circle, or wave-ridge, 
 of which on this plan the equations are, 
 
 CXLVIII. . . Sj/oP +1 = 0, 2p2 + («2 - c«) Svip - (a^ -f c2) Svop = ; 
 and because 
 
 CXLIX. . . (vo + vi) = - a-2 (vo + vi), (p (x/o -vi)=- c-2 (v^ - vi), 
 and therefore, with the value CXIII. of Cq, 
 
 CL. . . (To = - a'^c^VQ = J ((a2 -f c"-) vq - (»2 - c^)vi), 
 the second equation CXLVIII. represents (comp. CX.) the diacentric sphere, 
 
 CLL . . p2 = S(rop, or CLI'. . . S(Top-» = 1, 
 which passes through the wave-centre o, and of which the ridge here considered is a 
 section. The diameter of that ridge may thus be shown again to have the value 
 CXII. ; and it may be observed that the circle is a section also of the cone, 
 
 CLII. . . Sj/opSo-op = - p2, or CLir. . . SvopSo-op-^ = - 1. 
 
 (64.) It was shown in (17.) that the vibration Sp, at any point of the wave- 
 surface, or at the end of any ray p, is perpendicular to (p-^p, as well as to p. by II. ; 
 and is therefore tangential to the variable ellipsoid LIX., as well as to the wave itself. 
 Hence it is easy to infer, that this vibration must have generally the direction of the 
 auxiliary vector w, because not only S/iw = 0, by XXXIX., but also Suxp-^p 
 = Sp(p-^(o = Spv = 0, by XXII. and XXXVII. Indeed, this parallelism of Sp to to 
 results at once by XXXVIL from XII. 
 
 (65.) If then we denote by d'p an infinitesimal vector, such as pdp, which is tan- 
 gential to the wave, but perpendicular to the vibration dp, the parallelism dp \\ (a 
 will give, 
 
752 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 CLIII. . . d'p = ftSp II /ifc> -i- p, because CLIII'. . . Sp/tw = 0; 
 whence CLIV. . . SpS'p = 0, S'Tp = 0, or CLV. . . Tp = r= const., 
 
 for this new direction S'p of motion upon the wave. 
 
 (Q6.) And thus (or otherwise) it may be shown, that the Orthogonal Trajecto- 
 ries to the Lines of Vibration (17.) are the curves in which the Wave is cut by 
 Concentric Spheres, such as CLV. ; that is, by the spheres p2 ^ ;.2 _ q^ Jjj -y^hich the 
 radius r is constant for any one, but varies in passing from one to another. 
 
 (67.) The spherical curves (r), which are thus orthogonal to what we have 
 called the lines (K) of vibration, are sphero-conics on the wave ; either because each 
 such curve (r) is, by XXVIIL, situated on a concentric and quadric cone, namely, 
 
 CLVL .. O = Sp(0+r-2)-ip; 
 or because, by XXVII., it is on this other concentric quadric, 
 
 CLVIL ..-l = Sp(0-l+r2)-lp• 
 (68.) It is easy to prove (comp. LXXV.)) that, for any real point of the wave, 
 r^ cannot be less than c2, nor greater than fl2 j and that the squares of the scalar 
 semiaxes of the new quadric CLVII. are, in algebraically ascending order, r'^ —a^, 
 ,•2 _ 52^ r2 - c2 ; so that this surface is generally an ht/perboloid, with one sheet or 
 with two, according as r > or < 6. 
 
 (69.) And we see, at the same time, that the conjugate hyperboloid, 
 
 CL VIII . . . + 1 = Sp (0- » + r2)-i p, 
 
 which has two sheets or one, in the same two cases, r>h, r<b, and has (in descend- 
 ing order) the values, 
 
 CLIX. . . ^2 _ r-i, 62 _ y2^ c2 - r«, 
 
 for the squares of its scalar semiaxes, is confocal with the generating ellipsoid 
 XXIX. : so that the quadric CLVII. itself is the conjugate of such a confocal. 
 
 (70). To form a distinct conception (comp. (67.)) of the course of a curve (r) 
 upon the wave, it may be convenient to distinguish the^ve following cases : 
 
 CLX. . . (a)..r = «; {(3) . .r<a, >h', (y)..r = ft; (J) . . r < 6, >c; (e)..r = c. 
 
 (71.) In each of the three cases (a) (y) (t), the conic (r) becomes a circle, in 
 one or other of the three principal planes : namely the circle (a), for the case (a) ; 
 (b) for (y) ; and (c) for (c). 
 
 (72.) In the case (j3), the curve (r) is one of double curvature, and consists of 
 two closed ovals, opposite to each other on the wave, and separated by the plane (a), 
 which plane is not (really) met, in any point, by the complete sphero-conic (r) ; and 
 each separate oval crosses the plane {V) perpendicidarly, in two (real) points of the 
 ellipse (b), which are external to the circle (6) : while the same owaZ crosses a/so the 
 plane (c) at right angles, in some two real points of the ellipse (c). 
 
 (73.) Finally, in the remaining case {d), the ovals are separated by the plane 
 (c), and each crosses the plane (b) at right angles, in two points of the ellipse (6), 
 which are interior to the circle (6) ; crossing also perpendicularly the plane (a), in 
 two points of the ellipse (a). 
 
 (74.) Analogous remarks apply to the lines of vibration (h); which are either 
 the ellipses (a) (6) (c), or else orthogonals to the circles (a) (6) (c), and generally 
 to the sphero-conics (r), as appears easily from foregoing results. 
 
CHAP. III.] fresnel's theory. 753 
 
 (75.) It may be here observed, that when we only know the direction (U/x), 
 but not the length (T^), of an index-vector fi, so that we have two parallel tangent 
 planes to the wat^e, at one common side of the centre, the directions of the vibrations 
 dp differ generally for these two planes, according to a law which it is easy to as- 
 sign as follows. 
 
 (76.) The second values of fi and Sp being denoted by p,^ and dp,, we have, by 
 the equation IX. of the index- surface, tliese two other equations : 
 
 CLxi. . . = S/i (0-' - /x-2)-i p ; cLxr. ..o=Sp (0-1 - pr^y^p -, 
 
 of which the difference gives, suppressing the factor p^'^ — p.'^, 
 
 CLXIL ..0 = Sp (0-' -/i,-2)-i (0-1 _^-2)-i^ ; 
 or CLXII'. . . = S (0-1 - p-^yi p (0-1 - p-^y^p, 
 
 because (0"^ — /^t/^)"^ as a functional operator, is self- conjugate, so that p may be 
 transferred from one side of it to the other ; just as, U v = ^p be such a self-conju- 
 gate function of p, then v''' = Sp(pp = Sp<pv = Sp02p, &c. 
 (77.) But, by VIII., we have the parallelisms, 
 
 CLXiii. . . dp II (0-1 - p-^y^p ; CLXiir. . . dp, II (0-1 - /i/2)-v ; 
 
 hence, by CLXII'., we have the very simple relation, 
 CLXIV. . .Sdpdp=0; 
 
 that is, the two vibrations, in the ttvo parallel planes, are mutually rectangular. 
 
