Daughter ox Stuart Smith silv. ^ ,? 4 »3 VI PREFACE. 'I regret that I have so imperfectly fulfilled this last request of my revered frieud. When it was made I was already engaged, along with Sir W. Thomson, in the laborious work of preparing a large Treatise on Natural Philosophy. The present volume has thus been written under very disadvantageous circumstances, espe- cially as I have not found time to work up the mass of materials which I had originally collected for it, but which I had not put into a fit state for publication. I hope, however, that I have to some extent succeeded in producing a thoroughly elementary work, intelligible to any ordinary student; and that the numerous ex- amples I have given, though not specially chosen so as to display the full merits of Quaternions, will yet sufficiently shew their admir- able simplicity and naturalness to induce the reader to attack the Lectures and the Elements ; where he will find, in profusion, stores of valuable results, and of elegant yet powerful analytical investiga- tions, such as are contained in the writings of but a very few of the greatest mathematicians. For a succinct account of the steps by which Hamilton was led to the invention of Quaternions, and for other interesting information regarding that remarkable genius, I may refer to a slight sketch of his life and works in the North British Beview for Sej)tember 1866. ' It will be found that I have not servilely followed even so great a master, although dealing with a subject which is entirely his own. I cannot, of course, tell in every case what I have gathered from his published papers, or from his voluminous correspondence, and what I may have made out for myself. Some theorems and processes which I have given, though wholly my own, in the sense of having been made out for myself before the publication of the Elements, I have since found there. Others also may be, for I have not yet read that tremendous volume completely, since much of it bears on developments unconnected with Physics. But I have endeavoured throughout to point out to the reader all the more important parts of the work which I know to be wholly due to Hamilton. A great part, indeed, may be said to be obvious to any one who has mastered the preliminaries ; still I think that, in the PREFACE. vn two last Chapters especially, a good deal of original matter will be found. ' The volume is essentially a toorhlng one, and, particularly in the later Chapters, is rather a collection of examples than a detailed treatise on a mathematical method. I have constantly aimed at avoiding too great extension ;. and in pursuance of this object have omitted many valuable elementary portions of the subject. One of these, the treatment of Quaternion logarithms and exponentials, I greatly regret not ha\ing given. But if I had printed all that seemed to me of use or interest to the student, I might easily have rivalled the bulk of one of Hamilton's volumes. The beginner is recommended merely to read the first five Chapters, then to work at Chapters VI, YII, VIII (to which numerous easy Examples are appended). After this he may work at the first five, with their (more difiicult) Examples ; and the remainder of the book should then present no difficulty. ' Keeping always in view, as the great end of every mathematical method, the physical applications, I have endeavoured to treat the subject as much as possible from a geometrical instead of an analy- tical point of view. Of course, if we premise the properties of i,j, k merely, it is possible to construct from them the whole system^ ; just as we deal with the imaginary of Algebra, or, to take a closer analogy, just as Hamilton himself dealt with Couples, Triads, and Sets. Tliis may be interesting to the pure analyst, but it is repulsive to the physical student, who should be led to look upon ij, h from the very first as geometric realities, not as algebraic imaginaries. ' The most striking peculiarity of the Calculus is that muUipli' cation is not generalhj commutative, i.e. that (p- is in general different from rq, r and q being quaternions. Still it is to be remarked that something similar is true, in the ordinary coordinate methods, of operators and functions : and therefore the student is not wholly unprepared to meet it. No one is puzzled by the fact that log.cos.^ * This has been done by Hamilton himself, as one among many methods he has employed ; and it is also the foundation of a memoir by M. Allegret, entitled Esmi mr le Cnkul des Quaternions (Paiis, 1862). via PREFACE. is not equal to cos.log.a?/ or that vZ-f- is not equal to -j-y/y- itiXy (loo Sometimes^ indeed, this rule is most absurdly violated, for it is usual to take cos^a? as equal to (cos«)^, while qos~'^x is not equal to (cosa?)"-^. No such incongruities appear in Quaternions; but what is true of operators and functions in other methods, that they are not generally commutative, is in Quaternions true in the multipli- cation of (vector) coordinates. ' It will be observed by those who are acquainted with the Cal- culus that I have, in many cases, not given the shortest or simplest proof of an important proposition. This has been done wdth the view of including, in moderate compass^ as great a variety of methods as possible. With the same object I have endeavoured to supply, by means of the Examples appended to each Chapter, hints (which will not be lost to the intelligent student) of farther develop- ments of the Calculus. Many of these are due to Hamilton, who, in spite of his great originality, was one of the most excellent examiners any University can boast of. ' It must always be remembered that Cartesian methods are mere particular cases of Quaternions, where most of the distinctive fea- tures have disappeared; and that when, in the treatment of any particular question, scalars have to be adopted, the Quaternion solution becomes identical with the Cartesian one. Nothing there- fore is ever lost, though much is generally gained, by employing Quaternions in preference to ordinary methods. In fact, even when Quaternions degrade to scalars, they give the solution of the most general statement of the problem they are applied to, quite inde- pendent of any limitations as to choice of particular coordinate axes. ' ^ There is one very desirable object which such a work as this may possibly fulfil. The University of Cambridge, while seeking to supply a real want (the deficiency of subjects of examination for mathematical honours, and the conseqflent frequent introduction of the wildest extravagance in the shape of data for "Problems"), is in danger of making too much of such elegant trifles as Trilinear PKEFACE. IX Coordiuates, while gigantic systems like Invariants (which^ by the way, are as easily introduced into Quaternions as into Cartesian methods) are quite beyond the amount of mathematics which even the best students can master in three years' reading. One grand step to the supply of this want is^ of course, the introduction into the scheme of examination of such branches of mathematical physics as the Theories of Heat and Electricity. But it appears to me that the study of a mathematical method like Quaternions, which, while of immense power and comprehensiveness, is of extraordinary sim- plicity, and yet requires constant thought in its applicationSj would also be of great benefit. With it there can be no " shut your eyes, and write down your equations/' for mere mechanical dexterity of analysis is certain to lead at once to error on account of the no\^lty of the processes employed. 'The Table of Contents has been drawn up so as to give the student a short and simple summary of the chief fundamental for- mulae of the Calculus itself, and is therefore confined to an analysis of the first five [and the two last] chapters. ' In conclusion, I have only to say that I shall be much obliged to any one, student or teacher, who will point out portions of the work where a difiiculty has been found ; along with any inaccuracies which may be detected. As I have had no assistance in the revision of the proof-sheets, and have composed the work at irregular in- tervals, and while otherwise laboriously occupied, I fear it may contain many slips and even errors. Should it reach another edition there is no doubt that it will be improved in many important par- ticulars.' To this I have now to add that I have been equally surprised and delighted by so speedy a demand for a second edition — and the more especially as I have had many pleasing proofs that the work has had considerable circulation in America. There seems now at last to be a reasonable hope that Hamilton's grand in- vention w411 soon find its way into the working world of science, to which it is certain to render enormous services, and not be laid X PREFACE. aside to bo imoarthed some centuries hence b}^ some grubbing antiquary. It can hardly be expected that one whose time is mainly en- grossed by physical science, should devote much attention to the purely analytical and geometrical applications of a subject like this ; and I am conscious that in many parts of the earlier chapters I have not fully exhibited the simplicity of Quaternions. I hope, however, that the corrections and extensions now made, especially in the later chapters, will render the work more useful for my chief object, the Physical Applications of Quaternions^ than it could have been in its first crude form. I have to thank various correspondents^ some anonymous^ for suggestions as well as for the detection of misprints and slips of the pen. The only absolute error which has been pointed out to me is a comparatively slight one which had escaped my own notice : a very grave blunder, which I have now corrected, seems not to have been detected by any of my correspondents, so that I cannot be quite confident that others may not exist. I regret that I have not been able to spare time enough to re- write the work ; and that, in consequence of this, and of the large additions which have been made (especially to the later chapters), the whole will now present even a more miscellaneously jumbled appearance than at first. It is well to remember, however, that it is quite possible to make a book too easy reading, in the sense that the student may read it through several times without feeling those difficulties which (except perhaps in the case of some rare genius) must attend the acquisition of really useful knowledge. It is better to have a rough climb (even cutting one^s own steps here and there) than to ascend the dreary monotony of a marble staircase or a well-made ladder. Royal roads to knowledge reach only the par- ticular locality aimed at — and there are no views by the way. It is not on them that pioneers are trained for the exploration of unknown regions. But I am happy to say that the possible repulsiveness of my PREFACE, xi early chapters cannot long be advanced as a reason for not at- tacking this fascinating subject. A still more elementary work than the present will soon appear, mainly from the pen of my colleague Professor Kelland. In it I give an investigation of the properties of the linear and vector function, based directly ujDon the Kinematics of Homogeneous Strain, and therefore so different in method from that employed in this work that it may prove of interest to even the advanced student. Since the 'appearance of the first edition I have managed (at least partially) to effect the application of Quaternions to line, surface, and volume /integrals, such as occur in Hydrokinetics, Electricity, and Potentials generally. I was first attracted to the study of Quaternions b}' their promise of usefulness in such applications, and, though I have not yet advanced far in this new track, I have got far enough to see that it is certain in time to be of incalculable value to physical science. I have given towards the end of the work all that is necessary to put the student on this track, which will, I hope, soon be followed to some purpose. One remark more is necessary. I have employed, as the positive direction of rotation, that of the earth about its axis, or about the sun_, as seen in our northern latitudes, i.e. that opposite to the direc- tion of motion of the hands of a watch. In Sir W. Hamilton's great works the opposite is employed. The student will find no difficulty in passing from the one to the other ; but, without pre- vious warning, he is liable to be much perplexed. With regard to notation, I have retained as nearly as possible that of Hamilton,, and where new notation was necessary I have tried to make it as simple and as little incongruous with Hamil- ton's as possible. This is a part of the work in which great care is absolutely necessary ; for, as the subject gains development, fresh notation is inevitably required ; and our object must be to make each step such as to defer as long as possible the revolution which must ultimately come. Many abbreviations are possible, and sometimes very useful in private work ; but, as a rule, they are unsuited for print. Every xn PREFACE. analyst, like every short-hand writer, has his otvn special con- tractions ; but, when he comes to publish his results, he ought invariably to put such devices aside. If all did not use a com- mon mode of public expression, but each were to print as he is in the habit of writing for his own use, the confusion would be utterly intolerable. Finally, I must express my great obligations to my friend M. M. U. Wilkinson of Trinity College, Cambridge, for the care with which he has read my proofs, and for many valuable sug- gestions. P. G. TAIT. College, Edinbokgh, October 1873. CONTENTS. Chapter I. — Vectors and their Composition 1-22 Sketch of the attempts made to represent geometrically the unaginary of algebra. §§ 1-13. De Moivre's Theorem interpreted in plane rotation. § 8. Curious speculation of Servois. § 11. Elementary geometrical ideas connected vrith. relative position. § 15. Definition of a Vectoe. It may be employed to denote translation. § 16. Expression of a vector by one symbol, containing implicitly three distinct numbers. Extension of the signification of the symbol = . § 18. The sign + defined in accordance with the interpretation of a vector as representing translation. § 19, Definition of - , It simply reverses a vector. § 20. Triangles and polygons of vectors, analogous to those of forces and of simul- taneous velocities. § 21, When two vectors are imrallel we have a = xjS. § 22. Any vector whatever may be expressed in terms of three distinct vectors, which are not coplana, by the formula which exhibits the three numbers on which the vector depends, § 2.3. Any vector in the same plane with a and /3 may be written p=^xa + y^. §24. The equation zr = p, between two vectors, is equivalent to three distinct equations among numbers. § 25. The Commutative and Associative Laws hold in the combination of vectors by the signs + and - . § 27. The equation p — x^, where /) is a variable, and /3 a fixed, vector, represents a line drawn through the origin parallel to /3. p= a + a;j3 is the equation of a line drawn through the extremity of a and parallel to ^. § 28. p = ya + x^ represents the plane through the origin par.allel to a and /3. § 29. XIV CONTENTS. The condition that p, a, /3 may terminate in the same line is pp + qa + r^ -(\, subject to the identical relation p + q^ + r = 0. Similarly Tpp + qa + r^ + sy = 0, with p + q + r+s = 0, is the condition that the extremities of four vectors lie in one plane. § 30. Examples with solutions. § 31. Differentiation of a vector, when given as a function of one number. §§ 32-38. If the equation of a curve be p = ^{s) where s is the length of the arc, dp is a vector tangent to the curve, and its length is ds. §§ 38, 39. Examples with solutions. §§ 40-44. Examples to Chapter 1 22-24 Chapter II. — Products and Quotients of Vectors . . . 25-46 Here we begin to see what a quaternion is. When two vectors are parallel their quotient is a number. §§ 45, 46. When they are perpendicular to one another, their quotient is a vector per- pendicular to their plane. § 47, 72. When they are neither parallel nor perpendicular the quotient in general involves four distinct numbers — and is thus a Quateenion. § 47. A quaternion regarded as the operator which turns one vector into another. It is thus decomposable into two factors, whose order is indifferent, the stretching factor or Tensor, and the turning factor or Versor. These are denoted by Tq, and Uq. § 48. The equation ^ = qa gives - = 1, or /3a~i = q, but not in general a-i^ = 5. § 49. q or |3a~^ depends only on the relative lengths, and directions, of /3 and o. §50. Recipi'ocal of a quaternion defined, 2= -gives -or 5!=^, T.q-^ = Yq^ ■U.q-^ = (Uq)-\ §51. Definition of the Conjugate of a quaternion, Kq={Tq)^q-\ and qKq = Kq.q = {Tq)". § 52. Representation of versors by arcs on the unit-sphere. § 53, Versor multiplication illustrated by the composition of arcs. § 54, Proof that K (qr) = Kr . Kq. § 55. Proof of the Associative Law of Multiplication p.qr = pq.r. §§57-60. [Digression on Spherical Conies. § 59*.] CONTENTS. XV Quaternion addition and subtraction are commutatlte. § 61. Quaternion multiplication and division are distributive. § 62. Composition of quadrmital versors in planes at right angles to each other. Calling them i, j, h, we have P = j2 = jt2 = - 1, ij=z -ji = Ic, jk ^-Tcj-i, li= - ih =j, ijk^-1. §§64-71. A unit-vector, when employed as a factor, may be considered as a quadrantal versor whose plane is perpendicular to the vector. Hence the equations just written are true of any set of rectangular unit-vectors i, j, k. § 72. The product, and the quotient, of two vectors at right angles to each other is a third perpendicular to both. Hence Ka = — ct, and (Taf = aKa=-aK §73. Every versor may be expressed as a power of some unit-vector. § 74. Every quaternion may be expressed as a power of a vector. § 75. The Index Law is true of quaternion multiplication and division. § 76. Quaternion considered as the sum of a Scalar and Vector. q^^^x + y^Sq-l-Vq. §77. Proof that SKq = Sq, VA'q. = - Vq, § 79. Quadrinomial expression for a quaternion q — v; + ix +ji/ + kz. An equation between quaternions is equivalent to four equations between numbers (or scalars). § 80. Second proof of the distributive law of multiplication. § 81. Algebraic determination of the constituents of the product and quotient of two vectors. §§ 82-84. Second proof of the associative law of multiplication. § 85. Proof of the formulae Sa^ — S^a, Fa/3 = - 7/3a, aj3 = K^a, S.qrs = S.rsq = S.sqr, S.aj3y = S.^ya = S.ya^ = - S.ay^ = &c. §§ 86-89. Proof of the formulae V. a V^y = ySa^ - ^Sya, V. a^y = aSpy - ^Sya + ySa^, V.a^y = V.ypa, V. Fa|3 Vy5 = aS.^yd - ^S.ayS, = SS.a^y-yS.a^d, dS.a^y = aS.^yd + ^S.ya5 + yS.a^5, = VapSyd+ VpySa8+ VyaS^d. §§ 90-92. Hamilton's proof that the product of two parallel vectors must be a scalar, and that of perpendicular vectors, a vector; if quaternions are to deal with space indifferently in all directions. § 93. Examples to Chapter II 46, 47 xvi CONTENTS. Chapter III. — Interpretations and Transformations of Quaternion Expressions 48-67 If 6 be the angle between two vectors, a and (3, we have S^ = Z^~ cos e, Sa^ =-TaT^ cos 6, a Ta TV^ = ^ sin 9, y Fa/3 = TaT^ sin 6. a Ta Applications to plane trigonometry. §§ 94-97. ^fa/3 = shews that a is perpendicular to /3, while Fa/3 = 0, shews that a and ^ are parallel. S.a^y is the volume of the parallelepiped three of whose conterminous edges are a, /3, 7. Hence S. a^y = shews that a, j3, y are coplanar. Expression of S.a^y as a determinant. §§ 98-102. Proof that ( ^2)2 = (Sqf + ( T Vq)% and T{qr) = Tq Tr. % 103. Simple propositions in plane trigonometry. § 104. Proof that - a^a'^ is the vector reflected ray, when j3 is the incident ray and a normal to the reflecting surface. § 105. Interpretation of a/37 when it is a vector. § 106, Examples of variety in simple transformations. § 107. Introduction to spherical trigonometry. §§ 108-113. EepresentatioD, graphic, and by quaternions, of the spherical excess. §§ 114, 115. Loci represented by different equations — points, lines, surfaces, and solids. §§ 116-119. Proof that r'^ (r^Y^ 3-1 - U{rq + KrKq). § 120. Proof of the transformation BiQUATEKNIONS. §§ 123-125. Convenient abbreviations of notation. §§ 126, 127. Examples to Chaptee III 68-70 Chapter IV. — Differentiation of Quaternions .... 71-76 Definition of a differential, where dq is any quaternion whatever. We may write clFq =f(q, dq), where / is linear and homogeneous in dq ; but we cannot generally write dFq^i(q)dq. §§128-131. CONTENTS. xvii Definition of the differential of a function of more quaternions than one. d{qr) = qcb' + dq.r, but not generally d{qr) = qdr + rdq. § 132. Proof that --^ = S-!-, Tp p fP=v'IP,&c. §133. Ip p Successive differentiation; Taylor's theorem. §§ 134, 135. If the equation of a surface be J'ip) = c, the differential may be written Svdp = 0, where v is a vector normal to the surface. § 137. Examples to Chapter IV 76 Chapter V. — The Solution op Equations of the First Degree. 77-100 The most general equation of the first degree in an unknown quaternion q, may be written 2 V. aqh + S .cq = d, where a, b, c, d are given quaternions. Elimination of Sq, and reduction to the vector equation 'iJ.. Proof that m, whose value may be written as S.(t>'^.(p'lJL\iJ., whence the final form of solution w(0-i = /«, — ??(j^ + ^-. §§ 147, 148. Examples. § 149-161. b xviii CONTEXTS. The fundamental cubic -g^)lc=0, gi-g2^ f=^{gi + gz), ^ = -iigi-g2)- Another transformation is (pp = aaVap + hPSPp. §§167-169. Other properties of (p. Proof that 'Sp{<}> + g)~^p = 0, and Sp (

''^p = ghp''. Proof that when (p is not self-conjugate S'./35o = 0, where ^ is a vector force, a its point of application. Hence the usual six equations in the form 2i3 = 0, 2Fa/3 = 0. Central axis, &c. §§ 373-378. For the motion of a rigid system 2-S'()Ha-/3)5a = 0, whence the usual forms. The equation 2g = 5<^-'(r/-i75), where 7 is given in terms of t and g if forces act, but is otherwise constant, contains the whole theory of the motion of a rigid body with one point fixed. Reduction to the ordinary form dt _ dw _dx _dif _ dz Y^WX'T' Z ' Here, if no forces act, W, X, Y, Z are homogeneous functions of the third degree in w, x, y, z. Equation for precession. §§ 379-401. General equation of motion of simple pendulum. Foucault's pendulum. §§ 402-405. b 3 XX CONTENTS. Problem on reflecting surfaces. § 406. Fresnel's Theory of Double Refraction. Various forms of the equation of Fresnel's Wave-surface ; ^.p(,^-p=)-V=-l, T(p-^-<^-i)-ip = 0, l=-pp' + (T±S)V\pV^ip, The conical cusps and circles of contact. Lines of vibration, &c. §§ 407-427. Electrodynamics. The vector action of a closed circuit on an element of current a^ is proportional to FajjS where ^Vada rdUa P = r Vada _ rdUi I To? ~ I ~^ the integration extending round the circuit. Mutual action of two closed circuits, and of solenoids. Mutual action of magnets. Potential of a closed circuit. Magnetic curves. §§ 428-448. Physical applications of d . d d dx dy dz Effect of V on various functions of p = ix +jy + kz. 2 Vp=-3, VTp=Up, vrp=-—. VSap=~a, vVap=2a. Applications of the theorem S.SpVy^ = S-S.av^ . §§ 449-457. Jp -'P Farther examination of the use of V as applied to displacements of groups of points. Proof of the fundamental theorem for comparing an integral over a closed surface with one through its content jyifS.Vads =f/S.arvds. Hence Green's Theorem. Limitations and ambiguities. §§ 458-476. Similar theorem for double and single integrals fS.adp =ffS.VvVads. Applications of these to distributions of magnetism, and to Ampere's Directrice. Also to the Stress-function. §§ 477-491. £-ScrV/(p)= /(p + , Jattter sense, the expression a + b V —I implicitly contains the direction, as well as the length, of this line ; since, as we see at once, the direction is inclined at an ang-le tan~^- to the ° a axis of .<', and the length is \/a^-\-h'^. 6.] Operating on this symbol by the factor \/— 1, it becomes — h-\-a's/—\ ; and now, of course, denotes the point whose x and y coordinates are —h and a ; or the line joining this point with the origin. The length is still 's/a^ + b', but the angle the line makes with the axis of a? is tan~^ (~ 7) ' which is evidently 90° greater than before the operation. 7.] De Moivre's Theorem tends to lead us still farther in the same direction. In fact, it is easy to see that if we use, instead of \/— 1, the more general factor cosa4- v — 1 sina, its effect on any line is to turn it through the (positive) angle a in the plane of^, y. [Of course the former factor, ' \/ — 1 , is merely the par- ticular case of this, when a = - •] Thus (cos a + y — 1 sin a) (a + (5 \/— 1) = a cos a — b sin a + v— 1 {a sin a-^b cos a), by direct multiplication. The reader will at once see that the new form indicates that a rotation through an angle a has taken place, if he compares it with the common formulae for turning the co- ordinate axes through a given angle. Or, in a less simple manner, thus — Length = \/{a cos a — '^sina)^ + («sin a + (5» cos a)^ = \/a- -vb^ as before. 12.] VECTORS, AND THEIR COMPOSITION. Inclination to axis of ^ h tan a-\ — , a sm a + 6 cos a , , a = tan~i 5-^ — = tan ^ a cos a — <5 sm a , i , 1 tana b a = a + tan ^ - • 8.] We see now, as it were, loliy it happens that (cos a+ \/— 1 sin a)™ = cos rna-\- J —\ sin ma. In fact, the first operator produces m successive rotations in the same direction, each through the angle a; the second, a single rotation through the angle ma. 9.] It may be interesting, at this stage, to anticipate so far as to state that a Quaternion can, in general, be put under the form i\^(cos^ + OTsin^), where i\^ is a numerical quantity, Q a real angle, and This expression for a quaternion bears a very close analogy to the forms employed in De Moivre's Theorem ; but there is the essential difference (to which Hamilton's chief invention referred) that ot is not the algebraic y/ —I, but may be cmy directed unit-line what- ever in space. 10.] In the present century Argand, Warren, and others, extended the results of Wallis and De Moivre. They attempted to express as a line the product of two lines each represented by a symbol such as a + 5 \/^. To a certain extent they succeeded, but sim- plicity was not gained by their methods, as the terrible array of radicals in Warren's Treatise sufficiently proves. 11.] A very curious speculation, due to Servois and published in 1813 in Gergonne's Annates, is the only one, so far as has been discovered, in which the slightest trace of an anticipation of Quaternions is contained. Endeavouring to extend to space the form a + b's/^l for the plane, he is guided by analogy to write for a directed unit-line in space the form j9 cos a 4- 2" cos jQ + r cos y, where a, /3, y are its inclinations to the three axes. He perceives easily that jh 9., ^ m^^st be non-reals : but, he asks, '^ seraient-elles imaginaires reductibles a la forme generale yi + ^V— 1?" This he could not answer. In fact they are the i,j, k of the Quaternion Calculus. (See Chap. II.) 12.] Beyond this, few attempts were made, or at least recorded, in earlier times, to extend the principle to space of three dimensions ; B 2 4 QUATERNIONS. [ 1 3- and^ thoug-h many such have been made within the last forty years, none, with the single exception of Hamilton's, have resulted iii simple, practical methods ; all, however ingenious, seeming- to lead at once to processes and results of fearful com- plexity. For a lucid, complete, and most impartial statement of the claims of his predecessors in this field we refer to the Preface to Hamilton's Lectures on Quaternions. 13.] It was reserved for Hamilton to discover the use of \/— 1 as a geometric realitij, tied down to no particular direction in space, and this use was the foundation of the singularly elegant, yet enormously powerful, Calculus of Quaternions. While all other schemes for using sf^l to indicate direction make one direction in space expressible by real numbers, the re- mainder being imaginaries of some kind, leading in general to equations which are heterogeneous ; Hamilton makes all directions in space equally imaginary, or rather equally real, thereby ensuring to his Calculus the power of dealing with space indifferently in all directions. In fact, as we shall see, the Quaternion method is independent of axes or any supposed directions in space, and takes its reference lines solely from the problem it is applied to. 14.] But, for the purpose of elementary exposition, it is best to begin by assimilating it as closely as we can to the ordinary Cartesian methods of Geometry of Three Dimensions, which are in fact a mere particular case of Quaternions in which most of the distinctive features are lost. We shall find in a little that it is capable of soaring above these entirely, after having employed them in its establishment; and, indeed, as the inventor's works amply prove, it can be established, atj initio^ in various ways, without even an allusion to Cartesian Geometry. As this work is written for students acquainted with at least the elements of the Cartesian method, we keep to the first-mentioned course of exposition ; especially as we thereby avoid some reasoning w^hich, though rigorous and beautiful, might be apt, from its subtlety, to prove repulsive to the beginner. We commence, therefore, with some very elementary geometrical ideas. 15.] Suppose we have two points A and B in space, and suppose A given, on how many numbers does -S's relative position depend ? If we refer to Cartesian coordinates (rectangular or not) we find 19-] VECTOKS, AND THEIR COMPOSITION. 5 that the data required are the excesses of ^'s three coordinates over those of A. Hence three numbers are required. Or we may take polar coordinates. To define the moon's position with respect to the earth we must have its Geocentric Latitude and Longitude, or its Right Ascension and Dechnation, and, in addition^ its distance or radius-vector. Three again. 16.] Here it is to be carefully noticed that nothing has been said of the acUial coordinates of either A or B, or of the earth and moon, in space ; it is only the relative coordinates that are contemplated. Hence any expression, as AB, denoting a line considered with reference to direction as well as length, contains imj^licitly three numbers, and all lines parallel and equal to AB depend in the same way upon the same three. Hence, all lines which are erpial and parallel may he represented hy a common symbol, and that sytnbol contains three distinct numbers. In this sense a line is called a \'ECTOR, since by it we pass from the one extremity. A, to the other, B ; and it may thus be considered as an instrument which carries A to B : so that a vector may be employed to indicate a definite translation in space. 17.] We may here remark, once for all, that in establishing a new Calculus, we are at liberty to give any definitions whatever of our symbols, provided that no two of these interfere with, or contradict, each other, and in doing- so in Quaternions simplicity and (so to speak) naturalness were the inventor's aim. 18.] Let AB be represented by a, we know that a depends on three separate numbers. Now if CB be equal in length to AB and if these lines be parallel, we have evidently CD = AB = a, where it will be seen that the sign of equality between vectors contains implicitly equality in length and parallelism in direction. So far we have extended the meaning of an algebraical sj'mbol. And it is to be noticed that an equation between vectors, as contains three distinct equations between mere numbers. 19.] We must now define + (and the meaning of — will follow) in the new Calculus. Let A, B, C be any three points, and (with the above meaning of = ) let AB=a, BC= 13, AC=y. If we define + (in accordance with the idea (§ 16) that a vector represents a translation) by the equation a + /3 = y, 6 QUATERNIONS. [20. or AB + BC = AC, we contradict nothing that precedes, but we at once introduce the idea that vectors are to be compounded, in direction and magnitude, like simultaneous velocities. A reason for this may be seen in another way if we remember that by addinrj the differences of the Cartesian coordinates of A and " B, to those of the coordinates of B and C, we get those of the coordinates of A and C. Hence these coordinates enter linearly into the expression for a vector. 20.] But we also see that if C and A coincide (and C may be any point) i^ = 0, for no vector is then required to carry A to C. Hence the above relation may be written^ in this case, AB + BA = 0, or, introducing, and by the same act defining, the symbol — , BA=-AB. Hence, ike symbol —, applied to a vector, simply shows that its direction is to be reversed. And this is consistent with all that precedes ; for instance, AB + BC = IC, and AB = AC-BC, or =AC+CB, are evidently but different expressions of the same truth. 21.] In any triangle, ABC, we have, of coarse, AB + BC+ CA = 0; and, in any closed polygon, whether plane or gauche, AB + BC+ + YZ+ZA = 0. In the case of the polygon we have also AB + BC+ + rZ=AZ. These are the well-known propositions regarding composition of velocities, which, by the second law of motion, give us the geo- metrical laws of composition of forces. 22.] If we compound any number of parallel vectors, the result is obviously a numerical multiple of any one of them. Thus, if A, B, C are in one straight line, BC= xAB; where a; is a number, positive when B lies between A and C, other- wise negative : but such that its numerical value, independent of sign, is the ratio of the length of BC to that of AB. This is 25.] VECTORS, AXD THEIR COMPOSlTIOJTr^^,. ^^ 7 at once evident if AB and BC he commensurable ; and is easily extended to incommensurables by the usual rechictio ad absurdum. 23.] An important, but almost ob^4ous, proposition is that anj/ vector may he resolved^ and m one vxnj only, into three components parallel respectively to any three given rectors, no tioo of ivhich are parallel, and icliicli are not pjarallel to one pAane. Let OA, OB, OC be the three fixed vectors, OP any other vector. From P draw PQ parallel to CO, meeting- the plane BOA in Q. [There must be a definite point Q, else PQ, and therefore CO, would be parallel to BOA, a case specially excepted.] From Q draw QR parallel to BO, meeting OA in R. Then we have OP = OR ^- RQ ^ QP {^ 2\), and these components are respectively parallel to the three given vectors. By § 22 we may express OR as a numerical multiple of OA, RQ of OB, and QP of OC. Hence we have, generally, for any vector in terms of three fixed non-coplanar vectors, a, /3, y, OP = p = xa + y^ + zy, which exhibits, in one form, the t/iree numbers on which a vector depends (§ 16). Here x, y, z are perfectly definite, and can have but single values. 24.] Similarly any vector, as OQ, in the same plane with OA and OB, can be resolved into components 0^, RQ, parallel re- spectively to OA and OB ; so long, at least, as these two vectors are not parallel to each other. 25.] There is particular advantage, in certain cases, in employ- ing a series of three mutually perpendicular unit-vectors as lines of reference. This system Hamilton denotes by i,j, h. Any other vector is then expressible as p = xi-^yj + zk. Since i, j, k are unit- vectors, x, y, z are here the lengths of con- terminous edges of a rectangular parallelepiped of which p is the vector-diagonal ; so that the length of p is, in this case, Let trr = i^ + vJ+C^ be any other vector, then (by the proposition of § 23) the vector equation p = zr obviously involves the following three equations among numbers, «^ = i, .y = V, ^ = C 8 QUATERNIONS. [26. Suppose i to be drawn eastwards, j northwards, and k upwards, this is equivalent merely to saying that if two j)oints coincide, they are equally to the east {or west) of any third jmnt, equally to the north [or south) of it, and equally elevated above {or dejrressed below) its level. 26.] It is to be carefully noticed that it is only when a, /3, y are not coplanar that a vector equation such as or A-a-^y^ + zy = ^a^i)^+Cy, necessitates the three numerical equations •■^ = ^, >/ = -n, z = (. For, if a, /3_, y be coplanar (§ 24), a condition of the following form must hold y = aa + b^. Hence p = {x + za)a-\- {y + zh) ^, ^=(f+C^)a + (r, + C^)/3, and the equation p = ct now requires only the tioo numerical conditions x + za = ^^Ca, y-\-zi = r] + (b. 27.] The Commutative and Associative Laws hold in the combination of vectors by the signs + arid —. It is obvious that, if we prove this for the sign + , it will be equally proved for — , because — before a vector (§ 20) merely indicates that it is to be reversed before being considered positive. Let A, B, C, B be, in order, the corners of a parallelogram ; we have, obviously, JB = BC, AB - BC. And AB+BC= AC= AB+BC= BC-\-AB. Hence the commutative law is true for the addition of any two vectors, and is therefore generally true. Again, whatever four points are represented by A, B, C, B, we have AB = AB-i-BB = AC+CB, or substituting their values for AB, BB, AC respectively, in these three expressions, AB + BC+CB = AB + {BC+CB)= {IB + BC) + CB. And thus the truth of the associative law is evident. 28.] The equation p — r^a where p is the vector connecting a variable point with the origin, j3 a definite vector, and x an indefinite number, represents the straight line drawn from the origin parallel to jQ (§ 22). 30.] VECTORS, AND THEIll COMPOSITION. .. 9 The straight line drawn from A, where OA =. a, and parallel to p, has the equation fj = a + .T/S , (1) In words, we may pass directly from to P by the vector OP or p ; or we may pass first to A, by means of OA or a, and then to P along a vector parallel to /3 (§ 16). Equation (1) is one of the many useful forms into which Quater- nions enable us to throw the general equation of a straight line in space. As we have seen (§ 25) it is equivalent to f/iree numerical equations ; but, as these involve the indefinite quantity x, they are virtually equivalent to but i/co, as in ordinary Geometry of Three Dimensions. 29.] A good illustration of this remark is furnished by the fact that the equation p — ua + x(3, which contains two indefinite quantities^ is virtually equivalent to only one numerical equation. And it is easy to see that it re- presents the plane in which the lines a and /3 lie ; or the surface which is formed by drawing, through every point of OA, a line parallel to OB. In fact, the equation, as written, is simply § 24 in symbols. And it is evident that the equation P = y + ya + a?/3 is the equation of a plane passing through the extremity of y, and parallel to a and (3. It will now be ob\nous to the reader that the equation P =7^l'^l+/'2«2+ = ^i'"^ where a^, Og, &c. are given vectors, and 2\,P2, &c. numerical quan- tities, rejoresents a straight line i^ Pi,P2J ^^- ^^ linear functions of 0716 indeterminate number ; and a plane, if they be linear expres- sions containing tioo indeterminate numbers. Later (§31 (l)), this theorem will be much extended. Again, the equation p_^,^^^^^.^ refers to anj/ point whatever in space, provided a, /3, y are not coplanar. {Ante, § 23). 30.] The equation of the line joining any two points A and B, where OA = a and OB — /3, is obviously p = a + x{l3 — a), or p = l3+9/{a-fi). These equations are of course identical, as may be seen by putting 1 — _y for X. 10 QUATERNIONS. [3 1. The first may be written p + (x—l)a — xl3 = 0; or Pp-\- qa. + rf3 = 0, subject to the condition p-L q-^r = identically. That is — A homogeneous linear function of three vectors, equated to zero, expresses that the extremities of these vectors are in one straight line, if the sum of the coefficients he ulenticctUy zero. Similarly, the equation of the plane containing the extremities A^ B, C of the three non-coplanar vectors a, /3, y is p = a + ;r(/3-a) + J/(y-/3), where x and y are each indeterminate. This may be written ;jp + ^a + f/3 + 5y = 0, with the identical relation li + q-^-r + s = 0. which is the condition that four points may lie in one plane. 31.] We have already the means of proving, in a very simple manner, numerous classes of propositions in plane and solid geo- metrv. A very few examples, however, must suffice at this stage ; since we have hardly, as yet, crossed the threshold of the subject, and are dealing with mere linear equations connecting two or more vectors, and even with them 7ve are restricted as yet to ojierations of mere addition. We will give these examples with a painful minute- ness of detail, which the reader will soon find to be necessary only for a short time, if at all. (fl.) The diagonals of a parallelogram bisect each other. Let ABCD be the parallelogram, the point of intersection of its diagonals. Then AO^OB = lB = WC= Bb+OC, which gives AO-OC= BO-OB. The two vectors here equated are parallel to the diagonals respect- ively. Such an equation is, of course, absurd unless (1) The diagonals are parallel, in which case the figure is not a parallelogram ; (2) Id = OC, and BO = OB, the proposition. (b.) To shoto that a triangle can be constructed^ ivhose sides are parallel, and equal, to the bisectors of the sides of any triangle. Let ABC be any triangle, Aa, Bb, Cc the bisectors of the sides. I.] VECTOES, AND THEIR COMPOSITION. 11 Then Aa = AB+Ba = AB + ^ BC, m - - - = BC+ \ CA, a - - - = CA + ^AB. Hence A^ + Td + C^=%(AB + BC+C^i) = ; which (§21) proves the proposition. Also I^ = AB+^BC = AB-\{CA + lB) = \{IB-Cl) = \{lB+AC), results which are sometimes useful. They may be easily verified by producing" Aa to twice its length and joining the extremity with B. {¥.) The bisectors of the sides of a triangle meet in a jwint, which trisects each of them. Taking A as origin, and putting a, (3, y for vectors parallel, and equal, to the sides taken in order BC, CA, AB ; the equation of Bb is {§28(1}) p = y + x(y+ ^) = (l+.r)y+-/3. That of Cc is, in the same way, At the point 0, where Bb and Cc intersect, p = (l+^)y+-/3=-(l+^)/3-|y. Since y and /3 are not parallel, this equation gives l+a? = -|, and - = _(i+y). From these a; = 3^ = — |, Hence AO = \ (y-/3) = | 2^. (See Ex. {Ij).) This equation shows, being a vector one, that Aa passes through 0, and that JO: Oa\:2\l. {c) If OA = a, 0B=^, 0C=aa+bl3, be three given co-jolanar vectors, and the lines in- dicated in the fgnre he drawn, the points a^,t)^,c^ lie in a straight line. 12 QUATERNIONS. [31. We see at once, by the process indicated in § 30, that ~ , Ua — • 1—^ \—a Oc = a + b Hence we easily find I— a — 2b Ob Oa^ Ob^ = a a Oc^ = — aa-{^bj3 l — 2a — b ^~ b—a These give -{\-a-2b)Oi^^{\-2a-b)Ob^-{b-d)Oc^ = 0. But —{l-a — 2b) + {l — 2a — b) — (b—a)=0 identically. This, by § 30, proves the proposition. {(I.) Let OA = a, OB = ^, be any two vectors. If MP be jm rail el to OB j and OQ, BQ, be drawn parallel to AP, OP resj)ectively ; the locus of Q is a straight line ]}arallel to OA. _ Let 031 = ea. Then_ AP = e~la + a;j3. Hence the equation of OQ is p = ?/{e- la + x^) ; and that oi BQ is ^B W p = (3 + z{ea + xl3). At Q we have, therefore, x?/ = 1 + ::x, ) ^{e-\) = ce.S These give x?/ = e, and the equation of the locus of Q is p = efi + /a, i. e. a straight line parallel to OA, drawn through N in OB pro- duced, so that ON -.OB:: 031: OA. _CoR. If BQ meet 3IP in q,¥q = ^; and if AP meet NiQ in p, Qp=a. _____ Also, for the point B we have jjR = AP., QR = Bq. Hence, if from any tivo points, A and B, lines be draivn intercepting a given length Pq on a given line 3Iq; and if, from B their point of intersection, Bp be laid off ■= PA, and BQ — qB ; Q and p lie on a fixed straight line, and the length of Qp is constant. (e.) To find the centre of inertia of any system. If OA = a, OB = Oj, be the vector sides of any triangle, the vector from the vertex dividing the base AB in C so that I-] VECTORS, AND THEIR COMPOSITION. 13 BC:CA:: m : m^ ma + m-f^tti m -\- m^ For AB is a^ — a, and therefore AC is [a^ — a). m + m^ Hence 6C = OA + AC = a + IS («j — a) This expression shows how to find the centre of inertia of two masses ; m at the extremity of a, m^ at that of a^ . Introduce ni.^ at the extremity of a.^, then the vector of the centre of inertia of the three is, by a second application of the formula, ^ma + m-, a,\ -) + m^ Go {m+m{) (- w^ + m-^ m a + w?j^ 0;^ + m^ a.2 {in + ;«i) + w^2 m + ?«j + ;i>/2 For any number of masses, expressed generally by m at the extre- mity of the vector a, we have the vector of the centre of inertia This may be written 2 m (a— [3) = 0, Now aj^ — (3 is the vector of m^ with respect to the centre of inertia. Hence the theorem. If the vector of each element of a mass, drawn from the centre of inertia, he increased in length in projjortion to the mass of the element, the sum of all these vectors is zero. (/",) We see at once that the equation where ;! is an indeterminate number, and a, (3 given vec- tors, represents a parabola. The origin, 0, is a point on the curve, ^ is parallel to the axis, i. e. is the diameter OB drawn from the origin, and a is OA the tangent at the origin. In the figure QP=at, 0Q = ^- 14 QUATERNIONS. [3 1. The secant joining the points where t has the values t and f is represented by the equation , = „, + ^ +,(„,-+ ^'-a<-^) (5 30) Put f= i, and write x for x{t' —t) [which may have any value] and the equation of the tangent at the point {t) is rut X — —I, a — — J or the intercept of the tangent on the diameter is —the abscissa of the point of contact. Otherwise : the tangent is parallel to the vector a-\-^t or ai5 + /3^^ or at+ ^ + ^ or OQ+OF. But TF =z TO + OP, hence TO = OQ. (j/.) Since the equation of auy tangent to the parabola is p = al + -^ + x{a+(3i), let us find the tangents which can be drawn from a given point. Let the vector of the point be p=pa + q(3 (§ 24). Since the tangent is to pass through this point, we have, as con- ditions to determine t and x, i-{-x =: p, -^xt = r, by equating respectively the coefiicients of a and j8. Hence ^ = P+ ^P' — 2q. Thus, in general, tivo tangents can be drawn from a given point. These coincide if pi —. ^q • that is, if the vector of the point from which they are to be drawn is «2 p -pa^-q^ =pa^ —{3, i. e. if the point lies o?i the parabola. They are imaginary if 2q>_p'^, i. e. if the point be P =i^«+(^ +r)(3, 2 r being posiiive. Such a point is evidently within the curve, as at 2 n, where OQ = ^/3, Qp = pa, PR = rfi. 31.] VECTORS, AND THEIR COMPOSITIOlSr. 