A MANUAL OF QUATERNIONS. A MANUAL OF QUATERNIONS BY CHARLES JASPER JOLY MA. D.Sc. F.R.S. Sec.R.IA. FELLOW OF TRINITY COLLEGE, DUBLIN" ANDREWS' PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF DUBLIN AND ROYAL ASTRONOMER OF IRELAND ^ OF THE UNIVERSITY $0ub0n MACMILLAN AND CO., Limited NEW YORK : THE MACMILLAN COMPANY 1905 1A1 OLASGOW: PRINTED AT THK UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CO. LTD. PREFACE. Readers of the Life of Sir William Rowan Hamilton will recollect that he undertook the publication of a book on quaternions to serve as an introduction to his great volume of Lectures. This Manual of Quaternions was intended to occupy about 400 pages, but while the printing slowly pro- gressed it grew to such a size that it came to be regarded by its author as a " book of reference " rather than as a text-book, and the title was accordingly changed to The Elements of Quateimions. By a curious series of events one of Hamilton's successors at the Observatory of Trinity College has felt himself obliged to endeavour to carry out to the best of his ability Hamilton's original intention. And on the centenary of Hamilton's birth a Manual of Quaternions is offered to the mathematical world. Last year I was called upon by the Board of Trinity College to assist in the examination for Fellowship. I had long ago recognized that another work on quaternions was required, and this want was forciblj^ brought home to me by my new duties. A mathematician, whose time is limited, is frightened at the magnitude of Hamilton's bulky tomes, although a closer acquaintance with the Elements would reveal the admirable lucidity and the logical completeness of that wonderful book, and although the Lectures have a charm all their own. The student wants to attain, by the shortest and simplest route, to a working knowledge of the calculus; he cannot be expected to undertake the study of quaternions in the hope of being rewarded by the beauty of the ideas and by the elegance of the analysis. And for his sake, though with reluctance I must confess, I have abandoned Hamilton's methods of establishing the laws of quaternions. 158427 vi PREFACE. By a brilliant flash of genius Hamilton extended to vectors Euclid's conception of ratio. A quaternion is the mutual relation of two directed magnitudes with respect to quantity and direction as a ratio is the mutual relation of two undirected magnitudes with respect to quantity. From this enlarged view of a ratio, the calculus of quaternions is deve- loped in the Elements. But the way is long and winding, and after much labour, I found I could not greatly shorten it or make it much less indirect. I therefore adopted another plan. The two cardinal functions of two vectors are Sa/3 and Ya/3. These functions may be defined by the statements that —Sa^ is the product of the length of one vector into the projection of the other upon it, and that Ya^ is the vector which is perpendicular to a and to ' /3, and which contains as many units of length as there are units of area in the parallelogram determined by a and /3. Both these functions enjoy some of the properties of an algebraic product. They are distributive with respect to each of the vectors. I The product of the vector a into /S may be defined to be J the sum of these functions, This is a quaternion — the sum of a scalar and a vector. A product of a pair of vectors is distributive but not commuta- tive. It is now necessary to define the product of a quaternion iq) into a vector (y), and we say that it is the sum of the product of the scalar (Sq) into y and the product of the vector (Vq) into y, or that g.y = Sg.y-hVg'.y. From these principles it follows almost immediately that quater- nion multiplication is associative as well as distributive. Division is seen to be deducible from multiplication, and on p. 12 we arrive at the important result that every function of quaternions formed by ordinary algebraic processes is a quaternion, scalars and vectors being considered to be special cases. What we may call the grammar of the subject may be said to terminate on p. 20, the laws of combination of quaternions having been established, the five special symbols S, V, K, T and U I PREFACE. vii having been defined and tlieir chief properties explained, various constructions for products and quotients having been made, and the non-commutative property of multiplication having been illustrated by conical rotations and otherwise. In the succeeding chapters, I have not scrupled to introduce, either in the articles in small type or in the worked examples in small type, illustrations of the applications of quaternions to subjects that can hardly be supposed to be familiar to the beginner in mathematics. It is suggested in the table of con- tents that these more difficult portions should be omitted by a beginner at tirst reading. The book is, however, primarily intended for those who commence the study of quaternions with a fair knowledge of other branches of mathematics; in other words, it is written for the majority of those at present likely to read quaternions because, as yet, the subject is not generally taught in elementary classes. On the other hand, I have abstained from printing examples of an artificial nature, and I have avoided unnecessary difficulties. Although this book may be regarded as introductory to the works of Hamilton, it may also to some extent be considered as supplementing them. Many of the results contained in it have appeared only in the publications of learned societies, and many others are believed to be novel. It is possible, therefore, that this volume may be found to have some points of interest for the advanced student of quaternions. He will find, for example, that quaternions lend themselves to the treatment of projective geometry quite as readily as to investi- gations in mathematical physics and in metrical geometry. By means of a somewhat elaborate table of contents, modelled on those prefixed by Hamilton to his Lectures and Elements, and by the aid of a full index and numerous cross references, I trust that the contents of this book will be found to be fairly accessible to the casual reader as well as to the systematic student. It must be remembered, however, that the objects of a work of this nature are to introduce a subject of the highest educational value, and to develop a powerful and comprehen- sive calculus. Such ends can be attained only by illustration and by suggestion, and it is not easy to tabulate methods of investigation. viii PREFACE. . It would be impossible to overestimate what I owe to Hamilton's Lectures on Quaternions (Dublin, 1853) and to his Elements of Quaternions (London, 1866, 2nd edition, in two volumes, with notes and appendices by C. J. Joly, London, 1899, 1901). The admirable Elementary Treatise on Qua- ternions (3rd edition, Cambridge, 1890), by the late Professor P. G. Tait — who has done so much for quaternions by his classical applications of Hamilton's operator V — has also been very useful. Other writers to whom I am indebted are referred to in the text.* I am glad to have this opportunity of offering my thanks to my respected friend, Benjamin Williamson^ Esq., F.R.S., Senior Fellow of Trinity College, Dublin, for his great kindness in assisting me with a considerable portion of the proofs. I am also indebted to him for the uninterrupted encouragement he has given me, alike privately and in his official capacity as a member of the governing body of Trinity College, in my attempts to render Hamilton's work more widely known. CHAELES JASPER JOLY. The Observatory, DuNsiNK, Co. Dublin, 1st Jan., 1905. *The Bibliography by Dr. Macfarlane, published by the International Association for the promotion of the Stud}' of Quaternions and Allied Systems of Mathematics {Dublin, 1904), renders unnecessary any detailed list of works on quaternions. CONTENTS. CHAPTER I. THE ADDITION AND SUBTRACTION OF VECTORS. RT. PAGE 1. Definition of a vector. I 2. Sum of two vectors. Commutative property. . - - - 1 3. Addition of vectors is associative and commutative. - - - 2 4. Null sums of vectors. _.- 2 5. Use of the sign -. - 2 6. Multiplication of vectors by scalar coefficients. - - - - 3 7. Unit of length. Tensor and versor of a vector, - - - - 4 a=Ta.Ua. 8. Resolution of a vector along three given vectors. . _ _ 4 CHAPTER H. MULTIPLICATION AND DIVISION OF VECTORS AND QUATERNIONS. 9. Definition of the scalar function SafS of two vectors. - - - 6 The doubly distributive property, S2a2/3 = 22Sa/?. 10. Definition of Ya/S. The doubly distributive property, - - 7 V2al^^ = 22Va^. 11. The product of two vectors defined to be - - -^ - - 8 a^ = Sa^ + Vtt^ (b) The doubly distributive property, "" ^ 2a2/?=22a^. The relations, af3 + ^a = 2Sa(3y a(3- f3o = 2Ya(3. X CONTENTS. ART, PAOK 12. A quaternion is defined to be the sum of a scalar and a vector, - 9 q = Sq + Yq; and is expressed as a product of a pair of vectors, q = af3. Addition of quaternions. 13. The product of a quaternion and a vector is defined to be dis- 9 tributive with respect to the scalar and vector of the quaternion, or qy = {Sq + Yq)y = Sq.y + 'Vq.y ; and products of vectors are shown to be distributive and to be interpretable by formula (b), 14. Laws of three mutually rectanglar unit vectors, i, j, k, deduced 10 from formula (b). Multiplication of i, j, k is associative. Hamilton's formula, ii=f=]c^=ijk= - 1 (a) 15. Multiplication of quaternions is associative. - - - - 11 16. Division reduced to multiplication. Eeciprocal of a vector, - 11 _j _ 1 _ a ^ ~a-""Ta2' ^-i = a-iy-i/3-ia-i if Q = afSy8. Every function of quaternions is a quaternion. 17. The conjugate, tensor, versor, angle, plane and axis of a qua- 12 ternion, Kq=^Sq-yq, qKq = {Tqy==Kq.q, q = Tq.Uq, Sq = Tq COS Lq, TYq = Tq^inLq. Example giving functions for the quadrinomial form, q = to + ix +jy + kz. 18. A quaternion expressed as a quotient of vectors, - - - 13 OB q = — , Lq = LAOB, ^ oa' ^ The ratio of two vectors, Hamilton's extension of Euclid's con- ception. 19. A quaternion as an operator. Eiffect of a vector, of a versor. - 14 20. Construction for the product of two quaternions. - - • 14 The relation K(r^) = K^K/'. CONTENTS. xi ART. PAGE 21. Spherical representation of multiplication. Vector-arcs. - - 16 Commutative quaternions are coplanar. Square of right versor is negative unity. Conical rotation, qpq-\ 22. The laws of combination of the five symbols, - - - - 19 S, V, K, T and U. Remarks on notation. Biquaternions. Examples to Chap. II. - - 21 I CHAPTER III. FORMULAE AND INTERPRETATIONS DEPENDING ON PRODUCTS OF VECTORS. 23. Vector as directed area. Moment of force. - - - - 23 24. Volume of parallelepiped = iSa^y. ------ 23 25. The formula, 24 V.aV^y = ySa^-/3S7a. 26. Resolution of a vector along three lines, 24 pSa/3y = aS/Syp + fSSyap + ySaf^p, pSa/3y = V^y Sap + YyaS/Bp + Va^Syp. 27. Resolution of a vector along and perpendicular to a given vector, 25 p = A . A- V = AS A- V + A VA- ip. 28. T((3-ay=TI^' + 2Saf3 + Ta^ - - - 25 Fundamental formulae of a plane triangle. Examples to Chap. III. 26 CHAPTER IV. APPLICATIONS TO PLANE AND SPHERICAL TRIGONOMETRY.^ 29. Coplanar versors. De Moivre's theorem, - - - - - 27 •2a U^' = cos A + / sin J. = a ' . Quaternion as a power of a vector q = a*. Roots of a quaternion. Exponential, power and logarithm of a -. '^ quaternion. *The beginner may pass at once to Chapter V, xii CONTENTS. ART, PAGK 30. Spherical trigonometry. The fundamental trigonometrical rela- 29 tions for a spherical triangle deduced from the identity 31. The quaternion f3a~^y and its interpretation. The area of a 31 spherical triangle is twice the angle of the quaternion Examples to Chapter IV., 33 CHAPTER Y. GEOMETRY OF THE STRAIGHT LINE AND PLANE.* 32. Relations between a straight line and a plane. The method of 35 equations : S(/3-y)a = for a plane, V(/o- 8)^ = for a line. The method of indeterminates : /) = y + ar for a plane, p~8i- ^t for a line, T being an arbitrary vector subject to an implied condition and t being an arbitrary scalar. 33. The line through two points and a plane through two points. - 37 34. The plane through three points. Plane satisfying given con- 38 ditions, 35. Intersections of planes and conditions of intersection. - - 39 36. A pair of lines. Shortest vector perpendicular. Points of 39 closest approach. Examples, complex, congruency, ruled surface. 37. Anharmonics, 41 38. Symmetrical relations for a tetrahedron. - - - - - 42 39. Relations connecting five vectors. 43 40. Hamilton's anharmonic coordinates. ------ 43 Examples to Chapter V. - - . 45 *The last three Articles may be omitted at first reading. CONTENTS. xiii CHAPTER VI. THE SPHERE * ART. PACK 41. Equation of a sphere, intercepts on a line, tangent cone, poles 49 and polars. 42. Two spheres, radical plane, angle of intersection. - - - 50 43. Three spheres, radical axis. Four spheres, radical centre. - - 51 44. Inversion. 52 45. Examples relating to a tetrahedron and a sphere. - - - 53 46. Product of vector sides of inscribed polygon. .... 55 47. Inscription of polygons. 56 Examples to Chapter VI. - - -- - - - - 68 CHAPTER VII. ^- DIFFERENTIATION, t 48. Differentiation of a vector which involves a single variable 62 parameter. op = p = , d"P=-Si/d>, where dp, d'p, and d"p are three non-coplanar differentials of p. Introduction of Hamilton's operator V, Vd'pd"p . dP+ Yd"pdp . d'P+ Vdpd'p . d'7^ ^ „ Sdpd pd p 65. The result of operating by V on a quaternion function of p is - V.i^p = lim-[dvi^p, where dv is an outwardly directed element of vector area of any small closed surface surrounding the extremity of p and where y is the volume included by the surface. Illustrations from fluid motion. Equation of continuity. 56. For any small closed circuit at right angles to an arbitrary vector Uv, i r V(UvV) . /^p = lini -^ \ dpFp, where A is the area enclosed by the circuit and dp a directed element of length of the circuit. Condition for perfect differential, VVo- = Oif So-dp = dP. 57. Analytical expressions for V ; one being The operator V may be treated as a symbolic vector ; both when q is the operand and when it is a constant quaternion. 58. Given p = F(q), to find dq in terms of dp. Particular cases solved, the general theory being explained in Art. 150. Differential of 7^*^ root of quaternion. 59. Successive differentiation and development. Taylor's theorem for quaternions, f(q+p) =fq + T dA + f-g ^^f^ + ^^^-J where d^ ^p. /(p + ar) = e-sajv./(p). CONTENTS. XV ART. PAGB 60. Successive differentiations corresponding to different differentials 79- d^-, d'q, etc., of q, 61. Stationary value of a scalar function of p with or without 80 equations of condition. Examples to Chapter VII. 82. CHAPTER YIII. LINEAR AND VECTOR FUNCTIONS. 62. Symbolic definition of linear vector function, - - - - 88" (I>(a + (3) = (t>a + ct>f3, S', (fip — X'SXp + fjbSixp + v'Svp, (f>'p = XSX.'p+iJLSixp + vSv'p. 63. Considered geometrically the equation - . . - , - 90 (T = p establishes a linear transformation from vectors p to vectors (T, equal vectors becoming equal vectors. 64. To pass from vectors o- to vectors p there is the inverse trans- 90 formation y3 = <^-io- or mp = \l/o; where the scalar m and the linear vector function ^ depend only on direct operations of (f)'. fVa/3 = V(^'a<^'/3, mSaf3y = S'l3cl>'y, 65. Cases of exception. The auxiliary function x? and the in- 91 variants m' and m", m" = Sa^y = S'a(fi'f3y, The symbolic relations m = (f>yfr = \jr^. 66. The symbolic cubic - - -* 9a. (f)^ — mfcfy^ 4- 7n'^ - m = 0, and the latent (scalar) cubic g^ - rti'g^ + m'g — m = 0. The axes y^, y^ and yg of (^ and the latent roots g^^ g<^ and g^. {-9i)i't>-92)p=493-9i)(9A-92)y3, {-9iM-92M -9s)p=0 if p=.^yi+.yy2+273- A^es of a function and those of its conjugate determine sup- plemental triangles on a unit sphere. Special cases, indeterminate and coincident axes. xvi CONTENTS. ART. PAGE 67. The self -con jugate part and the spin-vector e, - - - - 96 The axes of a self -con jugate function are mutually rectangular and the latent roots are real. 68. The relations 97 €, 77i' = M' — €^, m'—M". 69. Geometrical interpretation of the vanishing of invariants. - 98 70. The square root of a linear vector function. . . . . 99 71. Miscellaneous theorems relating to linear functions. - - - 100 The canonical reduction of a pair of functions. Examples to Chapter VIII. 102 CHAPTER IX. QUADRIC SURFACES. 72. Equation of the general central quadric, 106 S/)<^/o= — 1. 73. Lengths of radii. Asymptotic cone. Tangent cone. Normals. 107 74. Pole of plane; Reciprocal of quadric. 109 75. Principal axes of section. 110 76. Conjugate radii. 112 77. Cyclic planes. Hamilton's cyclic form, - - - - - 113 S/5c^/o =gp^ + 2S\pSiip = - 1. 78. Tangent right cylinders. Hamilton's focal form, - - - 115 Sp<^p = 6(S/)U^)2 + a(VpUa)2= - 1. 79. Generators of a quadric, - - - - - - - - 116 p = ± 'V . a~id)a 4- i/a. .60. Non-central quadrics. 117 ^1. Quadric cones. Sphero-conics. Hamilton's proofs of the 118 associative principle. i82. Confocal quadrics, 121 Sp{(f> + :v)-^p=-l. i83^ Tangent cones to confocals. - 123 CONTENTS. PAGE 84. The elliptic equation of confocals - - - - - - 124 /t)=V{(<^ + ^)(<^+y)(<^ + 2)}€, where 6^ = 0, Sc(^e = 0, S€<^-€= -1. Examples on surface of centres, umbilical generators, etc. Examples to Chapter IX. - - - 126 CHAPTER X. GEOMETRY OF CURVES AND SURFACES. (i) Metrical Properties of Curves. 85. The method of emanants. The vector of rotation (i) of the 131 emanant (77) at a point (p) of a curve, dUry _ (It; UryTdp %Td/)' The vector curvature of a curve, dUdp ^ dy dp dpTdp' V The vector torsion, ^ dUYdpdV _ &p " • U VdpdVTdp ~ ^ ^f"^ Ydpd^p The vector twist of a curve, 0) = vector curvature + vector torsion . 86. The unit vectors, a tangent, P principal normal, y binormal, 134 -J =c, 7 = vector curvature; ^ = a,a = vector torsion; a ^' y '^ = a^a + Cjy = vector twist, where suffixes denote differentiation with respect to the arc. Expansion of vector to point of curve in terms of arc. 87. The developables connected with a curve. General expressions 135 I for their planes, lines and cuspidal edges. (ii) Ruled Surfaces. 88. Ruled surface regarded as generated by a moving emanant line. 137 The rate of translation of the emanent is joi, where p is the pitch or parameter of distribution, Udp dpUry ^~^ I ~^ dU77* "Vector equation of line of striction, h CONTENTS. PAGE 89. Normal and tangent plane for ruled surface. The involution for 139 perpendicular tangent planes, Qc . Qc' = +jt?2, (Uqc 4- Uqc' = 0). (iii) Curvature of Surfaces. 90. Curvature of projection of curve. Vector curvature of curve 141 traced on surface resolved into component curvatures in"^and at right angles to tangent plane ; dUd/) ^ Sdvdp Ig vdUdp dp d/ovTd/o V d/o = curvature of normal section + geodesic curvature. 91. Surface represented by /(p)=co?is^., 142 d//) = 7iSvd/3, di/ = <^/o, = ^' if n is constant. In general Sv€ = 0, where € is spin- vector of ^. 92. Equation for principal curvatures, 142 Tangents to lines of curvature, Tjl(<^o-^iTv)-ii^, T,\\{cj>,-C^Tv)-^v. Curvature of normal section through dp, C=CiCo&^l + C2^u)^l if 'Udp = TiCosZ + T2sin^. Surfaces generated by normals. 93. Second method for curvature. Measure of curvature, - - 144 ^^ YdUvd^Uv ^1^2- Ydpd'p • Gauss's theorem of the linear element. 94. Kinematical method. Moving system of tangents and normal. 145 Examples on geodesies, etc. (iv) Families of Curves and Surfaces. 95. Family of curves, 148 p = r){t; a, b, c, ...). Curves touching given surface or intersecting given curve. 96. Diflferential equation of surfaces met in n consecutive points by 148 curves of the family. 97. Equation of family of surfaces, 149 /(p; a, 6, c,...)=0. Genesis of partial differential equations. 98. Analogue of Charpit's equations. - - 151 CONTENTS. xix CHAPTER XL STATICS. .RT. * PAGE 99. The vectors a being drawn to points of application of the forces jS, 1 56 2Va^ = />t = resultant moment at o, 2^ = A = resultant force ; p is pitch of resultant wrench, CT the vector perpendicular on central axis, y the vector to Hamilton's centre, 2a^ is total quaternion moment. 100. Reduction of system of forces to two forces. - - - - 158 101. Astatics. The linear function - 159 (^p = 2aS^/3. ^ For astatic equilibrium 0=0, A = 0. Arrangements of central axes relative to the forces and relative to the system of points of application. |j02. Composition of wrenches. Equation of three-system of screws. 163 The cylindroid. Resolution of wrench into components on six given screws, 103. Example on equilibrium of heavy chain] lying on a surface 166 and acted on by any forces. CHAPTER XII. FINITE DISPLACEMENTS. i04. Rotation followed by displacement. 168 Composition of successive displacements. Small displacements. Screw of the displacement. 105. Twist- velocity of body. Instantaneous screw. The general 169 equation of relative velocity. Fixed and moving axes. Acceleration. 106 Rotation depending on two parameters. 173 107. Examples on the applications of the relation. - - - - 173 W h2 XX CONTENTS. CHAPTER XIIL STRAIN. '' ART. PAGE 108. Homogeneous strain. Vectors p changed to a, where - - 177 (r=(f>p, m>0. m Strain ellipsoid T. 111. Displacement along and at right angles to p , - - - - 180 8=cr — p = p{Sp~^2P = 0. 116. Central forces, 186 P=i, ypi=o, ypp=^. Law of nature. Circular hodograph. Examples on moving J orbits, etc. ^ 117. Constrained motion. - - - 189 118. Brachistochrone. 192 CONTENTS. CHAPTER XV. DYNAMICS. ART, PAGE 119. For any number of particles, the reactions cancel in the equations 194 where M is total mass, p vector to centre of mass, Q moment of momentum at origin, f resultant force, y] resultant moment at origin. Energy equation. 120. Moment of momentum with respect to moving point and with 195 respect to centre of mass. 121. Case of rigid body. Equations of motion, - - - - 196 Energy equation. 122. Instantaneous twist- velocity produced by impulsive wrench 200 acting on free body. Energy equation. 123. Case of constrained body. Reciprocal screws. Evoked and 204 \ reduced wrenches. CHAPTER XVI. THE OPERATOR V. (i) The Associated Linear Functions. 124. The invariants and auxiliary functions for - - - - 211 cfia= — SaV . cr, ^'a = - VSao", in which o- is a vector function of p. When o- denotes the velocity of the extremity of p, the rates of change of a line-element (d/o), a surface-element (dv) and a volume-element dv are Dt.dp = (fidpy D« . dv = x'dv, Dt.dv= m"dv. The quaternion invariant m"-2€= -Vo-; m" = — SVo- = divergence^ 2e = Wo- = curl. The auxiliary function i/^, ^y = _ ^VV V'So-o-'y, ir'y = - ^Sy VV . Ycrfr' ; \- ^4 and the invariants m' - 2(/)€ = - iVVWo-o-', m = |S VVV'So-crV'. xxii CONTENTS. (ii) Integration Theorems. ART. PAGE 125. The transformations 215 fdv.q = jVq.dv, jdp.q = JY{dv .V). q. Cases (a) of discontinuity ; (b) when q is multiple-valued ; (c) when q becomes infinite ; (d) of multiply-connected region. (iii) Inverse Operations. 126. Interpretations for the functions 218 p=V~^q and r—V~^q where Vp = q and V^^r=q, deduced from the identity ^yjf Vp'-dv' y^f dv' .p' ""jATrTip'-p) ^j47rT{p'-p)' (iv) Spherical Harmonics. 127. Expansion in terms of spherical harmonics. - - . . 222 The fundamental theorems. (v) Various expressions for V. 128. Expressions for V and V^ in terms of arbitrary differentials of p. 225 Case in which p is given as a function of thi'ee parameters. Examples on systems of equipotential surfaces, etc. (vi) Kinematics of a deformahle system. 129. Eate of change of quantity q associated with point moving 228 with velocity o*, T>tq =q- ScrV . q. The relations T)lqdv) = (D,g -f m'q) . dy, D«(SCTdv) = St^dv, D.StTTd/) = Sgdp, where ^ = TIT - V V Vo-CT - o-S VCT, g = CT - VSo-CT - Vo-V V^TT. The voluminal, areal and linear equations of continuity (De-fm")g = 0, 12 = 0, g = 0. Euler's and Lagrange's methods. 130. Flow of a vector CT along a curve, and rate of change of flow, - 231 F=-jsz^dp, T>tF=-fsg^dp. Circulation of the vector for a closed curve, (7=- [strrd/3=-Jscodi/; D,(7= - Jsgdp= - Jswdv, (o = VVct. CONTENTS. xxiii ART. PAGE rFlux of a vector TS through a Surface 131. Expression for vector tJ in the form - - - - - 233 CT = VP+Vry + V/i; where \^^R = 0. Irrotational distribution if 97 = ; no divergence if P=0. Transformation relating to vortex motion. 132. Transformations effected by means of the invariant property 235 of V. jv5.div= jpSVvj . dv - [pSdvtrr = | J/oY V?rr . dv - 1 Lvdi/cr, and other analogous relations. (vii) Equations of motion of a deformable system. 133. Equations of motion for Euler's method, - - - - - 236 D«o-=^+c-i.^V, C77 + 2e = 0; and for Lagrange's method. Equations of continuity. 134. Determination of stress function ^ for viscous fluid and for 238 isotropic solid. 135. Rate of change of kinetic energy. 239 136. Dissipation function for viscous fluid. - - - - - 240 Example on motion of solid in liquid. 137. General case of elastic solid. Equation of motion, - - . 242 D,2(9=^ + c-i$V. Quaternion statement of Hooke's law, . *a = e(a, V, (9) = 0(a, 0, V). Energy function. Elastic constants. 138. Equation of vibrations of elastic solid, 247 cd=e(V, V, 6). Equation for plane wave moving with wave-velocity f, oe=e(i, i e). Three plane polarized waves propagated in direction TJv with vibrations parallel to axes of function 6(Uv, Uv, a). Wave- velocity surface. Internal conical refraction. Wave-surface as envelope of S^ = l or S/>t/o + l = 0. Ray-velocity p, cp= -0 (vd, UO, i). xxiv CONTENTS. (viii) Electromagnetic Theory. ART. PAGE 139. The circuital laws, 249 jSrjdp = ljSydv, jSedp=-hyAy. . ■] where t] is magnetic and e electric force ; y electric and y, magnetic current, and u velocity of light. Differential equations of field, where 8 is electric displacement, ^ magnetic induction ; t electric, t^ magnetic, conduction current ; e density of elec- trification carried with velocity v and e^ density of magneti- fication carried with velocity v,. Meaning of e and t] ; e< and 7]t are total forces ; Ci and 7)i are impressed forces ; and €« = € + €» = € + eic + eid, rit = ri + r(i = 7]->irr]ic + r]ii. Conduction current equations fc = $(€ + €ic), ty=*X^ + W- Displacement and induction equations, 140. Activity of impressed electric and magnetic forces. Evoked 251 mechanical force and stress on element of medium. Joulian waste of energy due to resistance Stored energy, electric ( IF) and magnetic ( W) ; W= - |S8<^-i8, IF, = - i^/34>r^l3. Radiation of energy. The Poynting vector uYerj. Determination of evoked mechanical force (^) and of stress function ^«. Across an arbitrary vector-area /x, the stress is ^,fji=^fiS8(^-^8 - (^-iSSS/x+^/xS^!i 1 1 v=^Y8/3, € = -V/3/), rj = -Yp8. Determination of the vectors when one is given. Pair of plane polarized waves with given direction of wave- or of ray- velocity. Relations connecting the vectors depending on the two waves. Construction for the vectors by means of two quadric surfaces. Conical refraction. Wave- and ray-velocity surfaces. CHAPTER XVIL PROJECTIVE GEOMETRY. 143. A quaternion {q) represents a point {Q) loaded with a weight 85- ; 263 g = S^.(l-h||)=Sg.(l-fOQ) = Sg.Q. ^ The sum of weighted points is their centre of mass loaded with the sum of their weights. 144. The combinatorial functions ----... 264 («, 5) = 6Sa-aS6; [a, 6] = V . V«V6 ; [«, 6, c] = (a, 6, c)-[6, c]Sa-[c, a]S6-[a, 6]Sc ; («, h, c) = S[a, 6, c] ; (a, h, c, o?) = Sa[6, c, 0?]. Symbol of plane [a, 6, c] ; principle of reciprocity. 145. The equations - -- - - -- - - - 266 {q^ a, 6] = and {q, a, 6, c) = represent the line ah and the plane abc. The plane 85-^ = and its reciprocal with respect to the unit sphere S . ^^ = 0. Formulae of reciprocation, ([a6c]; [_abd]) = \ah]{abcd); [[a6c] ; [ahd]'\= - {ah){ahcd). 146. The relations connecting five points, 268 a {hcde) + h {cdea) + c{deah) + d{eahc) + e {abed) = ; e{abcd) = [bcd]Sae - [acd]Sbe + [abd]Sce - [abc]Sde. 147. Combinatorial functions. Construction and development of 270 these functions. XXVI CONTENTS. ART. 148. The general linear transformation in space, - . - - Determination of transformation converting five given points into five others. 149. When / transforms points, /'-I transforms planes. - - 150. Inversion of a linear function. The auxiliary functions, i^, (7, H \ the invariants n^ n', n", n'". 151. The united points, lines and planes. 152. The self -con jugate and the non-conjugate parts of a function, - /o=i(/+A /=!(/-/)• The equations of the general quadric and of the general linear complex are respectively The equations of the polar plane of a and of the plane contain- ing the lines of the complex through h are respectively and the equations of the reciprocals of the quadric and of the complex are ^qfo-'q=o, ^pfr'q=o. Nature of united points of /^. Common self-conjugate tetrahedron of two quadrics Sqf,q = 0, Sqf,q^O, is determined by united points oi f2~Yv Examples on generalized confocals, etc. 153. Square root of linear quaternion function. . _ _ . Beduction of function to form f=M, where fsHff)K ftft=^' Further reduction of /f. ft=fuU where /,2=l, f^=r{ ) Transformations converting one quadric into another, etc. Curve of intersection of two quadrics. Examples on lines traced on surfaces, etc. 154. Invariants of linear transformations and of quadric surfaces. - 155. Numerical characteristics of curves represented by - iViViV^^^ and by {{ViP^PzV^Vd)^^ \ number of points represented by {VxPi) = ^ and by {{{VinVzP^Vr,'p^)) = % where p„ is homogeneous and of order m„ in q. CONTENTS. xxvii ART. PAGE 156. The general surface. The relations of reciprocity, - - - 293 dp = (m-l)f4q, dq = (n-l)fpdp, /p/g = l. ^ Conjugate tangents, asymptotic lines, generalized curvature. 