 (78.) The following quite different method has however the advantage of not 
 only proving anew this known relation of rectangidarity, but also of assigning qua- 
 ternion expressions for the two directions separately : and, at the same time, that 
 of leading easily to what appears to be a new and elegant Geometrical Construction, 
 simpler in some respects than the known one, which can indeed be deduced from it. 
 
 (79.) By the first principles of Fresnel's theory (comp. (3.)), the vibration (dp^, 
 on any one tangent plane to the wave, is situated in the normal plane (through p"), 
 which contains the direction (dt) of the elastic force ; that is to say, we have the 
 Equation of Complanarity, 
 
 CLXV. . . S/i ^p ^f = 0. 
 
 (80.) We have then, by II. and V., the system of the two equations, 
 
 CLXVI. . . ^pdp = 0, Spdp(p-^dp = ; 
 
 comparing which with the equations of the same form, 
 
 SvT = 0, Sj/r0r = 0, 410, V. VI. 
 
 we derive at once the following Construction, lohich may also be expressed as a The- 
 orem : — 
 
 ^^ At cither of the two points Q of the Reciprocal Ellipsoid XXX., the tangent 
 plane at which is parallel to that at the given point P of the Wave, the tangents to 
 the Lines of Curvature on the Ellipsoid are parallel to the tangents to the Lines of 
 Vibration on the Wave ;" namely, to one at that given point p itself, and to another 
 at the other point p', on the same side of the centre, at which the tangent plane is 
 parallel to each of the two others above mentioned. 
 
 5 D 
 
754 ELEMENTS OF QUATRRNIONS. [bOOK III. 
 
 (81.) Thus for each of the two points p, p' the line of vibration is parallel to 
 one of the lines of curvature at Q ; and it is evident, from what precedes, that the 
 other of these last lines has the direction of the corresponding Orthogonal (66.) at 
 p or p' : nor is there any danger of confusion. 
 
 (82.) As regards quaternion expressions, for the two vibrations on a given wave- 
 front, the sub-article, 410, (8.), with notations suitably modified, shows by its for- 
 mulae XIX. XXII. that we have here the equations, 
 
 CLXVII. . . O^SfiSpvoSp VI 
 
 = S/x dp vo Si'i Sp + Sp, dp vi Sj'o fV» 
 and CXVIII. . . ^p 11 VYfir^ ± UV/nn, 
 
 if Vo, vi be, as in earlier formulae of the present Series 422, the ci/clic normals of the 
 reciprocal ellipsoid, which are often called the Optic Axes of the Crystal. 
 
 (83.) And hence ma}' be deduced the known construction, namely, that " for 
 any given direction of wave-front, the two planes of polarization, perpendicular 
 respectively to the two vibrations in Fresnel's theory, bisect the two supplementary 
 and diedral angles, which the two optic axes subtend at the normal to the front :" 
 or "that these planes of polarization bisect, internally and externally, the angle be- 
 tween the two planes, fxVQ and pv\. 
 
 (84.) It may not be irrelevant here to remark, that if fi and ju, be any two in- 
 dex-vectors, which have (as in (76.)) i\\Q same direction, but not the same length, the 
 equation LXIV. enables us to establish the two converse relations: 
 
 CLXIX. . . abcTfx^ = (S/i0iLt)-i ; CLXIX'. . . abcTfi = (Sfi,iPfi,)X 
 
 (85.) Either by changing a, b, c, 0, p, to a'^, b'^, c^, (p~\ p, or by treating the 
 form LXIII., in (19.), of the Equation of the Wave, as we have just treated tlfe 
 form LXIV., of the equation o( Index Surface, in the same sub-article (19.), we see 
 that if p and p, be any two condirectional rays (Up, = Up), then, 
 
 CLXX. . . (a6c)-iTp, = {^p^'^p) K or, abcTp,-^ = (Sp0- V)^ ; 
 and CLXX'. . . («6c)-iTp = S {p,r^p,y', or, a&cTp-i = (Sp^0-'pj^. 
 
 (86.) A somewhat interesting geometrical consequence may be deduced from 
 these last formulae, when combined with the equation LIX. of that variable ellipsoid, 
 Sp0"ip= h^, which cuts the wave in a line of vibration (A). For if we introduce this 
 symbol h* for Sp0-'p, and write r, instead of Tp^ to denote the length of the second 
 ray p, the first equation CLXX. will take this simple form, 
 
 CLXXl. . .r^ = abch-2, 
 
 which shows at once that r, and // are together constant, or together variable ; and 
 therefore, that "a Line of Vibration on one Sheet of the Wave is projected into an 
 Orthogonal Trajectory to all such Lines on the other Sheet, and conversely the latter 
 into the former, byJLhe Vectors p of the Wave :" so that one of these ^«»o curves would 
 appear to be superposed upon the other, to an eye placed at the Wave- Centre o. 
 
 (87.) The visual cone, here conceived, is represented by the equation CLVI., 
 with some constant value of r ; and as being a surface of the seconrf degree, it ought 
 to cut the wave, which is one of the fotirth, in some curve of the eighth degree ; or in 
 some system of curves, ^^ hich have the product of their dimensions equal to eight. 
 
CHAP. III.] fresnkl's theory. 755 
 
 Accordingly we now see that the complete intersection, of the cone CLVI. with the 
 wave, consists of two curves, each of the fourth degree ; one of these being, as in 
 (67.), a complete sphero-conic (r), and the other a complete line of vibration (A): 
 a new geometrical connexion being thus established between these two quartic 
 curves. 
 
 (88.) As additional verifications, we may regard the three prijicipal planes, as 
 limits of the cutting coves ; for then, in the plane (a) for instance, the circle («) and 
 the ellipse («), which are (in a sense) projections of each other, and of which the 
 latter has been seen to be a line of vibration, are represented respectively by the two 
 
 equations, 
 
 CLXXII. . . r = a, and CLXXII'. . . 6c = A2, 
 
 in agreement with CLXXI. ; and similarly for the two other planes. 
 