15 [Ji.) Calling- the values of / for the two tang-ents found in {g) fi and t^ respectively, it is obvious that the vector join- ing' the points of contact is which is parallel to f -{ f. or, by the values of i^ and t.^ in (//), Its direction, therefore, does not depend on q. In words, If pairs of tangents be drawn to a parabola from points of a diameter produced, the chords of contact are parallel to the tangent at the vertex of the diameter. This is also proved by a former result, for we must have OT for each tangent equal to Q 0. (i.) The equation of the chord of contact, for the point whose vector is p=pa + ql3, 81 ^ is thus p = a/i+ ~^-\-^{a+p(3). Suppose this to pass always through the point whose vector is p = aa + b(3. Then we must have or fi = p± Vp' — 2pa + 2b. Comparing" this with the expression in {g), we have q =pa — b; that is, the point from which the tangents are drawn has the vector p =pa + [pa — b) ft — —]jji-\p{a + aft), a straight line (§ 28 (1)). The mere form of this expression contains the proof of the usual properties of the pole and polar in .tlie parabola ; but, for the sake of the beginner, we adopt a simpler, though equally general, process. Suppose a = Q. This merely restricts the pole to the particular diameter to which we have referred the parabola. Then the pole is Q, where p = bjS; and the polar is the line TU, for which p = -bft-\-pa. 16 QUATERNIONS. [3 1, Hence the polar of an?/ point is parallel to the tangent at the extremity of the diameter on which the 'point lies, and its intersection with that diameter is as far beyond the vertex as the iwle is within, and vice versa. (J.) As another example let us prove the following theorem. If a triangle he inscribed in a parabola, the three points in 2vhich the sides are met by tangents at the angles lie in a straight line. Since is any point of the curve, we may take it as one corner of the triangle. Let / and t^ determine the others. Then^ if CTi, OTo, -33-3 represent the vectors of the points of intersection of the tangents with the sides, we easily find *•'' (a+|/3). These values give tt^ -CTo := ; a. n^-t 2t-t^ t^-t^ Also — V- 7—^ — \7 = identically. t^ t tt^ "^ Hence, by § 30, the proposition is proved. (/(•.) Other interesting examples of this method of treating curves will, of course, suggest themselves to the student. Thus p = a cos t-{- fi sin t or p = a«+/3 \/l — «^ represents an ellipse, of which the given vectors a and /3 are semi- conjugate diameters. Again, p = at-\-- or p = a tan x + 13 cot x o evidently represents a hyperbola referred to its asymptotes. But, so far as we have yet gone with the explanation of the calculus, as we are not prepared to determine the lengths or in- clinations of vectors, we can investigate only a very small class of the properties of curves, represented by such equations as those above written. x30 ] VECTORS, AND THEIR COMPOSITION. 17 (l.) We may now, in extension of the statement in § 29, make the obvious remark that p = S7;a is the equation of a cirn-e in space, if the numbers Pi,jJ.2, &c. are functions of one indeterminate. In such a case the equation is sometimes written p = <^ (0- But, [^ Pi,j)o, &c. be functions of f/ro indeterminates, the locus of the extremity of p is a surface ; whose equation is sometimes written p = t, 35.] It is obvious that, in the present case, because the vectors involved in are constant, and their numerical mult'ipUers alone vary, the expression (j){t + bt) is, by Taylor's Theorem equivalent to m.^^^qp^. Hence, dc^i^) ^, J'^ii) i^t)'. op = j^ Ot-\ -j^ -;;—;; 1- &C. dt ' dt^ 1.2 And we are thus entitled to write, when bt has been made inde- finitely small, Limit (^) ='!^='^P=ct:.\t). In such a case as this, then, we are permitted to differentiate, or to form the differential coefficient of, a vector, according to the ordinarv rule^ of the Differential Calculus. But great additional insight into tlie process is gained by apijlying- Newton's method. 35.] Let OP be _ P = <\>{t), and Oq^ p = 4^{t + dt), where dt is any number whatever. The number t may here be taken as representing time, i. e. we may suppose a point to move aloni^- the curve in such a way that the value of t for the vector of point P of the curve denotes the interval which has elapsed (since a fixed epoch) when the moving point has reached the extremity of that vector. If, then, dt represent any interval, finite or not, we see that OQ, = cl.{t+dt) will be the vector of the point after the additional interval dt. T,S.~\ VECTORS, AND THEIK COMPOSITION. 19 But this, in g-eneral, gives us little or no information as to the velocity of the point at F. We shall get a better approximation by halving- the interval df, and finding- Q.y, where 0Q^= (lf{t+\dt), as the podtion of the moving- point at that time. Here the vector virtually described in ^dt in FQo- To find, on this supposition, the vector described in df, we must double I^Q-;,} ^i^d we find, as a second approximation to the vector which the moving point would have described in time dt, if it had moved for that period in the direction and with the velocity it had at P, Tq, = 2 PQ,, = 2 {OQ2- OP) = 2{(ji{t+\dt)-4>{i)}. The next approximation gives Pq,= 3PQ,= 3{6Q,-0P) = 3{(P{i + idt)-(t>{t)}. And so on, each step evidently leading- us nearer the sought truth. Hence, to find the vector which would have been described in time dl had the circumstances of the motion at P remained undisturbed, we must find the value of df> = Tci = ^...«.'|c/>(^ + Ut)-{t)]^. We have seen that in this particular case we may use Taylor's Theorem. We have, therefore, = c}>'{l)di. And, if we choose, we may now write '! = *-». 37.] But it is to be most particularly remarked that in the whole of this investigation no regard whatever has been paid to the magnitude of df. The question which we have now answered may be put in the form — A point descrides a given curve in a given manner. At any point of its path ifs motion suddenly ceases to be accelerafed. What space will it describe in a definite interval? As Hamilton well observes, this is, for a planet or comet, the case of a ' celestial Atwood's machine.' 38.] If we suppose the variable, in terms of which p is expressed, to be the arc, s, of the curve measured from some fixed point, we find as before , ,,^, ,, cb'(t)ds a p = (p [t] dt = (p'{s)ds. c 2 ds dt 20 QUATERNIONS. [39. From the very nature of the question it is obvious that the leng-th of dp must in this case be ds. This remark is of importance, as we shall see later ; and it may therefore be useful to obtain afresh the above result without any reference to time or velocity. 39.] Following- strictly the process of Newton's Vllth Lemma, let us describe on Pq.y an arc similar to PQ^, and so on. Then obviously, as the subdivision of ds is carried farther, the new arc (whose length is always ds) more and more nearly coincides with the line which expresses the corresponding- approximation to dp. 40.] As a final example let us take the hyperbola P = at + -. Here dp = (a — -^) dt. This shews that the tangent is parallel to the vector at • In words, if the t^ector {from the centre) of a jioint in a hyperbola be one diagonal of a parallelogram, two of whose sides coincide loilh the asymptotes, the other diagonal is ptarallel to the tangent at the point. 41.] Let us reverse this question, and seek the envelope of a line which cuts off from two fixed axes a triangle of constaftt area. If the axes be in the directions of a and /3, the intercepts may evidently be written at and -- . Hence the equation of the line is (§30) . p^=.at-{x{— — atj- The condition of envelopment is, obviously, (see Chap. IX.) dp = 0. This g-ives =\a—x (.^^ + «) [ '^^ + (l^ °0 ^^^ *" Hence {\—x)dt — t dx = 0, , X ,, dx and — — di+= 0. * We are not here to equate to zero the coefficients of dt and dx ; for we must remember that this equation is of the form where p and q are numbers ; and that, so long as a and & are actual and non-parallel vectors, the existence of such an equation requires J) = 0, 9 = 0. 43-] VECTORS, AND THEIR COMPOSITION. 21 From these, at once, ^ = h^ since dx and di are indeterminate. Thus the equation of the envelope is B p = at+\ (y — a^) the hyperbola as before ; a, fi being- portions of its asymptotes. 42.] It may assist the student to a thorough comprehension of the above process, if we put it in a slightly different form. Thus the equation of the enveloping line may be written p = at{l-a;) + l3-, which gives rfp = = ad {/ (1 — «)) +/3^/ (-) • Hence, as a is not parallel to /3, we must have d{Hi-x)) = 0, ^'^(^) = 0; and these are, when expanded, the equations we obtained in the preceding section. 43.] For farther illustration we give a solution not directly em- ploying the differential calculus. The equations of any two of the enveloping lines are p = at + CO (- atji p =at^ + Xi(j-at^), t and /j being given, while x and x^ are indeterminate. At the point of intersection of these lines we have (§ 26), t{\-x) = t^{\-x^, \ X _Xi > These give, by eliminating x-^^ t{\-x) -t^(\- j-x), or X = t t h + t Hence the vector of the point of intersection is ait^ + 13 22 QUATERNIONS. [44. and thus, for the ultimate intersections, where .^ y = 1, /3 P=K«^ + 7) as before. Cor. (1). If tf^ = I, a + /3 P = '-] or the intersection lies in the diagonal of the parallelogram on a, ft. Cor. (2). If t^ =: mt, where f/i is constant, mta + — P = But we have also x = m+ 1 1 Hence t/ie locus of a point which divides in a given ratio a line cutting off a given area from tivo fixed axes, is a hyperbola of which these axes are the asymptotes. Cor. (3). If we take tt-^ (t+t-i) = constant the locus is a parabola ; and so on. 44.] The reader who is fond of Anharmonic Ratios and Trans- versals will find in the early chapters of Hamilton's Elements of Quaternions an admirable application of the composition of vectors to these subjects. The Theory of Geometrical Nets, in a plane, and in space, is there very fully developed ; and the method is shewn to include, as particular cases, the processes of Grassmann's AusdehmingsJehre and Mobius' Barycentrische Calcnl. Some very curious investigations connected with curves and surfaces of the second and third orders are also there founded upon the composition of vectors. EXAMPLES TO CHAPTER I. 1. The lines which join, towards the same parts, the extremities of two equal and parallel lines are themselves equal and parallel. [Euclid, I. xxxiii.) 2. Find the vector of the middle point of the line which joins EXAMPLES TO CHAPTER I. 23 the middle points of the diagonals of any quadrilateral, plane or g-aucl'.e, the vectors of the corners being- given ; and so prove that this i^oint is the mean point of the quadrilateral. If two opposite sides be divided proportionally, and two new quadrilaterals be formed by joining" the points of division_, the mean points of the three quadrilaterals lie in a straight line. Shew that the mean point may also be found by bisecting the line joining the middle points of a pair of opposite sides. 3. Verify that the property of the coefficients of three vectors whose extremities are in a line (§ 30) is' not interfered with by altering the origin. 4. If two triangles ABC, abc, be so situated in space that Aa, Bb, Cc meet in a point, the intersections o^ AB, ah, of BC, he, and of CA, ca, lie in a straight line. 5. Prove the converse of 4, i. e. if lines be drawn, one in each of two planes, from any three points in the straight line in which these planes meet, the two triangles thus formed are sections of a common pyramid. 6. If five quadrilaterals be formed by omitting in succession each of the sides of any pentagon, the lines bisecting the diagonals of these quadrilaterals meet in a point. (H. Fox Talbot.) 7. Assuming, as in § 7, that the operator cos 0+ v— 1 sin d turns any radius of a given circle through an angle in the positive direction of rotation, without altering its length, deduce the ordinary formulae for cos (A + B), cos (A — B), sin (A + B), and sin (A — B), in terms of sines and cosines of A and B. 8. If two tangents be drawn to a hyperbola, the line joining the centre with their point of intersection bisects the lines joining the points where the tangents meet the asymptotes : and the tangent at the point where it meets the curves bisects the intercepts of the asymptotes. 9. Any two tangents, limited b}^ the asymptotes, divide each other proportionally. 10. If a chord of a hyperbola be one diagonal of a parallelogram whose sides are pai-allel to the asymptotes, the other diagon;d passes through the centre. 1 1 . Shew that p = x^ a + //'^ l3 + {x+ i/f y is the equation of a cone of the second degree, and that its section by the plane ^; a + ^ /3 + r y j^ + (7 + r 24 QUATERNIONS. is an elli])se which touches, at their middle points, the sides of the triangle of whose corners a, (3, y are the vectors. (Hamilton, Elements, p. 96.) 12. The lines which divide, proportionally^ the pairs of opposite sides of a gauche quadrilateral, are the generating' lines of a hyper- bolic paraboloid. {Ibid. p. 97.) 13. Shew that p = x^a+i/^^ + z^y, where x + y-\-z = 0, represents a cone of the third order, and that its section by the plane _ pa + q l3 + ry ~ p+q+r is a cubic curve, of which the lines pa + qi^ p = — , KC. p + q are the asymptotes and the three (real) tangents of inflexion. Also that the mean point of the triangle formed by these lines is a conjugate point of the curve. Hence that the vector a + ^ + y is a conjugate ray of the cone. {Idid. p. 96.) CHAPTER 11. PRODUCTS AND QUOTIENTS OF VECTORS. 45.] We now come to the consideration of points in which the Calcuhis of Quaternions differs entirely from any previous matlie- matical method ; and here we shall g-et an idea of what a Qua- ternion is, and whence it derives its name. These points are fundamentally involved in the novel use of the symbols of mul- tiplication and division. And the simplest introduction to the subject seems to be the consideration of the quotient, or ratio, of two vectors. 46.] If the g-iven vectors be parallel to each other, we have already seen (^^ 22) that either may be expressed as a numerical multiple of the other ; the multiplier being- simply the ratio of their lengths, taken positively if they are similarly directed, nega- tively if they run opposite ways. 47.] If they be not parallel, let OA and OB be drawn parallel and equal to them from any point ; and the question is reduced to finding the value of the ratio of two vectors drawn from the same point. Let us try to find u]ion hoio 7nany distinct mmibers this ratio depends. We may suppose OA to be changed into OB by the following processes. 1st. Increase or diminish the length of OA till it becomes equal to that of OB. For this only one number is required, viz. the ratio of the lengths of the two vectors. As Hamilton remarks, this is a positive, or rather a signless, number. 2ud. Turn OA about until its direction coincides with that of OB, and (remembering the effect of tlie first operation) 26 QUATEKNlOxNS. [48. we see that the two vectors now coincide or become identiciil. To specify this operation three more numhers are required, viz. fioo angles (such as node and inclina- tion in the case of a ])lanet's orbit) to fix the plane in which the rotation takes place, and one angle for the amount of this rotation. Thus it appears that the ratio of two vectors, or the multiplier rec{uired to change one vector into another, in general depends upon four distinct numbers, whence the name QUATERNiojf. The particular case of perpendicularity of the two vectors, where their quotient is a vector perpendicular to their plane, is fully con- sidered below; §§ 64, 65, 72, &c. 48,] It is obvious that the operations just described may be performed, with the same result, in the opposite order, being per- fectly independent of each other. Thus it appears that a quaternion, considered as the factor or agent which changes one definite vector into another, may itself be decomposed into two factors of which the order is immaterial. The stretching factor, or that which performs the first operation in § 4 7, is called the Tensor, and is denoted by prefixing T to the quaternion considered. The turning factor, or that corresponding to the second operation in § 47, is called the Versor, and is denoted by the letter ?7 prefixed to the quaternion. 49.] Thus, if OA = a, OB = /3, and if q be the quaternion which changes a to /3, we have 13 = qa, which we may write in the form a = q, or 13a ^ = q, if we agree to dejine that 13 — .a = /3a \ a = (3. a Here it is to be particularly noticed that we write q tjefore a to signify that a is multiplied by q, not q multiplied by a. This remark is of extreme importance in quaternions, for, as we shall soon see, the Commutative Law does not generally apply to the factors of a product. We have also, by §§ 4 7, 48, q = Tq Uq = Uq Tq, 5<-] PKODUCTS AND QUOTIENTS OF VECTORS. 27 where, as before, Tq depends merely on the relative lengths of a and /3, and Uq depends solely on their directions. Thus, if a^ and /3| be vectors of unit length parallel to a and /3 respectively. r4 = i. «i U /3i U (3 As will soon be shewn, when a is perpendicular to {3, the versor of the quotient is quadrantal, i. e. it is a unit-vector. 50.] We must now carefully notice that the quaternion which is the quotient when /3 is divided by a in no way depends upon the absolute lengths, or directions, of these vectors. Its value will remain unchanged if we substitute for them any other pair of vectors which (1) have their lengths in the same ratio, (2) have their common plane the same or parallel, and (3) make the same angle with each other. Thus in the annexed figure if, and only if. OB V2 (1) Oi^i _ OB O^A^ OA (2) plane AOB parallel to plane Aj^O^B^, (3) LAOB = ZA^O^B^. [Equality of angles is understood to include similarity in direction. Thus the rotation about an upward axis is negative (or right-handed) from OA to OB, and also from 0^ A^ to 0^ 2?^.] 51.] The Beciprocal of a quaternion q is defined by the equation, Hence if we must have For this gives q- — qq- q - = ^y, ( a /3= qa, a _ 1 '^~q J./3 = q-'^.qa, = r'- and each member of the equation is evidently equal to a. 28 QUATERXIOXS. [5^ Or, we may reason thus, q chang-es OA to OB, (f'^ must therefore change OB to OA, and is therefore expressed by- (§ 49). The tensor of the reciprocal of a quaternion is therefore the reciprocal of the tensor ; and the versor differs merely by the reversal of its representative angle. The versor, it must be remembered, gives the plane and angle of the turning — it has nothing to do with the extension. 52.] The Conjugate of a quaternion q, written Kq, has the same tensor, plane, and angle, only the angle is taken the reverse way. Thus, if OA, OB, OA', lie in one plane, and if 0A'= OA, and IA'0B=IA0B, we have OB ,0B ■ , f r -^=^ = a, and^=r- = coniuo-ate oi 6r = Kq. OA ^ OA' 11 By last section we see that Kq = [Tqfq-\ Hence 1^1 = ^^ • ? = (^?)^- This proposition is obvious, if we recollect that the tensors of q and Kq are equal, and that the versors are such that either annuls the effect of the other. The joint effect of these factors is therefore merel}' to multiply twice over by the common tensor. 53.] It is evident from the results of § 50 that, if a and /3 be of equal length, their quaternion quotient becomes a versor (the tensor being unity) and may be represented indifferently by any one of an infinite number of arcs of g'iven length lying on the circumference of a circle, of which the two vectors are radii. This j's of considerable importance in the proofs which follow. Thus the versor -^=r may be represented in magnitude, plane^ and direction (§ 50) by the arc AB, which may in this extended sense be written AB. And, similarly, the versor OB, OA, IS repre- sented by A^^B, which is equal to (and measured in the same direction as) AB if ZJ^O^i = AAOB, i.e. if the versors are equal, in the quaternion meaning' of the word. 56.] PRODUCTS AND QUOTIENTS OF VECTOUS. 29 54.] By the aid of this process, when a versor is represented as an arc of a great circle on the unit-sphere, we can easily prove that quaternion nniHijjlication is not gene rail 1/ commutative. Thus let q be the versor AB or -_=- • _ _ . OA Make BC = AB, (which, it must be remembered, makes the points A, B, C lie in one great circle) ^ then q may also be represented by ^=r- • In the same way any other versor r ^ ^ ryD f)V may be represented by BB or BE and by -^=- or — ^ • OB OB The line OB in the figure is definite, and is given by the inter- section of the planes of the two versors ; being the centre of the unit-sphere. Now rOD = OB, and qOB = 00, Hence qr OB = OC, OC '— - or qr = -z=:r > and may therefore be represented by the arc I)C of ^ OB I . a great circle. But rq is easily seen to be represented by the arc AE. For qOA= OB, and rOB = OB, whence rq OA =- OB, and rq = -=^ • ^ OA Thus the versors rq and qr, though represented by arcs of equal length, are not generally in the same plane and are therefore un- equal : unless the planes of q and ;• coincide. Calling OA a, we see that we have assumed, or defined, in the above proof, that q.ra = qr.a and r.qa = rq.a when qa, ra, q.ra, and r.qa are all vectors. 55.] Obviously CB is Kq, Bl) is Kr, and CB is K{qr). But CB = BB.CB, which gives us the very important theorem K{qr) - Kr.Kq, i.e. the conjzigate of the product of tic quaternions is the product of (heir conjugates in inverted order. 56.] The propositions just proved are, of course, true of quater- nions as well as of versors ; for the former involve only an additional 30 QUATEl^NIONS. [57. minieritiil factor which has reference to the leng-th merely, and not the direction, of a vector (§ 48). 57. J Seeing- thus that the commutative law does not in general hold in the multiplication of quaternions, let us enquire whether the Associative Law holds. That is, \i 2h ?j ^' ^^ three quaternions, have we ^^_^^. _^5^_,.9 This isj of course, obviously true if 7;, q, r be numerical quantities, or even auy of the imaginaries of algebra. But it cannot be con- sidered as a truism for symbols which do not in general give M = ^P- 58.] In the first place we remark that jh ^} and r may be con- sidered as versors only, and therefore represented by arcs of great circles^ for their tensors may obviously (§ 48) be divided out from both sides, being commutative with the versors. Let AB =p, ^=CA = q,andFIJ = r. Join BC and produce the great circle till it meets EF in H, and make KE = FB = r, and HG=CB = pq (§ 54). »r ^ ,, Join GK. Then ^,.^'-^''''^^~7fK--J} , KG - HG . KH = pa . r. ^rvr 1--<'T^^^ •'^oin FIJ and produce it to (" /^^X^T^N meet AB in if. Make \^ ^mC. ^ ^=^. ^^ -j^'^jf^ and join i\\f. Then LN=^3m.Z3I = p.qr. Hence to shew that p. qr = jjq. r all that is requisite is to prove that FN, and KG, described as above, are equal arcs of the same great circle, since, by the figure, they are evidently measured in the same direction. This is perhaps most easily effected by the help of the fundamental properties of the curves known as Spherical Conies. As they are not usually familiar to students, we make a slight digression for the purpose of proving these fundamental properties ; after Chasles, by whom and Magnus they were discovered. An independent proof of the asso- ciative principle will presently be indicated, and in Chapter VII we shall employ quaternions to give an independent proof of the theorems now to be established. 59.*] Def, a spherical conic is the curve of intersection of a cone of the seco7i(l degree with a spJiere, the vertex of the cone being the centre (f the sphere. 59-] rK'JDUCTS AND QUOTIENTS OF VECTORS. 31 Lemjlv. If a one have one series of circular sections, it has another series, and any two circles belonging- to different series lie on a sphere. This is easily proved as follows. Describe a sphere. A, cutting' the cone in one circular section, C, and in any other point whatever, and let the side Oj)P of the cone meet A in j), P ; P being a point in C. Then PO-Oj) is constant, and, therefore, since P lies in a plane, p lies on a sphere, a, passing through 0. Hence the locus, c, of p is a circle, being the intersection of the two spheres A and a. Let OqQ be any other side of the cone, q and Q being points in c, C respectively. Then the quadrilateral q Q Pp is inscribed in a circle (that in which its plane cuts the sphere A) and the exterior angle at p is equal to the interior angle at Q. If OL, OM he the lines in which the plane POQ cuts the cy die planes (planes through parallel to the two series of circular sections) they are obviously parallel to pq, QP, respectively ; and therefore iLOp = LOpq = lOQP = 13I0Q. Let any third side, OrP, of the cone be drawn, and let the plane OPE cut the cyclic planes in 01, Om respectively. Then, e\ddently, Z lOL = Z qpr, Z MOm = Z QPP, and these angles are independent of the position of the points ^y and P, if Q and R be fixed points. In a section of tlie above diagram by a sphere whoso centre is 0, IL, Mm are the great circles which repre- sent the cyclic planes, PQJl is the spherical conic which rej)resents the cone. The '"^ point P represents the line OpP, and so with the others. The propositions above may now be stated thus Arc PL = arc MQ ; and, if Q and R be fixed, Mm and /L are constant arcs whatever be the ])Osition of P. 32 QUATERNIONS. [60. 60.] The application to § 58 is now obvious. In the figure of that article we have FE=KH, E1J = CA, HG=6b, L3I = FB, Hence Z, C, G, D are points of a spherical conic whose cyclic planes are those of AB^ FE. Hence also KG passes through 7y, and with LM intercepts on AB an arc equal to AB. That is, it passes through N, or KO and LN are arcs of the same great circle : and they are equal, for G and L are points in the spherical conic. Also, the associative principle holds for any number of quaternion factors. For, obviously, qr.st = qrs.t = &c., &c., since we may consider cp' as a single quaternion, and the above proof applies directly. 61.] That quaternion addition, and therefore also subtraction, is commutative, it is easy to shew. For if the planes of two quaternions, q and r, intersect in the line OA, we may take any vector OA in that line, and at once find two others, OB and OC, such that OB^ qOA, and OC = r OA. And (q + r)OA = OB-\-OC =OC+OB = ir+q)6A, since vector addition is commutative (§ 27). Here it is obvious that {q^r) OA, being the diagonal of the parallelogram on OB, OC, divides the angle between OB and OC in a ratio depending solely on the ratio of the lengths of these lines, i. e. on the ratio of the tensors of q and r. This will be useful to us in the proof of the distributive law, to which we proceed. 62.] Quaternion multi- plication, and therefore di- vision, is distributive. One simple proof of this depends on the possibility, shortly to be proved, of representing ati^ quaternion as a linear function of three given rect- angular unit-vectors. And when the proposition is thus established, the associative principle may readily be deduced from it. But we may employ for its proof the properties of Spherical 6^.~\ PRODUCTS AND QUOTIENTS OP VECTORS. 33 Conies already employed in demonstrating- the truth of the asso- ciative principle. For continuity we give an outline of the proof by this process. Let £A, CA represent the versors of q and r, and Ic the great circle whose plane is that of 7;. Then, if we take as operand the vector OA, it is obvious that U{q + r) will be represented by some such arc as DA where £, D, C are in one great circle ; for (q + r) OA is in the same plane as q OA and rOA, and the relative magnitudes of the arcs BB and DC depend solely on the tensors of q and r. Produce DA, DA, CA to meet dc in b, d, c respectively^ and make m = BA, Fd= DA, Gc=CA. Also make h^ = dh = cy = p. Then E, F, G, A lie on a spherical conic of w^hich DC and be are the cyclic arcs. And, because b^ = db = cy, ^E, bF, yG, when produced, meet in a point H which is also on the spherical conic (§ 59"^). Let these arcs meet DC in /, L, K respectively. Then we have JH=M = P Uq, LH=Fh =2iU{q + r), KII = Gy = p Ur. Also LJ ^ DB, and KL = CD. And, on comparing the portions of the figure bounded respectively by IIKJ and. by ACB we see that (when considered with reference to their effects as factors multiplying OH and OA respectively) p U{q + r) bears the same relation to p Uq and^j Ur that U{q + r) bears to Uq and Ur. But T{q + r)U(q + r) = q + r = TqUq + TrUr. Hence T{q + r)p U{q + r) = Tq.p Uq + Tr.p Ur ; or, since the tensors are mere numbers and commutative with all other factors, i^(?+n — pq-Vpr- In a similar manner it may be proved that {q-\-r)p = qp + rp. And then it follows at once that {p + q){r->rs) = j^r +ps + qr -f qs. 63.] By similar processes to those of § 53 we see that versors, and therefore also quaternions, are subject to the index-law at least so long as ?i/ and n are positive integers. D 34 QUATERNIONS. [64. The extension of this property to negative and fractional ex- ponents must be deferred until we have defined a negative or fractional power of a quaternion. 64.] We now proceed to the special case of quadrantal versors, from whose properties it is easy to deduce all the foregoing results of this chapter. These properties were indeed those whose in- vention by Hamilton in 184 3 led almost intuitively to the esta- blishment of the Quaternion Calculus. We shall content ourselves at present with an assumption, which will be shewn to lead to consistent results ; but at the end of the chapter we shall shew that no other assumption is possible, following for this purpose a very curious quasi- metaphysical speculation of Hamilton. 65.] Suj)pose we have a system of three mutually perpendicular unit-vectors, drawn from one point, which we may call for short- ness 7, /, K, Suppose also that these are so situated that a positive (i. e. left-handed) rotation through a right angle about I as an axis brings J to coincide with K. Then it is obvious that positive quadrantal rotation about / will make K coincide with /; and, about K, will make 7 coincide with 7. For definiteness we may suppose 7 to be drawn eastioards, J north- wards, and K ujnvards. Then it is obvious that a positive (left- handed) rotation about the eastward line (7) brings the northward line (7) into a vertically upward position {K ) ; and so of the others. 66.] Now the operator which turns 7 into TT is a quadrantal versor (§ 53) ; and, as its axis is the vector 7, we may call it i. Thus ~-h or K=:iJ. (1) J Similarly we may put -^^ = /, or I =1 j K, (2) and — = k, or J = kl. (3) [It may here be noticed, merely to shew the symmetry of the system we are explaining, that if the three mutually perpendicular vectors 7, 7, K be made to revolve about a line equally inclined to all, so that 7 is brought to coincide with 7, 7 will then coincide with K, and iT with 7: and the above equations will still hold good, only (1) will become (2), (2) will become (3), and (3) will become (I)-] Q7.'\ By the results of § 50 we see that -7_ K IT - J' 69.] PRODUCTS AISTD QUOTIENTS OF VECTORS. 35 i. e. a soutliward unit-vector Lears the same ratio to an upward unit- vector that the Letter does to a northward one ; and therefore we have -/ = I. or Similarly and -./ = iK. = j, or -K = jl; = k, or —1= kJ. (4) (5) (6) K -K I -I ~T 68.] By (4) and (1) we have -J =iK=i{iJ) = i'~J. Hence i^ = — 1 (") And, in the same way, (5) and (2) g-ive P=-h (8) and (6) and (3) k'^ =-\ (9) Thus, as the directions of /, •/, K are perfectly arbitrary, we see that the square of ever ^ quadrantal versor is negative nniti/. Though the following proof is in principle exactly the same as the foregoing, it may perhaps be of use to the student, in shewing him precisely the nature as well as the simplicity of the step we have taken. Let ABA' be a semicircle, whose centre is 0, and let OB be perpendicular to AOA'. Then -^ > = q suppose, is a quadrantal ^'^ 61' versor, and is evidently equal to -= ; §§ 50, 53. _ _ _ ' ^ OA' OB OA' _ Hence q"^ = -= =^ = -^= = — i . ^ OB OA OA 69.] Having thus found that the squares of /, J, k are each equal to negative unity ; it only remains that we find the values of their products two and two. For, as we shall see, the result is such as to shew that the value of any other combination whatever of i, j, k (as factors of a product) may be deduced from the values of these squares and products. Now it is obvious that D 2 36 QUATERNIONS. [70. (i. e. the versor which turns a westward unit- vector into an upward one will turn the upward into an eastward unit) ; or K = j{-I)=-jI* (10) Now let us operate on the two equal vectors in (10) by the same versor^ i, and we have IK = i {—J I) = —^jl- But by (4) and (3) iK = -J =:-kI. Compai'ing these equations, we have -ijl = -kl; ox, by § 54 (end), ij = 1-, ^ and symmetry gives jk = i, \ (11) M=j.\ The meaning of these important equations is very simple ; and is, in fact^ obvious from our construction in § 54 for the multiplication of versors; as we see by the annexed fig-ure, where we must re- member that ?', j, k are quadrantal versors whose planes are at right angles, so that the figure represents a hemisphere divided into quadrantal tri- angles. Thus, to shew that ij = h, we have, being the centre of the sphere, i\^, E, S, IF the north, east, south, and west, and ^the zenith (as in § 65) ; whence ij OIF = iOZ = OS ^ Jc OF. 70.] But, by the same figure, iON=OZ, _ _ _ whence jiON = jOZ = OE = -0 IF =-Jc ON. 71.] From this it appears that ji z=-k, \ and similarly kj =^ — t, > (12) ik = — y, ) and thus, by comparing (11), V = -J^' = ^y ) jk=-kj=i,\ ((11), (12)). ki z=—ik=j. ) * The negative sign, being a mere numerical factor, is evidently commutative with j ; indeed we may, if necessary, easily assure ourselves of the fact that to turn the negative (or revei-se) of a vector through a right (or indeed any) angle, is the same thing as to turn the vector through that angle and then reverse it. 74-] PRODUCTS AND QUOTIENTS OF VECTOKS. 37 These equations, along- with ^^=/ = Z•^=_l ((7), (8), (9)), contain essentially the whole of Quaternions. But it is easy to see thatj for the first group, we may substitute the single equation ijJc = -\, (13) since from it, by the help of the values of the squares of i, j, k, all the other expressions may be deduced. We may consider it proved in this way, or deduce it afresh from the figure above, thus kON= 6W^, jkON= jOTF= OZ, ijk ON = ij 0TF= iOZ= 0S = - ON. 72.] One most important step remains to be made, to wit the assumption referred to in § 64. "We have treated i,J, k simply as quadrantal versors ; and /, •/, K as unit- vectors at right angles to each other, and coinciding with the axes of rotation of these versors. But if we collate and compare the equations just proved we have (/= h (11) IJ= K, (1) \Jl=-K (12) \jI=-K, (10) with the other similar groups symmetrically derived from them. Now the meanings we have assigned to i,j, k are quite inde- pendent of, and not inconsistent with, those assigned to /, •/, A. And it is superfluous to use two sets of characters when one will suffice. Hence it appears that i,j, k may be substituted for /, J", K; in other words, a unit-vector v:7ien employed as a factor may he con- sidered as a quadrantal versor udiose plane is perpendicular to the vector. This is one of the main elements of the singular simplicity of the quaternion calculus. 73.] Thus the product, and therefore the rfiotient, of two perpen- dicular vectors is a third vector perpendicular to both. Hence the reciprocal (§ 51) of a vector is a vector which has the opposite direction to that of the vector, and its length is the re- ciprocal of the length of the vector. The conjugate (§ 52) of a vector is simply the vector reversed. Hence, by § 52, if a be a vector {Taf = aKa = a{-a)= -a^. 74.] We may now see that every versor may he represented hy a pov:er of a unit-vector. 38 QUATERNIONS. [75- For, if a Ijc any vector perpendicular to i (which is au_^ definite unit-vector), ia, = fi, is a vector equal in length to a, but perpendicular to both i and a ; i^ a = — a, m i^a = —ia = — /3, i'^a = — i^ = —Pa = a. Thus, by successive applications of i, a is turned round i as an axis throug-h successive right angles. Hence it is natural to define i^ as a versor which turns any rector j^erjiendicular to i through m right angles in the j)Ositive direction of rotation about i as an axis. Here m may have any real value whatever, whole or fractional, for it is easily seen that analogy leads us to interpret a negative value of fn as corresponding to rotation in the negative direction. 75.] From this again it follows that any quaternion may he expressed as a poiver of a vector. For the tensor and versor elements of the vector may be so chosen that, when raised to the same power, the one may be the tensor and the other the versor of the given quaternion. The vector must be, of course, perpendicular to the plane of the quaternion. 76.] And we now see, as an immediate result of the last two sections, that the index-law holds with regard to powers of a quaternion (§ 63). 77.] So far as we have yet considered it, a quaternion has been regarded as the product of a tensor and a versor : we are now to consider it as a sum. The easiest method of so analysing it seems to be the following. B Let represent any quaternion. Di BC perpendicular to OJ, produced if neces- sary. Then, §19, OB = OC+CB. But, § 22, 6C = xOA, where a? is a number, whose sign is the same as that of the cosine of Z AOB. Also, § 73, since CB is perpendicular to OA, CB = y6A, where y is a vector perpendicular to OA and CB, i.e. to the plane of the quaternion. OB _ xOA + yOA U2~ OA Hence = x + y. 79-] PRODUCTS AND QUOTIENTS OF VECTOKS. 39 Thus a quaternion, in general, may be decomposed into the sum of two parts, one numerical, the other a vector. Hamilton calls them the SCALAE, and the vector, and denotes them respectively by the letters S and F prefixed to the expression for the quaternion. 78.] Hence q = Sq+ Vq, and if in the above example OB 53=^_^ then 6B=0C+CB = Sq . OA + Fq . OA *. The equation above gives 0C= Sq.OA, CB=Vq.6A. 79.] If, in the figure of last section, we produce BC to D, so as to double its length, and join OD, we have, by § 52, -^^ = Kq = SKq + FKq : OA .-. 01) = 0C+C1) = SKq.OA+ VKq.OA, Hence 6C = SKq.OA, and 01) = VKq.OA. Comparing this value of OC with that in last section, we find SKq ■=■ Sq, ( 1 ) or the scalar of the conjugate of a quaternion is equal to the scalar of the quaternion. Again, CT) = —CB\>^ the figure, and the substitution of their values gives YKq = — Vq, ('-) or the vector of the conjugate of a quaternion is the vector of the quaternion reversed. We may remark that the results of this section are simple con- sequences of the fact that the symbols S, F, K are commutative f. Thus SKq = KSq = Sq, since the conjugate of a number is the number itself; and rKq = KVq = -Vq (§73). * The points are inserted to shew that S and V apply only to q, and not to qOA. \ It is curious to compare the properties of these quaternion symbols with those of the Elective Symbols of Logic, as given in Boole's wonderful treatise on the Zaws of Thouyht; and to think that the same grand science of mathematical analysis, by processes remarkably similar to each other, reveals to us truths in the science of position far beyond the powers of the geometer, and truths of deductive reasoning to which unaided thought could never have led the logician. 40 QUATERNIONS. [8o. Again, it is obvious that ^Sq = S^q, ^Vq = Flq, and thence 2Kq = K'2q. 80.] Since any vector whatever may be represented by where x, y, z are numbers (or Scalars), and i, j\ k may be any three non-coplanar vectors, §§ 23, 25 — thoug-h they are usually under- stood as representing a rectangular system of unit-vectors — and since any scalar may be denoted by w ; we may write, for any quaternion q, the expression q ■=. w + xi + j/j + zk (§ 78). Here we have the essential dependence on four distinct numbers, from which the quaternion derives its name, exhibited in the most simple form. And now we gee at once that an equation such as where q'= uf ■\-x'i-\-y'j-\-z^k, involves, of course, theybz^r equations w' =. IV, x'= X, y'-=y, ' z' •=. z. 81.] We proceed to indicate another mode of proof of the dis- tributive law of multiplication. We have already defined, or assumed (§ 61), that — I — = > a a a or /3a-i+ya-^ = (/3 + )/)a-\ and have thus been able to understand what is meant by adding two quaternions. But, writing a for a~^, we see that this involves the equality from which, by taking the conjugates of both sides, we derive a'(/3' + /) = «'y3' + aV (§55). And a combination of these results (putting l3 + y for a in the latter, for instance) gives + y)(/3'+/) = (/3+y)^'+(/3 + y)/ = j8^' + y/3' + /3y' + y/ by the former. Hence t/ie distributive j^rinciple is true in the multiplication of vectors. It only remains to shew that it is true as to the scalar and 83.] PRODUCTS AND QUOTIENTS OF VECTORS. 41 vector parts of a quaternion, and then we shall easily attain the general proof. NoWj if a be any scalar, a any vector, and q any quaternion, {a -\-a) q = aq + aq. For, if /3 be the vector in which the plane of q is intersected by a plane perpendicular to a, we can find other two vectors, y and b, in these planes such that y /3 {a + a)q = /3' wri «/3 + y /3 «/3 + y aB And, of course, a may be written — ; so that 13 b /5_^y /3_^y /3 = aq + aq. And the conjugate may be written /(a' + a') = qV + qa (§ 5o). Hence, generally, (rt + a)(^ + /3) = <7^ + «j8 + ^a + a/3; or, breaking up a and 5 each into the sum of two scalars, and a, ^ each into the sum of two vectors, K + «2 + ai + a.)(^Ji + ^, + /3i + ^2) = («'i + ^2) ih + ^2) + i^i + ^'.) (/3i + ^2) + (^1 + h) («i + «2) (by what precedes, all the factors on the right are distributive, so that we may easily put it in the form) = {a^ + oi) (^1 -\-(3^) + {a, + ai) (b., + fi^) + {a^ + a^) (^1 + A) + («2 + a2)(i2 + ^2)- Putting a^ + a^= p, a.^ -^a.^= q, ^^4-/3^ = r, i^g + /Sg = /, z, f f f X ,y ,z. We have ^ = qa, or x'i + i/j + //?; = (o) + £i + rj/ + (k) {xi +ijj + zk) = -{^x^m + C^) + {(^x-\-riz-Cy)i + {w7/+C^-^z)j+{coz + ^?/—,^x)k, as we easily see by the expressions for the powers and products of hj, ^, given in § 71. But the student must pay particular attention to the order of the factors, else he is certain to make mistakes. This (§ 80) resolves itself into the four equations 0= ix + r]?/ + Cz, x' ■=■ (ax +r}Z — (?/, /=o}y-iz +Cx, z = ujz + ^j/-r]X. The three last equations give ^'p' + ^/ + ^/ = w {x" + J/^ + z"^), which determines co. Also we have, from the same three, by the help of the first, which, combined with the first, gives yz'—zy zx'—xz' xy —yx' '' and the common value of these three fractions is then easily seen to be 1 It is easy enough to interpret these expressions by means of ordinary coordinate geometry: but a much simpler process will be furnished by quaternions themselves in the next chapter, and, in giving- it, we shall refer back to this section. 85.] The associative law of multiplication is now to be proved by means of the distributive (§81). We leave the proof to the student. He has merely to multiply together the factors lo^x'i-^yj-^zk, w' + xi + yj+z'k, and w" + x''i + yj +z''k, as follows : — First, multiply the third factor by the second, and then multiply the product by the first ; next, multiply the second factor by the 8S.] PKODUCTS AND QUOTIENTS OF VECTORS. 43 first and employ the product to multiply the third : always re- membering" that the multiplier in any product is placed before the multiplicand. He will find the scalar parts and the coefficients of i,j, k, in these products, respectively equal, each to each. 86.] With the same expressions for a, /3, as in section 83, we have a^ = {xi + 1/j + zh) {x'i + y'j + z'k) — _ [xx' + yy + zz) -f (y/ - zy) i + (za;' — xz)j 4 [xy —yx') L But we have also i3a = - (^r/ + yy +zz)- (y/ - z/) i - {zx' - xz)j - (.r/ - yx) k The only difference is in the si(jn of the vector parts. Hence Sa^ = S(ia, ( 1 ) ral3=-r^a, (2) also a/3 + /3a = 2/Sa/3, (3) a^-fia = 2Va(i, (4) and, finally, by § 79, al3 = K^a (5) 87.] If a = /3 we have of course (§25) X — X, y = y, z = z', and the formulae of last section become a/3 = (3a = a- = —{x'^ + y^ -\- Z') ; which was anticipated in § 73, where we proved the formula ^Tay =-a', and also, to a certain extent, in § 25, 88.] Now let q and ;• be any quaternions, then S.qr = S.{Sq+rq){Sr-{-rr), = S.{Sq Sr + Sr .Vq + Sq. Vr + Vq Fr), = SqSr + SFqrr, since the two middle terms are vectors. Similarly, S.rq = Sr Sq + SFr Vq. Hence, since by (1) of § 86 we have SVqVr =:i SnVq, we see that S.qr = S.rq, (1) a formula of considerable importance. It may easily be extended to any number of quaternions, because, r being arbitrary, we may put for it rs. Thus we have S.qrs = S.rsq, = S.sqr by a second application of the process. In words, we have the theorem — tke scalar of the product of any number of given quaternions depends only tijwn the cyclical order in v)hich they are arranged. 44 QUATERNIONS. [89. 89.] An important case is that of three factors, each a vector. The formula then becomes S.a(3y = S.jBya = S.-yafB. But S. a/3y = Sa{Sl3y+ V ^y) = SaV^y, since aS(iy is a vector, = -Saryf3, by (2) of §86, = -Sa{Sy^+Fyl3) = — S.ayj3. Hence i/ie scalar of the product of three vectors changes sign when the cyclical order is altered. Other curious propositions connected with this will be g-iven later, as we wish to devote this chapter to the production of the fundamental formulae in as compact a form as possible. 90.] By (1) of § 86, 2F/3y = /3y-y3. Hence 2FaFl3y = Fa {^y — y^) (by multiplying both by a, and taking the vector parts of each side) = F(afty + ^ay — (Bay — ayfi) (by introducing the null term fSay—^ay). That is 2FaFfiy=F.{al3 + j3a)y—F{l3Say + l3Fay + Say.(3i- Fay. (3) = F{2Sal3)y-2Fl3Say (if we notice that F. Fay.^ = — FjSFay, by (2) of § 86). Hence FaFlSy = ySaj3 — l3Sya, (1) a formula of constant occurrence. Adding aS^y to both sides we get another most valuable formula F.a^y = aS^y — l3Sya + ySai3; (2) and the form of this shews that we may interchange y and a without altering the right-hand member. This gives F. ajBy = F. yj3a, a formula which may be greatly extended. 