157. Poles and polars. Operator D analogous to V. Aronhold's 296 notation. Examples on Jacobians ; surface through points on given surface where tangent touches in four consecutive points. Examples to Chapter XVII. ------- 300 CHAPTER XVIII. HYPERSPACE. 158. Extension of formulae to any number of variables. - - - 303 Definition of product of two vectors in w-space ; a^ = V2a^ + VoaA where Vga/? is vector-area of parallelogram, and Vo<*yS is Sa^. Multiplication in general defined to be associative and dis- tributive. Product of m vectors, ttiOg . . . a^ = ( V^ + V^_2 + etc.) aitta . . . o^. Expansion of Vpaia2 . . . Om- 159. Sum of area vectors not generally an area vector but the 306 analogue of an angular velocity. Rotation in /i-space. ). Symbols of points and flats in hyperspace. Formulae for 308 projective geometry. Index, 310 CHAPTER I. THE ADDITION AND SUBTKACTION OF VECTORS. Art. 1. A right line, AB, considered as having not only length but also direction, is said to be a vector. The direction of the vector AB is that of the point B as viewed from A, and the vector BA is the opposite of AB, being equal to it in length but having the opposite direction. All equal right lines AB, A'B', etc., which have the same direction are equal vectors.*!* Y Art. 2. The sum obtained by adding the vector BC to AB is denoted by BC + AB, and is defined to be the vector AC. Thus symbolically (fig. 1), BC + AB = AC. Fig. 1. Fig. 2. Completing the parallelogram, ABCD, the definition of addition gives likewise the equation (fig. 2) DC + AD = AC or AB + BC = AC, because the vectors DC and AD are respectively equal to AB and BC. Thus the sum of two vectors is independent of the order I * Following the example of Hamilton in his Lectures on Quaternions and in his Elements of Qiiaternions, the table of contents of this volume is amplified into an analysis or commentary to which it may be useful occasionally to refer, t It sjiems to be an unnecessary complication to print a bar (ab) over the letters which represent a vector ab. Hamilton sometimes uses the notation ab to re- present the length of the vector ab. J.Q. A 2 ADDITION OF VECTORS. [chap. i. in which they are added, or the addition of two vectors is a commutative operation.* Art. 3. The sum obtained by adding any vector CD f to the sum of AB and BC (fig. 3), is the sum of CD and AC, or the vector AD. But AD is likewise the sum of AB and BD, that is, the sum of AB and the sum of BC and CD. And by completing Fig. 3. the parallelogram of which BD is a diagonal and BC and CD are sides, it appears that AD is also the sum of BC and the sum of AB and CD. In other words, the same vector is obtained by adding any one of the three vectors, AB, BC and CD, to the sum of the other two. This vector sum AD is consequently inde- pendent of the order in which the component vectors are taken and of the mode in which they are grouped. The same process applies in general, and the addition of vectors is an associative and a comm^utative oiJeration. It is associative inasmuch as the vectors may be grouped into partial sums in any way ; and it is commutative because the order in which the vectors are taken is immaterial. Art. 4. Any number of vectors being arranged as the succes- sive sides AB, BC, etc., of a polygon, their sum is the vector AD drawn from the initial point of the first to the terminal point of the last. If the polygon happens to be closed, the sum is a vector of zero length, or simply zero. Thus, in particular, AB4-BA = 0, AB-}-BC-}-CA = 0, AB + BC -f CD -f- D A = 0. Art. 5. It is natural, in accordance with the equations just given, to introduce the sign — , and to write BA=-AB, ■^ In certain systems of vector analysis, the word vector is used in a different sense, and a vector cannot be determined without reference to its position. The commutative law then ceases to be obeyed. An example of non-commutative addition will be found in Art. 21, p. 16, t In every case, unless the contrary is expressed or implied, the vectors with which we deal are not necessarily parallel to a plane. ART. 6.] SCALAR COEFFICIENTS. 3 or to agree that the sign — prefixed to a vector shall convert it into its opposite (Art. 1). Hence the subtraction of one vector from another may be regarded as equivalent to the addition of the opposite of the first vector to the second. Subtraction of vectors is thus included in addition. As we can now interpret — AB, it is convenient to use a single symbol to denote a vector. We shall follow Hamilton's admir- able notation, and shall employ the small letters of the Greek alphabet to represent vectors, using, as a general rule, the earlier letters a, /3, y, etc., for given or constant vectors, and p or cr for variable vectors. Art. 6. The sum of two equal vectors is a vector of the same direction and of twice the length. It is natural to write, as in algebra, ^ . o . . ^ 2a = a-{-a, Sa = a + a + a, etc., and generally, at least when n is an integer, if the vectors ^ and a have the same direction while the length of ^ is n times that of a. This result may be extended to the case in which n is fractional or incommensurable by a process identical with similar extensions in elementary algebra. The last article affords the interpretation to be a dopted when n is negative ; and when n is complex (n' + s/^^n''), the difficulties of interpretation are of the same nature as in ordinary algebra^ and need not be discussed here. Further, it is natural to say that the coefficient n results from the division of the vector ^ by the parallel vector a, and we shall therefore write n = ~, or 71 = |8 -^ a, or n = /3 : a, a as a consequence of ^ = na. Also, conversely, whenever the quotient of two vectors is an algebraic quantity or a scalar* we infer that the vectors are parallel, and that they have the same or opposite directions according as that scalar is positive or negative. Again, if n is an integer and if a and /5 are any two vectors^ the laws of addition give and by a process of induction this relation may be extended to *Thfe Jjvord 'scalar,' synonymous with algebraic quantity, was employed by Hamilton because such a quantity may be conceived to be constructed by "com- parison of positions upon one common scale (or axis)." Elements, Art. 17. ADDITION OF VECTORS. [chap. I. More the case in which n is fractional or incommensurable, generally, if x, y and z are any scalars, z {xa + y^) = zxa + zyfi, so that the multiplication of vectors by scalars is a distributive operation. Art. 7. In the calculus of quaternions a unit of length is selected to which the lengths of all vectors are referred. The tensor of a vector a is the number of units contained in its length, and is denoted by the symbol Ta. Thus the tensor is a positive or " signless " number, at least when the vector is real,* and in particular, Ta = T(' — a") In general, if -ti is a real scalar, T7la = 7^Ta if ti>0; Tna= -nTa ii n<0. Hamilton also uses the notation Ua to denote a vector of unit length having the same direction as a, and he calls Ua the versor of the vector a. Since the direction of — a is opposite to that of a, Ua=-U(-a), and, more generally, \Jna = 'Ua if '^^>0; JJna= —JJa if nii the scalars x, y and z are the quotients of parallel vectors, a; = OA':OA, 2/ = 0B':0B, = OC':OC; and it is further evident that this construction is unique. It may happen that some or all of these three scalars are negative, or some may be zero, but these cases can present no difficulty. Ex. 1. Find the vector oc to a point which divides ab in a given ratio. Ttt AC CB AC + CB R-a v-a la + mB~\ Here — =-^ = ^- =9- — =l or 7= , ^ . L ml l + m l+m m ' ' l + m J Ex. 2. If weights I, m and n are placed at a, b and c, find their centre of mass. [The extremity of the vector (la + 7nl3 + 7iy) :{l+m + n\ supposed to be coinitial with a, and y.] Ex. 3. Prove that the mean centre of a tetrahedron is (a) the intersection of bisectors of opposite edges ; (6) the intersection of lines joining the vertices to the mean points of the opposite faces. Show that the former lines bisect one another, and that the latter quadrisect one another. Ex. 4. Prove that the vectors ±a±/3±y when drawn through a common point terminate at the vertices of a parallelepiped. Ex. 5. Discuss the arrangement of the extremities of the sixteen coinitial vectors ±a±^±y±8. Consider the points with reference to the extremities of ±a, etc., and with reference to one of the points, the extremity of a + jS 4- 7 + 8 for example. Ex. 6. Prove that four arbitrary vectors are connected by a linear relation, , in , , js- r. ' aa + b/i-\-cy + d8=0. ;. Ex. 7. If three vectors are linearly connected, or if they are coplanar. ■ Ex. 8. If aoA + 6oB+coc=0, a + b + c=0, the points a, b, c are coUinear. Ex. 9. If aoA + 6oB + coc + c?OD = 0, a + b + c-\-d=:0, the points a, b, c, d are coplanar. CHAPTER II. MULTIPLICATION AND DIVISION OF VECTORS AND OF QUATEENIONS. Art. 9. The product of the length of one vector (a) into the length of the projection of another (/3) upon it is denoted by the expression ^ ^ — t^ap, and this function Sa/3 of two vectors is called the scalar of a/3. By similar triangles it follows that (fig. 5) Sa/3 = S/3a, Fig. 5. Fig. 6. and because the sum of the projections of any number of vectors on any line is the projection of their sum, it appears that (fig. 6) Sa (j3 + y) = Sa/3 4" Say ; and therefore the function is a doubly distributive function, or SSa2/3 = 22Sa/3. If the vectors a and y are at right angles, Say = 0, and conversely. An equation such as Sa^ = SyS implies that the projection of a on /3 multiplied by the length of fi is equal to the projection of y on ^ into the length of S. ART. 10.] SCALAR AND VECTOR. Art. 10. A unit of length having been assumed, let a vector be drawn at right angles to two given vectors a and ^ so that rotation round this vector from a to ^ is positive,* and let the length of this vector be numerically equal to the area of the parallelogram determined by a and p. This vector is denoted by the symbol y^/^, and is called the vector of a/3. If the vectors are taken in the reverse order, \^a has the same length as Va^S, but the direction is opposite, the rotation being now reversed, so that If an equation such as Va^ = \yS . Fig. 7. exists, the vectors a, ^8, y and S must all be parallel to the same ^lane; the areas of the parallelograms determined by a and ^ and by y and S must be equal, and the sense of rotation from a to /3 must be the same as that from y to ^ (fig. 7). Like Say8, the function Yafi is a doubly distributive function. If /3' is the component of the vector ^ at right angles to a it is obvious that Va/3 = Va^', Fig. 8. and the tensor of Va^S is equal to the product of the tensors of a and of P' (tig. 8). ■^ The convention respecting rotation which is here adopted is the opposite of that employed by Hamilton. The axis of a rotation is taken to be in the direction of the advance of a right-handed screw turning in a fixed nut, and this system is now known as the right-handed system of rotation (Clerk Maxwell, Electricity and Macfnetism, Art. 23). On the other hand Hamilton calls his system right-handed, but he takes as the axis the direction from blade to handle of a turn screw when screwing a right-handed screw iiUo a nut {Lectures, Art. 68, Elements, note to Art. 293), and accordingly some little care is necessary in comparing Hamilton's demonstrations with those of the present volume. Tait uses the modern right- handed system in his quaternion writings. 8 ^ MULTIPLICATION OF QUATERNIONS. [chap. ii. If y8' and y are the components of ^ and y at right angles to a, and in the plane of the paper while a is drawn upwards at right angles to the plane (fig. 9), the vectors ^a^' and Nay will ,r Fig. 9. lie in the plane of the paper, at right angles respectively to /3' and y'. But TVa/3' : T^' = TVay : Ty' = Ta, and consequently the triangles OB'C' and OB^C^ are directly similar. Hence OC^ is at right angles to OC' and TOC^ : TOC' = Ta. Consequently OC^ = Va(/3' + y ) = OB^ + B^C, = Va^' + Vay . In this relation we may replace ^' and y by /5 and y, so that Va(/3 + y) = Va^ + Vay; V(/3 + y)a = V^a + Vya, «, j8, and y being three arbitrary vectors. We have now V2a2^ = 22Va/3 for any number of vectors, since in particular for four vectors, V(a + ^)(y4-^) = V(a + /3)y + V(a + /8)^=Vay + V/3y + Va^ + V/5^. If Vaj8 = without having either a or /3 zero, the vector a must be parallel to /3, for the area of the parallelogram deter- mined by a and ^ must vanish. Art. 11. The product of the vector a into /3 is defined by the equation, a^ = Sa^ + Va/3, (b) and because it is the sum of two doubly distributive parts, it is likewise doubly distributive, or The product /3a is not generally equal to a/3. In fact /3a = Sa/3~Va^ because Sa^ = S^a, Va/3= -V^a. ART. 13.] PRODUCT OF TWO VECTORS. 9 Thus multiplication of vectors is not commutative. We speak of a/3 as the product of ^ by a, or the product of a into /3. Adding and subtracting the expressions for the two products a^ and y8a, we find Art. 12. The sum of a scalar and a vector is called a quater- nion because it involves four independent numbers, such as the scalar and the three coefficients of the vector when resolved along three given directions (Art. 8). Thus the product of a pair of vectors is a quaternion, and conversely, every quaternion may be expressed as a product of a pair of vectors. If q is a quaternion, if Sg is its scalar part and \q its vector part, so that if a and /3' are two vectors at right angles to one another and to Vg, so that Ya/S' = Yq; and if /3 — /3' is the vector parallel to a, for which Sa(/8 — /3') = Sg, then we have Vg = Va/3 because Va(^-/3') = 0; Sq = Sa^ because Sa/3' = 0, and therefore q = afi, or the quaternion has been reduced to the product of a pair of vectors. Scalars and vectors may be regarded as simply degraded cases of quaternions. The sum of any number of quaternions we define to be the sum of their scalar parts plus the sum of their vector parts. Addition of scalars is associative and commutative, and likewise addition of vectors (Art. 3). It follows that addition of quaternions is associative and commutative. Art. 13. We next define the product of a quaternion and a vector to be distributive with respect to the scalar and the vector of the quaternion. Thus yq = y{Sq-{-Yq)-=ySq-\-yYq, qy = {Sq + Yq)y = Sq .y-{-Yq.y. The products yYq and Yq . y fall under formula (b), and we define that multiplication of a scalar and a vector is commutative, so that ySq = Sg . y. Thus we can interpret expressions such as a . /8y or a^ . y (the product of a into the product /5y and the product of the pro- duct a^ into y), and we see that they are distributive with respeijt to the three vectors, so that la . l/31y = SSSa . /3y, 2aS/3 . Zy = Ula/S . y. 10 MULTIPLICATION OF QUATERNIONS. [chap. ii. We shall now prove that the products are associative, so that we may omit the points, and to this end we shall consider the laws of combination of three mutually rectangular unit-vectors, i, j and k. Art. 14. Let any three mutually rectangular unit-vectors, iy j and k, be drawn so that rotation round i from j to k is positive. According to the usual convention, if i and j are in the plane of the paper, k will be directed vertically upwards, and it is seen at Fig. 10. once that rotation round j from k to i, and also round k from i to j is positive (Fig. 10). We have then, because the vectors are mutually perpendicular and of unit length, Sjk = Ski = Sij = 0; Si2 = S/ = SF=-l; (Art. 9) Yjk = i, Yki=j, Yij = k; Ykj= —i, Yik= —j, Yji— —k; (Art. 10) and by formula (b) it follows at once that i^ =^'2 = 7(;2 = _ 1 ^ jk = i= — kjj ki =j = — ik, ij = k= —ji. . . . (c) Let us now, as in the last article, form the ternary products of these vectors. We have by the relations just given i ,jk = i A= —l=k. k = ij , k = ijk, i.f=-i=+k.j::=ij.j = ij'\ the points being omitted as they are seen to be unnecessary. Similarly, for every ternary product of i, j and k, the points may be shown to be unnecessary. For quaternary products, let i, k,\, /ul each denote some one of the three symbols i, j, k, then I . kX/ul = 1 . k . X/x = IK . Xjui = f /c. X . /x = ik\ . fji = ikX/ul, because, for example, i . /c. X/x is a ternary product, as X/ul must be :th dijy ±A^ or — 1. In this way all products of the symbols i, j, k are seen to be associative. ART. 16.] ASSOCIATIVE PROPERTY. 11 It may be a useful exercise to show that the associative law- enables us to deduce all the relations (c) from Hamilton's funda- mental formula (a), o -o 7 2 • -y 1 / x For example, i . ijk = —i gives jk = i. Ex. 1. Prove that ijk~jki—kij= —l=—kji= —jik= - ikj. Ex. 2. If the symbols i, j, k obey the laws,* i2=j2=k2=+l; jk=i, ki=j, ij=k; kj=-i, ik=-j, ji=-k, prove that their multiplication is dissociative. [i2.j = +j but i.ij=i.k=-j.] i Art. 15. We can now show that multiplication of vectors is associative. Let any three vectors, a, /3 and y be expressed in terms of i, j, k, so that a = xi-{-yj+zk, P = xi-\-y'j + z% y==x"i-\-y"j-{-z"k. By Art. 13, h ■ « . /5y = SSSo^i . y'jz"k = ^.^Ixy'z'i . jk = Y11xy'z"ijk, "^ ap . 7 = S22 xiy'j . z"k = ^lllxy'zf'ij . k = ^^Ixy'zijk, so that a. /3y = al3 . y = a^y, and similarly for all products of higher orders. Hence multiplication of quaternions is associative, for a qua- ternion may be expressed as the product of a pair of vectors. It now appears (compare Art. 13) that the product of any number of vectors taken in any given order is a definite quaternion. Art. 16. The division of vectors may be reduced to multi- plication. By formula (b) the square of a vector is a2=S.a2=-(Ta)2; so that a.7^2=l> and thus it appears that — a : (Taf is the reciprocal of the vector o, say a~^ or -. The vector a'^ is opposite to a in direction, * Mf . Oliver Heaviside bases his vectorial Algebra on these laws. Prof. Knott {Recent innovations in Vector Theory, Proc. R.S.E., 1892-3) draws attention to papers written by the Rev. M. O'Brien in the years 1846-52, in which the square of a vector is taken to be positive. 12 MULTIPLICATION OF QUATERNIONS. [chap. ii. and its tensor is the reciprocal of that of a. We can therefore interpret products such as ^a-i and a'^/^, and the first of these we shall call the quotient of /3 by a, and denote it by o — or B:a. The reciprocal of any product of vectors is the product of their reciprocals taken in the reverse order. For if we have QQ' = 1 in virtue of the associative law. Similarly, the reciprocal of a product of quaternions is the product of the quaternions taken in the reverse order. Hence every quotient of vectors or of quaternions is a quaternion ; and more generally every com- bination of quaternions by the processes of addition, subtraction, multiplication and division is a quaternion. Ex. 1. Prove that Ex. 2. Distinguish between the expressions y a « ya [These maj^ be written Sy~-^^a~^ and 8j8a~^y~\] Ex. 3. Prove that ay a ya ' ^."•Z Art. 17. The conjugate ILq of a quaternion q is defined by the relation Kq = Sq-Yq. ^(jT -^ If then q = a^, we have Kq = l3a (Art. 11), and -J&ri qj^q = ^^^^ = ^2^2 = Kgg = Ta^T^"^ (Art. 16). The products of the tensors of the vectors into which a quaternion is resolvable is therefore independent of any particular selection of the vectors since Sg and Yq are independent of any particular pair of vectors ; and the square of this product is qKq = (Sq-^yq)(Sq--Yq) = (Sqf^{Vqf = Kqq = (Tqf, if we call this constant product of tensors, the tensor (Tq) of the quaternion. ART. 18.] TENSOR AND VERSOR. 13 Again, g = a^ = Ta.Ua.T^.U/3 = TaT^.UaU^ = Tg.Ug and Vq = UaU/3 is called the versor of the quaternion. If ir— Lq is the angle between the vectors a and S, which is less than two right angles and measured from a to /3, we see by the definitions of Sg and Vg that (Arts. 9 and 10) S^ = Tq cos L g, T Vg = Tg sin l q. The angle z.g is called the angle of the quaternion, and is independent of any particular set of vectors a, /5. A plane at right angles to Vg is called the plane of the quaternion and JJYq is called the axis. Ex. 1 . Prove that Kg' =w — ix —jy - kz^ VYq = { + ix+jy + hz) : >J{ + x^+y^+z^), Vq = (w+ix+jy+kz) :^(w^+a;'^+y^-\-z^), ^^where q = w + ix +jy + kz. Write down the analogous functions of Kg- in terms of x, y, z Ex. 2. and w. Ex. 3. Ex. 4. Prove that a-ij8 = K . /?a-i. What is the nature of g^ if g* = Kg- ? If ^=-K^ Art. 18. We can always reduce a quaternion to a quotient of vectors (Arts. 12, 16), and write ^^OB ^ TOB ^, UOB ~0A' <1^ m TOB ^^ Sg a ui\. * lUA - UOA' ^ OA the line BA' being drawn perpendicular to OA. OA' ^^ A'B , A^g=--, z.g=AOB, OA^ Thus the shape of the triangle AOB is constant for a given quaternion. From this point of view, a quaternion is called by Hamilton a ratio of vectors, as it depends on their relative magnitudes and on their relative directions. 14 MULTIPLICATION OF QUATERNIONS. [chap. II. It is not difficult to show that the conjugate (see Fig. 12) ^ OB' .„ OB ^^ = 0A '^ ^ = 0A' for q + Kq = 2Sq, q--Kq = 2Yq. The triangle AOB' is inversely similar* to AOB. Fig. 12. Art. 19. Conversely, if the product qa is a vector ^, it is evident that a and /3 are both at right angles to Yq. And if a is any vector at right angles to Yq, qa is a vector making a constant angle (^q) with a, and having its length Tg times that of a. In other words, regarding the quaternion as an operator, it turns vectors in its plane through a given angle, and alters their lengths in a given ratio. In particular we may regard a vector as turning vectors at right angles to it through a right angle, and altering their lengths proportionately to its own. The versor JJq turns vectors in its plane through the angle Lq but leaves their lengths unaltered. The tensor Tq alters the lengths of all vectors in a given ratio. The total effect produced by g on a vector in its plane may be considered to be effected in two stages or at once as indicated by the relation l3==qa = Tq.\Jq.a = Vq.Tq.a. Art. 20. The results of articles 18, 15 and 16 afford an ex- tremely elegant construction for the product of two quaternions q and r. Take any vector OB along the line of intersection of the planes of the two quaternions. Make the triangle BOG in ■* Hamilton uses tlie phrases direct similitude and inverse similitude in the sense that two directly similar figures in a plane appear to have the same shape ; while of two inversely similar figures one has the same shape as the reflection of the other in a mirror. ART. 20.] CONSTRUCTION FOE PRODUCT. 15 the plane of r similar to the triangle determined by r (Art. 18) ; make AOB in the plane of q similar to the triangle of q ; then, by the associative principle (Fig. 13) ^ 0A\ OB OA / ,-^B Fig. 13. Fig. 14. If the triangles BOA' and C'OB are respectively coplanar with and similar to AOB and BOC, the second product is (Fig. 14) _OAY_aA/ 0B\ ^^ ""OC'V OB 'OCV* Ex. 1. Prove that K(r^) = K5'Kn [Take c, on oc and a, on oa so that c^ob and boa, are inversely similar to BOG and AOB, and the triangle a^oc, is inversely similar to coa. Art. 18.] Ex. 2. The product of the conjugates of any number of quaternions is the conjugate of their product in reverse order. [By Ex. 1 , K (p . ^r) = K (^r) . Kp, etc.] Ex. 3. Show that ^PlP2Ps ' . -Pn = h [P1P2 '"Pn + ^pJ^P.. -1 • • • K^l], ypiPiPs ...Pn = i[PlP2 '"Pn - Kp„Kp„_ 1 . .. K^i]. If ttiao ...an are ?2. vectors, and if Ila = aic'2 . . . a„, Il'a = a,ia„_i . . . ai> Ex. 4. show that Ex. 5. Ex. 6. sna=|na+^(-)"n'a, vna=|na-^(-)"n'a. Prove that Qpq = Sqp ; TVpq = TY qp ; Lpq == Lqp. Prove that pK.q + ^Kp = 2S . pK.q = 2S . qY.p. Ex. 7. Prove that the tensor of a product of any number of quaterniona is independent of their order. 16 MULTIPLICATION OF QUATERNIONS. [chap. ii. Ex. 8. Prove that the versor of a product of any number of quaternions is the product of the versors taken in the same order. Ex. 9. Show that three quaternions cannot in general be reduced simultaneously to the forms Ex. 10. Prove that the scalar of a product of any number of quaternions is unchanged when the quaternions are cyclically transposed. Ex. 11. Prove that the tensor of the vector part of a product of quaternions remains unchanged for cyclical transposition. Ex. 12. Prove the identity {ww' — xx' — yy' — zz')^ + {wx' + w'x •\-yz' — y'zf + {wy' + w'y + zx' — z'x^ + {ivz' + w'z + xy' — x'y^ ^{w'^ + x^+y'^+z^){w"^+x'^+y'^ + z'^). [See Ex. 1 of this series and Ex. 1, Art. 17. This identity is of historical interest as regards the discovery of quaternions. See Graves's Life of Sir William Rowan Hamilton, vol. ii., p. 437.] Art. 21. The multiplication of versors, to which the multipli- cation of quaternions may be reduced by separating the tensors, admits of a simple spherical representation. Fig. 15. A versor is represented by a directed great circle arc belonging to a definite great circle (the plane of the versor) and having a definite length (the angle of the versor). From the figure iA,. op. A~V~^- Also z./xA = ^, where is the angle between the mirrors, and 2LfjLX = 20.] Ex. 10. Given three lines intersecting in a point, it is required to draw three planes, each through one of the lines, so that the lines of intersection in one plane may be equally inclined to the contained line. When is the problem indeterminate ? [Let a, (3, y be the vectors of the given lines. The sought lines of inter- section are VySay, Vy^a, Vay^. Compare Art. 31, p. 31.] Art. 22. The laws of combination of the five symbols S, V, K, T and U may be summarized in the symbolical multiplication table : S V K T U s s s T su V V -V vu K s -V 1 T KU T +s TV T T — U — uv UK — u to be read from the left. For example, the tensor of the vector of a quaternion is TVg; the scalar of the vector is 0; the tensor of the scalar is +Sg according as Sq is positive or negative. A positive scalar may be regarded as the quotient of two vectors having the same direction; for a negative scalar the directions are opposite. Hence we may write \JSq = + 1 according as Sg is positive or negative. The versor of a zero quaternion must be regarded as arbitrary, unless we know a law according to which the quaternion diminished indefinitely. TVq = l=JJTq for all quaternions. The versor of the conjugate and the conjugate of the versor of a quaternion are easily seen to be equal to one another and to the reciprocal of the versor. The symbols T and U are not distributive like the symbols S, K and V. Apart from change of sign, it is easy to see that the only new combination arising from further repetition of the symbols is TYJJq ( = sin Lq). It is necessary to make some convention concerning the notation to be employed when we wish to denote for example the square of the scalar of a quaternion q or the scalar of the square of the 20 MULTIPLICATION OF QUATERNIONS. [chap. ii. quaternion. There can be no mistake if we employ brackets and write (Sqf for the square of the scalar and S(q^) for the scalar of the square, and whenever there is the least fear of confusion brackets should be used. One of the great advantages of quaternions is the extreme brevity of the notation. Another and still greater advantage is its great explicitness, and this should never be sacrificed for the sake of a few brackets. Hamilton writes S . q^ for the scalar of the square and Sg^ for the square of the scalar whenever there is no fear of confusion, and he uses the notation V . q^ and Yq^ in a similar sense and in conformity with the established notation d . x^ and dx^ for the differential of x"^ and for the square of the differential of x. Some eminent authorities, Tait for instance, in conformity with the notation cos^ x = (cos x)^, write S^g instead of Sq^, though in strictness this would mean S . Sg ( = Sg). But considering the enormous care Hamilton took with his notation we prefer to abide by his convention. No confusion can arise with respect to T . g2 or Tg^ or (Tg)^, for the tensor of the square is the square of the tensor, and similarly U. g^ = Ug^ = (Ug)^ and K . g2 = (KqY = Kg^. The expression S^ . g means the product of Sp into g, and it is well when possible to write this in the equivalent form qSp, while S .pq is the scalar of the product pq, but if the expressions are at all complicated, it is safer to write {Sp)q and S(pq). An imaginary quaternion where p and q are real quaternions and where V — 1 is the imaginary symbol of algebra regarded as a scalar commutative with all quaternions, is called a biquaternion by Hamilton. Similarly he calls imaginary vectors (a+>/ -I . f3) bivectors and imaginary scalars, biscalars. No ambiguity attaches to S$ = S^+\/^S^, or to V^ = VjD + \/^Vg, and the only ambiguity in T^ is one of sign, and this Hamilton removes as follows. He writes T^=^+x/^.y, where x and y are real scalars and where x is positive, and in order to determine x and y he employs the relation (Art. 17) (T02 ^qKQ =pKp-qKq + s/^^ipKq + qKp), or {TQf = Tjo2 - Tj2 + 2>J^S . pKq =x^-y^ + 2 V^ . xy, observing that qKp=K.,pKqf so that the imaginary part of (T^)^ may be written 2^^S.pKq, or sV^^is . qKp. Equating reals and imaginaries we find, from x^-y^ = Tp^-Tq^ and xy = S.pKq, that the real positive value of ^ is a:={^(Tp2_T^2) + |-i(T^2_T^2)2 + (s.pKy)2]i|i ART. 22.] THE SYMBOLS S, V, K, T, U. 21 It may happen that T(^^) is -T^T^' instead of +TQTQ' where Q and Q' are biquaternions. In other particulars ambiguity does not arise. The tensor of a biquaternion may vanish, and in this case we have an equation such as where Q' = K^ without having either Q or Q' zero. The conditions are Tp'^=Tq^ and S.pKq = 0, and when these are satisfied, the biquaternion Q is called by Hamilton a nullifier. A few examples will be found in Chap. IV. ; and the Lectures on Quaternions (Arts. 669-675), from which this account of biquaternions has been taken, may be consulted with advantage.