 (89.) It was an early result of the quaternions, that an ellipsoid with its centre 
 at the origin might be adequately represented by the equation (comp. 281, XXIX., 
 or 282, XIX.), 
 
 CLXXIII. . . T (i/5 + pk) = k2- i2, if Ti > T/c ; 
 
 or, without ani/ restriction on the two vector constants, t, k, by this other equa- 
 tion, * 
 
 CLXXIir. . . T (tp + Pk)2 = (/c2 - t2)2. 
 
 (90.) Comparing this with Sptpp= 1, as the equation XXIX. of the Generating 
 Ellipsoid, we see that we are to satisfy, independently of p, or as an identity, the re- 
 lation (comp. 336) : 
 
 CLXXIV. . . (/e2 - i5)2 Sp^p = (ep + pk) {pi + Kp 
 
 = (i2-f k2) p + 2SipKp; 
 
 which is done by assuming (comp. again 336) this cyclic form for <p, 
 CLXXV. . . (k2 = t2)2 0p = (i2 + k2) p + 2V/cpt 
 
 = (t - K)2p + 2tS/cp + 2KStp ; 
 or as in (24.) comp. 359, III. IV., 
 
 0p =gp + VXpX; Sp^p =^p2 + SXpVp = 1 ; LXXII. LXXIII. 
 
 * This equation, CLXXIII', or CLXXII., which had been assigned by the 
 author as a form of the equation of an ellipsoid, has been selected by his friend 
 Professor Peter Guthrie Tait, now of Edinburgh, as the basis of an admirable 
 Paper, entitled : " Quaternion Investigations connected with Fresnel's Wave-Sur- 
 face," which appeared in the May number for 1865, of the Quarterly Journal of 
 Pare and Applied Mathematics ; and whicli the present writer can strongly re- 
 commend to the careful perusal of all quaternion students. Indeed, Professor Tait, 
 who has already published tracts on o^/ter applications of Quaternions, mathematical 
 and physical, including some on Electro-Dynamics, appears to the writer eminently 
 fitted to carry on, happily and usefully, this new branch of mathematical science: 
 an(ilikely to become in it, if the expression may be allowed, one of the chief succes- 
 sors to its inventor. 
 
756 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 •with expressions for the constants^, X, \', which give, hy LXXVI., the following 
 values fpr the scalar semiaxes,* 
 
 CLXXVI. . . a=Tt + T*c; b = ^l " '' ; c = Ti-T,f; 
 
 T(i-k) 
 
 whence conversely, 
 
 CLXXVII. ..Ti = ^±l T/c=^:i^; T (c - fc) = - ; &c. 
 
 (91.) Knowing thus the form CLXXV. of the function 0, which answers in the 
 present case to the given eqxiation CLXXIII. of the generating ellipsoid, there 
 would be no difficulty in carrying on the calculations, so as to reproduce, in connexion 
 with the two constants i, /c, all the preceding theorems and formulfe of the present 
 Series, respecting the Wave and the Index- Surface. But it may be more useful to 
 show briefly, before we conclude the Series, how we can pass from Quaternions to 
 Cartesian Co-ordinates, in any question or formula, of the kind lately considered. 
 
 (92.) The three italic letters, ijk, conceived to be connected by thQ four funda- 
 mental relations, 
 
 i2=ji = k^=ijk = -l, (A), 183, 
 
 were originally the only peculiar symbols of the present Calculus ; and although 
 they are not now so much used, as in the early practice of quaternions, because cer- 
 tain general signs of operation, such as S, V, T, U, K, have since been introduced, 
 yet they (the symbols ijk) may be supposed to be still familiar to a student, as links 
 between quaternions and co-ordinates. 
 
 (93.) We shall therefore merely write down here some leading expressions, of 
 which the meaning and utility seem likely to be at once perceived, especially after 
 the Calculations above performed in this Series. 
 
 (94.) The vector semiaxes of the generating ellipsoid being called a, (3, y (comp. 
 (40.) (42.)), we may write, 
 
 CLXXVIII. . . a = ia, (3 =jb, y = kc; 
 CLXXIX.. . 0p = a-iSa-ip + /3-iS/3-V + r'Sy-V = 2:a-iSa-V = - Si^-^^; 
 CLXXX. . . Sp<pp = 2 (Sa-V)2 = 2:a-2a;2 ; CLXXXI. . . Sp^-ip = ^a^x^ ; 
 CLXXXII. . . (0 + P = Sa (a-2 + e) Sa-^p ; 
 
 * The reader, at this stage, might perhaps usefully turn back to that Construc- 
 tion of the Ellipsoid, illustrated by Fig. 53 (p. 226), with the Remarks thereon, 
 which wei-e given in the few last Series of the Section II. i. 13, pages 223-233. It 
 will be seen there that the three vectors, i, k, i— k, of which the lengths are ex- 
 pressed by CLXXVII., are the three sides, cb, ca, Ab, of what may be called the 
 Gejierating Triangle x^c in the Figure; and that the deduction CLXXVI., of the 
 three semiaxes, abc,- from the two vector constants, i, k, with many connected 
 results, can be very simply exhibited by Geometry. The whole subject, of the equa- 
 tion T (tp + p/c) = k2 - 1^ of the ellipsoid, was very fully treated in the Lectures; 
 and the calculations may be made more general, by the transformations assigned in 
 the long but important Section III. ii. 6 of the present Elements, so that it seems 
 unnecessary to dwell more on it in this place. 
 
CHAP. III.] GEN. LAWS OF REFLEXION AND REFRACTION. 757 
 
 CLXXXIII. . . (^ + e)-ip = 2a (a -2 + e)-> Sa-ip ; 
 CLXXXIV. . . if r« = Tp2 = Sa:2, then v = r-2(^ + r-2)-ip 
 
 = r-2S-~I^=-2i^- 
 
 fl2a;3 ^2a;2 J2y2 c2y2 
 
 CLXXXV. . . for Wave, = Sou = S = + -^^— + ^ • 
 
 ' ^ r2_a3 r2_^2 r2-62^r2-c2' 
 
 or CLXXXVI. . . 1 = - S|Ow = - Sp^u = - Su? 
 
 a;2 a;2 v« 
 
 + -^^^0 + 
 
 r1 _ ^3 r2 - a2 y2 _ 62 
 
 and the Index-Surface may be treated similarly, or obtained from the Wave by 
 changing abc to their reciprocals. 
 
 423. As an eighth specimen of physical application we shall in- 
 vestigate, by quaternions, MacCullagh's Theorem of the Polar Plane,* 
 and some things therewith connected, for an important case of inci- 
 dence of polarized light on a biaxal crystal : namely, for what was 
 called by him the case of uniradial vibrations. 
 