91.] We have also TFayS Fyh = - FFyb Fa^ by ( 2) of § 8 6 : = bSyFal3 — ySbFa^ = bS.ajBy — yS.a/Sb, = - ^Sa Fyb + aSfB Fyb = - /3^. ayS + aS. (Syb, all of these being arrived at by the help of § 90 (1) and of § 89 ; and by treating alternately T a/3 and Fyb as simj)le vectors. Equating two of these values, we have bS.afty = aS.ftyb-\-fiS.yah + yS.al3b, (3) 93-] PEODUCTS AND QUOTIENTS OF VECTORS. 45 a very useful formula, expressing any vector whatever in terms of three given vectors. 92.] That such an expression is possible we knew already by §23. For variety we may seek another expression of .a similar character, by a process which differs entirely from that employed in last section, a, ^, y being any three vectors, we may derive from them three others Va^, V^y, Vya ; and, as these will not generally be coplanar, any other vector 8 may be expi-essed as the sum of the three, each multiplied by some scalar (§ 23). It is required to find this ex- pression for S. Let h =^ X Fa(3 -\-y V^y + z Vya. Then 8yh = xS.yaji = xS.ajSy, the terms in ?/ and z going out, because SyFlBy = S.ypy = SjSy^ = y^S[3 = 0, for y- is (§ 73) a number. Similarly SjBb = zS.[3ya = zS.ajSy, and Sab =j/S.a/3y. Thus bS.ajSy = FajSSyb + V^ySah + VyaS^h (4) 93.] We conclude the chapter by shewing (as promised in § 04) that the assumption that the product of two parallel vectors is a number, and the product of two perpendicular vectors a third vector perpendicular to both, is not only useful and convenient, but absolutely inevitable, if our system is to deal indifferently with all directions in space. We abridge Hamilton's reasoning. Suppose that there is no direction in space pre-eminent, and that the product of two vectors is something which has quantity, so as to vary in amount if the factors are changed, and to have its sign changed if that of one of them is reversed ; if the vectors be parallel, their product cannot be, in whole or in part, a vector inclined to them, for there is nothing to determine the direction in which it must lie. It cannot be a vector parallel to them ; for by changing the sign of both factors the product is unchanged, whereas, as the whole system has been reversed, the product vector ought to have been reversed. Hence it must be a number. Again, the product of two perpendicular vectors cannot be wholly or partly a number, because on inverting one of them the sign of that number ought to change; but inverting one of them is simply equivalent to a rotation through two right angles about the other, and (from the symmetry of space) ought to leave the number 46 QUATERNIONS. unchang-ed. Hence the product of two perpendicular vectors must be a vector, and a simple extension of the same reasoning shews that it must be perpendicular to each of the factors. It is easy to carry this farther, but enough has been said to shew the character of the reasonino-. EXAMPLES TO CHAPTER II. 1 . It is obvious from the properties of polar triangles that any mode of representing versors by the sides of a triangle must have an equivalent statement in which they are represented by angles in the polar triangle. Shew directly that the product of two versors represented by two angles of a sf)herical triangle is a third versor represented by the su])plemeni of the remaining angle of the triangle; and determine the rule which connects the directions in which these angles are to be measured. 2. Hence derive another proof that we have not generally • m = qp- 3. Hence shew that the proof of the associative principle, § 57, may be made to depend upon the fact that if from any point of the sphere tangent arcs be drawn to a spherical conic, and also arcs to the foci, the inclination of either tangent arc to one of the focal arcs is equal to that of the other tangent arc to the other focal arc. 4. Prove the formulae 2S.a^y = a^y — y^a, 2 r.a(3y = ai3y + y(3a. 5. Shew that, whatever odd number of vectors be represented by a, 13, y,&ce., we have always F.apyb€= r. (by 13a, F.a^ybeCrj = F.i](ebyl3a, &c. 6. Shew that S. Fafi F(3y Fya = - (S.a^yY, F. Fal3 F^y Fya = Fa^ {y^Sa^ - Sl3ySya) + , and F. ( Fa^ F. F^y Fya) = {^Say- aS(3y) S.a^y. 7. If a, /3, y be any vectors at right angles to each other, shew that (a3 + /33 + y 3) s.a^y = a* F/3y + ft* Fya + y^ Fa{3. EXAMPLES TO CHAPTER II. 47 8. If a, /3, y be non-coplanar vectors, find the relations among the six seahirs, x,y, z and ^, r], (, which are implied in the equation xa+i/^ + zy = ^ Vjiy + rj Vya -\- C /''a/3. 9. If a, /3, y be any three non-coplanar vectors, express any fourth vector, 8^ as a linear function of each of the following- sets of three derived vectors, F.yajS, V.a^y, J .^ya, and V.Va^r^yVya, V.V^yVyaVali, F. Fyara(3T'[Sy. 10. Eliminate p from the equations Sap = a, SjBp = b, Syp = c, Sbp = d, where a, /3, y, b are vectors, and a, 6, c, d scalars. 11. In any quadrilateral, plane or gauche, the sum of the squares of the diagonals is double the sum of the squares of the lines joining the middle points of opposite sides. CHAPTER III. INTERPRETATIONS AND TRANSFORMATIONS OF QUATERNION EXPRESSIONS. 94.] Among the most useful eliaracteristics of the Calcuhis of Quaternions, the ease of interpreting- its formulae g-eometrically, and the extraordinary variety of transformations of which the simplest expressions are susceptible, deserve a prominent place. We devote this Chapter to some of the more simple of these, to- o-ether with a few of somewhat more complex character but of constant occurrence in geometrical and physical investigations. Others will appear in every succeeding Chapter. It is here,, perhaps, that the student is likely to feel most strongly the peculiar difficulties of the new Calculus. But on that very account he should endeavour to master them, for the variety of forms which any one formula may assume^ though puzzling- to the beginner, is of the most extraordinary advantage to the advanced student, not alone as aiding him in the solution of complex questions, but as affording an invaluable mental discipline. 95.] If we refer again to the figure of § 77 we see that OC = OB cos AOB, CB = OB sin AOB. Hence, if OJ = a, OB = (3, and I AOB = 6, we have OB = Tl3, OA = Ta, OC = T^ cos 6, CB = Tj3 sin 0. Hence ^^ = -— = -^^ cos0. a UA 1 a Similarly TF^ = § = |^ sinO. 97-] INTERPRETATIONS AND TRANSFORMATIONS. 49 Hence, if e be a unit- vector perpendicular to a and ^, or UOA « we have A - = ^^ sin ^.e, a la 96.] In the same way we may shew that Sa^ =-TaTl3cos9, TVafi = Ta r/3 sin d, and ra/3 = ^a T'/S sin d.r] where 77 = VVafi = UF-- a Thus l//e scalar of t/ie product of two vectors is the continued product of their tensors and of the cosine of the supplement of the contained angle. The tensor of the vector of the product of two vectors is the con- tinued product of their tensors and the sine of the contained angle ; and the versor of the same is a tmit-vector perpendicular to both, and such that the rotation about it from the first vector (i. e. the multiplier) to the second is left-handed or positive. Hence TFaj3 is double the area of the triangle two of whose sides are a, j3. 97.] (a.) In any triangle ABC we have IC = lB + £G. Hence AC^ = SAC AC = S.AC{AB + BC). With the usual notation for a plane triangle the interpretation of this formula is — b"^ = —be cos A—ab cos C, or b = a cos C+ c cos A. (b.) Again we have, obviously, VABAC= rlB{AB^BC) = rlBBC, or cb sin A = ca sin B, , sin A sin B sin C whence = — - — = a e These are truths, but not truisms, as we might have been led to fancy from the excessive simplicity of the process employed. E 50 QFATEENIONS. [98. 98.] From § 96 it follows tliat, if a and ^ be both actual (i. e. real and non-evanescent) vectors, the equation /Sa/3 = shews that cos ^ = 0, or that a is perpendicular to {3. And, in fact, we know already that the product of two perpendicular vectors is a vector. Again, if jr^i^ ^ 0^ we must have sin 6 = 0, or a is parallel to (3. We know already that the product of two parallel vectors is a scalar. Hence we see that SajB = is equivalent to a = Vy(i, where y is an undetermined vector ; and that Va^ = is equivalent to a = xjS, where x is an undetermined scalar. 99,] If we write, as in § 83, a = ix +j// + hz, /3 = ix'+jy'^hz\ we have, at once, by § 86, Sa^ = —xx'—yy' — zz' X x' y y' z -rr' ( r + — ^ + t) \ r r r r r r ^ where r = Vx^ -\- y'^ -\- z'^ , r'= \/x"^ + y"^ +/^. „ Ayz'—zy . zx' — x/ . xy —yx' ,1 Also Va^ = rr' \ ^ r' ^ + > — J + -=^—f— k \ • ( rr rr rr ) These express in Cartesian coordinates the propositions we have just proved. In commencing the subject it may perhaps assist the student to see these more familiar forms for the quaternion expressions ; and he will doubtless be induced by their appearance to prosecute the subject, since he cannot fail even at this stage to see how much more simple the quaternion expressions are than those to which he has been accustomed. 10.0.] The expression S.aBy may be written S{fa^)y, because the quaternion a/3y may be broken up into (^a/3)y + (ra/3)y of which the first term is a vector. I02.] IXTERPKETATIONS AXD TRANSFOEMATIONS. 51 But, by § 96, S ( Fal3) y = TaT^ sin 6 Sriy. Here Ti] = 1, let (/> be the angle between r/ and y, then finally -5'.a/3y =—TaTfi Ty sin cos (/.. But as rj is perpendicular to a and /3^ I'y cos is the length of the perpendicular from the extremity of y upon the plane of a, (3. And as the product of the other three factors is (§ 96) the area of the parallelogram two of whose sides are a, (3, we see that the mag- nitude of -iS'.a/Sy, independent of its sign, is Me volume of the parallel- ejnped of toh'ick three coordinate edges are a, jB, y, or six times the volume of the pyramid which has a, (3, y for edges. 101.] Hence the equation S.alSy = 0, if we suppose a, (3, y to be actual vectors, shews either that sin 6 = 0, or cos (f) = 0, i. e. t/vo of the three vectors are parallel, or all three are parallel to one ptlane. This is consistent with previous results,, for if y ^ p(3 we have S.aj3y =pS.a{3' = ; andj if y be coplanar with o, j3, we have y = pa + q[3, and S.al3y = S.a^{pa + qj3) = 0. 102.] This property of the expression S.a^y prepares us to find that it is a determinant. And, in fact, if we take a, ^ as in § 8 3, and in addition ^ ^ ^y/ _^jy^ ^ ^.//^ we have at once S. a(3y = - x" ( j// - zf) -y" [zx - xz) - z" [xy -yx'\ = — X y z X y z' X y z The determinant changes sign if we make any two rows change places. This is the proposition we met with before (§ 89) in the f^^^ &.a\^y = -S.(3ay = S.^ya, &c. If we take three new vectors a^ = ix ■\-jx' -I- hx" , yi = /.:;+//+/.•/', we thus see that they are coplanar if a, (3, y are so. That is, if S.al3y =: 0, then S.a^(3^ y^ = 0. E 2 '^2 52 QUATERNIONS. [l03- 103.] We have, by § 52, {Tqf = qKq = {Sq+ Vq) {Sq- Vq) (§ 79), = {Sqf-iVqf by algebra, = {Sqf + [TVqf (§73). If §- = a/3, we have Kq — I3a, and the formula becomes a/3.i3a = a^/S^ = {S ajBf - {Fafi)^ . In Cartesian coordinates this is More generally we have {T(qr)y = qrK{qr) = qr Kr Kq{^ 55) = {Tqf {Trf (§52). If we writ« ^ _ ^ _,. ^ _ ^ ^2^ ^jy ^ j^^, r = iv' + j3 — w' + ix' -\-j/ + kz' ; this becomes (2^,2 ^^,2 ^_^2 _,_ „2) (^^'2 _^ ^'2 _^y2 ^ .'2) = {i.vio' —ocx' —yy —zz'f + (2^0?' + w'a? -\-yz'—z/) -\- [wi/' + ?^'^ + 5;^ — xz'Y + (^^/ + w'z + a^y' — T/x'Y, a formula of algebra due to Euler. 104.] We have, of course, by multiplication, (a + /3)2 = a2+a/3 + /3a + i32 = a'' + 2Sap + ^^ (§86 (3)). Translating into the usual notation of plane trigonometry, this becomes c"- =0^-2 ah cos C+ b\ the common formula. Again, r(a + ,S) (a-/3) = - ^0^3+ V^a = -2 Ta^S (§ 86 (2)). Taking tensors of both sides we have the theorem, the parallelogram whose sides are parallel and equal to the diagonals of a given paral' lelogram, has double its area (§ 96). Also /S'(a + /3)(a-/3) = a2-^2^ and vanishes only when a^ = /3", or Ta ■= TjS ; that is, the diagonals of a parallelogram are at right angles to one another, ivhen, and only when, it is a rhomhiis. Later it will be shewn that this contains a proof that the angle in a semicircle is a right angle. 105.] The expression p = al3a~^ obviously denotes a vector whose tensor is equal to that of /3. But we have S.jSap = 0, so that p is in the plane of a, /8. Also we have Sap = Sa(3, I06.] IXTERPRETATIONS AND TRANSFORMATIONS. 53 SO that fi and p make equal ang-les with a, evidently on opposite sides of it. Thus if a be the perpendicular to a reflecting- surface and /3 the path of an incident ray, p will be the path of the re- flected ray. Another mode of obtaining- these results is to expand the above expression, thus, § 90 (2), p = 2a-^Sa^ — l3 = 2a-^Sal3 — a-^ {SajS + Va^) = a-\Sal3—Fal3), so that in the fig-ure of § 77 we see that if OA = a, and 0£ = fi, we have OD = p = alSa"^. Or, again, we may get the result at once by transforming the equation to U- = U -• a (3 106.] For any three eoplanar vectors the expression p = a/3y is (§ 101) a vector. It is interesting to determine what this vector is. The reader will easily see that if a circle be described about the triangle, two of whose sides are (in order) a and /3, and if from the extremity of (3 a line parallel to y be drawn again cutting the circle, the vector joining the point of intersection with the origin of a is the direction of the vector a/3y. For we may write it in the which shews that the versor T-) which turns j3 into a direction parallel to a, turns y into a direction parallel to p. And this ex- presses the long-known property of opposite angles of a quadri- lateral inscribed in a circle. Hence if a, j3, y be the sides of a triangle taken in order, the tangents to the circumscribing circle at the angles of the triangle are parallel respectively to a/3y, /3ya, and ya/3. Suppose two of these to be parallel, i. e. let a/3y = xjSya = way (3 (§ 90), since the expression is a vector. Hence /3y = xyp, which requires either (C= I, ryl3 = or y || jS, a case not contemplated in the problem ; or x = -\, Sl3y=0, 54 QUATERNIONS. [lO?- i. e. tlic triang-le is right-angled. And geometry shews us at once that this is correct. Again, if the triangle be isosceles, the tangent at the vertex is parallel to the base. Here we have a?/3 = a(3y, or x{a + y) = a{a + y)y ; whence x = y^ = o?, or Ty — Ta, as required. As an elegant extension of this proposition the reader may prove that the vector of the continued product afiyh of the vector-sides of a quadrilateral inscribed in a sphere is parallel to the radius drawn to the corner {a, h). 107.] To exemplify the variety of possible transformations even of simple expressions, we will take two cases which are of frequent occurrence in applications to geometry. Thus T{p + a) = T{p-a), [which expresses that if 62= a, 0A'= —a, and OP = p, we have AF = A'F, and thus that P is any point equidistant from two fixed points,] may be written (p + aY = (p — a)^ or p^ + 2Sap + a'^ = p^ — 2Sap + a- (§ 104), whence Sap = 0. This may be changed to ap+ pa =i 0, or ap-\-Kap = 0, SU^ = 0, a or finally, TFUf^=l, all of which express propei-ties of a plane. Again, Tp = Ta may be written T - = 1 , o. {p + ay —2Sa{p + a) = 0, p = {p + ay^a{p + a), S{p + a){p — a) = 0, or finally, T.{p + a){p-a) = 2TVap. 109-] INTERPRETATIONS AND TRANSFORMATIONS. 55 All of these express properties of a sphere. They will be in- terpreted when we come to geometrical applications. 108.] We have seen in § 95 that a quaternion may be divided into its scalar and vector parts as follows : — a a a 1 a where 6 is the angle between the directions of a and [3, and e = UP- is the unit- vector perpendicular to the plane of a and /3 so situated that positive (i. e. left-handed) rotation about it turns a towards fi. Similarly we have (§ 96) a/3 = Sal3 + Vafi = TaTl3{ — cose + (sm 6), 6 and e having the same signification as before. 109.] Hence, considering the versor parts alone, we have U- = cos (9 H- e sin ^. a Similarly U^ = cos ^ -f e sin (/) ; P (f) being the positive angle between the directions of y and j3, and e the same vector as before, if a, (3, y be coplanar. Also we have ^^ = cos (^ + (^) + € sin (0 + (^). a But we have always ^ • - = - , and therefore pa a pa a or cos {(j) + 6) + e sin {({) + 6) = (cos ^ + e sin (f)) (cos ^ -f e sin 6) = cos (f) cosO — sin ^ sin 6 + e (sin ^ cos ^ -f cos (psind), from which we have at once the fundamental formulae for the cosine and sine of the sum of two arcs, by equating separately the scalar and vector parts of these quaternions. And we see, as an immediate consequence of the expressions above, that cos m d + €s\nm6 = (cos 6 + e sin 0)"' if m be a positive whole number. For the left-hand side is a versor which turns through the angle mO ai once, while the right-hand 56 QUATERNIONS. [ 1 I O. side is a versor which effects the same object by m successive turn- ings each through an angle Q. See § 8. 110.] To extend this proposition to fractional indices we have only to write - for ^, when we obtain the results as in ordinary n trigonometry. From De Moivre's Theorem, thus proved, we may of course deduce the rest of Analytical Trigonometry. And as we have already deduced, as interpretations of self-evident quaternion trans- formations (§§ 97, 104), the fundamental formulae for the solution of plane triangles, we will now pass to the consideration of spherical trigonometry, a subject specially adapted for treatment by qua- ternions ; but to which we cannot afford more than a very few sections. (More on this subject will be found in Chap. X, in con- nexion with the Kinematics of rotation.) The reader is referred to Hamilton's works for the treatment of this subject by quaternion exponentials. Ill,] Let a, /3, y be unit- vectors drawn from the centre to the corners J, 5, C of a triangle on the unit-sphere. Then it is evident that, with the usual notation, we have (§ 96), 8a^ = — cos c, Sl3y = — cos a, Sya = — cos 6, TFal3= sine, TV^y = sin a, TVya= sin ^. Also UFal3, UVfiy, UVya are evidently the vectors of the corners of the polar triangle. Hence S. UFa^ UF^y = cos B, &c., TF.UFa^UF^y = sinB, &c. Now (§ 90 (1)) we have SFaj3 Fj3y = S.a F. (3 F^y = -Sa^S^y + ^^Say. Remembering that we have SFa^F^y = TFal3TFl3yS.UFa^UF(3y, we see that the formula just written is equivalent to sin a sin c cos B = — cos a cos c + cos b, or cos b = cos a cos c + sin a sin c cos B. 112.] Again, F.FafiF^y z= -fiSa^y, which gives TF. Fa^F^y = S.a^y = S.aF(3y = S./SFya = S.yFa(3, or sin a sin c sin B = sin a sin/)„ = sin b sin p^ = sin c sin^;,, ; where j)^ is the arc drawn from A perpendicular to BC, &c. 113.] INTEKPRETATIONS AND TKANSFOKMATIONS. 57 Hence sin^„ = sin c sin B, sin a sin c sin^, sin 5 sin^^, = sin a sin B. 113.] Combining the results of the last two sections, we have Ta/S . F^Y = sin a sin c cos B — ^sina sin c sin 5 = sin« sin c {cos B—jS sin B). Hence ?7. Fa/3 Ffiy = (cos B- (3 sin ^) , and £/■. Fy^ F(3a = (cos ^ + /3 sin B) . These are therefore versors which turn the system negatively or positively about OB through the angle B. As another instance, we have sin 5 1 tan B = cos 5 _ TFJ'a^FjBy ~ S.Fa(3Fi3y FJ'alBF^y ~ ' S.Fal3Fl3y Say + SafiSliy The interpretation of each of these forms gives a different theorem in spherical trigonometry. Again, let us square the equal quantities r. a/3y and aSjiy — /SSay + ySa(3, supposing a, /3, y to be any unit-vectors whatever. We have - ( F.alByf = S'^i3y + S-ya + ^-a/3 + 2S^ySyaJSa^. But the left-hand member may be written as TKa^y-S-^.a^y, whence 1 - S^-.a^iy = S-t3y + S^ya + S-a^^ + 2S^ySyaSa^, or 1 — cos-« — cos-3 — cos-f + 2 cos a cos b cos c = sm~a sm'^j)^ = &c. = sm^a sin^(5 sin- 6' = &c., all of which are well-known formulae. Such results may be multiplied indefinitely by any one who has mastered the elements of quaternions. 58 QUATERNIONS. [114. 114.] A curious proposition, due to Hamilton, gives us a qua- ternion expression for the splier'ical excess in any triang-le. The following proof, which is very nearly the same as one of his, though by no means the simplest that can be given, is chosen here because it incidentally gives a good deal of other information. We leave the quaternion proof as an exercise. Let the unit-vectors drawn from the centre of the sphere to A, B, C, respectively, be a, (3, y. It is required to express, as an arc and as an angle on the sphere, the quaternion The figure represents an orthographic projection made on a plane perpendicular to y. Hence C is the centre of the circle JDUe. Let the great circle through A, B meet BEe in E, e, and let BE be a quadrant. Thus ^ represents y (§ 72). Also make EJ' = AB=(3a-\ Then, evidently, BF= /3a- V. which gives the arcual representation required. Let BE cut Ee in G. Make Ca — EG, and join B, a, and a, F. Obviously, as B is the pole of Ee, Ba is a quadrant ; and since EG = Ca, Ga = EC, a quadrant also. Hence a is the pole of BG, and therefore the quaternion may be represented by the angle BaF. Make Cb = Ca, and draw the arcs Pa/3, Pba from P, the pole of AB. Comparing the triangles Eba and eaft, we see that Ea = e^. But, since P is the pole of AB, Ffia is a right angle : and therefore as Fa is a quadrant, so is F/B. Thus AB is the complement of Ea or j3e, and therefore ^3 _ 2AB. 115.] INTERPEETATIOXS AXD TRANSFORMATIOXS. 59 Join dA aud produce it to c so that Ac = bA; join c, P, cuttino* A£ in 0. Also join c, B, and B, a. Since P is the pole of AB, the ang-les at are right angles ; and therefore, by the equal triangles baA, coA, we have aA = Ao. But a^ = 2AB, whence oB = B(3, and therefore the triangles coB and BajB are equal, aud c, B, a lie on the same great circle. Produce cA and cB to meet in H (on the opposite side of the sphere). II and c are diametrically opposite, and therefore cP, produced, passes through H. Now Pa = Ph = PH, for they differ from quadrants by the equal arcs fl/3, ba, oc. Hence these arcs divide the triangle Hab into three isosceles triangles. But IPHb + IPHa = LaHb = Ibca. Also lPab = 'n—l cab — Z PaH, LPba = lPab = i:-l. cba - IPbH. Adding, 2 Z Pab = 2~— Lcab — Icha— /. bca = 77 — (spherical excess of abc). But, as ZFaj3 and LBae are right angle Sj we have angle of /3a~V = LFaB = /,3ae = ZPab = - — ^ (spherical excess of a be). [Numerous singular geometrical theorems,, easily proved ab initio by quaternions, follow from this : e. g. The arc AB, which bisects two sides of a spherical triangle abc, intersects the base at the distance of a quadrant from its middle point. All spherical tri- angles, with a common side, and ha^nng their other sides bisected by the same great circle (i. e. having their vertices in a small circle parallel to this great circle) have equal areas, &c., &c.] 115.] Let Oa = a, Ob = ^' , Oc = /, and we have @*(7)*(7)* = 5-'^-^ = Ca.BA But FG is the complement of BF. Hence the a7i(jle of the quaternion „' 1 o' i ./ j 60 QUATERNIONS. [u6. is half the sjiherical excess of the triangle ichose angular 2^oints are at the extremities of the unit-vectors a, ^\ y . [In seeking" a purely qviaternion proof of the preceding" proposi- tionSj the student may commence by shewing that for any three unit- vectors we have « v « The angle of the first of these quaternions can be easily assigned ; and the equation shews how to find that of ^oT^y. But a still simpler method of proof is easily derived from the composition of rotations.] 116.] A scalar equation in p, the vector of an undetermined point, is generally the equation of a surface; since we may sub- stitute for p the expression _ ^,^^ wbere x is an unknown scalar, and a any assumed unit-vector. The result is an equation to determine x. Thus one or more points are found on the vector xa. whose coordinates satisfy the equation ; and the locus is a surface whose degree is determined by that of the equation which gives the values of x. But a vector equation in p, as we have seen, generally leads to three scalar equations, from which the three rectangular or other components of the sought vector are to be derived. Such a vector equation, then, usually belongs to a definite number oi points in space. But in certain cases these may form a line, and even a surface, the vector equation losing as it were one or two of the three scalar equations to which it is usually equivalent. Thus while the equation ap = B gives at once ^ ^ ^-i^^ which is the vector of a definite point (since we have evidently /Sa/3 = 0) ; the closely allied equation jr _ o is easily seen to involve g^o _ q and to be satisfied by p^ a'^ji + xa, whatever be x. Hence the vector of any point whatever in the line drawn parallel to a from the extremity of a~^/3 satisfies the given equation. 117.] Again, Tap .Vp^ = {Va^f is equivalent to but two scalar equations. For it shews that Vap I 1 9-] INTERPKETATTONS AXD TRANSFORMATIONS. 61 and T/Sp are parallel, i.e. p lies in the same plane as a and j3, and can therefore be written (§ 24) p = xa+y;3, where x and j/ are scalars as yet undetermined. We have now Tap = y Fa^, Vp^= xFajB, which, by the g-iven equation, lead to xj/ = I, or ^ = -, or finally p = xa+ 'J3 ', which (§ 40) is the equation of a hyperbola whose asymptotes are in the directions of a and /3. 118.] Again, the equation V.Va^rap = 0, though apparently equivalent to three scalar equations, is really equivalent to one only. In fact we see by § 9 1 that it may be written —aS.afip = 0, whence, if a be not zero, we have S.ajBp = 0, and thus (§101) the only condition is that p is coplanar with a, /3. Hence the equation represents the plane in which a and ^ lie. 119.] Some very curious results are obtained when we extend these processes of interpretation to functions of a quaferniou q = w + p instead of functions of a mere vector p. A scalar equation containing- such a quaternion, along with quaternion constants, gives, as in last section, the equation of a surface, if we assign a definite value to w. Hence for successive values of w, we have successive surfaces belonging to a system ; and thus when w is indeterminate the equation represents not a surface, as before, but a volime, in the sense that the vector of any point within that volume satisfies the equation. Thus the equation {Tqf = a^-, or IV— p'^ = a-, or {Tpf = a''-w\ represents, for any assigned value of w, not greater than a, a sphere whose radius is l/a'^ — w-. Hence the equation is satisfied by the 62 QUATERNIONS. [l20. vector of any point whatever in tlie volume of a sphere of radius a, whose centre is orig-in. Again, by the same kind of investigation, where ^ = w + p, is easily seen to represent the volume of a sphere of radius a described about the extremity of ^ as centre. Also S{rf)= —a^ is the equation of infinite space less the space contained in a sjDhere of radius a about the origin. Similar consequences as to the interpretation of vector equations in quaternions may be readily deduced by the reader. 120.] The following transformation is enuntiated without proof by Hamilton {Lectures, p. 587, and Elements, p. 299). ^.-1(^2^2)1^-1 _ U{rq-^KrKq). To prove it, let r-\r''cff-q-^ = t, then Tt = \, and therefore Kt = r ^ ; But {r'-q^f = rtq, or r^q^ = rtqrtq, or rq = tqrt. Hence KqKr = t-^KrKqt-'^, or KrKq = tKqKrt. Thus we have u(^rq ± KrKq) = t U{qr ± KqKr) t, or, if we put s = U{qr + KqKr), Ks — + tst. Hence sKs = {Tsf = 1 = + stst, which, if we take the positive sign, requires st= +1, or t = ±s-'^=i ± UKs, which is the required transformation. [It is to be noticed that there are other results which might have been arrived at by using the negative sign above ; some in- volving an arbitrary unit-vector, others involving the imaginary of ordinary algebra.] 121.] As a final example, we take a transformation of Hamil- ton's, of great importance in the theory of surfaces of the second order. 12 1.] INTEKPEETATIONS AND TRANSFORMATIONS. G3 Transform the expression in which a, ^, y are any three mutually rectangular vectors, into tlieform .T{ip + pK)'-' \ K- — r ' which involyes only two vector-constants, t, k. {T{,p^pK)Y = {Lp+pK){pi + Kp) (§§52,55) = (r + K^ )p2 + (tpKp + pKpt) = (t2+x2)p2 + 2.S.tpKp = (t — k)'P" + 4 SipSKp. Hence (5ap)^ + (S'/3yi Syy^ 7. If, in § 102, a, /3, y be three mutuall}' perpendicular vectors, can anything be predicted as to a^, /S^, y^ ? If a, /3, y be rectangular 7inii vectors, what of a^, ^-^, y-^? EXAMPLES TO CHAPTER III. 69 8. If a, /3, y, a, /3', y' be two sets of rectangular unit- vectors, shew that Saa = Syfi'S^y -S^^'Syy, &c., &c. 9. The lines bisecting" pairs of opposite sides of a quadrilateral are perpendicular to each other when the diagonals of the quadri- lateral are equal. 10. Shew that {a.) S.q''= 2S'^q-T% (6.) S.q^=Shj-3SqT'Fq, (c.) a2/32y2+^^.a^y = FKa(3y, (d.) S{r.a^yV.^yaV.ya^) = ^Sa^S^ySyaS.al3y, {e. ) r.q^= (3 &-'~q-r- Vq) Vq, (/) qUYq-^ = -Sq.UFq + TFq; and interpret each as a formula in plane or spherical trigonometry. 11. If g' be an undetermined quaternion, what loci are repre- sented by (a.) {qa-^)^ = -a\ (b.) {qa-Y=^\ (c.) S.{q-a)^=a\ where a is any given scalar and a any given vector ? 12. If ^ be any quaternion, shew that the equation is satisfied, not alone by Q= +q but also, by Q= ± V^(Sq.Urq-TFq). (Hamilton, Lectures, p. 673.) 13. Wherein consists the difference between the two equations 2 r-^=l, and r^)=-i? a ^a^ What is the fuE interpretation of each, a being a given, and p an undetermined, vector ? 14. Find the full consequences of each of the following groups of equations, both as regards the unknown vector p and the given vectors a, /3, y : — _ Sap = 0, Sap = 0, («.) Tf^ ~ ' (^.) S.alSp = 0, (c.) S.afip = 0, ^■^yP = ''' Sl3p =0; S.apyp=0. 15. From §§ 74, 109, shew that, if e be any unit-vector, and m mi: . m-n any scalar, €™ = cos — + e sm — - • 70 QUATERNIONS. Hence shew that if a, /3, y be radii drawn to the corners of a tri- angle on the unit-sphere, whose spherical excess is m right angles, a-f-/3 y + g ^ + y^^,» 13 + y' a + li'y + a Also that, if A, B, C be the angles of the triangle, we have 2C 2B 2^ y-^ p^a" = — 1. 16. Shew that for ani/ three vectors a, fi, y we have ( Ua^f + ( U^yf + ( Uayf + ( U.ajiyf + 4 Uay.SUa^SU^y = - 2. (Hamilton, Elements, p. 388.) 17. If a^, fi^2 5 ^3 5 ^ ^^6 ^^y four scalars, and p-^, p^, p^ any three vectors, shew that {S.p,p,p,)^ + {2.a,rp,p,Y+x%2rp,p.;)'-x%l.a,ip,-p,)y^ + 2n(a;2 + Spip.^ + a^a,^) = 2U(x^ ■+ p^) + ^Ua^ -|-2{(a;2+«,2 + p^2) {{Vp^p^)'' + 2a.^a^{x^ +Sp.^p^)-xHp2-Pz?)] > where Yla^ = a^a.^a^. Verify this formula by a simple process in the particular case «j = «2 = ^3 = ^ = 0- {Ihkl) CHAPTER IV. DIFFERENTIATION OF QUATERNIONS. 128.] In Chapter I we have already considered as a special case the differentiation of a vector function of a scalar independent variable : and it is easy to see at once that a similar process is applicable to a quaternion function of a scalar independent variable. The differential, or differential coefficient, thus found, is in general another function of the same scalar variable ; and can therefore be differentiated anew by a second, third, &c. application of the same process. And precisely similar remarks apply to partial differentia- tion of a quaternion function of any number oi scalar independent variables. In fact, this process is identical with ordinary differ- entiation, 129,] But when we come to differentiate a function of a vector, or of a quaternion, some caution is requisite ; there is, in general, nothing which can be called a differential coefficient ; and in fact we require (as already hinted in § 33) to employ a definition of a differential, somewhat different from the ordinary one but, coinciding with it when applied to functions of mere scalar variables. 130,] If r = F{q) be a function of a quaternion q, dr = dFq = ^Jl {F{q + '^j^)-F{q)], where ?e is a scalar which is ultimately to be made infinite, is defined to be the differential of r or Fq. Here dq may be any quaternion whatever^ and the right-hand member may be written /., 7 x where / is a new function, depending on the form of F\ homo- geneous and of the first degree in dq ; but not, in general, capable of being put in the form {(ri)dq. 72 QUATERNIONS. [l SI- IS 1. J To make more clear these last remarks, we may observe that the function fi j \ thus derived as the differential of F{q), is distributive with respect lock. That is f{q,r + s)=f{r^,r)-vAq,s), r and s being any quaternions. For f{q,r+.,)=J^^n(F{q+'^)-F(q)) = /(?.»■)+/(?. 4 And, as a particular case, it is obvious that if x be any scalar f{q, A1-) = x/{q, r). 132.] And if we define in the same way dF{q,r,s ) as being the value of ^.»ln^+?"-+T-+V )-^(^'^''- )}' where q, r, s, ... dq, dr, ds, are any quaternions whatever ; we shall obviously arrive at a result which may be written /(q, r, s, ...dq, dr, ds, ), where y is homogeneous and linear in the system of quaternions dq, dr, ds, and distributive with respect to each of them. Thus, in differentiating any power, product, &c. of one or more quater- nions, each factor is to be differentiated as if it alone were variable ; and the terms corresponding to these are to be added for the com- plete differential. This differs from the ordinary process of scalar differentiation solely in the fact that, on account of the non-com- mutative property of quaternion multiplication, each factor must in general be differentiated in situ. Thus d(qr) = dq.r+qdr, but not generally = rdq + qdr. 133.] As Examples we take chiefly those which lead to results which will be of constant use to us in succeeding Chapters. Some of the work will be given at full length as an exercise in quaternion transformations. (1) m = -p'. The differential of the left-hand side is simply, since 7^p is a scalar, 2TpdTp. 1 33-] DIFFERENTIATION. 73 That of p2 is ^^u((p + ±)'-p^) = 2Spdp. Hence Tp dTp = —Spdp, or dTp =-S.Up dp = S-^, Up dTp r, dp Ip p (2) Again, p = TpUp dp = dTp.Up + TpdUp, dp dTp dUp whence -^ = -yr. — I- -~- p Ip Up Hence dUp _ ^dp W^ J' This may be transformed into V—~- or , j &c. (3) {Tqf = qKq 2TqdTq = d{qKq) = ^^n\{q + '^-^) K{q + '^) - qKq'\ = l.ni&^^l,dqKdq), = qKdq + dqKq, = qKdq + K{qKdq) (§55), = 2S.qKdq = 2S.Kqdq. Hence dTq = SMKqdq ^ S.Uq'^dq since Tq = TKq, and C/X^ = Uq-^. If ^ = p, a vector, Kq = Kp — —p, and the formula becomes dTp=z-S.Updp, as in (1). ^ ' Tq q But dq = TqdUq + UqdTq, , . , . ^?^ dTq dUq which gives -^=^ + -^5 whence, as S— = -~ > q Tq we have y -^ = -— ^ • 74 QUATERNIONS. [ 1 34. (4) 'l(f) = ^.'^((j = by, (^V = ^V, <^V = ^V. (pk = c-k, <\>'k = c^k, (p''k = c''k. Hence, putting separately /, y, k for p in the equation (1) of last section, we have _^c _ x-\-yaP- ■^za'^, -}/' =x^-i/b'^+zb^, — c*^ = x-\-yc^ +zc^. Hence a^, b-, c- are the roots of the cubic which involves the conditions y = ayj^+b^c^ +c''a% Thus, with the above value of 0, we have 142.] Putting ^-^ a in place of p (which is any vector whatever) and changing the order of the terms, we have the desired inversion of the function in the form aWc^-'^a = {aH'' -\-Ji^c- + c'a^) {p)= 03p_(^2^^2_^c2) 02p_^(,,2^2 ^ ^2^2 ^ ^2^2) ^pp^aWc^^ p = ((^_a2)(c|>-^2)(^_,2)|p (2) This last transformation is permitted because is commutative with scalars like a^, i. e. (f>(a'^p) = a^cpp. Here we remark that (by § 1 40) the equation F.p(f)p = 0, or (})p = [IP, where g is some undetermined scalar, is satisfied, not merely by every vector of null-length, but by the definite system of three rect- angular vectors Ai, Bj, CIc whatever be their tensors, the corre- sponding particular values of ^ being a"^, h"^, -'^Fkfx\ hence we have «20~^FXjii = F^'A(^Vj (4) and the problem of inverting ^ is solved. I47-] SOLUTION OF EQUATIONS. 81 146.] It remains to find the value of the constant m, and to express the vector F(f/\(j/iji. as a function of TA/x. Operate on (4) by S.cf/v, where v is any vector not coplanar with A. and /j,, and we get mS4'v(\>-^VKii = mS.vcfxp-U'^AiM (by (3) of § 144) = mS.X^xv = S.(l>X^'ix.^'v, or S.(\)X(\>ix(i>v m = S.Xixv (5) [That this quantity is independent of the particular vectors A, fx, v is evident from the fact that if y = j)\ + qix+rv, fx=jJj^K + q^iJ. + rj^v, and v=P2^ + q2l^ + r.^v be any other three vectors (which is possible since A_, /x, v are not coplanar); we have 0V = ^0'a -j- ^0V + ^'(i>'i'> &c., &c. ; from which we deduce iS.(|)'A>V>'^' = P ? r S.(f)'\(j)'lX(j)'v, Pi (Lx n Ih % ^2 id S.\'}jfv=. P 9. r S.Xixv, Px (h ^\ Pi fh ^2 so that the numerator and denominator of the fraction which ex- presses m are altered in the same ratio. Each of these quantities is in fact an Invariant, and the numerical multiplier is the same for both when we pass from any one set of three vectors to another. A still simpler proof is obtained at once by writing A +j)^ for A in (5), and noticing that neither numerator nor denominator is altered.] 147.] Let us now change to (fi+ff, where ^ is any scalar. It is evident that ^' becomes (f/+ff, and our equation (4) becomes nigi+ff)-''rX[x= r{'+ff)X{ct>'+ff)lJi{'+ff)v '^^ - S.X,xv = m + m^ff + m.^ff^+ff^ Q 82 QUATERNIONS. [H^- is what m becomes when <\> is changed into + ^; m\ and m^ being two new scalar constants whose values are '"' = s7Kirv ' S. {\ii(i/v + 0' Ajuv + X^'ixv) ^o, = '■ e^;^ o.Kixv If, in these expressions, we put \ +j>;/:a for A, we find that the terms in 2^ vanish identically ; so that they also are invariants. Substi- tuting for m^, and equating the coefiicients of the various powers of^ after operating on both sides by (\>-\-g, we have two identities and the following two equations, ^2 = ^ + X' [The first determines x, and shews that we were justified in treat- ing r((|)'Aju + A(^V) as a linear and vector function of F.Aju,. The result might have been also obtained thus, S.Xx^'Xyi = S.X(p'X[j. =—S.XfxrXiJ., S.ixx J^XjjL = S.fxXcf/ix = — S.fjic}) TA/x, S.vx^^h"- = S.{v(})'XlX + vX(t)'fJL) = vi^^SXixv — S.X[X(pi'v = S.v{m.J^Xix—(^VXy); and all three (the utmost generality) are satisfied by 148.] Eliminating x from these equations we find m-^^ = (^ {m.^ — Say + /3-i^/3y-y) = a-^8^-^y ^^-^ Sor^y-ySa^^^-^:] The conclusion that ^ o _ q in this case, is not necessarily true if Sa^ = 0. But then the original equation becomes aSl3p + ^Sap = y, which is consistent with S.a^y = 0. This equation gives + /3 Sl3y SajS Say a^ by comparison of which with the given equation we find Sap and Sjip. The value of p remains therefore with one indeterminate scalar. 154.] Uscaniple II. Let = 0, That is, = -^} ey + xe, = €~^y + X€, as before. Our warrant for putting xe, as the equivalent of 0~^ is this : — The equation ^^2^. _ q may be written r.eVfa =0 = ere- - eSea. Hence, unless o- = 0, we have or || e = a?e. 160.] Example IV. As a final example let us take the most general form of 0, which, as will be soon proved, may be expressed as follows : — (^p = aSl3p + a-^Sl3ip + a2Sj3.^p = y. Here (1/ p ^= ftSap + ^-^^Sa^p + fi^^a^p, and, consequently, taking a, a^ , a.2, which are in this case non- coplanar vectors, for A, ju, v, we have u = S.{l3Saa + jS^Sa^^a + (S.SaoO) (SSaa^ + (B^Sa^a^ + . . .) {(SSaa^ + .... S.aa^a2 S.I3M, Saa Sa^a Sa.20. Saa-j^ Sa^a^ Sa^o.^ Saa^ Sa-^a2 Sa2a.2 S.aa^a^ = ?^hh(ASaa + A,Sa,a + A2Sa2a), 88 QUATERNIONS. [l6o. where A = Sa-^a-iSa^a^^ — Sa^a-^Sa^a^ = — S. Va^a^ Va^a^ Ai ^ /SagOj/SaOg — Saa-^Sa20,2 = — S. Va.2 a T a^a.^ A^ = 8aa^SaT^a2 — Sa-^a^Saa^ = — 8. Vaa^ fa^a^. Hence the value of the determinant is — (SaaS. Va^a^ VayO.^ + Sa-^aS. Va^a f a^a^ -\- Sa^aS. Vaa^ Va-^a^ = — S.a{J a^a^S.aa-^a^ {by §92 {Z)]=. —[S.aa-^a^^. The interpretation of this result in spherical trigonometry is very interesting-. (See Ex. (6) p. 68.) By it we see that m = ~ S.aa-^a^S.^fi-^p^' Similarly^ ^l = ^ SSjx {fiSaa-i + ^^Sa^a^ + ^2^^^'^}) {^^aa^ + ^i8a-^a2 + ^^^'^t'^-z) + &c.] = X [S.al3j3i (8aa^Sa^a2 — Sa^a-i^Saa^) + ) ^ p (S.ajBjS^S.a V.a^ ^^aoa, + ) o.aa-^a2 + 8.a2 ( V\^^^8. Vaa^ Vaa^ + )] ; or, taking the terms by columns instead of by rows, = — ~ \8. J^jSjSi {aS. Vaa-^^ Va^a.^ + 0^6'. Vaa-^^ Va2 a + Og -S'. Vaa-^ Vaa^ 8.aa^a2 " -, = — 7^ \ 8.V33-,(Vaa-,S.aa-,a2J + ...... 1, o.aa^a2 = —8{ Faa^ Fl3(3^ + T\a2 T% f32 + Va./i 7^21^). Again, »?2 = -„ • 8[aa-^^ {^8aa2 + ^iSa-ya2+ ...) + a2a{l3Saai + ...) + a-^a2{(iSaa+ . ■ •)! or^ grouping as before, = -^^ — 8 \^j3 (Vaai8aa2 + Va20.8aa^ + Va-^a28aa) 4- . . ], = ^^Sli^{^S.aa,a.^+ ] (§92(4)), = /S (a/3 + aj/3i + 02^2)- And the solution is, therefore, (f)~^y8.aaia28.j3(3iP2 = pS.aa-^a28.^^il32 — ylS, Faa^ V^^^ + P2 = S.p^4)^p.^, or = S.p.^fp^p^, because (p is its own conjugate. But 0V2 = ^iP2> <^VI =fflpl> and therefore 9i9oSpxP2 = ffl^Pi'^-z = ffi'^PiPi > which, as ^^ and ff.2 are by hypothesis different, requires ^PiP-i = 0. Similarly ^^^2^3 — ^' '^PsPi = ^• If two roots be equal, as ff2> ffs^ "^^ ^^^^^ have, by the above proof, Sp-ip^ = and Spip^ = 0. But there is nothing farther to determine P2 and pg , which are therefore an^ vectors perpendicular to p^. If all three roots be equal, ever^ real vector satisfies the equation {4>-ff)p = 0. 164.] Next, as to* the realif^ of the three directions in this case. Suppose "2— hp2+ff2^2- Operating on these by S.a.^, S.po respectively, and subtracting the results, remembering our condition as to the nature of (p S(T^(bp2 — ^P2^^2} we have ^^■j.i'^l +P2) = ^• But, as (To and p^ are both real vectors, the sum of their squares cannot vanish. Hence //.^ vanishes, and with it the impossible part of the I'oot. 165.] When ^ is self-conjugate, we have shewn that the equa- tion ^3 — m.^ff'^ + mj^ff — m = has three real roots, in general different from one another. Hence the cubic in may be written 167.] SOLUTION OF EQUATIONS. 91 and in this form we can easily see the meaning of the cubic. For, let Pj , P2 5 P3 be three vectors such that {(p-ffi)Pi = 0, ((p-ffdP'Z = 0, {+g)FAfx; a formula which will be found to be of considerable use. 172.] SOLUTION OP EQUATIONS. 93 171.] Again, by § 147, Similarly -j- S.p{(p + //)~V = y *^p^~V + '^PXP + V"^- Hence That is, the functions !^^.p((^+^)-ip, and '^S.picfy + Ay-'p are identical, i. e. iv/ie?i equated to constants represent the same series of surfaces, not merely when ff = ^^ but also, whatever be (/ and k, if they be scalar functions of p which satisfy the equation mS.pifj'^p = cjlip^. This is a g-eneralization, due to Hamilton, of a singular result ob- tained by the author*. 172.] The equations S.p{<^+0Y^9 = ^.\ ,.. >s.p(c^+/^)-V = oJ are equivalent to mSp(p~^p+ffSpxp+ff'^p^ = 0, mSpcl)~^p + hSpx^p + /i^p2 = 0. Hence m{\—x)Sp^-''^p-\-{g — hx)Spxp + {g" — h'^x)p'^ = 0, whatever scalar be represented by x. That is, the two equations (1) represent the same surface if this identity be satisfied. As particular cases let ( 1 ) X ■= \, in which case Sp-\p+g + h=^ 0. (2) y — hx^=-0, in which case or mSp'~^(f)''^p — ff/i = 0. 2 2 m(l-^^)Sp(}>-^p+(ff-A^^)Spxp = 0, or m {h+g) Sp^~'^p -f gh Spxp = . * Note on the Cartesian equation of tlie Wave-Surface. Quarterly Math. Journal, Oct. 1859. 94 QUATERNIONS. [l73- 173,] In various investigations we meet with the quaternion q = a(t)a + ^y — Sya - Sacj^y) + y {Sa(f)l3 — Sj3(})a) = aSl3{(t) — (p')y + fiSy{(l) — (f>'}a + ySa{(t) — 4)')^ = aS.IBey + {BS.yea + yS.ae^ = — {aSae + jBS/Se -\- ySye) := e. [We may note in passing that this quaternion admits of being expressed in the remarkable form . d ^ d d where V = a-r- + /3^-+y-Y-> dx dij dz and p = ax + p7/ + yz. We will recur to this towards the end of the work.] Many similar singular properties of in connection with a rect- angular system might easily be given ; for instance, T[a r(})(3(})y + ^ F(f>y(})a + y Vt^iaip^) = m V{a(^'-'^a + /3(/)'-i/3 + y^'" V) = ^>^ T. V0'- V = 4>^ > which the reader may easily verify by a process similar to that just given, or (more directly) by the help of § 145 (4). A few others will be found among the Examples appended to this Chapter. 174.] To conclude, we may remark that as in many of the immediately preceding investigations we have supposed to be self-conjugate, a very simple step enables us to pass from this to the non-conjugate forai. For, if ^' be conjugate to ^, we have Sp(f)'(T = Sa(f)p, and also Spifya = Sa-cp^p. 1 77-] SOLUTION OF EQUATIONS. 95 Adding, we have iSp(0 + (/)') 0- = Scr{(j) + (f)')p; SO that the function {(p + (f)') is self-conjugate. Again, Spcpp ^ Spcfy'p, which gives Sp{(l) — (f/)p = 0. H ence [(f) — (j)')p = Fep, where, if be not self-conjugate, e is some real vector, and therefore Thus evetj non-conjugate linear and vector function differs from a conjugate function solely hy a term of the form r^p. The geometric signification of this will be found in the Chapter on Kinematics. 175.] We have shewn, at some length, how a linear and vector equation containing an unknown vector is to be solved in the most general case; and this, by § 138, shews how to find an unknown quaternion from any sufficiently general linear equation containing it. That such an equation may be sufficiently general it must have both scalar and vector parts : the first gives one, and the second three, scalar equations ; and these are required to determine com- pletely the four scalar elements of the unknown quaternion. 176.] Thus Tq^a being but one scalar equation, gives q =. aUr, where r is any quaternion whatever. Similarly Sq ^= a gives ■ q = a + d, where 9 is any vector whatever. In each of these cases, only one scalar condition being given, the solution contains three scalar in- determinates. A similar remark applies to the following : TT'q = a gives q = x + ad, and SUq = cos a, 2a gives q — xQ'' , in each of which x is any scalar, and Q any unit vector. 177.] Again, the reader may easily prove that 96 QUATERNIONS. [ 1 78. where a is a g-iven vector, gives, by putting Sq = x, Vaq = (3 + xa. Hence, assuming Saq = y, we have aq = y + a;a + /3, or q = x + j/a''^ + a~^fi. Here, the given equation being equivalent to two scalar con- ditions, the solution contains two scalar indeterminates. 178.] Next take the equation Faq = p. Operating by S.a"^, we get Sq = /Sa-i/3, so that the given equation becomes ra{Sa-^l3+rq) = l3, or FaFq = ^ — aSa-'^l3 = aVa~'^^. From this, by § 158, we see that Fq=z a-^{x + aFa-'^l3), whence q = /Sa~^/3 + a~^ {x + a Fa~'^^) = a-i(/3 + i»), and, the given equation being equivalent to three scalar conditions, but one undetermined scalar remains in the value of q. This solution might have been obtained at once, since our equation gives merely the vector of the quaternion aq, and leaves its scalar undetermined. Hence, taking x for the scalar, we have aq ■=■ Saq + Faq = x-{-^. 179.] Finally, of course, from aq = j3, which is equivalent to four scalar equations, we obtain a definite value of the unknown quaternion in the form q = a-i/3. 