* Ex. 1. Prove that combinations of the symbols prefixed to q lead to one or other of the following : S^, V^, K^, Tq, Vq ; TV^, SU^, VU^, TVU^, QJq)-\ VYq. Ex. 2. Express these functions in terms of x, y, 0, w^ ^, j and h. (See Ex. 1, Art. 17, p. 13.) Ex. 3. Express these functions in terms of the tensor, axis and angle of the quaternion. Ex. 4. Show that the vectors \JYpq and UV , \]p\Jq are identical. Ex. 5. If a, B and y are vectors, prove that V is a redundant symbol in S.aV./3y. ^ Ex. 6. Find the diflference of the expressions S . pqr and S . pY . qr. c Ex. 7. If VYp = VL>, prove that Sp = 0. Ex. 8. What inference can be drawn from the equation Yq=Y\Jql and what from Yq = JJq ? Ex. 9. Prove that T(y + /5)>±(Ty-T^) unless JJy=-V(3, and find the relation in the exceptional case. Ex. 10. Show that Tq + Tp>T(q+p) unless q=xp, x>0. Ex. 11. Show that Tq + Sq>0 unless L + CBM being A + B + B>L + A^BM = A + B -fB^CN + A^CN. Moreover, if we join PC^, the angle PC^A will be right, C^ being the middle of the base of the isosceles triangle APB; and the angle C^PA will be equal to L^^a'^, for it is J^BPA or J ML or A^B^, since by the equality of the small triangles MA^ = A^N and NB^ = B^L. Hence by the construction of Ex. 5, Art. 30, the angle C^PA represents ^^a,"^ and AC^P represents y^, so that C^AL repre- sents p or /3^a^~^y^, and therefore z.p=^/3,a,-iy. = J(A + B + C), UV^ = a (IV.) Again we have this remarkable transformation by (i.). I ART. 31.] SPHERICAL EXCESS. 33 so that for the new quaternion, ^'-©'(i)'©' -■) ^p' = J2 = KA + B + C-7r), VYp' = a, (VII.) if 2 is the spherical excess of the triangle ABC, because Lp'= Lpa~'^= Lp — EXAMPLES TO CHAPTER IV. Ex. 1. If a is a unit vector at right angles to /?, show that where tc is a scalar. Ex. 2. If a, /3 and y are unit vectors, mutually at right angles, ^^ Ex. 3. Given two sets a, f3, y and a', /5', y' of mutually rectangular unit rectors in the same order of rotation, so that a'==+(3'y' if a=+^y, show that we may connect the two sets by the series of relations (1) yi = y, ai = acosVr + ^sin-\/r, ^j = - a sin i/r + /? cos -^/^ ; (2) /^2 = A) 72 = 7iCOS^ + ajsin ^, 03= -yisin ^+aiCOS ^ ; (3) y' = y.2^ a = a^ cos + ^2 ^i^ ^1 (^' — ~ ^2 ^i^ <^ + i^2 ^^^ 4* > and draw a figure to exhibit the Eulerian angles \/r, $ and (p. Ex. 4. The conical rotation q( )q~^ which converts the first set of vectors of the last example into the second is determined by the versor q — co^^6 cos i(<^ + V^) + 7 cos I ^ sin ^ (^ + i/r) + a sin ^ ^ sin J(<^ - i/r) + /3 sin ^ ^ cos |(^ - V^) (see Tait's Quaternions, Art. 373); while other expressions for the same versor are i = y + a-^ V(y-^)a, ...(vi.) since the line being now parallel to a is perpendicular to the plane. The vector perpendicular from the point D on the plane is DM = />t-^= -a-^S((5-y)a = aSa-^DC, (VII.) and it will be noticed that we may directly obtain the vectors DM and CM by resolving the vector DC along and perpendicularly the vector a. (Art. 27.) If in (iv.) we replace a by ^, we find the foot of the perpen- dicular from the point C on the line to be the extremity of the vector ON = i. = ^-/3-iS((5-y)^ or ,. = y + /3-iV(y-(5)/3, (vill.) because now the plane is perpendicular to the line. The vector perpendicular is CN = ^-iV(y-^)/3 = ^-iV/3CD (IX.) In general the normal to the plane (ii.) makes with the line an angle determined by cosa=SU^, or sine = TVU^, or tan0=-?^; ...(x.) a a D V /5a and if we are required to draw a plane through the point C making a given angle with the line, we have U^ = cos OJJa + sin OVra ; while Va = cos ^U^ + sin 0Ut/3, . . .(xi.) if the line is to be drawn inclined at a given angle to the plane. In these equations the vector r is arbitrary, subject to the implied conditions, which are Sra = and St/3 = respectively. Ex. 1. Two objects, b and c, are observed from the origin of the vector a to be in the directions ij/5 and Uy, and from the extremity of a to be in the directions U^' and Uy' ; prove that the vector bc is and point out the conditions implied in this expression. [For the point b we have ^U/? = a + ?/U/5', and therefore ^VU^^' = VaU^'.] Ex. 2. Four points a, b, c, d are viewed from a fifth point p. Prove that they appear to form a parallelogram abcd if U(Upa + Upc) = U(Upb + Upd) ; a rectangle if Upa + Upc = Upb + TJpd ; and a square if in addition SUpa . pb = SUpb . pc. ART. 33.] PERPENDICULAES. 37 [The first condition requires the diagonals ac and bd to appear to bisect one another. The second requires that they should also appear to be equal, and the third imposes the additional condition that adjacent sides should appear to be equal.] Ex. 3. Find the equation of the locus of a point equidistant (1) from two fixed points, (2) from two fixed planes. Ex. 4. The extremity of the vector p is projected from the extremity of the vector a into a point on the plane SA/3 4-l=0. Prove that this point lies at the extremity of the vector VAVap+( /o-a) • SA(a-p) • Art. 33. The equation of a plane tlirough the points C, C', and of a line through D, D', are respectively, S(^-y)(y^y)a = and Y {p- S){6' - S)^0; ........(I.) or S(py + yy + y»a = and V(p^ + ^(5' + (5» = ; (ii.) or the plane being determined by the condition that the vectors CP and CC' shall be coplanar with some fixed vector a, and the line requiring that DP shall be parallel to DD'. The various expressions given in the last article may be modi- fied to suit the present case by replacing a and /3 by V(y' — y)a and S' — S respectively. The plane through CCf parallel to the line DD' is S{p-yXy'-Y)iS'-S) = 0, (iv.) because the normal to the plane must be perpendicular to the line, so that SY{y -y)a .(S'-S) = 0, or a = x{y-y)-\-y(S'-S), where x and y are certain scalars which disappear on substituting in (I.). If a plane can be drawn through CC' perpendicular to DD', the equation YY(y-y)a.(S'-S) = 0, requiring S(y -y)(^'-o) = 0, must be satisfied. We may, without loss of generality, take a to be perpendicular to CO', and as it easily appears that the plane for which in addi- tion Sa(^' — ^) = is most inclined to the given line, we can verify that the minimum value of where the vector a is regarded as variable, and that the plane -'• SV(p-y)(y'-y)V(y'-y)(^'-5) = ....(VI.) is most inclined to the given line. 38 THE STRAIGHT LINE AND PLANE. [chap. v. Art. 34. The equation of a plane through three given points, A, B, C, is SpY{^y-\-ya-ha/3) = SaSy, (l.) for the condition that PA, PB and PC should be coplanar reduces to this expression ; and in this equation V(/5y + ya 4- a/3) repre- sents double the vector area of the face ABC, while — Sa/3y is the volume of the parallelepiped having three conterminous sides, OA, OB, OC (Art. 24). The equation may be taken as asserting that if through the boundary of a vector area determined by Y(/3y-\-ya + al3) we draw vectors equal and parallel to OP (P being any point in the plane), the volume of the solid thus constructed is equal to that of the parallelepiped (Art. 23). Writing for brevity, the equation of a plane in the form the vectors SXp = l, (ii.) ^ = S-X-\S\S-l) = \-^VXS-h\-\ and DM-X-i-X-^SX^ (ill.) 4re respectively the vector to the foot of the perpendicular from a point D on the plane, and the vector-perpendicular from the same point. To find a plane equally inclined to three given lines OA, OB and OC, we have cos . TX = - SXUa = - SXU^ = - SXUy, so that (Art. 26) UX . secO. SUa^y= - V(U/!^y + Uya + Ua/3), sec ^= -TV(U^y + Uya + Ua^)(SUa/3y)-\ and the equation of the plane is Syo V(U/3y + Uya + Va/3) = const, or Syo V(^yTa + yaT/3 + a/3Ty) = const A plane equally inclined to the faces of the pyramid OABC is represented by Sp(aTV^y + /STYya + yT Va/3) = const ; a plane cutting off equal areas on its faces is Sp(UY/3y + U Vya + U Va/5) = const, while the equations of the planes cutting off equal intercepts from the edges and from the normals to the faces have been already found. Ex. 1. Find a plane equally inclined to the bisectors of the angles of the faces of the pyramid oabc. SX/o = l, SiuLp = l is YpY\^ = /ui'-\, or P = ^^^-y^ ;•••(!.) ART. 36.] INTERSECTIONS OF PLANES. 39 Ex. 2. The planes through an edge and through the bisector of the angle of the opposite face intersect in a line. Ex. 3. Find the equation of the plane bisecting the angle between a pair of faces. Ex. 4. Find the equation of a plane through an edge and normal to the opposite face, and prove that three such planes intersect in a line. Art. 35. The line of intersection of the planes SX/0=1, S/X/9 = l and that of the planes S\p = l, Syu/o = is VpVA/a = yu, or yO = ^^. Three planes SXp = I, Sjmp = m, Si/p = n intersect in the point pS\jULv = Y{liuLv{-mvX-\-n\iui); (ii.) and the condition that the planes should intersect in a line is Y(lijLv+mv\-\-nXiuL) = 0, (iii.) if I, m and n are not all zero. If they are all zero, the condition is SXiuLv = (IV.) Four planes intersect in a point if the condition S(//)tj/C7 — mXi/trr + tiXyuCT— p\/ii/) = (v.) is satisfied, the equation of the fourth plane being Spz:^ =p. The conditions of intersection (iii.) and (v.) may be replaced by the pairs of simultaneous equations x\ + yiuL-\-zv = 0, xl-^yin + zn = 0; (vi.) and x\-{-yiuL-\-zv-\-wTo = 0, xl-\-ym-\-zn+ivp = (vii.) respectively, the compatibility of the equations (vi.) or (vii.) being equivalent to (ill.) or (v.). Art. 36. Given a pair of lines V(/o-y)a = 0, or p^y + ta', and V(p-y )a =0, or p = y-\-fa, (l.) the vector from a point P on the first to a point P' on the second is TF' = y-y + fa-ta (ll.) If it is possible to select the scalars t and f so that this vector may vanish, the lines intersect and the condition of their inter- section is - '4 S . PP' Va'a = 0, or 8(7 - y)a a = 0, (ill.) P and P' being arbitrary points on the lines. .(IV.) 40 THE STRAIGHT LINE AND PLANE. [chap, v. Resolving the vector PP' into two components, parallel and perpendicular to the vector No! a, which is at right angles to the directions of the two lines, pp' = VaaS(VaaO-^PP' + VaaT.(VaaO-'PP^ = Vaa'S,^, + aS^,PP' - a'S v,^,PP', V aa \ aa \ aa and substituting from (ii.) on the right, PP' = Vaa'SV^ + a(Sv^(y-y)-0 Vaa \ Vaa ' ' / Thus the line joining the arbitrary points has a fixed com- ponent perpendicular to the directions of the two lines, and suitably selecting the scalars t and t' in (iv.) we see that ( (V.) OP„' = y' + a'SY^,(y'-y) J are respectively, the vector-perpendicular to the two lines, or the vector shortest distance from the first line to the second, and the vectors from the origin to the feet of this shortest vector — the points Pq and P^'. Ex. 1. Verify that PoPo' = oPo'-oPo in equation (v.). Ex. 2. Draw a line through a point (e) to intersect two given lines V(p-y)a = 0, V(p-y')a' = 0. [The line is parallel to Y . V(€ - 7)aV(c - y')a\ See (iii.).] Ex. 3. The locus of a line which intersects three given lines is repre- sented by S.V(p-y)aV(p-7')a'V(/)-/)a" = 0. (a) Reduce this equation to the form XY=ZW, where X, 1", Z and W are planes. Ex.4. Writing o-^Vpip^j T=A>2-Pi, prove that o- and t are merely multiplied by a scalar, if for p^ and p^ are substituted the vectors to any two points on the line of their extremities. {a) Conversely, given any two vectors, a- and t, satisfying the relation ScrT=0, show how they determine a line parallel to t. (6) In this notation any two lines may be denoted by the symbols (o-, t) and (o"', t'). Prove that the lines intersect if Sa-T' + So-'T=0. (c) Any scalar relation homogeneous in the pair of vectors o- and r imposes a single condition on a line. ART. 37.] ANHAEMONICS. 41 {d) If the planes SAip + l=0, SA2P+1=0 contain the extremities of th& vectors p^ and po, show that where u is some scalar. (e) Hence any relation homogeneous in the pair of vectors a- and t when equated to zero may be expressed in the forms /(cr,T) = 0, f(VpiP2,P2-pi)=0, /(A2-A1, -VAiAo) = 0. (/■) According as the equation /(cr, t) = is equivalent to one, two or three scalar equations, it represents a complex, a congruence or a regulus of right lines, and the constituents of the vectors a- and t, when resolved along three mutually rectangular directions, are Pliicker's coordinates of a line. (See Salmon, Geometry of Three DiTnensionSy Chap. XIII., Section 11.) (g) The lines of a complex /(cr, t) = (/ being now a scalar function), which pass through a point, the extremity of the fixed vector pi, generate a cone /(VpiT,T) = 0; and the lines which lie in a fixed plane, SAip + l=0, envelope the cone whose vertex is the origin and which is the reciprocal of the cone /(o-, -VAiO-) = 0. Art. 37. The vector to any point on the line joining two given points A and B is o'-"'-^ (.-. t being a variable scalar. If P^ and P^ are any two points on the line, their vector distance is ^i''^ 14.^^ 1+^^ (I4.^^)(l + y ~(l+g(]+y"V"7 and the anharmonic ratio of any four collinear points is ^ ^^ ' *' - p^p^ . p^p^ - (t,-x,)(t, - tj - ('"•> In particular (APBP-)-^^ -i)(o-n-? ^'^-^ More generally, the anharmonic ratio of any four points Qij Q9, Qs and Q^ collinear with any two points P', P", of the range, «^-^2|±-£, , «,^,.^,.|=mza. ,,, The two ranges (i.) and (v.) are homographic. Ex. 1. If the range apbp' is harmonic, prove that — + — =— or -I_ + -i_=__2 AP AP' ab' p~a p -a~ /3 — a 42 THE STRAIGHT LINE AND PLANE. [chap. v. Ex. 2. Any two homographic ranges situated on a common line, _ aa + th(B t _Gy + tdS ^~ a + tb ' P~ c + id' may be simultaneously reduced to the forms, € + 87] , €€ + sri ''==1+7' ''=^+7' Ex. 3. Show that the vectors c and t) satisfy the equation, ^ «c;(a-€)(8-€)-6c(^-€)(y-c) = 0. " -^ Art. 38. In many problems relating to a tetrahedron, it is convenient to have the equations expressed in a symmetrical manner, and some of the following relations will be found occasionally useful. If the vectors X, fx, v and To are the vector areas of the faces of a tetra- hedron ABCD we may write v = Y(a^ + f38 + 8a), Tn=-Y{af3 + f3y + ya). j ^ '^ These vectors are independent of the origin, and their sum is zero, or SA = A + /x + v + n=0 (ii.) Again, if I, m, n, p are the sextupled volumes of the pyramids subtended at the origin tjy the four faces, l = S(3y8, m=-SayS, 7i = Sa/38, p=-Saf3y; (m.) and their sum is the sextupled volume of the tetrahedron, or ^l = l-{-7n-\-n+p = v, (iv.) and is independent of the origin. Also, ^la = la + m/3 + ny-\-p8^0 (v.) Changing the origin to the extremity of the vector w, and putting a' = a-co, etc., the volumes subtended by the faces at the new origin are ," = S/5y8' = S(/5-a>)(y-o>)(8-w), etc., or r = l — So)Xy m'=m — &o)iJLf yi=7i — So}v, p'=p — S(t)7n (vi.) But still (by v.), 2ra' = 0-2(^-StoX)(a-a)) = 2^a + (o2^-2aSwA + Sw2A, and this reduces by former results to the new relation, (o^l + ^aSo)X = 0, (vii.) which holds for all vectors w. Operating on this by Sw', we may write the result in the form, S(o(a>'2^-}-2ASaw') = ; and, because w is arbitrary, the part within brackets must vanish. But a>' is also arbitrary, and accordingly, for all vectors oj, we have w2^ + !2ASwa = (viii.) Again, it is easy to see that :2aA = aA + ^/x + yi' + 8^=-3i? = :SAa; (ix.) and, for verification, it is sufficient to take the terms in a^y, which are a V/3y - /3 Vay + y Va^ = - 3^. The sum 2a A is independent of the origin. On the whole, we have 2A = 0; 1,1 = V ; 2^a = 0; — wv = 2 ASwa = 2ttS(oA ; — 3v = 2aA = 2Aa. ...(x.) Airr. 40.] TETRAHEDRON. 43 It is sometimes convenient to employ the vector perpendiculars from the vertices on the opposite faces instead of the vector areas. If a,, ;8^, y, and 8, are these vectors, it is easily seen that v = a^\ = j3,fx=^y^v = 8p, (xi.) because, in fact, the equation of the face bcd may be written Sp\ = l, or S(p-a)X = v, or S(p-a)a-^ = l. Thus (x.) gives ^~ = 0,yi = V', ^la = 0; -(ov = 2iSa>a = 2aS-; -3 = 2- = 2ia....(xil.) a, ' a, a, a, a, Ex. 1, Prove that the vector sides of the tetrahedron are given in terms of the vector areas of the faces by the relations YXp.=.{y-8)v; YXv=-(fi-S)v; YXr^=(/3-y)v; YiJLV = {a-B)v; V/xtTT = - (a - y) y ; Yvm = (a-/3)v; and show how to connect the rule of signs with that for the expansion of a determinant of the fourth order. Ex. 2. Show that ^ixvzu = SXv^y = SA/>tt7 = - SXfMV = v'^. Ex. 3. Given the magnitudes of the areas of the faces of a tetrahedron, show that the directions of the normals UA, Uju, and Uv to three of the faces must satisfy the relation TW = TX:^ + T/X2 + Ti/2 _ 2TfxvSVfx,v - 2Ti' ASU vX- 2TA/xSUAiit. Art. 39. Any five vectors are connected by relations of the form aa + bfS + cy+dS+ee^O, where a + b + c+d+e=0 ', (i.) and if the vectors are drawn from a common origin o, and terminate at the five points a, b, c, d, e, a:b:c :ci?:e = (BCDE) :-(acde) :(abde) :-(abce) :(abcd), (ii.) where (abcd) is the volume of the tetrahedron determined by the four points A, b, c, d. To prove this, remark that if a{a-€) + b(f3-€) + ('{y-e) + d(8-e) = 0, the ratios of the four scalars a, b, c and d have the values defined by equation a. The locus of the centres of the chords is derived from (iv.) by putting J (t^ -\-t^a = p — P, and is given by ^j4='' ^"'"-^ J.Q. D 50 THE SPHERE. [chap. vi. which represents a sphere on BE as diameter. For it expresses that the projection of BE on BP is equal to BP, so that the angle BPE is right. Taking the harmonic mean of the vector intercepts tohe p — ^, we have by (iii.) and (iv.), j Q + J,)^ = ^' '^^ S(p-e)(/3-e) + «^ = (IX.) is the locus of its extremity — the polar plane of the point B. Art. 42. Any two spheres, p^-2Sap + l = 0, p2_2S/5yQ + m = 0, (l.) intersect in the plane, 2S(a-p)p = l-m: (ii.) and if P is any point on the second sphere and P' any point in this radical plane, the power of the first point P with respect to the first sphere is (Art. 41 (m.)), i Tp^ + 2Sap-l = 2S(a-l3)p-l-]-m = 2S(a-P)(p-p),...(Uhy or twice the projection of PP' on the line of centres into the ^ distance between the centres. The spheres cut at an angle determined by . ^_ l-\-m-2SaP ^- ~V{(T«' + 0W2 + m)}' ^^^'^ since if a and b are their radii, a^-\-h^ — 2ab cos = T(a — /3y. For further investigation, the origin should be taken at the intersection of the line of centres with the radical plane. A variable sphere cuts two given spheres at constant angles, prove that it cuts an infinite number of spheres at constant amgles. Let the sphere (i.), determined by /3 and m, be the variable sphere, and let it cut the spheres {a, I) and (a', V) at the angles Q and 0\ Assume that it cuts the sphere {a, I") at the angle 0'\ Then the third of the equations, i Z4-m-2Sa/3 = 2a6cos0; r+m-2Sai8 = 2a'6cos0'; I" + m - 2Sa''^ = 2a"b cos &' , analogous to (iv.), must be equivalent to a linear combination of the other two. Multiply by scalars, x, y and z', add and separately equate to zero the coefiicients of the variables, m, /9 and b, and xl+yl' + zl" = 0) x + y + z = 0; xa-\-ya+za =0; xa cos + ya' cos 0' H- xa" cos 0'' = 0. ART. 43.] COAXIAL SPHERES. 51 The first, second and third show that the sought sphere (a\ V) must be coaxial with the given spheres, and we have, in fact, on eHmination of x, y and z, a"(^-zo+a(r-r)+aXr-o=o, a'cos e'\l - V) + a cos e{V - n + a'cos e\i" -0 = 0. Substituting for a" its value, ^(^a"^-\-l"), the equation cos e"J{T[a{V - n + aXr - IW + l'Xl - IJ}^ + a cos e(V - r) + a' COS e\V' -0 = becomes a quadratic, which gives two values of V for each value of cos Q". One sphere only is cut at right angles because the condition becomes linear in I", Ex. Reduce the equations of a pair of spheres to the form, p2-2?^Stt/) + ^=0; p2_2^;Sa/o + ^ = 0, where Ta = l* {a) Prove that all spheres of the family obtained by giving various values to 7^ in p2 _ 2^^Sa/> + ^ = intersect in a common circle. {b) Examine the condition for the reality of the circle, and show that whether real or imaginary, it lies in a real plane. (c) If the circle is imaginary, there are two real point spheres of the family. Find them. {d) The spheres of the doubly infinite family p^-2S(3p-l = 0, S^a=0, formed by giving all possible values to the vector /3, cut the spheres of the family (a) at right angles. Art. 43. Given any three spheres, p^-2Sap-hl = 0, p^-2S/3p + m = 0, p^^2Syp + n = 0;...(l.) the radical planes of each pair intersect in the line, 2Sap-l = 2S/3p-m = 2Syp-n; (ii.) ov p = i (lY/3y + m Vya + nVa^XSa/Sy) " ^ + tY{py + ya + a^). (ill.) If the origin is taken on this line, l = m = n; and if it is taken where the line intersects the plane of centres ABC, the equations of the spheres may be reduced to the type, p2-2SAC/) + ^ = 0, Sku = 0, (IV.) the vector v being fixed, but k being susceptible of various values. The spheres of this family (iv.) of given radius (a) have their centres on a fixed circle, It is ^asy to verify that the radical axes of every three out of four given spheres intersect in a point. This point is the radical 52 THE SPHERE. [chap. vi. centre of the four spheres, and is situated at the extremity of the ^~2^SaV(/5y + y^+W the fourth sphere being p^ — 2SSp +p = 0. It may be verified that if in this equation p and 8 are rendered arbitrarily variable, we fall back on the radical axis of three spheres. If, in addition, y and n are arbitrary, the same equation represents the radical plane of two. For example, we may put 8=xa+^f^ + zy, where x, y and z are arbitrary. Ex. 1. Find the locus of the centre of a sphere cutting three spheres orthogonally. [Let 8 and jo determine the sphere whose centre is sought, and let the three spheres belong to the family (iv.). The condition ^ + jo - 2SSk = must be satisfied by three values of the vector k. Hence p= ~l, 8 || v, and the locus is the radical axis.] Ex. 2. Find a sphere cutting four spheres orthogonally. Ex. 3. If four spheres are mutually orthogonal, their centres determine a tetrahedron self -con jugate to a sphere. [Let the spheres be referred to their radical centre. The conditions are ^ = Sa^ = Say = Sa8 = S/?y = S/3S = Sy 8, and the centres are conjugate in pairs to the sphere p^ = l.] The Method of Inversion. Art. 44. We have seen that represents a vector having its tensor reciprocal and its direction opposite to the tensor and the direction of the vector p (Art. 16). . Hence more generally if CT' = p' -y= - R%p^y)-'= - RK CV-\ (I.) P and P' are inverse points with respect to the sphere, centre C and radius R, for UCP' = UCP, TCP'TCP = i22 The inverse of the sphere T(yo — a) = a is T(y-„- R^ \ n./ NO ^-02Q« — y , __^^ a, or T(a-yf-2R^S p.yj -> - -V- // — ^_^'T(p-y)2 -' The symbol T prefixed to the scalar on the right is intended to \ show that it is to be taken positively. Thus, to invert the given sphere into a sphere of radius b, we have aR^ h— + rn/ _ — .2_ 2 ^ccording as T(a — y)> or and from these 7=^1^. and ^=^-|^(T(a-^)-(±6-an There are two real positions for the centre, but there may be only one positive value of H?.'\ Ex. 3. Invert a system of coaxial spheres into concentric spheres. [A system of coaxial spheres p^ — '2,w'^ap + l=Q inverts into a system of spheres having their centres on the line locus, ^ f Y^-22vSya + r If this is independent of ?^, it is easy to see that y'^ — l = 0, y||a, or y=±as/—l. The centre of the inverted spheres is ±a\^ — It ^aR^ :>/ -I. Ex.4. Prove that ^g ^(3 zy p = i- 4 represents a sphere through the four points a, b, c and d. [Invert with respect to the point d.] Art. 45. The following examples relating to a sphere and a tetrahedron are easily solved by the formulae x. or xii. of Art. 38, or by the method of Art. 39. Ex. 1. Determine the sphere through a, b, c and d. [The vector k to the centre is k= —\v~^^\(^= —\Yi\.~^d?^ and the squared radius is i?2= - v-^S^a^- ^^"^(^Xa^)^.] Ex. 2. Given four spheres having their centres at a, b, c and d, and their radii equal to a, 6, c, o?, find their radical centre. [If w ^s the vector to the radical centre, and if A = (w — a)^ + a^, we have co= -|v-i2X(a2 + a2), A = v-^2^(a2 + a2)+i^-2(2A.(a2 + a2))2.] 64 THE SPHEEE. [chap. vi. Ex. 3. Describe a sphere to cut four spheres orthogonally. Ex. 4. Describe a sphere to cut four given spheres at given angles. [Here there are four equations of the form (K-af + a^- 2aR cos 6 + R^ = 0. Multiplying by the scalars I and the vectors A and forming the sums, the equations, v{K^ + R^) + ^l{a^ + a^)-2Ri:iacose=0; 2KV + ^X{a^ + a'^)-2R'2XacosO=0, are obtained. Substitution for k in the first gives a quadratic in R. For the origin at the radical centre, the equations are, Rm'ZXa cos Oy + v^-2Rvi:iacoiiO + hv^=0 ; Kv = R^Xaco&e.] Ex. 5. To invert four spheres into four others of given radii. [If a', b', c\d' are the radii which the inverted spheres are required to have, and if the vector i terminates at the centre of inversion, t2 - 2Sta ^o?^d^±-,R^ = ^. ( Ex. 2 of last Article.) Taking the origin at the radical centre, ^ ' a ' a These lead to a quadratic in R^ for each set of signs.] Ex. 6. Find the equation of a sphere touching the four faces of a tetrahedron. [0 = y+r2±TA; = 'y/< + r2±aTA.] Ex. 7. Find the condition that five points a, b, c, d, e should lie on a sphere. [In the notation of Art. 39, p. 43, this is aa^ + ^/^^ + cy^ + ^S^ + ee^ = 0, or oa2(bcde) - ob2(acde) + oc2(abde) - od2(abce) + oe2(abcd) = 0.] Ex. 8. If five spheres are orthogonal to a sphere, prove that Pa (bcde) - Pb (acde) + Pc ( abde) - p^ (abce) + Pg (abcd) = 0, where a, b, c, d, e are centres of the spheres and where p^, Pb, Pc, Pdj and Pb are the powers of any point with respect to the five spheres. Ex. 9. If five spheres cut a sixth at the angles ^, Q\ etc., prove that the radius {K) of the sixth is given by the relation 2pa (bcde) = '2,R^a cos ^(bcde). Pa being defined as in the last Example, and a, &, c, c?, e being the radii of the five spheres. Ex. 10. Find the equation of a sphere in anharmonic coordinates. [Compare Art. 40, p. 43. The imaginary cone standing on the circle at infinity is T/)2 = 0, or i2 = STa2aV-22Sa/5a6A7/ = 0, and a sphere is 12 + '^ax2lx =0.] Ex. 11. Prove that the equation of the sphere circumscribing the tetrahedron abcd is in anharmonic coordinates, 2T(a-/5)2a6.r3/ = 0. ART. 46.] INSCKIBED POLYGON. 55 Art. 46. The product of the successive vector sides of a poly- gon of odd order inscribed in a sphere is a tangential vector at the initial point of the polygon ; and if the number of sides is even, the product is a quaternion whose vector part is parallel to the vector radius to the initial point The centre of the sphere being O, and A^, A^ being successive vertices, the isosceles j-triangle A^Afi is inversely similar to AjAgO, and therefore (Art. 18, p. 14), Thus, if OAj — a^, OA2 = a2, etc., A^A2 = yi, A2A3 = y2, etc., «2 = - yi«iyi " ^ «3 = - y-i^ij-i " ^ = + r2yi«iyi " V2 " ^ ^^c. ; and generally, the polygon being closed so that a^+i = ai, «l = (-)''2«l9'"^ where q = ynyn-i -" y^yi (i-) For an odd number of sides, gai + aig = 0, or aiSg + SaiVg = 0, or Sg = 0, Vg±ai; ...(ii.) and for an even number, qa^ — a^q = 0, orY .a^q = 0, or Yq\\a^ (ill.) In the first case {n odd), the product is a vector, and is perpen- dicular to ttp or parallel to a tangent at A^. In the second case {n even), the product is a quaternion having its vector part parallel to a^ In connection with this article and its examples. Art. 296 of the Elements of Quaternions should be consulted. Ex. 1. The equation of the sphere through four given points a, b, c, d may be written in the form S(p-a)(a-/3K/?-r)(7-S)(S-p) = 0. . Ex. 2. The normal at the point p on this sphere is parallel to V(p-a)(a-^)(/i-y)(y-p); and the vector CT being variable, S(^-p)(p-a)(a^/?)(/5-y)(y-p) = is the equation of the tangent plane at p. Ex. 3. The equation of the circle abc is V(p-a)(a-/?)(^-y)(y-p) = 0, and the tangent to the circle at the point p is V(C7-p)(p-a)(a-/:?)(^-p) = 0. [The vector part of a product of an even number of coplanar vectors is perpen4icular to their plane, being a product of half the number of coplanar quaternions. Therefore when the points are coplanar the expression for the normal vector in Ex. 2 must vanish, as this vector cannot be perpendicular to the plane. The equation is also susceptible of geometrical interpretation.] 