 (1.) Let homogeneous light in air (or in a vacuum), with a velocityf taken for 
 unity, fall on a plane face of a doubly refracting crystal, with such a polarization 
 that only one refracted ray shall result ; let p, p', p" denote the vectors of ray-velo- 
 city of the incident, refracted, and reflected lights respectively, p having the direc- 
 tion of the incident ray, prolonged within the crystal, but p" that of the reflected 
 ray outside ; and let jx be the vector of wave-slowness, or the index-vector (comp. 
 422, (1.)), for the refracted light : these /o2<r vectors being all drawn from a given 
 point of incidence o, and p.\ like p', being within the crystal. 
 
 (2.) Then, by a?/J wave theories of light, translated into the present notation, 
 
 we have the equations, 
 
 L..p2 = S/p'=p"2 = ~l; 
 
 IL . . p" = - I'pv-i, with II'. . . v = ju' - p, 
 
 where v is a normal to the face ; whence also, 
 
 in...p" = pS^-2/S-^; 
 II - p "^ ix'-p 
 
 IV. . . p" + p = 2i, if IV'. . . t = v-^Yp.'p = v-^Yvp ; 
 
 and V. . . p"-p = -2i'Spi^-i=-2i/-iSpv; 
 
 * See pp. 39, 40 of the Paper by that great mathematical and physical philo- 
 sopher, '• On the Laws of Crystalline Refiexion and Refraction,'^ already referred 
 to in the Note to page 737 (Trans. R. L A., Vol. XVIIL, Part I.). 
 
 f Of course, by a suitable choice of the units of time and space, the velocities and 
 slownessses, here spoken of, may be represented by lines as short as may be thought 
 convenient. 
 
 "l These equations may be deduced, for example, from the principles of Huy- 
 ghens, as stated in his Tractatus de Limine (Opera reliqua, Amst, 1728). 
 
758 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 80 that the three vectors, p, /x', p'\ terminate on one right line, which is, perpendi- 
 cular to the face of the crystal : and the bisector of the angle between the first and 
 third of them, or between the incident and reflected rays, is the intersection i of the 
 plane of incidence with the same plane face. 
 
 (3.) Let r, r', r" be the vectors of vibration for the three rays p, p', p", con- 
 ceived to be drawn from their respective extremities ; then, by all* theories of tan- 
 gential vibration, we bave the equations, 
 
 VI. ..Spr = 0; VII. .. S/i'T' = 0; VIII. . . SpV'= ; 
 
 to which Mac Cullagh adds the supposition (o), that the vibration in the crystal is 
 perpendicular to the refracted ray : or, with the present symbols, that 
 
 IX. . . Sp'r' = Oj whence X. . . r'|| V/t'p', 
 
 the direction of the refracted vibration r being thus in general determined, when 
 those of the vectors p' and fx are given. 
 
 (4.) To deduce from r' the two other vibrations, r and r", Mac Cullagh as- 
 sumes, (6), the Principle of Equivalent Vibrations, expressed here by the formula, 
 
 XI. . . r-r'+r" = 0, 
 in virtue of which the <Aree vibrations are parallel to one common plane, and the re- 
 fracted vibration is the vector sum (or resultant) of the other two ; (c), the Principle 
 of the Vis Viva, by which the reflected and refracted lights are together equal to the 
 incident light, which is conceived to have caused them ; and (d), the Principle of 
 Constant Density of the Ether, whereby the masses of ether, disturbed by the three 
 lights, are simply proportional to their volumes : the two last hypothesesf being 
 here jointly expressed by the equation, 
 
 XII. . . Sv Cpt^ - p't'^ + p"t"2) = 0. 
 
 (5.) Eliminating p" and /' from XII. by V. and XL, r^ goes off; and we find, 
 
 with the help of I. and II'., the following linear equation in r, 
 
 r Svo' Sou' 
 
 XlIL..2S- = l4-^ = -^, if Xlir. .. v' = u'-p'; 
 
 r' Svp Spv '^ 
 
 a secoMC? such equation is obtained by eliminating p" and r" by III. and XL from 
 VIIL, and attending to I. VI. VII., namely, 
 
 XIV. . . 2Spj/S/r= (p2 -^'2) Spr'= - S/v'Spr' ; 
 and a third linear equation in r is given immediately by VI. 
 
 * The equations VI. VII. VIIL hold good, for instance, on Fresnel's principles; 
 but Fresnel's tangential vibration in the crystal has a direction perpendicular to that 
 adopted by Mac Cullagh. 
 
 t In the concluding Note (p. 74) to this Paper, Professor Mac Cullagh refers to 
 an elaborate Memoir by Professor Neumann, published in 1837 (in the Berlin Trans- 
 actions for 1835), as containing precisely the same system of hj'pothetical principles 
 respecting Light. But there was evidently a complete mutual independence, in the 
 researches of those two eminent men. Some remarks on this subject will be found 
 in the Proceedings of the R. I. A., Vol. I., pp. 232, 374, and Vol. IL, p. 96. 
 
CHAP. III.] MAC CULLAGH*S POLAR PLANE. 759 
 
 (6.) Solving then for r, by the rules of the present Calculus, this system of the 
 three linear and scalar equations VI. XIIL XIV., we find for the incident vibration 
 the following vector expression^* 
 
 XV. . . r = ^^§^' ; or XV'. . . 2rS|0v = r'Spv' - v'^pr' ; 
 Iiopv 
 
 and accordingly it may be verified by mere inspection, with the help of VII. and IX., 
 that this vector value of r satisfies the three scalar equations (5.). And when the 
 incident vibration has been thus deduced from the refracted vibration r', the reflected 
 vibration r" is at once given by the formula XL, or by the expression, 
 
 XVI.. . r"=r'-r; 
 
 (7.) The relation XV'. gives at once the equation ofcomplanarity, 
 
 XVII. . . Si/'rr' = 0, or the formula XVIII. . . / - p' |1 1 r, r' ; 
 
 if then a plane be anywhere so drawn, as to be parallel (4.) to the three vibrations 
 r, r', r", it will be parallel also to the line p! — p', which connects two correspond- 
 ing points^ on the wave and index surface in the crystal : but this is one form of 
 enunciation of Professor Mac Cullagh's Theorem of the Polar Plane^ which theorem 
 is thus deduced with great simplicity by quaternions, from the principles above sup- 
 posed. 
 