180,] Before taking leave of linear equations, we may mention that Hamilton has shewn how to solve any linear equation con- taining an unknown quaternion, by a process analogous to that which he employed to determine an unknown vector from a linear and vector equation ; and to which a large part of this Chapter has been devoted. Besides the increased complexity, the peculiar fea- ture disclosed by this beautiful discovery is that the symbolic equation for a linear quaternion function, corresponding to the cubic 183.] SOLUTION OF EQUATIONS. 97 in (^ of § 162, is a biquadratic, so that the inverse function is given in terms of the first, second, and third powers of the direct function. In an elementary work like the present the discussion of such a question would be out of place : although it is not very difficult to derive the more general result by an application of processes already explained. But it forms a curious example of the well-known fact that a biquadratic equation depends for its solution upon a cubic. The reader is therefore referred to the Elements of (Quaternions, p. 491. 181.] The solution of the following frequently-occurring par- ticular form of linear quaternion equation aq -Jrqh = c, where a, b, and c are any given quaternions, has been effected by Hamilton by an ingenious process, which was applied in § 133 (5) above to a simple case. Multiply the whole bi/ Ka, and into b, and we have T'-a.q + Ka.qb = Ka.c, and a(jlj ■\- (jtr- — cb. Adding, we have ^ {T'^a + ^2 + '>-'8a.b) = Ka.c + c/j, from which q is at once found. To this form any equation such as a'qV + c'qcV = e' can of course be reduced, by multiplication by c'^ and into b'~'^. 182.] As another example^ let us find the difi'erential of the cube root of a quaternion. If •,3 _ ^■^ = r we have q-dq + qilq.q + dq-q"^ = dr. Multiply by q, and into q~^, simultaneously^, and we obtain q^dq.q'^ + q^dq + qdq.q = qdr.q~'^. Subtracting this from the preceding equation we have dq.q^ — (f'dqxf'^ = dr — qdr.q''^, or dq.q^ — q^dq = dr.q — qdr, from which dq, or d (r^), can be found by the process of last section. The method here employed can be easily applied to find the difi'erential of any root of a quaternion. 183.] To shew some of the characteristic peculiarities in the solution even of quaternion equations of the first degree when they are not sufficiently general, let us take the very simple one aq = qb, and give every step of the solution, as practice in transformations. II 98 liUATEENIONS. [l^S- Apply Hamilton's process (§ 181)^ and we g-et r^a.q = Ka.qb, qlr' = acjh. These give q {T^a ■\-W'- 2bSa) = 0, so that the equation gives no real finite value for q unless T'a + d^-2ljSa = 0, or b = Sa + (BTra, where fi is some unit-vector. By a similar process we may evidently shew that a = Sd + aWb, a being another unit-vector. But, by the given equation, Ta = Tb, or S''a + T^Va = SH + TWb; from whichj and the above values of a and b, we see that we may write Sa Sb Wa = Wb=''^'''^'^'''- If, then; we separate q into its scalar and vector parts, thus q = r + p, the given equation becomes (a + a)(/ + p) = (r + p)(a + /3) (1) Multiplying out we have r{a — l3) = pfi — ap, which gives S{a—j3)p=0, and therefore p = Vy (a — (3), where y is an undetermined vector. We have now r{a — (3) = pji — ap = Fy{a-l3).i3-ary{a-(i) ^y{Safi+l)-{a-l3)Sliy + y{l+Sal3)-{a-l3)Say = -{a-l3)S{a + ^)y. Having thus determined r, we have ,j=S{a + (3)y+ry{a-l3) 2^/=-(a + /3)y-y(a + /3) + y(a-/3)-(a-/3)y = —2ay—2y(3. Here, of course, we may change the sign of y, and write the solution of aq = qb in the form q = ay + yj3, where y is any vector, and a = UVa, IS = UFb. 185.] SOLUTION OF EQUATIONS. 99 To verify this solution, we see by (1) that we require only to shew that aq = qji. But their common vahie is evidently — y + ayl3. It will be excellent practice for the student to represent the terms of this equation by versor-arcs, as in § 54, and to deduce the al)ove solution from the diagram directly. He will find that the solution may thus be obtained almost intuitively. 184.] No g-eneral method of solving quaternion equations of the second or higher degrees has yet been found ; in fact, as will be shewn immediately, even those of the second degree involve (in their most general form) algebraic equations of the sixteenth degree. Hence, in the few remaining sections of this Chapter we shall con- fine ourselves to one or two of the simple forms for the treatment of which a definite process has been devised. But first, let us consider how many roots an equation of the second degree in an unknown quaternion must generally have. If we substitute for the quaternion the expression 10 -\- ix -\- ji/ -\- kz (§80), and treat the quaternion constants in the same way, we shall have (§ 80) four equations, generally of the second degree, to determine w, «, y, z. The number of roots will therefore be 2* or IG. And similar reasoning shews us that a quaternion equation of the »^th degree has m'^ roots. It is easy to see, however, from some of the simple examples given above (§§ 175-178, &c.) that^ unless the given equation is equivalent to four scalar equations, the roots will contain one or more indeterminate quantities. 185.] Hamilton has effected in a simple way the solution of the quadratic ^2 _ ^^ ^ ^^ or the following, which is virtually the same (as we see by taking the conjugate of each side), q^ = aq + h. He puts q — l{a + w + p), where to is a scalar, and p a vector. Substituting this value in the first equation, we get cr + (w + pY + 2 loa + ap + pa = 2 {a^ + wa + pa) + 1 h, or (yW + pY + ap — pa = a- + 4 h. If we put Va = a, S (a- + 4*5') = c, V{a- + Ah) = 2 y, this becomes {w + p)'^ + 2Fap = c+2y; II 2 100 QUATERNIONS. [l86. which, by equating separately the scalar and vector parts, may be broken np into the two equations w'^ + p'^ = c, F{w + a)p = y. The latter of these can be solved for p by the process of § 156, or more simply by operating- at once by S.a which gives the value of S{7v + a)p. If we substitute the resulting value of p in the former we obtain, as the reader may easily prove, the equation The solution of this scalar cubic gives six values of w, for each of which we find a value of p, and thence a value of q. Hamilton shews (Lectures, p. 633) that only two of these values are real quaternions, the remaining four being biquaternions, and the other ten roots of the given equation being infinite. Hamilton farther remarks that the above process leads, as the reader may easily see, to the solution of the two simultaneous equations ^^,. _^^ qr = -d; and he connects it also with the evaluation of certain continued fractions with quaternion constituents. (See the Miscellaneous Ex- amples at the end of the volume.) 186.] The equation q"^ = aq + ql^, though apparently of the second degree, is easily reduced to the first degree by multiplying b^, and into, q~'^, when it becomes 1 = q~'^a-\-hq~'^, and may be treated by the process of § 181. 187.] The equation f = aqb, where a and 1/ are given quaternions, gives q{aql)) = {aqb)q', and, by § 54, it is evident that the planes of q and aqb must coincide. A little consideration will shew that the solution depends upon drawing two arcs which shall intercept given arcs upon each of two great circles ; while one of them bisects the other, and is divided by it in the proportion of )ii : 1 . EXAMPLES TO CHAPTER V. 101 EXAMPLES TO CHAPTER V. 1 . Solve the following' equations : — (a.) F.aplB = F.ay^. {b.) ap^p = papl3. (c.) ap + pl3 = y. {(l) S.a(3p + (SSap — a V(3p = y. (p=U (L) Spcp'^p = — l. ^ : , /, > , {c.) s.p{4>^-p^^)-^p = -u ' 'KnV. {d.) Tp = T4Up. , ^,,^ ; ^iyr\ 3. li X,iJ.,v be ani/ three non-coplanar vectors, and q = Fixv.(f)\+ Fv\.(j)iJi+ Fkp,.(t)v, shew that q is necessarily divisible by S.Xpiv. Also shew that the quotient is ^2 — 2 €, where Fep is the non-commutative part of (pp. Hamilton, Elements, p. 442, 4. Solve the simultaneous equations : — Sap = 0, 1 ^'''> S.ap4>p = 0.) Sap =0,| ^^ Spc}>p =0.) ^ '^ /S.atp^p = 0. ) 5. If 0p = ^(3Sap+ Frp, where r is a given quaternion, shew that m = 2 {S.a^a^a^S.lB^PM + ^^i^ ^\<^2 • ^'^M + Sr^S.ajir - 2 (/Sar^/3r) + /Sr:ZV2, and w?(^-V = 2(raia2'S./32/3i(r) + '2F.aF{Fl3(j.r) + FarSr- FrScrr. Lectures, p. 561. 102 QUATERNIONS. 6. If [.PQ~} denote P^ — ^2^, Uw~\ > - {pfp') + E^'?] '^'^ + Ip^'] ^i + \JiP^ ^'>'' and {pqn) „ S.jilqrs]; show that the following relations exist among any five quaternions =p (qrst) + q{rsfjj) + r{sfjjq) + s{fpqr) + t{pqrs), and q{prst) =[_rsf\Sjjq — [sfjj~]Srq + [_fp7-]Ssq —[prs'jSfq. Elements, p. 492. 7. Shew that if i3y{/l3 + (t>y^y) is an invariant. [This will be immediately seen if we write it in the form 6 = F.(f)V^p, which is independent of the directions of a, ^, y. But it is good practice to dispense with V.] If (Pp = 2r]SCp, and \l/p = 2ri-^SCip, shew that this invariant may be expressed as Shew also that cj)\l/p — \j/(()p = V6p. The scalar of the same quaternion is also an invariant, and may be written as —^^^Siqii^^SCCi 8. Shew that if (j)p = aSap + /3Sj3p + ySyp, where a, 13, y are any three vectors, then — (f)~''-pS^.al3y = aiSayp + l3^Sl3j^p + yiSy-^p, where a^ = F^y, &c. 9. Shew that any self-conjugate linear and vector function may in general be expressed in terms of two given ones, the expression involving terms of the second order. Shew also that we may write + ^ = «(CT + .T)2+^(CT+.r)((o+^) + c(co+^)^ where a, /j, c, x, ij, z are scalars, and ct and w the two given func- tions. What character of generality is necessary in tzr and w ? How is the solution affected by non-self-conjugation in one or both ? 10. Solve the equations : — (a.) q^ =5^?'+ioy. {h) (f =2q + i. {c.) qaq = hq + c. [d.) aq = qr = rh. EXAMPLES TO CHAPTER V. 103 11. Shew that ^V^(^p = mVVfp-'^p. 12. If (/) be self-conjugate, and a, /3, y a rectangular system, S .FacparfBcpfBFycf)-/ = 0. 13. ci)\(/ and xfrcf) give the same values of the invariants m, m^, m^. 14. If ^' be conjugate to (p, (fxp' is self-conjugate. 1 5. Shew that ( Fad)- + ( F/30y + ( Fy6f = 20- if a, /3, y be rectangular miit-vectors. 16. Prove that V^- {(l>—g)p =—pV'^g + 2Vg. 17. Solve the equations : — {a.) (j)^ = zt; (b.) (f) + x = ^,) where one, or two, unknown linear and vector functions are given in terms of known ones. (Tait, Proc. S. S. E. 1870-71.) 18. If ^ be a self-conjugate linear and vector function, ^ and rj two vectors, the two following equations are consequences one of the other, viz. : — ^ F.-qcjyr] From either of them we obtain the equation This, taken along with one of the others, gives a singular theorem when translated into ordinary algebra. What property does it give of the surface S.p(pp(p^p = 1 ? CHAPTER VI. GEOMETRY OF THE STKAIGHT LINE AND PLANE. 188.] Having, in the five preceding Chapters, given a brief exposition of the theory and properties of quaternions, we intend to devote the rest of the work to examples of their practical appli- cation, commencing, of course, with the simplest curve and surface, the straight line and the plane. In this and the remaining Chapters of the work a few of the earlier examples will be wrought out in their fullest detail, with a reference to the first five whenever a transformation occurs ; but, as each Chapter proceeds, superfluous steps will be gradually omitted, until in the later examples the full value of the quaternion processes is exhibited. 189.] Before proceeding to the proper business of the Chapter we make a digression in order to give a few instances of applications to ordinary plane geometry. These the student ma}^ multiply in- definitely with great ease. (a.) Euclid, I. 5. Let a and /3 be the vector sides of an iso- sceles triangle ; /3 — a is the base, and Ta = Tf3. The proposition will evidently be proved if we shew that a(a-^)-i=V/3(^-a)-i (§ 52). This gives a(a-/3)-^= (/3— a)-i/3, or {l3 — a)a = l3{a—(3\ or -a^ = -/3K {h.) Euclid, I. 32. Let ABC he the triangle, and let AC AB = '^ Ij-- - ..I 189.] GEOMETRY OF STRAIGHT LINE AND PLANE. 105 where y is a unit-vector perpendicular to the plane of the triangle. If / = 1, the angle CAB is a right angle (§ 74). Hence A = l-{^ 74). Let B = m- , C = n-- We have UlC^ymlB, UCB=zy^UCA, UBA=zy'"'UBC. Hence UBA = y™. y\ f UAB, or — 1 = yl+'n + n. That is l+m + n =2, or A + B+C = TT. This is, properly speaking, Legendre^s proof ; and might have been given in a far shorter form than that above. In fact we have for any three vectors whatever^ 13 y a which contains Euclid's proposition as a mere particular case. (c.) Buclid, I. 35. Let /3 be the common vector-base of the parallelograms, a the conterminous vector-side of any one of them. For any other the vector-side is a + .r/3 (§ 28), and the proposition appears as TF(3{a + x(3) = Tri3a (§§ 96, 98), which is obviously true. {(I.) In the base of a triangle find the point from which lines, drawn parallel to the sides and limited by them, are equal. If a, l3 be the sides, any point in the base has the vector p = (1— .r)a-f x(3. For the required point il-a:)Ta = xTl3 which determines x. Hence the point lies on the line p=^{Ua+Ul3) which bisects the vertical angle of the triangle. This is not the only solution, for we should have written T{\-x)Ta = TxT/3, instead of the less general form above tv/iick tacitly assumes that 1 — x and X are positive. We leave this to the student. 106 QUATERNIONS. [19O. ((?.) If perpendiculars be erected outwards at the middle points of the sides of a triangle, each being- proportional to the corresponding side, the mean point of the triangle formed by their extremities coincides with that of the original triangle. Find the ratio of each perpendicular to half the corresponding side of the old triangle that the new triangle may be equilateral. Let 2 a, 2(3, and 2(a + /3) be the vector-sides of the triangle, i a unit-vector perpendicular to its plane, e the ratio in question. The vectors of the corners of the new triangle are (taking the corner opposite to 2/3 as origin) Pi = a + eia, P2 = 2a + f3 + eili, Pa = a + (3 — ei(a + l3). From these HPi + P2 + P^) = H4a+2/3) = i (2a+ 2 (a + /3)), which proves the first part of the proposition. For the second part, we must have T{p2 — Pl) = ^iP3 — p2) = ^(Pl — Ps)- Substituting, expanding, and erasing terms common to all, the student will easily find 3e^ = 1 . Hence, if equilateral triangles be described on the sides of any tri- angle, their mean points form an equilateral triangle. 190.] Such applications of quaternions as those just made are of course legitimate, but they are not always profitable. In fact, when applied to plane problems, quaternions often degenerate into mere scalars, and become (§33) Cartesian coordinates of some kind, so that nothing is gained (though nothing is lost) by their use. Before leaving this class of questions we take, as an additional example, the investigation of some properties of the ellipse. 191.] We have already seen (§31 (k)) that the equation p = a cos ^4-/3 sin ^ represents an ellipse, 6 being a scalar which may have any value. Hence, for the vector-tangent at the extremity of p we have dp . ^ _ • CT = -J- =—a?,mQ + li cos 6, do which is easily seen to be the value of p when 6 is increased by - • Thus it appears that any two values of p, for which 6 differs by 1 94-] GEOMETIIY OF STEAIGHT LINE AND PLANE. 107 - , are conjugate diameters. The area of the parallelog-ram circum- scribed to the ellipse and touching it at the extremities of these diameters is, therefore, by § 96, ^.TVp-Y- = 'iTF {acos6 + ^sh\e)(—asm + l3cose) (Id = iTFafB, a constant, as is well known. 192.] For equal conjugate diameters we must have T{acos9 + l3&me) = T( — asme + ftcose), or (a2— /32)(cos-^-sin2^) + 45ay3cos0sin0 = 0, a- — p The square of the common length of these diameters is of course 2 because we see at once from § 1 9 1 that the sura of the squares of conjugate diameters is constant. 193.] The maximum or minimum of p is thus found; dTp I r, dp = - (_(a2— /3^)cos0sin^ + -Sa/3(cos2^— sin^^)). For a maximum or minimum this must vanish *, hence tan 26= -^— ^, , a'^ — p" and therefore the longest and shortest diameters are equally inclined to each of the equal conjugate diameters. Hence, also, they are at right angles to each other. 194.] Suppose for a moment a and (3 to be the greatest and least semidiameters. Then the equations of any two tangent- lines are p = acosd +/3sin0 + a:'( — asin^ +{3 cos 6), p = acos^i + i3sin0i + a'i( — asin0j + /3cos^i). If these tangent-lines be at right angles to each other ^( — asin^-(-/3cos0)( — asin^i + iScos^i) = 0, or a^ sin d sin d-^ + (3- cos 6 cos 6^ = 0. dTp ^ J ^ <^P n * The student must carefully notice that here we put -— - = 0, and not -^- = 0. A little reflection will shew him that the latter equation involves an absurdity. 108 QUATERNIONS. [ 1 9 5 • Also, for their point of intersection we have, by comparing coeffi- cients of a, /3 in the above values of p, cos^— ^sin^ = cos^j— a?jsin^i, QinO + x cos 6 = sin 6^ + x^ cos 6^ . Determining x^ from these equations, we easily find the equation of a circle ; if we take account of the above relation between 6 and 6-^. AlsOj as the equations above give x = —x-,^, the tangents are equal multiples of the diameters parallel to them ; so that the line joining the points of contact is parallel to that joining the extremities of these diameters. 195.] Finally, when the tangents p = a cos 9 + /3 sin ^ +x { — a sin 6 + {3 cos 6), p = a cos ^j + ^ sin ^1 + a?i ( — a sin 9i + (3 cos 6^), meet in a given point p = aa + d/S, we have a = cos 9 — x sin 9 = cos 9^ — a?^ sin 9^, h = sin 9-^x cos 9 = sin 9-^ + x-^ cos 9-^ . Hence x'^ =. a^ •{■h^ — \ =. x\ and a cos 5 + 5sin^ = l = a cos 9^-\-h sin 9-^ determine the values of 9 and x for the directions and lengths of the two tangents. The equation of the chord of contact is p =z y{a COS 5 + /3 sin ^) + ( 1 —y) (a cos ^j + /3 sin 9^. If this pass through the point p = 7Ja + ^^, we have ji; = y cos^ + (l— _^)cos^j, q = j/sin0 + (l— j/)sin^i, from which, by the equations which determine 9 and 9^ , we get a2) + lq = 7/+l—y = 1 . Thus if either a and b, or ^; and 5-, be given, a linear relation con- nects the others. This, by § 30, gives all the ordinary properties of poles and polars. 196.] Although, in §§ 28-30, we have already given some of the equations of the line and plane, these were adduced merely for their applications to anharmonic coordinates and transversals ; and not for investigations of a higher order. Now that we are prepared to determine the lengths and inclinations of lines we may investigate these and other similar forms anew. 200.] GEOMETRY OF STRAIGHT LINE AND PLANE. 109 197.] The equation of the imlejinite line draion through the origin 0, of lohich the vector OA, = a, forms a part, is evidently p = xa, or p II a, or " Vap = 0, or Up = Ua ; the essential characteristic of these equations being" that they are linear, and involve one indeterminate scalar in the value of p. We may put this perhaps more clearly if we take any two vectors, (3, y, which, along with a, form a non-coplanar system. Operating with S.FajB and S.Vay upon any of the preceding equa- tions, we get S.ajip = 0,| and S.ayp = 0.) Separately^ these are the equations of the planes containing a, fi, and a, y ; together, of course, they denote the line of intersection. 198.] Conversely, to solve equations (1), or to find p in terms of known quantities, we see that they may be written S.pVa^ = 0,^ S.pVay = 0,3 so that p is perpendicular to Ta/S and Tay, and is therefore parallel to the vector of their product. That is, pII r.Va^Vay, II — aS.ai^y, or p = xa. 199.] By putting p—^ for p we change the origin to a point B where OB = — /3, or BO = /3 ; so that the equation of a line parallel to a, and passing through the extremity of a vector /3 drawn from the origin, is p — [3 = xa, or p = /3 + xa. Of course any two parallel lines may be represented as p ■= [3 +xa, p = ^i+Xj^a; or Va{p-^) = 0, ra(p-/3j) = 0. 200.] The equation of a line, draivn thronyh the extremit// of ^, and meeting a perpendicularly, is thus found. Suppose it to be parallel to y, its equation is p = /3 + xy. To determine y we know, first, that it is perpendicular to a, which gives Say = 0. 110 QUATERNIONS. [2OI. Second/^, a, /J, and y are in one plane, which gives S.a^y = 0. These two equations g-ive y^V.aVaP, whence we have p = l3 + xa Ea/3. This might have been obtained in many other ways ; for instance, we see at once that /3 = a- la/3 = a-^Sal3 + a-'^FajB. This sliews that a~^Va^ (which, is evidently perpendicular to a) is coplanar with a and /3, and is therefore the direction of the re- quired line ; so that its equation is p = l3+^a-'^ral3, the same as before if we put — yf^-^ for x. la 201.] By means of the last investigation we see that — a-^Val3 is the vector perpendicular drawn from the extremity of /3 to the line p = xa. Changing the origin, we see that -a-Wa{^-y) is the vector perpendicular from the extremity of /3 upon the line p = y + ^a. 202.] The vector joining B (where OB = /3) with any point in p •= y ■\- Xa is y + a?a— /3. Its length is least when f/T(y + a?a— /3) = 0, or xS'a(y + a?a — /3) = 0, i. e. when it is perpendicular to a. The last equation gives xd^+Sa{y-^) = 0, or xa =—a~''-Sa{y—f3). Hence the vector perpendicular is y-(3-a-'Sa{y-l3), or a-ira(y — /3) =— a-ira(/3— y), which agrees with the resvilt of last section. 203.] To find the shortest vector distance between two lines p = /3 + xa, and pi = /3i + x^a^ ; 204-] GEOMETRY OF STRAIGHT LINE AND PLANE. Ill we must put (lT{p — p^ = 0, or S(p-pj){dp-dpi) = 0, or S{p — p{){adiC — aidu\) = 0. Since x and x^ are independent, this breaks up into the two con- ditions Sa{p—Pi)=0, %(p-pi) = ; proving the well-known truth that the required line is perpendicular to each of the given lines. Hence it is parallel to Vaa-^^ , and therefore we have p — p^ = l3 + xa — f3^ — x^ai = j/raa^ (1) Operate by S.nai and we get This determines j/, and the shortest distance required is Tip-P,) = n^jVaa,) = '^fy^^ = TS.{Uraa,){,3-l3,). IJYote. In the two last expressions T before S is inserted simply to ensure that the length be positive. If '5'.aai(/3 — /3i) be negative, then (§89) S.ai^a{l3 — ^^) is positive. If we omit the T, we must use in the text that one of these two ex- pressions which is positive.] To find the extremities of this shortest distance, w^e must operate on (1) with S.a and Sm^. We thus obtain two equations, which determine x and x-^ , as ?/ is already known. A somewhat different mode of treating this problem will be dis- cussed presently. 204.] In a ffiveti tetrahedron to find a set of rectangular coordinate axes, such that each axis shall pass throiujh a pair of opposite edges. Let a, /3, y be three (vector) edges of the tetrahedron, one corner being the origin. Let p be the vector of the otigin of the sought rectangular system, which may be called i, y, h (unknown vectors). The condition that /, drawn from p, intersects a is 8.tap = ^ (1) That it intersects the opposite edge, whose equation is ^ = f3 + x{^ — y), the condition is S.i{i3-y){p-[3) = 0, 01' &'{(/3-y)p-/3y} = (2) There are two other equations like (1), and two like (2), which can be at once written down. 112 QUATERNIONS. [205. Put j3 — y=zai, y — a = /3i, a — l3 = yi, ^I3y = a^, Vya = jS,^, FajB = y^, J\a = 03 , FfB^lB = /33 , Vy-i^y = y^ ; and the six become S.iap = 0, S.ittyp — Sia^ = 0, S.jl3p = 0, S.j0,p-Sj^., = 0, S.kyp = 0, S.^Yip — Sky.^ = 0. The two in i give i \\ aSa2p — p{Saa2 + Sa^p). Similarly, J II /3%p-/5(^/3/3, + xS'/Sgp), and /c \\ ySy,j-p{Syy, + Sy^p). The conditions of rectangularity, viz., Sij = 0, Sjk = 0, Ski = 0, at once give three equations of the fourth order, the first of which is =: SajS Sa.2p Sj3.2P — Sap Sa.^p {S(3j32 + SfS^p) — S^p S/S.j^p [Saa.^ + Sa^p) + p2 (Saa^ + Sa,p) ( W2 + ^M- The required origin of the rectangular system is thus given as the intersection of three surfaces of the fourth order. 205.] The equation Sap = imposes on p the sole condition of being perpendicular to a ; and therefore, being satisfied by the vector drawn from the origin to any point in a plane through the origin and perpendicular to a, is the equation of that plane. To find this equation by a direct process similar to that usually employed in coordinate geometry, we may remark that, by § 29, we may write p = xji +yy, where /3 and y are any two vectors perpendicular to a. In this form the equation contains two indeterminates, and is often useful ; but it is more usual to eliminate them, which may be done at once by operating by S.a, when we obtain the equation first written. It may also be written, by eliminating one of the indeterminates only, as Vjip = ya, where the form of the equation shews that Saji = 0. Similarly we see that Sa{p — I3) = represents a plane drawn through the extremity of /3 and perpen- dicular to a. This, of course, may, like the last, be put into various equivalent forms, 206.] The line of intersection of the two planes S.aip-^) =0,1 and S.a^{p — ^i) = oj ^^ 208.] GEOMETRY OF STRAIGHT LINE AND PLANE. 113 contains all points whose value of p satisfies both conditions. But we may write (§ 92), since a, a^, and Faa^ are not coplanar, pS.aa^Faa^ = raaiS.aaiP+ T .aj aa^Sap + T J {aa^) aSa-^^p, or, by the given equations, — pT~ Vaa^ = F.a^ Vaa^SaH + V. F{aa^) a^/S^ + X Faa-^, (2) where .f, a scalar indeterminate, is put for S.aa^p which may have any value. In practice, however, the two definite given scalar equations are generally more useful than the partially indeterminate vector-form which we have derived from them. When both planes pass through the origin we have /3 = /S^ = 0, and obtain at once p ■=xFaa as the equation of the line of intersection. 207.] The plane xiasuncj through the origbi, and through the line of intersection of the two planes (1), is easily seen to have the equation Sa-j^f3iSap — Sa^Saip= 0, or S{aSaj^(3i — a^Sai3)p — 0. Por this is evidently the equation of a plane passing through the origin. And^ if p be such that Sap = Sal3, we also have Sa-^p = Sa^iS-^^, which are equations (1). Hence we see that the vector aSa^jBi — a-j^SajB is perpendicular to the vector-line of intersection (2) of the two planes (1), and to every vector joining the origin with a point in that line. The student may verify these statements as an exercise. 208.] To fad the vector-perpendicular from the extremitij of ^ on the plane Sap = 0, we must note that it is necessarily parallel to a, and hence that the . value of p for its foot is p _ ^ _j_ ^^^ where xa is the vector-perpendicular in question. Hence Sa{(3 + .ra) = 0, which gives xa- = — Sa0, or .ra = — a^^SafS. Similarly the vector-perpendicular from the extremity of /3 on the plane Sa{p-y) = may easilv be shewn to be -a-'Sa{f3-y). I 114 QUATERNIONS. [209. 209.] The equation of the plane which passes through the extremities of a, li, y may be thus found. If p be the vector of any point in it, p — a, a— (3, and jB — yMe in the plane, and therefore (§ 101) S.{p-a){a-(3){(3-y) = 0, or Sp ( Fa0 + Ff3y + Fya) — S.af3y = 0. Hence, if b = x{ Fa/3 + F(3y + Fya) be the vector-perpendicular from the origin on the plane containing the extremities of a, (3, y, we have b = {Fa0+ Fj3y-\-Fya)-^S.al3y. From this formula, whose interpretation is easy, many curious pro- perties of a tetrahedron may be deduced by the reader. Thus, for instance, if we take the tensor of each side, and remember the result of § 100, we see that T{Fal3+F^y+Fya) is twice the area of the base of the tetrahedron. This may be more simply proved thus. The vector area of base is \F{a-0){y-j3) =-\{Fal3+Fl3y^Fya). Hence the sum of the vector areas of the faces of a tetrahedron, and therefore of any solid whatever, is zero. This is the hydrostatic proposition for solids immersed in a fluid subject to no external forces. 210.] Taking any two lines whose equations are p = /3 + .ra, p = jB^ + x^a-^, we see that S.aa-^^ip — h) = is the equation of a plane parallel to both. Which plane, of course, depends on the value of 8. Now if 8 = /3, the plane contains the first line; if 8 = 13-^, the second. Hence, \^ i/Faa^ be the shortest vector distance between the lines, we have s.aa^ [^-^^-^Vaa,) = 0, or T{yFaa^ = m(/3-/3i) UFaa^, the result of § 203. 211.] Find the eqimtion of the plane, passing through the origin, which makes equal angles with three given lines. Also find the angles in question. Let a, ^, y be unit-vectors in the directions of the lines, and let the equation of the plane be Sbp — 0. 2 13-] GEOMETRY OF STRAIGHT LINE AND PLANE. 115 Then we have evidently Sab = Sl3b = Syb = .r, suppose, where — -^^ is the sine of each of the required angles. But (§92) we have bS.a/Sy = a;{Fal3+ F^y + Vya). Hence S.p ( Ta^ + Vfiy + Tya) = is the required equation ; and the required sine is S.a^y ~ T{ra^+Vi3y+rya)' 212.] Find the locus of the middle points of a series of straight lines, each parallel to a given plane and having its extremities in tioo fixed lines. Let Syp = be the plane, and ^ _ ^3 _,_ ^a, p = I3i + ^r^ a^ , the fixed lines. Also let x and j\ correspond to the extremities of one of the variable lines, ct being- the vector of its middle point. Then, obviously, 2^ = l3 + xa + fS^+x^a^. Also SyiiB — ^i + xa — x^ai) = 0. This gives a linear relation between x and x-^ , so that, if we sub- stitute for Xj^ in the preceding equation, we obtain a result of the form r^ = b+xe, where b and e are known vectors. The required locus is, therefore, a straight line. 213.] Three planes meet in a point, and through the line of inter- section of each pair a plane is drawn perpendicular to the third ; prove that, in general, these pjlanes pass through the sa^ne line. Let the point be taken as origin, and let the equations of the planes ^e Sap = 0, Sf3p = 0, Syp = 0. The line of intersection of the first two is || FafB, and therefore the normal to the first of the new planes is F.yFa^. Hence the equation of this plane is S.pF.yFa[3 = 0, or SiSpSay — SapSlBy = 0, and those of the other two planes may be easily formed from this bv cyclical permutation of a, f3, y. I 2 116 QUATERNIONS. [214. We see at once that an}' two of these equations give the third by addition or subtraction, which is the proof of the theorem. 214.] Given any number of j^oints A, B, C, Sfc, whose vectors {from the origin) are a^, a^, Og, 8fc.,find the j}l(ine through the origin for tohich the sum of the squares of the perjpendicwlars let fall iipon it from these points is a 7naximum or minimum. Let SsTp = be the required equation, with the condition (evidently allowa1)le) Ttn- = 1. The perpendiculars are (§ 208) — ot'^/Sctoi, &c. Hence HxS'^tn-a is a maximum. This gives ^.S-sjaSadsT ^ ; and the condition that -57 is a unit-vector gives S-arduT =■ 0. Hence, as dzs- may have any of an infinite number of values, these equations cannot be consistent unless IS.O/iS'ats" = 3737, where x is a scalar. The values of a are known, so that if we put 2.a and whose ^1—n-"' centre is at i?. where OB = ^ , a definite point in the line OA. 1 —n^ 222.] Ifi7t any line, OP, drawn from the origin to a given jdane, OQ be taken such that OQ.OP is constant, fnd the locus of Q. Let Sap = a be the equation of the plane, zr a vector of the required surface. Then, by the conditions, TW Tp = constant = b^ (suppose), • and U-sT = Up. b^U^ b^^ trova these p = —^ — = 5-- I -or xs Substituting in the equation of the plane, we have aiz^-^-F-SavT = 0, which shews that the locus is a sphere, the origin being situated on it at the point farthest from the given plane. 223.] Find the locus of points the sum of the squares of whose dis- tances from a set of given points is a constant quantity. Find also the least value of this constant, and the corresponding locus. Let the vectors from the origin to the given points be Oj, a^, a„, and to the sought point p, then -c2 = (p_aJ-4-(/3 — 02)- + -f (p-aj^ = V — 2^/3 2a +2 (a^). Otherwise fp_?-«)^ = _ ^^ + ^("^) + 2^ , ^ n ^ n n^ the equation of a sphere the vector of whose centre is — j i.e. whose centre is the mean of the system of given points. Suppose the origin to be placed at the mean point, the equation becomes • /^s , v („2\ P'=- Z^ (for2a=0,§31(.)). 226.] THE SPHERE AND CYCLIC CONE. 123 The rig-lit-liand side is negative, and therefore the equation denotes a real surface, if ^2 ^ 2Ta^ as might have been expected. Wlien these quantities are equal, the locus becomes a point, viz. the new origin, or the mean point of the system. 224.] If we differentiate the equation Tp = Ta we get Sfidp = 0. Hence (§ 137), p is normal to the surface at its extremity, a well- known property of the sphere. If OT be any point in the plane which touches the sphere at the extremity of p, -sr— p is a line in the tangent plane, and therefore perpendicular to p. So that Sf) (ot — p) = 0, or Szsp = — Tp^ = a^ is the equation of the tangent plane. 225.] If this plane pass through a given point B, whose vector is /3, we have S/3p = a^. This is the equation of a plane, perpendicular to /3, and cutting from it a portion whose length is Tl3 ' If this plane pass through a fixed point whose vector is y we must have Sl3y = a-, so that the locus of /3 is a plane. These results contain all the ordinary properties of poles and polars with regard to a sphere. 226,] A line drawn parallel to y, from the extremity of (3, has the equation p _ S-\-xy. This meets the sphere ^z _ ^2 in points for which x has the values given by the equation fB'' + 2xSf3y + x^y^ = aK The values of x are imaginary, that is, there is no intersection, if a2y2 ^ /-2^3y < 0. The values are equal, or the line touches the sphere, if a'-y-'+ni3y = 0, or S'^jSy = y" (j3^ — a^). This is the equation of a 'cone similar and similarly situated to the cone of tangent-lines drawn to the sphere, but its vertex is at the centre. That the equation represents a cone is obvious from the 124 QUATERNIONS. [227. fact that it is homogeneous in ly, i.e. that it is independent of the length of the vector y. [It may be remarked that from the form of the above equation we see that, if x and x' be its roots, we have {xTy){x'Ty) = a^-^\ which is Euclid, III, 35, 36, extended to a sphere.] 227.] Find the locus of the foot of the perpendicular let fallfrofn, a given point of a sphere on any tangent-plane. Taking the centre as origin^ the equation of any tangent-plane may be written ^^^ ^ „2_ The perpendicular must be parallel to p, so that^ if we suppose it drawn from the extremity of a (which is a point on the sphere) we have as one value of ot CT = a-\-xp. From these equationsj with the help of that of the sphere 2 2 9 = a > we must eliminate p and x. We have by operating on the vector equation by S.'sr ■37^ = Sa'!iT-{-xSrs'p = Sa^ + xa^. T_ OT — a a^ Osy — a) Hence p ■= = — 5 — ^ X OT^ — oata- Taking the tensors^ we have (ct^ — SaiirY = a^('5T — a)^, the required equation. It may be put in the form and the interpretation of this gives at once a characteristic property of the surface formed by the rotation of the Cardioid about its axis of symmetiy. 228.] We have seen that a sphere, referred to any point what- ever as origin, has the equation T{p-a) = Tl3. Hence, to fnd the rectangle under the segments of a chord drawn through any point, we may put p = xy; where y is any unit-vector whatever. This gives x'y'^-2xSay-Va^ = ^''-, and the product of the two values of x is /o2 2 P —o. „ „,, 231.] THE SPHERE AND CYCLIC COXE. 125 This is positive, or the vector-chords are drawn in the mme direc- tion, if T(3- there is no point which satisfies the conditions. 230.] Describe a sjjhere, loith its centre in a given line, so as to jjass through a given point and touch a given plane. Let xa, where x is an undetermined scalar, be the vector of the centre, r the radius of the sphere, /3 the vector of the given point, and Syp = a the equation of the given plane. The vector-perpendicular from the point xa on the given plane is (§208) {a-xSya)y'\ Hence, to determine x we have the equation T.{a-x8ya)y-^ = T{xa-^) = r, so that there are, in general, two solutions. It will be a good exercise for the student to find from this equation the condition that there may be no solution, or two coincident ones. 231.] Describe a sphere tvhose centre is in a given line, and which passes through tv)0 given points. Let the vector of the centre be xa, as in last section, and let the vectors of the points be ^ and y. Then, at once, Ti^y-xa) = T{l3-xa) = r. Here there is but one sphere, except in the particular case when we have Ty = TjB, and Say = SajB, in which case there is an infinite number. 126 QUATERNIONS. [232. The student should carefully compare the results of this section and the last, so as to discover why in g-eneral two solutions are possible in the one case, and only one in the other. 232.] A sphere touches each of two straight lines, which do not meet : find the locus of its centre. We may take the origin at the middle point of the shortest dis- tance (§ 203) between the given lines, and their equations will then be p = a + i»j8, P=— a + .ri/3i, where we have, of course, Sa^ = 0, Sap^ = 0. Let o- be the vector of the centre, p that of any point, of one of the spheres, and r its radius ; its equation is Tip-a) = r. Since the two given lines are tangents, the following equations in x and a?i must have pairs of equal roots, T{a^ x(3-(t) = r, T{-a^-x^(3^-a) = r. The equality of the roots in each gives us the conditions S'^fia- =^\{a-^)-+r^), Eliminating r we obtain /3-2^2^,,_/3-2^2^^^ ^ (a_(r)2-(a + (T)2 =_4^a(T, which is the equation of the required locus. [As we have not, so far, entered on the consideration of the qua- ternion form of the equations of the various surfaces of the second order, we may translate this into Cartesian coordinates to find its meaning. If we take coordinate axes of x, y, z respectively parallel to /3, ^1, a, it becomes at once {x 4- miff — (j/ + mxf = j)z, where m and p are constants ; and shews that the locus is a hy- perbolic paraboloid. Such transformations, which are exceedingly simple in all cases, will be of frequent use to the student who is proficient in Cartesian geometry, in the early stages of his study of quaternions. As he acquires a practical knowledge of the new calculus, the need of such assistance will gradually cease to be felt.] Simple as the above solution is, quaternions enable us to give one vastly simpler. For the problem may be thus stated — Find the locus of the point whose distances from ttco given lines are equal. 2 34-] THE SPHERE AND CYCLIC CONE. 127 And, with the above notation, the equality of the perpendiculars is expressed (§ 201) by TV. (a - 0-) t//3 = TV. (a + 0-) f//3i , which is easily seen to be equivalent to the equation obtained above. 233.1 Two spheres being given, shew that spheres which cut them at given angles cut at right angles another fixed sphere. If c be the distance between the centres of two spheres whose radii are a and b, the cosine of the angle of intersection is evidently 2ab Hence, if a, Oi, and p be the vectors of the centres, and a, a^, r the radii, of the two fixed, and of one of the variable, spheres ; A and A^ the angles of intersection, we have (p — of + a^ + r^ = 2 ar cos A, {p — a-^^+a'l + r^ = 2 a^r cos A^. Eliminating the first power of r, we evidently must obtain a result such as (^p_^)2^52_j_^2 _ 0, where (by what precedes) /3 is the vector of the centre, and b the radius, of a fixed sphere (p_/3)^ + ^'^ = 0, which is cut at right angles by all the varying spheres. By effect- ing the elimination exactly we easily find b and /3 in terms of given quantities. 234.] To inscribe in a given sphere a closed 2)olygon, plane or gauche, lohose sides shall be parallel respectivelij to each of a series of given vectors. Let Tp = 1 be the sphere, a, ^, y, , rj, 6 the vectors, n in number, and let Pi,P2, Pn5 be the vector-radii drawn to the angles of the polygon. Then p^ — Pi = -^i") &c., &c. From this, by operating by S.{p2+pi), we get Also = rap2 — Vapj^ . Adding, we get = ap., + Kap^ = ap2 + Pi a- Hence P2 — — a~^pia. [This might have been written down at once from the result of §105.] Similarly P3 = — Z^" V2/3 = /3~^ "~^ Pi «A &c. Thus, finally, since the polygon is closed, P-. + i = Pi = (-)"^-''r' f3-'a-^p,al3 11O. 128 QUATERNIONS. [235. We may suppose the tensors of a, /3 v, to Ije each unity. Hence, if , a = 0/3 yi0, we have a~'^ = 6~^ ?/~^ /3~^ a~^, which is a known quaternion ; and thus our condition becomes Pi = (-)"«- V]«- This divides itself into two cases, according as n is an even or an odd number. If n be even^ we have «Pi - Pi^f- Removing the common part p^Sa, we have Fp^Fa = 0. This gives one determinate direction, ^f T'a, for p■^ ; and shews that there are two, and only two, solutions. If % be odd, we have ap = p a. which requires that we have Sa = 0, i. e. a must be a vector. Hence Sap^^ = 0, and therefore p^ may be drawn to any point in the great circle of the unit-sphere whose poles are on the vector a. 235.] To illustrate these results^ let us take first the case of 71 = 3. Here we must have S.alSy = 0, or the three given vectors must (as is obvious on other gi'ounds) be jjarallel to one plane. Here a/3y, which lies in this plane, is (§ 106) the vector-tangent at the first corner of each of the inscribed tri- angles ; and is obviously perpendicular to the vector drawn from the centre to that corner. If n = 4, we have p^ ]| r. aiByb, as might have been at once seen from §100. 236.] Hamilton has given {Lectures, p. 674) an ingenious and simple process by which the above investigation is rendered ap- plicable to the more difficult problem in which each side of the inscribed polygon is to pass through a given point instead of being parallel to a given line. His process depends upon the integration of a linear equation in finite differences. By an immediate appli- cation of the linear and vector function of Chapter V, the above solutions may be at once extended to any central surface of the second order. 237.] To fad the equation of a cone of rerolutlon. whose vertex is the origiu. 240.] THE SPHERE AND CYCLIC CONE. 129 Suppose a, whore 2a = 1 , to be its axis, £incl e the cosine of its semi-vertical angle ; then, if p be the vector of any point in the cone, SaUp = +e, or S^ap = —e^p^. 238,] Change the origin to the point in the axis whose vector is xa, and the equation becomes { — x + Sa-sj)^ = — e-(.-i^a + Tn-)^. Let the radius of the section of the cone made by Sar^ := retain a constant value b, while so changes ; this necessitates SB _ so that when x is infinite, e is unity. In this case the equation becomes ^2^_^^^2^j2 _ q, which must therefore be the equation of a circular cylinder of radius h, whose axis is the vector a. To verify this we have only to notice that if OT be the vector of a point of such a cylinder we must (§ 201) have TFavT = 6, which is the same equation as that above. 239.] To finely generaUj/, the equation of a conewhicJi has a circular section : — Take the origin as vertex, and let the circular section be the intersection of the plane Sap = 1 with the sphere (passing through the origin) p2 = Si3p. These equations may be written thus, SalJp = ^-, -Tp = SfiUp. Hence, eliminating Tp, we find the following equation which Up must satisfy— SaUpS(3Up =-1, or p'^ — SapSjBp = 0, which is therefore the required equation of the cone. As a and (3 are similarly involved, the mere/orm of this equation proves the existence of the subcontrary section discovered by Apol- lonius. 240.] The equation just obtained may be written 8MaUpS.Ui3Up = --^—, 130 QUATERNIONS. [24 1. or, since a and /3 are perpendicular to the cyclic arcs (§ 59*), sin^; sin7j'= constant, where p and j/ are arcs drawn from any point of a spherical conic perpendicular to the cyclic arcs. This is a well-known property of such curves. 241.] If we cut the cyclic cone by any plane passing through the origin, as Syp = 0, then Vay and Vjiy are the traces on the cyclic planes, so that p = xUVay+i/UVjiy (§ 29). Substitute in the equation of the cone, and we get —x''--f + Pxi/ = 0, where P is a known scalar. Hence the values of x and y are the same pair of numbers. This is a very elementary proof of the proposition in § 59*, that PL = 3IQ (in the last figure of that section). 242.] When x and y are equal, the transversal arc becomes a tangent to the spherical conic, and is evidently bisected at the point of contact. Here we have This is the equation of the cone whose sides are perpendiculars (through the origin) to the planes which touch the cyclic cone, and from this propei-ty the same equation may readily be deduced. 243.] It may be well to observe that the property of the Stereo- graphic projection of the sphere, viz. that the projection of a circle is a circle, is an immediate consequence of the above form of the equation of a cyclic cone. 244] That §239 gives the most general form of the equation of a cone of the second order, when the vertex is taken as origin, follows from the early results of next Chapter. For it is shewn in § 249 that the equation of a cone of the second order can always be put in the form 2^.SapSi3p + Ap^ = 0. This may be written Spcpp = 0, where cf) is the self-conjugate linear and vector function (/)p = 2 r. ap(3 + [A + 2: So.li)p. By § 168 this may be transformed to and the general equation of the cone becomes (7J - SAp.) p2 + 2SAp Sp.p = , which is the form obtained in § 239. 247-] THE SrHERE AND CYCLIC CONE. 131 245.] Taking the form Spcf^p = as the simplest, we find by differentiation Sdp(j)p + Spdp = 0, and S.ipKp = 0, may degenerate into pairs of i:)lanes ? EXAMPLES TO CHAPTER VII. 133 9. Find the locus of the vertices of all right cones which have a common ellipse as base. 10. Two right circular cones have their axes parallel, shew that the orthogonal projection of their curve of intersection on the plane containing their axes is a parabola. 11. Two spheres being given in magnitude and position, every sphere which intersects them in given angles will touch two other fixed spheres and cut a third at right angles. 12. If a sphere be placed on a table, the breadth of the elliptic shadow formed ])y rays diverging from a fixed point is independent of the position of the sphere. 1 3. Form the equation of the cylinder which has a given circular section, and a given axis. Find the direction of the normal to the subcontrary section. 