56 THE SPHEEE. [chap. vi. Ex. 4. The product of four successive vector sides of a quadrilateral inscribed to a circle is a positive or negative scalar according as the quadrir lateral is crossed or uncrossed. [Use the relation U . ^ — = ± U — , which asserts that the angles abc, adc BC DC are equal or supplementary.] Ex. 5. The " anharmonic function of four points in space " being defined by the equation , . AB CD (abcd) = — • , ^ ^ BC DA examine the nature of this quaternion when the four points are coney clic. Ex. 6. Prove that the anharmonic functions of any four points in space satisfy the relations (abcd) + (acbd) = 1 , ( abcd) . ( adcb) = 1 ; and that (abcd) = K -r-7j C B where b', c' and d' are the inverse points of b, c and d with respect to the point A. [Note that a-^ - ^"^ = a-^ . (/5 - a) y8-\] Ex. 7. If (oABc)=— 1, prove that ob-^ = ^(oa"^ + oc-^). Ex. 8. Inscribe a polygon to a sphere, given the directions of the sides of the polygon. [Here J5q is given, q denoting the quaternion in the text ; and (ii.) and (ill.) show that the vector to the first corner is _L VU^', or else || ± VIJ^'.] Ex. 9. For the gauche quadrilateral oabc, which may always be con- ceived to be inscribed in a determined sphere, we may say that the angle of the quaternion product, L (oa . ab . bc . co), is equal to the angle of the lunule, bounded by the two arcs of small circles cab, ocb ; with the same construction for the angle of the anharmonic L (oabc), or L (oa : ab . bc : co). {Elements, Art. 296 (15).) Ex. 10. Let ABCD be any four points in a plane or in space, connected by four circles, each passing through three of the points ; then, not only is the angle at a, between the arcs abc, adc, equal to the angle at c, between cda and CBA, but also it is equal to the angle at b, between the two other arcs BCD and BAD, and to the angle at d, between the arcs dab, dcb. {Elements, Art. 296 (18).) Ex. 11. The vector part of the product of four successive sides of a gauche quadrilateral inscribed in a sphere is equal to the diameter drawn to the initial point of the polygon, multiplied by the sextuple volume of the pyramid, which its four points determine. {Elements, Art. 296 (43).) 'Art. 47. To inscribe a polygon in a sphere so that its sides may pass through given points. Let the unit of length be selected equal to the radius of the sphere. Let the centre be taken as origin, and let p, p^, p^, ... Pn{ = p) be the vectors to the vertices, while /3p fi^y ••• ^n are the ART. 47.] INSCRIPTION OF POLYGON. 57 vectors to the fixed points. The rectangle under the segments of the chords through ^^ is (/>-A)(pi-/3i)=i+A^; (I-) so that ft= -§^= -^^' if Pi = fiv 9x = l ("•) Again, P2 ~ Pi V2 ~ H2P and it is easy to see that, in general, Pm — ^mp ["'V^') ^m^^ Pm^m-i'T\ ) Pm-vj Finally, P = {-Yfzfp ^^ Pn = P- Pn^P. ?» = ? (v.) Two cases now arise according as n is odd or even. In the first place, if n is odd, remembering that p^= — 1, pp-\-pp = pqp-q = p(qp + pq); or pSp + Spp = p{pSq+8pq); or, separating the scalar and the vector parts, Spp + Sq = and Spq-Sp = (vi.) Introducing the imaginary of algebra, these may be combined into the single relation, S(p + s/^l)(q + J-lp) = (VII.) The equations (vi.) give a line locus for p which intersects the sphere in two points — real or imaginary — which satisfy the conditions. In the second place, if n is even, pp-pp = pqp-\-q = p(qp-pq); or V . pVp = p V . Vgp. Adding to each side x — Spj^, we have YpYq + Yp — p-^x=—xp; and this gives SYpYq = — xSpYq on operating by SYq. Hence, p(Yq + x)= -Yp-x-'^SYpYq, as we see by adding SpYq to each side. Thus, as appears on taking the tensor, remembering that Tp^ = l. This quadratic in x^ has one negative root. The other root is positive, and tliere are thus two real values for x, and two real points satisfying the conditions. 68 THE SPHERE. [chap. vi. We have now to determine jp and q. Multiply jpm in equation (iv.) hy s/ —\ and add it to q^, and = (/3^ + (-W^)(g,n-i + x/-li)m-i). This gives at once, on referring to (ii.), ...(/3,+(-)V-l)(ft-x/-l)v/-l J""^ and the real and imaginary parts of this product are q and _p. A quaternion of the form g + -v/ — 1 . p is called by Hamilton a bi-quaternion. (Compare Art. 22, p. 20.) Ex. Show that in the notation of this article [Multiply q + sj — Ip into K.q + J — lKjp and separate the real and the imaginary parts.] EXAMPLES TO CHAPTER VI. Ex. 1. The sphere which has its centre at the origin, and has the vector OA, or a, with a length Ta = a, for one of its radii, may be represented by any one of the following equations : p a p + a p + a p + a \ a a/ T{p-ca) = T{cp-a), which are transformations one of the other, and each of which exhibits some geometrical property of the surface. Ex. 2. The circle which has its centre at the origin, which lies in the plane Sap = 0, and which has Ta for its radius, is represented by the equation a/ Ex. 3. If ^ is a variable parameter, in absolute magnitude not greater than unity, the equations s^=^, fv^y=^2-i, a \ a/ represent a system of circles which generate a sphere. Ex. 4. The equation of the sphere through the four points o, a, b, c may be written in the forms S (OA . AB . BC . cp . po) = ; a^SfSyp + fS^Syap + y^SafSp = p^Sa/Sy ; SC/3-1 - a-i)(7-i - a-i)(/)-i - a-i) = 0. Ex. 5. If we project the variable point p of a sphere into points a\ b\ c^ on the three given chords oa, ob, oc by three planes through that point p parallel to the planes boc, coa, aob, we shall have the equation Op2 = OA . OA^ + OB . ob' + oc . 00" I ART. 47.] EXAMPLES. 59 Ex. 6. The expression* p = r]<^j*kj~^k~* or p= r]i^j^k'^~^, in which r is a given scalar, ^, j^ k mutually rectangular unit vectors, while s and t are parameters, represents a sphere concentric with the origin. The expression may also be put under the form and it may be expanded as follows : p=r{{i cos tir +j sin tir) sin stt + ^ cos stt }. (a) Show how to establish the first form of the expression by the properties of conical rotations. Ex. 7. Show that the equation ,2 fw + p- ay in which w is a real scalar capable of receiving any value consistent with the reality of the vector /a, represents the portion of the plane S(/) — a)y8=0 included within the sphere T.{p — a) = Tfi. Ex. 8. The equation t T (w + /a) = 1 , in which /a is a real variable vector and w a real variable scalar, represents the region enclosed by the sphere T/) = l. Ex. 9. A sphere passes through the intersection of the planes SAp=0, Sfxp = 0, Sv/t) = 0, which cut off caps the sum of whose areas is equal to ^Tra^. Show that the locus of the centre is represented by 3T/32 + T/o . S (U A + U/x + Uv)/D = a2. Ex. 10. The centre of a sphere of constant radius a describes a circle of radius b concentric with the origin and in the plane Sa/o = 0, Ta=l. The equation of the surface generated may be written T(±bV.a-'Yap-p) = a', or . 2bTYap = ± (ly + U^ - a?) ; or 462(Sa/3)2 = 462T/32_(T/32 + 52_a2^2. or 4a2T/)2 - 462(Sa/o)2 = (T/32 -b'^ + a^f ; g^ /)-a(a2 -62)1 ^^5. P + a(a^-b^)i a or p= ±bJJ . a~^VaT + aUT (r a variable vector). (a) Taking f3 and y, two auxiliary unit-vectors perpendicular to one another and to a, show that a^Tp2 _ h^(SapY = a2(Syp)2 + Sp(a/? + « VP^^) Sp(a/3 - a JW^^), and prove that each of the planes Sp(al3±as/b^'a^)=0 touches the surface in two points and cuts it in a pair of circles. ■^ExsCmples 1-6 are taken from Hamilton's Elements of Quaternions. t This and the last example are to be found in Hamilton's Lectures on Quaternions, Art. 679. 60 THE SPHERE. [chap, vi.j Ex. 11. If p and q are variable quaternions, while a and /8 are given vectors, show that OY=p=-pap-^ + qfiq-^ represents the shell included between the spheres T/o=Ta + T/?, T/)--=T(Ta-T/?). (a) If y is a third given vector, and if a and h are given scalars, the point p terminates on the circle of intersection of the spheres T(ap-y) = T(a-6)A T(6/)-y) = T(a-6)a, " when the quaternions p and q are connected by the relation apap~^ + hqfBq~^ = y. ih) When the relation Vy {apap-'^ + bq/Sq-^) = connects p and q, the locus of p is the surface 4(Spyf{abTp^ + (a - 6)(aTa2 - ^T^^)} = Ty2{ {a + b)Tp^ + {a- b){Ta^ - T^2) p. (c) If the condition Sy (apap-^ + bq/Sq-^) = is satisfied, the point p must render the expression 4(S/)y)2{a6T/o2 + (a-&)(aTa2-6T/52)} + {a- 6)2Ty 2 (T/)^ + Ta* + T/?* - 2Ta2/52 _ 2T /3^p^ - 2Tp^a^), less than zero. Ex. 12. The bars ab, bc and cd are connected by universal joints at B and c, and also to two fixed points a and d. If p is a point fixed in bc, and if we write p = AP = AB + ?*BC, p' = FT) = tl'BC + CT), U + ^l'=l, where u is a given scalar, and also AB=pap~^, BC=q/3q~^, CD = ryr-^, da = 8, where a, /?, y, 8 are given vectors and p, q and r variable quaternions, prove that r, .1 P'u\p^ + u^(3^ - a2) - pu{p'^ + u'^fS'^ -y^) + t ^^^ - 2mc'Ypp' ' t being a scalar, and hence show that the inequality f^— ^Tuu'Ypp' determines the region within which the point p must lie. (a) If the bar bc remains parallel to the fixed vector ^, the locus of p is the intersection of the spheres (^p-u^y = a\ {p'-u(Sf = y\ {b) In this case the locus of the bar bc is the cylinder (c) When the quadrilateral abcd is coplanar and when the motion is confined to the plane abcd, find equations of the form ART. 47.] EXAMPLES. 61 for the path of any point of a plane lamina attached to bc, t being a constant unit- vector perpendicular to the plane abcd, and /(^, y) being a scalar function of x and y. Ex. 13. Solve the equation -J- + J: \ I_=0 p-a /)-/? p-y p-8 (a) If /)', a', f3' and y' are the vectors from the point d, the extremity of the vector 8, to the inverses of the extremities of p, a, /? and y with respect tOD, 111 rp-a p- 15 p-y Hence deduce the relations p'-li_ y'-l3! _p'-YJY-p!\^ p'-a! p'-y' y'-a! \y'-a'J (b) Solve similarly the quaternion equation —+—,-- ^=0 q-a q—b q-c q-d by assuming {q-d){q'-d) = {a-d){a'-d) = {b-d){b'-d) = {c-d){c'-d) = l. -^ (Eobert Eussell.) CHAPTER VII. DIFFERENTIATION. Art. 48. The equation OT = p = (n.) and for the sake of argument we shall suppose that the para- meter t represents the time, so that P is the position of a moving point at the time t, and P' its position at the time t\ Fig. 23. Writing PQ' = ^^=M*1^), („,) it is apparent that had the point passed from P to V, in the time ART. 48.] DIFFERENTIAL OF VECTOR TO CURVE. 63 t' — t, not along the curve and with varying velocity, but along the chord and with uniform velocity, and that had it continued to move uniformly along the production of the chord, it would have reached the point Q' in unit time. In a similar manner the point Q" would have been reached in unit time had the point moved uniformly along the chord PP"' in the time in which it had described the curve and had its motion been continued along the chord without alteration. In the limit PQ represents rigorously the velocity at the point P, in magnitude and direction, for Q is the position the point would have reached in unit time had it left the curve at the point P, preserving unchanged the velocity it actually possessed at that point. The equations t' = t * ~t h=0 1^ = lim«(,!,(« + i)-^(0 .(IV.) are equivalent modes of expressing the limit to which we advance ; the third being perhaps in closest agreement with the illustration. It is usual to write PQ = dMi) = ^'(0 (V.) as an abbreviation for the limit. The vector (f){t) is the derivative, the derived or the differential coefficient of the vector function ^(i) of the scalar t, and the differential of (f>{t) corrcvsponding to any scalar differential d^ of t is d.0(O = lim7i,L(^ + -)-0(O) = ^XO-d^. -. (VI.) 7i=a3 \ \ n/ / This is a vector tangential to the curve and of length propor- tional to the differential d^ which may be large or small. If t is the arc of the curve, the vector (l>\t) is of unit length, for in this case we may consider t to represent the time for unit and uniform velocity along the curve. If 0X0 = ^' ^^® extremity of the vector OP = ^ {t) is a cusp or stationary point. Ex. 1. The curve /o = acos^ + ^sin^ represents an ellipse of which a and /5 are conjugate radii. [The vector p' = ^=-a&mt-\-fico8t = aco^i^ + tj + fi&m{^-\-tj is the radius conjugate to p.] ^ Ex. 2. The parallelogram determined by conjugate radii of an ellipse is constant ki area. • [Vpp' = Va/5.] 64 DIFFERENTIATION. [chap. vii. Ex. 3. How is the point at the extremity of the vector cos^(t + t') n sin ^(t + t') related to the points t and t' on the ellipse ? Ex. 4. The curve p = at'^ + 2^t + y is the trajectory of a point moving with uniform acceleration. Ex. 5. What is the curve Investigate its properties. Ex. 6. A helix is represented b}^ /o = a cos t + f3 sin t + y^, the vectors a, /3 and y being mutually rectangular, and the tensors of a and /3 being equal. Determine all particulars. Ex. 7. A conic is represented by the equation at'^-h2(3t + y ^ at^ + 2bt+c' Its centre is at the extremity of the vector _ ac — 2/36 + yg "" 2{ac-b^) ' [The curve meets an arbitrary plane in two points. Find the pole of a chord, and in particular of the chord at infinity.] Ex. 8. The equation YpaYfSp = (Vaf3y represents a plane curve — a hyperbola of which a and f3 are the asymptotes. Ex. 9. Write the equation of the conic of Ex. 7 in a vector form' independent of the parameter, Akt. 49. A vector function of two parameters, t and u, P=^=0, or S(p-)v=^0, if v\\Y(f>',, (IV.) and the vector p is normal to the surface. The equation of the normal is V(yo-0)V^>, = O, or Y(p-f/>)v = 0, or p = ^-^xv (v.) Ex. 1. If <^(^) is a function of a single parameter, the equation p = cf>(t)-\-u'(t) represents a developable surface. [This surface is generated by the tangent lines to the curve p—' + uj)").' or Y', and is independent of u. The tangent plane is S(p-(f>-u<^')Y'(f>"=0, or 8{p-cf))(f>'(f>"=0, and as this is independent of ii, it touches the surface all along the generator determined by t Conceive the tangent plane to roll over the surface and the successive generators to become attached to it, the surface will be unfolded or developed in the moving plane.] Ex. 2« The equation in which a is a constant vector, represents a cylinder standing on the curve p = (fi(t) and having its generators parallel to a. The equation p = ic{t) + a represents a cone standing on the same curve and having its vertex at the extremity of a. Ex. 3. Find the locus of a line joining corresponding points on two homographically divided lines ab and cd. [The surface is p^^^±i^t±^ if p=<^, pJ.l±^ are the l + t+s{l + t7ii) ^ 1 + ^ l-\-tm homographically divided lines. This is a hyperboloid of one sheet.] Ex. 4. Show that the variable line determines homographic divisions on the lines ac and bd. Ex. 5. Find the scalar equation of the locus of Example 3, and show that it may be reduced to the form xr=zw, where X, Y, Z and W are planes. Ex. 6. Find the locus of a line similarly dividing two given lines ab and CD. Art. 50. The equation ^'^ 9-=-^{t,n,v\ (I.) in which t, u and v are variable parameters, may at pleasure be J.Q. E 66 DIFFERENTIATION. [chap, viu regarded as determining (i.) a singly infinite family of surfaces, for example, the surfaces found by assigning various but constant values to v ; (ii.) a doubly infinite family of curves, for example, t variable, u and v constant ; (ill.) any point in space, for we can in general find one or more sets of values of t, u, v corresponding to an arbitrary vector p. The scalars t, u, v are curvilinear coordinates of the extremity of the vector p. Differential of a quaternion function. Art. 51. The differential of a quaternion function of a quaternion is defined by the equation d.F(q) = limjl{F(q + '^)-Fq]^f(dq), (l.) or d.F{q) = lim l{F(q+hdq)-Fq}=f(dq). a definition in complete agreement with the results of Art. 48. The function f(dq) is a linear and distributive function of the differential dg, while it also in general involves the quaternion q in its constitution. To prove this proposition, observe that if r and s are any two quaternions, f{r+s)^\imn[F{q + ''-^)-Fq] or simply f{r+s)=f{r)+f{s) (ll.) As a corollary, f(xr) = xf{r), (lil.> if X is any scalar. As an example, d.,^=lim^4(,+J)-,^} = lim^4,H,.5-.J,+(-t^,^} and thus d.q'^ = q.dq-\'dq .q .(iv.) ART. 52.] QUATERNION FUNCTION. 67 There is a notable difference between the differential of a function of a single scalar and a function of a quaternion, which is clearly illustrated by this example. In general, from a differ- ential of a function of a single scalar d . F{x), we can form a differential coefficient , , which is absolutely independent of d X- d a^ daj. Thus, -A — = ^x, but — T^ = g'H-dg .q.dq-'^ is not indepen- dent of dg. And this, which is a consequence of the non- commutative law of multiplication, is really quite in keeping with the ordinary theory, for if F{x, y) is a function of two independent scalars x and y, we cannot form a complete differential coefficient from d . F(xy) =—dx-\-—dy, where dx and dy are arbitrary, though we can of course form the partial differential coefficients 7^ and ;r— . We must remember that a Zx ^y quaternion is a function of four numbers, and that a differential dq is susceptible of a quadruply infinite system of values. As a second example, d.q-^^—q-'^ .dq.q-^, (v.) for d.q-'^ — \\mn\(q-\ — dq) — g"^[ Ex. 1. Prove that d.S^ = Sd^, dV^=Vd^, dK^=Kd^. U [Note that these symbols are distributive, or that S(2'+%-idg')=Sg'+w-iSd5'.] Ex. 2. If V is a vector function of a variable vector p, and if dv = <^p £v show that '=^ •d'p-^' and interpret the meaning of this relation. Ex. 4. Compare an element of vector area with the corresponding element into which it is changed by inversion. [The elements are Vdpd'p and R^p-* . p-^Vdpd'p . p.] Ex. 5. Prove that ^-. (a)dVyq = V^.Vyq. (b) dyVq = Y(v^.Vqy (c) dsu^=s(v^.u^). (d) dLq = s{^--^y Ex. 6. The vector a being constant, prove that d . qaq-^ = 2V . Vd^^-i . qaq-^ = 2q(Y . Yq-^dq . a)q~K Ex. 7. Prove that da* = d.rAogTa + |Ua) a% where a is a constant vector and a: a variable scalar ; and that da- = ^.S^.a* + V^.Va^ a a where .v is constant and a variable. Art. 54. If P is any scalar function of a variable vector p, a differential of P is connected with the corresponding differential of p by a relation of the form '4 dP=-.SKl/0 (I.) the vector v being a function of p but independent of dp. 70 DIFFERENTIATION. [chap. vii. The rate of variation of P along any direction a (Ta = 1), may- be written in the form d.P=-Sm, (II.) it being understood that the suffix a attached to d signifies that the corresponding differential of p is dp = a (III.) This rate of variation as expressed by (ii.) is the projection of the vector v along the vector a, and consequently the rate of variation of P is maximum along the vector v, being then equal to Tj/, while it is zero along any direction normal to v. Having given the variations of P along three non-coplanar directions, or what is equivalent, having given the differentials dP, d'P and d"P of P corresponding to three non-coplanar differentials dyo, d'p and d!' p of p, we can determine the vector v. We have in fact dP=-S,.d/o, dT=-Si.d>, d''P=-SvdV, (IV.) and by the fundamental formula of Art. 26, p. 24, we find Vd>dV . dP + Vdydp . dT + Vdyodp . d"P . """ Sdyod>d> ^ ^^ Thus it appears that the vector v is derived from P by means of the differentiating operator -_ vd>d> . d + vd>dp . d^ + vdpdy . d-- ^~~ sdpd>dv ' ^^^'^ in which dp, d'p and d!' p are any three non-coplanar differentials of p, and in which d, d' and d'' are the corresponding symbols of differentiation. Ex.1. Prove that VSa/)=-a, VT/o=+U/D, VTVa/3=+UVa/o.a, VT(p - a)-i = - U(/> - a) . T(p - a)-2. [These follow from the relation dP= - SdpyP.] Ex. 2. Show that SaV.Tp-i=-Sa/3.T/9-3, S)8VSaV . Tp-i = 3SapS/3p . Tp-- + Sa^ . Tp-% SyVS^SVSaV . Tp-i = - 3 . bSapS^pSyp . Tp-^ - 32S^y Sa/) . T/^-^ Art. 55. The form of the expression found in the last article for VP suggests a new view of the subject which is applicable in the general case when P is a vector or even a quaternion function of p. Suppose a parallelepiped constructed having its edges equal to any three vectors dp, d'p and d"p, and having its ART. 55.] THE OPERATOR V. 71 centre at the extremity of p. If we suppose the vectors arranged in positive order of rotation (compare Art. 24), Yd' p6!'p is the outwardly directed vector area of the face having its centre at the extremity of p-{-hdp ; and — Yd'pd"p is likewise the outwardly directed area of the face, centre p - ^dp. Also — Sdpd'pd"p is the volume of the parallelepiped. Let F(p) be any function of p, scalar, vector or quaternion, then the sum of the products oi the outwardly directed vector faces into the value of F(p) at their middle points is Yd' pd"p.F(p + Jdp) + Vd^d^. F(p+id'p) +Ydpd'p.F(p+ Jd» - Vd>dV. F(p - \dp) -Yd'' pdp.Fip -id'p) - Ydpd'p.F(p- ld"p), (I.) and the quotient of this sum b}^ the volume of the parallelepiped is ^Yd'pd"p.{{F{p + idp)-^F(p-^idp)} ^Sdpd'pd"p ^ -^ Each edge being diminished in the ratio - , the quotient becomes .-^SYd>dv{4 + i^dp)^4^^^dp)} _ --n-''Sdpd'pd"p ^ '^ So that when n increases without limit, or when the parallele- piped whose edges are -d/o, - d'p, - d"p decreases without limit, Ifh 'Yh 71/ the limiting value of the quotient (iii.) is (compare Art. 51 (i.)) ZYd>dV..{4-FA^d,)-4^i^dp)} Sdpd'pd"p = ^ ^Yff/:^^^ ==v.Fp (IV.) Sdpd'pd"p ^ ^ ^ Thus V .F(p) is the limit of the ratio which the sum of the products of the outwardly directed faces of a parallelepiped into the mean values of F(p) over the faces bears to the volume of the parallelepiped. And the vectors dp, d'p, d"p being arbitrary, the result is independent of the shape of the parallelepiped. Take the case in which F(p) is a vector function (d3. is the outwardly directed vector element of area. Since it has appeared that these results are independent of the shape of the parallelepiped, it follows that they are true for any closed surface formed of a single sheet, and we have iimiM(e)=v.i'(p). (VI.) where div is an outwardly directed element of vector area of the surface, and where v is the volume, the limit being arrived at when the surface becomes indefinitely small. Art. 56. Towards further elucidation of the operator V, con- sider the analogous integral taken round the vector sides of a parallelogram, having its centre at the extremity of the vector p. Circuiting in the positive direction and forming the product of the vector sides into the corresponding values of F{p) at their middle points, the sum is dp . F(p - JdV) + d> . F{p + idp) - dpF{p + id» - d'pFip - idp). Collecting terms and dividing by the area of the parallelogram, the result is dp .{F(p+ jdp) - F(p - jdp)} - dp{F(p + |d» - F(p - jd'p }} TYdpd'p ART. 56.] CIECUIT THEOREMS. 73 Now let the parallelogram be indefinitely diminished by replacing dp and d'p by - dp and - d'p, and we have in the limit, ^^ n-n'Ydpd'p _ d'p.dFp-dp.d'Fp . TVdpd> • ^ ^^ But this is equal to ^ {V . Vdpd>(Vd>d> . d . 4- VdV/) . d^ . + Vdpdy . d" . )}Fp -Sdpd>d'VTVdpdV «S=QO v/ because Y(ydpd'p . Vd'pd'p)= — d'pSdpd'pd'p, etc., so that the integral is I^MV_V).^, = y(U..V).^, (n.) if Ui/ = UVdpd'p is the normal to the area about which the direction of circuiting is positive. As in the last article, we have for any plane closed curve without loops, )imlM(£)=V(U..V).ii'(p), (III.) ^1 dp being now a vector element of arc of the curve and A being its scalar area. In particular for a vector function {a) of p, we have separately lim I^ = S(VU.V . 0-), Km 1^^ = V(VU.V . o-)....(iv.) It is obvious on using the expanded form of V that we may write S(yui/V . (7)=s(Ui.vvo-)=sUi/V(7, (v.) or that we may in this relation at least treat V as a vector in combination with other vectors, it being understood that V operates on a but not on Ui/. This result leads us back to an interpretation of War analogous to the interpretation of VP in Art. 54. We have SU.VV(7 = lim I^, (VI.) or th.e limit of the ratio which the integrated component of a- along^the arc of a plane curve ( — f Sdpcr) bears to the area of that curve, is equal to the component ( — SUr/Y V(t) of the vector Wo- 74 DIFFERENTIATION. [chap. vii. along the positive normal to the plane. This is a maximum and equal to TWo- when the plane is at right angles to U Wo- ; it vanishes when the plane is parallel to that direction. If SdpG- is the differential of P (some scalar function of p), the integral jSd/oo- depends merely on the limits between which the integral is taken (leaving aside cases in which singularities occur), and is in fact P(p2)^P(pi) if the integration extends from p^ to /Og. For any small closed circuit therefore thie integral vanishes, the initial and final points of the path of integration being coincident, and therefore VV(7 = 0, if So-d/) = dP (VII.) Conversely, if VV(r = 0, we must have Sa-dp the differential of a scalar P ; for in this case the integral round any small closed circuit vanishes, or what is equivalent, the integral from pi to p.2 is equal and opposite to the integral back by another path from p^ to p^, or again, the integral from p^ to p^ is indepen- dent of the path. These results will be extended to the general case of curves which are not small. At present we remark that VVVP = 0, or YV^P = 0, or V^P = scalar, (viii.) if P is a scalar function of p, is involved in equation (vii.). Art. 57. It is useful to express the operator V in various forms. If, for example, as in Art. 50, we suppose the vector p to be expressed in terms of three parameters u, v and w, and if we write ^P='^'^^='pA'^' d> = |^dv = P2dv, d''p = :^dlV = p^dlU, (I.) the symbols of differentiation d, d' and d'' refer respectively to 11, V and tv, so that symbolically d = —.du, d' = — .di;, d" = :— .div (il.) ^u dv dw ^ ^ On this understanding, equation (vi.). Art. 54, becomes V= n (HI.) ^PlP2P3 If the parameters are so selected that the derived vectors Pi, p2 and yQg are always mutually perpendicular, the sj'^mbols V and S in (ill.) become superfluous, and the expression for V reduces to the simple form, ART. 57.] CANONICAL FORM OF V. 75 If the vector p is expressed in terms of the Cartesian coordinates x, y and z, so that p = ix-^jy-\-kz, we have /Oi = ^' P2—J^ P3 — h and ^=v.+4+4 <-) This last form may be regarded as the canonical form of the operator. We have, for example, when q is the operand, and we shall write so that in combination with its operand V acts as a vector in combination with a quaternion. Again if a is a constant quaternion, we have symbolically, an operand being understood, V.a = m;|-+ja;|- + /ca|- = V.Sa4-SVa + VVVa, ox cy oz ' "P^ /-^ 7^ ^x ^dy dz y-if and in combination with a quaternion, not the operand, V still plays the role of a vector. In combination with itself V-V-. = V.('^iH-i|+/4f) ^y ^^''^^^'^^z^'^^^y-^^'^yk ^x^ ^f" ^z^ -^^ -^^ and generally in all combinations V may be treated as a symbolic vector. Of course some little care is necessary when V is ex- pressed in the general form, but it is precisely of the same kind as the care required to distinguish between 76 DIFFERENTIATION. [chap. vii. Ex. 1. Show that iiq=W+ iX+j ¥+ kZ, 'dx 'dy dz \ 'dx By dz / + ■('^+'^^-'^]+k(— + — - — ). \^y ^ "dxl x'dz 'dx 9y / Ex. 2. Verify that V.Vo- = V2.o-=-(^,+|i+^,)o-, where cT = iX+jY+kZ. Ex. 3. Prove that Vp= -3 ; VVA/3 = 2A ; VUp= -2Tp-i ; Vp-i = Tp-2 j V2.T(p-A)-i=0 if /o is not equal to A; V=^TVAp= -T(VA-V)-M V2 log TVAp = ; VyTp = -f'Tp - 2Tp-i/Tp. Tt. , -,., 2V^yVAa V/?AVAa-VaAVA/?-] [For example, VVAp= f^= -^ ^^^ ^.J Ex. 4. Prove that VAV . /o = - 2A ; VVYAV . P= - AV^P^- VSAV . P. Ex. 5. Show that {aV + Va)q = 2SaV . q, (uV - Va) ^ = 2 VaV • q. |_±lere (aV + Va^.g^- ^ sdpd'pd> •'^^- "^^Sdpd'pd^^ ^J Ex. 6. If P and Q are scalar functions of p, show that V.P$ = VP.(2 + V^.P. Ex. 7. If jP and g- are quaternion functions of /), show that where the suffix is intended to denote that the affected symbols are not to be operated on by V. Ex. 8. Interpret the expressions YVV.PQ', SVVV'.PQ'R', where the accents indicate that a marked symbol is to be operated on by the correspondingly marked V. [If P and Q are scalars, the first expression is V(VP)(V^), or ^^('dPdQ_dPdQ\ \di/ dz dz dy /' This last expression is also true when P and Q are quaternions.] Ex. 9. Find an expanded form for V^ . PQ. Ex. 10. Find the expression for V in terms of the usual r, and <^ coordinates. [Use the relation (iv.).] ^^"Ex. 11. Show that g' . V = - K , VKg- where V operates on q in situ. [It is sometimes convenient to place the operator to the right of the operand.] Ex. 12. If fn(p) is any homogeneous function of p of the order n which O vanishes under the operation of V^ the function Tp~^"~^./„(/o) will vanish \ under the same operator. ART. 68.] ROOT OF QUATERNION. 77 [Expressing that V^(Tp^. /*,») = 0, we may write this relation in the form (\/ + Vy.(Tp'"'. fn)=0, provided we remove the accents after the operation. This expands iiito Vn>"^./; + 2SV'T/)'-V./„ + T/3"". V%=0, and observing that SpV .fn= -nfn because /„ is homogeneous in p, we easily find the equation reduces to m{2n + l'+m) = 0. This result is of importance in the theory of spherical harmonics.] Art. 58. Given a quaternion function p = F(q) of another quaternion q, we have seen how to express d^ in terms of dq (Art. 51). It is a more difficult problem to express dq in terms of dp, and we postpone the general method of solution for the present.