 .(8.) For example, if we suppose that op and oq, in Fig. 89, represent the re- 
 fracted ray p', and the index vector p,' corresponding, and if we draw through the 
 line PQ a plane perpendicular to the plane of the Figure, then the plane so drawn 
 will contain (on the principles here considered) the refracted vibration t, and will 
 be parallel to both the incident vibration t and the reflected vibration t" ; whence 
 the directions of the two latter vibrations may be in general determined, as being 
 aXm perpendicular XQi'^eciivQly to i\\e incident a\^6.reflected rays, p and p" : and then 
 the relative intensities (Tr^, Tr'2, Tr"^) of the three lights may be d duced from the 
 relative amplitudes (Tr, Tr', Tr") of the three vibrations, which may them- elves be 
 found from the three complanar directions, by a simple resolution of one line r' into 
 two others, of Avhich it is the vector sum, as if the vibrations were forces. 
 
 (9.) The equations ir. IV. V. and XIII'. enable us to express the four vectors, , 
 fi'i=p + v), i(=p - v-^Spp), p" (= p - 2 v-'Si/p), and p' (= p + r - v), in terms 
 of the three vectors p, v, v\ which are connected with each other by the relation, 
 
 XIX. . . t(=p-v-iSi^p), p"(=p-2v-iSj/p), and p'(=p + v-v'), 
 XIX. . . j/2+2Si/p=S»/'(p + i'), because XIX'. . . Srp'=S (v'- j/)p, 
 
 * The expressions XV. XVI. enable us to determine, not only the directions Ur, 
 Ur" of the incident and reflected vibrations, but also their amplitudes Tr, Tr", or 
 the intensities Tr2, Tr"^ of the incident and reflected lights, for any given or assumed 
 amplitude Tr' of the refracted vibration, or intensity Tr'2 of the refracted light, 
 after having determined the direction Ur' of the refracted vibration by means of the 
 formula X. 
 
760 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 as in XIIL, or because /Lt'2 - p2 = S/aV) by I. and XIII'.; and with which r' is 
 connected (VII. and IX.), by the two equations, 
 
 XX. .. S(p + v)r' = 0, and XXI. . . Si/'r' = ; 
 while T and r" are connected with the same three vectors, and with r', by the rela- 
 tions VI. VIII. XI. XIII., wliich conduct, by elimination of r", to the following 
 system (comp. (5.)) of three linear and scalar equations in r, 
 
 XXII. . . Spr = 0; 2Sj^pSrr = Sj/'(p+ v) Sj// ; 2Sj/|oSr'-ir = Sa/> ; 
 
 and therefore to the vector expression, 
 
 2rSr|0 = VjOJ/V, as in XV. 
 
 (10.) By these or other transfomations, there is no difficulty in deducing this 
 new equation, in which <u may be any vector, 
 
 XXIII. ..VvV{(p-o>)r-(p'-w)r' + (p"-a;)rr}r' = 0; 
 and conversely, when w is thus treated as arbitrary, the formula XXIII., with the 
 relations (9.) between the vectors p, p', p", v, v\ fi\ but without any restriction (ex- 
 cept itself) on r, r', r", is sufficient to give the two vector equations, 
 
 XI. . . r-r'+ r" = 0, and XXIV. . . pr - p'r' + p"r" = £cv- ' + y, 
 in which 
 
 XXV. . . iK = Sr (pr — p'r' + p'V") = Si/j/V', and XXI. . . y = S (pr - pW + p"r") ; 
 and which conduct to the two scalar equations (among others), 
 
 XXVII. . . Sk ipT - p'r' + p"r") = 0, if XXVII'. . . Skv = 0, 
 and XXVIII. . . Sj/p (Spr - Sp'V") = Si^p'S/r' ; 
 
 so that if we now suppose the equations VI. VIII. IX. to be given, the equation 
 VII. will /o//ow, by XXVIII. ; while, as a case of XXVII., and with the significa- 
 tion IV. or IV'. of t, we have the equation, 
 
 XXIX. . . Si (pr - p'r' f p"r") = 0. 
 
 (11.) And thus (or otherwise) it may be shown, that the three scalar equations 
 •VI. VIII. IX., combined with the one vector formula XXIII., which (on account of 
 the arbitrary w) is equivalent io five scalar equations, are sufficient to give the same 
 direction of r', and the same dependencies of r and r" thereon, as those expressed by 
 .the equations X. XV. XVI. ; and therefore (among other consequences), to the for- 
 mula XII. and XVII. 
 
 (12.) But the equations VI. VIII. IX. contain what may be called the Princi- 
 ple of Rectangular Vibrations (or of vibrations rectangular to rays)', and the for- 
 mula XXIII. is easily interpreted (416.), as expressing what may be termed the 
 Principle of the Resultant Couple : namely the theorem, that if the three vibrations 
 (or displacements), r, r', r", be regarded as three forces, rt, k't', r"t", acting at the 
 ends of the three rays, p, p', p'', or or, or', or" (drawn in the directions (1.) from 
 the point of incidence o), then this other system of three forces, rt, — r't', r"t" (con- 
 ceived as applied to a solid body), is equivalent to a single couple, of which the plane 
 is parallel (or the axis perpendicular) to the face of the crystal. 
 
CHAP. Ill] PRINCIPLE OF EQUIVALENT MOMENTS. 761 
 
 (13.) It follows then, by (10.) and (11.), that from these two principles,* (I.) 
 and (II.), we can infer all the following : 
 
 (III.) t\\Q Principle of Tangential Vibrations (or of vibrations tangential to the 
 waves) ; 
 
 (IV.) the Principle of Eqvivalent Vibrations (^4:.) ; 
 
 (V.) the Principle of the Vis Viva, as expressed (in conjunction with that of the 
 Constant Density of the Ether) by the equation XII. ; 
 
 (VI.) the Principle (or Theorem) of the Polar Plane; 
 
 And (VII.) what may be called the Principle of Equivalent Moments,f namely 
 
 * The word " Principle" is here employed with the usual latitude, as representing 
 either an hypothesis assumed, or a theorem deduced, but made a grozmd of subsequent 
 deduction. The principle (I.) of rectangular vibrations coincides, for the case of an 
 ordinary medium, with the principle (III.) of tangential vibrations ; but, for an ex- 
 traordinary medium, except for the case (not here considered) oi ordinary rays in an 
 vniaxal crystal, these two principles are distinct, although both were assumed by 
 Mac Cullagh and Neumann. The present writer has already disclaimed (in the Note 
 to page 736) any responsibility for the physical hypotheses ; so that the results given 
 above are offered merely as instances of mathematical deduction and generalization 
 attained through the Calculus of Quaternions. 
 