11. Given the base of a sphc^rical triangle, and the product of the cosines of the sides, the locus of the vertex is a spherical conic, the poles of whose cyclic arcs are the extremities of the given base. 15. (Hamilton, Bishop Law's Premium Ex., 1858.) {/(,) What property of a sphero-conic is most immediately in- dicated by the equation ^■^^^= 1? a p {h.) The equation {VXpf + {Siipf = also represents a cone of the second order ; A is a focal line, and /x is perpendicular to the director-plane cor- responding. (c.) Wliat property of a sphero-conic does the equation most immediately indicate ? 16. Shew that the areas of all triangles, bounded by a tangent to a spherical conic and the cyclic arcs, are equal. 17. Shew that the locus of a point, the sum of whose arcual dis- tances from two given points on a sphere is constant, is a spherical conic. 18. If two tangent planes be drawn to a cyclic cone, the four lines in which they intersect the cyclic planes are sides of a right cone. 19. Find the equation of the cone whose sides arc the intersections of pairs of mutually perpendicular tangent planes to a given cyclic cone. 134 QUATERNIONS. 20. Find the condition that five g-iven jwints may lie on a sphere. 2 1 . What is the surface denoted by the equation where p = a?a +y/3 + zy^ a, /3, y being given vectors^ and x, y, z variable scalars ? Express the equation of the surface in terms of p, a, /3; y alone. 22. Find the equation of the cone whose sides bisect the angles between a fixed line and any line, in a given plane, which meets the fixed line. What property of a spherical conic is most directly given by this result ? CHAPTER VIII. SURFACES OF THE SECOND ORDER. 248.] The general scalar equation of the second order in a vector p must evidently contain a term independent of p, terms of the form S.apb involving p to the first degree, and others of the form S.apbpc involving p to the second degree, «, h, c, &c. being constant quater- nions. Now the term S.apb may be written as SpFiJja), or as S. {Sa + Va) p {Sb -\-7b) = Sa Sp Vb + Sb Sp Fa + S.prb Va, each of which may evidently be put in the form Syp, where y is a known vector. Similarly * the term S.apbpc may be reduced to a set of terms, each of which has one of the forms Ap'^, [SapY, SapSfBp, the second being merely a particular case of the third. Thus (the numerical factors 2 being introduced for convenience) we may write the general scalar equation of the second degree as follows : — 22.SapS0p + Ap^ + 2Syp = a (1) 249.] Change the origin to D where OD = b, then p becomes p 4- 8, and the equation takes the form 2 2 . Sap S^p + Ap^-^2 l(Sap SjSb + S/Bp Sah) + 2 A Sbp + 2 Syp + 2 S . SahS(ih 4- Ah'^ + 2 Syh— C = ; from which the first power of p disappears, that is the surface is referred to its centre, if 2(a/S/38 + i3/Sa§) + J8 + y = 0, (2) * For S.aphpc = S.capip = S.a'php = {2Sa'Sb-Sa'h)p'^ + 2Sa'pSbp; and in particular cases we may have Va' = Vb. 136 QUATERNIONS. [25O. a vector equation of the first dcg-'rec^ which in general gives a single definite value for 8, by the processes of Chapter V. [It would lead us beyond the limits of an elementary treatise to consider the special cases in which (2) represents a line^ or a plane, any point of which is a centre of the surface. The processes to be employed in such special cases have been amply illustrated in the Chapter re- ferred to.] With this value of S, and putting D = C-2Syb-Ab^-2^.SabSl3b, the equation becomes 2'2.SapSl3p + Ap^ = D. If D = 0, the surface is conical (a case treated in last Chapter) ; if not, it is an ellipsoid or hyperboloid. Unless expressly stated not to be, the surface will, when D is not zero, be considered an ellip- soid. By this we avoid for the time some rather delicate con- siderations. By dividing by D, and thus altering only the tensors of the constants, we see that the equation of central surfaces of the second order, referred to the centre, is (excluding cones) 22{SapSl3p)+ffp^ = 1 (3) 250.] Differentiating, we obtain 2 2 { Sadp Sl3p + Sap SjSdp } + 2 (/Spdp = , or S.dp{l{aBl3p + (3Sap)+ffp} = 0, and therefore, by § 13 7, the tangent plane is Si-^-p) maS(3p + l3Sap)+^p} = 0, i.e. S.,^{l(aSf3p-hfBSap)+//p} = I, by (3). Hence, if v = l{aSf3p + j3Sap) + ffp, (4) the tangent plane is -6Vot = 1, and the surface itself is Srp = 1. And, as v~^ (being perpendicular to the tangent plane, and satis- fying its equation) is evidently the vector-perpendicular from the origin on the tangent plane, v is called the vector of proocimiiy . 251.] Hamilton uses for />, which is obviously a linear and vector function of p, the notation (/;p, <^ expressing a functional operation, as in Chapter V. But, for the sake of clearness, we will go over part of the ground again, especially for the benefit of students who have mastered only the more elementary parts of that Chapter. We have, then, ^^ = l.{aSiip + jiSap)-\-gp. 2 53-] SURFACES OF THE SECOND ORDER. 137 With this definition of (p, it is easy to see that (a.) (ji{p-[~a) = {dp). (d.) SiTcpp = ^(SacrS(3p + Sl3(TSap)+ffSp(T = Sp(pa; or is, in this ease, self-conjugate. This last property is of great importance. 252.] Thus the general equation of central surfaces of the second degree (excluding- cones) may now be written Sp4^p=l (1) Differentiating, Sdp(f)p-{-Spd<})p = 0, which, by applying (pdp = 0, and therefore, as in § 250, though now much more simply, the tang-ent plane at the extremity of p is S{^—p)(pp = 0, or S-sTcpp = Sp(f)p = 1 . If this pass through A{OA = a), we have Sa(f)p = 1 , or, by {d.), Spcfja = 1, for all possible points of contact. This is therefore the equation of the plane of contact of tang-ent planes drawn from J. 253.] To find the envelop'wg cone zoJiose vertex is A, notice that {Sp(l,p- 1 ) Jri:'{Spa^-\f = 0, and^ is found. Then our equation becomes {Sp(f>p—l) {Sacfya- l) — {Sp(f)a—lf = 0, (1) which is the cone required. To assure ourselves of this, transfer the origin to A, by putting p + a for p. The result is, using (a.) and {d.], {Spcj^p + 2 Sp(jm + Sa4m - 1 ) {Sa^m— 1 ) — {Sp(l)a + Smjia - 1 )- = 0, or Spcpp {Sa(pa - 1 ) — (Spcjmf = 0, which is homogeneous in Tp^, and is therefore tlic equation of a cone. 138 QUATERNIONS. [254. Suppose A infinitely distant, then we may put in (1) xa for a, where x is infinitely great, and, omitting- all but the higher terms, the equation of the cylinder formed hy tangent lines parallel to a is {Sp(f)p — l)Sa(}}a — (Spcpa)^ = 0. 254.] To study the nature of the surface more closely, let us J! lid the locus of the mi (Idle points of a sj/siem of paralleJ chords. Let them he parallel to a, then, if ct be the vector of the middle point of one of them, ■^■\-oca and ct — o^a are simultaneous values of p which ought to satisfy (1) of§ 252. That is xS.(CT + ira)(|)(OT + 5'a) = 1. Hence, by {a.) and (^/.), as before, Srs<\>ts! -\- oc^Sa^a. = 1, /S't!7(/)a = (1) The latter equation shews that the locus of the extremity of ct, the middle point of a chord parallel to a, is a plane through the centre, whose normal is 0a ; that is, a plane parallel to the tangent plane at the point where OA cuts the surface. And (^/.) shews that this relation is reciprocal — so that if /3 be atiy value of -cr, i, e. be any vector in the plane (1), a will be a vector in a diametral plane which bisects all chords parallel to /3. The equations of these planes are S^(j)a = 0, S-sx(t)/3 = 0, so that if r.cpcKpiS = y (suppose) is their line of intersection, we have Sycpa = = Sacpy, \ Sy4>(3 = = Sf3a = = Sa(f)l3. ) Hence there is an infuiite numher of sets of three vectors a, /3, y, such that all chords imrallel to any one are bisected hy the diametral jdane containing the other two. 255. J It is evident from § 23 that any vector may be expressed as a linear function of any three others not in the same plane, let then p = cca-\-yP + zy, where, by last section, Sac^ji = -iS/S^a = 0, Sa(f)y = Sycpa = 0, S/Bcpy = SycjilB = 0. And let Sa(f)a = Ij ) Sy(f)y = 1, ) so that a, 13, and y are vector conjugate semi-diameters of the surface we are ena-aired on. 2 57-] SURFACES OF THE SECOND ORDER. 139 Substituting- the above value of p in the equation of the surface, and attending- to the equations in a, /3, y and to {a), [h.), and {d.), we have Spc^^p = S{xa +^/3 + zy) (/) {xa +y/3 + zy), = x"^ + 1/'^ -\- z^ = 1. To transform this equation to Cartesian coordinates, we notice that X is the ratio which the projection of p on a bears to a itself, &c. If therefore we take the conjugate diameters as axes of ^, ??, C) ^md their lengths as a, h, c, the above equation becomes at once P ,,2 ^2 - + - -t- — - = 1 a" b^ c^ the ordinary equation of the ellipsoid referred to conjug-ate diameters. 256.] If we write — ^/^ instead of 0, these equations assume an interesting form. We take for granted, what we shall afterwards prove, that this halving- or extracting the root of the vector func- tion is lawful, and that the new linear and vector function has the same properties [a.), (Jj.), (c), (d.) (§ 251) as the old. The equation of the surface now becomes Sp\j/"p = —1, or S^p\j/p = — 1 5 or, finally, T\}/p = 1 . If we compare this with the equation of the unit-sphere Tp=l, we see at once the analogy between the two surfaces. T//e spliere can he changed into the ellipsoid, or vice versa, hj/ a linear deformation of each vector, the o2)erator being the function i// or its inverse. See the Chapter on Kinematics. 257.] Equations (2) § 254 now become Sa^^^ = == *9x//ai///3, &c., (1) so that y^a, \(/(3, ^y, the vectors of the unit-sphere which correspond to semi-con fig ate diameters of the ellipsoid, form a rectangular system. We may remark here, that, as the equation of the ellipsoid referred to its principal axes is a case of § 255, we may now suppose ?", /, and T • 11 . . x'^ y'^ z"^ k to have these directions, and the equation is -s-^-T5"^ — 7= ^> . , . . . a^ 0^ c- which, m quaternions, is We here tacitly assume the existence of such axes, but in all cases, by the help of Hamilton's method, developed in Chapter V, we at once arrive at the cubic equation which gives them. 140 QUATERNIONS. [258. It is evident from the last-written equation that , 'Sip jSjp kSkp and V^^=-(— + "r + ~7~>'' which latter may be easily proved by shewing- that yj/^y =-(t)p. And this expression enables us to verify the assertion of last section about the properties of \|/. As Sip=—a; kc, x,y,z being- the Cartesian coordinates referred to the principal axes, we have now the means of at once transform- ing any quaternion result connected with the ellipsoid into the or- dinary one. 258.] Before proceeding to other forms of the equation of the ellipsoid, we may use those already given in solving a few problems. Fmd the locus of a jioint when the perpendicular from the centre on its polar plane is of constant length. If OT be the vector of the point, the polar plane is Sp(pnT = 1 , and the length of the perpendicular from is jf— (§208). Hence the required locus is or S-sTy}, and, that the first power of p may disappear, {b-a) = e^^Sl3b-^'^b), a linear equation for 8. To solve it, note that Sa^ = 0, operate by S.I3 and we get (1 -e^/J^ + e^-i3'-)Sf3b = Sf3b = 0. 26 1.] SURFACES OF THE SECOND ORDER. 141 Hence 8 — a=— e-/3^8, a or b = Referred to this point as origin the equation becomes which shews that it belongs to a surface of revohition (of the second order) whose axis is parallel to (3, as its intersection with a plane Sf3p = a, perpendicular to that axiS;, lies also on the sphere ,^ e'-a- e'^f3'"a^ p^ " TT^^ ~ (i + ^W ' In fact, if the point be the focus of any meridian section of an oblate spheroid, the line is the directrix of the same. 260.] A si^here, passing through the centre of cm ellipsoid, is cut bi/ a series of spheres tohose centres are on the ellipsoid and which j^ass through the centre thereof; find the envelop of the planes of inter- section. Let (p — a)- = or be the first sphere, i.e. p2_2/Sap = 0. One of the others is fj^~2Sjyp = 0, where Sus-cp'uy = 1. The plane of intersection is *S'(ot- — a) p = 0. Hence, for the envelop, (see next Chapter,) StjJ (A 37 =1 0, ) , , , ^ , J- where ct = j/ct, OCT- p = 0, ) or (})r^ = xp, {Fx=0}, i.e. OT = xcp^^p. Hence x^Sp(f)~'^p = I, } and xSp(f)'''^p = Sap, ) and, eliminating x, Spcp-'^p = {Sap)'^, a cone of the second order. 261.] From a point in the outer of two concentric ellipsoids a tan- gent cone is dravm to the inner, find the envelop of the jjlane of contact. If ASOTt|)CT = 1 be the outer, and Sp\\/p = 1 be the inner, and ^/r being any two self-conjugate linear and vector functions, the plane of contact is /Sotv/^P = 1 . Hence, for the envelop, /S'tn-'x/^p = 0, ) S-!!T(P'S7 =: 0, ) 142 QUATERNIONS. [262. therefore (^ot = u'-^p, or CT = X(j)~^\ljf). This gives xS.\\/fj^~^\\fp = !> ) and a;^S.\j/p(l)~'^\}/p = I, ) and therefore, eliminating x, S.y\rp(l)~^\\fp := 1 , or S.p\f/(f>~'^\}/p = 1, another concentric ellipsoid, as \}f(f)'^\l/ is a linear and vector func- tion =:x suppose ; so that the equation may he written Spxp = 1 . 262.] Find the locus of intersection of tamjent ])lanes at the extre- mities of conjugate diameters. If a, /3, y be the vector semi-diameters, the planes are S-sT\\f'~a = — 1 , ^ SxtT\lf^^ = — I, ^ S-sT\p'-y = — 1 , ) with the conditions § 257. Hence —\l/rnS.\j/a\pl3\}/y = \/aot = v//a + \///3 + \/^y, by § 92, therefore T\j/t^ — v 3, since \/^a, \|//jj -^y form a rectangular system of unit- vectors. This may also evidently be written Ssnp'^TjT = — 3, shewing that the locus is similar and similarly situated to the given ellipsoid, but larger in the ratio Vs : 1 . 263.] Fi7id the locus of the intersection of three spheres ivhose dia- meters are semi-conjngate diameters of an ellipsoid. If a be one of the semi-conjugate diameters Saxj/^a := — 1. And the corresponding sphere is p^ — Sap =■ 0, or p^ — S\lfa\l/~^p = 0, with similar equations in /3 and y. Hence, by § 92, \l/-^pS.\}/a^^\l/y = —if'^p = p'ii'a + ^l^ + ^y), and, taking tensors, T-^'^P = VSTp'^, or Txjf-'^p-^ =V3, or, finally, Sp\l/-'^p = — 3 p*. This is Fresnel's Surface of Elasticity in the Undulatory Theory. 264.] Before going farther we may prove some useful properties of the function <^ in the form we are at present using — viz. I Sip jSjp kSkp ^P = -Z^- + ^72" + ^:2~ • 265.] SURFACES OF THE SECOND OlIDER. 143 We have p = —iSlp— jSjp—kSkp, and it is evident that t . 1 Hence ., iSip J^Jp kSkp <})~^p = cr'lSlp + h'^jSjp + c'^kSkp, Also and so on. Again, if a, /3, y be aii^ rectangular unit-vectors {Slaf {Sjaf {Skaf oaoa = — 3 1 T^ f- — s — ' ir b^ c^ But as we have Again, &c. = &c. {8lpf + {Sjpf + {8kpY=-p\ Sa(pa + Sj3(l)f3 + Sy(py = —^ + ,o + ^ ASia S4amy=s.(^ + ■■■)(-? ^■■■K9 + ---) 7i' ■iSly S/a, Sja, Ska I SljB, SJl3, Skfil ^h> %'y^ ^h \ i^U'c' Sia Sja Ska — 1 a^ ^- 6'^ cirh'^c'^ Slf3 SJf3 Sk^ Sly Sjy Sky a^ b'^ 6"'^ And so on. These elementary investigations are given here for the benefit of those who have not read Chapter V. The student may easily obtain all such results in a fiir more simple manner by means of the formulae of that Chapter. 265.] Find the locus of intersection of a rectang^dar splem of three tangents to an ellipsoid. If OT be the vector of the point of intersection, a, /3, y the tangents, then, since -ot + xa should give equal values of x when substituted in the equation of the surface, giving • S^iji + xa) (}) (ct + .I'a) = 1 , or x^Sa(()a + 2xSvT(^a + (/Sot^ot— 1 ) = 0, we have (*S'CT(/)a)^ = Sac^a {S-^(\)Vj — 1). Adding this to the two similar equations in /3 and y {Sa4>^f + {S^^^f + {,Syi\>V7f = {Sa4>a + >S'/3c/)/3 + Syc^y) {S^c}>:^ - 1 ), 144 QUATERNIONS. [266. or -(0-.-)2 = (i + 1 + ^) {S^,— f — oacpa Hence the required plane passes through the extremity of oacpa and those of two other vectors similarly determined. It therefore passes through the point whose vector is , - aSa(f)Z7 + l3S'/3(l)-!Ar + ySy(f)-ST ~ " Sa4>a + xS'/i0/3 + -^'y/^y or = ^+^^ (§173). Thus the first part of the proposition is proved. 268.] SURFACES OF THE SECOND ORDER. 145 But we have also tjt = — (cj) + -^^ 0, whence by the equation of the ellipsoid we obtain the equation of a concentric ellipsoid. 268.] Find the direct ions of the three vectors lohich are jparallel to a set of conjugate diameters in each of tioo central surfaces of the second degree. Transferring the centres of both to the origin, let their equations l^e Sp(\>p = 1 or 0, and Sp\l/p= 1 ov 0. S ^ If Qj l3, y ])e vectors in the required directions, we must have (§254) Sa(p(3 = 0, Sa\lr(3 = 0, \ Sl34>y = 0, S^xlfy=0,i (2) Sycfya = 0, Syxj/a — 0. ) From these equations (pa \\ J'lSy \\ y\/a, &c. Hence the three required directions are the roots of r.## = o (3) This is evident on other grounds, for it means that if one of the surfaces expand or contract unforinlj/ till it meets the other, it will touch it successivel// at points on the three sought vectors. We may put (3) in either of the following forms — F.pcp-'xlfp=0,l or r.px/A-i^p = J ^ ^ and, as ^ and xj/ are given functions, we find the solutions by the processes of Chapter V. [_Note. As (f)~^\}/ and i//~^^ are not, in general, self-conjugate functions, equations (4) do not signify that a, /3, y are vectors parallel to the principal axes of the surfaces S.p(lj~^\l/p =: 1, S.p\l/~^-f)p=0, or F4>p^p=0, as we might have seen without analysis. The locus of the centres is given by the equation {x},-f(P)p-fa = 0, where/' is a scalar variable. 269.] Find the equation of the elUpsoid of loh'ich three conjncjate semi-diameters are given. Let the vector semi-diameters be a, j3, y, and let Spcpp = 1 be the equation of the ellipsoid. Then (§ 255) we have Sa(Pa = 1, Sa(f)l3 = 0, Sl3 + h)-^p; and that to another of the series, if it passes through the common point whose vector is p, is there But ^'.(0 + /.)-V( + /^x)-V = ^■P f^^.L^i. P and this evidently vanishes if h and //^ are different, as they must be unless the svirfaces are identical. 272.] To Jind the conditions of similarity of two central surfaces of the second order. Referring them to their centres, let their equations be Hp<^p =1.) , . Sp^'p= \.\ ^'^ Now the obvious conditions are that the axes of the one are pro- portional to those of the other. Hence, if ^3 _j_ m^g"^ + m^ g + m = 0, ) /•^ + p = (2) This shews that t/ie maxinmni or minimum vector^ the normal at its extremity^ and the perpendicular to the plane of section^ lie in one plane. It also shews that there are but two vector-directions which satisfy the conditions, and that they are perpendicular to each other, for (2) is satisfied if ap be substituted for p. We have now to solve the three equations (1) and (2), to find the vectors of the two (four) points in which the ellipse (1) intersects the cone (2), We obtain at once (^p = xl .<\r^aVap. Operating by S.p we have 1 = xp-Sa(l)~^a. Sp(l)~^a Hence ^^^ = ^"".^0^^ or p = g^(i-p^^)-^«; (3) from which S.a (1 -p^(t>y^a = ; (4) a quadratic equation in p^, from which the lengths of the maximum and minimum vectors are to be determined. By § 147 it may be written ^np'^Sacp-'^a— p^S.aim.^—cj)) a+a'^ = (5) [If we bad operated by S.cp-'^a or by S.(t)-'^p, instead of by S.p, we should have obtained an equation apparently different from this, but easily reducible to it. To prove their identity is a good exercise for the student.] Substituting the values of p^ given by (5) in (3) we obtain the vectors of the required diameters. [The student may easily prove directly that (1— pf^)^^a and (1 — /3|(/>)"^a 276.] SURFACES OF THE SECOND ORDEK. 149 are necessarily periiendieular to each other, if both he perpendicular to a, and if pf and p| be different. See § 271.] 274.] By (5) of last section we see that 2 2 2 _ " Hence the area of the ellipse (1) is irTa Also the locus of normals to all diametral sections of an ellipsoid, whose areas are equal, is the cone Sa4>~^a =■ Co?. When the roots of (5) are equal, i.e. when {in.^c? -8a<\>af = 4 ma^8a<^-^ a, (6) the section is a circle. It is not difficult to prove that this equation is satisfied by only two values of JJa, but another quaternion form of the equation gives the solution of this and similar problems by inspection. (See § 275 below.) 275.] By § 168 we may write the equation Sp(j)p = 1 in the new form S.Kp\xp-\- pp^ = 1, where 7; is a known scalar, and A and /x are definitely known (with the exception of their tensors, whose product alone is given) in terms of the constants involved in (/>. [The reader is referred again also to §§ 121, 122.] This may be written 2SXpSixp + {j)-SKix)p'' = \ (1) From this form it is obvious that the surface is cut by any plane perpendicular to A or /m in a circle. For, if we put 8\p = «, we have 2aSixp + (p — Skfx)p^ = 1, the equation of a sphere which passes through the plane curve of intersection. Hence A and /x of § 168 are the values of a in equation (6) of the preceding section. 276.] All?/ two circular sections of a central surface of the second order, tvliose planes are not parallel^ lie on a sphere. For the equation {S\p—a) {Sp-p — b) = 0, where a and b are any scalar constants whatever, is that of a system of two non-parallel planes, cutting the surface in circles. Eliminating the product S\pSpp between this and equation (1) of last section, there remains the e(piation of a sphere. 150 QUATERNIONS. [277, 277.] Tofuid the generating lines of a central surface of the second order. Let the equation be 8p<^p = 1 ; then, if a be the vector of any point on the surface^ and 37 a vector parallel to a generating line, we must have p ■=. a^ .TOT for all values of the scalar x. Hence 8{a-\- x-si) (p{a + xuy) = 1 , which gives the two equations Sa(f)^ = 0, ) >,9ot0ot =0.5 The first is the equation of a plane through the origin parallel to the tangent plane at the extremity of a, the second is the equation of the asymptotic cone. The generating lines are therefore parallel to the intersections of these two surfaces, as is well known. From these equations we have y0OT = FaOT where ^ is a scalar to be determined. Operating on this by S.fi and S.y, where /3 and y are any two vectors not coplanar with a, we have /SOT(y(^^+ra/3) = 0, S^{f/(Py-rya)=:0 (1) Hence S.(f)a {?/rp[3 + Va/B) {^(f)y— Vya) = 0, or my'^S.a^y — Sa^aS.a^y = 0. Thus we have the two values ^ ^ m m belonging to the two generating lines, 278.] But by equation (1) we have ^OT = r.(y0/3 + Fa/B) {>/y- Vya) = mi/'^ (p-^ FjBy +i/V.(j)a Ffiy — cuS.a Vjiy ; which, according to the sign of y, gives one or other generating line. Here T/Sy may be any vector whatever, provided it is not per- pendicular to a (a condition assumed in last section), and we may write for it 6. Substituting the value of ^ before found, we have ^OT = (^-^ ^ — aSaO ± sj— T'fpae, = T.d>aV.a(h-^0 + >J— FihaO, T r _ -v ^^^ 7 278.] SURFACES OF THE SECOND ORDER. 151 or, as we may evidently write it, = (}>-^{r.aF(Pa6)±^—F(t>ae (2) 1 m Put r = f^cpaO, and we have zur = d)-i Far + ^/ — t, ~ ^ m with the condition 8T<^a = 0. [Any one of these sets of vakies forms the complete sohition of the problem ; but more than one have been given, on account of their singular nature and the many properties of surfaces of the second order which immediately follow from them. It will be excellent practice for the student to shew that xl/9=U(^F.(j)aFa(f)-^e±^— F(pae) is an invariant. This may most easily be done by proving that F.\}/d\}re^ = identically.] Perhaps, however, it is simpler to write a for FjBy, and we thus obtain _ , 1 ZST = — Payacha + sJ — f awa. ^ m [The reader need hardly be reminded that we are dealing with the general equation of the central surfaces of the second order — the centre being origin.] EXAMPLES TO CHAPTER A^Il. 1 . Find the locus of points on the surface Sp(pp =^ 1 where the generating lines are at right angles to one another. 2. Find the equation of the surface described by a straight line which revolves about an axis, which it does not meet, but with which it is rigidly connected. 3. Find the conditions that Sp(})p = 1 may be a surface of revolution, with axis parallel to a given vector. 4. Find the equations of the right cylinders which circumscribe a given ellipsoid. 5. Find the equation of the locus of the extremities of perpen- diculars to central plane sections of an ellipsoid, erected at the 152 QUATERNIONS. centre, their leng-tlis being the principal semi-axes of the sections. [Fresnel's Wave-Surface. See Chap. XI.] G. The cone touching* central plane sections of an ellipsoid, which are of equal area, is asymptotic to a confocal hyperboloid. 7. Find the envelop of all non-central plane sections of an ellip- soid whose area is constant. 8. Find .the locus of the intersection of three planes, perpendicular to each other, and touching", respectively, each of three confocal surfaces of the second order. 9. Find the locus of the foot of the perpendicular from the centre of an ellipsoid upon the plane passing- throug"h the extremities of a set of conjugate diameters. 10. Find the points in an ellipsoid where the inclination of the normal to the radius-vector is greatest, 1 1 . If four similar and similarly situated surfaces of the second order intersect, the planes of intersection of each pair pass through a common point. 12. If a parallelepiped be inscribed in a central surface of the second degree its edges are parallel to a system of conjugate dia- meters. 13. Shew that there is an infinite number of sets of axes for which the Cartesian equation of an ellipsoid becomes x^ +?/'^ +z^ = e^. 14. Find the equation of the surface of the second order which circumscribes a given tetrahedron so that the tangent plane at each angular point is parallel to the opposite face; and shew that its centre is the mean point of the tetrahedron. 15. Two similar and similarly situated surfaces of the second order intersect in a plane curve, whose plane is conjugate to the vector joining their centres. 16. Find the locus of all points on Sp(j)p = 1 , where the normals meet the normal at a given point. Also the locus of points on the surfoce, the normals at which meet a given line in space. 17. Normals drawn at points situated on a generating line are parallel to a fixed plane. 18. Find the envelop of the planes of contact of tangent planes drawn to an ellipsoid from points of a concentric sphere. Find the locus of the point from which the tangent planes are drawn if the envelop of the planes of contact is a sphere. EXAMPLES TO CHAPTER VIII. 153 1 9. The sum of the reciprocals of tlie squares of the perpendiculars from the centre upon three conjugate taug-ent planes is constant. 20. Cones are drawn, touching- an ellipsoid, from any two points of a similar, similarly situated, and concentric ellipsoid. Shew that they intersect in two plane curves. Find the locus of the vertices of the cones when these plane sec- tions are at rig-ht angles to one another. 21. Find the locus of the points of contact of tangent planes which are equidistant from the centre of a surface of the second order. 22. From a fixed point A, on the surface of a given sphere, draw any chord AD ; let I)' he the second point of intersection of the sphere with the secant BB drawn from any point B ; and take a radius vector AE, equal in length to BB' , and in dh-ection either coincident with, or opposite to, the chord AB : the locus of B is an ellipsoid, whose centre is A, and which passes through B. (Hamilton, Blemeiifs, p. 227.) 23. Shew that the equation /2 (g2 _ 1 ) (g _^ Sua) = (SapY - 2eSap8a'p + [Sa'pf + ( 1 - e^) p'', where e is a variable (scalar) parameter, and a, a unit- vectors, repre- sents a system of confocal surfaces. {Ibid. p. 611.) 24. Shew that the locus of the diameters of Sp(f)p = 1 which are parallel to the chords bisected by the tangent planes to the cone Sp\}/p = is the cone S.p(f)\l/~^(})p = 0. 25. Find the equation of a cone, whose verte.K is one summit of a given tetrahedron, and which passes through the circle circum- scribing the opposite side. 26. Shew that the locus of points on the surface Sp4>p = 1, the normals at which meet that drawn at the point p = '^, is on the cone S.{p — 7^) (fi^cjyp = 0. 27. Find the equation of the locus of a point the square of whose distance from a given line is proportional to its distance from a given plane. 28. Shew that the locus of the pole of the plane Sap =1, with respect to the surface Sp(pp =1, 154: QUATERNIONS. is a sj)here, if a be subject to the condition Sa(j)~"a = C. 29. Shew that the equation of the surface g-enerated by lines drawn through the orig-in parallel to the normals to Sp(f)~^p = 1 along its lines of intersection with is ■sj^—kSvj{(p + /c)-'^r^ = 0. 30. Common tangent planes are drawn to ^ 2S\pSixp + {p — SXp.) p2 = 1 , and Tp = h, find the value of h that the lines of contact with the former surface may be plane curves. What are they, in this case, on the sphere ? Discuss the case of p'^ — S^Xp. = 0. 31. If tangent cones be drawn to Spcj,^p= 1, from every point of Sp(f)p = 1, the envelop of their planes of contact is Sp(t)^p = 1 . 32. Tangent cones are draw^n from every point of S (p — a) (p{p — a.) = n^, to the similar and similarly situated surface Sp4>p = 1, shew that their planes of contact envelop the surface {Sa(l>p-\f = n^Sp(l)p. 33. Find the envelop of planes which touch the parabolas p =. ai^ + fit, p = aT- + yr, where a, ^, y form a rectangular system, and t and t are scalars. 34. Find the equation of the surfiice on which lie the lines of contact of tangent cones drawn from a fixed point to a series of similar, similarly situated^ and concentric ellipsoids. 35. Discuss the surfaces whose equations are SapSfSp = Syp, and S^ap + S.a^p = 1, 36. Shew that the locus of the vertices of the right cones which touch an ellipsoid is a hyperbola. 37. If oj, a2, 03 be vector conjugate diameters of Spcpp = 1, where (j)^ — ot^ 0^ -(- w/^c^ — m = , shew that S (a2) = -^ , 2 ( Va. a,f = - '-"^ , S'^.a.a.a^ = - — , and -(f/>'i)" = ^'?2' CHAPTER IX. GEOMETRY OF CURVES AND SURFACES. 279.] We have already seen (§ 3 1 {I)) that the equations p = "s is the reciprocal of the radius of ahsolute cnrrature at the point s. 283.] Thus, if OP = (/)* be the vector of any point P of the curve, and if C be the centre of curvature at P, we have s and thus OC = (bs rr- ^ (P's is the equation of the locus of the centre of curvature. Hence also F. (^/s ([/^s or (p's (j/'s is the vector perpendicular to the osculating plane ; and T^^ii>^sU4.\.) is the tortuositj/ of the given curve, or the rate of rotation of its osculating plane per unit of length. 284.] As an example of the use of these expressions let us fuel the curve whose curvature and tortuositj/ are tjoth constant. We have curvature = Tc\>"s = Tp"= c. Hence (f)'s(}j"s = p p" = ca, where a is a unit vector perpendicular to the osculating plane. This gives p.(j pp +p -=c\— = cc.^L/p =c.^p , if Cj represent the tortuosity. Integrating we get p'p''= c^p+/3, (1) where /3 is a constant vector. Squaring both sides of this equation, we get ^2 = c? _ /3- - 2 c^S^p (for l)y operating with S.p' upon (1) we get +Cj^ = Sf3p'), or Tf3= Vc^Vf.. 285.] GEOMETKY OF CURVES AND SURFACES. 157 Multiply (1) l^y p , remembering" that and we obtain — //' = — q + piB, or^ by integration, p = c-^s—pji-\-a, (2) where a is a constant quaternion. Eliminating" /, we have — p" =—Cy-\- c^^ti — p/3- + a/3, of which the vector part is p" — pl3^ = — c?i.y/3— VajB. The complete integral of this equation is evidently P = icos. sTl3 + ri sin. sTf3--^^{c,s(3+ Fa ^), (3) ^ and ?; being any tw^o constant vectors. We have also by (2), SjSp = CyS + Sa, which requires that SjB^ = 0, Si3)] = 0. The farther test, that Tp'= 1, gives us - 1 = Tf3^{^'-sm\sTi3 + ifcosKsTi3-2Sir^sm.sTf3cos.sTi3)- -^^ • C" -|- Ci This requires^ of course, C -f- 6'J SO that (3) becomes the general equation of a helix traced on a right cylinder. (Compare § 31 (w),) 285.] The vector perpendicular from the origin on the tangent to the curve p — ^kg is, of course, ' , Fp'p, or p' Vpp' P (since p' is a unit vector). To find a common propertij of curves whose tangents are all equi- distant' from the origin. Here Trpp = c, which may be written —p"—S'^pp^=c^ (1) This equation shews that, as is otherwise evident, ever^ curve on a sphere lohose centre is the origin satisfies the condition. For ob- viously — p^ = c^ gives Spp = 0, and these satisfy (1). 11 Spp' does not vanish, the integral of (1) is ^ff^—^2^s, (2) an arbitrary constant not being necessary, as we may measure s from any point of the curve. The equation of an involute which commences at this assumed point is -in- = p — Sp . 158 QUATERNIONS. [286. This gives IW^ = ly + ^2 _^ 2 sSpp' = Tp''^H'-2s^iy-c\ by(l), = c\ by (2). This includes all curves whose involutes lie on a sphere about the origin. 286.] Find the locus of the foot of the perpendicular drawn to a tangent to a right helix from a point in the axis. The equation of the helix is s ^ . s p = a cos — h p sin — f- ys, a a where the vectors a, (3, y are at right angles to each other, and Ta = T!i = h, while aTy = Va^^^. The equation of the required locus is, by last section, IJT = p'Vpp' s a'^-h'^ . s. . . s a^-¥ s. l'^ = a (cos- H ^— 5sin-j + /3(sin — ^cos - ) + 7-;^*. ^ a a^ a^ v a a^ a^ ^ a^ This curve lies on the hyperboloid whose equation is as the reader may easily prove for himself. 287.] To find the least distance between consecutive tangents to a tortuous curve. Let one tangent be ot = p-\-xp\ then a consecutive one, at a distance hs along the curve, is The magnitude of the least distance between these lines is, by §§203,210, 8. (y%S + P" ^ + r^^ + ..)Ur.p\p'^p''ls^p"'^-^^ + ...) Is'' 12 .pp p ~ TVp'p^bs if we neglect terms of higher orders. It may be written, since p'p' is a vector_, and Tp = 1, —s.wrp'p": B„t (5,33,.)) ^ = rW;'a. = ^^,,'V.> // /f/ Hence ^,S.Up'Tp'p' 289.] GEOMETRY OF "CURVES AND SURFACES. 159 is the small ang-le, b(j), between the two successive positions of the osculating- plane. [See also § 283.] Thus the shortest distance between two consecutive tangents is expressed by the formula ^^t) ^s- 12r ' where r, = J^, , is the radius of absolute curvature of the tortuous curve. 288.] Let us recur for a moment to the equation of the parabola (§31(/.)) . /3/^ p = at + —^ ' ,dt Here p = {a + ftt)j^. whence, if we assume Saji = 0, ds at from which the length of the arc of the curve can be derived in terms of t by integration. d-t (dt.^ Again, 9" = (« + /^O ;p + /3 ("^ J • dH _ d 1 _ ^ dt S.l3{a + I3l) ^^* di'~Ts'T{a+l3f)~'^ ds T{a^^lf ' „ {a + ^t)7a^ Hence p =- y(„^^^)4 ' and therefore, for the vector of the centre of curvature we have (§ 283), ^^^f_^i^ _(a2+^2^2)2(_^^2^^^2^)-l^ which is the quaternion equation of the evolute. 289.] One of the simplest forms of the equation of a tortuous curve is ^t^ yt"^ where a, /3, y are any three non-coplanar vectors, and the numericat factors are introduced for convenience. This curve lies on a para- bolic cylinder whose generating lines are parallel to y ; and also on cylinders whose bases are a cubical and a semi-cubical parabola, their generating lines being parallel to /3 and a respectively. We have by the equation of the curve , / yt^\ dt IGO QUATERNIONS. [29O. from which, l)y 7'/=l, the length of the curve can be fbimd in terms of t ; and from which p" can be expressed in terms of s. The investigation of various properties of this curve is very easy, and will be of great use to the student. \_Note. — It is to be observed that in this equation t cannot stand for 5, the length of the curve. It is a good exercise for the student to shew that such an equation as or even the simpler form involves an absurdity.] 290.] The equation p = (/.'e, where is a given self-conjugate linear and vector function, t a scalar variable, and e an arbitrary vector constant, belongs to a curious class of curves. We have at once -j- — y\} log (/)6, where log <\> is another self-conjugate linear and vector function, which we may denote by x- These functions are obviously commu- tative, as they have the same principal set of rectangular vectors, hence we may write dp fPp which of course gives -j-^ = x^P, &c., since x does not involve t. As a verification, we should have = (14 8/X+ Y^\'+ )P = e"xp, where e is the base of Napier's Logarithms. This is obviously true if 't<\>"t. (See § 283.) To find the form of the section of the surface made hy a normal plane through a point in the curve. The equation of the surface is CT = p + <9p' 4- — p" + &c. +^ (p'+ V + &c.). The part of ct— p which is parallel to p' is -/^(..-p)p'=-/(-(.+*0-p''^(y + ^) + --); therefore ot — p = Jp' + ( -^ + xs) ?" —(-^ -^ -^) P ^p'p" + • • • • 297-] GEOMETRY OF CURVES AND SURFACES. 163 And, when A = 0, i.e. in the normal section, we have approximately oe = —s, so that OT — p = — ^P'~Y^' ^PP"' Hence the curve has an equation of the form (T = ^-^a + i-'^/S, a semicuhical parabohi. 295.] A Geodetic line is a curve drawn on a surface so that its osculating plane at any point contains the normal to the surface. Hence, if v be the normal at the extremity of p, p and p" the first and second differentials of the vector of the g-eodetic, S.vp'p" ^ 0, which may be easily transformed into r.vdUp'= 0. 296.] In the sphere Tp = a we have V II P> hence S.pp'p"-= Q, which shews of course that p is confined to a plane passing through the origin, the centre of the sphere. For a formal proof, we may proceed as follows — The above equation is equivalent to the three Sep = 0, SQp'= 0, Sdp''= 0, from which we see at once that is a constant vector, and therefore the first expression, which includes the others, is the complete in- tegral. Or we may proceed thus — =-pS.pp'p''+p''s.p'y'= r.rpp'Vpp"= r.Fpp'dFpp', whence by § 133 (2) we have at once UVpp'= const. = 6 suppose, which gives the same results as before. 297.] In any cone we have, of course, Svp = Oj since p lies in the tangent plane. But we have also Svp = 0. Hence, by the general equation of § 295, eliminating v we get = S.pprp'p"= SpdUp by § 133 (2). Integrating C = Sp Up' -fsdp Up' = Sp Up' + JTdp. The interpretation of this is, that the length of any arc of the geo- detic is equal to the projection of the side of the cone (drawn to its M 3 164 QUATERNIONS. [298. extremity) upon the tangent to the geodetic. In other words^ when the cone is developed the geodetic becomes a straight line. A similar result may easily be obtained for the geodetic lines on any develop- able surface whatever. 298.J To find the shortest line connecting ttvo points on a given surface. Here / Tdp is to be a minimum^ subject to the condition that dp lies in the given surface. Now a \Tdp =fbTdp = -f^^ = -fs. Udpdbp = -IS. Udphp'] + fs.bpdUdp, where the term in brackets vanishes at the limits, as the extreme points are fixed, and therefore bp = 0. Hence our only conditions are S.bpdUdp = 0, and Svbp = 0, giving V.vdUdp = 0, as in § 295. If the extremities of the curve are not given, but are to lie on given curves, we must refer to the integrated portion of the ex- pression for the variation of the length of the arc. And its form S. Udp hp shews that the shortest line cuts each of the given curves at right angles. 299.] The osculating plane of the curve p=^4>t is S.4>'t(i)''t{^-p) = 0, (1) and is, of course, the tangent plane to the surface p = (]it + u(j)'t (2) Let us attempt the converse of the process we have, so far, pursued, and endeavour to find (2) as the envelop of the variable plane (1). Differentiating (1) with respect to t only, we have By this equation, combined with (1), we have or ZT = p + u(f)^= (f) + 7((f/, which is equation (2). 300.] This leads us to the consideration of envelops generally, and the process just employed may easily l)e extended to the problem 302.] GEOMETRY OF CURVES AND SURFACES. 165 of finding the enveloj) of a series of surfaces whose equation contains one scalar parameter. When the given equation is a scalar one, the process of finding the envelop is precisely the same as that employed in ordinary Cartesian geometry, though the work is often shorter and simpler. If the equation he given in the form p =i\lf{t, u, v), where \// is a vector function, t and u the scalar variables for any one surface, v the scalar parameter, we have for a proximate surface Pi = '^ ih ^ ^1 J ^i) = P + V^'i ^^ + Vu ^^^ + i^'v ^^• Hence at all points on the intersection of two successive surfaces of the series we have \j/\ bt + \}/\( bu +\j/\. 8y = 0, which is equivalent to the following scalar equation connecting the quantities t, n, and v ; S.\l/\\jr\,, \j/\. = 0. This equation, along Avith P = ^{t, u, v), enables us to eliminate t, n, v, and the resulting scalar equation is that of the required envelop. 301.] As an example, let us find the envelop of the osculating plane of a tortuous cm-ve. Here the equation of the plane is (§ 299), S.(XiT — p) (f)' t (f)'^ t = 0, or ■^iT = (f)t + x4)'t+ ?/(f>"t = \l/{x,^,t), if p = (pt be the equation of the curve. Our condition is, by last section, or S.ct>'t 0'7 [07 + iP0'7 + 7/ "'t] = 0, or ijS4't(p"t(l)''"t=0. Now the second factor cannot vanish, unless the given curve be plane, so that we must have ^ = 0, and the envelop is ct = 0^5 + xcf/t the developable surface, of which the given curve is the edge of regression, as in § 299. 302.] When the equation contains two scalar parameters its differential coefficients with respect to them must vanish, and we have thus three equations from which to eliminate two numerical quantities. 166 QUATERNIONS. [SOj- A very common form in which these two parameters appear in quaternions is tliat of an unknown unit-vector. In this case the problem may he thus stated — Find the envelop of the surface tchose scalar equation is jpfp ^^n _ q where a is subject to the one condition Ta = 1. Differentiating" with respect to a alone, we have Svda = 0, S'ada = 0, where y is a known vector function of p and a. Since da may have any of an infinite number of values, these equations shew that Fav = 0. This is equivalent to two scalar conditions only, and these, in addi- tion to the two given scalar equations, enable us to eliminate a. With the brief explanation we have given, and the examples which follow, the student will easily see how to deal with any other set of data he may meet with in a question of envelops. 303.] Find the envelop of a plane lohose distance from the orifiu is constant. Here Sap =—c, with the condition Ta = \. Hence, by last section, Fpa = 0, and therefore p = ca, or Tp = c, the sphere of radius c, as was to be expected. If we seek the envelop) of those onltj of the planes ivhich are parallel to a given vector j3, we have the additional relation Safi = 0. In this case the three differentiated equations are Spda = 0, Sada = 0, S^da = 0, and they give S.a^p = 0. Hence a = U.^F^p, and the envelop is TF^p = cTfi, the circular cylinder of radius c and axis coinciding with /3. By putting Sa^ = e, where e is a constant different from zero, we pick out all the planes of the series which have a definite in- clination to /3, and of course get as their envelop a right cone, 304] The equation S^ap+2S.al3p = d represents a parabolic cylinder, whose generating lines are parallel to the vector aFajB. For the equation is of the second degree, and 305.] GEOMETEY OF CURVES AND SURFACES. 167 is not altered by increasing p by the vector xa Faj3 ; also the surface cuts planes perpendicular to a in one line_, and planes perpendicular to Fa,3 in two parallel lines. Its form and position of course depend upon the values of a, (3, and d. It is required to find its envelop M /3 and h be constant^ and a be subject to the one scalar condition Ta= 1. The process of § 302 gives, by inspection, pS'ap-{- Vfip = xa. Operating by S.a, we get S^ap + S.a[3p =—x, which gives S.ajBp = x -[ b. But, by operating successively by S. V^p and by S.p, we have {VjipY' = xS.a^p, and {p^—x)Sap = 0. Omitting, for the present, the factor Sap, these three equations give, by elimination of x and a, {V^pY = p^P^^bi which is the equation of the envelop required. This is evidently a surface of revolution of the fourth order whose axis is /3 j but, to get a clearer idea of its nature, put C^ p~^ = •ar, and the equation becomes ( Vj^sif = c* + brs'^, which is obviously a surface of revolution of the second degree, referred to its centre. Hence the required envelop is the reciprocal of such a surface, in the sense that the rectangle iinder the lengths of condirectional radii of the two is constant. We have a curious particular case if the constants are so related that ^ + /3- = 0, for then the envelop breaks up into the two equal spheres, touching each other at the origin, p^ _ ^ ^y3p^ while the corresponding surface of the second order becomes the two parallel planes ^a-^jy _ _|. ^2^ 305.] The particular solution above met with, viz. 8ap = 0, limits the original problem, which now becomes one of finding the envelop of a line instead of a surface. In fact this equation, taken in conjunction with that of the parabolic cylinder, belongs to that generating line of the cylinder which is the locus of the vertices of the principal parabolic sections. 168 QUATERNIONS. Our equations become 2 /S. a/3/3 = h, Sap = 0, Ta =zl; whence FfSp = xa, g"iving- i» = — ■S.a^p =—2' and thence Tm = \; [506. so that the envelop is a circular cylinder whose axis is ^. [It is to be remarked that the equations above require that xSa/3 = 0, so that the problem now solved is merely that of the envelop}' of a parabolic cylinder ivliicli rotates ahout its focal line. This discussion has been entered into merely for the sake of explaining a peculiarity in a former result, because of course the present results can be obtained immediately by an exceeding-ly simple process.] 306.] The equation SapS.a/3p = a^, with the condition Ta = 1, represents a series of hyperbolic cylinders. It is required to find their envelop. As before, we have pS.al3p+ FjBpSap = xa, which by operating- by S.a, S.p, and S. F(3p, g-ives 2a'^ = —X, p'^S.ajBp = xSap, {r^pfSap=xS.a^p. EHminating a and x we have, as the equation of the envelop^ Comparing this with the equations p^ = -2a\ and {rf3py = ~2a\ which represent a sphere and one of its circumscribing- cylinders, we see that, if condirectional radii of the three surfaces be drawn from the origin, that of the new surface is a geometric mean be- tween those of the two others. 307.] Fi?id the envelop of all sptheres which touch one given line and have their centres in another. Let p = ^-\-yy be the line touched by all the spheres, and let xa be the vector of the centre of any one of them, the equation is (by § 200, or § 201) y\p-xaf =-[r.y{^-xa))\ 308.] GEOMETRY OF CURVES AND SURFACES. 