* However, there are a few cases in which the problem can be solved directly, such as to find the dififerential of the square root of a quaternion. Here p = q^ or p^ = q, (i.) so that pdp-\-dp.p = dq (ii.) Multiply this by Kp and into p, and two relations equivalent to {11.) are obtained, Kp .p . dp-\-Kp . dp . p = Kp . dq ; p. dp .p-^^dp .p^ = dq . p. (ill.) Adding, we have d^ . (Tp^ + 2pSp +p^) = Kp.dq-\-dq.p because p + Kp = 2Sp, Kp.p = Tp^; or 4 . dp . pSp = Kp .dq-}-dq .p because Tp^ = (Spf^(Ypy, p^ = (Spy-\-2Sp .Yp-\-{Ypf; and hence ^^^ Kq^dq.q-^ + dq ^^^^ 4S^2 As another example, under which this might have been in- cluded, to find the differential of the n^^ root of a quaternion (n being an integer), we have p^qn, q—p^^ dg = dp._p'*-^-|-i5.dp.jp*'-2+...+_p"-^dp. (v.) Multiply dq into p and subtract the product pdq, and dq.p^pdq — dp.q^qdp, or V . Vdg Vp = V . Vdp Vg. (vi.) Thus, with an indetermined scalar x, ^,, Y.YdqYp^-x . Y.YdqYp , /^ , . ^ \ / x Vdp = Y^^^^ or dp= ^^ ^ + (^Sdp+^|(vii.) Turning to (v.), we have on substitution from (vii.), dg = 71 .^"-^SdpH- Vdp .j^'^-^+p . Vdp . p"-^+ . . . +j9'^-^Vdp = n.p^-^Sdp+^,np^^-^ + '^^^^f^ '"'$ x(^"-^ + Kp.i>»-H(Kp)2._p«-2 + ...-h(Kp)»^-^),...(viii.) I * See Art. 150, p. 273. 78 DIFFERENTIATION. [chap. vii. because q and p are commutative in order of multiplication, and because ap — Kp . a, or a (Sp + Vp) = (Sp — Yp) a if SaY^ = 0, the vector V . YdqYp . (Yg)'^ being perpendicular to Yp. Again, ^ +jvp.^ +etc._ ^__j^^ _^,^^ since ^ and K.p are commutative in multiplication, and the expression (viii.) reduces further to dq = n.p^'-'Sdp + ^,np^-' + ^ '^^^^^ ^ (ix.) Thus we have by (vii.) and (ix.) on elimination of x '^^■" Yq V nqYp)^ nq ' ^"^'^ and the sought differential dp is expressed in terms of ^, q and dq. diflferential oifq in the form d.fq-^Aiq, dq\ (i.) ;ion of q and of dq, linear in the latter, the tressed by d',fq=f2(q. dq)+Mq, d^q), (ll.) where /2(g', dq) is homogeneous and quadratic in dq., A similar process holds in general, and in particular if dq is constant, sa that d^q = 0, d^q—Oj etc., we have d-./5' = d./^_ife, dq)=Mq, dq) (ill.) Suppose that f{q) and its successive differentials up to the m*** are finite for finite differentials of q^ and consider the function F{x)=Ag+^)-f(q)-^.A{q,p)-^^Mg,p)...-0^f„,,(q,p), (ir.) in which .r is a scalar and q and p are two quaternions. Differentiating with respect to ^, and leaving p and q constant, we find by the general relation (in.), -^=fi{q + ^p, p) -Mq, p)-\-fi{q, p)...-^^^^:^fm-i{q, p\ ^'^F(x) X x^~^ -^^ -f2{q+^p, p)-f2(q, p)- i-fsiq, p)"--c^^;—^fm-i(q, p), (^ Art. 59. Writing the first differential oifq in the form to indicate that it is a function of q and of dq, linear in the latter, the second differential may be expressed by ...(V.) i d'^-^F(x) g^m-l ^fm-l{q + OCp, p) -fra-l{q, p), ^Jn =Mq+^p,p). Putting a;=0 in (iv.) and (v.), we see that F{x) and its successive deriveds up to the order m-l vanish when ^ = 0, and consequently n^)=h:iMq. pHr^\ (VI.) ART. 60.] TAYLOR'S SERIES. 79^ where r„i is some quaternion function of :r, q and py and where by (v.) lim -'^;^=\im{f,n{q,p)+r,n)=fm{q,p) (vii.> By taking x small enough it is consequently possible to render r,» infinitely small in comparison with /,„ (5', jo), or I lim ^/!'*" , = Q (viii.> Replacing .vp by p in (iv.), what we have proved is that where r^ is a function of q and jo, which becomes evanescent in comparison with fmiq, p) for sufficiently small tensors of p. This theorem is what Hamilton calls " Taylor's Series adapted to quaternions." In certain cases, for a large value of w, the term ^{U{q.p) + rjf becomes negligible, and we may write the expansion in the usual symbolic form, , 2 /(^+ji,)=ey(^)=/(^) + -./j(^,^) + j-2/2(g',^)+etc.; d^=p, (x.) or more explicitly for a vector variable, - /(/t> + C7) = e-s«^./(p)=/(p)-istJV./p4-|^ .(StJV)2./(p)/etc (xi.) Art 60. Instead of differentiating a second time with the same char- acteristic d, let the diflferential of d/(^)=/i(?, ^q) be taken for a new characteristic, d' corresponding to the differentials di'q and d'd^' of q and dq. The result may be written d'd./(?)=/i(g, d%)+/2(^, dV, d^), (I.) where in full, f / 1 \ "^ f^{q, d'q, dq) = \imn\^fi{q-\--d'q, dqj-f^{q, d^)j. (11.) Reversing the order of differentiation, dd'./(?)=/i(9, dd'q)^-f^{q, dq, d'q) (ill.) We shall now prove the relation Mq^ '>\ «)=/2(?» «> ^X (iv.) where r and s are any two quaternions replacing dq and d'q in the functions whach occur in (i.) and (m.). We have by (11.), and from symmetry this is equal to f^iq^ s, r). I «0 DIFFERENTIATION. [chap. vii. More generally, ifhy siiccessive differentiation of a function f(q), a function f„(q, rj, r2, ... r„) is constructed^ the order in which tlie quaternions r^, rg, ...r^ are grouped ainoncf themselves is iminaterial. In virtue of (iv.), it appears that d'd./(^)-dd'./(^)=/;(^, d'dg-dd'^); (V.) and in general this difference vanishes if, and only if, d'd5' = dd'g'. Ex. 1. If ^ is a scalar function of p, and if d$ = Si/dp, di/ = <^d/o, show that the function ^ is self -con jugate, or that Sa<^/3=S^<^a, where a and (^ are any two vectors. [This is a particular case of (iv.). Compare Art. 51, Ex. 2, and Art. 62.] Ex. 2. If 1^1 and v^ are duwy two vector functions of the vector p ; if dv| = (j^i (d/a) and dv2 = <^2(^p)) ^^^ i^ ^ operates on all functions of p on its right, show that SviV . SvijV . - SvaV . SviV . = S(<^iVo - 2Vi) V . ; or in other words prove that the two operators produce the same effect on any function of p. ^ Ex. 3. *lt p, q and r are any three quantities or operators, not necessarily ^ commutative in order of operation or multiplication, show that [[P, qy]Hlq, r]p] + [[r, p]q]^0 where [p, q]=pq-qp, [[p, q], ^J=[p, qy-r[p, q]. Ex. 4. If p and q are any two quantities or operators, show that ^ e-'ipe^=p+^ + :^-{- ^ ^^ 3 + etc., where Pn=[Pn-iiq]; and hence prove the equation connecting operators, where Vi and V2 are any given functions of p, where Vg is a determinate function of p and where V operates only on functions on its right. Art. 61. To find a stationary value of the scalar function f(p)> whenever a stationary value exists, we equate to zero the first differential d/(p) = S.dp (I.) of f{p) for all differentials dp. This requires the vector v to be zero, for otherwise Svdp cannot be zero for every differential d;9, and the stationary values are obtained by substituting in f(p) ihe vectors p which satisfy the equation v = (II.) If the stationary value is subject to the condition 9(p) = 0,. (III.) where g(p) is a given scalar function of p, the differential dp is no longer arbitrary, and the conditions are df(p)=^Spdp = 0, dg(p) = S\dp=^0, (iv.) ART. 61.] STATIONARY VALUES. 81 where X is a new vector function of p defined by the nature of the function g{p). Considered geometrically the condition (ill.) requires the vector p to terminate on a certain surface and con- strains the differential dp to be tangential to the surface as expressed by SXdp = 0. The function f{p) has a stationary value if d/(/o) vanishes for every differential dp at right angles to X. In other words we must have v parallel to X, or ^+a:;X = 0, or Vi/X = 0, (v.) where a; is a scalar multiplier. The solutions of (ill.) and (v.) afford vectors p which render f{p) stationary in value. Again if there are two equations of condition, g{p)^0, h(p) = 0, (VI.) the differential of dp consistent with these conditions must satisfy dg(p) = S\dp = 0, dh(p) = Sjmdp = 0, (vii.) so that dp II VX)U, and if in addition f(p) is stationary in value so that d/(/o) = 0, or Si/d/o = 0, we must have v coplanar with X and JUL, or \v+xX-\-yiii = 0, or Sv\jul = 0, (viii.) where x and y are two scalar multipliers. Here the three vanishing scalar functions of p, g(p) = 0, h(p) = and Sj/X/x = 0, serve to determine a certain number of vectors p as vectors to the points of intersection of three known surfaces, and substitu- tion of any one of these vectors in f{p) will give a stationary value. For the solution of the equations, no general rule can be laid down. Sometimes, indeed most frequently, it is more convenient to deal with the equations (v.) and (viii.) involving x and y rather than with the results of elimination of these scalars. To examine the nature of the stationary values of f(p), it is necessary to proceed to second differentials. For example when there are two equations of condition, we have in addition to (vii.) (compare Art. 51, Ex. 2, Art. 60, Ex. 1), d^g(p) = SXd^p + Sdp(l>^dp = 0, d:%(p) = SjULd^p-hSdpcl>^,dp = 0, (ix.) where 0^ and ^^^ are two linear vector functions determined by the functions g{p) and h(p), and we must consider the sign of dJ(p) = Si^d^p-\-Sdp^dp, (X.) when appropriate values of p and dp are substituted therein. By adding the equations (ix.) multiplied by x and y to this we havej by (viii.) d!/*(/o) = Sdp(^+fl;^^-h2/^Jdp, where dp || VXyu, (xi.) J.Q. F 8? DIFFERENTIATION. [chap. vii. the scalars x and y being given by (viii.) in terms of v, X and yu by means of the relations Yiulp = xY\iul, Yi/\z=yY\iuL, in which we suppose the appropriate value of p to be substituted. For the negative sign, f{p) is a maximum, while it is a minimum if the sign is positive. In like manner, when there is only one equation of condition, we find dJ(p) = Sdp((p-{-x)dp, where SXdp = 0, p-\-x\-0, (xii.) and if dy(yo) is positive for every dp perpendicular to A the function f(p) is a minimum ; if dy(/o) changes sign for some vectors dp perpendicular to X, the function is merely stationary ; if dy(yo) is constantly negative for the differentials dp, the function is a maximum. Ex. 1. Find the stationary values of T/o, subject to the condition, (p-a)2 + a2 = 0. [Here dTp= -SU/3dp = 0, • where dp satisfies S(p-a)d/) = 0, so that JJpWp — a, or pl|a, or p = .ra say, and the condition gives (:r-l)2a2 + a2^0, or ^ = l±aTa-i, so that p = a ± alJa.] Ex. 2. Find the stationary values of Tp when (p — ay + a^ — 0, S/3p — 0. EXAMPLES TO CHAPTER VII. Ex. 1. If op=/o = a'^, Ta=l, Sa^ = 0, the locus of the point p will be the circumference of a circle, with o for centre, and ob ( = /?) for radius, and in a plane perpendicular to oa ( = a). Ex.2. If OF = p = Y .a* f3, 7 = oc = Vaj8, Ta = l, the locus of p is an ellipse, with its centre at o, and with ob and oc for its major and minor semiaxes. Ex. 3. If under the same conditions as in Ex. 2, OB' = ^' = a~WafS, OP' = p' = a~Wap, the locus of p' is a circle with ob' and oc for two rectangular radii. The equation of the circle may be written p' = a*fi\ Ex. 4. If op=/o = a'y8, Saf3=0y the locus of p is a logarithmic spiral with o for its pole. Ex. 5. If op=/o=V . a*/3, the locus of p is an elliptic logarithmic spiral — a plane curve which may be projected into an ordinary logarithmic spiral. Ex. 6. The equation p = cta + a'ft with Say8 = 0, Ta=l, represents a helix, while the locus of the perpendiculars to the axis of the helix which intersect the curve is represented by p = cta+ua^l3, where uis sl variable scalar.* •^ These Examples are taken from the Elements of Quaternions, Art. 314. ART. 61.] EXAMPLES. 83 Ex. 7. If we project the ellipse p = a cos x + 13 sin x on a plane at right angles to the vector A, the vectors a and /5 will project into the principal semiaxes of the projection provided S.YXaVXfS^O. (a) They will project into equi-conjugate radii if TVAa=TVA/5. (6) If SaUA=±V[i(/52-a2)±V{i(/3^-aT + (Sa/3)n], S/3UA= Tv/[i(a^-i8^)± V{i()82-a2)2 + (Sa/?)2}], the ellipse will project into a circle — one of four, of which two are imaginary, (c) The squared radii of the circles of projection are -Ka' + i82)T^/{i(/32-a2)2 + (Sa^)2}, the upper sign corresponding to the real circles. Ex. 8. A circle of radius ±n~'^Tf3 rolls on a circle of radius T^ and centre o, and carries with it a point p at a distance lT/3 from its centre. The locus of the point p is represented by OP = p = (l+9i-i)a«j8-^a*^, Ta = l, Sa/3 = 0. (a) Prove that dp=^(l+%)a(/)-a'^)d^, and assign the geometrical interpretation. (b) If the variable scalar t represents the time, the equation of the hodo- graph * is p= i;r(l +n)a{n-^a'/3 - Wi+")«/3), and show that this curve may be generated by a point carried by one circle rolling on another. (c) Show that the condition for a cusp on the path of the point p is and discuss fully the nature of this equation. (d) Prove that the vector of acceleration of the point p for uniform motion of the circle is p=^7r\l+n){{2 + n)a'f3-(l+n)p}, and determine the condition that the acceleration may momentarily vanish. (e) The condition for an inflexion is found by expressing that Udp is stationary or that Vd/od^p = 0, and it may be reduced to l^n\l +n)-ln{2+ n) Sa'^ + 1 = 0. (/) Show that the inflexions lie on the circle J (3 + 7^)(l+7^)-W(3 + 2n) \^ ^~1 n(2+n) J ■^^• {2+n) Ex. 9. Under the same conditions, what curve, or rather what system of curves for various values of the scalar I is represented by p = j8^ + ^a'/3? * Th^ hodograph of an orbit is the locus of the extremity of a vector drawn from a fixed point to represent the velocity of the moving body. 84 DIFFERENTIATION. [chap. vii. Ex. 10. (a) If oq = cf){t) and OQ' = i/r(?*) are the equations of any two curves the relation Td.{t) = Td.ylr(u) is equivalent to a diflferential equation connecting the parameters so that corresponding values of the parameters in an integral determine equal arcs measured from fixed points on the curves. (b) If the condition (a) is satisfied, the quaternion d.;8)'*(l+/8a-')"*; and find the development to the third order when TfB is small in comparison with Ta. Ex. 17. Supposing the earth to describe a circular orbit round the sun, show that the parallactic ellipse of a fixed star is represented by t7=-V.y*ao-i.Uo- where o* and y^'a are the heliocentric vectors to the star and to the earth respectively. (a) Show also that UVo-y.Tao-i and U . o-Vo-y . TaSyo-i are the principal vector radii of the parallactic ellipse. Ex. 18. If V is the (scalar) velocity of light and p the velocity of the earth in its orbit, the aberration of a star is represented by U(yU(r+p)-Uo-. {a) The earth's orbit being supposed circular, the aberrational ellipse is given by trr = - v-hiVV . y'+^ao- . Uo- where u is the scalar velocity of the earth. Ex. 19. Assuming the effect of refraction to be K times the tangent of the zenith distance, show that a star in the direction of the unit vector a- appears to be in the direction of the vector where k is the unit vector directed to the zenith. Ex 20. If p is a point in a body attached at b and c by universal joints to two. bars ba and cd having fixed universal joints at a and d, show that the mo'tion of the point p is subject to the conditions implied in the equations XP = p=pap~^-\-qeq~^, VD = p = lyr'^ + qrjq~\ 86 DIFFERENTIATION. [chap. vii. where a, y, e and rj are fixed vectors and where p, q and r are variable quaternions ; prove that the envelope of the point may be determined by identifying the equations SYdqq-^ . Yqeq-^p = 0, SYdqq-^ . Yqrjq-^ p' = 0; and show that these conditions require the five points abpcd to be coplanar. Ex. 21. If Sordp becomes the differential of a scalar function of p when multiplied by a suitable factor, show that So-Vcr=0. Ex. 22. If dv is the directed element of a surface at the extremity of the vector p, the element of solid angle it subtends at the origin is Ex. 23. Show that d.e^=(^Sdq + S^.Yqy + Y .^.Ye<^. . . Ex. 24. The differential of a function of the vectors p and o-, cr being a function of /o, may be written in the form * d . P= - Sdp(Vp - Vp'So-'Vo-) . P where Vp and Vo- operate respectively on p and on o- as explicitly involved in P, and where Vp' operates on p as involved in cr', the accents being removed after the performance of the indicated operations. (a) If P is a scalar function of p and cr, and if cr is a function of p which renders P constant, VpP-Vp'So-'VaP=0. (6) If the same function o- renders constant another scalar function Q of p and cr, the relation (P,Q) = S.VVerPV,rQVV^ where (P, Q) = S(VpPV^Q- VpQV^P) must be satisfied. And if o- can be derived from a scalar function of p by the operation of V, we must have (P, Q) = 0. (c) If Ai, /xj, A2 and /xg ^i'® ^^7 vector functions of p and cr, the operator S(AiVp+/XiVa-)S(A2Vp+/x2V<.)-S(A2Vp + /x2Va)S(AiVp+jaiV«.) reduces to the form ^{X^^p + ixi^^a)- (d) If Pv denotes the operator S (VpPVo- - Vo-PVp), we have PvQ=-QvP = (P,Q), where P and Q are scalar functions ; and if E is any third scalar function, the expression PvQv.R-QvPv.R=Pv(Q, R)+Qv(R, P)=(P, (Q, R))+(Q, (R, P)) does not involve the second deriveds of R. (e) Hence (P, (Q, R))+(Q, (R, P))+(R, (P, Q))^0 ; and the operator (P, Q)v = PvQv — Qvl*v- * Compare Jacobi's method of solution of partial differential equations and Lie's work on Pfaff's Equation. ART. 61.] EXAMPLES. 87 Ex. 25. Bright curves are seen on a surface owing to light reflected by scratches on the surface from a source at A to an eye at B. If the scratches are represented by putting u = const, in the equation of the surface p=(t, u\ show that the equation of the curves may be found by combining the equation of the surface with the result of expressing that T(c^-a) + T(c^-/3) is a minimum with respect to t. (a) If the equation of the surface is/p=0 and if Fp = 2i is the equation of a family of surfaces through the scratches, the bright curves are given by fp = 0, SVf^F{V(p-a) + V{p-f3)}=0. (b) The bright lines due to the grooves made in turning a surface of revolution {Tp=fSkp) lie on the surface Skp{V(p-a) + V(p-/3)} = 0; and meridian grooves on the same surface give rise to bright curves on the SYkpiVp + kfSkp){V(p -a) + V(p -/3)]=0. Ex. 26. The differential of T(p — a) corresponding to a given differential of p ceases to be determinate when p comes to coincidence with a unless we know a law according to which p tends to coincide with a. CHAPTER VIII. LINEAR AND VECTOR FUNCTIONS. Art. 62. A vector function of a vector, distributive with respect to that vector, is called a linear vector function. Thus if (li(a + l3) = cpa + /3, S0a = O, S^/3 = 0, (l.) for all vectors a and ^8, the function is linear and vector. As a corollary to the equations of definition 4)(xa) = x^a (II.) if X is any scalar. Given the vectors a=a, ^' = ^13, y=0y, (m.) the results of operating by cp on any three given and non- coplanar vectors, the function ^ is determinate ; for by (i.) ^^- s^;^^; ' ^'"-^ since pSa^y = XaS/3yp for any arbitrary vector p. With a new signification of the vectors, a, I3\ y, a, /S, y, any linear function may be reduced to the trinomial form, ^p — a Sap + /3'S/3/o + ySyp, ( V.) in which either set of vectors a, /S', y or a, /8, y may be arbitrarily assumed. For if we resolve ^p along three fixed vectors a, ^' , y\ the coefficients in the resolution must be scalar and distributive functions of p ; that is, they must be of the form Sap, S)8/o, Sy/o. If, on the other hand, we assume a, ^ and y, the set (Xy /3' and y follow, being 0V/3y : Sa/3y, etc. Thus in any case, the general linear function is seen to involve nine constants, the nine constituents of three vectors a, /5 and y, or a, ^' and y. For arbitrary vectors, a and /3, if Sa0/3 = S/3^'a, (VI.) ART. 63.] LINEAR VECTOR FUNCTION. 89 the function . The conjugate for the trinomial form (v.) is 0V = aSa> + ^S^/) + ySy> (VII.) Ex. 1. Given (T = p = a'Sap + fS'Sf^p + y'Syp, show that p = <^-io- = (V/3ySy8y(r + VyaSy'ttV + Va/JSa'^'o-) : (Sa'/3ySaj8y). Ex. 2. Show that Va/o/? is a linear vector function of p, and find its conjugate. Ex. 3. Is aTp a linear vector function of p Art. 63. From a geometrical point of view the equation (r = /3 if p = a + t^: (7 = 0a + ^/5 + U0y if p = a + t/3 + Uy ...(ll.) — consequences of the formula of definition (Art. 62 (i.)). The plane whose equation is S(/o~a)/3y = becomes S(o--^a)0/50y = O ; (ill.) and the vector area Ya^ transforms into Ycjyacj)^: (iv.) while the volume Sa)8y becomes S^a^/5^y (v.) Ex. 1. Verify that S^a^^_S^a;^<^y' SajSy ~ Sa'^^y' ^ "'"''' where a, ^, y and a, ^', y' are any two sets of non-coplanar vectors, i Ex. 2. Prove that ya(3 + Ycl>y8=Ycfi€cf>C if Ya(3+Yy8 = Y€C. [Take a along the edge of the planes of a/? and of yS, and reduce Va/? and Vy8 to Ya'/3' and Va'y', etc.] E:^.'.3. Prove that V^a^^ is a linear vector function of YafS. [This is practically included in the last example. Verify by the trinomial form.] 90 LINEAR VECTOR FUNCTION. [chap. viii. Art. 64. There is an inverse transformation which converts vectors o- into vectors p, so that p = (l)-'^a- if o- = 0/o; (I.) and we propose to investigate this transformation. Writing 0p = o- = yX^, (ll.) the conditions of perpendicularity of the vectors a-, X and cr, jul give SX0p = O, Syu0yo = O, or Sp^'X = 0, Sp'X'. Ex. 3. Show that yjr' is the conjugate of -yfr if yjr'Yaf3=y(f)acf)/3. [Expand SVy8V^a<^/?, and prove that it is equal to SVa^V<^'y^'8.] Ex. 4. Show that volumes are altered in the ratio 7n:l by the trans- formation produced by . [mSa^ = SacfiflS = Scfy'a^l^fS = S . f <^'a/3.] Ex. 5. Follow in detail the geometrical meaning of the transformation employed in deducing mp — ^cr from G = cf)p. [See Art. 63 (iv.) and Art. 150.] Art. 65. The transformation in the last article fails in one case — if m is zero. In that case the vectors cr are all coplanar, the volume of any parallelepiped formed by them being zero (Ex. 4, Art. 64); and because in general myo = ^o- if (7 = 0/o, in this particular case, the function i/r destroys every vector in the plane. To cover this case, consider the general transformation for an arbitrary function (p, cr = (' '^c)/ul(c'^c = V^c^o ( V. ) 92 LINEAR VECTOR FUNCTION. [chap. viii. or m + m'c + m'c^ + c^ = (0 + c)(Vr + cx + c^) = V-0 + c(Vr + x0)+o2(x + 0) + c^ and therefore equating the coefficients of c on each side m = 0V^ = i/r0; m' = x/. + 0x = \^ + X^; ^^" = + x; •••(^1-) it being understood that these equations denote that equal results are obtained by operating with right or left hand numbers on an arbitrary vector. One of the transformations most frequently required in quaternions is to invert a function 04-c, or to replace an equation o- = (0 + c)/o by mcp = yfrc(T\ and in general the process, due to Hamilton, as given in the text is the shortest and most certain. We first calculate V(^' + c)X(0' + c)yot and express it in terms of VXyu. Then we either calculate rric from (il.), or it is sometimes better to calculate it directly from (v.), namely from In particular mp = \lra-, m'p = \j/-p + x^' rjri'p — xp + (T if cr = ; Xp = 2 Va Va> ; mp = ^fp = SSa/^ySy '/?'a' . p.] Ex. 2. Find the auxiliary functions for cf>p = YXpp. [Find ^c and i/^c for AS/xp + /^S/aA = <^o/o.] Ex. 3. Solve the equations o- = VaV^/) and (r = Yap by the general method, and directly. Ex. 4. Express t/^c' and Xc' in terms of \jrc and Xc- Ex. 5. Construct a linear vector function which renders four given vectors parallel to four others. [The data are 8\\ 8', and the function is where c is arbitrary.] ART. 66.] LATENT AND SYMBOLIC CUBICS. 93 Ex. 6. Prove that m'Yap = Y'a'f3 + Ycj>'af3 + <^Va<^'/3 ; m"YafS = Ycf>'aJ3 + ya'^ + 4>Va^. [See equation (vi.). These relations are often useful.] Ex. 7. Prove that m'Va/? = VV^ai/r^ ; yfrYa\fr'/3 = mYcf>'af3 ; Ycf>'afi = Vai/r' /?. Ex. 8. Prove that the equation p = { + t)-^a, or Y((f>p-a)p = 0, a being a fixed vector and t a variable scalar, represents a twisted cubic. [Show that it cuts an arbitrary plane in three points.] Art. 66. From tlie equations of tlie last article connecting ' + ^'2 ; = m - m:' + rn"(j)' - (j>'^. (ii.) These may be proved by reflecting that Sa02^ = S0'a95>/3 = S/30' V etc. ; so that for example Sax/3 = Sa (m" - ^)^ = S^(m'' - ^0 a = S/^x a, and from the third and fourth of these we have {m" — (f))a = ^a because /3 is perfectly arbitrary. Let g^, 02 and g^ be the roots of the scalar cubic, = 7n — m'g + rri'g'^ —g^ = 0; (ill.) so that m = g^g,_g^, m! ^g^g^+g^^-^-g^g,^, m^'^g^+g^+g^. ...(iv.) This scalar cubic is called the latent cubic of the function, and its roots are the latent roots of the function 0. We may now write the symbolic cubic (I.) satisfied by the function (p in the form i^-9i)(^-g2)i-gs)^ = Sa{—gi)a is perpendicular to the vector {(p—g2){^—9B)^' The vectors a and ^ being both arbitrary, it follows that one or other of the vectors {'— gi) a or {(l> — go){(i> — g^)^ must be parallel to a fixed direction. But {(/>' — gi) a is not generally parallel to a fixed direction whe^i'the vector a is arbitrary, for if it were we should have 94 LINEAR VECTOR FUNCTION. [chap. viii. where a and /3 are quite arbitrary ; or symbolically, \^-5^iX+5'i' = 0, or (-gs) = 0, (ra.) utilizing (i.) and (iv.), and replacing tyi" —g^ and "in' — g^m" -\-g^ by their values, g^-^-g^ ^^^ g29z- ^^ ^^^® case, which is quite special, the symbolic cubic of the function degrades into a quad- ratic (VII.). We conclude therefore that the product of a pair of factors of (v.) operating on an arbitrary vector reduces it to a fixed direction, and writing (-gs)(i>-gi)p II 72; (0-5^1X9^-^2^1173 (VIII.) the directions of the vectors y^, y^, y^ are fixed and are called the axes of the function ^. We have by (v.), 07i = ^i7p 072 = 5^272' 4>yz = 9zYz'^ (ix.) and these vectors are generally distinct if the latent roots 9v 92') 93 ^^® unequal, and they are also generally non-coplanar. Resolving then any vector p along y^ y^ and yg we have P = ^7i + 2/72 + ^73; (X.) (-9i)p=yi92-9i)y2+<9s-9i)y3'^ (-9s)p==^i9i-93)yi+yi92-93)y2'^ (-93)p = H9i-92)(9i-9z)yi', (-9s)(-gi)p=y(92-93)(92-9i)y2'^ {—g-dp is coplanar with the pair of axes y^ and yg, and if y^ is the axis of the conjugate function corresponding to the root g-^, it follows from the equation ^pW-9i)yi=^^ = ^yi{-9i)p (XI.) that the vector y^ is perpendicular to the plane of {(t>—g-^p, and in particular to the vectors yg and yg. If vectors are drawn from the centre of a sphere along the axes of a function and of its conjugate, the two spherical triangles the two sets of axes deter- mine are supplemental. Conceive the function to undergo continuous variation so that two latent roots, g^ and g^, approach coincidence. The corresponding axes approach and ultimately coincide, but their plane is still determinate being perpendicular to y^. Similarly all three axes may coincide in a line perpendicular to that in which the three axes of the conjugate simultaneously coincide. I ART. 66.] ROOTS AND AXES. 95 We shall give an illustration of a function having three equal roots. Let a = ^, ^y8 = y, (f>y = 0, then 0^/5 = 0, ^^a = and generally (/y^p = 0, but (p^p and (pp are not zero. The function is (pp = (/3S/3y/) + ySyap) : Sa/3y, and (p^p = yS/3yp : Sa/3y — yfrp. A totally different class of functions is characterized by the equivalent conditions that the axes are indeterminate or that the function satisfies a symbolic quadratic and not a cubic (compare (vil.)). If 72 and y^ are two different axes corresponding to the same root g^, the function c^—g^ destroys every vector in the plane of 72 and y^, and the function is of the form p=92P+(yi-92)yi^y2y3p • ^71727^ ; and (i>-9i)(-gi)[{^-g^)^{(t>'-g^)] is real and must produce a real vector from a real vector ; but {-9i)[{^'-92)-{^-9^)'\ is imaginary and produces an imaginary vector from a real vector. Ex. 1. Every function coaxial with a given function //"/o=--0, or Sppcf)^p = Oy represents the three planes through pairs of axes of <^. 96 LINEAR VECTOR FUNCTION. [chap. viii. Ex. 5. In general, if {(f)^ +x.p, ({>^p and (f>^p in the relation pSXfjLv = XSfxvp + fxSvXp + vSXjxp. Art. 67. Combining a function and its conjugate by way of addition and subtraction we obtain two more functions, */o = J(0 + 0')/o and Ve/o = J(0-0O/3 (l-) To justify the form attributed to the second function, observe *^** Sp('P-X.YfjLv + 4>fx. YvX -f- cf)v . V A/x) : SXfiv, g'2 = ( A . Y<^ixv + IX . Ycf)vcf)\ + V . Y^af3y = 0, S4>^f3ya=0, Bifida ft = 0, and S<^a<^^/3<^*y = 0, etc. See the next Article and the Appendix to new edition of Elements of Qitatermo7is, vol. ii., note v.] Art. 70. The square root of a linear vector function may be defined as a linear vector function, which, operating twice in succession on any vector, produces the same effect as the given function. Writing then <^^ for the square root of the function , we have, if y^, y^ and y^ are the axes of have equal roots, and let it have indeterminate axes, so that {'^—g\){pcyx-\-yy^ = ^ where x and y are arbitrary, gx=gi being the repeated root. A square root of the function may have three distinct 111 roots -Vgx-) —gxtg^- III this case there is an infinite number of square roots, because we may select any vector 0Gy^-\-yy^ to be an axis of ^ corresponding to +^1^, and any other vector xfy^ -^y'y^ "^^7 he selected as the axis corresponding to -^j^. For real square roots, the three roots g^^ g^ and ^3 must of course be positive. The following resolution of a linear function <^ and its conjugate is , sometimes useful — for example, in the theory of strain. It is due to Tait, to whom is also due the conception of the square root of a linear vector function. Let i, jj h be the mutually rectangular axes of the self -con jugate function <^^', and let a^, W, c^ be its roots. Eeducing <^ to the trinomial form (Art. 62), ct>p = aiSi'p + bjSfp + ckSk'p, (11.) where i', /, ^ are to be determined, we have 'i=—a{', <^'j=-hj' and ^'k=—ck'. These give 'i=aH. Consequently we must have i'^= —1, S^y = S^T = 0, and in fact i\j\ k' form a mutually rectangular unit system* j,of vectors. Thus in particular cf)i'=—ai, and cf)cf>i'= —a^i= +aH\ and thus it follows that ^', / and k' are the axes of the new self -con jugate function <^'4>'f.q-^pq and 'p = q .