 t In a very clear and able Memoir, by Arthur Cayley, Esq, (now Professor 
 Cayley), " On Professor Mac CuUagh's Theorem of the Polar Plane," which was 
 read before the Royal Irish Academy on the 23rd of February, 1857, and has been 
 printed in Vol. VI. of the Proceedings of that Academy (pages 481-491), this name 
 "principle of equivalent moments," is given to a statement (p. 489), that "the- 
 moment of R't' round the axis AH, is equal to the sum of the moments oi Rt and 
 R"t" round the same axis" ; the line AH being (p. 487) the intersection of the 
 plane of incidence with the plane of separation of the two media, that is, with the 
 face of the crystal : while Pt, R't', R"t'' are lines representing (p. 488) the three 
 vibrations (incident, refracted, and reflected), at the ends of the i/iree rays AR, AR' 
 AR", which are drawn from the point of incidence A, so as to lie, all three (p. 487), 
 within the crystal. And in fact, if this statement be modified, either by changing 
 the sign of the moment of R"t" (p. 491), or by drawing the reflected ray AR", like 
 the line or" of the present investigation in the air (or in vacuo), instead of prolong- 
 ing it backwards within the biaxal crystal, it agrees with the case XXIX. of the 
 more general formula XXVI I., which is M^qM included in what has been called above 
 the Principle of the Resultant Couple. In venturing thus to point out, as the sub- 
 ject obliged him to do, what seemed to him to be a slight inadvertence in a Paper of 
 such interest and value, the present writer hopes that he will not be supposed to be 
 deficient in the admiration, (long since publicly expressed by him), which is due to the 
 vast attainments of a mathematician so eminent as Professor Cayley, 
 
 Since the preceding Series 423, including its Notes (so far), was copied and sent 
 to the printers, the writer's attention has been drawn to a later Paper by Mac Cul- 
 lagh (read December 9th, 1839, and published in Vol. XXI., Part I., of the Trans- 
 actions of the Royal Irish Academy, pp. 17-50), entitled '■'■An Essay towards a 
 Dyn&mical Theory of crystalline Reflexion and lief •action ;" in which there is given 
 at p. 43) a theorem essentially equivalent to the above-stated "Principle of the 
 
 5 E 
 
762 ELEMENTS OF QUATERNIONS. [bOOK III. 
 
 theorem that the Moment of the Refracted Vibration (r't') is equal to the Sum of 
 the Moments of the Incident and Reflected Vibrations (Rxand r"t"), with respect to 
 any line, which is on, or parallel to, the Face of the Crystal. 
 
 [It appears by the Table of Initial Pages (see p. lix.), that the Author had in- 
 tended to complete the work by the addition of Seven Articles.] 
 
 Resultant Couple," but expressed so as to include the case where the vibrations are 
 not uniradial, so that the double refraction of the crystal is allowed to manifest itself. 
 Mac CuUagh speaks, in his enunciation of the theorem, of measuring each ray, in the 
 direction of propagation : which agrees with, but of course anticipates, the direc- 
 tion of the reflected ray, adopted in the preceding investigation. The writer believes 
 that subsequent experiments, by Jamin and others, are considered to diminish much 
 the physical value of the theory above discussed. 
 
39 rAXEUI-iOSTER E.OW, E.G. 
 
 London: JanKar f/ 1S70. 
 
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INDEX. 
 
 Acton's Modern Cookery 20 
 
 Alcock's Residence in Japan 17 | 
 
 Allies on Formation of Christendom 15 
 
 Alpine Guide (The) 17 
 
 Althjlus on Medical Electricity 10 
 
 Andeews's Life of Oliver Cromwell 3 
 
 Arnold's Manual of English Literature . . 5 
 
 Aenotx's Elements of Physics 8 
 
 Arundines Cami 18 
 
 Autumn Holidays of a Country Parson .... 6 
 
 Ayee's Treasury of Bible Knowledge 15 
 
 Bacon's Essays by Whately 5 
 
 Life and Letters, by Spedding . . 4 
 
 Works 5 
 
 Bain's Mental and Moral Science 7 
 
 — on the Emotions and Will 7 
 
 on the Senses and Intellect 7 
 
 on the Study of Character 7 
 
 Ball's Guide to the Central Alps 16 
 
 Guide to the Western Alps IG 
 
 Guide to the Eastern Alps 16 
 
 Baenard's Drawing from Nature 12 
 
 Bayldon's B/ents and Tillages 13 
 
 Beaten Tracks 16 
 
 Becker's Charicles and Galltis 18 
 
 Benfey's Sanskrit-English Dictionary 6 
 
 Black's Treatise on Brewing 20 
 
 Rlackley's Word-Gossip 7 
 
 German-English Dictionary . . 6 
 
 Blaine's Rural Sports 19 
 
 Veterinary Art 19 
 
 Bourne on Screw Propeller 13 
 
 's Catechism of the Stoam Engine . . 13 
 
 Examples of Modern Engines . . 13 
 
 ■ Handbook of Steam Engine .... 13 
 
 Treatise on the Steam Engine. . . . 13 
 
 Improvements in the Steam- 
 
 Eugine 13 
 
 Bowdler's Family Shakspeare 18 
 
 Brande's Dictionary of Science, Literature, 
 
 and Art 9 
 
 Bray's (C.) Education of the Feelings .... 7 
 
 Philosophy of Necessity 7 
 
 On Force 7 
 
 Browne's Exposition of the 39 Articles .... 14 
 
 Buckle's History of Civilisation 2 
 
 Bull's Hints to Mothers 20 
 
 Maternal Management of Children . . 20 
 
 BuNBEN's Ancient Egypt 3 
 
 God in History 3 
 
 Memoirs 4 
 
 BUNSEN (E. De) on Apocrypha 15 
 
 's Keys of St. Peter 15 
 
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 Burke's Vicissitudes of Families 4 
 
 BuRroN's Christian Church 3 
 
 Vikram and the Vampire 17 
 
 Cabinet Lawyer 20 
 
 Calvert's Wife's Manual 15 
 
 Cates's Biographical Dictionary 4 
 
 Cats and Farhe's Moral Emblems 12 
 
 Changed Aspects of Unchanged Truths .... 6 
 
 Chesney's Euphrates Expedition 17 
 
 Indian Polity 2 
 
 Waterloo Campaign 2 
 
 Child's Physiological Essays 11 
 
 Chorale Book for England 11 
 
 Clough's Lives from Plutarch 2 
 
 CoBBE's Norman Kings 3 
 
 CoLENSO (Bishop) on Pentateuch and Book 
 
 of Joshua 15 
 
 Commonplace Philosopher in Town and 
 
 Country C 
 
 Conington's Chemical Analysis 9 
 
 Translation of Virgil's ^neid 19 
 
 CoNTANSEAu's Two French Dictionaries . . 6 
 CONYBEARE andHowsoN'sLifc and Epistles 
 
 of St. Paul 14 
 
 Cook's Acts of the Apostles 14 
 
 Voyages 4 
 
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 Copland's Dictionary of Practical Medicine 11 
 