169 or^ putting for simplicity, but without loss of generality, Ty=\, Sai3 = 0, -S/3y = 0, so tliat /3 is the least vector distance between the given lines, and, finally, p'^-l^^- ^ccSap = x- S^ay. Hence, by § 300, —2Sap = 2xS~ay. [This gives no definite envelop if Say =■ 0, i.e. if the line of centres is perpendicular to the line touched by all the spheres.] Eliminating x, we have for the equation of the envelop >SV + /S'2ay(p2-/3-) = 0, which denotes a surface of revolution of the second degree, whose axis is a. Since, from the form of the equation, Tp may have any magnitude not less than TjB, and since the section by the plane Sap = is a real circle, on the sphere p2-/32 = 0, the surface is a hyperboloid of one sheet. [It will be instructive to the student to find the signs of the values of g^, g^^ g^ as in § 165, and thence to prove the above con- clusion.] 308.] As a final example let us find the envelop of the hyperbolic cylinder SapSi3p—c = 0, where the vectors a and /3 are subject to the conditions Ta = Tl3=l, Say = 0, *S'/38 = 0, y and 8 being given vectors. [It will be easily seen that two of the six scalars involved in a, [3 still remain as variable parameters.] We have Sada = 0, Syda = 0, so that da = xVay. Similarly d^=ijr^h. But, by the equation of the cylinders. Sap Spd^ + Spda S/3p = 0, or ^SapS.ftbp +xS.aypSl3p = 0. Now by the nature of the given equation, neither Sap nor S^Sp can vanish, so that the independence of da and ^/3 requires S.ayp = 0, S.^bp = 0. 170 QUATERNIONS. [309- Hence a =U.y Vyp, /3 =U.h Fbp, and the envelop is T.VypFbp — cTyh = 0, a surface of the fourth order^ which may be constructed by laying off mean proportionals between the lengths of condirectional radii of two equal right cylinders whose axes meet in the origin. 309.] We may now easily see the truth of the following general statement. Suppose the given equation of the series of surfaces, whose envelop is required, to contain m vector^ and n scalar, parameters ; and that the latter are subject to 7; vector, and q scalar, conditions. In all there are Zm +11 scalar parameters, subject to Zp-\-q scalar conditions. That there may be an envelop we must therefore in general have {2m + n) — {d2^^q) = ^> or =2. In the former case the enveloping surface is given as the locus of a series of curves, in the latter of a series o^ points. Differentiation of the equations gives us 3p + q+l equations, linear and homogeneous in the 3m + w differentials of the scalar parameters, so that by the elimination of these we have one final scalar equation in the first case, hvo in the second ; and thus in each case we have just equations enough to eliminate all the arbitrary parameters. 310.] To fnd the locus of the foot of the perpeyuVicular drmvn from the origin to a tangent plane to any surface. If Svdp = be the differentiated equation of the surface, the equation of the tangent plane is S{<:T — p)v=0. We may introduce the condition Si'p = 1 , which in general alters the tensor of v, so that v"'^ becomes the required vector perpendicular, as it satisfies the equation Stjjv = 1 . It remains that we eliminate p between the equation of the given surface, and the vector equation The result is the scalar equation (in ot) required. For example, if the given surface be the ellipsoid Sp(})p = 1, we have ot""^ = /' = (pp, 3 1 3-] GEOMETRY OF CURVES AND SURFACES. 171 so that the required equation is « ^sj-i 0-1 uj-i = 1, or SsT(f)~^:zr = zy^, which is Fresnel's Surface of Elasticity . (§263.) It is well to remark that this equation is derived from that of the reciprocal ellipsoid Spcjy^'^p = 1 by putting- ■sx~'^ for p. 311.] To find the reciprocal of a given surface tvith respect to the unit sphere tvhose centre is the origin. With the condition Spv = 1, of last section, we see that — j; is the vector of the pole of the tangent plane S{^ — p)v = 0. Hence we must put zj=i —v, and eliminate p by the help of the equation of the given surface. Take the ellipsoid of last section, and we have ST = —4>P, SO that the reciprocal surface is represented by SsT(f)~'^:^ = 1. It is obvious that the former ellipsoid can be reproduced from this by a second application of the process. And the property is general, for Spv =^ 1 gives, by differentiation, and attention to the condition Svdp = 0, the new relation Spdv = 0, 60 that p and v are corresponding vectors of the two surfaces : either being that of the pole of a tangent plane drawn at the extremity of the other. 312.] If the given surface be a cone with its vertex at the origin, we have a peculiar case. For here every tangent plane passes through the origin, and therefore the required locus is wholly at an infinite distance. The difficulty consists in Spif becoming in this case a numerical multiple of the quantity which is equated to zero in the equation of the cone, so that of course we cannot put as above Spv ^ 1 . 313.] The properties of the normal vector v enable us to write the partial differential equations of fomiHes of surfaces in a very simple form. Thus the distinguishing property of Cylinders is that all their 17'2 QUATERNIONS. [SH- generating lines are parallel. Hence all positions of v must be parallel to a given plane — or Sav = 0, which is the quaternion form of the well-known equation ,dF clF (IF ^ I -J- +m~j- +«^- = 0. ax ay dz To integrate it, remember that we have always Svdp = 0, and that as v is perpendicular to a it may be expressed in terms of any two vectors, /3 and y, each perpendicular to a. Hence v = xj3 + yy, and xS^dp+1/Sydp =: 0. This shews that S^p and Syp are together constant or together variable, so that Sfip =/[Syp), where/' is any scalar function whatever. 314.] In Surfaces of Revolution the normal intersects the axis. Hence, taking the origin in the axis a, we have S.apv =. 0, or V = xa + yp. Hence xSadp + ySpdp = 0, whence the integral Tp =f{Sap). The more common form, which is easily derived from that just written, is TFap = F{Sap). In Cones we have Svp = 0, and therefore Svdp = S.v{TpdUp+ UpdTp) = TpSvdUp. Hence SvdUp = 0, so that V must be a function of Up, and therefore the integral is /(?//>) = 0, which simply expresses the fact that the equation does not involve the t«nsor of p, i. e. that in Cartesian coordinates it is homogeneous. 315.] If equal lengths he laid off on the normals drawn to any surface, the new surf ace formed by their extremities is normal to the same lines. For we have ot = p + a Uv, and Svdjs = Svdp-{-aSvdUv = 0, which proves the proposition. Take, for example, the surface Sp(})p = I ; 3 1 7-] GEOMETRY OF CURVES AND SURFACES. 173 the above equation becomes so that p = (|^ + i) ^, and the equation of the new surface is to be found by eliminating- ;™— (written x) between the equations 2 and -^=84 (xcf) + 1 )-^ct0 (a-^ + 1 )- V. 316.] It appears from last section that if one orthogonal surface can be drawn cutting a given system of straight lines, an inde- finitely great number may be drawn : and that the portions of these lines intercepted between any two selected surfaces of the series are all equal. Let p = a+XT, where cr and r are vector functions of p, and x is any scalar, be the general equation of a system of lines : we have St dp = = S{p — a-)dp as the differentiated equation of the series of orthogonal surfaces, if it exist. Hence the following problem. 317.] It is required to find the criterion of integraUliti/ of the equation Svdp = ;..(!) as the complete differential of the equation of a series of surfaces. Hamilton has given {Elements, p. 702) an extremely elegant solu- tion of this problem, by means of the properties of linear and vector functions. "We adopt a different and somewhat less rapid process, on account of some results it offers which will be useful to us in the next Chapter ; and also because it will shew the student the connection of our methods with those of ordinary differential equa- tions. If we assume Fp ^ C to be the integral, and apply to it the very singular operator de- vised by Hamilton, _ . f/ . d , d ~ dx dy dz , „-n .(IF JF ,dF we have VF =?-^+;-^-(-/?r^-- dx dy dz 174 QUATERNIONS. [3 1 8. But p = ix -\-ji/ + kz, whence dp = idx-{-jdt/-\-kdz, , ,„ dF ^ dF , dF ^ air^T. and = di = — dx + -^ d// + -^dz =— SdpVF. dx dy dz Comparing with the given equation, we see that the latter repre- sents a series of surfaces if v, or a scalar multiple of it, can he ex- pressed as S/F. If V = VF, aPF d^F d^F. we have v. = v^^= -(^, + ^ + ^) . a well-known and most important expression, to which we shall return in next Chapter, Meanwhile we need only remark that the last-written quantities are necessarily scalars, so that the only requisite condition of the integrability of (1) is FVv= (2) If V do not satisfy this criterion, it may when multiplied by a scalar. Hence the farther condition FV {wv) = 0, which may be written FvVw — wFVv = (3) This requires that SvVv = (4) If then (2) be not satisfied, we must try (4). If (4) be satisfied w will be found from (3) ; and in either case (1) is at once integrable. [If we put du = cf)dp where (^ is a linear and vector function, not necessarily self-con- jugate, we have by § 173. Thus, if be self-conjugate, e = 0, and the criterion (2) is satisfied. If cj) be not self-conjugate we have by (4) for the cri- terion Sev = 0. These results accord with Hamilton's, lately referred to, but the mode of obtaining them is quite different from his.] 318.] As a simple example let us first take lines diverging from a point. Here i' 1| /?, and we see that if y = p Vz;=-3, so that (2) is satisfied. And the equation is Spdp = 0, whose integral Tp = C gives a series of concentric spheres. 3 1 9-] GEOMETRY OF CURVES AND SURFACES. 175 Lines perjyenilicular to, and intersecting, a fixed line. If a be the fixed lino, ^ any of the others, we have S.a^p — 0, Sa^ = 0, S^dp = 0. Here v \\ af ap, and therefore equal to it, because (2) is satisfied. Hence S.dpaVap = 0, or S.VapVadp = 0, whose integral is the equation of a series of right cylinders r-Fap = a 319.] To find the orthogonal trajectories of a series of circles tcltose centres are in, and their j)lanes perpendicular to, a given line. Let a be a unit-vector in the direction of the line, then one of the circles has the equations Tp = G,\ Sap = C , S where C and C are any constant scalars whatever. Hence, for the required surfaces v\\d^p\\ Vap, where d-^p is an element of one of the circles, v the normal to the orthogonal surface. Now let dp be an element of a tangent to the orthogonal surface, and we have Svdp ■= S.apdp = 0. This shews that dp is in the same plane as a and p, i.e. that the orthogonal surfaces are planes passing through the common axis. [To integrate the equation S.apdp = evidently requires, by § 3 1 7^ the introduction of a factor. For r V Fap = r {i Vai +j Vaj + ]c VaJc) = 2 a, so that the first criterion is not satisfied. But S.rapVWap = 2S.aVap = 0, so that the second criterion holds. It gives, by (3) of § 317, V.VwVap-\-2wa = 0, or pSaVw — aSpVw -\-2toa= 0. That is SaVw = 0, | SpViv = 2w. J These equations are satisfied by 1 y^ap But a simpler mode of integration is easily seen. Our equation may be written P ■ Up ^ ^ 176 QUATERNIONS. [320. which is immediately integrable, (3 being an arbitrary but constant vector. As we have not introduced into this work the logarithns of ver- sors, nor the corresponding" angles of quaternions, we must refer to Hamilton's Elements for a farther development of this point.] 320.] To find the orthogonal trajectories of a given series of sur- faces. If the equation Fp = C, give Svdp = 0, the equation of the orthogonal curves is Vvdp = 0. This is equivalent to two scalar differential equations (§ 197), which, when the problem is possible, belong to surfaces on each of which the required lines lie. The finding of the requisite criterion we leave to the student. Let the surfaces he concentric spheres. Here p2 ^ q^ and therefore Vpdp = 0. Hence Tp^ dUp = - Up Vpdp = 0, and the integral is Up = constant, denoting straight lines through the origin. Let the surfaces he spheres touching each other at a common point. The equation is (§ 218) Sap~^ = C, whence F.papdp = 0. The integrals may be written S.a^p = 0, p^+hTFap = 0, the first (/3 being any vector) is a plane through the common dia- meter ; the second represents a series of rings or tores (§323) formed by the revolution, about a, of circles touching that line at the point common to the spheres. Let the surfaces he similar, similarly situated, and concentric, sur- faces of the second order. Here Spxp = C, therefore ^XP^^P — ^• But, by § 290, the integral of this equation is p = e'^e where

/ may have any value whatever : so that to specify a particular surface we must have a relation between x and x^ . By the help of this, x^ may be eliminated from (1), which then takes the usual form of the equation of a surface p = (f) {x, y). Or we may operate on (1) by V.{a-\-x^—%-^—x^^^, so that we get a vector equation equivalent to two scalar equations (§§ 98, 116), and not containing y. From this x and a\ may easily be found in terms of p, and the general equation of the possible surfaces ma}^ be written /(a?, .r^) = 0, where f is an arbitrary scalar function, and the values of x and x-^ are expressed in terms of p. This process is obviously applicable if we have, instead of two straight lines, any two given curves through which the line must pass ; and even when the tracing line is itself a given curve, situated in a given manner. But an example or two will make the whole process clear. 326.] Suj)2)ose the moveable line to be restricted by the condition that it is always parallel to a fixed plane. Then, in addition to (1), we have the condition Sy{a^-\-X-^^^ — a — X^) = 0, y being a vector perpendicular to the fixed plane. We lose no generality by assuming a and a^, which are any 327.] GEOMETRY OP CURVES AND SURFACES. 181 vectors drawn from the origin to the fixed lines, to be each per- pendicular to y ; for, if for instance we could not assume Sya = 0, it would follow that Syji = 0, and the required surface would either be impossible, or would be a plane, cases which we need not con- sider. Hence x^Sy^^—xSy^ = 0. Eliminating- a\, by the help of this equation, from (1) of last section, we have , r,\ r-, ^ z' ■ ,d ^y^ \ Operating by any three non-coplanar vectors and with the charac- teristic S, we obtain three equations from which to eliminate x andy. Operating by S.y we find Syp = xSj3y. Eliminating x by means of this, we have finally which appears to be of the third order. It is really, however, only of the second order, since, in consequence of our assumptions, we have y^aa^ \\ y, and therefore Syp is a spurious factor of the left-hand side. 327.] Le( the fixed lines he jperpendicidar to each other, and let the moveable line pass through the circumference of a circle, tvhose centre is i?i the common perpendicular, and tvhose plane bisects that line at right angles. Here the equations of the fixed lines may be written p = a + x(3, p =—a + A\y, where a, (3, y, form a rectangular system, and we may assume the two latter to be unit-vectors. The circle has the equations Equation (1) of § 325 becomes Hence Sar^p = y—{\ —y) = 0, or y — \. Also p2^-a2 = (2y-l)2a2-a;y-«f (l-3^)^ or 4^2 = (a;2_i_a;f), so that if we now suppose the tensors of ^ and y to be each 2a, we may put x = cos 6, x-^ = sin 6, from which p = {2y —l)a+ yfi cos + {1 —y)y sin 9 ; M^ 11 ^"^f" M ^'^P -^a^ 182 QUATERNIONS. [328. For this very simple case the solution is not better than the ordinary Cartesian one; but the student will easily see that we may by very slight changes adapt the above to data far less sym- metrical than those from which we started. Suppose, for instance, /3 and y not to be at right angles to one another ; and suppose the plane of the circle not to be parallel to their plane, &c., &c. But farther, operate on every line in space by the linear and vector function , and we distort the circle into an ellipse, the straight lines remaining straight. If we choose a form of {p + bp)= 1. Hence 2Sbphp = (1) Now (f)p is normal to the tangent plane at the extremity of p, so that if t denote the distance of the point p + bp from that plane t =-SbpU(l)p, and (1) may therefore be written 2tT(i>p-r-bpS.UbpUbp = 0. But the cm-vature of the section is evidently •^ T^P ' or, by the last equation, -^^is.mp4>ubp. In the limit, bp is a vector in the tangent plane ; let ct be the vector semidiameter of the surface which is parallel to it, and the equation of the surface gives T'^ot-^.^/ct^^/ct = 1, so that the curvature of the normal section, at the point p, in the direction of tsr, is 1 329.] GEOMETRY OF CURVES AND SURFACES. 183 direcilij as the perjioulicular from the centre on the tarujent plane, and inversely as the square of the sewddiameter parallel to the tangent line, a well-known theorem. 329.] By the help of the known properties of the central section parallel to the tang-ent plane^ this theorem g-ives us all the ordinary properties of the directions of maximum and minimum curvature, their being at right angles to each other, the curvature in any normal section in terms of the chief curvatures and the inclination to their planes, &c., &c., without farther analysis. And when, in a future section, we shew how to find an osculating surface of the second order at any point of a given surface, the same properties will be at once established for surfaces in general. Meanwhile we may prove another curious proj^erty of the surfaces of the second order, which similar reasoning extends to all surfaces. The equation of the normal at the point p-\-hp in the surface treated in last section is OT = p + 8/9 + a'0(p + 8p) (1) This intersects the normal at p if(§§ 203, 210) S.bp(j)p(j)bp = 0, that is, by the result of § 273, if bp be parallel to the maximum or minimum diameter of the central section parallel to the tangent plane. Let (Ti and o-g be those diameters, then we may write in general hp =pa-^ + qa2, where 79 and q are scalars, infinitely small. If we draw through a point P in the normal at p a line parallel to a-j, we may write its equation w =: p + a(f)p 4- yo"! . The proximate normal (1) passes this line at a distance (see § 203) S . {a(l)p — bp) UVa^ cf) {p ^- bp), or, neglecting terms of the second order, mjr . {apS.(f)p(T^(f)a^ + aqS^pa^cjia^ + q S.a^a^^p). 1 y Cifpp The first term in the bracket vanishes because o-j is a principal vector of the section parallel to the tangent plane, and thus the expression becomes ^ a „, \ Hence, if we take a = Ta'j, the distance of the normal from the new line is of the second order only. This makes the distance of P from the point of contact TffypTal, i.e. the principal radius of curvature 184 QUATERNIONS. [So^- along the tangent line parallel to o-g- That is, tke groujp of normals draion near a point of a central surface of the second order pass ulti- mately through tivo lines each parallel to the tangent to one principal section, and passing through the centre of curvature of the other. The student may form a notion of the nature of this proposition by con- sidering a small square plate, with normals drawn at every point, to be slightly bent, but by different amounts, in planes perpendicular to its edges. The first bending will make all the normals pass through the axis of the cylinder of which the plate now forms part ; the second bending will not sensibly disturb this arrangement, except by lengthening or shortening the line in which the normals meet, but it will make them meet also in the axis of the new cylinder, at right angles to the first. A small pencil of light, with its focal lines, presents this appearance, due to the fact that a series of rays originally normal to a surface remain normals to a surface after any number of reflections and refractions. (See § 315). 330.] To extend these theorems to surfaces in general, it is only necessary, as Hamilton has shewn, to prove that if we write dv = 0f/p, ^ is a self -conjugate function ; and then the properties of ^, as ex- plained in preceding Chapters, are applicable to the question. As the reader will easily see^ this is merely another form of the investigation contained in §317. But it is given here to shew what a number of very simple demonstrations may be given of almost all quaternion theorems. The vector v is defined by an equation of the form dfp = Svdp, where y is a scalar function. Operating on this by another inde- pendent symbol of differentiation, 8, we have hdfp = Shvdp-\-Svhdp. In the same way we have dbfp = Sdvhp -\- Svdbp. But, as d and b are independent, the left-hand members of these equations, as well as the second terms on the right (if these exist at all), are equal, so that we have Sdvbp = Shvdp. This becomes, putting dv = dp, and therefore hv = (f)bp, Sbp(})dp = Sdp(f)bp, which proves the proposition. 2,33-] GEOMETRY OF CURVES AND SURFACES. 185 331.] If we write the differential of the equation of a surface in the form d/p = 2Svdp, then it is easy to see that /{p + dp) =fp+ 2Svdp + Sdvdp + &c., the remaining terms containing as factors the third and higher powers of Tdp. To the second order, then, we may write, except for certain sing^ular points, = 2Svdp-\- Sdvdp, and, as before, (§ 328), the curvature of the normal section whose tangent line is dp is 1 „ dv Yv Tp' 332.] The step taken in last section, although a very simple one, virtually implies that the first three terms of the expansion of f{p-\-dp) are to be formed in accordance with Taylor's Theorem, whose applicability to the expansion of scalar functions of quater- nions has not been proved in this work,, (see § 135); we therefore give another investigation of the curvature of a normal section, employing for that purpose the formulae of § (282). AVe have, treating dp as an element of a curve, Svdp = 0, or, making s the independent variable, Svp'= 0. From this, by a second diiferentiation, o -=- p +ovp =0. ds The curvature is, therefore, since v \\ p" and ^//= \, Tp"=-^s'^p'^ = ^S~,2iB before. Iv dp Iv dp 333.] Since we have shewn that dv ■= (jidp where is a self-conjugate linear and vector function, whose con- stants depend only upon the nature of the surface, and the position of the point of contact of the tangent plane ; so long as we do not alter these we must consider as possessing the properties explained in Chapter V. Hence, as the expression for Tp" does not involve the tensor of dp, we may put for dp any unit-vector r, subject of course to the condition Svt = (1) And the curvature of the normal section whose tangent is r is 186 QUATERNIONS. [334- If we consider the central section of the surface of the second order S-sT(p:ff + Tv = 0, made by the plane Sv!:t = 0, we see at once that the curvature of the given surface along the normal section touched hy r is inverseli/ as the square of the parallel radius in the auxiliary surface. This, of course, includes Euler's and other well-known Theorems, 334.] To find the directions of maximum and minhnuni curvature, we have xS'r{vT) = 0, or S.v/(j)T := 0. Hence the sections of greatest and least curvature are perpendicular to one another. We easily obtain, as in § 273, the following equation S.v{(1) + St(I)t)-^v = 0, whose roots divided by Tv are the required curvatures. 335.] Before leaving" this very brief introduction to a subject, an exhaustive treatment of which will be found in Hamilton's Elements, we may make a remark on equation (1) of last section S.VT(f)T = 0, or, as it may be written, by returning' to the notation of § 333, S.vdpdv =0. This is the general equation of lines of curvature. For, if we define a line of curvature on any surface as a line such that normals drawn at contiguous points in it intersect, then, hp being an element of such a line, the normals zx = p-\-xv and ct = p + hp-^y {v-\-hv) must intersect. This gives, by § 203, the condition as above. ^-^P'^" = ^' EXAMPLES TO CHAPTER IX. 187 EXAMPLES TO CHAPTER IX. 1 . Find the leng-th of any arc of a curve drawn on a sphere so as to make a constant angle with a fixed diameter. 2. Shew that, if the normal plane of a curve always contains a fixed line, the curve is a circle. 3. Find the radius of spherical curvature of the curve p = (fyt. Also find the equation of the locus of the centre of spherical curvature. 4. (Hamilton, Bishop Lato's Premium Examination, 1854.) (a.) If p be the variable vector of a curve in space, and if the differential ch be treated as = 0^ then the equation dT(p-K) = obliges K to be the vector of some point in the normal plane to the curve. (b.) In like manner the system of two equations, where (Ik and (Pk are each =0, dT{p-K) = 0, cPT{p-k) = 0, represents the axis of the element, or the right line drawn through the centre of the osculating chcle, per- pendicular to the osculating plane. {c.) The system of the three equations, in which k is treated as constant,, clT{p-K) = 0, (PT{p-k) = 0, cPT{p-k) = 0, determines the vector k of the centre of the osculating sphere. [d.) For the three last equations we may substitute the follow- ing : S.{p — K)dp = 0, S.{p-K)d^p + dp^ = 0, S.ip-K)d^p + 3S.dpd^p = 0. ((?.) Hence, generally, whatever the independent and scalar variable may be^ on which the variable vector p of the curve depends, the vector k of the centre of the oscu- lating sphere admits of being thus expressed : _ 3 F. dpd^pS.dpd^p - dp"- V. dpd'^p "-P^ S.dpd^pd^y 188 QUATERNIONS. (/.) In general, cl{d-^r.dpUp) = d{Tp-^V.pdp) = Tp-'> ( 3 F. p dp S.p dp - p2 r.p d'-p); whence, 3r.pdpS.pdp-p"F.pd'p = p^Tpd{p-^r.dpUp); and^ therefore, the recent expression for k admits of being thus transformed, _ dphljdp-^r.d^pUdp ) "-P'^ SJ-^pd^pUdp { + k)' V = 1 , where /i is an arbitrary parameter. 17. Shew how to find the equation of a tangent plane which touches a surface along a line^ straight or curved. Find such planes for the following surfaces Sp4)p = 1, Sp{cf>-p^)-^p=l, and {p'"-a^- + lj'-y- + 4.{a^"p'- + //'S'-ap) = 0. 18. Find the condition that the equation S{p + a)(f)p = 1 , where is a self-conjugate linear and vector function, may represent a cone. 19. Shew from the general equation that cones and cylinders are the only developable surfaces of the second order. 20. Find the equation of the envelop of planes drawn at each point of an ellipsoid perpendicular to the radius vector from the centre. 2 1 . Find the equation of the envelop of spheres whose centres lie on a given sphere, and which pass through a given point. 22. Find the locus of the foot of the perpendicular from the centre to the tangent plane of a hyperboloid of one, or of twO;, sheets. 190 QUATERNIONS. 23. Hamilton, Bishop Law's Premium Examination, 1852, (rt.) If p be the vector of a curve in space, tlie length of the element of that curve is Tdp ; and the variation of the length of a finite arc of the curve is hfTdp ^ -fSUdphdp = - ASUdpbp+fSdUdpbp. {b.) Hence, if the curve he a shortest line on a given surface, for which the normal vector is v, so that Svhp = 0, this shortest or geodetic curve must satisfy the differential equation, VvdUdp = 0. Also, for the extremities of the arc, we have the limiting equations, SUdpQ 8po = ; SUdp^ bp-^ = 0. Interpret these results. (f.) For a spheric surface, Fvp=0, pdUdp = 0; the integrated equation of the geodetics is p Udp = w, giving Sz^p = (great circle). For an arbitrary cylindric surface, Sav — 0, adUdp = ; . the integral shews that the geodetic is generally a helix, making a constant angle with the generating lines of the cylinder. {d.) For an arbitrary conic surface, Svp=z 0, SpdUdp = ; integrate this differential equation, so as to deduce from it, TVpUdp = const. Interpret this result ; shew that the perpendicular from the vertex of the cone on the tangent to a given geo- detic line is constant ; this gives the rectilinear develop- ment. When the cone is of the second degree, the same property is a particular case of a theorem respecting confocal surfaces. {e.) For a surface of revolution, S.apv = 0, S.apdUdp = ; integration gives, const. = S.apUdp = TFapSU [Vap.dp) ; the perpendicular distance of a point on a geodetic line from the axis of revolution varies inversely as the cosine of the angle under which the geodetic crosses a parallel (or circle) on the surface. EXAMPLES TO CPI AFTER IX. 191 {/.) The differential equation, S.apdUdp = 0, is satisfied not only l)y the geodetics, but also by the circles, on a surface of revolution ; give the explanation of this fact of calculation_, and shew that it arises from the coinci- dence between the normal plane to the circle and the plane of the meridian of the surface. {(/.) For any arbitrary surface, the equation of the geodetic may be thus transformed^ S.vdpcPp = ; deduce this form, and shew that it expresses the normal property of the osculating plane. {//.) If the element of the geodetic be constant, dTdp = 0, then the general equation formerly assigned may be reduced to F.vd'^p = 0. Under the same condition, d^p = —v~^Sdvdp. (/.) If the equation of a central surface of the second order be put under the form fp =. \, where the function f is scalar, and homogeneous of the second dimension, then the differential of that function is of the form dfp = 2S.vdp, where the normal vector, v = (})p, is a dis- tributive function of p (homogeneous of the first dimen- sion), dv = d(f)p =:cf)dp. This normal vector v may be called the vector of proximity (namely, of the element of the surface to the centre) ; because its reciprocal, v~^, represents in length and in direction the perpendicular let fall from the centre on the tangent plane to the surface. (/I-.) If we make S(j<^p =f{a^p), this function/" is commutative with respect to the tivo vectors on which it depends, y (p, ct) =f{(T^ p) ; it is also connected with the former function/, of a single vector p, by the relation,_/(/j, p) =-fp : so thaty]b = 6/)c/)p. fdp = Sdpdv ; dfdp = 2S.dvdrp ; for a geodetic, with con- stant element, dfdp „ 6?y 2fdp V this equation is immediately integrable, and gives const. =Tv's/{fUdp) = reciprocal of Joachimstal's pro- duct, PI). (/.) If we give the name of " Didonia'' to the curve (discussed by Delaunay) which, on a given surface and with a given perimeter, contains the greatest area, then for 192 QUATERNIONS. such a Didouian curve we have by quaternions the formula, fS. Uvclphp + c h/Tdp = ; where c is an arbitrary constant. Derive hence the differential equation of the second order, equivalent (through the constant c) to one of the third order, c'^dp = V.UvdUdp. Geodetics are, therefore, that limiting case of Didonias for which the constant c is infinite. On a plane, the Didonia is a circle, of which the equation, obtained by integration from the general form, is p = ■sT + cUvdp, CT being vector of centre, and c being radius of circle. (m.) Operating by S. Udp, the general differential equation of the Didonia takes easily the following forms : c-^Tdp = S{UvdpJUdp); c-^Tdp^ = S{Uvdp.d^p); • c-''Tdp^ = S.Uvdpd^p; Uvdp (n.) The vector w, of the centre of the osculating circle to a curve in space, of which the element Tdp is constant, has for expression, dp^ " = ^ + ^- Hence for the general Didonia, J— p)~ Uvdp . ' vdp [o.) Hence, the radius of curvature of any one Didonia varies, in general, proportionally to the cosine of the inclination of the osculating plane of the curve to the tangent plane of the surface. And hence, by Meusnier^s theorem, the difference of the squares of the curvatures of curve and surface is con- stant; the curvature of the surface meaning here the reciprocal of the radius of the sphere which osculates in the reduction of the element of the Didonia. (jo.) In general, for any curve on any surface, if f denote the vector of the intersection of the axis of the element (or ^(co-p)-^ EXAMPLES TO CHAPTER IX. 193 the axis of the circle osculating- to the curve) with the tangent plane to the surface^ then ^~^^ S.vdpcPp' Hence, for the general Didonia, ^ith the same significa- tion of the symbols, i= p — cUvdp; and the constant c expresses the length of the interval p — $, intercepted on the tangent plane, between the point of the curve and the axis of the osculating circle. (q.) If, then, a sphere be described, which shall have its centre on the tangent plane, and shall contain the osculating circle, the radius of this sphere shall always be equal to c. (r.) The recent expression for ^, combined with the first form of the general differential equation of the Didonia, gives di = -cVdUvUdp ; Vvdi = 0. (s.) Hence, or from the geometrical signification of the con- stant c, the known property may be proved, that if a developable surface be circumscribed about the arbitrary surface, so as to touch it along a Didonia, and if this developable be then unfolded into a plane, the curve will at the same time be flattened (generally) into a circular arc, with radius = c. 24. Find the condition that the equation Sp{cp+f)-'p=l may give three real values of/ for any given value of p. If/ be a function of a scalar parameter ^, shew how to find the form of this function in order that we may have ^,,_d^i d^i d^_^ Prove that the following is the relation between/ and $, . f 4f _ f ¥ ^ V(^l +/) (i7. +/) (>3 +/) ^ ^'^f in the notation of § 147. 25. Shew, after Hamilton, that the proof of Dupin's theorem, that "each member of one of three series of orthogonal surfaces cuts each member of each of the other series along its lines of curvature," may be expressed in quaternion notation as follows : o 194 QUATERNIONS. If Svdp = 0, Svdp = 0, S.vvdp = be integrable, and if Svv = 0, then Fvdp = 0, makes S.vvdv = 0. Or, as follows, If SvVv = 0, S/Vv = 0, . Sv''Vv''= 0, and r.vvV'= 0, then S./'{SvV.v) = 0, . . d . d J d where V = t-^ — \- ? -i — I- «-• -4- • dx dy dz 26. Shew that the equation Vap = pV^p represents the line of intersection of a cylinder and cone, of the second order, which have /3 as a common generating line. 27. Two spheres are described, with centres at A, B, where 62= a, OB = j3, and radii a, h. Any line, OPQ, drawn from the origin, cuts them in P, Q respectively. Shew that the equation of the locus of intersection of JP, PQ has the form V{a + aU{p-a)) {l3 + bU(p-^)) = 0. Shew that this involves S.a/Bp = 0, and therefore that the left side is a scalar multiple of F.aj3, so that the locus is a plane curve. Also shew that in the particular case Fal3 = 0, the locus is the surface formed by the revolution of a Cartesian oval about its axis. CHAPTER X. KINEMATICS. 336.] When a point's vector, p, is a function of the time t, we have seen (§36) that its vector- velocity is expressed by -j- or, in Newton's notation, by p. That is, if p = (pi be the equation of an orbit, containhig (as the reader may see) not merely the form of the oriit, but the law of its description also, then p = <^'t g-ives at once the form of the Koclograph and the law of its de- scription. This shews immediately that the vector-acceleration of a iwint^s motion. drp ^ "^' P' is the vector-velocity in the hotlogniph. Thus the fundamental pro- perties of the hodograph are proved almost intuitively. 337.] Changing" the independent variable, we have dp ds , P = Ts~di = 'P' if we employ the dash, as before, to denote ~j- • This merely shews, in another form, that p expresses the velocity in magnitude and direction. But a second differentiation gives p = Vp + V^p . This shews that the vector-acceleration can be resolved into two components, the first, vp', being in the direction of motion and equal in magnitude to the acceleration of the velocity, v ov -j- ; (It the second, v'^p\ being in the direction of the radius of absolute o 1 196 QUATERNIONS. [338. curvature, and having for its amount the square of the velocity multiplied by the curvature. [It is scarcely conceivable that this important fundamental pro- positioiij of which no simple analytical proof seems to have been obtained by Cartesian methods, can be proved more elegantly than by the process just given.] 338.] If the motion be in a plane curve, we may write the equation as follows^ so as to introduce the usual polar coordinates, r and Q, ^J p = ra^'fi, where a is a unit-vector perpendicular to, /3 a unit-vector in, the plane of the curve. Here, of course, r and 9 may be considered as connected by one scalar equation ; or better, each may be looked on as a function of t. By differentiation we get 26 29 p = ra'^ ji + rdaa'^ ^, which shews at once that /' is the velocity along, rd that perpen- dicular to, the radius vector. Again, 29 29 which gives, by inspection, the components of acceleration along, and perpendicular to, the radius vector. 339.] For uniform acceleration in a constant direction, we have at once, p = a. "Whence p = at + (3, where 13 is the vector-velocity at epoch. This shews that the hodograph is a straight line described uniformly. o-t^ Also p = — +(3t, no constant being added if the origin be assumed to be the position of the moving point at epoch. Since the resolved parts of p, parallel to /3 and a, vary respect- ively as the first and second powers of t, the curve is evidently a parabola (§31 (/)). But we may easily deduce from the equation the following result, T{p + \^a-^^) = -SUa (p + ^ a-i ) , the equation of a paraboloid of revolution, whose axis is a. Also S.ajip = 0, 34^-] KINEMATICS. 197 and therefore the distance of any point in the path from the point — |^a~^/3 is equal to its distance from the line whose equation is /o = —a ^-\-xafa^. Thus we recognise the focus and directrix property. 340.] That the moving- point may reach a point y we must have, for some real value of t, Now suppose 7^/3, the velocity of projection, to be given =.v, and, for shortness, write w for Uji. Then y 33 - ^2 ^ vtT^. Since T-sr = 1 , we have ?^ - (v^ - Say) f" + ly = 0. Tlie values of i''^ are real if {v'-Sayy-Ta'Ty'- is positive. Now, as TaTy is never less than Say, it is evident that v^ — Say must always he positive if the roots are possible. Hence, when they are possible, both values of f^ are positive. Thus we hhYB Jbur values of t which satisfy the conditions, and it is easy to see that since, disregarding the signs, they are equal two and two, each pair refer to the same path, but described in opposite directions between the origin and the extremity of y. There are therefore, if any, in general two parabolas which satisfy the conditions. The directions of projection are (of course) given by the corresponding values of ct. 341.] The envelop of all the trajectories possible with a given velocity, evidently corresponds to {v^-Sayf-Ta^Ty^ = 0, for then y is the vector of intersection of two indefinitely close paths in the same vertical plane. Now v^ - Say = TaTy is evidently the equation of a paraboloid of revolution of which the origin is the focus, the axis parallel to a, and the directrix plane at a distance 7=- • Ta All the ordinary problems connected with parabolic motion are easily solved by means of the above formulae. Some, however, are even more easily treated by assuming a horizontal unit-vector in 198 QUATERNIONS. [342. the plane of motion^ and expressing /3 in terms of it and a. But this must be left to the student. 34.2.] For acceleration directed to or from a fixed point, we have, taking that point as origin, and putting P for the magnitude of the central acceleration, P =FVp. Whence, at once, Fpp = 0. Integrating, Vp'p =: y = a constant vector. The interpretation of this simple formula is — first, p and. p are in a plane perpendicular to y, hence the path is in a plane (of course passing through the origin) ; second, the area of the triangle^ two of whose sides are p and p is constant. [It is scarcely possible to imagine that a more simple proof than this can be given of the fundamental facts, that a central orbit is a plane curve, and that equal areas are described by the radius vector in equal times.] 343.] When the lazo of acceleration to or from the origin is that of the inverse square of the distance, we have p_ M Tp^' where /x is negative if the acceleration be directed to the origin. Hence p = ^ . The following beautiful method of integration is due to Hamilton. (See Chapter IV.) Generally, dJJp ^ _ U p Tpp _ Up.y dt Tp"" ~ Tp^ therefore py = —u —rr ' dt and py =: e — p-Up, where e is a constant vector, perpendicular to y, because Syp = 0. Hence, in this case, we have for the hodograph, p = ey"-*^ — p.Up.y^. Of the two parts of this expression, which are both vectors, the first is constant, and the second is constant in length. Hence the locus of the extremity of p is a circle in a plane perpendicular to y (i.e. parallel to the plane of the orbit), whose radius is ^ > and whose centre is at the extremity of the vector ey~^. [This equation contains the whole theory of the Circular Hodo- 345-] KINEMATICS, 199 graph. Its consequences are developed at length in Hamilton's Elements7\ 344.] We may write the equations of this circle in the form (a sphere), and Sy'p = (a plane through the origin, and through the centre of the sphere). The equation of the orbit is found by operating by F.p upon that of the hodograph. We thus obtain or y^ =S€p + fji.Tp, or fxTp = Se{y^r'^-p); in which last form we at once recognise the focus and directrix property. This is in fact the equation of a conicoid of revolutioii about its principal axis (e), and the origin is one of the foci. The orbit is found by combining it with the equation of its plane, Syp = 0. We see at once that y- e~^ is the vector distance of the directrix . Te from the focus ; and similarly that the eccentricity is — ) and the . -2p.y' ^ maior axis -^ ^ • p? + e^ 345.] To take a simpler case : let the acceleration vary as the dis- tance from the origin. Then p = ±m^p, the upper or lower sign being used according as the acceleration is from or to the centre. This is (-j-^ + m^^ p = 0. Hence p = ag™' + /3£-"'*; or p = a cos mt + ^ sin mt, where a and j3 are arbitrary, but constant, vectors; and g is the base of Napier's logarithms. The first is the equation of a hyperbola (§ 31, >?;) of which a and /3 are the directions of the asymptotes ; the second, that of an ellipse of which a and fi are semi-conjugate diameters. Since p = m {a£'"*-/3£-"''}, or = m {—a sin mt + (3 cos mt}, the hodograph is again a hyperbola or ellipse. But in the first case it is, if we neglect the change of dimensions indicated by the 200 QUATERNIONS. [346. scalar factor m, conjugate to the orbit ; in the case of the ellipse it is similar and similarly situated. 346.] Ag-ain, let the acceleration he as the inverse third power of the distance^ we have uJJp P = Tp^ Of course, we have, as usual, Vpp = y- Also, operating by S.'p, Spp = IxSpp Tp'' ' of which the integral is •9 p^ p2 = the equation of energy. Again, Spp = p'' Hence Spp -\- p'^ = C, or Spp = Ct, no constant being added if we reckon the time from the passage through the apse, where Spp = 0. We have, therefore, by a second integration, p2 = a2 + c" (1) [To determine C", remark that PP= Ot + y, or /3V = <^^^'-/- But jO-/3^ = C(P-—p. (by the equation of energy), = C'^e-J.CC'-v., by (1). Hence CC = /^t-y^.] To complete the solution, we have, by § 133, p dUp d Up p dt ^ ^' dt ^ ^ where /3 is a unit-vector in the plane of the orbit. But V^- = -\- P P Hence log -~ = 13 ~ 'J Cf' + C The elimination of t between this equation and (l) gives Tp in terms of Up, or the required equation of the path. We may remark that if 9 be the ordinary polar angle in the orbit, Uo }og^ = eUy. 348.] KINEMATICS. 201 Hence we have 6 = —Ty I y^ — -^ j / J L/C -f" ^ ( and r^=-{Ct^ + C), ) from which the ordinary equations of Cotes' spirals can be at once found. [See Tait and Steele's Dynamics of a Particle, third edition, Appendix (A).] 347.] To find the conditions that a given curve may he the hodo- graph corresponding to a central orhit. If OT be its vector, given as a function of the time, fixdt is that of the orbit ; hence the requisite conditions are given by Y-^jsfvjdt ■=. y, where y is a constant vector. We may transform this into other shapes more resembling* the Cartesian ones. Thus Virfrndt = 0, and VTzfxsdt-^ Fctct = 0. From the first f-sydt = ccir, and therefore xV-stvj ■= y, or the curve ya jilane. And X Vmis + 7 tJTCT := ; or eliminating x, yVi^is = —{V-sr-ss-Y. Now if v' be the velocity in the hodog-raph, R its radius of curva- ture, ^/ the perpendicular on the tangent ; this equation gives at once hv'= U'^r-, which agrees with known results. 348.] The equation of an epitrochoid or hypotrochoid, referred to the centre of the fixed circle, is evidently p =1 ai '" a-\-hi '^ a, where a is a unit-vector in the plane of the curve and i another perpendicular to it. Here w and co^ are the angular velocities in the two circles, and t is the time elapsed since the tracing point and the centres of the two circles were in one straight line. Hence, for the length of an arc of such a curve, s =.fTpdt =fdt\/ {(j>''^a^ + 2 (OM^ab cos {(o — o)^) t + o}j^b-}, Ttj / (/ — 7XO 7 ICOS^ I 0)— CO, ,) = dtV< ((»a + (a-.iY + 4:0)0), ab\ . „ — tt-^^C J ( ' — '■ ' sm-^ I 2 3 which is, of course, an elliptic function. 202 QUATERNIONS. [349. But when the curve becomes an epicycloid or a hypocycloid, cofi^Wji^ = 0, and which can be expressed in finite terms, as was first shewn by Newton in the Principia. The hodograph is another curve of the same class, whose equa- tion is 2(o< 2coi« p = i(a(oi '" a + ^coji " a); and the acceleration is denoted in magnitude and direction by the vector 2(o< 2(01 < p =—a Suda = — Sad-ST. ) d(T = Vaa satisfies them all, by (2), and we have thus the proposi- tion that ani/ small displacement of a rigid hody about a fixed point is equivalent to a rotation. 351.] To represent the rotation of a rigid hodi/ about a given axis, through a given finite angle. Let a be a unit-vector in the direction of the axis, p the vector of any point in the body with reference to a fixed point in the axis, and Q the angle of rotation. Then p = a~^Sap -\- a^'^Vap, = — aSap — a Vap. The rotation leaves, of course, the first part unaffected, but the second evidently becomes — a"^ aVap, or —a Vap cos 6 -f- Vap sin 6. Hence p becomes Pj = — aSap — a Vap cos 6 -\- Tap sin 6, .6 . e^ . e .6. = (^cos- + asm-jp(^cos- — (?sin-j, • ^ 6 If ~ jf = a pa 204 QUATERNIONS. [352. 