(yp.q-^ ', (m.) and from these we also deduce qpq-^ = 4>'.{<^^')~^p , (IV.) In like manner we may prove that p=p-^.{<^'4>Yp.p, y.ppp-^; (v.) and thus we can reduce the effect of a function (^^ to a rotation preceded or followed by the operation of a self -con jugate function. Art. 71. We add one or two miscellaneous propositions respecting two or more functions. The functions (ficf), and /7=^7j 0-) and thus the functions have the same roots and the axes (y) of <;6,<^ are deducible from those of cfxf)^ by operating with (/>,. In particular <^,~i<^^, has the same symbolic cubic as cfi, and thus any peculiarity in the nature of one function occurs also in that of the other. Any two functions may be reduced simultaneously and generally in one way to the forms cf)p = aSXp + /SSfxp + ySvp ; <^,p = aaSA/3 + 6^S/x/) + cyS»//o (ii.) Assuming the possibility of the reduction, it appears that cf>^YfjLv=acfiYfjiv = aaSXfxv, etc., and thus the vectors VA/x, etc., are the axes of the function ^~V/ ^^^ ^j ^> ^ are its roots. If both functions are self-conjugate, we must have VaA-l-V/3/x-fVyi/ = 0, aYaX + bYl3p, + cYyv = 0, YaX^Y/3f,_Yyv ^ b — c c — a a-b ' and therefore for self-conjugate functions <^p = ASA/o -I- /xS/Ap + ySvp, (fi,p = aXSXp + bfjiSfxp + cvSvp, (iii.) and further it is evident that SYfxv(f>YvX = 0, SYfiv(j>JvX = 0, etc. It is sometimes necessary to invert the function cfi -f tcfi^, and the auxiliary t/t of this function is defined by irYaf^ = Y(cf>' + tcfi;)a{' + tcf>;)/3 = fYaf3 + t^Yal3 + t'^irya(3 (iv.) where ^Ya/3 = Y'a;/3 + Y;acf,'(3 (v.) The invariant mt is , 7^ , 7 ^2 , ^s / \ ^ mt=m + lti-l,t^ + m^t^ (vi.) * We must have i'j'k'= - 1 =ijk, but this can always be secured by attributing proper signs to a, 6, c. If i'fk' were +1, we should not be able to rotate the vectors into ijk, for qiq~^ . qjq~^ . qkq~^ = q .ijk.q~^= - 1. ART. 71.] A PAIR OF FUNCTIONS. 101 where m and m, are the third invariants of ^ and ^, and where 2 and ^, are the two new invariants ^Sa/3y = 2S<^a<^/?<^,7, l,SafSy = ^Scf>acl>,l3,y (vii.) Ex. 1. The locus of axes of the functions (f) + t, where Ms a scalar parameter is the cubic cone [If p is an axis cfip + t({),p=gp. The surface represents a cone, as it is inde- pendent of Tp.] Ex. 2. The axes of functions of the family <^ + ^<^, form co- residual triads on the cubic cone. [The quadric cone SXpcf)p = in which X is arbitrary cuts the cubic in the three axes of <^ and again in three lines in which it cuts SXp + t^)p = passes through the axes of <^ + 1(^^ and through the three lines above mentioned, so that these three lines are the residuals of every triad of axes (Salmon's Higher Plane Curves, Art. 154). For other properties see Quaternion Invariants of Linear Vector Functions, Proc. P.I.A., 1896.] Ex. 3. Prove that the invariants I and l^ are merely multiplied by a scalar when and <^, are replaced by <^i^<^2 ^^^ ^i^/<^2- [The scalar is the product of the third invariants of <^i and ^2- ^h^^ ^^^7 general invariantal property leads to many theorems. See Phil. Trans., vol. 201, Part VIII., sections iii. and x.] Ex. 4. Prove that the function ^Va/5=V{(r(^2^ ', and more generally if cr is connected with p by the chain of relations Pi il Vip<^2pJ P2 II ^i^Spl^iPv ••• O" II ^{p, t) is a linear and vector function of p and also a function of the scalar t, the equation Yp{p,t) = represents a cone whose order is the number of values of t which satisfy ^X4>'{X,t)4>'{<\>'{X,t\t} = 0, X being any constant vector. Ex. 7. The equation V(p-a)(p,0 = O represents a surface which meets an arbitrary right line V(p-j8)y = in as many points as there are values of t which satisfy S(i8 - a)y<^(A 0Sr<^(i8, t){y, t) = S{^-a)y(y, t)S(fi-a)i^, t)(y, t). 102 LINEAR VECTOR FUNCTION. [chap. viii. EXAMPLES TO CHAPTER VIII. Ex. 1. Find the auxiliary functions X and t/t and the invariants of the function p — 1,mYaYpa. Ex. 2. Invert the function cfip + YaYpa where Jp = 'Ya-Wap, and prove that the spin- vector is €-^Va~i^'a. (a) Show that the auxiliary \}r function of ^^ + <^ i^ expressible in either of the forms ^raSa-^ p + c(Xp-Y'aY a-' p) + c^p or (V^ + cX + c2)p-V + c)-^a. Ex. 4. If 'l3Yap + aS/SpSafS and m-S/3(t)Xf^ + ^/34>aSal3. Ex. 6. li (f>c = Yap.f3 = x'y.yap.l3-Y.Yap.cl,(3. (a) Show that the conjugate of this linear function of p is V . (f>'Yf3p . a, and prove that the spin-vector is ^'Ya/3-aSef3 M^here € is the spin -vector of (^. (6) Show that the auxiliary -xj/- function is aS^/oSax/r^. (c) If Y .cj>Yap. /3=cr, show that p=xa-(j)'(T{Sa\lr(3)~^ where x is an arbitrary scalar. Deduce this result by the aid of the implied relations Sap<^'o-=0, S/?(r=0. Ex. 8. Prove that V .a(Sa/?7)-i where a, /? and y are arbitrary vectors. (a) Show that V . V-(p) = - 2a . V<^'^f y . (Sa/^y)-i. (b) Express these quaternions in terms of the scalar invariants and the spin-vectors. ART. 71.] EXAMPLES. 103 Ex. 9. Three lines are defined by the pairs of vectors (o-j, t^), (o-g, T2), (0-3, Tg) as- in Art. 36, Ex. 4, show that any line which is met by all the transversals of the given lines may be represented by (T = Ti, 0-2 = <^'''2) ^3 = ^'''3' (a) The transversals of the same set of lines may be represented by cr' = — <^'t' where St'<^'t' = 0, the function ' being the conjugate of ( )- Vp( ) has a zero root, the locus of the lines is found to be S(p-e)(p-e) = m-hSe . ED. (a) Show that corresponding faces of the tetrahedron determine with the point E tetrahedra having a common ratio of volumes. (b) If the lines joining corresponding vertices are generators of the same system of a hyperboloid, it is possible to find four scalars I, m^ n, p so that ■'> l{a' -a) + m(l3' - l3) + 7i('/ -y)-hp(8' -S) = ; IVaa +mYl3l3' + nYyy' +p8S'==0. 104 LINEAE VECTOR FUNCTION. [chap. viii. (c) These scalars are independent of the origin, and if the origin is taken at the point e, we shall have la + m/3 + ny+p8 = 0, la +771/3' + nY + p8' = for an arbitrary pair of tetrahedra, while if the lines joining the vertices are generators of the same system of a hyperboloid, we shall have in addition lY aa' + mY/3(3' + 7iVyy' + pY88' = 0. Ex. 14. Identify the expressions where ^ is a scalar variable, and show how to express the function <^, and the vectors A and /x in terms of the vectors a, ^, y and S, and the scalars a, h, c and d. Ex. 15. Of what nature are the curve loci p = {cfi + t)-\a + t/3) and p = (p = 0, SC7<^CT = may, when (/> is self-conjugate, be expressed in the form Sa\l/aSpcf)p - mSa/)2 = 0. (a) In the same case, S(f)paV . Ya(j>p = YacfiYacfip = pSaxfra — i/^aSap. (b) And moreover * S<^paAS<^pa/x = S/x^pSVaA^Vaju,-}- SA"0-/xSa/)2 — Sayj/^pSXpSap — SaxfrXSppSap + Sa^aSXpSpp. ^ These examples are quaternion equivalents of the transformations in Arts. 383, 385 and 390 of Salmon's Higher Plane Curves. CHAPTER IX. QUADRIC SURFACES. Art. 72. If f{p, p) is a homogeneous, rational and integral scalar function of the second order in a variable vector p, so that /(a + i/3, a + t^)=f(a, a) + t(f{a, /8)+/(;8, a))+m^, /S), ...(l.) where a and /3 are arbitrary vectors, the equation /(/o,/o) = const (II.) represents a surface of the second order, referred to its centre as origin. For by (i.) we find a quadratic in t which determines two points in which an arbitrary line p = a-\-t^ cuts the surface ; and on putting a = 0, the roots of the quadratic are equal and opposite, showing that every chord through the origin is bisected at that point. The coeflScient of t in (i.) is linear and homogeneous both in a and in /3, and as it involves these vectors symmetrically we may write f(a,^)+f{/3, a) = 2Sa^/3 = 2S^95>a (ill.) where is a self -conjugate linear vector function. Thus the equation of the central quadric is expressible in the form f(p, p) = Sp^yo = const (IV.) Without loss of generality we may suppose the constant incor- porated in p=-l, (V.) in which, as we have said, (p is self -conjugate. Of course, and without gain of generality, we may suppose ^ not to be self- conjugate in (v.), for the spin- vector automatically disappears from an equation of this form (Art. 67) ; but this is very likely to l«ad to mistakes in further developments, and it adds needless complexity. ART. 73.] QUADRIC SURFACES. 107 Art. 73. Equation (v.) of the last article gives VSUyQ0UyO=-l or -SUyO^Up = ^2 = ^ W if r is the length of the central radius parallel to JJp. For a closed quadric, an ellipsoid or sphere, r^ is always positive, as every line through the centre meets the closed surface in real points. For a hyperboloid, the radius becomes infinite for an edge of the cone SVp(f>Vp = or Sp(l>p = 0, (II.) the asymptotic cone of the surface. The sign of the expression r~2 or —SJJ pcpUp changes on passing through a zero value, and the expression remains with changed sign until it passes again through a zero value. So on one side of the cone Sp^/o = 0, lines meet the hyperboloid in real points, and on the other side the points are imaginary and the corresponding vectors are of the form p = ij — lp\ (Uyo = Uyo', Tp^fJ — ITp), where p is a real vector. The vectors p terminate on the quadric Sp(t>p= +1 (III.) — the conjugate of the quadric Spcpp— —1. For the sake of brevity we shall write generally r- for the square of the length of the radius whether that square be positive or negative, the interpretation in the latter case being that just given. An arbitrary right line p — a + t^ cuts the quadric Sp(pp= —1 in the points determined by the roots t of the quadratic Saa-^2tSa(p/3 + r'S/3/3 = 0, (v.) for if a is the vector to any point in this plane, the roots of (iv.) are equal and opposite. If the diametral plane of jS contains the vector a, that of a contains /5 in virtue of the self -conjugate property of 0, for then Sa95>/3 = S/3^a = (VI.) T^^ equation has equal roots if Sfi

/3 = (XI.) These two equations jointly represent the section of the quadric by the plane Syo0/5 = 2^S/30/3, (xii.) and the centre of the section is the origin of vectors p\ or the extremity of the vector t/3. Hence the locus of centres of sections by planes parallel to (v.) is the line through the centre parallel to 8, as indeed might have been proved directly from (v.). The section (xi.) is similar to the parallel central section of the quadric, for if r' is the radius of the section parallel to p' and r that of the quadric, -/2SUp>U/>' = 5 = l+«2S;8^/3 = l-M.' (XIII.) if h' is the radius of the quadric parallel to /?. The equation of the normal to the quadric at the extremity of the vector a is p = a + X(pa, or Y{p — a)(j)a = 0; (xiv.) and the normals which pass through a given point /3 are six in number and are determined by the equation ^ = p + X(pp, or \{^-p)(f>p = 0, and S/o^yQ= -1. ...(xv.) To solve these equations we have ^ = (l + ic0)-i/3, where S^0(l+cc0)-2^= -1, (xvi.) because ^pcpp = — 1 , and on inversion we find a sextic equation n X. MIT. 74.] TANGENTS AND NORMALS. 109 Ex. 1. Prove that the rectangle under the intercepts from the extremity of a on the line p = a + tB is (Ta2-a'2)fe'2a'-2 where a' and b' are the central radii parallel to a and (3. [t^t^TI3' = (Sacl>a+l) : SU^<^U/?.] Ex. 2. The ratio of the rectangles un^er the intercepts of lines drawn from a fixed point is independent of the position of the point, and is equal to the ratio of the squares of parallel central radii. Ex. 3. Chords drawn through a point are divided harmonically by the quadric and the polar plane of the point. [Put = -i + -;| where L and U are the roots of the quadratic (iv.).] p-a 13 ^ Ex. 4. Find the central vector perpendicular on the tangent plane at any point, and obtain the locus of the feet of central perpendiculars, or the central pedal surface. [CT=-(<^a)-i; a=-cf>-^To-'; SC]r-i<^-icy-i= - 1.] Ex. 5. Prove that the central pedal surface is the inverse of the reciprocal quadric. Ex. 6. Prove that the ratio of the perpendiculars from a point a and from the centre on the polar plane of b is equal to the ratio of the perpen- diculars from B, and from the centre on the polar plane of a. Ex. 7. Find the locus of the poles of tangent planes to the surface S/o(/)i/o= - 1 with respect to the surface S/o<^2P= ~ 1- Ex. 8. Find the pedal surface for an arbitrary point. Ex. 9. The feet of the normals which pass through a given point are the intersections of a twisted cubic with the quadric. [Compare (xv.) and Art. 65, Ex. 8, p. 93.] Ex. 10. The normals through a given point lie on a quadric cone S{p- I3)(f)l34>p=^, and the feet of the normals lie on the cone S^pcf>p = 0. (a) Both these cones have edges parallel to the three axes. Ex. 11. Find the condition of the intersection of normals at two points a and /?. Ex. 12. Find the equation of the polar plane of a to the quadric Sp(f>i(f)2P= - Ij <^i^2 being the product of two linear functions. [Note that i ^^ *^^ conjugate of <^i<^2-] Ex. 13. Prove that the polar line of p = a-\-t(3 with respect to the quadric S/)(^/3 = — 1 is _4>(3 + s Art. 74. The central plane SXp = is the diametral plane of chords parallel to ^"^X, as appears on comparison with (v.) of the last article. The locus of the centres of sections by planes parallel to SXp = is the right line Vp^-iX-0 (I.) \ no QUADEIC SURFACES. [chap. ix. The vector to the pole of the plane (Art. 73 (ix.)) SXp=-l is (jy-'^X; ....(II.) and the plane touches the quadric if (Art. 73 (x.)) SX^-'^X^-l, (III.) and as .\ varies this is the tangential equation of the quadric. But SX/o=— 1 is the polar* plane of the extremity of X with respect to the unit sphere, T^=l or p^=z —1, and the equation (ill.) may therefore be regarded as that of the reciprocal of the quadric with respect to the unit sphere. The vector to the centre of the section by SXp= — 1 is by (i.) . 0-^X / X -sx^v <'^-> ■ the tensor being determined so that this vector may terminate in the plane SXp = — 1 ; and on comparison with (xiii.) of the last article, the ratio of the radii is given by r'^ 1+SX^-iX r^~ SX0-1X *■ •-' Ex. 1. By direct comparison of SA/)+l = with (xii.) of the last article, find the vector (iv.) of the present. Ex. 2. Find the reciprocal of the surface with respect to an arbitrary sphere. Ex. 3. Find the lines in which the plane S\p = cuts the cone Sp^p = ; and show that they are parallel to YXTa^-l)-^\] (VI.) and because SXa = 0, we have if Ta^ = r^, SX(0r2-l)-iX = O or r'SXxl^\-r^S\x\ + \^ = (vii.) using the formula of inversion (Art. 65). Thus a quadratic in 7^^ is obtained and substitution of its roots in {(pr^ — iy^X gives the directions of the vectors required. The principal axes of a surface are normal to the tangent planes at their extremities, so that Yp(l)p = (vm.) for a principal axis. These are the axes y^, y^, y^ of the function (p. Ex. 1. Find the maximum and minimum radii in a central section. [Here SA/) = 0, Spcf>p= —I, Tp = max., and on differentiation, SXdp = 0, S(f>pdp = 0, S/)d/) = 0, so that the three vectors A, cf^p and p are coplanar, or {<^-\-x)p=yX. Operating by Sp, we fall back on (vi.). Ex. 2. Find the maximum and minimum radii of the quadric, and show that their directions are the solutions of Yp)2 = (Sa/3y)2 (iV.) as appears on substituting p = ]SaS/3y/o : SajSy in Sp(pp = — 1 and attending to the conditions. Writing (compare Art. 70) a = -^a, ^ = ^~^/3:, y^fS' (v.) it appears by (i.) that the vectors a\ /3' and y are mutually perpendicular, and because a, 13 and y terminate upon the surface Spp=gp^ + 2SXpSiixp= -1 (i.) or to determine g, X and /x so that for all vectors p, ^p=gp + XSyOtp + yuSXyO (II.) It follows that (l> — g must reduce every vector to a fixed plane, that of X and ju. The scalar g must therefore be one of the latent roots of 0, say g=g2, and in terms of the axes, M^-92)P= -(gi-92K^^py-(9z-92)(^kpy = ^^^p^f^p (ra) because (pp= — ig^Sip —jg<^jp — kg^Skp. Thus ' -^ t\ = Jg^ - g^i + Jg^ - gjc, 2^ " V = V^g " 9^^ " "^93 " 92^ (l^-) J.Q. H 114 QLTADRIC SURFACES. [chap. ix. where t is arbitrary. The transformation is real only if yi>92>93 or gC>92>9i (^^O The cyclic planes SXyo = 0, SjULp = cut the surface in circles of radius (73, ^^^ these circles are real only if f/g^O. The planes S\p-\-l = 0, S/A/3 + m = cut the surface in circles lying on the sphere Sp(Pp-\-l-2(S\p-\-l)(Siixp + m) = 0, or gp^-2Sp(lfjL + m\)-2lm + l=0. In nearly every problem relating to quadrics some valuable information will be gained by throwing the equation into the cyclic form or into the focal form of the next article. This transformation is not generally of any great difficulty. Ex. 1. Reduce a quadric to the form T(p-ay = e{Skp + l)(Sfxp + l). [This gives Dr. Salmon's focal property. The locus of the extremity of the vector a is a hyperbola — the focal hyperbola, and this depends on equation (iv.).] Ex. 2. Prove that the roots for Hamilton's cyclic form are g, g + SXfji + TXfji, ^ + SA/x-TA/z. Ex. 3. Any two circular sections of opposite systems lie on the same sphere. Ex. 4. If a quadric is a surface of revolution, for all vectors p. [The self -con jugate function <^ has two equal roots (c) and (Art. 66 (xii.), p. 95) V((^-c)a((/)-c)^ is identically zero for all vectors a and /3, or yjrp-cXp + c^p — 0.] Ex. 5. If for all vectors p SpXpi^p = 0, or Sp(j)p(ly^p = 0, or Spcf>p^p = 0, the quadric is of revolution. Ex. 6. From a fixed point a, on the surface of a given sphere, draw any chord AD ; let d' be the second point of intersection of the same spheric surface with the secant bd drawn from a fixed external point b ; and take a radius vector ae, equal in length to the line bd', and in direction either coincident with, or opposite to, the chord ad : the locus of the point e will be an ellipsoid, with a for its centre, and with b for a point of its surface. [^Elements of Quaternions^ Art. 217 (6). See also Lectures, Art. 465. If c is the centre of the sphere, the isosceles triangle acd gives — = K — , or CD = - AD~^ . CA . AD = - AE'^ . CA . AE, and therefore DB = CB + AE~1.CA. AE = t+p-lKp ART. 78.] CYCLIC AND FOCAL EQUATIONS. 115 if CB = 6, AE = p, CA = K. By the property of the sphere d'b . db = cb^-ca^ = k^- t^, and by the construction T> = TD'B = T(t2 - k^) . Tdb-i, or T(pt + Kp) = T{i^ - k^). Squaring both sides, we have Tp^T{i^ + k^) + 2SplKkp = T(l^ - k^^, which reduces immediately to Hamilton's cyclic form.] Ex. 7. Conceive two equal spheres to slide within two cylinders of revolution, whose axes intersect each other, in such a manner that the right line joining the centres of the spheres shall be parallel to a fixed right line ; then the locus of the varying circle in which the two spheres intersect each other will be an ellipsoid, inscribed at once in both the cylinders. [Hamilton, Lectures, Art. 496. Taking the spheres to be T(p-ta) = b, T(p-tB) = b, where a, /S and b are given and where ^ is a variaole scalar, we find on elimination of t, (p2 + 62)(a-2 - /3-^)(a2 - f3') = 2S(a-i - f3-')pS{a - fS)p.] Art. 78. To find the right circular tangent cylinders of a quadric, observe that if the vertex of the tangent cone (Art 7S (viii.)) passes off to infinity, the equation of the tangent cylinder parallel to a is (Sp^p + l)Sa^a-(S^0a)2 = O (l.) A right circular cylinder parallel to a and of radius Ta~^ is represented by TVap = l, or (Vapf + 1 = 0, .: (ll.) and identifying this with (i.) we have to satisfy Sp^pJ^if + i^apf (III.) for all vectors p, or what is equivalent we must identify 0aS^_ ^^^y This is identical for p = a; and for p = ^a we have ,2 =92-9i-9z^ which is Hamilton's focal form, if we remark that by (vi.) and (vii.) 7 T(0a)" Sa(h^a 0= Q . = - 1^^=92- 9i-9z- If a = ix-\-kz we have by (vi.) and (vii.) x^-\-z^=g^, 9iX^-\-9b^^==939v and a^iJd^l^+jJ^MEM (IX.) Art. 79. To find the generators of a quadric, we express that when we substitute p + ta in its equation, the equation is satisfied for all values of t Thus Sp-'^a-^-Sa-^(j>-^a-'^Saa^-ya, (ill.) it being implied by the form of this equation that Sa"-^0a = O. Generators of one system correspond to the sign +, and those of the other system to the sign — . Ex. 1. Prove that generators of opposite systems intersect. Ex. 2. Find the locus of the feet of central perpendiculars on the generators. [From the equation p= ± a/ a~i(^a we find a 1| ypp, and substitution in Saa = gives a quartic cone which intersects the quadric along the locus.] Ex. 3. Prove that the locus of intersections of generators which cut at right angles is the intersection of a sphere with the quadric. [Note that a central plane parallel to a tangent plane cuts the asymptotic cone in lines parallel to the generators.] I ART. 80.] GENERATORS AND CENTRES. 117 Ex. 4. The locus of intersections of generators which cut at a given angle is tanw=2^^^— ^; Spcf,p + l=0. [See Ex. 4, Ai-t. 74.] Art. 80. When the equation of a quadric is given in the form Sp0/,-2S€pH-^ = O, (I.) in order to find its centre, or centres, we may replace the equation by S(/Q-ft))95)(yo~ft)) + 2S(/o-w)(0ft)-e) + Sft)0a)-2Seft) + ^ = O, ...(ll.) and if w terminates at a centre the part linear in p — w vanishes, and o) is a solution of the equation corresponding to its zero root, and the length of ^w being indeterminate. If m vanishes, the function \/r vanishes identically since cp is self -conjugate (Art. 67), and in fact is of the form —aiSip. If x^ is not zero, the line of centres is at infinity since (v.) can only be satisfied for infinite values of oo. If however x^~^' ^^^ solutionis m"o. = e + xa), x^ = ^. •• (vi-) and the surface is a pair of parallel planes. More simply when (j)(jo= — aiSico = € and x^ = ae + aiSie = 0, equation (ill.) becomes aSiw = Sie. In the case of the paraboloid, equation (v.) without the condition xj/^e = 0, or m'w = xe + uk, \jre\\k, (pk = .....(vil.) is the equation of the axis, remembering that \/yot) || k — uk where u is an indeterminate scalar. We have in fact on operating by (p, on\Sl% = sin^a ; (vi.) since (SU. eyo)2 + (TVU. e/o)-= 1. (Compare Elements of Qua- ternions, Art. 360.) This may also be written in the form S(Ue — cos(xU?;)yo= ±sm.aTWjrip, (vil.) ART. 8-2.] CONFOCALS. 121 so that the sine of the arc between a point and a focus is propor- tional to the sine of the perpendicular on a directrix arc. Many interesting examples and illustrations will be found in the ElementSy Book II., Chap. III., Sections 1 and 2, and in Art. 306, and also in the sixth of the Lectures on Quaternions. Ex. 1. Through three given points on the surface of a sphere, it is required to draw a sphero-conic so that a given great circle shall be one of its cyclic arcs. [If yj, y^ and ys are the vectors to the three given points, it is necessary to find /3 so that S/3pSap + p^ = may be satisfied on replacing p by y^, y2 and yg, a being a given vector. The vector f3 is given by i8Syiy,y3=-2Vy2y3(SayrrM Ex. 2. Find the relations between the cyclic normals of a cone and its focal lines. [Identifying (vi.) with the second form of (ii.), the required relations are easily, obtained.] Ex. 3. Prove that S . V . Ya^YSeY . Y/SyYepY . YySYpa = represents the cone which has five edges parallel to five given vectors, a, ft, y, 8, €, and show that the form of the equation furnishes a proof of Pascal's property of the hexagon inscribed to a conic. {Lectures on Quaternions^ Art. 442.) CONFOCAL QUADRICS. Akt. 82. Quadrics of the family Sp(cj>-hx)p= -1, : (I.) in which ic is a variable parameter, are called concyclic, as they have common planes of circular section (Art. 77). The reciprocal system of quadrics Syo(0 + ^)-V= -1 (II.) is called a confocal system. Because we may write (ii.) in the form Sp(\lr+xx+x^)p = — (m-[-m'x-\-mV-{-x% (iii.) it appears that three quadrics (ii.) pass through an arbitrary point; and reciprocally, three quadrics (i.) touch an arbitrary plane. Also one quadric (i.) passes through an arbitrary point, and one quadric (ii.) touches an arbitrary plane. Confocal quadrics cut at right angles. Let x, y and z be the parameters of the three quadrics which pass through an arbitrary point (a). Then = Sa(^-\-x)-^a-Sa{

+a^)-^p + BSp{cj>+^)-^p+CSp{ct^ + z)- [We may employ the method of partial fractions, and treat <^ as a scalar, it being commutative with scalars and with cf) + ^v, etc.] Ex. 2. If ^'j y and z are the parameters of the confocals through the extremity of the vector p, the expressions Sp. ( + zr^p, are respectively equal to T(cb-\-y)~^p^, zero, and — y-^ z-x Ex. 3. Prove that Jx-y:\J{4>-]ry)-^p, sj. are the principal axes of the central section of the quadric x made by the plane parallel to the tangent plane at p. Ex. 4. Find the centres of curvature at a point on the quadric .r, and prove that they are the poles of the tangent plane to x with respect to the confocals y and z. [If y is the vector to a centre of curvature, two consecutive normals intersect at its extremity, or y — p + t{<^ + x)~'^p is stationary when p and t vary. Therefore [1 + ^ ((^ + x)-^] dp + (^ + .r-)"^ p^t = 0, or ((^ + JP + dp + pd^ = 0, or dLp + {(ji + x + t)-^pdit=-0. Operate with S(^ + ^)~V, and Sp(^ + .r)-i(^ + ^' + ^)-ip = 0, and on comparison with (iv.) the roots of this quadratic in t are seen to he y — x and z-x. Therefore y = {^-\-y){4) + x)-^p^ y' = {cfi + z){cf) + x)-^p are the vectors to the two centres. Observe that dp is also tangential to the quadric z. Compare Art. 87, Ex. 1, p. 136, for the method employed.] Ex. 5- If X, y and z are the parameters of the three confocals through the extremity of the vector p, prove that x+y+z= —m" — p^ ; yz + zx+xy = m' + SpXp ; xyz= —m — Spyj/p. Ex. 6. Prove that the plane SAp + l = touches a confocal at the extremity of the vector p = A-i(VA<^A-l); and show that the locus of points of contact for a system of parallel jjlanes is a rectangular hyperbola. RT. S3.] NORMALS AT A POINT. 123 Ex. 7. Prove that the locus of points of contact of planes through a line is a twisted cubic. [Put for A in the last example {\ + tfi){\-{-t)~^ and verify that an arbitrary plane meets the curve in three points.] Ex. 8. The locus of the poles of a plane with respect to a system of confocals is a right line. Ex. 9. The locus of the poles of planes through a given line is a hyper- bolic paraboloid. [p = {(f) + u)(X-\-tiJL)(l-\-t)~^ is the locus of a line dividing two given lines similarly.] Ex.10. The plane S/oAc^A = is the locus of poles of planes perpendicular to A. Art. 83. In many investigations relating to the confocals through a given point, the extremity of the vector a, it is convenient to employ the vectors A = ((^ + .^)-ia, /x = ( + z)-^a, (i.) which when originating at the centre terminate at the reciprocals of the three tangent planes. These vectors ai'e of course normal to the three confocals. We have then a = {cfi+a;)k = { + i/)[x = ((^-\-z)v, SAa = S/>ta = Sva= - 1 ; (ii.) and because these equations give -l=S/x(<^+.r)A = SA(<^+y)^, or (.^'-^)SA/x = 0, it follows that S/>n/ = Si/A = SA/>t = 0, (ill.) or confocals cut at right angles. We also have from the same equations X = fji + {i/-x){(fi + x)-^fi, etc., (iv.) so that fi^ + (i/-x)Sfji{(f)-\-x)-^fi = 0, (y-.r)Sv(<^ + ;r)-V = 0, or {x-7/)-^=-SVfx{cf) + x)-^Vfx=+SU\{. Hence as trr varies subject to the condition S?7a) = Saa>= —1, we find by (viii.) that S(t7-a)((9 + ?*)-Hn7-a) = 0, or 8t(0-{-u)-^t = ,...r. (x.) is the equation of the tangent cone from a to the confocal w, referred in the first case to the centre of the quadrics and in the second to the extremity of a. The form of the equations shows that the tangent cones drawn from a point are confocal. They intersect in pairs along any line through the point, for (x.) may be replaced by Sr(f0 + ^*X^ + ^2)^^O, (XI.) and may be regarded as a quadratic determining the quadrics touched by a given line (Ut= const.); and they intersect at right angles by the general property of confocals. We can thus determine the two quadrics touched by an arbitrary line. Ex. 1. Prove that (f + uXe + 11^) p = (f-\-toX-^ u^) p + ya{(ji + u) Yap. Ex. 2. A right line defined by the vectors o- and t of Art. 36, Ex. 4, touches the confocals whose parameters are the roots of the equation, Ex. 3. The lines through a given point touching confocals with a given sum of parameters, generate the reciprocal of the tangent cone to a fixed confocal. [The cone of the lines is St{6 - m'^ - a^ - v)t = 0, if v is the sum of the parameters.] Ex. 4. If V and v are the vectors to the reciprocals of the tangent planes of the confocals u and u' at the points a and b, and if t is the vector ab, [Here t = (<^ + ^c') v' — ((f) + u) v. This is Gilbert's theorem.] Ex. 5. If the points a and b are both points of contact of the line with the quadrics, 81/1/' = 0, Sv(^i/' + l=0. Art. 84. There is a third general method which is often useful for dealing with the properties of confocals. Writing the equations of the three confocals through a point in the forms T(<^+^)V = ], T(<^+^)V = 1, T(c/> + .)V = l, (I.) we are led to assume /o=v^{(+-^)(<^+y)((^+2)}€ (II.) as an expression for the vector to the point of intersection. The square roots (<^ + ^)^, etc., are commutative, and, accordingly, on substitution in Sp((^ + ^)-V=-l, we find -l = S€{cf>-h^j)((f> + z)€ = S€(j>h + {^/ + z)Se(ti€ + 9/z€'^ (iii.) ART. 84.] ELLIPTIC COORDINATES. 