 Cotton's Introduction to Confirmation. . . . 14 
 
 CouLTHAET's Decimal Interest Tables 20 
 
 Counsel and Comfort from a City Pulpit . . 6 
 
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 Aryan Mythology 3 
 
 Tale of the Great Persian War 2 
 
 Tales of Ancient Greece 18 
 
 — (H.) Ancient Parliamentary Elections 1 
 
 History of the Reform Bills .... 1 
 
 ■ Whig and Tory Administrations 1 
 
 Cresy's Encyclopaedia of Civil Engineering 13 
 
 Critical Essays of a Country Parson 6 
 
 Crowe's History of France 2 
 
 Culley's Handbook of Telegraphy 12 
 
 Cusack's History of Ireland 2 
 
 Dart's Iliad of Homer 19 
 
 D'AuBiGN^'s History of the Reformation in 
 
 the time of Calvin 2 
 
 Davidson's Introduction to New Testament 14 
 
 Dayman's Dante's Divina Commedia 19 
 
 Dead Shot (The), by Marksman 19 
 
 De la Rive's Treatise on Electricity 8 
 
 Denison's Vice-Regal Life 1 
 
 De Tocqueville's Democracy in America . 2 
 DOBSON on the Ox 19 
 
22 
 
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 Life of Gibson 11 
 
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 Edwards's Shipmaster's Guide 20 
 
 Elements of Botany 9 
 
 Ellicott's Commentary on Ephesians .... 14 
 
 Destiny of the Creature 14 
 
 Lectures on Life of Christ .... 14 
 
 Commentary on Galatians .... 14 
 
 . Pastoral Epist. 14 
 
 Philippians,&c. 14 
 
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 Essays and E«views 15 
 
 EwALD's History of Israel 14 
 
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 Felkin on Hosiery & Lace Manufactures. . 13 
 
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 Five Years in a Protestant Sisterhood 14 
 
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 Short Studies 6 
 
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 Home Politics 2 
 
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 Gray's Anatomy 10 
 
 Gee FNHOW on Bronchitis 10 
 
 Geove on Correlation of Physical Forces . . 8 
 
 Gueney's Chapters of French History .... 2 
 
 Gwilt's Encyclopaedia of Architecture .... 12 
 
 Hare on Election of Representatives 5 
 
 Haetwig's Harmonies of Nature 9 
 
 Polar World 9 
 
 Sea and its Living Wonders .... 9 
 
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 Holmes's Surgical Treatment of Children . . 10 
 
 System of Surgery 10 
 
 Hooker and Walker-Aenott's British 
 
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 How we Spent the Summer 16 
 
 Howaed's Gymnastic Exercises 11 
 
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 Northern Heights of London. ... 17 
 
 Rural Life of England 17 
 
 Visits to Remarkable Places .... 17 
 
 Hughes's Manual of Geography 8 
 
 Hume's Essays 7 
 
 Treatise on Human Nature 7 
 
 Humphreys's Sentiments of Shakspeare . . 12 
 
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 Ingelow's Poems 18 
 
 Story of Doom 18 
 
 Mopsa 18 
 
 Instructions in Household Matters 20 
 
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 Legends of the Madonna 12 
 
 Legends of the Monastic Orders 12 
 
 Legends of the Saviour 12 
 
 Johnston's Geographical Dictionary 8 
 
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 on Types of Genesis 15 
 
 Kalisch's Commentary on the Bible 5 
 
 HeV)rew Gi ammar 5 
 
 Keith on Destiny of the World 14 
 
 Fulfilment of Prophecy 14 
 
 Kerl's Metallurgy, by Crookes and 
 
 ROHRIG 13 
 
 Kesteven's Domestic Medicine 11 
 
 KiEB Y and Spence's Entomology 9 
 
 Landon's (L. E. L.) Poetical Works 18 
 
 Latham's E nglish Dictionary 5 
 
 River Plate 8 
 
 Lawloe's Pilgrimages in the Pyrenees .... 16 
 
 Lecky's History of European Morals 3 
 
 Rationalism 8 
 
 Leighton's Sermons and Charges 14 
 
 Leisure Hours in Town 6 
 
 Lessons of Middle Age 6 
 
 Letheby on Food 20 
 
 Lewes's Biographical History of Philosophy 3 
 
 Lewis's Letters 4 
 
 LiDDELL and Scott's Greek-Eng'ish Lexicon 6 
 
 Abridged ditto 6 
 
 Life of Man Symbolised 11 
 
 Mariraret M. Hallahan 14 
 
 LiNDLEY and Moore's Treasury of Botany 9 
 
 Lindsay's Evidence for the Papacy 14 
 
 Longman's Edward the Third 2 
 
 Lectures on History of England 2 
 
 Chess Openings 20 
 
 Lord's Prayer Illustrated 11 
 
NEW WORKS PUBLISHED BY LONGMANS and CO. 
 