352.] Hence to compound two rotations about axes which meet, we may evidently write, as the effect of an additional rotation ^ about the unit- vector ^, i. _*_ Hence p.2 ^= ti"" a" pa " ^ " . If the /3-rotation had been first, and then the a-rotation, we should have had 1 ^ _i _i and the non-commutative property of quaternion multiplication shews that we have not, in general, If a, j3, y be radii of the unit sphere to the corners of a spherical triang-le whose angles are - > -^ > -^ ' ^^'^ know that \p ^ y-^ ^^a'" = — 1. (Hamilton, Lectures, p. 267.) Hence ^''a'^=-y"", ^± t and we may write P^ — J ^ 91^ ■> or, successive rotations about radii to tivo corners of a spherical triangle, and through angles double of those of the triangle, are equivalent to a single rotation about the radius to the third corner, and through an angle double of the exterior angle of the triangle. Thus any number of successive ^;«V(? rotations maybe compounded into a single rotation about a definite axis. 353.] When the rotations are indefinitely small, the effect of one is, by § 351, p^ = p^oJap, and for the two, neglecting products of small quantities, P2 = p + arap+ br^p, a and b representing the angles of rotation about the unit-vectors a and /3 respectively. But this is equivalent to P2 = p + ^ {<\oi + 6/3) TU{ e . f . . ,\ — sm— sm-sm -cos^ — sm — cos-sm ' smy cosii^) 2 2 2 2 2 2 ^ ,/<^ e . f . 4> 9 yj/ a + k ( cos - cos - sm — + sm ^ cos - cos - cos t^ + sm ^ sm - sm ^ sm dsin\l/+ sm ^ sm - cos - sm^cosi/^ 1 2 2 2 2 ^ J 2 2 2 2 2 2 2 2 which is, of course, essentially unsymmetrical. 208 QUATERNIONS. [oS^- 358.] To find the usual equations comiectivg xjf, 6, <^ with the an- gular velocities about three rectangular axes fixed in the hodi/. Having the value of q in last section in terms of the three angles, it may be useful to employ it^ in conjunction with equation (3) of § 356, partly as a verification of that equation. Of course, this is an exceedingly roundabout process, and does not in the least re- semble the simple one which is immediately suggested by qua- ternions, "We have 2q = ^q — {oi^OA + m.^OB + (i^^OC} q, whence '^ff^^i = q~^ {^i OA + o^.^ OB + o)^ OC} q, or 2q = q{i(iiiy+joi.2-{-ku>.^). This breaks up into the four (equivalent to three independent) equations d , — ^ ■ G ■ ^2 2^2 2^2 2 d / . (b — \/r . ^\ 2 — I sm ^--^- sm - ) dt V 2 2^ (b + yl/ 6 . (b + xl/ e (b — \l/ . 6 = coi cos ^ — COS CO9 sm — - — cos - + coo cos — sin - ? 1 2 2^2 2^2 2 d / (b — \U . 6\ 2 — ( cos ~—^ sm - ) dt V 2 2>' . (b + xlf e (b + xp- 6 . (b — ylr . 6 = (0, sin — cos - + CO, cos cos - — coo sin sm - > 1 2 2 2 2^ 2 2 d , . Cb + \lf (9^ 2 ^ ( sin ^ cos - ) dt V 2 2>' (b — \l/ . 6 . (b — ylr . 6 (b + \lr 6 = — Wj cos ~ — sm - + 0)., sm sm - + CO3 cos — — ^ cos - • iii ^ di dt d dt From the second and third eliminate ^ — \}/, and we get by in- spection ^ n / ■ J, . J,N ^ ^ cos - . y = (wj sm 9 + Wg cos 0) cos - > 2 <^ or ^ = Wj sin (^ + w^ cos 0. (1) Similarly, by eliminating d between the same two equations, . e- ■, .6 ^ e . ^ 6 sm -( X = sin 6 2' COS/; y = sin 6 2 cosy. z = sin 6 2' COSi^. * Camh. and Duh. Math. Journal. Vol. i. (1846.) P 210 QUATERNIONS. [S^O. From these we pass at once to Rodrigues' subsidiary formulae, K = -^ = sec^ - J X = — = tan - cos /, w 2 '^' &c. = &c. 360.] By the definition of Homogeneous Strain, it is evident that if we take any three (non-coplanar) unit- vectors a, /3, y in an un- strained mass, they become after the strain other vectors, not neces- sarily unit- vectors, a^, /S^, y^. Hence any other given vector, which of course may be thus ex- pressed, p = xa -\- y^ ■\- zy, becomes p^^ = a?ai4-^^i+ ^yi, and is therefore known if a^, ^-^, y^ be given. ♦ More precisely pS.a^y = aS.f3yp + l3S.yap + yS.al3p becomes p■^^S.a^y == cjypS.afty = aj^S.(3yp-\- ^^S.yap + y-^S.a^p. Thus the properties of cf), as in Chapter V, enable us to study with great simplicity strains or displacements in a solid or liquid. For instance^ to find a vector whose direction is unchanged lij the strain, is to solve the equation Vp^p = 0, or (T = 8a<^p, Sp\f/(r = Scnf/p. But for the compound strain we have Pi = XP = f^P} and we have not generally Spxa- = Saxp. For Sp\l/cf)a = S(T(p\l/p, by (I), and \l/(f) is not generally the same as = ^+r.^{ ), = q^{ )q-\ or 4> = ^^.q{ )q-\ where e is a vector, and q a quaternion (which may obviously be regarded as a mere versor). That this is possible is seen from the fact that involves nine independent constants, while \/^ and ot each involve six, and e and q each three. If 0' be the function conjugate to 0, we have 0'=^-r.e( ), so that • 2\|/- = + 0', and 2V.i[ ) = — 0', which completely determine the first decomposition. This is, of 565.] KINEMATICS. 213 course, perfectly well known in quaternions, but it does not seem to have been noticed as a theorem in the kinematics of strains that there is always one, and but one, mode of resolving- a strain into the geometrical composition of the separate effects of (1) ^ imre strain, and (2) a rotation accompanied by uniform dilatation perpendicular to its axis, the dilatation being- measured by (sec. Q—\) where Q is the angle of rotation. In the second form (whose solution does not appear to have been attempted), we have = q^{ )q~\ where the pure strain precedes the rotation, and from this 0'=^.^-i( )q, or in the conjugate strain the rotation (reversed) is followed by the pure strain. From these 4y'(f) = ^.q-^q^i )q-^)q and CT- is to be found by the solution of a biquadratic equation^. It is evident, indeed, from the identical equation S.acf)^(f)p = S.p'( ) = q^^q-H )q)q-\ or ^-1 (00V) ^ = ^2 {q-^pq) = 0'0 {q^ ^pq), which shew the relations between 00', 0'0, and q. To determine q we have (pp.q = q^p * Suppose the cubic in ct to be write o) for -^)aip = Spcj>x. Hence the angle between the planes S(p = 0, and Srjp = 0, which is eos~ '^(—S.UCUri), becomes eos-\-S.UP = P + l^^(^P> where Sa^ = 0. The vectors which are unaltered in length are given by Tp, = Tp, or 2 S^pSap + ^"^S^ap = 0, which breaks up into S.ap = 0, and xS/3(2^ + /32a) = 0. The intersection of this plane with the plane of a, /3 is perpen- dicular to 2/3 + ^^a. Let it be a + i»/3, then S.{2ji + l3'-a){a + X0) = O, i.e. 2x—\ = 0. Hence the intersection required is 368.] KINEMATICS. 215 For the axes of the strain, one is of course a/3, and the others are found by making T(f)Up a maximum and minimum. Let p = a + wf3, then p-^= (f)p = a + xfi—^, A '^Pi and -=- = max. or mm.. Tp 1 g-ives x^—x+-^ = 0, from which the values of x are found. Also^ as a verification, ^ -iS.(a + a?i/3)(a + a?.2/3) = — I + jS^WiiJC^, and should be = 0. It is so, since, by the equation, _ 1 Again S{a + {x^-\)i3} {a + (Xo-l)i3} = -I + (3^(V^w.,-{Xi + x.;)+ 1}, which ought also to be zero. And, in fact, sPi + Xo= 1 hy the equa- tion ; so that this also is verified. 368.] We regret that our limits do not allow us to enter farther upon this very beautiful application. But it may be interesting here, especially for the consideration of an?/ continuous displacements of the particles of a mass, to in- troduce another of the extraordinary instruments of analysis which Hamilton has invented. Part of what is now to be given has been anticipated in last Chapter, but for continuity we commence afresh. If Fp = C (1) be the equation of one of a system of surfaces, and if the difierential of (l)be Svdp= 0, (2) v is a vector perpendicular to the surface, and iis length is inversely proportional to the normal distance between two consecutive surfaces. In fact (2) shews that v is perpendicular to dp^ which is any tangent vector, thus proving the first assertion. Also, since in passing to a proximate surface we may write Svhp = bC, we see that F{p + v-'^bC) = C + bC. This proves the latter assertion. It is evident from the above that if (1) be an equipotential, or an isothermal, surface, — i; represents in direction and maynitude the force at any point or the jiux of heat. And we have seen (§ 317) that if . d . d , d dx dy dz 216 QUATERNIONS. [3^9- (P (P (P ffivino" V= •' — } ^ ^ f&2 ^7^2 ,/22 then y = V Fp. This is due to Hamilton {Lectures on Quaternions, p. 611). 369.] From this it follows that the effect of the vector operation V, upon any scalar function of the vector of a point, is to produce the vector tvJiich represents in magnitude and direction the most rapid change in the value of the function. Let us next consider the effect of V upon a vector function as o- = i^+jr]-^kC *■ We have at once v/» d;// dz^ ^dz dy^ '' and in this semi-Cartesian form it is easy to see that : — If o- represent a small vector displacement of a point situated at the extremity of the vector p (drawn from the origin) /SVo" represents the consequent cubical compression of the group of points in the vicinity of that considered, and FVo- represents twice the vector axis of rotation of the same group of points. Similarly S.v= - (^ +,| + 4) = -^" or is equivalent to total differentiation in virtue of our having* passed from one end to the other of the vector cr. 370.] Suppose we fix our attention upon a group of points which originally filled a small sphere about the extremity of p as centre, whose equation referred to that point is T<^ = e (1) After displacement p becomes p + o-, and, by last section, p + co becomes p + w + cr — (6'coV)(t. Hence the vector of the new surface which encloses the group of points (drawn from the extremity of P + 0-) is 0)1 = CO — (*S'a)V)o- (2) Hence to is a homogeneous linear and vector function of co^ ; or (I) = ^^\, and therefore^ by (1), l^^w^ = e, the equation of the new surface, which is evidently a central surface of the second order, and therefore, of course, an ellipsoid. We may solve (2) with great ease by approximation, if we re- member that Ti^T is very small, and therefore that in the small term we may put Wj for co ; i.e. omit squares of small quantities; thus CO = Wj + (r(dz is a differential of three independent variables. Hence the vector axis of rotation (jt dy, 'iTy-Th)''^'" vanishes by the vanishing- of each of its constituents, or r.Vcr = 0. Conversely, if there be no rotation., the displacements are in the direction of and proportional to, the normal vectors to a series of surfaces. For = r. dp V.Va- = (SdpV) a - VSadp, where, in the last term, V acts on o- alone. Now, of the two terms on the right, the first is a complete differ- ential, since it may be written —Ddpcr, and therefore the remaining- term must be so. Thus, in a distorted system, there is no compression if SVa = 0, and no rotation if F.Vcr = ; and evidently merely transference if a = a = a constant vector, which is one case of Vo- = 0. In the important case of a = eVFp there is evidently no rotation, since Vo- = eV^Fp is evidently a scalar. In this case, then, there are only translation and compression, and the latter is at each point proportional to the density of a distribution of matter, which would give the potential Fp. For if r be such density, we have at once V-Fp = ^Tir*. 372.] The Moment of Inertia of a body about a unit vector a as axis is e^ddently MF- = — 2 w ( Vapf, where p is the vector of the portion m of the mass, and the orig-in of p is in the axis. * Pfoc. R. S. E., 1862-3. 218 QUATEENIONS. [37 -■ Hence if we take kTa = e^, we have, as locus of the extremity of a, ife* =—^m{Vapf = MSa^a (suppose), the raomental ellipsoid. If -53- be the vector of the centre of inertia, a the vector of m with respect to it, we have p = ot + o- ; therefore Mk^ = - 2 »^ { ( Futtt)^ + ( Faa)^ } = -M{ VaTT^f + MSa4>^a. Now, for principal axes, k is max., min., or max.-min., with the condition q2 j ^ Thus we have Sa{T;sFa'UT—(l)-^^a) = 0, Saa = ; therefore —(p^a + sTFa^ = pa = Pa (by operating- by So). Hence (cf)^ + k- + ct^) a = + ■srSa-ar, (1) determines the values of a, ^^ being found from the equation Sm{(l) + k'^ + vy^)-^^= 1 (2) Now the normal to Sa (0 + /^2 ^ ot^^-i^. _ ^^ ^3^ at the point o- is (^ + y^^ + ct^)~^ a. But (3) passes through — ot, by (2), and there the normal is (c^+z^H' .2W1 tiJ, which, by (1), is parallel to one of the required values of a. Thus we prove Binet's theorem that the principal axes at any p>oint are normals to the three surfaces, confocal with the momental ellipsoid, which pass through that point. EXAMPLES TO CHAPTER X. 1. Form, from kinematical principles, the equation of the cycloid ; and employ it to prove the well-known elementary properties of the arc, tangent, radius of curvature, and evolute, of the curve. 2. Interpret, kinematically, the equation p = aUi^t-'p), where |3 is a given vector, and a a g-iven scalar. Shew that it represents a plane curve ; and g-ive it in an in- tegrated form independent of t. EXAMPLES TO CHAPTER X. 219 3. If we write rs = ^t — p, the equation in (2) becomes j3 — zj = all-sT. Interpret this kinematically ; and find an integal. What is the nature of the step we have taken in transforming from the equation of (2) to that of the present question ? 4. The motion of a point in a plane being given^ refer it to (a.) Fixed rectangular vectors in the plane. (d.) Rectangular vectors in the plane, revolving uniformly about a fixed point. ((?.) Vectors, in the plane, revolving with different, but uni- form, angular velocities. (<^.) The vector radius of a fixed circle, drawn to the point of contact of a tangent from the moving point. In each case translate the result into Cartesian coordinates. 5. Any point of a line of given length, whose extremities move in fixed lines in a given plane, describes an ellipse. Shew how to find the centre, and axes, of this ellipse ; and the angular velocity about the centre of the ellipse of the tracing point when the describing line rotates uniformly. Transform this construction so as to shew that the ellipse is a hypotrochoid. 6. A point, A, moves uniformly round one circular section of a cone; find the angular velocity of the point, a, in which the o-enerating line passing through A meets a subcontrary section about the centre of that section. 7. Solve, generally, the problem of finding the path by which a point will pass in the least time from one given point to another, the velocity at the point of space whose vector is p being expressed by the given scalar function y^. Take also the following particular cases : — (a.) /p = a while Sap> 1, fp=b while Sap < 1 . (b.) fp = Sap. ((,.) fp ^ _p2_ (Tait, Trans. R. S. E., 1865.) 8. If, in the preceding question, /Jb be such a function of Tp that any one swiftest path is a circle, every other such path is a circle, and all paths diverging from one point converge accurately in another. (Maxwell, Cam. and Dub. Math. Journal, IX^p. 9.) 220 QUATERNIONS 9. Interjpret, as results of the composition of successive conical rotations, the apparent truisms - ^ ^= 1 y 13 a and °^^ -1^=1- Kid y p o- (Hamilton, Lectures, p. 334.) 10. Interpret, in the same way, the quaternion operators J = (6e-')*(€f-')*(f6-')*, -d .=(7/ar0*(|)*(^)*- (liii-) 11. Find the axis and angle of rotation by which one given rect- angular set of unit-vectors a, /3, y is changed into another given set ttj, /3i, yi. 12. Shew that, if 0p = p+ Vep, the linear and vector operation <^ denotes rotation about the vector e, together with uniform expansion in all directions perpendicular to it. Prove this also by forming the operator which produces the expansion without the rotation, and that producing the rotation without the expansion ; and finding their joint effect. 13. Express by quaternions the motion of a side of one right cone rolling uniformly upon another which is fixed, the vertices of the two being coincident. 14. Given the simultaneous angular velocities of a body about the principal axes through its centre of inertia, find the position of these axes in space at any assigned instant. 15. Find the linear and vector function, and also the quaternion operator, by which we may pass, in any simple crystal of the cubical system, from the normal to one given face to that to an- other. How can we use them to distinguish a series of faces be- longing to the same zone ? 16. Classify the simple forms of the cubical system by the properties of the linear and vector function, or of the quaternion operator. 17. Find the vector normal of a face which truncates symmetri- cally the edge formed by the intersection of two given faces. 18. Find the normals of a pair of faces symmetrically truncating the given edge. EXAMPLES TO CHAPTER X. 221 19. Find the normal of a face which is equally inclined to three given faces. 20. Shew that the rhombic dodecahedron may be derived from the cube, or from the octahedron, by truncation of the edges. 2 1 . Find the form whose faces replace, symmetrically^ the edges of the rhombic dodecahedron. 22. Shew how the two kinds of hemihedral forms are indicated by the quaternion expressions. 23. Shew that the cube may be produced by truncating the edges of the regular tetrahedron. 24. Point out the modifications in the auxiliary vector function required in passing to the pyramidal and prismatic systems re- spectively. 25. In the rhombohedral system the auxiliary quaternion operator assumes a singularly simple form. Give this form, and point ovit the results indicated by it! 26. Shew that if the hodograph be a circle^ and the acceleration be directed to a fixed point ; the orbit must be a conic section, which is limited to being a circle if the acceleration follow any other law than that of gravity. 27. In the hodograph corresponding to accelerationy*(i)) directed towards a fixed centre, the curvature is inversely as l)'^f{L). 28. If two circular hodographs, having a common chord, which passes through, or tends towards_, a common centre of force, be cut by any two common orthogonals, the sum of the two times of hodo- graphically describing the two intercepted arcs (small or large) will be the same for the two hodographs. (Hamilton^ Elements, p. 725.) 29. Employ the last theorem to prove, after Lambert, that the time of describing any arc of an elliptic orbit may be expressed in terms of the chord of the arc and the extreme radii vectores. 30. If ([{ ) q"^ be the operator which turns one set of rect- angular unit- vectors a, /3, y into another set a^, /S^, y^, shew that there are three equations of the form CHAPTER XL PHYSICAL APPLICATIONS. 373.] We propose to conclude the work by giving a few in- stances of the ready applicability of quaternions to questions of mathematical physics, upon which^ ever* more than on the Geo- metrical or Kinematical applications, the real usefulness of the Calculus must mainly depend — except, of course, in the eyes of that section of mathematicians for whom Transversals and Anharmonic Pencils, &c. have a to us incomprehensible charm. Of course we cannot attempt to give examples in all branches of physics, nor even to carry very far our investigations in any one branch : this Chapter is not intended to teach Physics, but merely to shew by a few examples how expressly and naturally quaternions seem to be fitted for attacking the problems it presents. We commence with a few general theorems in Dynamics — the formation of the equations of equilibrium and motion of a rigid system, some properties of the central axis, and the motion of a solid about its centre of inertia. 374.] When any forces act on a rigid body, the force /3 at the point whose vector is a, &c., then, if the body be slightly displaced, so that a becomes a + 8 a, the whole work done is This must vanish if the forces are such as to maintain equilibrium. Hence i/te condition of equilibrium of a rigid bodt/ is 2 S^ha = 0. For a displacement of translation ba is an^ constant vector, hence 2/3=0 (1) For a rotation-displacement, we have by § 350, e being the axis, and Te being indefinitely small, ha = T ea, 377-] PHYSICAL APPLICATIONS. *■ 223 and ^S.IBFea = -^S.^FajB = S.e2{7'a(3) = 0, whatever be e, hence S . Fa/3 = (2) These equations, (1) and (2), are equivalent to the ordinary six equations of equilibrium, 375.] In general, for any set of forces, let 2/3 = /3i, 2, Fa/3 = Oi, it is required to find the points for ivhlch the couple a^ has its axis coincident with the resultant force /3^. Let y be the vector of such a point. Then for it the axis of the couple is 2.F(a-y)/3 = ai-ry/3i, and by condition x^^ = a^—J y/3i . Operate by S^-^ ; therefore X^\ = Sa^^^, and ry/3i = a^ -/3r'^ai/3i = -f^i ^al/3^^ or y = rai/3i-i+y/3i, a straight line (the Ce?itral Axis) parallel to the resultant force. 376.] To find the points about which the couple is least. Here ^(ai— Fy^J = minimum. Therefore S. {a^- Yy^^ Y^^y^ 0, where y is any vector whatever. It is useless to try y — /Sj, but we may put it in succession equal to a^ and Ya^^i. Thus ^.yF./3iFai/3i = 0, and ( Ya^ ^^f - fi\ S.y Ya^ l3^ = 0. Hence y = x Ya^ /3^ + y/3i , and by operating with S.Ya-^^^, we get ^{Ya,P,Y=x{Ya,^,f, Hi or y =Fai/3r'+y/3i, the same locus as in last section. 377.] The couple vanishes if a,-Yyl3, = 0. This necessitates xS'a^/3i = 0, or the force must be in the plane of the couple. If this be the case, y = ai/3i-i+a;/3i, still the central axis. 22|. * QUATEKNIONS. [378. 378.] To assign the values of forces i, |i, to act at e, e^, and be equivalent to the given system. Hence r,i + re,{l3^- i) = a„ and ^ = {( — ei)'^ {ai-Tei(3i) + x{€-e{). Similarly for f^. The indefinite terms may be omitted^ as they must evidently be equal and opposite. In fact they are any equal and opposite forces whatever acting in the line joining the given points. 379.] For the motion of a rigid system, we have of course ^S{md—ft)ba= 0, by the general equation of Lagrange. Suppose the displacements 8a to correspond to a mere translation, then ba is anj/ constant vector, hence I. {ma- 13) = 0, or, if ai be the vector of the centre of inertia, and therefore a^1,m = 2 ma, we have at once a-^^m—^^ = 0, and the centre of inertia moves as if the whole mass were concen- trated in it, and acted upon by all the applied forces. 380.] Again, let the displacements 8 a correspond to a rotation about an axis i, passing through the origin, then 8a = Via, it being assumed that It is indefinitely small. Hence -^S.^Vaima-^) = 0, for all values of e, and therefore ^.Fa{ma-j3) = 0, which contains the three remaining ordinary equations of motion. Transfer the origin to the centre of inertia, i.e. put a = 0^ + -:^, then our equation becomes 2r(a, +CT)(«iai + Mzij- — /3) = 0. Or, since ^msr = 0, 2^-57(^^-/3) + Fai(ai2m-2^) = 0. But a\ 2^ — 2/3 = 0, hence our equation is simply 2r^{mi^-l3) = 0. Now 2FW/3 is the couple, about the centre of inertia, produced by the applied forces ; call it (f), then ^mF'sji& = (f) (1) 383.] PHYSICAL APPLICATIONS. . 225 381 .] Integrating once, -LmVtJjir = y+fit>(U (2) Again, as the motion considered is relcdive to the centre of inertia, it must be of the nature of rotation about some axis, in general variable. Let e denote at once the direction of, and the angular velocity about, this axis. Then, evidently, •nr = Few. Hence, the last equation may be wi'itten ^musVixn = y+ycfxlf. Operating by S.(, we get 2/«(ret^)2 = S:y + Sef(l>cU (3) But, by operating directly by 2fSeU upon the equation (1), we get ^m ( Fex^f =-h- + 2fSi4>dt (4) (2) and (4)x}ontain the usual four integrals of the first order. 382.] Wlien no forces act on the body, we have <^ == 0, and therefore ^m^Ve-ar = y, (5) S«m2 = 2m{r€r;jf = -/i\ (6) and, from (5) and (6), Sey =—/i^ (7) One interpretation of (6) is, that the kinetic energy of rotation remains unchanged : another is, that the vector e terminates in an ellipsoid whose centre is the origin, and which therefore assigns the angular velocity when the direction of the axis is given ; (7) shews that the extremity of the instantaneous axis is always in a plane fixed in space. Also, by (5), (7) is the equation of the tangent plane to (6) at the extremity of the vector e. Hence the ellipsoid (6) rolls on the plane (7). From (5) and (6), we have at once, as an equation which e must satisfy, y^2.?n{re^y = —P {^.m^Fe^f. This belongs to a cone of the second degree fixed in the body. Thus all the ordinary results regarding the motion of a rigid body under the action of no forces, the centre of inertia being fixed, are deduced almost intuitively : and the only difficulties to be met with in more complex properties of such motion are those of integration, which are inherent to the subject, and appear whatever analytical method is employed. (Hamilton, Proc. R. I. A. 1848.) 383.] Let a be the initial position of ct, q the quaternion by which the body can be at one step transferred from its initial posi- tion to its position at time t. Then qaq ^ Q q-^yq 226 QUATERNIONS. [384. and Hamilton's equation (5) of last section becomes '2.mqaq~^V.€qaq~^ = y, or ^.mq {a/^.a^"^ (q—q'^(qa^]q~^ = y. Let (jyp = ^.m{aSap — a^p), (1) where (^ is a self-conjugate linear and vector function, whose con- stituent vectors are fixed in the body in its initial position. Then the previous equation may be wi'itten q(p{q-^eq)q-'^ = y, or (i>{q'-^€q) = q-'^yq. For simplicity let us write "' ll\ ; '^^ Then Hamilton's dynamical equation becomes simply r)=C (3) 384.] It is easy to see what the new vectors 7j and ( represent. For we may write (2) in the form y = qCq ^ ; 3 from which it is obvious that -q is that vector in the initial position of the body which, at time t, becomes the instantaneous axis in the moving body. When no forces act, y is constant, and C is the initial position of the vector which, at time t, is perpendicular to the invariable plane. 385.] The complete solution of the problem is contained in equa- tions (2), (3) abovcj and (4) of §^356 *. Writing them again, we have qr] = H, (4) yq = qC, (2) 4>r] = (- (3) We have only to eliminate f and r], and we get 2q = qcl>-^{q-'yq), (5) in which q is now the only unknown ; y, if variable, being supposed known in terms of q and t. It is hardly conceivable that any simpler, or more easily interpretable, equation for q can be presented * To these it is unnecessary to add T5 = constant, as this constancy of Tq is proved by the/o?'j)i of (4). For, had Tq been variable, there must have been a quaternion in the place of the vector rj. In fact, ^{Tqf = 2S.qKq = {TqfSr]^0. 387.] PHYSICAL APPLICATIONS. 227 until symbols are devised far more comprehensive in their meaning- than any we yet have. 386.] Before entering into considerations as to the integration of this equation, we may investigate some other consequences of the group of equations in § 385. Thus, for instance, differentiating (2), we have and, eliminating q by means of (4), whence f = V^r] + ^"^ yq ; which gives, in the case when no forces act, the forms c=nr'c (6) and (as C= p = -/i\ evidently belongs to an ellipsoid, of which the radii-vectores are inversely as the square roots of the moments of inertia about them ; so that, if i, J, k be taken as unit- vectors in the directions of its axes respectively, we have Si(f)i = — A, ^ 8jj = -B, (8) Sk^k = — C, J A, B, C, being the principal moments of inertia, Consequently (f)p = -{AiSip + BJSjp + aS^p] (9) Thus the equation (7) for j] breaks up, if we put 7] = ico-i^ + jM^ + kuy^, into the three following scalar equations A(ai+ {C — B) (j>2^(ii^ = 0, \ Ba)2 + (A — C) cOgCOj = w C(o^ + {B — A) 0)1^(1)2 = 0, ) Q 2 228 QUATERNIONS. [388. which are the same as those of Euler. Only, it is to be understood that the equations just written are not primarily to be considered as equations of rotation. They rather express, with reference to fixed axes in the initial position of the body, the motion of the extremity, co^, (o.^, Wg, of the vector corresponding to the instan- taneous axis in the moving" body. If, however, we consider coj, ca^, ^3 as standing" for their values in terms of iv, a?, y, 2; (§ 391 below), or any other coordinates employed to refer the body to fixed axes, they are the equations of motion. Similar remarks apply to the equation which determines (, for if 37.: 3) we put ( = /sTj + jTSTr, + k (6) may be reduced to three scalar equations of the form . 1 1 X 388. J Euler's equations in their usual form are easily deduced from what precedes. For, let whatever be p ; that is, let <|> represent with reference to the moving principal axes what (f) represents with reference to the principal axes in the initial position of the body, and we have <|>e = q(P {q-^ kq) q'^ = qcf) {->)) q'^ = qCq-' =qr{(ri)q-'^ = -r.qr]4>{r])q-^ = -r.qi^q-^q(p{q-'^€q)q-'^ = -r.6<|>e, which is the required expression. But perhaps the simplest mode of obtaining this equation is to start with Hamilton's unintegrated equation, which for the case of no forces is simply S.mFwOT = 0. But from ct = Few we deduce ot = Fcct + Fcct = CTe^ — eSetjT-r Vezr, SO that E.T^z (FeOT/SfOT — CCT^ + CT'S'eCT) = 0. If we look at equation (1), and remember that (j) differs from simply in having ct substituted for a, we see that this may be written f^gc|,e + «t>e = 0, 390.] PHYSICAL APPLICATIONS. 229 the equation before obtained. The first mode of arriving- at it has been given because it leads to an interesting set of transformations, for which reason we append other two. By (2) y = ,xr^ therefore = qcf^.qC(r^+qCr^ — qCq-'^qq~^, or qtq~'^ = 2F.yFqq-'^ = Fye. But, by the beginning of this section, and by (5) of § 382, this is again the equation lately proved. Perhaps, however, the following is neater. It occurs in Hamil- ton's Elements. By (5) of § 382 (t>e = y. Hence <|)e = — (i>e = —2. to (tir Few + ot TfOT) = — 'Si.mvjSe'm = — F.e^.m-srSe-aT = — re<|>e. 389.] However they are obtained, such equations as those of § 387 were shewn long ago by Euler to be integrable as follows. Putting 2/o}^(a2(»^(U = s, we have Ao,^^ = Aa^'~ + {B - C) s, with other two equations of the same form. Hence ds so that t is known in terms of s by an elliptic integral. Thus, finally, r) or ( may be expressed in terms of t ; and in some of the succeeding investigations for q we shall suppose this to have been done. It is with this integration, or an equivalent one, that most writers on the farther development of the subject have commenced their investigations. 390.] By § 381, y is evidently the vector moment of momentum of the rigid body ; and the kinetic energy is But S^y = S.q-'^eqq-'^yq = Sr]C, so that when no forces act But, by (2), we have also TC=Ty, or Tcprj:=Ty, so that we have, for the equations of the cones described in the 230 QUATERNIONS. [39 1. initial position of the body by 77 and (, that is, for the cones de- scribed in the moving- body by the instantaneous axis and by the perpendicular to the invariable plane, This is on the supposition that y and k are constants. If forces act, these quantities are functions of f, and the equations of the cones then described in the body must be found by eliminating ( between the respective equations. The final results to which such a process will lead must, of course, depend entirely upon the way in which t is involved in these equations, and therefore no general statement on the subject can be made. 391.] Eecurring to our equations for the determination of q, and taking first the case of no forces, we see that, if we assume rj to have been found (as in § 389) by means of elliptic integrals, we have to solve the equation _ g^-jf that is, we have to integrate a system of four other differential equations harder than the first. Putting, as in § 38 7, rj = ico^ +J(^2 + ^^s ■> where Wi, w^.^ Wg are supposed to be known functions of t, and q = w^ix^jt/-^ kz, , . . . \ ^ dw dx dy dz this system is - ^/^ = ^-= = ^=- =: -— = -^ , •^ 2 W X Y Z * To get an idea of the nature of this equation, let us integrate it on the supposi- tion that 77 is a constant vector. By dififerentiation and substitution, we get Hence q=Q,cos^ t+ Q.sin^ t. Substituting in the given equation we have Tt} {- Q, sin^t+Q.cos-^tj = ^Q, cos -^ i + Q^sin -|- tjrj- Hence Tr].Q^=Q,rj, which are virtually the same equation, and thus r=Q:(c .-^t+UrjSin — t^ And the interpretation of ^ ( ) q-^ will obviously then be a rotation about 77 through the angle tTrj, together with any other arbitrary rotation whatever. Thu^ any posi- tion whatever may be taken as the initial one of the body, and Qi ( ) Qi ^ brings it to its required position at time i = 0. 392.] PHYSICAL APPLICATIONS. 231 where W ■= — m^oc — cn^,?/ — «3 z, X =■ CrijW + COgy — Wg^^, Y =: (ii^lO -\- (i)-^ Z — (li^X, Z =■ CO^W -\- (i}2X — (^>i^ ', or, as suggested by Cayley to bring out the skew symmetry, X= . (0^9/ — COg^ + COjW, Y=z —COoX . +(i)iZ + (02^iV, Z= OioX — COj^ . +CO3W, W ■= — Wj^a? — CO2J' — Wo^ Here, of course^ one integral is ?(j^ +a^^ +^^ + *^ = constant. It may suffice thus to have alluded to a possible mode of solution, which, except for very simple values of r], involves very great diffi- culties. The quaternion solution, when r] is of constant length and revolves uniformly in a right cone^ will be given later. 392.] If, on the other hand, we eliminate ??, we have to inte- grate q(p-'^{q-'^yq)= 2q, so that one integration theoretically suffices. But, in consequence of the present imperfect development of the quaternion calculus, the only known method of effecting this is to reduce the quaternion equation to a set of four ordinary differential equations of the first order. It may be interesting to form these equations. Put q = w + ix+ J^ + kz, y = ia -\-jh + kc, then, by ordinary quaternion multiplication, we easily reduce the given equation to the following set : (It dw _ dx _dy ^dz T'^F~T~ Y~Y' where JF=- -xn-yB — z€ or X= . T/dL—zB + WE, X = 20^+y€—zB Y=—x€ . +z^-{-tvB, Y = wJB + z'^—x(S:, Z— x'^—y% . +w€:, Z = 2v(!L + xm-y^ W^-x^-y^-z^iL . , and % — —{p, {lo^ —x"^ —y'^-z"') -\-2x{ax^-hy^cz) + 2w (bz — cy)\ B = -^\h{w'^—x'^—y''-—z'^)-{-2y{ax-^hy-\-cz)-\-2to{cx—az^, C = -Yr\c{w'^—x'^—y'^ — z'^)^-2z{cix-\-hy-\-cz)-^2w{ay—hx)'], 232 QUATERNIONS. [393- IF, X, Y, Z are thus homogeneous functions of w, x, y, z of the third degree. Perhaps the simplest way of obtaining' these equations is to trans- late the group of § 385 into w, x, y, z at once, instead of using the equation from which ^and t] are eliminated. We thus see that -p. , -^ , 7- One obvious integral of these equations ought to be w^ + x"^ +y^ +z^ = constant, which has been assumed all along. In fact, we see at once that wJF+xX+?/Y+zZ= identically, which leads to the above integral. These equations appear to be worthy of attention, partly because of the homogeneity of the denominators JF, X, Y, Z, but particularly as they afford (what does not appear to have been sought) the means of solving this celebrated problem at one stejo, that is, without the previous integration of Euler's equations (§ 387). A set of equations identical with these, but not in a homogeneous form (being expressed, in fact, in terms of k. A, /x, y of § 359, instead of to, X, y, z), is given by Cayley {Camh. and Duh. Math. Jotirnal, vol, i. 1846), and completely integrated (in the sense of being re- duced to quadratures) by assuming Euler's equations to have been previously integrated. (Compare § 391.) Cayley's method may be even more easily applied to the above equations than to his own ; and I therefore leave this part of the development to the reader, who will at once see (as in § 391) that %, 33, ffl: correspond to co^, cog, Wg of the rj type, § 387. 393. J It may be well to notice, in connection with the formulae for direction cosines in § 359 above, that we may write ^ = —rla ii'^^ + x'^ —y'^ — z^)+2b (xy + tvz) + 2c {xz—wy)'], 38 = -r^ \^2 a (xy — ivz) -{^ 6{w^ — x'^+y^—z^) + 2 c {yz + tox)'], (iL — -T^[2a{xz-\- wy) + 2b (yz—wx) + c iv)"- —x"^ —y'^ + z^)]. These expressions may be considerably simplified by the usual assumption, that one of the fixed unit-vectors {i suppose) is perpen- dicular to the invariable plane, which amounts to assigning defi- nitely the initial position of one line in the body ; and which .gives the relations ^ = o c = 396.] PHYSICAL APPLICATIONS. 233 394.] When forces act, y is variable, and the quantities a, b, c will in general involve all the variables zo, x, y, z, t, so that the equations of last section become much more complicated. The type, however, remains the same if y involves t only ; if it involve q we must differentiate the equation, put in the form y = 2?0(?-^?)?-\ and we thus easily obtain the differential equation of the second order ^ = 4F.^(^(^-i^)^-i + 2^(^ (T.^-i^)^-! ; if we recollect that, because q~^q is a vector, we have S.q-''q= {q-^f. Though remarkably simple, this formula, ifi the present state of the development of quaternions, must be looked on as intractable except in certain very particular cases. 395.] Another mode of attacking- the problem, at first sio-ht entirely different from that in § 383, but in reality identical with it, is to seek the linear and vector function which expresses the Homogeneous Strain which the body must undergo to pass from its initial position to its position at time t. Let^ -CT =r ya, a being (as in § 383) the initial position of a vector of the body, OT its position at time t. In this case x i^ 3, linear and vector function. (See § 360.) Then, obviously, we have, -sr^ being the vector of some other point, which had initially the value a^, xSra-CTj = S.)(a)^a■^^ = Saa-^, (a particular case of which is T^ = Txa = To) and FctOTj = ^•X'^X'^i — X ^<^o.^ • These are necessary properties of the strain-function x, depending on the fact that in the present application the system is rigid. 396.] The kinematical equation ■air = Fe-BT becomes ^a = F. e^a (the function x being formed from x by the differentiation of its constituents with respect to t). Hamilton's kinetic equation '^i.Tn-srV^'UT = y, becomes S.wx«^-fX« = 7- 234 QUATERNIONS. [397- This may be written or S.m(a^.ax'e — x~^f -o^) = X~^Yj where \ is the conjugate of x- But, because S.xa\a-f^ = Saa-,^ , we have Saa-^ = S.a\\a^, whatever be a and a^ , so that x'= X~^- Hence 2.?«(a/S'.ax~^e — x~^e.a-) = x~V> or, by§ 383, X"'f = X^V- 397.] Thus we have, as the analogues of the equations in §§ 383, 384, ^-1^ ^ ^^ x''y=C, and the former result x^a — V. exa becomes xa = F. XVX^ = X ^^"• This is our equation to determine x? V being supposed known. To find r] we may remark that ^f? = C and C = x-'y- But XX"'« = «» so that XX~^« + XX"^a = 0. Hence C = —x~^'k\~^v or (prj ^= — Vr](f)ri. These are the equations we obtained before. Having found rj from the last we have to find x from the condition X~^X« =^V- 398.] We might, however, have eliminated rj so as to obtain an equation containing x alone, and corresponding to that of § 385. For this purpose we have r] = (p~'^C= 4>~^x~^y} so that, finally, X~^X« = ^•<^~^X~^y«5 or X~^« = ^-X"^ a which may easily be formed from the preceding equation by putting X~^a for a, and attending to the value of x~^ given in last section. 40I.] PHYSICAL APPLICATIONS. 235 399.] We have given this process, thoug-h really a disguised form of that in §§ 383, 385, and though the final equations to which it leads are not quite so easily attacked in the way of integration as those there arrived at, mainly to shew how free a use we can make of symbolic functional operators in quaternions without risk of error. It would be very interesting, however, to have the problem worked out afresh from this point of view by the help of the old analytical methods : as several new forms of long-known equations, and some useful transformations, would certainly be obtained. 400.] As a verification, let us now try to pass from the final equation, in x alone, of § 398 to that of § 385 in q alone. We have, obviously, OT = qaq-'^ = xa, which gives the relation between q and x- [It shews, for instance, that, as S.jSxa = S.axjS, while /S./Sxa = S.jBqaq"^ = S.aq~'^j3q, we have x'/3 = q~^/3q, and therefore that xx'^ = ^{l~^^l)9~^ — ^^ or x' = X"^5 ^s above.] Differentiating, we have qaq~^ — qaq~^qq~^ = xa. Hence X^^X*'^ = q~^qa — aq'^q = 2F.Fiq-^)a. Also (f)-^ x~ V = ^~^ ir^ yi\ so that the equation of § 398 becomes 2 r. r(^-i^) a = r. e = y = 331 - Simplifying the value of by assuming that the earth has two principal axes of equal moment of inertia, we have Be — {A—B)aSae = vector-constant + 3 J/ (^—^) / "f^ "^ (U. This gives Sae = const. = 12, whence e = — Ha + ad, so that, finally, BVad-A£ld = j^{A-B)FapSap. The most striking peculiarity of this equation is that the form of the solution is entirely changed, not modified as in ordinary cases of disturbed motion, according to the nature of the value of p. Thus, when the right-hand side vanishes, we have an equation which, in the case of the earth, would represent the rolling of a cone fixed in the earth on one fixed in space, the angles of liot/i being exceedingly small. If p be finite, but constant, we have a case nearly the same as that of a top, the axis on the whole revolving conically about p. 405.] PHYSICAL APPLICATIONS. 237 But if we assume the expression p = r (J cos rjif + k sin mt), (which represents a circular orbit described with uniform velocity,) a revolves on the whole conically about the vector i, perpendicular to the plane in which p lies. {Trans. B. S. E., 1868-9.) 402. ~| To form the equation of motion of a simple pendulum, taking account of the eartKs rotation. Let a be the vector (from the earth's centre) of the point of suspension, A its inclination to the plane of the equator, a the earth's radius drawn to that point ; and let the unit-vectors i, j, k be fixed in space, so that i is parallel to the earth's axis of rotation ; then, if co be the angular velocity of that rotation a = a [/ sin A + {j cos lat + k sin wt) cos A] ( 1 ) This gives a = « co ( — j sin wt + k cos oot) cos A = ioTia (2) Similarly d = (nVid = — a)^(a— 6ri sin A) (3) 403.] Let p be the vector of the bob vi referred to the point of suspension, R the tension of the string, then if a^ be the direction of pure gravity m\d + p) =-mgUa^-RUp, (4) which may be written rpd+rpp = ^ra,p (5) To this must be added, since r (the length of the string) is constant, Tp = r, (6), and the equations of motion are complete. 404.] These two equations (5) and (6) contain every possible case of the motion, from the most infinitesimal oscillations to the most rapid rotation about the point of suspension, so that it is necessary to adapt different processes for their solution in different cases. We take here only the ordinary Foucault case, to the degree of approximation usually given. 405.] Here we neglect terms involving w^. Thus we write a = 0, and we write a for a^, as the difference depends upon the ellipticity of the earth. Also, attending to this, we have p= a + OT, (7) a where by (by (6)) Sa^ = 0, (8) and terms of the order ot^ are neglected. 238 QUATERNIONS. [405. With (7), (5) becomes Faar = — Va-sr : a a so that, if we write - = w^, (9) we have Fa(iJ3- + n^'sr) = (10) Now, the two vectors ai — a sin \ and Via have, as is easily seen, equal tensors ; the first is parallel to the line drawn horkontally northwards from the point of suspension, the second horizontally eastwards. Let, therefore, ta = x{ai — as\n\)-\-yVia, (11) which [x and y being very small) is consistent with (6). From this we have (employing (2) and (3), and omitting co^) ■u: = ga{ai — asinX)+ j^Fia— a?£o sin AFm— yo) (a— a^sin A), OT = ^(ae — a sinA)+J/Tia— 2^cosinXFm— 2j^co(a— «isin A). "With this (10) becomes Va \x (fl^— a sin A.) + y Via— 2xu> sin A Fia—2y(o{a—ai sin A) + n^x{ai— a sink) + }i'^yria] = 0, or, if we note that V.aVia = a (a^ — a sin A), {^—x — 2y sinX + n^y)a {ai — asink) = 0. This gives at once x + n^x+2u)ys\nk = O,'! , , y + n-y — 2 (act sin A=0,3 which are the equations usually obtained ; and of which the solution is as follows : — If' we transform to a set of axes revolving in the horizontal plane at the point of suspension, the direction of motion being from the positive (northward) axis of x to the positive (eastward) axis of y, with angular velocity H, so that X = ^cos Q,t—ri sin 12/, y ■=■ ^ sin Q.t-\-r] cos il t, and omit the terms in Q? and in (dI2 (a process justified by the results, see equation (15)), we have (^■+w2^)cosX2^-(^4-^«2??)sinI2;f-2j^(X2— cosinA) = 0,^ (^*+^i^£)sinia2! + (T+?i2^)cosIli{+2£c(X2 — wsinA) = 0.) ^ So that, if we put 12 = w sin A, (15) we have simply . ^+%^^ = 0, ) ^ + ^^2^ = 0,3 ^ ^ the usual equations of elliptic motion about a centre of force in the centre of the ellipse. {Troc. 7?. S. E., 1869.) l\ ("' 407-] PHYSICAL APPLICATIONS. 239 406.] To construct a reflecting surface from which rays^ emitted from a point, shall after 7'eflection diverge imiformly, hut horizontally. Using the ordinary property of a reflecting surface, we easily obtain the equation 8. dpi- -) p = 0. V p ^ By Hamilton's grand Theory of Systems of Bays, we at once write down the second form 7> — T (/3 + a Vap) = constant. The connection between these is easily shewn thus. Let ct and T be any two vectors whose tensors are equal, then = 2CTT-1 (1 +S'UTT-'^), whence, to a scalar factor ^r^^, we have (•sr\a r + 'sr Hence, putting ct = ?7(/3 + aFap) and r = Up, we have from the first equation above S.dplUp+ UifS + aFap)'] = 0. But d{fi-\-aVap) = aVadp = — dp — aSadp, and S.a{l3 + a Vap) = 0, so that we have finally S.dpUp-S.d{^ + arap)U(^ + aVap) = 0, which is the difi'erential of the second equation above. A curious particular case is a parabolic cylinder, as may be easily seen geo- metrically. The general surface has a parabolic section in the plane of a, /3 ; and a hyperbolic section in the plane of /3, a(3. It is easy to see that this is but a single case of a large class of integrable scalar functions, whose general type is S.dp(^^ = o, V p ^ the equation of the reflecting surface ; while S{(T — p)da = is the equation of the surface of the reflected v/ave : the integral of the former being, by the help of the latter, at once obtained in the form Tp ± T{a-p) = constant*. 407.] We next take Fresnel's Theory of Bouhle Refraction, but * Proc. R.S.E., 1870-71. 240 QUATERNIONS. [408. merely for the purpose of shewing- how quaternions simpHfy the processes required, and in no way to discuss the plausibility of the physical assumptions. Let tvT be the vector displacement of a portion of the ether, with the condition ^^ = — 13 (1) the force of restitution, on Fresnel's assumption, is t {a~iSi-s7 + h^jSjtJT + c'^kSk-us) = i + v'-)-^a = - ^2 ((/, + t;2)-i «, because ^ is self-conjugate, we have v{4> + v^Y^a = ^^^^- This gives at once, by rearrangement, v{4> + v'')-^a = ((^_p2)-ip. Hence ((^_p2) ip = ___ . Operating by S.p on this equation we have Sp{-^ + /i)-^P = 0, if mSp(f>p = ghp'^ = — li. Thus the new form is Sp{(p-^-mSp(f>p)-'^p = (1) Here m = ^,^ ^ , Spcfyp = a'^x^ + b^^^ + c^z^, and the equation of the wave in Cartesian coordinates is, putting rf = a^x^ + 6^f+c^z\ x^ y'^ z"^ •^ -9.9 ..9 T^ ..9Z9 ..9 " ^2^2 — ^2 c^a^—r\ a'^b^—r\ 412.] By means of equation (1) of last section we may easily prove Pliicker's Theorem. The Wave-Surface is Us orvn reciprocal with resjoect to the ellipsoid whose equation is Sp(f)^p = —J— ' 4 1 3-] PHYSICAL APPLICATIONS. 