125 This is identically satisfied, for the confocal x as well as for the confocals y and 2;, if €2 = 0, S€<^e = 0, Se<^2^=-1; (iv.) or what is equivalent, if €2=0, S€X€ = 0, S€Vre=-l; (v.) that is, if € is the vector to a point of intersection of three known surfaces, one of which is of course imaginary. Therefore (11.) coupled with the con- ditions (iv.) or (v.) is the vector to a point of intersection of the three confocals ; and allowing any two of the parameters, y and 2, in (11.) to vary, the vector equation represents the surface x ; if only one parameter {x) varies, the equation represents the curve of intersection of the confocals y and z. Again, we may differentiate p, regarded as a function of x, y and z, as given by equation (11.) just as if <^ were a scalar, and we have '^''=*-(^+^Ty+^.)''= ("•) and the method easily lends itself to the treatment of lines traced on a quadric surface. Ex. 1. Prove that the vectors (cf>+x)-^pj {4>+y)~^p, {cf) + z)-'^p are mutually rectangular, and that the squares of their tensors are (z-x)(x-y) (x-y)(y-z) (y-z)(z-x) where m(x)=m + m'x + m"x^ + a^, and where x, y and z are the parameters of the confocals through the extremity of p. [Using (11.), we have Sp{-hy)-\(p + z)-^p = S€(+y){4> + z)e. This is reduced by replacing y by x+y-x, etc., to S€( + x)~^€ multiplied by a factor. On inversion of (^ + .r)~i the rest follows.] Ex. 2. Find Tp2 in terms of x, y and z. [x-{-y + z+m"=Tp\] Ex. 3. Express the vector e in terms of the roots and axes of cfy. Ex. 4. Prove that Tdp2=i2- ^^~^y-^~-^W . Ex. 5. Prove that p = ((f) + u){(j)+x)~^(+y)^(<^ + z)^€ is the equation of a tangent to the curve of intersection of the quadrics y and z ; u being alone variable. [Use (VI.).] Ex. 6. Prove that p = ((f)+x)~^(cf)+y)\(f> + z)h is the equation of the surface of centres of the quadric x — the locus of the principal centres of curvature — when y and z vary. (See Art. 82, Ex. 4.) Ex. 7. Find the lengths of the principal radii of curvature in terms of X, y and z. Ex.^ 8. The imaginary right line, t variable, " * p = (-\-x)K is an umbilical generator of the quadric x. 126 CONFOCAL QUADRTCS. [chap. ix. [It is evidently a generator of the quadric, and parallel to a line to a circular point at infinity for T(^+^)^€ = 0. That is, it is one of the eight generators through the four points in which the imaginary circle at infinity cuts the quadric. But the tangent plane at an umbilic cuts the surface in a point circle — or a pair of these imaginary generators. See Art. 67, Ex. 1, p. 96.] Ex. 9. Find the locus of a point through which two of the three inter- secting confocals coincide. Show that it is a developable surface generated by the tangent lines to the curve [This is the locus of the umbilical generators of the system, or the circum- scribing developable.] Ex. 10. The focal conies are double curves on this developable. [Put t equal —g^, —g^ or —g^ in the equation of Ex. 8, and we get a plane curve in one of the principal planes. For t= -g^we have Sp(-g,r^p = ^€(cl,-g,){4> + x)e=-l, S^> = 0. The conic is double on the developable because a double sign is lost owing to the destruction of the component of the vector normal to the plane.] Ex. 11. If a is a constant vector, and x, y variable scalars, the equation represents a quadric surface, <\> being a self-conjugate function. [Assume the equation of the quadric to be S/o(a(/)''^ + 6<^ + c)/) + l = 0, and determine the constants a, h and c] Ex. 12. Prove that the imaginary vector e of equation (iv.) satisfies the relation v — 1 . e = Ve^c, EXAMPLES TO CHAPTER IX. Ex. 1. Three right lines through a common point are mutually at right angles. If the first and second move in the planes SA/3 = and S/>t/) = respectively, the third describes the cone SVA/oV/x;o = 0. Ex. 2. The cone SazSajSa ^ SfSiSfSjSfSk SyiSyjSyk _ Sap "^ Sf3p ■*" Syp contains the six unit vectors ^', y, k and a, ^, y, the vectors of each set being mutually perpendicular. Ex. 3. If the cone Spcj)p = has three mutually rectangular edges, the condition m"=0 must be satisfied ; if it touches three mutually rectangular planes, m' = 0. Ex. 4. The four cones of revolution which touch the planes SA/o = 0, S/x/o = 0, Syp = are represented by T . Yp-Wpl, ± YfivTX (S A/zi/)-i = 1 ; and the cones of revolution through the three lines VA/D=0, V/A/3 = 0, Yvp = are represented by T . p-'^Sp2 ± V/xvTA (SXfxv)'^ = 1. ART. 84.] UMBILICAL GENERATORS. 127 Ex. 5. Three points fixed on a line move in given planes. Find the locus of a fourth point fixed on the line, and show that it is represented by an equation of the form T{aYfxvSXp + 6y i/AS/xp + cYXfxSvp) = 1 . Ex. 6. Interpret the equation T/3-^Yl3p = eTX-^SXp as determining the locus of a point moving in accordance with a certain law in relation to a given line and a given plane. Ex. 7. The polar planes of points situated on certain fixed lines cut a quadric in circles. Ex. 8. Find the locus of the centre of a sphere which rolls along two straight wires. Ex. 9. Determine the locus of the vertex of a right cone standing on a given ellipse of which a and jS are the principal vector radii. Ex. 10. A plane cuts a constant volume from a pyramid having its vertex at the centre of a quadric. Find the locus of the pole of the plane with respect to the quadric. Ex. 11. Find a tangent plane to a quadric which along with three Ynutually conjugate planes passing through the centre forms a tetrahedron of minimum volume. Ex. 12. Find the locus of the point of intersection of three mutually perpendicular planes each of which touches one of three given confocal quadrics. Ex. 13. Find the locus of the foot of the central perpendicular on a plane through the extremities of three mutually conjugate radii of a quadric. Ex. 14. Find the locus of intersection of tangent planes at the ex- tremities of three mutually conjugate radii of a quadric. . Ex. 15. Find the locus of a point whence three mutually perpendicular tangent lines can be drawn to a quadric. Ex. 16. Find the locus of a point whence three tangent lines can be drawn to a quadric so as to be parallel to three mutually conjugate radii. Ex. 17. Show that the equation '2p p 4>^p Spp p^ i4>pf determines the directions of the radii of the quadric Spcfip + 1 = which are most or least inclined to the corresponding normals. Solve this equation. Ex. 18. Through the extremity of the vector a mutually perpendicular lines are drawn to cut a quadric. Prove that -m" 1 . 1 1 1 + Sacf>a x^X2 where x^ and .r^ are the intercepts on one of the lines. Ex. 19. From a point on the quadric S/3<^/3+l = 0, the extremity of the vector a, mutually rectangular lines are drawn to terminate on the surface. The plane through their extremities passes through the extremity of the Vtector 2 + l=0. Ex. 21. If UV^' is a fixed vector y, eliminate the scalar t and the variable part of q from the relation and discuss the locus represented by t,=t(/j+„M^). Ex. 22. The vectors a, f3 and y being unit and mutually rectangular, show that the condition that T'JJ4>a + Yf3cl>'Vp = by the plane SA/) + l=0 is equal to the section of the quadric Sp+l)-A'(SA/)+l)][/x(S/x> + l)-/i'(S/x/o + l)]=0 may be generated by the intersection of two perpendicular planes each of which contains a fixed line. Ex. 30. Prove that the foci of central sections of the quadric Sp(f>p-{-l =0 generate the surface p^ (Ypcf^pY ^ Sp(f>p SY pcfipcfiY p(f)p " 0. Ex. 31. The envelope of a sphere which passes through the centre of a quadric and which cuts it in a pair of circles is a quartic surface touching the quadric along a sphero-conic. Ex. 32. Quadrics similar to Sp$p + l=0 are described on a system of parallel chords of Sp(f>p-\-l=0 as diameters. Prove that the envelope of these quadrics is also a quadric, and find its equation. Ex. 33. Prove that v 2 o v . 2m' _ where pn is the vector to the foot of a normal from the extremity of the vector a to the surface Sp^p + 1 = and where m! and m are the second and third invariants of the function p + 1=0 which enjoy the property that the normals at their extremities intersect. E?;.* 39. The locus of the centres of chords at whose extremities the normal^ intersect and which are parallel to a fixed direction t is the right line SpT = 0, Spii^T^O. J.Q. I 130 QUADRIC SURFACES. [chap. ix. Ex. 40. Prove that the squared radii of the circular sections of the quadric gp^ + 2S\pSfMp + l=0 which pass through the extremity of the vector (X are g-^ + X-^(SkafgYg-' and g'^ + fM-^Sfia)Wg-' where g, g' and g" are the latent roots of the linear function determining the quadric. Interpret these results. Ex. 41. Determine the spheres cut in diametral planes by a quadric. Ex. 42. If planes through an edge (/a) of the cone S/)^p = afld through the vectors a and ^ respectively meet the cone again in edges coplanar with the vector y, show that S(pSacl>a-2aSpclia){pSf3cf>/3-2/3Sp(t>l3)y = 0, and reduce this by the aid of the equation of the cone to i Sp(j^aSfSy + Spa - Spcl>ySa(f>f3 = 0. \ Ex 43. Using the notation of Art. 38, p. 42, show that if a translation represented by the vector o> will carry the tetrahedron abcd so that it becomes inscribed to the quadric Spcfip +1 = 0, we shall have (0 = ^v-i cf>-^^XSaa ; S2 ASa(^a<^-»2 ASac^a + 4v2^Sa, a scalar u, a quaternion q and a pair of vectors k and K so that the conditions ^~^{p-K) = {^ + uy'^(qp'q~^ -K') = a unit vector may be satisfied when p and p' terminate at corresponding vertices of the tetrahedra in their initial positions. (6) If (j> is the linear vector function defined by the relations ^{a-^) = a'-^\ (/? -8) = ^'-8', <^(y-8) = y-8', we find that ^*-i = <^'<^-l, and q()q-^ = {(j>'4>y^cfi'. (c) Also in the notation of Art. 38, k and u are given by 2v. (y is a unit vector along the emanant vector rj drawn from P, while P'H' = UV is a unit vector along the emanant rj' drawn from P'. The vector PH'' is drawn equal to P'H'. R..£l- Ur7 Fig. 27. In the limit the quaternion .(I.) JJrjT(p'-p) PH.TPP' is a vector perpendicular to fj and to r]' so that rotation round it from rj to rj' is positive, the angle of the quaternion (the exterior angle at H) being ultimately equal to a right angle. The tensor of this vector is ultimately equal to the ratio of the circulaj- measure of the angle HPH" (the angle between »; and rj') to the ''arc of the curve, and thus the vector represents in magnitude and direction the rate of rotation in question. In 132 METRICAL PROPERTIES OF CURVES. [chap. x. terms of the differential of JJrj and the corresponding differential of yo( = OP), the vector of rotation is the second form of the expression for the vector being deduced from (iv.), Art. 53, p. 68, and the third form resulting from the consideration that If, in particular, we replace the vector ;; by dp, a vector tangential to the curve, we have for the vector of rotation of the tangent, or the vector ciirvature at P, dIZd£_ydV 1 _ VdpdV dp ~ dpTdp~ Tdp=^ ' ^ ^^ for in accordance with the foregoing this vector represents in magnitude and direction the rate of bending of the curve at the point P, the bending taking place in the plane through P at right angles to this vector.* In the case of a plane curve this vector curvature is always parallel to a fixed direction — that of the perpendicular to the plane, but in the general case the direction of the vector is continually changing. The plane through P to which it is perpendicular, or the plane of the bending at P, is the osculating plane of the curve at P. To investigate the rate of rotation of the osculating plane as we pass along the curve, or, what is equivalent, the rate of rotation of the normal UVdpd'^p to that plane (compare the third form of (iii.))> we have by (ii.), dUVdpdV _ VdpdV J _ d3p UVdyodV . Tdp ~ • VdyodV 'Tdp~^^P' ^ VdpdV •" '^ ^^ since d Vdyod^p = Vdyod^p. This is the vector torsion of the curve at P. It gives in magnitude and direction the rate of rotation of the osculating plane, and we see (what is geometrically obvious) that the osculating plane rotates about the tangent line (Udp). * The phrases vector curvature and vector torsion correspond to Hamilton's vector of curvature and vector of second curvature. We shall see what advantage results from considering an angular velocity to be a vector on the plan of this article, and the present case is quite analogous. It is easier in Quaternions to represent the primary characteristics of a curve, the curvature and the torsion, by vectors than to represent the somewhat artificial and indirect conception of an osculating circle or radius of torsion. The theory of emanant lines has been worked out by Hamilton {Elements of Quaternions, Art. 396). ART. 85.] VECTOR CURVATURE AND TORSION. 133 The vector curvature and the vector torsion may be com- pounded into a single rate of rotation . = V^+Ud,.s4^, (V, which may perhaps be called the vector twist of the curve. This rotation produces the same effect on the tangent line and on the osculating plane as the vecior curvature and the vector torsion respectively, for the former vector is at right angles to the osculating plane and the latter is parallel to the tangent line, and we do not here consider the rotation of the osculating plane in its plane or the rotation of the tangent line round itself. If the equation of the curve is given in the form considered in Art. 48, that is if p is given as a function of a parameter t, the expression (v.) may be written in the form ^ = -^+U^'S/^ (VI.) where p\ p and p" are the successive deriveds of p with respect to the parameter. If the arc of the curve is taken as the independent variable, and if p^, p^, /03, etc., denote the successive deriveds of p with respect to the arc, the relations (compare Art. 48, p. 63) Tpi=l, S/Oi/02 = 0, SyOiyO3 + p/ = 0, ctc, (VII.) found by equating to zero the successive deriveds of Tp^, serve to simplify the various formulae. Thus (v.) becomes <«> = /0i/02 + /0iS-^ (VIII.) P\P2 Ex. 1. Show how to connect the deriveds of p taken with respect to t and with respect to s. ds „ /d«\2 dh ^ -] /=/>id7' P=PAdt) ^^idT^' ^*^J Ex. 2. Show that the tangent line and the osculating plane of any curve may be written respectively in the forms, '^ = p-\-xp\ '^=^p-\-xp'-\-yp", X and y being variable scalars. Ex. 3. The tangent line and osculating plane of the twisted cubic may be expressed by respectively, a being a constant vector and <^ a given linear vector function. Ex. 4. Calculate the vector w for the helix trr = a {i cos t -^j sin ^) + kbtf I, j and k being mutually rectangular unit vectors. 134 METRICAL PROPERTIES OF CURVES. [chap. x. Ex. 5. Find the centre of the osculating circle of a curve, [The vector to the centre from the point on the curve has the same direction as Yp'p'Tp'-^ . JJp\ and its tensor is the reciprocal of that of this vector.] Art. 86. The important relations (ii.) and (iv.) of the last article enable us to reduce every affection of the curve to a function of the unit vectors a = Udyo, y = UVdpdV, l3 = VYdpd^p\Jdp,..: (I.) of the scalars TVdpdV ^_g cJBp ^1- Td/>3 ' ''i-^Vd^dV and of the deriveds of these scalars with respect to the arc. We notice first that a, (3 and y form a mutually rectangular unit system so that a/S = y, /3y = a, ya = (3. The scalars a^ and Cj are the ordinary scalar torsion and curvature respectively, and partly for the sake of symmetry we regard them as the deriveds -r-^, ^ oi two angles a and c. The angle a is the total angle through which the osculating plane has turned about the tangent line in passing from some initial point P^ on the curve to the point P. In like manner c is the total or integrated angle through which the tangent line has turned in the osculating plane from P^ to P. The vector a is along the tangent, /3 along the principal normal and y along the bmormal to the curve. Denoting still deriveds with respect to the arc s by suffixes, the fundamental formulae, (ii.) and (iv.) of the last article, give in accordance with (i.) and (ii.) of the present, the simple relations «i Yi ^1 ■ / X = ^iy> ^ = a.a, ^ = a.a-{-c.y = o), (III.) a y p or a^ — c^P, l3i = a-^y — c^a, yi=—a^a, (iv.) or simply rj^ = Ycotj, (v.) if r] stands for a, jS or y. The formulae in a and y are translations of the formulae of the last article. The formula in fi is derived from these by aid of the relation /5 = y«. To express the successive deriveds, with respect to the arc, of the vector to any point on the curve in terms of a, 13, y and of the scalars a^, c^ and the deriveds a^, c^, etc., of these scalars, we have P3 = a2 = ^iCi + /5c2 = ^C2 + (yai-aCi)Ci, | P4 = i^Cg + 2(yai - aCi)c2 + (ya^ - ac2)c^ -^W + ^i^)^i ; ART. 87.] KINEMATICAL METHOD 135 and in general we shall find the ti*^ derived to be of the form pn=aAn-{-l3Bn-\-yCn. (VII.) where An, Bn and (7„ are certain scalars {not the n^^ deriveds of scalars A, B, G, however). We may remark that the deriveds of highest order of Cj and a^ occur in yo„ in the term ^c^ + yotn-iCi, as we see from (vi.). Thus, as we have asserted, every affection of the curve may be expressed in terms of a, jS, y of a^ and C-^, and of the deriveds of these^ scalars. (See Appendix. Elements, Vol. ii.) Art. 87. The developables connected with the curve may all be investigated in one common way. The vector rj and the scalar e being in some way variable with a point on a curve, a plane of any developable connected with the curve is expressible by an equation of the form ^{Tn-p)rj = e, (I.) V5 being the variable vector to a point in the plane, and p being the vector to the point P on the curve to which the plane corre- sponds. The equation of a successive plane is of the form ■pi I ^ . ., S(CT-yo),;-e + ds.^-(S(trr-yo)»7-e) = 0, (ii.) e, r] and p being regarded as functions of the arc s, but zs being independent of s. Thus two successive planes intersect in the line of intersection of the first plane and of the plane determined by equating to zero its derived with respect to s. The inter- section of the plane (f.) and its consecutive is accordingly the line Common to (l.) and to the plane I S(Tn-p)t]^ = Sat] + e^, (III.) i/j and e^ being the first deriveds of rj and e. This line of the developable is also given by the vector equation (Art. 35 (i.), p. 39), where t is a, variable parameter. In the same way, equating to zero the second derived of (i.) with respect to s, S(T^-p)r]2=2Sar]^ + SPrj.C^ + e^, (V.) and combining this with (ill.) and (l.), we have the point of intersection of three successive planes of the developable. 136 METEICAL PROPERTIES OF CURVES. [chap. x. This point is on the cuspidal edge of the developable, and it corresponds to the point P on the curve. More generally if in (vi.) we allow the arc to vary, we have the equation of the cuspidal edge of the developable. In particular, the polar developable corresponds to »? = a, e = ; while >7 = /8, 6 = gives the rectifying developable; and »y = y, e = is the tangent line developable. It is shorter in many cases to treat the developables ah initio rather than to substitute in the general formulae (iv.) and (vi.). Ex. 1. The vectors from a point on the curve to the centres of the osculating circle and sphere are respectively ^ and -+7 3- . — • Cj Ci 'da Cj [These expressions follow from consideration of the polar developable. Or the first is geometrically obvious, and it is also evident that the centre of spherical curvature lies on the polar line, TS = p-\-^-\-xy, which is by geometry the locus of points equidistant from three consecutive points on the curve. To determine x we may express that V5 is the vector to a point which is momentarily stationary as we pass along the curve. Thus d^=0 = a+5"^ + ^£(i)-^^a,+g.y, and therefore ^=„^.^,g)- We must remember that x is not here a function of s. ^x is some small scalar. See the next example.] Ex. 2. For a spherical curve [In this case we can determine x so that the vector in the last example terminates at a fixed point in the centre of the sphere containing the curve, and now ^x : ds is the derived of x with respect to . + d(o):a and (w + da>) : (a + da) are equal to the second order of small quantities, and hence show that when the developable is flattened out the curve becomes a right line, so that it is a line of shortest distance (or a geodesic) on the developable.] ART. 88.] THE DEVELOPABLES. 137 Ex. 5. If the ratio of curvature (scalar) to torsion is constant, the curve is a geodesic on a cylinder. [If Til) . cos H = Ui, Ta>sin^ = Ci, the angle H is here constant, and equations (m.), Art, 86, give d(a cos ZT+'y sin JI) = 0, or on integration U(o = >[:, a constant vector. The rectifying developable is therefore a cylinder.] Ex. 6. Show how to determine the curves for which the ratio of curvature to torsion is constant. [By the last example we have a^ = ya sin H . Tw = V^a . Tw. If d^ = Tw . d«, we have, on changing the variable from s to t, a' = V^a, and on differentiating, a" = Yka' = kYka= -a-^S^a= -a + kcosH; A2 or -=-2(a-^cosi7) + (a-^cosZr) = 0. The integral of this equation is a — k cos H=kcos t + fi sin t, and as we must have S^a=— cos^ and Ta = l, it appears that A and /x must be perpendicular to one another and to k, and that their tensors must be equal to sin IT. Thus a = ^ cos H+ sin II{i cos t-^j sin t), and on integrating again 'n5={ads=pQ + kscosII-{-sin H . [(^cos^^-^*sin?)ds, where p^ is a vector constant of integration.] Ex. 7. Find the conditions that the unit vectors (a, ^, y) of one curve may remain constantly inclined to those (a', /3', y') at corresponding points of another. [We must have wds = w'ds', or ada + yd?S ^V^^ -- (H-) Putting in these p'^ p-]rdp, rj' = r}-{-dr] and proceeding to the limit having divided Q'Q by T(p' — p), we find ; ^ = V,d,S^ = ,S^ = ^S^^ = p,,....(nr.)- Tdyo Yridr] I Vrj dJJr} ^ ^ ^ by Art. 85 (ii.); and neglecting a vanishing term in the expression for OQ, the various transformations being easy consequences of the formula just cited, and p being a scalar defined by Udyo _ ^ dp _ ^ dpJJt] [rjrj-^~ dVr] The vector pi represents the rate of translation of the emanant line as it passes through successive positions, this vector being the ratio of the shortest distance between consecutive positions to the arc ds of the curve. In other words, the emanant may be supposed to pass from one position to a consecutive in virtue of a rotation ids about the shortest distance QQ' coupled with a translation QQ' = ptds along that shortest distance. Or again p is the ratio of the shortest distance to the angle between the consecutive lines. The quantity p is usually called the para- meter of distribution of the ruled surface, though the theory of screws would offer the more suggestive term pitch, because the transference of the generator from one position to the consecutive is in the language of the theory of screws effected by a twist about the screw coaxial with the shortest distance and of pitch p. The point Q, the extremity of the vector (iv.), is the point of closest approach of successive generators; and as s varies Q describes the line of striction of the ruled surface. For a developable, this coincides with the cuspidal edge, and p vanishes. Ex. 1. Prove that the line of striction and the parameter of distribution of the surface generated by the principal normals of a curve are Ex. 2. The tangent to the line of striction of this surface is parallel to and the shortest distance between consecutive generators is parallel to w. [ ART. 89.] PITCH. LINE OF STRICTION. 139 Ex.3. If r) = aco8l + (3smlcosm + ysmlsmm, prove that the condition that the euianant line tj should generate a developable is sin^. d(a + wi)-cos^sin7ndc = or sin? = 0. [By (v. ) if JO = 0, Sarjdr) = 0.] Ex. 4. Prove that no line except a in the plane of a and /? can generate a developable ; that the only developables generated by lines in the plane of a and y are the tangent-line and the rectifying developables ; and that any line whatever in the plane of (3 and y is capable of generating a developable. [For the plane of a and JS, ^ = or m = 0, and m=0 is impossible if a varies. For the plane of a and y,l = or m = -. If m = -, we find t/ = U(o since sin I .da = cosl .dc. If ^ = ^, we have a series of developables CT=p + ^(/5cos(a-ao)-y sin(a-ao)) ; and their cuspidal edges are trr = p + ^ - '^^ tan (a - ao), Qq being an arbitrary constant.] Ex. 5. Prove that the curves ^x trr=p+"-'^ tan(a-ao) are the evolutes of the curve ^=/o, and that they lie in the polar developable. Ex. 6. If the emanant is perpendicular to the tangent, prove that _ T/Cjcosm ^ _ a^+ m^ where 17 = /5 cos m + y sin 7w. Akt. 89. The normal to the ruled surface Tn = p-\rUri (I.) at any point ct is parallel to p:=Yr}{dp + udf]), (II.) this vector being perpendicular to every tangential vector dz:^ = dp'{-udr}-\-r]du (iii.) The tangent plane is S(7;y-p)Yr](dp + udf]) = 0, (iv.) and as it generally involves u, it varies from point to point along the generator. Moreover, since it involves u linearly, the an- harmonic of four tangent planes is equal to the anharmonic of the four corresponding normal vectors (ii.), or of the four cor- responding points of contact (i.), (Art. 37, p. 41). Expressing that the tangent planes at two points u and u' on the saijue generator are perpendicular, we have a relation Si/|/' = 0, or SYr}(dp-{-udri)Yrj(dp-hu'drj) = 0, (V.) 140 EULED SUEFACES. [chap. x. which determines an involution between the corresponding points u and u. This may be thrown into the form because (SAm"^)'-T(Xm"')'= -T(VX^-i)2. Comparing with equation (iv.) of the last article, it appears that the point Q in which the generator meets the line of striction is the centre of the involution, and that the foci are imaginary. If C and C' are the two points u and u, it is not difficult to see that this equation (vi.) is equivalent to QC.QC'=+p^ (VII.) QC and QC' being vectors, and because their product is positive, they must be oppositely directed. That the quantity on the right in (v[.) reduces to Trj-^^p^ follows most easily by taking the arc as the independent variable, and then = _^-iSUd^.ri ^y (v-) of the last article. Ex. 1. If the tangent planes of a ruled surface touch the surface all along the generators, the surface must be a developable or a cylinder. [The direction of the normal must be independent of u. This requires d7;||r;, that is, dUi7 = 0, or else dpH?;, or the line is a tangent to the curve ^ = p.] Ex. 2. If for any point p = the tangent plane touches all along the generator. [A generator of this kind is said to be tarsal. A ruled surface has in general a definite number of torsal generators.] Ex. 3. The point q being on the line of striction, prove that the tangent of the angle between the tangent planes at q and at any point c on the same generator is TCQ tan A=^. P Ex. 4. Prove that the vector velocities of the points c and c' are at right angles, and compare their magnitudes. [The vector velocity of c is i{q,c+p). See Art. 88.] Ex. 5. Prove that the vector to a point on the line of striction of the quadric S/3<^p + l = 0, and the corresponding parameter of distribution are respectively where Sr]r]=0 as well as S7](fi7j = 0.] ART. 90.] NORMAL AND GEODESIC CURVATURE. 141 (iii) Curvature of Surfaces. Art. 90. Projecting a curve on any plane, normal to the fixed vector k, the curvature of the projection is (Art. 85 (iii.), p. 132) d.Vd.k-Wkp _ Y,k-''Ykdp.k-Wkd^p _ k-^Skdpd^p d.k-^Ykp ~ T(k-Wkdpf ~ TiYkdpf dUdp Tdp3 -/c b/. ^^^ T(Ykdpr ^^ or the curvature of the projection is the projection of the cur- vature into the cube of the cosecant of the angle between the tangent to the curve and the normal to the plane of projection. If the plane of projection is parallel to the tangent, the pro- jection of the curvature is the curvature of the projection. Resolving the vector curvature of a curve traced on a surface into its components perpendicular to and along the normal j/, we have . dUdp^.-iV.dUdp ,g^dUd£ dyo dp dp ^ ^ and since Si/dyo = 0, the first component is, by what we have just proved, the curvature of the projection of the curve on the normal plane {l.vdp) to the surface through the tangent line, and the second is the curvature of the projection on the tangent plane. Remembering that Svdp — 0, and that its derived is also zero, or Si/d'^yo= — Sdt/dp, the first component admits of the trans- formations dp ^ ^^ ^ dp.v.l^dp dp.v.Tdp ^ ^ The last of these shows that the component is the same for all curves traced on the surface, provided they have a common tangent line dp, dv being a linear function of dp ; and thus in particular it is the curvature of the normal section of the surface through dp. This is Meusnier's theorem — the magnitude of the curvature of the normal section is that of the oblique section into the cosine of the angle between their planes. The second component is, as we have already shown, the cur- vature of the projection of the curve on the tangent plane, or it is the rate of bending of the curve round the normal (or in the tangent plane). It vanishes for a geodesic — the straightest curve on the surface between a pair of points — for such a curve can have no component of bending in the tangent plane ; and it is called the geodesic curvature of the curve. The difierential equatipn of a geodesic is therefore SixipdUdyo = 0, or ^vdpd^p = (iv.) 142 CUEVATUKE OF SURFACES. [chap. x. The normals to the surface along the curve trace out a ruled surface, and by Art. 88 the equation of the line of striction and the value of the parameter of distribution are \ '^='^+'^4^' P-^yl^r (V.) The tangent planes along the curve generate a developable. This and its cuspidal edge are respectively represented by ^ , ^r J ^ , Vi/di^Sdpdj/ ■■ 5 , - ;. I Art. 91. , If fp is any scalar function of p, and if we write d/yO = 7lSl/dyO, dj/=0d/), (i.) the function ^ is self-conjugate when n is independent of p or when it is a function * of fp. Let dp and d'p be any two independent differentials of p so that d'dyo = dd'yo, d'dfp = ddfp (ii.) We find on expansion by (i.) if dn = S(Tdp, d'dfp = nScpd'pdp + nSpd'dp + Scrd'pSi/dp, 1 ddfp = nS(pdpd'p + nSpdd'p + Sa-dpSvd'p ; and by (ii.) these expressions give j Sdp(no-V).