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 _ — Gardening 13 
 
 Plants 9 
 
 LoWKDES's Engineer's Handbook 12 
 
 Lyra Eucharistica 16 
 
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 Messianica ^6 
 
 Mystica 16 
 
 Mabeldean 17 
 
 Macaulat's (Lord) Essays 3 
 
 — History of England . . 1 
 
 Lays of Ancient Rome 18 
 
 ■ Miscellaneous Writings 6 
 
 -^H:^ Speeches 5 
 
 Works 1 
 
 Macfaeren's Lectures on Harmony 11 
 
 Mackintosh's Scenery of England and 
 
 Wales 8 
 
 MACLEOD'S Elements of Political Economy 4 
 
 Dictionary of Political Economy 4 
 
 Elements of Banking 19 
 
 Theory and Practice of Banking 19 
 
 McCuLLOCH's Dictionary of Commerce 20 
 
 Geographical Dictionary 8 
 
 Maguiee's Life of Father Mathew 4 
 
 Manning's England and Christendom .... 15 
 
 Maecet on the Larynx 10 
 
 MAESHALii's Physiology 11 
 
 Marshman's History of India 2 
 
 Life of Havelock 4 
 
 Marttneau's Endeavours after the Chris- 
 tian Life 16 
 
 Maetineau's Letters from Australia 16 
 
 Massey's History of England 1 
 
 Massingberd's History of the Reformation 8 
 
 Matheson's England to Delhi 16 
 
 Maunder's Biographical Treasury 4 
 
 Geograuhical Treasury 8 
 
 Historical Treasury 3 
 
 Scientific and Literary Treasury 9 
 
 ' Treasury of Knowledge 20 
 
 Treasury of Natural History . . 9 
 
 Mauet's Physical Geography 8 
 
 May's Constitutional History of England.. 1 
 
 Melville's Digby Grand 18 
 
 General Bounce 18 
 
 ■ Gladiators 18 
 
 •— Good for Nothing 18 
 
 .. Holmby House 18 
 
 Interpreter 18 
 
 . Kate Coventry 18 
 
 Queen's Maries 18 
 
 Mendelssohn's Letters 4 
 
 Menes and Cheops 7 
 
 Meelvale's (H.) Historical Studies 2 
 
 (C.) Pall of the Roman Republic 3 
 
 . Romans under the Empire 3 
 
 . '■ Boyle Lectures 3 
 
 Meerifield and EvEES's Navigation .... 7 
 
 Miles on Horse's Foot and Horse Shoeing. 19 
 
 on Horses' Teeth and Stables 19 
 
 MiLi. (J.) on the Mind 4 
 
 Mill (J. S.) on Liberty 4 
 
 England and Ireland 4 
 
 Subjection of Women 4 
 
 on Re pres!*ntative Government 4 
 
 oil Utilitarianism 4 
 
 '8 Dissertations and Discussions 4 
 
 Political Economy 4 
 
 Mill's System of Logic 4 
 
 Hamilton's Philosophy 4 
 
 Inaugural Address at St. Andrew's . 4 
 
 Millee's Elements of Chemistry 9 
 
 • Hymn Writers 15 
 
 Mitchell's Manual of Assaying 13 
 
 Monsell's Beatitudes 16 
 
 His Presence not his Memory. . 16 
 
 ' Spiritual Songs ' 16 
 
 Mooee's Irish Melodies 18 
 
 LallaRookh 18 
 
 Journal and Correspondence .... 8 
 
 Poetical Works 18 
 
 (Dr. G.) Power of the Soul over 
 
 the Body 15 
 
 Moeell's Elements of Psychology 7 
 
 Mental Philosophy 7 
 
 Muller's (Max) Chips from a German 
 
 Workshop 7 
 
 Lectures on the Science of Lan- 
 guage 5 
 
 (K. O.) Literature of Ancient 
 
 Greece 2 
 
 MUECHISON on Continued Fevers 10 
 
 on Liver Complaints 10 
 
 Muee's Language and Literature of Greece 2 
 
 New Testament Illustrated with Wood En- 
 gravings from the Old Masters 11 
 
 Newman's History of his Religious Opinions 4 
 
 Nichols's Handbook to British Museum.. 20 
 
 Nightingale's Notes on Hospitals 20 
 
 NiLSSON's Scandinavia 9 
 
 Noethcote's Sanctuary of the Madonna . . 14 
 
 NoRTHCOTT on Lathes and Turning 12 
 
 Noeton's City of London 17 
 
 Odling's Animal Chemistry 10 
 
 Course of Practical Chemistry . . 10 
 
 Manual of Chemistry 9 
 
 Lectures on Carbon 10 
 
 Outlines of Chemistry 10 
 
 Our Children's Story 18 
 
 Owen's Comparative Anatomy and Physio- 
 logy of Vertebrate Animals 9 
 
 Lectures on the Invertebrata 8 
 
 Packe's Guide to the Pyrenees 17 
 
 Paget's Lectures on Surgical Pathology . . 10 
 
 Pereiea's Manual of Materia Medioa 11 
 
 Perkins's Italian and Tuscan Sculptors .. 12 
 
 Phillips's Guide to Geology 8 
 
 Pictures in Tyrol 16 
 
 Piesse'S Art of Perfumery 13 
 
 Chemical, Natural, and Physical Magic 13 
 
 Pratt's Law of Building Societies 20 
 
 Peendeegast's Mastery of Languages .... 6 
 
 Prescott's vScripture Difficulties 16 
 
 Peoctgr's Handbook of the Stars 7 
 
 Saturn 7 
 
 Pyne's England and France in the Fifteenth 
 
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 Quarterly Journal of Science 9 
 
24 
 
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 Reillt's Map of Mont Blanc 16 
 
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 Religious Republics *. 15 
 
 Riley's Memorials of London 17 
 
 RiVEKs's Rose Amateur's Guide 9 
 
 RoBBiNS's Cavalry Catechism 19 
 
 Rogees's Correspondence of Greyson ...... 7 
 
 Eclipse of Faith 7 
 
 Defence of Faith 7 
 
 Essays from the Edinburgh Re- 
 
 view 6 
 
 Reason and Faith € 
 
 Roget's Thesaurus of English AVords and 
 
 Phrases 5 
 
 Roma Sotterranea 17 
 
 RoNALDS's Fly-Fisher's Entomology 19 
 
 Rowton's Debater 5 
 
 RussEiiL on Government and Constitution 1 
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 Samuelson's German "Working Man 17 
 
 Sandaes's Justinian's Institutes 5 
 
 SCHEFFLEB on Ocular Defects 11 
 
 Scott's Lectures on the Fine Arts 11 
 
 — Albert Durer 11 
 
 Seebohm's Oxford Reformers of 1498 2 
 
 Sewell's After Life 17 
 
 Glimpse of the "World 17 
 
 History of the Early Church .... .'J 
 
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 Passing Thoughts on Religion . . 15 
 
 Preparation for Communion 15 
 
 Principles of Education 15 
 
 Readings for Confirmation 15 
 
 Readings for Lent 15 
 
 Examination for Confirmation .. 15 
 
 Stories and Tales 17 
 
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 Seymoue's Pioneering in the Pampas 16 
 
 Shaftesbury's Characteristics 7 
 
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 Invocation of Saints 16 
 
 Short's Church History 3 
 
 Smart's Walker's English Pronouncing 
 
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 •Smith's ( Southwood) Philosophy of Health 20 
 
 (J.) Paul's Voyage and Shipwreck 14 
 
 (Sydney) Miscellaneous Works .. 6 
 
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 Stanley's History of British Birds 9 
 
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 Stirling's Secret of Hegel 7 
 
 Stonehenge on the Dog 19 
 
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 Strickland's Tudor Princesses 4 
 
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