243 The equation of the plane of contact of tangents to this surface from the point whose vector is p is S'ard)^ p = — , — • The reciprocal of this plane, with respect to the unit-sphere about tlie origin, has therefore a vector a where a = 's/fiitj)^ p. Hence p = —y— (b~^(r, and when this is substituted in the equation of the wave we have for the reciprocal (with respect to the unit-sphere) of the reciprocal of the wave with respect to the above ellipsoid, S.(T ((p Su(f) ^ ~^—p~^)~^P~^- Finally, we have for the required equation, by (4), ^p-i(r^-p-rv = o, or, by a transformation already employed, Sp{ct>-p^)-^p=-l. 415.] It may assist the student in the practice of quaternion analysis, which is our main object, if we give a few of these invest- igations by a somewhat varied process. Thus, in § 407, let us write as in § 168, aHSm-^h'^jSj^ + c^kSk^sT = X'xS/ot + |u'aSaV_/ct. We have, by the same processes as in § 407, S.'S7aySf/'ST + S.'sraix Sk'^ - 0. 4I7-] PHYSICAL APPLICATIONS. 245 (1) This may be written, so far as the generating lines we require are concerned, S.i!TaV.X.''^[/ = =1 S.XtxaX^'UTi/, since wa is a vector. Or we may write Equations (1) denote two cones of the second order which pass throug-h the intersections of (3) and (4) of § 407. Hence their in- tersections are the directions of vibration. 416.] By (1) we have Hence ctA'ot, a, \x are coplanar ; and, as ot is perpendicular to a, it is equally inclined to JX'a and F/x'a. For, if X, M, A be the projections of A', /, a on the unit sphere, BC the great circle whose pole is A, we are to find for the projections of the values of -57 on the sphere points F and P', such that if LP be produced till fq = LP, Q may lie on the great circle A3L Hence, evidently, CP = PB, and CF'=FB; which proves the proposition, since the projections of Fk'a and Vfxa on the sphere are points b and c in BC, distant by quadrants from C and B respectively. 417.] Or thus, S^jsa = 0, S.-^V.aK'^lJ.' ■=■ 0, therefore a?OT = V. a V. ak'-sri/, = — V. A'OTja' — aSa V. Wju,'. Hence {Sk'ix - a;) ot = (X' + aSak') Sf/^ + (/ + aSafx) Sk'zr. Operate by SX, and we have (x + Sk'aSix'a)Sk'^ = [X'2a2_^2Va]/S/xV Hence by symmetry, 246 QUATERNIONS. [4 1 8. 1 I ka I y [xa ^' =0, TJ'ixa and as S-ara = 0, 418.] The optical interpretation of the common result of the last two sections is that the planes of polarization of the two rays whose %oar,e-fronts are imrallel, bisect the angles contained hy planes jmssing through the normal to the wave-front and the vectors (optic axes) \% \x. 419.] As in § 409, the normal velocity is given by _ SW/a ~^' '^{T+SyVk'aVpfa' [This transformation, effected by means of the value of ct- in § 4 1 7, is left to the reader.] Hence, if r^, Vo be the velocities of the two waves whose normal is a, vl -vl = 2T. Vk'a T'lxa oc sin k'a sin \ia. That is, the difference of the squares of the velocities of the two waves varies as the product of the sines of the angles between the normal to the wave-front and the optic axes (A', \i). 420] We have, obviously, {T^-S^).rk'aJya = T^r. Vk'aiya = S.'^k'ix'a. Hence v^- =/ + {T±S).rk'aF,j.'a. The equation of the index surface, for which Tp = -, Up = a, V is therefore 1 = -7/^ + iT±S). Vk'p TYp. This will, of course, become the equation of the reciprocal of the index-surface, i.e. the wave-surface, if we put for the function (/> its reciprocal : i. e. if in the values of A', /, p' we put -, ^, - for a, b, c respectively. We have then, and indeed it mig-ht have been deduced even more simply as a transformation of § 409 (7), \ = -pp^^^{T±S).Vkpriip, as another form of the equation of Fresnel's wave. 424.] PHYSICAL APPLICATIONS. 247 If we employ the t, /c transformation of § 1 2 1, this may be written, as the student may easily prove, in the form 421.] We may now, in furtherance of our object, which is to g-ive varied examples of quaternions, not complete treatment of any one subject, proceed to deduce some of the properties of the wave- surface from the different forms of its equation which we have given, 422.] FresneVs constncction of the loave hy points. From § 273 (4) we see at once that the lengths of the principal semidiameters of the central section of the ellipsoid Sp(p''^p = 1, by the plane Sap = 0, are determined by the equation /S.a ((/)-! -p-2)-ia= 0. If these lengths be laid oflP along a, the central perpendicular to the cutting plane, their extremities lie on a surface for which a-=Up, and Tp has values determined by the equation. Hence the equation of the locus is . .. Sp(d>-^-p-^)-'^p= 0, as in §§409, 414. ' ^^ ^ ^ ^ ' Of course the index-surface is derived from the reciprocal ellip- soid Sp(j)p = 1 by the same construction. 423.] Again, in the equation 1 =-pp^ + {T±S).rkprp.p, suppose Vkp = 0, or Ffxp = 0, we obviously have UX , Ufx P = + —7= OV p = ±-^r:J Vp ^ !> and there are therefore four singular points. To find the nature of the surface near these points put UK where Tct is very small, and reject terms above the first order m Ttz. The equation of the wave becomes, in the neighbourhood of the singular point, 2p8K'rn + S.^K\ VXp. = ±T. VK-tj TAju, which belongs to a cone of the second order. 424.] From the similarity of its equation to that of the wave, it 248 QUATERNIONS. [425. is obvious that the index- surface also has four conical cusps. As an infinite number of tangent planes can be drawn at such a point, the reciprocal surface must be capable of being" touched by a plane at an infinite number of points ; so that the wave-surface has four tangent planes which touch it along ridges. To find their form, let us employ the last form of equation of the wave in § 420. If we put TVLp=TrKp, (1) we have the equation of a cone of the second degree. It meets the wave at its intersections with the planes ^(t-K)p = +(K2_.t2) (2) Now the wave-surface is touched by these planes, because we cannot have the quantity on the first side of this equation greater in abso- lute magnitude than that on the second, so long as p satisfies the equation of the wave. That the curves of contact are circles appears at once from (1) and (2), for they give in combination p- ='tS{l-\-k)p, (3) the equations of two spheres on which the curves in question are situated. The diameter of this circular ridge is 2TF1K 1 , [Simple as these processes are, the student will find on trial that the equation Sp ((})~'^ — p~^)~''- p = 0, gives the results quite as simply. For we have only to examine the eases in wdiich —p~^ has the value of one of the roots of the symbolical cubic in (f)~^. In the present case Tp = d is the only one which requires to be studied.] 425.] By § 41 3, we see that the auxiliary vector of the succeed- ing section, viz. is parallel to the direction of the force of restitution, ^ot. Hence, as Hamilton has shewn, the equation of the wave, in the form Srp = 0, (4) of §414, indicates that fJ/e direction of the force of restitution is perpendicular to the ray. Again, as for any one versor of a vector of the w^ave there are two values of the tensor, which are found from the equation S.Up{4r'-p'^)-Hrp = 0, 429.] PHYSICAL AITLICATIONS. 249 we see by § 422 that the lines of vibration for a given plane front are parallel to the axes of any section of the ellipsoid S.p(f)~^ p =■ 1 made ly a plane parallel to the front ; or to the tangents to the lines of curvatnre at a point where the tangent plane is parallel to the ivave- front. 426.] Again, a curve which is drawn on the tvave-surface so as to touch at each point the corresponding line of vibration has -!+ (7-2)- V = 0, the equation of a cyclic cone, whose vertex is at the common centre of the sphere and the wave-surface, and which cuts them in their curve of intersection. {Quarterly Math. Journal, 1859.) 428.] As another example we take the case of the action of electric currents on one another or on magnets ; and the mutual action of permanent magnets. A comparison between the processes we employ and those of Ampere {Theorie des Phenomenes Mectrodynamiques, Sfc, many of which are well given by Murphy in his Electricity) will at once shew how much is gained in simplicity and directness by the use of quaternions. The same gain in simplicity will be noticed in the investigations of the mutual effects of permanent magnets, where the resultant forces and couples are at once introduced in their most natui'al and direct forms. 429.] Ampere's experimental laws may be stated as follows : I. Equal and opposite currents in the same conductor produce equal and opposite effects on other conductors : whence it follows 250 QUATERNIONS. [430. that an element of one current has no effect on an element of an- other which lies in the plane bisecting- the former at right angles. II. The effect of a conductor bent or twisted in any manner is equivalent to that of a straight one, provided that the two are traversed by equal currents, and the former nearly coincides with the latter. III. No closed circuit can set in motion an element of a circular conductor about an axis through the centre of the circle and per- pendicular to its plane. IV. In similar systems traversed by equal currents the forces are equal. To these we add the assumption that the action between two elements of currents is in the straight line joining them : and two others, viz. that the effect of auy element of a current on another is directly as the product of the strengths of the currents, and of the lengths of the elements. 430.] Let there be two closed currents whose strengths are a and a^ ; let a, a^ be elements of these, a being the vector joining their middle points. Then the effect of a on a^ must, when resolved along a^ , be a complete differential with respect to a (i.e. with respect to the three independent variables involved in a), since the total resolved effect of the closed circuit of which a is an element is zero by III. Also by I, II, the effect is a function of Ta, Saa, Saa^ , and SaUi^ » since these are sufficient to resolve a and a^ into elements parallel and perpendicular to each other and to a. Hence the mutual effect is aa-JJaf{Ta, Saa, Saa^, Saaj), and the resolved effect parallel to a^ is aa-^ SUtti Vaf. Also, that action and reaction may be equal in absolute magnitude, /'must be symmetrical in Saa and Saa-^. Again, a (as differential of a) can enter oyily to the first power, and must appear in each term of/ Hence f = ASa'a^ + BSaaSaa-^ . But, by IV, this must be independent of the dimensions of the system. Hence J is of — 2 and ^ of — 4 dimensions in Ta. There- fore 1 yfT- iASaaiSa'ay + I)Saa'S^aaj^} Id is a complete differential, with respect to a, if da = a'. Let la 433-] PHYSICAL APPLICATIONS. 251 where C is a constant depending on the units employed, therefore and the resolved eflPect ^^ C'^^/, -^^ — 7=r-^ (iS. FaaFaa-, + 2 -S'aa'iS'aa,). The factor in brackets is evidently proportional in the ordinary notation to sin 6 sin 6'cos co — ^ cos 6 cos 6\ 431.] Thus the whole force is Caa^a , 'S'^aa^ Ccta^a , S-aa as we should expect, d^a being = a^. [This may easily be trans- formed into 2Caa^Ua which is the quaternion expression for Ampere's well-known form.] 432.] The whole effect on Cj of the closed circuit, of which a is an element, is therefore Caa^ C a ASan^'^ Caflj f a Saa^ „ f Vaa 1 between proper limits. As the integrated part is the same at both limits, the effect is Caa, fVaa f dUa - -^F«,/3, where l^=Jj^=J-^' and depends on the form of the closed circuit. 433.] This vector (3, which is of great importance in the whole theory of the effects of closed or indefinitely extended circuits, cor- responds to the line which is called by Ampere '' dii-ectrice de V action electrodynamique.'''' It has a definite value at each point of space, independent of the existence of any other current. Consider the circuit a polygon whose sides are indefinitely small ; join its angular points with any assumed point, erect at the latter, perpendicular to the plane of each elementary triangle so formed, a vector whose leno-th is - j where w is the vertical ang-le of the tri- 252 QUATEKNIONS. [434- angle and r the length of one of the containing sides ; the sum of such vectors is the '•'■ directrice^^ at the assumed point. 434.] The mere Jbnn of the result of § 432 shews at once that if the element a^ he turned ahoui its middle poi^it, the direction of the resultant action is confined to the plane tohose normal is /3. Suppose that the element a^ is forced to remain perpendicular to some given vector 8, we have Sa^h = 0, and the whole action in its plane of motion is proportional to But r.hra^li=-a^S^h. Hence the action is evidently constant for all possible positions of a^ ; or The effect of any system of closed currents on an element of a con- ductor which is restricted to a given plane is [in that plane') independent of the direction of the element. 435.] Let the closed current be plane and very small. Let e (where Te = 1) be its normal, and let y be the vector of any point within it (as the centre of inertia of its area) ; the middle point of Cj being the origin of vectors. Let a=^ y-{-p; therefore a'= p, to a sufficient approximation. Now (between limits) fVpp'= 2yie, where A is the area of the closed circuit. Also generally fVypSyp = I (Syp Vyp + y V. yffpp) = (between limits) AyFyc. Hence for this case A ^^ ^ SyFye^ 3y/S'ye> - rpy^ y^^ Ty^ ) y 436.] If, instead of one small plane closed current, there be a series of such, of equal area, disposed regularly in a tubular form, let X be the distance between two consecutive currents measured along the axis of the tube; then, putting y'= xe, we have for the whole effect of such a set of currents on a^ 438.] PHYSICAL APPLICATIONS. 253 CAaa^ Fa-^y r -4. x = — - — rp 3 (between proper limits). If the axis of the tubular arrangement be a closed curve this will evidently vanish. Hence a closed solenoid exerts no influence on an element of a conductor. The same is evidently true if the solenoid he indefiyiite in both directions. If the axis extend to infinity in one direction, and y^ be the vector of the other extremity, the effect is CAaa-^ Va-^y^ ~2^^yl' and is ^exe^ore perpendicular to the element and to the line joining it with the extremity of the solenoid. It is evidently inversely as Ty% and directly as the sine of the angle contained betiveen the direction of the element and that of the line joining the latter with the extremity of the solenoid. It is also inversely as x, and therefore directly as the number of currents in a unit of the axis of the solenoid. 437.] To find the effect of the whole circuit whose element is a^ on the extremity of the solenoid, we must change the sign of the above and put a^ = y^'; therefore the effect is CAaa^ r rypyo 2x J Tyl ' an integral of the sj)ecies considered in § 432 whose value is easily assigned in particular cases. 438.] Suppose the conductor to he straight, and indefinitely extended in both directions. Let ho be the vector perpendicular to it from the extremity of the canal, and let the conductor be |1 r;, where Td = Trj =. 1 . Therefore y^ = hd + yr, (where y is a scalar), ^yoVo = ¥^"^0, and the integral in § 436 is .rri6 1 ' =w. The whole effect is therefore and is thus perpendicular to the plane passing through the conductor and the extremity of the canal, and varies inversely as the distance of the latter from the conductor. 254 QUATERNIONS. [439. This is exactly the observed effect of an indefinite straight current on a mag-netic polcj or particle of free magnetism. 439.] Siip^iose the conductor to be circular, and the pole nearly in its axis. Let EPD be the conductor, AB its axis^ and C the pole ; BC perpendicular to AB, and small in comparison with AE = h the radius of the circle. Let where Then AB be ar^, BC=hk, A'P = h[jx\hy) J iciiiJ '•GTn-' y CP =1 y = a^i-\- bk — h{jx-\-ki/). VyY And the effect on C ^- tP ' may easily be shewn to coincide with that given by Ellis {Camh. Math. Journal, iv. 95), though it seems to lose in simplicity and capability of interpretation by such modifications. 443.] The above formulae shew that the whole force exerted by one small magnet M, on the centre of inertia of another w, consists of four terms which are, in order, 1st. In the line joining the magnets, and inoportlonal to the cosine of their mutual inclination. 2nd. In the same line, and proportional to five times the product of the cosines of their respective inclinations to this line. 3rd and 4th. Parallel to {^} and proportional to the cosine of the M inclination of [ j to the joining line. All these forces are, in addition, inversely as the fourth power of the distance between the magnets. For the couples about the centre of inertia of 7n we have 1st. A couple tohose axis is perpiendicular to each magnet, and which is as the sine of their mutual inclination. 2nd. A couple ivhose axis is perpendicular to m and to the line joining the magnets, and whose inoment is as three times the product of the sine of the inclination of m, and the cosine of the inclination of M, to the joining line. In addition these couples vary inversely as the third power of the distance between the magnets. S 258 QUATERNIONS. [444. [These results afford a good example of what has been called the internal nature of the methods of quaternions, reducing-, as they do at once, the forces and couples to others independent of any lines of reference, other than those necessarily belonging to the system under consideration. To shew their ready applicability, let us take a Theorem due to Gauss.] 444.] If two small magnets he at right angles to each other, the moment of rotation of the first is anj^roximatel^ tivice as great tohen the axis of the second glasses through the centre of the first, as lohen the axis of the first passes through the centre of the second. In the first case e 1| ^±fi ; C' 2C' therefore moment = j-^T{^e^ — Zi€-^) = j^Tee^. In the second e^ || /3±e ; C therefore moment = y^r-^Tee-,. Hence the theorem. 445.] Again, we may easily reproduce the results of § 442, if for the two small circuits we suppose two small magnets perpendicular to their planes to be substituted. (3 is then the vector joining the middle points of these magnets, and by changing the tensors we may take 2 e and 2 e^ as the vector lengths of the magnets. Hence evidently the mutual effect which is easily reducible to as before, if smaller terms be omitted. If we operate with F. e^ on the two first terms of the unreduced expression, and take the difference between this result and the same with the sign of e^ changed, we have the whole vector axis of the couple on the magnet 2ei, w^hich is therefore, as before, seen to be proportional to 4 . 3rei/3/S'/3eN 446.] We might apply the foregoing formulae with great ease to other cases treated by Ampere, De Montferraud, &c. — or to two finite circular conductors as in Weber's Dynamometer — but in general the only difficulty is in the integration, which even in some of the simplest cases involves elliptic functions, &c., &c. {Quarterly/ Math. Journal, 1860.) 448.] PHYSICAL APPLICATIONS. 259 447.] Let F{y) be the potential of any system upon a unit particle at the extremity of y. F{y) = C (1) is the equation of a level surface. Let the differential of ( 1 ) be Sv(ly= 0, (2) then i; is a vector normal to (1), and is therefore the direction of the force. ^ But, passing- to a proximate level surface, we have Svhy = bC. Make by=xu, then —wTv" = bC, -^\=Tby\ Hence v expresses the force in magnitude also. (§ 368.) Now by § 435 we have for the vector force exerted by a small plane closed circuit on a particle of free magnetism the expression A ^ ZySye\ omitting the factors depending on the strength of the current and the strength of magnetism of the particle. Hence the potential, by (2) and (1), . r 1 /o V , ^SydySy^ ^ ASey oc ) Ty^ area of circuit projected perpendicular to y oc -^ •' cc spherical opening subtended by circuit. The constant is omitted in the integration, as the potential must evidently vanish for infinite values of Ty. By means of Ampere's idea of breaking up a finite circuit into an indefinite number of indefinitely small ones^ it is evident that the above result may be at once ex- tended to the case of such a finite closed circuit. 448.] Quaternions give a simple me- thod of deducing the well-known pro- perty of the Magnetic Curves. Let A, A' be two equal magnetic poles, whose vector distance, 2 a, is bi- sected in 0, QQ' an indefinitely small magnet whose length is 2p, where p= OP. Then evidently, taking moments, s 2 260 QUATERNIONS. [449- F (p+a)/ _ F{p — a)p T{p + af -- T{p-af' where the upper or lower sign is to be taken according" as the poles are like or unlike. Operate by S. Tap, Sap'(p + a)^—Sa(p + a)Sp'(p + a) , -^-u i — '—^ ^—= \o. = ± {same with —a}, l[p + aY or S.aV (-^)U{p + a) = ± {same with —a}, i.e. SadU{p + a) = ± SadU{p — a), Sa { U{p + a) + C/(p — a)} = const., or cos /.OAP + cos LOAF — const., the property referred to. If the poles be unequal, one of the terms to the left must be multiplied by the ratio of their strengths. 449.] If the vector of any point be denoted by p = ix-\-jy^'kz, (1) there are many physically interesting and important transformations depending upon the effects of the quaternion operator „ . d . d J d ,^. ^ = 'r.^'a,+'ii ('' on various functions of p. When the function of p is a scalar, the effect of V is to give the vector of most rapid increase. Its effect on a vector function is indicated briefly in § 369. 450.] We commence with one or two simple examples, which are not only interesting, but very useful in transformations. Vp= (i-^ +&c.)(y> + &c.) =— 3, (3) v.. = (4 +.0.) (.«-..^+.^)^= ^-|±^ = A = ^„ (^) V{Tpf = n{Tpf-'^VTp = n{TpY-^p; (5) 1 np {fpf'^~m - 1 /ip / \i and, of course, ^ 7^:^ =— rmAn+2 '' (^) I p Up , . whence, V ^^ ^~T^"^~T^' ^> and, of course, V^--- = — V;^ = (6)i Ip Ip" Also, Vp=-3 = TpV Up + VTp . Up = TpV Up- 1, •:• ^^^=-W ^" 453-] PHYSICAL APPLICATIONS. 261 451.] By the help of the above results, of which (6) is especially useful (though obvious on other grounds), and (4) and (7) very remarkable, we may easily find the effect of V upon more complex functions. Thus, V Sap = —V {ax -^ kc.) = —a, (1) VJ'ap = —Vrpa =—V{pa—Sap) = 3a — a = 2a (2) Hence ^ ^o.p 2a SpFap _ 2ap' + 3pFap _ ap'^ — 3pSap T^'^W~^W~~ Tp^ ^ Tp^ ^^^ Hence (-, -^ J ap p- Sahp — SSapSpbp Sahp SSapSpbp ^ Sap ^■^P^jy= ip =-W~~T^=W''^'^ This is a very useful transformation in various physical applica- tions. By (6) it can be put in the sometimes more convenient form S.bpV^ = bS.aVp^ (5) And it is worthy of remark that, as may easily be seen, —S may be put for F in the left-hand member of the equation. 452.] We have also Vr.(3py = V{ jiSyp - pSfiy + ySjip} = - y/3 + 3/S/3y - /3y = S^y. ( 1 ) Hence, if (p be any linear and vector function of the form <^p = a + ^r.^py + 7np, (2) i.e. a self-conjugate function with a constant vector added, then V(l)p = '2.Si3y—3))i = scalar (3) Hence, an integral of V(T = scalar constant, is or ■= , we have (^V = T — iVSra. Hence ^7 = 2 (^ + ^') ^ + 2 (<)^ ~ 0') ^ = t--[^(tV)o-+ VxSro-]-- r.TFVcr. The first three terms form a self-conjugate linear and vector func- tion of r, which we may denote for a moment by vst. Hence or, omitting f^ as above, ^ 2 Hence the deformation may be decomposed into — ( 1 ) the pure strain OT, (2) the rotation t -r^„ - V Vcr. 2 Thus the vector-axis of rotation of the group is 266 QUATERNIONS. [46, If we were content to avail ourselves of the ordinary results of Cartesian investigations, we mig-ht at once have reached this con- clusion by noticing that and remembering as in (§362) the formulae of Stokes and Helmholtz. 463.] In the same way, as di dr] dC ova = — y J- , ax cii/ clz we recognise the cubical compression of the group of points considered. It would be easy to give this a more strictly quaternionic form by employing the definition of § 458. But_, working with quaternions, we ought to obtain all our results by their help alone ; so that we proceed to prove the above result by finding the volume of the ellipsoid into which an originally spherical group of points has been distorted in time t. For this purpose, we refer again to the equation of deformation and form the cubic in according to Hamilton's exquisite process. We easily obtain, remembering that f^ is to be neglected"^, = <\>^-{2,-tSVa)i\>'^ + {2, — n8V(T)<\> — {\—mVa\ or = ((^-1)2 ((/)-! + ^.9V(r). The roots of this equation are the ratios of the diameters of the ellipsoid whose directions are unchanged to that of the sphere. Hence the volume is increased by the factor from which the truth of the preceding statement is manifest. * Thus, in Hamilton's notation, \, //, v being any three non-coplanar vectors, and m, rWi , mi the coefficients of the cubic, — mS Kfiv = S.^'X^'iKp'v = S.(\-tvS\a) (fj.-tVSn(7) (v-tVSvff) = S.{\-tVS\a){ynv-tViJ.VSvcT + tVi'VSn(r) = S.\iJ.v-tlS.fivVSKaJrS.v\VSfj.(r + 8.\nVSv(y] = S.\nv — tS.l\S.ixvV + fiS.v\V + i'S.\nV'\a = S.\/xv-tS.\fJtvSV(T. miS.\nv = S.\(p'ix(p'v + S.fi'ij. = S.Knv — tS.\fjiVSv(r + &c. = ZS.\ixv-tSV(TS.KiJi.v 465.] PHYSICAL APPLICATIONS. 207 464.] As the process in last section depends essentially on the use of a non-conjugate vector function, with which the reader is less likely to be acquainted than with the more usually employed forms, I add another investigation. Let ny = (t)T = T — fS{TV)o: Then r = 0- V = ^ + iS (ctV) a. Hence since if, before distortion, the group formed a sphere of radius 1, we have Tt = I, the equation of the ellipsoid is or •57^ + 2 tSr^VS^^ya = — 1 . This may be written -S.CTXOT = xS.CT (ot + tVS-sra + tS (otV) (t) = — \, where \ is now self-conjugate. Hamilton has shewn that the reciprocal of the product of the squares of the semiaxes is whatever rectangular system of unit-vectors is denoted by i,j, k. Substituting the value of \, we have — S.{i-irtVSi(T+tS{iSI)(T) 0' + &c.) {k-\-k(i.) = — S.(i + IVSia + tS {iV) a) {i + 2 tiSVa — tS (iV) a — iVSia) = 1 +2tSVa: The ratio of volumes of the ellipsoid and sphere is therefore, as before, 1 , = l-tSVa-. VI + 2tSV(T 465.] In what follows we have constantly to deal with integrals extended over a closed surface, compared with others taken through the space enclosed by such a surface ; or with integrals over a limited surface^ compared with others taken round its bounding curve. The notation employed is as follows. If Q per unit of length, of surface, or of volume, at the point p, Q being any qua- ternion, be the quantity to be summed, these sums will be denoted by //Qds and J//Qck, when comparing integrals over a closed surface with others through the enclosed space ; and by //Qds and /QTdp, when comparing integrals over an unclosed surface with others round its boundary. No ambiguity is likely to arise from the double use of 2G8 QUATERNIONS. [466. for its meaning- in any case will be obvious from the integral with which it is compared. 466. J We have just shewn that^ if a be the vector displacement of a point originally situated at then S.Va expresses the increase of density of aggregation of the points of the system caused by the displacement. 467.] Suppose, now, space to be uniformly filled with points, and a closed surface 2 to be drawn, through which the points can freely move when displaced. Then it is clear that the increase of number of points within the space 2, caused by a displacement, may be obtained by either of two processes — by taking account of the increase of density at all points within 2, or by estimating the excess of those which pass inwards through the surface over those which pass outwards. These are the principles usually employed (for a mere element of volume) in forming the so-called ^Equation of Continuity.' Let V be the normal to 2 at the point p, drawn outwards, then we have at once (by equating the two different expressions of the same quantity above explained) the equation ///S.Vads =//S.aUvds, which is our fundamental equation so long as we deal with triple integrals. 468.] As a first and very simple example of its use, suppose o- to represent the vector force exerted upon a unit particle at p (of ordinary matter, electricity, or magnetism) by any distribution of attracting matter, electricity, or magnetism partly outside, partly inside 2. Then, if P be the potential at p, and if r be the density of the attracting matter, &c., at p, Vo- =V^P = 4-n-r by Poisson's extension of Laplace's equation. Substituting in the fundamental equation, we have 4:7T///rds= 4:7x31 =//S.VPUvds, where Jf denotes the whole quantity of matter, &c., inside 2. This is a well-kno^vn theorem. 469.] Let P and P^ be any scalar functions of p, we can of course find the distribution of matter, &c., requisite to make either of them 47I-] PHYSICAL Ari'LICATIONS. 269 the potential at p ; for, if the neecssaiy densities be r and /\ re- spectively, we have as before Now V (P VPi) = VP VPj + PV^-P^, Hence, if in the above formula we put CT=PVP,, we obtain JffS.VPVP.ds = -fffPV'P,, we have (by the process of § 4 77, above) /Pdr=/{P,-S{rV)P,)dr = -/S.TVP^.dT. But r.Vr.rdr = drS.rV-rS.dTV, and dirSrV) = drS.TV + t S.drV . These give fPdT = -\{TSTV-r. Frdr V) Po = ds r. UvVPq, Hence, for a closed curve of any form, we have fPdp=f/dsr.Uvvp, from which the theorems of §§ 477, 479 may easily be deduced. 481.] Commencing afresh with the fundamental integral fffSV' J Ti Vpdp where dp is an element of a closed circuit, and the integration extends round the circuit. This may be written -jr.^dpv)\, so that its value as a surface integral is jjs ( UvV) V - ds -fJuvV'^ - ds. Of this the last term vanishes, unless the origin is in, or infinitely near to, the surface over which the double integration extends. The value of the first term is seen (by what precedes) to be the vector-force due to uniform normal magnetisation of the same surface. 2 485.] Also, since VUp = — -^ we obtain at once Tp 'iirpp=ii'-^^''^''^ Tp whence, by differentiation, or by putting p + a for p, and expanding in ascending powers of Ta (both of which tacitly assume that the origin is external to the space integrated through, i. e., that Tp nowhere vanishes), we have ^Uvds ^im=iY-^^F^-=\i!' Tp'' ~JJ Tp JJ Tp ' and this, again, involves 486.] The interpretation of these, and of more complex formulae of a similar kind, leads to many curious theorems in attraction and in potentials. Thus, from (1) of § 481, we have 488.] PHYSICAL APrUCATIONS. 277 which gives the attraction of a mass of density t in terms of the potentials of vohime distributions and surface distributions. Putting (T = ifi+Jh + ^h' this becomes JJJ ~Yp JJJ Tp' ~JJ Tp ' By putting (t = p, and taking the scalar, we recover a formula given above ; and by taking the vector we have r/yUvUpds = 0. This may be easily verified from the formula /Pdp =jy/Uv.VPds, by remembering that VTp ^= Up. Again if, in the fundamental integral, we put (J ■=. tip, .-e have flT-f^''--\[lTi = jj 'S.V.U^,s. 487.] As another application, let us consider briefly the Stress- function in an elastic solid. At any point of a strained body let A be the vector stress per unit of area perpendicular to i, p. and v the same for planes per- pendicular to j and Jc respectively. Then, by considering an indefinitely small tetrahedron, we have for the stress per unit of area perpendicular to a unit-vector oj the expression \Si(o + p.Sj(o + vSk(a =— ^co, so that the stress across any plane is represented by a linear and vector function of the unit normal to the plane. But if we consider the equilibrium, as regards rotation, of an infinitely small parallelepiped whose edges are parallel to i, j, k respectively, we have (supposing there are no molecular couples) r^'A+yV + zi-i') = 0, or 2Fi{Uv)rk+///r.pVHds = 0. This is easily done as follows : (1) gives S.V(p(T + S.aVu = 0, if, and only if, a satisfy the condition S.(}){V)(T = 0. Now this condition is satisfied if or = Kap where a is any constant vector. For S. ^«P + S.apVu) = 0, or f/dsS.ap(f) Uv -^-fffds S.apVu = . Multiplying by a, and adding the results obtained by making a in succession each of three rectangular vectors, we obtain the required equation. 490,] Suppose a to be the displacement of a point originally at p, then the work done by the stress on any simply connected portion of the solid is obviousl}^ jr=//S.(l>{Uv)<7d.s, 49I-] PHYSICAL APPLICATIONS. 270 because (^{Uv) is the vector force overcome per unit of area on the element (Is. This is easily transformed to w =f/fS.V(i)(T(h. 491.] In this case obviously the strain-function is Now if the strain be a mere rotation, in which case S.yvT^^T — xS.-saT = 0, wliatever bo the vectors ot and t, no work is done by the stress. Hence the expression for the work done by the stress must vanish if these conditions are fulfilled. Again, it is easily seen that when the strain is infinitely small the work must be a homogeneous function of the second degree of these critical quantities; for, if it exist, it is essentially positive. Hence, even when finite, the work on unit-volume may be ex- pressed as w = 2 . («s. xexe' - S^^l {S- xnxn - Smi), where e, € , r], r{, which are in general functions of a, become con- stant vectors if the stress is indefinitely small. When this is the case it is easy to see that, whatever be the number of terms under S, w involves twenty-one separate and independent constants only ; viz. the coefiicients of the homogeneous products of the second order of the six values of form for the values i, j, ^ of tn or r. Supposing the strain to be indefinitely small, we have for the variation of ^o, the expression hw = I.{S.b)(^(x^' + S.bxi\€){S.xr]XV—Srjri') + 2 ('^•X^X^' - '^"') {S.bxvxi + ^-^xixv)- Now, by the first equation, we have bx^ = — S(zT'V)b(T. Hence, writing the result for one of the factors only, the variation of the whole work done by straining a mass is 8 TF = bffjwch =fffbw ds = -^///ds{S.xrjxv-SW) {S.xe'S.{,V)bcT + S.x^S{eV)bcT}. Now, if we have at the limits ba = 0, i.e. if the surface of the mass is altered in a ffiven way, we have obviously, ///dsS.VTS{(V)ha = -///diS.b) F{a) =f[p + A) F{a) = F{a + V)f{p) ; of which a particular case is e^/i^)F{^) =/(.. + ±)Fij/) = F(s^+ !-)/(..). The modifications which the general expression undergoes, when yand i'^are not commutative, are easily seen. If one of these be an inverse function, such as_, for instance, may occur in the solution of a linear differential equation, these theorems of course do not give the arbitrary part of the integral, but they often materially aid in the determination of the rest. Other theorems, involving operators such as e^p'^, e*'-"''^, &c., &c. are easily deduced, and all have numerous applications. 493.] But there are among them results which appear startling from the excessively free use made of the separation of symbols. Of these one is quite sufficient to shew their general nature. Let P be any scalar function of p. It is required to find the difference between the value of P at p, and its Mea7i value throughout 494-] PHYSICAL APPLICATIONS. 281 a very small sphere, of radius r and volume ?;, which has the ex- tremity of p as centre. From what is said above, it is easy to see that we have the fol- lowing expression for the required result : — l///(,-»,v_,)p,,,, where o- is the vector joining- the centre of the sphere with the ele- ment of volume ch, and the integration (which relates to o- and th alone) extends through the whole volume of the sphere. Expanding the exponential, we may write this expression in the form higher terms being omitted on account of the smallness of r, the limit of Tcr. Now, symmetry shews at once that ///ack = 0. Also, whatever constant vector be denoted by a, ///{Sacrych = -aV//{S'~F" + (00' + ^lf

'^^°^^)' whence p = cos qi (cos qt + y sin qt) (3~ ^ (which may be verified at once by multiplication). Finally, taking the origin so that the constant of integration may vanish, we have 2p/3 = t+ — {sin 2 qt — y cos 2 ql), 2q which is obviously the equation of a cycloid referred to its vertex. The tangent at the vertex is parallel to /3, and the axis of symmetry to a. 503.] In the case of a chain hanging under the action of given forces Q = Pr, where F is the potential, r the mass of unit-length. Here we have also, of course, /Tdp = I, the length of the chain being given. It is easy to see that this leads, by the usual methods, to the equation ^ {{Pr + ?()p'} - rVP — 0, where n is a scalar multiplier. 504.] As a simple case, suppose the chain to be uniform. Then r may be merged in w. Suppose farther that gravity is the only force, then P = Sap, VP = —a, and -J- {{Sap + 7()p}+a = 0. Differentiating, and operating by Sp', we find S.pj^p'(^Sap+~^)+a^ = 0; - which shews that u is constant, and may therefore be allowed for by change of origin. 505.] PHYSICAL APPLICATIONS. 287 The curve lies obviously in a plane parallel to a, and its equation is {Sap)'^ + a' *- = const., which is a well-known form of the equation of the catenary. AVhen the quantity Q of § 496 is a vector or a quaternion, we have simply an equation like that there given for each of the con- stituents. 505.] Suppose F and the constituents of a to be functions which vanish at the bounding surface of a simply-connected space 2, or such at least that either F or the constituents vanish there, the others (or other) not becoming" infinite. Then, by § 467, ///dsS.V (Per) = f/ds FSa Uv = 0, if the integrals be taken through and over S. Thus ///dsS.aVF = -///ds FS.Va. By the help of this expression we may easily prove a very re- markable proposition of Thomson {Cam. and Buh. Math. Journal.^ Jan. 1848, or Repint of Fajoers on Electrostatics, § 206.) To sheio that there is one, and but one, solution of the equation SSIi/'Vii)- 4:71 r ichere r vanishes at an infinite distance, and e is any real scalar what- ever, continuous or discontinuous. Let V be the potential of a distribution of density r, so that VH = 4 7rr, and consider the integral That Q may be a minimum as depending on the value of u (which is obviously possible since it cannot be negative, and since it may have any positive value, however large, if only greater than this minimum), we must have = \hQ=-fffd^S.{e^Vu-Vv)Vhu = .///^^s S?« ^-"^ (e'-Vu-Vv), by the lemma given above, =fffdsbu {S.V {e^Vn)-iTTr}. Thus any value of u which satisfies the given equation is such as to make Q a minimum. But there is only one value of n which makes Q a minimum ; for, let Qi be the value of Q when ?/^ = U + (p is substituted for this value of u, and we have 288 QUATERNIONS. [505. Q^= -jjj ch{eV{u + 4>)--^Vv) The middle term of this expression may^ by the proposition at the beginning of this section, be written and therefore vanishes. The last term is essentially positive. Thus if ?^i anywhere differ from ti (except, of course, by a constant quan- tity) it cannot make Q a minimum ; and therefore 2* is a unique solution MISCELLANEOUS EXAMPLES. 1. The expression Fa^ Vyh + Fay FbjS + Fab F(3y denotes a vector. What vector ? 2. If two surfaces intersect along a common line of curvature, they meet at a constant angle, 3. By the help of the quaternion formulae of rotation, translate into a new form the solution (given in § 234) of the problem of inscribing in a sphere a closed polygon the directions of whose sides are given. 4. Express, in terms of the masses, and geocentric vectors of the sun and moon, the sun's vector disturbing force on the moon, and expand it to terms of the second order; pointing out the mag- nitudes and directions of the separate components. (Hamilton, Lectures, p. 615.) 5. If ^ = y^, shew that 2dq = 2dri = ^{dr + Kqdrq-'^)Sq-^ = \{dr + q-'^drKq)Sq-^ = {drq + Kqdr)q-'^{q-\-Kq)-^ = {drq + Kqdr){r + Tr)-'^ _ dr+ TJq-^drUq-^ _ drUq + JJq-^dr _ q-'^ {Uqdr + drlJq-'^) ~ Tq{Uq-^Uq-^) ~ q{Uq+Uq-^) Uq+Uq'^ _ q~^{qdr + Trdrq-'^) _ drUq+ Uq-'^dr _ dr Kq-^ ■\- q-'^ dr ~ Tq{Uq+Uq-^) ~ Tq{l + Ur) ~ 1+ Ur MISCELLANEOUS EXAJIPLES. 280 2rlq = ^ ,h- + r. Vdr ^ ^ I ry-i = I (h - r. Fdr -^ ^- 1 L-i /Ir „ ^^dr V dr „ -.ulr V . q q S ^ q q S ^ = drq-^ + r. Fq-^ Vdr (l + -^^-i) : and g-ive geometrical interpretations of these varied expressions for the same quantity. [Ihid. p. 628.) 6. Shew that the equation of motion of a homogeneous solid of revolution about a point in its axis, which is not its centre of gravity, is BTpp-A^p = Fpy, where X2 is a constant. {Trans. R. S. E., 1869.) 7. Integrate the differential equations : («.) ^ + «? = «, where a and b are given quaternions, and (f) and xj/ given linear and vector functions. (Tait, Proc. R.S.K, 1870-1.) 8. Derive (4) of § 92 directly from (3) of § 91. 9. Find the successive values of the continued fraction where i and J have their quaternion significations, and x has the values 1, 2, 3, &c, (Hamilton, Lectures, p. 645.) 10. If we have u^. = f^:—) c, where c is a given quaternion, find the successive values. For what values of c does u become constant ? {Ibid. p. 652.) 1 1 . Prove that the moment of hydrostatic pressures on the faces of any polyhedron is zero, (a.) when the fluid pressure is the same throughout, {b.) when it is due to any set of forces which have a potential. 12. What vector is given, in terms of two known vectors, by the relation p-^ = H«~' + /3"')? Shew that the origin lies on the circle which passes through the extremities of these three vectors. u 290 QUATERNIONS. 13. Tait, Trans, and P roc R.S.E., 1870-3. With the notation of §§ 467, 477, prove (a.) ///S{aV)Tds =//TSaUvds. {h.) 1^ S{pV)r=:-nT, ^n + Z)fffTds = -f/rSp Uvds. {c.) With the additional restriction V^r = 0, /fS.Uv{2np-\-{n + 2,)p^V).Tds = 0. {d.) Express the value of the last integral over a non- elosed surface by a line-integral. (g.) -fTdp=/fdsS.UvV = {L—Kfp+2l.SKp+2KSip. {h.) The last equation gives also, (^2_^2^2^^ (tVK2)/^' + 2npK. MTSCELLANEOrS EXAMPLES. 203 (/'.) With the same signification oC ;-, the differential equations of the elh'psoid and its reciprocal become Svdp = 0, S^xh' = 0. {j.) Eliminate p between the four scalar equations, Sop = f/, Sl^p = h, Syp = c, Sep = e. 20, Hamilton, Bishop Laio's Prem'wm Examination, 18G4. {a.) Let ^1^1, ^2-^2' ••■ ^^«-^n ^^^ ^"y given system of posited rig-ht lines, the 2n points being- all given; and let their vector sum^ AB = A,B, + A,B,+ ...^A,,B,„ be a line which does not vanish. Then a point //, and a scalar //, can be determined, which shall satisfy the quaternion equation, HA,.A,B,+ ...+ILl,,.A,,B,, = LAB; namely by assuming any origin 0, and writing, Qj^ ^ jr OA,.A,B,+ ... + OA,,.A,,B,, ^ A^B^ + ...+A„B,, ^^^^OA,.AB,+ ...^ A,B,+ ... ijj.) For any assumed point C, let q, = CA^.A,B,+ ...^CA,,.A,B,r, then this quaternion sum may be transformed as follows, Qc = q,i + CH.AB = {h + CH).AB ■ and therefore its tensor is TQc = {h.^ + CB^)KAB, in which AB and CH denote lengths. {c.) The least value of this tensor TQc is obtained by placing the point C at //; if then a quaternion be said to be a minimum when its tensor is such, we may write min, Qc = Qif= h.AB; so that this minimum of Qc is a vector. {J.) The equation TQc = c = any scalar constant > TQu expresses that the locus of the vai'iable point C is a spheric surface, with its centre at the fixed point H, and with a radius r, or CH, such that r.AB = {TQc--TQ,r-)'^ = {c^ - /i' . AB'")"^ ; 204 QUATERNIONS. so that //, as being- thus the common centre of a series of concentric spheres, determined by the g-iven system of right lines, may be said to be the Central Point, or simply the Centre, of that system. (e.) The equation TFQc = c-^ = any scalar constant > TQfj represents a right cylinder^ of which the radius divided by AB, and of which the axis of revolution is the line, fQc = q„ = h.AB; wherefore this last right line, as being the common axis of a series of such right cylinders, may be called the Central Axis of the system. (/.) The equation SQc = C, = any scalar constant represents a plane ; and all such planes are parallel to the Central Plane, of which the equation is sqc=o. {(/.) Prove that the central axis intersects the central plane perpendicularly, in the central point of the system. (//.) AMien the n given vectors A^B^, ... A^^B,^ are parallel, and are therefore proportional to n scalars, b^,...b,^, the scalar /i and the vector Qn vanish ; and the centre // is then determined by the equation 6^.RA^ + L,.EA.,+ ...+b,.HA, = 0, or by the expression, ^^^^ b,.OA^+...-\b,.OA„ ^ b^+...+b„ where is again an arbitrary origin. 21. Hamilton, BisJioj) Law^s Premium Examination, 18G0. (c^) The normal at the end of the variable vector p, to the surface of revolution of the sixth dimension, which is represented by the equation (p2_a^)3 = 27a2(p-a)^ ,..(a) or by the system of the two equations,, p-' — a- = 3t-a-, [p — a)- = t^a", (a j MISCELLANEOUS EXAMPLES. 295 and the tangent to the meridian at that point, are respectively parallel to the two vectors, V = 2{p — a) — tp, and T=2{l-2i){p-a) + i-'p; so that they intersect the axis a, in points of which the vectors are, respectively, JZTt' and ^^37)2172 • (b ) If dp he in the same meridian plane as p, then t(l-f){i-f)dp=3T(H, and ^^ = ^-. (r.) Under the same condition, {d.) The vector of the centre of curvature of the meridian, at the end of the vector p, is, therefore, dv\-'^ 3 V 6a-(4-/)p . dv.-^ 3 I. _ "^p-^'^^dp) =P-2Y^t- 2(1-0 (e.) The expressions in Example 38 give 9 ^ - 9a^l ;.9 hence {a-pf = - a2^^ and dp- = j— ^ ^//^^ ; the radius of curvature of the meridian is, therefore, E = T{a-p):=ltTa; and the length of an element of arc of that curve is ds = Tdp=3Ta(j--^ydL if.) The same expressions give ^(^rapf=-aH\\-t)\4-t)', thus the auxiliary scalar t is confined between the limits and 4, and we may write t = 2 vers^, where 6 is a real angle, which varies continuously from to 27r ; the recent expression for the element of arc becomes, there- fore, ds=^Ta.tde, and gives by integration s = 6Ta{e-sin6), if the arc s be measured from the point, say F, for which P = a, and which is common to all the meridians ; and the total periphery of any one such curve is = 127rra. 296 QUATERNIONS. {(J.) 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