(0o"'-C^r'Ti.-i)-i(^-/o + 9!>o"M = Si/0o~'»^(xil.) which pass through the extremity of p. The subject will be resumed in Art. 156, p. 295. Art. 93. The equation of the normal to a surface at the point p being TS^p — Xv, (i.) to find the condition that two successive normals should intersect, we express that the extremity of V5 is momentarily stationary and we have dtrr = = dyO — Xdv — V^X = dyO — iC^dyO — vdiX, (ii.) where die is some small scalar if dyo is small (see Art. 87, Ex. 1). The condition of intersection is therefore Sd^,/d^ = 0, (III.) and this is the differential equation of the lines of curvature. I Moreover we have from (ii.) dp\\{\-X(j>)-\ where Sr.(l-iC^)-ii/ = 0, (iv.) because Si/d/o = 0, and from these equations we can find the directions of the lines of curvature and the principal curvatures (7^ = a5j-^Ti/"^, G<2^ = x^~'^Tv~^ if x^ and x^ are the roots of the quadratic. More directly, we have for the vectors to the centres of curvature, xn^ = p-G^-^Vv, v5^ = p^Gf^'\Jv, (v.) and if dj/o and d^p are tangential to these lines, diyo = Oi-idiU,., d^p = G,^-^d^Vv', (VI.) and the measure of curvature, or the product of the principal curvatures, is t ART. 94.] MEASURE OF CURVATURE. 145 if dyo and d'p are arbitrary tangential vectors, as we may prove by supposing p and Ui^ expressed in terms of two parameters. The interpretation of this remarkable expression is that the small area determined on a unit sphere by lines drawn through its centre parallel to the normals round any small contour on the surface, bears to the area of the small contour a ratio equal to the product of the principal curvatures. If we suppose the vector to a point on the surface to be a function of two parameters t and u, and if we use upper accents to denote differentiation with respect to t and lower accents for differentiation with respect to u, we have dp = pdt-{-p^du, and Tdyo^ = edt^ + 2fdtdu -[- gdu\ if e=-p'\ f=-Sp'p^, 9=-p? (vm.) Writing also v = ^pp, , equation (ii.) becomes p'dt + pdu — x(v'dt + v^du) — pdx = 0, (ix.) and according as we eliminate x and da; or d^, du and dx we find the differential equation of the lines of curvature dt^Sppv-dtduS(pi^^ + py)v-{-du^Sp^v^v = 0, (x.) or the equation of the principal curvatures (C=x~'^Tv~'^) h- C^Ti;^-hCTi;S{p\ + p'p^)v-Sv\p = (XL) It is not difficult to see that we obtain for the measure of curvature the expressions CjOg . Tj/* = Si/p'Svp^^ — (Sj/^/)^ =SY.p"Y,p,-(v,p:f+,^(Sp"p-p;^y, (XII.) and that in terms of the deriveds of e, f and g, 2Y.p' = (e-2f)p+e'p,, 2V.^;= -g'p+e^p^, 2V.^.= -g.pH^f-9lp., ^{^p"p-p:') = ^-^f;+9\ „2==y2_g^. (xjjj) and hence it follows that the measure of curvature is an explicit function of the quantities e, f and g and of their deriveds, so that the measure of curvature depends only on the expression (viii.) of the square of a linear element. If then the surface undergoes any transformation in which the lengths of linear elements remain unchanged, the measure of curvature preserves a constant value. Art. 94. Tlie following kinematical method is often useful in investi- gating the geometry of a surface. Suppose the vector p to a point on the surface to be given in terms of two parameters, u and v, and let a unit vector a be drawn at the extremity of the vector p tangent to the curve J.Q. K 146 CURVATUEE OF SURFACES. [chap. x. u variable ; let y be the unit vector along the normal at the same point and let /?=ya be at right angles to both — a tangential vector. These three variable vectors may be supposed connected with three fixed unit vectors e, ^'j h by the relations qiq-^ = a, qjq-'^ = (3, qkq-^^y, (i.) so that the conical rotation represented by q would bring the vectors i,j, k into parallelism with a, /?, y. These relations being supposed to hold for all points on the surface, it follows that q must be a function of u and v. It will be proved in Art. 106, p. 173, that if ^ is any vector function of u and v, its differential is expressible in the form, dJ = V(co'd^ + o>4v)5 + d(^), (II.) where* wi'diU + oi^diV = '2Yd.qq~^ and d(^) = ad.r + /5d?/ + 7d2; if ^ = ax+(^y-\-yz^ while of course d^ involves differentials of a, (S and y. We shall write in terms of a, /?, y, (D'=aa' + ftb' + yc\ o>^ = aa, + ^6, + yc,, (m-) so that equation (v.), Art. 106, is equivalent to 'da' 9a, ,, , , 36' 36, , , Be' 3c ,, ,, . . -7^ — ^ = bc, — bfi ; 75 7=~ = ca, — ca ; ^ — P^ = ab, — an ; ....(iv.) &i) OU OV OIL OV OU these being the results of equating coefficients of a, ^, y in the equation cited : 3(0)') 3((o,) „ , ■ A ?5— ^= VCO W,. OV OU It will be sufficient for us to confine our attention to the case in which the curves u and v cut at right angles, so that ^ is tangent to v variable, since a is tangent to u variable. There is, however, no difficulty in taking the general case. We have then for the orthogonal curves, dp^AadiU + BfidiV and Tdp^ = A^dLu"^ + B'^dv^ (v.) so that AAu and B&v are elements of the arcs of these curves. The vector p being a function of u and v, we obtain additional relations connecting the six scalars a', 6', c', a^, 6„ c„ by expressing that ^=l,(^")=li(^^^=^.- ("•> Now, attending to (11.), we have for example, by (m.), da=V(a)'d2* + a),dv)a = (^c'-y6')dw + (/3c,-y6,)dv, (vii.) and the differentials of y8 and y are obtained by cyclically transposing a, /5, y, a\ 6', c', a^, 6„ c,. Hence (vi.) at once leads to the three relations ^ + i?c'=0, ^^-10=0, Ab, + Ba'^0 (viii.) obtained by equating the coefficients in |^.a + J(/3c,-76,) = '^/3+S(7a'-ao'). These three relations coupled with (iv.) give all that is necessary for the investigation. * Note that Vdg^-^ is not a perfect differential. ART. 94.] KINEMATICAL METHOD. 147 To ascertain the meaning of tlie scalars, observe that the vector curvatures of the curve u variable and v variable are (Art. 86, p. 134) 3a 1 _ fih'-\-yc' g/3 1 _ aa, + yc, , . du' Aa~ A ' dv'Bfi' B ' ^ '^ so that by what we have shown A~^h' is the curvature of the normal section through u variable and A~^c' is the geodesic curvature of the same curve. For any curve traced on the surface, if Ud/o = U(a^dM + /3-6dv) = acos^ + j8sin^, cosMs = ild2*, sinM« = 5dv, (x.) the vector curvature is - yJJdpl (a' sin I - b' cos l)—j- + (c(',sml-b^cos 0~"d~ n ••••(xi.) which follows easily on substituting for du and dv in d . Ud/3 = (^cosl-a sin I) dl + {(^{c'du + c^dv) — y {h'du + h^dv) ) cos I + (y {a'du + a^dv) - a{c'du + c,dv)) sin I. Thus the geodesic curvature depends simply on c', c,, and the rate of variation of the angle I which the curve makes with u variable. The normal curvature depends on the four quantities a', a,, 6', 6,. The relation (xi.) includes everything relating to the second differentials of the curve, and if we write for the curve a' = Ud/o, y' = U . dUdp.d/)"^, y'a' = /5', we may, for brevity, replace (xi.) by the relation y = y cos m + ya sin m, (xii.) and we may determine the torsion and everything depending on third differentials by differentiating once more. Ex. 1. Determine the equations of the lines of curvature, and prove Gauss's theorem that the measure of curvature depends on differentials of the line element. [If C and C, are the principal curvatures, p - C'~^y and p - C~^y are the vectors to the centres of curvature, and expressing that these are stationary for the moment, we have Aadu + BfBdv - C-\a{h'du + h^dv) - jB{a'du + a,dv))=0, and according as w^e eliminate the ratio du :dv ot C we have the equation of the lines of curvature, and the equation of the curvatures, Aa'du'^ + {Aa^-\-Bh')dudv + Bh,dv^ = 0, C^AB-C{b'B-a,A) + a%-a,b' = 0. By (iv.) and (viii.) we see that the product of the curvatures is a function of ^, J5 and their differential coefficients.] Ex. 2. Prove that when the curves u and v are lines of curvature, b'^C'A, a,^-CA a' = 0, b, = 0, c'=-5-i^, c, = ^-i^; and show that dC,_ {C'-C,) dB dC {C-C) dA du B du' dv A d^v' ' AB \ov\B ov J ou\A du / ) 148 CUEVATUEE OF SUEFACES. [chap. x. Ex. 3. If th( that in this case where G is the geodesic curvature of any curve, and I the angle it makes with the curve u variable. [Here c' = 0, so that A is independent of v, and by a change of the variable u we may put ^4 = 1,] Ex. 4. Prove that the total curvature of any portion of the surface is where d*S' is an element of the surface ; and where I is the angle the bounding curve makes with the curve u variable, O is the geodesic curvature of the bounding curve and ds an element of its length. (a) Examine the case in which the bounding curve is composed of geodesies. (iv) Families of Curves and Surfaces, Akt. 95. If p = tj(t; a,h, c, ...), (i.) where ;/ is a given function of a variable parameter t and of certain scalar constants a, b, c, etc., the equation represents a family of curves, any particular member of the family being determined by assigning fixed values to the constants a, h, c, etc. If there are n constants, the family is said to be -n-way, or to be of the n^^ order. The curves of the family which touch a given surface or inter- sect a given curve co^mpose a family of order n — 1. If the given curve is p = r]-^(t-^), the condition of intersection f](t; a, h, c, ...) = r}i{h) (n.) is equivalent to three scalar equations, so that on elimination of t and t^ from these, we are left with a scalar equation in the constants a, h, c, etc., and thus one of the constants may be expressed in terms of the remaining n—\. If the given surface is f(p) = 0, the conditions for contact are fM = 0, %^=0, (m.) and on elimination of t, a relation connecting the constants is obtained, so that a family of order n—1 touches the given surface. Art. 96. Expressing that an unknown surface f(p) = meets a curve of the family at the extremity of the vector p in n consecutive points we have Svrj''^ -{- 2Sr]" 2(>^X.) = 0, and the equation is obtained by solving for A (Art. 74, Ex. 3) from the first and second and substituting in the third.] Ex. 6. Find the differential equation of surfaces generated by similar and similarly situated curves. [Here a generating curve is p=K-\-aa(t) where k and a are constants to be eliminated and where a(^) is a given function of t.] Ex. 7. The differential equation of surfaces generated by equal and similarly situated ellipses is SYYa/S. I/. 0. VYa^. v = (Sai^2 + s^v2)t a and f3 being a pair of conjugate radii. Art. 97. As in the last article, being given the scalar equation of a family of surfaces involving n constants, fip; a,6,c, ...) = 0, ....(i.) we can determine the differential equation of a surface which at each point is touched by some member of the family in as many consecutive points as serve to eliminate the constants. If only one constant is involved, only one surface is touched at each point by a member of the family, and that is the envelope obtained as the locus of intersection of consecutive members by eliminating the constant a between . /(p,«) = and §^^) = 0.. (II.) 150 FAMILIES OF SURFACES. [chap. x. If two constants are involved, the conditions for contact with some unknown surface at the point p are v = xv^, f(p;a,b) = 0, (iii.) where v is the normal to the unknown surface and pq the normal to the surface of the family. The first equation, on elimination of the unknown scalar x, is equivalent to two scalar equations, and between these and the second we can eliminate a and 6, and we obtain the differential equation of the touched surface as a function of p and v, homogeneous in v. When the family contains three parameters, we express that the surfaces touch at two consecutive points, and we have V — xvq, (pdp = X(pQdp -\-dx.vQ = 0, f{p ; a, b, c) — 0, Si/dp = 0. (iv.) We can eliminate dp and replace the equations by v = xvq, Sv((I) — X(Pq)-'^v = 0, f{p; a, 6, c) = 0;.. (v.) and these equations are equivalent to five scalar equations from which to eliminate x, a, b and c. Observe that we find two directions dp for contact according as we substitute one or other of the values of x given by the scalar equation (v.) in the second equation (iv.) It is not hard to see that each additional condition of successive contact affords one additional scalar equation in x and the constants. In fact if we attend merely to the new unknowns d^p and d'^'x introduced in d^-\^dp — X(pQdp-\-dxi/Q) = and d"*"^Si/d/) = 0, we see that they occur in the forms d> + (0-ic0o)-\.d'"^ + etc. = O, Si/d> + etc. = ; and when we eliminate the vector d^p, the scalar d^x disappears also by (v.). The preceding vector condition d^ - X(l>dp - xcp^dp + dxi^o) = serves to eliminate d'^~'^x, and so on. The conditions of contact at n — 1 successive points serve to eliminate the n constants, and the result is the differential equation of surfaces touched at each point by some one member of the family in n—\ successive points. In particular, the equation is the differential equation of envelopes of the family obtained by replacing the n constants by arbiti^ary functions of a single constant. When the family of surfaces is given in terms of two para- meters t and u, — // 01 • // ^ /. \ r\n \ we have v^xYri'ri^, dv — (t>{ridt + r}^du) = xdYriri^-\-y^r{fj^dx,....{Yii.) and on direct elimination of d^, du and dx, Sv .W -xY{rl%-^r,'rt;)-][cpr,^--xN{riX + r;ri,;j\ = 0. ...(VIII.) ART. 98.] DIFFERENTIAL EQUATIONS. 151 The next differentiation introduces dH, d^u and d^x, and these being eliminated by an equation analogous to (viii.) we use (vii.) to get rid of d^, du and dx. Ex. 1. Prove that for the envelopes of a family of spheres, ±'UvR = p-K, where k is the vector to the centre of a sphere and R the corresponding radius. Ex. 2. The differential equation of envelopes of spheres of constant radius whose centres lie on a curve on the surface //o=0 is /(/o ± U vi?) = 0. Ex. 3. The differential equation of the envelopes of spheres having their centres on the ellipse p = a cos t + fSsint is, Ex. 4. Find the differential equation of developable surfaces. Ex. 5. Show how to find the differential equation of the envelopes of a surface carried parallel to itself. [Take p = 8+rj(t, u).] Ex. 6. Find the envelopes of a rotated surface. [Take p = q .7j(t,u).q~^.] Art. 98. A differential equation of the first order presents itself in the form w \ _ q /j \ homogeneous in p. For any variation of p and p subject to this condition, d.F(p,v) = STdp-{-S,^d, = 0, (ii.) where t and /ul are determinate functions of p and v. If the equation has a solution, there must be some scalar function of P^^^^^^^ d.fp^nSvdp, (III.) and for any arbitrary differentials of /o, if dn = Sa-dp, d'dfp = nSd'vdp + nSvd'dp + Sa-d'pSi/dp = ddyp = nSdpd'p + nSvdd'p + S(TdpSvd'p, so that (compare Art. 91) S (TidV - a-Svd'p + vSa-d'p) dp - nSd'pdp = ; (iv.) and this general relation must include (ii.) as a particular case. Hence for some differential d'p satisfying Spd'p = 0, we must have XT^nd'p+pScrd'p, XfA= -nd'p, (v.) and from this we have the equivalent of Charpit's equations "' * dp Ypdp o J /^ / \ ^= - v~' Si/dp = 0. (VI.) JUL \PT 152 CUEVES AND SUEFACES. [chap. x. EXAMPLES TO CHAPTEE X. Ex. 1. Determine the equations of the osculating circle and osculating helix of a curve in terms of the vectors a, /?, y and the scalars a-^ and c^ corresponding to the point of contact, and find the deviation of the curve from the circle or helix. Ex. 2. Show that the vector to a point on an ellipsoid may be expressed in the form p = acos^ + Tsin2^ where Tt = 6, SAt = 0,TA = 1, the vectors A and a being constant but t being variable. (a) A tangential vector is d/o = ( — a sin w + t cos u) d^i + At sin udt, and the equation of the tangent plane is Svp = b^SXa where v=VAT(asin?i-TCOs^). Ex. 3. The differential equation of a geodesic on the quadric Spp _ ^dpfdp dp' "^ dp.cf)p'=C.dpl (b) The geometrical interpretation is that PB is constant along the geodesic, where F is the central perpendicular on the tangent plane and where Z> is the diameter of the quadric parallel to the tangent to the geodesic. (Compare Ex. 14, p. 287.) Ex. 4.* A unicursal curve of order n is represented by an equation of the form <"„„„„ 'h/ ^^^ p = \(^Oi «!, a2"'<^nit, I) . («o, «i, a2...ajt, l)" and in general this equation may be transformed into P — Po + ^i jr f> and the curve may be described as the locus of the mean centre of corre- sponding points on n homographically divided lines. (a) The equation of the asymptotic tangent parallel to /?i is Ex. 5. Find expressions for the curvature and torsion of a line of curvature on a quadric in terms of the elliptic coordinates of Art. 84. Ex. 6. The vectors p = d(t) to points on a curve are transformed by the operation of a linear vector function (f>. Compare the curvature and torsion at corresponding points. *See Proc. P. I. A., 3rd Series, Vol. iv., 1897. ART. 98.] EXAMPLES. 153 Ex. 7. (a) If a, /?, y and 8 are vectors from a common origin to four points A, B, c and d, it is always possible to determine four scalars a, 6, c and d, so that aa + b0-\-cy + d8 = 0. (b) If the sum of these scalars is zero, the four points lie in a plane. (c) It is also possible to determine a second set of scalars so that a'a-i + 6'^-i + c'y-i + <^'S-i = 0. (d) If the sum of this new set vanishes, the points lie on a sphere passing through the origin. (e) The equation of this sphere may be written in the form Sp-i (iS-iy"' + r~'«~' + a-^/5-i) = Sa-i/?-iy-i. (/) If it is possible to determine a third set of scalars so that a"ai + b"l3^ + c"yi + d"8^ = 0, the four vectors are edges of the right circular cone SU/)(U .f3y + V. ya + U . a/5) = SU . af3y. (g) If the additional condition is imposed that the sum of the scalars of this third set vanishes, the four points lie on a surface whose equation may be written SA^* = 1, A being a constant vector. (h) Discuss briefly the nature of this surface. (Bishop Law's Premium.) Ex. 8. The differential equation of surfaces generated by lines of the complex (Art. 36, Ex. 4, p. 40) /((t,t) = may be found by eliminating o- and t between this equation and O- = V/0T, SvT = 0, St<^t = 0. (a) For the linear complex S{axT + (3r) = 0, the equation is (b) Lines common to the two linear complexes S(ao- + ^T) = 0, S(yo- + 6T) = 0, generate the surfaces whose differential equation is S.v(Vap + /?)(Vyp + S) = 0. (c) Find the differential equation of surfaces generated by lines of the congruency /(o-, t) = 0, S(ao- + ^T) = 0. Ex. 9. If the vector ^8 is a given function of a variable unit vector a, the equation Y(^_^)^_0 represents a congruency of right lines. (a) If d^ = ^da determine the meaning of the several terms in the equation <^da + .rda + ad.r = Pada. (b) A line of the congruency is intersected by consecutive lines at two focal points p = (^ + xa where .a? is a root of the quadratic ^a{(^-irx)-^a = 0, or Sa(.^2_^^Xo + Wa-Sea2 = 0, € being the spin- vector of <^ and ^^ being the self -con jugate part. 154 CURVES AND SURFACES. [chap. x. (c) The points of closest approach of consecutive rays to the ray p = l3 + xa lie between the extreme points determined by the condition that SUda^Uda may be a maximum, and the corresponding values of x are the roots of the quadratic Sa(^Q + ^)~ia = 0, or Sa(^'2 + ^Xo + >//'Q)a = 0. {d) The vectors of shortest distance at the extreme points between the ray a and its consecutives are mutually perpendicular ; and if these shortest vectors are parallel to the unit vectors a and a,, the extreme points are determined by ^' = Sa,<^a, and x=^a'4>a'. (e) If the vectors a, a' and a^ are in positive order of rotation so that a'a/ = a5 Sa'<^a,= - Sa^cf>a'= - Sea ; and if the shortest vector at the point corresponding to x makes the angle u with a' so that u^da = a cos u + a^ sin u, the scalars x and P are connected with x' and x^ by the relations, x=x'cos^u+x^sm^u, P=Sea + (x^-x')sinucosu. Ex. 10. A circle may be represented by means of a pair of vectors (/c, X) since its equations may be thrown into the form T(p-K) = TA, SA(/)-k) = 0; and an equation such as /(k, A) = 0, where / is a general function, may be regarded as representing a family of circles. (a) In like manner an equation such as /(a, f3,y)=0 where Sa^=0 represents a family of conies, y being the vector to the centre of one of the conies and a and ft being its principal vector radii. (Compare Ex. 11, p. 103.) Ex. 11. The general surface generated by a variable circle (k, A) may be represented by P = k + Xt where SAt = 0, Tt = 1, the vectors k and A being functions of a single parameter and the auxiliary vector T being arbitrary so far as the conditions allow. (a) If P is a scalar analogous to the parameter of distribution of a ruled surface, ^t xdA-dK F — = dK + d.AT. Hence dr= -75 r-, T Ft- A p_ S(dK-TdA )A_ S(dA+TdK)A SrdK ^ SrdA • (b) These expressions for P lead to four values of the vector t which determine points at which neighbouring elements of successive circles approach most closely or are most widely separated. (c) If successive circles intersect in one point T(VdAASdAA+Vd/i4-m", 2^ = X, (VI.) so that we have jui = Yi:a/3=p\ + Tn\, y = zn, + m"\-'^, (VII.) where /x is the resultant vector moment at the origin and where m" is minus the resultant virial at the same point. The plane of no virial is represented by S2(a-yo)^ = or SpX = m"; (viii.) and Hamilton's centre is obviously the intersection of this plane and the central axis. Ex. 1. Vectors (a) are drawn from a variable origin to the points of application of forces {0). The equation 2Vay8 = impHes equilibrium. [If the vectors Uq are drawn from a fixed origin to the points of appli- cation, we must have separately 2^ = 0, 2Vao/?==0 (Elements, Art. 416).] Ex. 2. Forces act at the vertices of a triangle, in its plane and pro- portional and perpendicular to the opposite sides. Prove that they are in equilibrium. [If a, /? and y are the vectors from a variable origin, the forces are v{0-y), ^{y-o-)) v(a-j8) where j/ is a vector perpendicular to the plane of the triangle. The moment formed as in the last example vanishes identically because Yavl3=Y/3va, etc.] Ex. 3. The conditions of equilibrium of a rigid body may be expressed by the equation 2S/5da = 0, which contains the principle of virtual velocities (Elements, Art. 416 (17)). [For any possible small displacement of the body da=S4-Va>a where 8 and (J) are arbitrary. Hence 2yS = 0, 2Va^=0.] Ex;. '4. The moment of the force ab about the line cd is six times the volume of the tetrahedron abcd divided by the number of units of length in CD. [The vector moment at the point c is V . ca . ab and the component along CD is - S(UcD . V . CA . ab) = - S . CD . ca . abT . CD~^] 158 STATICS. [chap. XI. Ex. 5. A force of unit intensity acts along the line Y{p~a)/3 = 0. Its moment about the line Y(p — a')/5' = is — S(a — a')U^/3'. Ex. 6. If three forces are in equilibrium, they must be in the same plane. [Operate on the condition V(p-a)^+V(/)-a')/3' + V(/o-a")y8" = by S(/o - a) and put p = a where we find S(a' — a) (a' - a")^" = 0. Ex. 7. If four forces are in equilibrium, their lines of action are generators of a hyperboloid. [One method of proof (Chap. VIII., Ex. 10, p. 103) is to express the four vector moments Va„/?„, etc., in terms of the four forces by means of a linear vector function, so that Va„/?„ = (/)^,i + w. The vector w is zero because 2Van^n = 0, 2^„=0, and therefore the equation of a line of action is p = fSnl3n-^ + j;fin- (See Art. 79, p. 116.)] Ex. 8. Eesolve a wrench into forces along the edges of a tetrahedron ABCD. [If fx is the moment and A the force of the given wrench at the fixed origin of vectors o, the moment at the point p is /x - V . OP . A = Sj^abV . pa . ab where t^B, etc., are scalars proportional to the forces along the edges. Take the point p at d, and ju,-Y.OD. A = ^AB- V. DA. ab4-«^bcY.db.bc + #caV.dc.ca serves to determine three of the unknown scalars. Operate by S . DC and ^AB.S.DA.DB.DC = S(/X- V.OD . A) DC, Or ^ab.(aBCD) = S .CD ./x + S.oc.od.A.] Art. 100. To reduce a system of forces to two forces, let JUL and X be the resultant couple at the origin and the resultant force of the system, and assume ^ = Va/3 + Va'/3', X = /3 + /3', (I.) where ^ and /3' are the two forces and a and a the vectors to their points of application. Hence /3' = X-^, M = V(a-aO/6 + Va'X; (ll.) and from the form of the second equation, it is obvious that if two of the unknown vectors a, a, /3 are suitably assumed, the third may be regarded as the vector to a point on a determinate line. But a condition must be satisfied, for on operating in turn by S(\ — /3) and S{a — a) we have S(X-/3)M = SXa^ and S(a-a)^-=SaaX (ill.) so that if any one of the three unknown vectors is assumed (say a) the other two may be regarded as terminating on definite planes. Suitably selecting either /3 or a in accordance with (ill.) (which is a consequence of (ii.)), the remaining vector is constrained by (ii.) to terminate on a line. Ex. 1. A rigid body is acted on by any number of forces. It is required to equilibrate the body by two forces whose points of application are situated on given lines. ART. 101.] EEDUCTION TO TWO FORCES. 159 [If ^ and ^' are the required forces and V(p-a)^ = 0, V(p-a')j8' = the equations of the given lines, we have where x and x' are scalars. Hence and this equation of condition establishes a homography connecting the points of application.] Ex. 2. A framework is composed of rods jointed by smooth hinges. Three of the rods, a^a^, A^Ag and A4A3 terminate at a point A4 and are acted on by given wrenches. Determine the reactions at the joints ; it being supposed that the three rods are not coplanar. [Let (/x,„i„, Xrnn) rcprescut the wrench applied to the rod a,„a„, the origin of vectors being taken as base-point, and let f^rnn be the reaction of the joint on the rod at the point a^. For equilibrium of the rod a^a^, M4i-^M4i + V(a4-^)^4i + V(ai-/,)/?i4=0, and putting p^a^, this gives /Z41 - VaiA4i + V(a4 - ai)/34i = 0, or, for some scalar x^^, /^4i = (/^4i - ^^1X41 + ^4i)(«4 - ai)~^ For equilibrium of the joint A4, we have Ai + A2 + As = ^j ^^ 2l^(/^4« - ^anA4n)(a4 - ttn)"^ = - ^Z-^^X^ni^i " ^n)"^ and from this vector equation the three scalars x^n can be found.] Ex. 3. A rigid body is in equilibrium under the action of an impressed system of forces (jjl, A) and the tensions of two strings a'a and b'b attached to points a' and b' in the body and to fixed points a and b. Show that the forces exerted by the strings on the body are represented by ,(„_„-)=.M±^', ^(0_^) = M + ^« where x, y and t are scalars which may be determined by expressing that the lengths of the lines a'a, b'b, a'b' and ab are given, and where a, ^, a' and /3' are the vectors from the base-points to the points a, b, a' and b'. (a) What condition is implied in these equations ? (5) If «, 6, G and d are the tensors of the vectors a'a, b'b, a'b' and ab, respectively, show that the scalar t satisfies the equation c = T{aU(/>i -I- Ay8 + + &U(/x -f- Aa -I- + d). Art. 101. The resultant quaternion moment (Art. 99 (vi.)) for an arbitrary base-point (the origin of the vectors a) of a system of forces (^8) acting at points fixed in a rigid body is the first quaternion invariant of the linear vector function 0/0 = 'EaS/Sp, (l.) the first scalar invariant of this function being minus the resultaat virial (w,'' = SSa/3), and double the spin-vector being the resultant vector moment (/x = 2Va^). * That is the invariant - (pi . i-\ = (v.) As one root of ^ is now zero, the function is reducible to the binomial form, and the auxiliary x/r function is of the type \frp = \SKp (VI.) where ac is a vector fixed in the body. The equation of the central plane is now 8/c/o = 0. * These are the rotations which convert i', f, k' of the article cited into + *> +i> +^'» +*> ~ij ~^; ~*j +ij -^; or -i, -j, +k. Compare the foot- note to the article cited. ART. 101.] ASTATICS. 161 In addition to the equations (v.) and (vi.) we have <^'A = VA/x and ^/x = ^'ju, = V/ix is double the spin-vector of i/r. These relations coupled with the expression fjL=p\ + Yrjk, (vTii.) for the moment in terms of the pitch p and the vector 97 from the astatic centre to a point on the central axis of the forces in any position enable us to deduce all the theorems of astatics. We first remark that the function (f>cf)' is Jived relatively/ to the body (or to the vectors a) and that the function Y7j\ = TYK\ or SY7]k'cf>YrjX = (VKX.y ; (x.) and therefore relatively to the forces the central Ojxes compose a coaxial family of similar elliptic cylinders whose linear dimensions are proportional to the cosine of the inclination (TVUkA) of the central plane to the axes whose direction (UA) is of course fixed relatively to the forces. The arrangement of the central axes in the body is determined by the equation 'X = YXY7jX (XI.) obtained by operating on (viii.) by VA and attending to (vii.). Taking the tensor TV^A=T<^'UA=v/(-SUA<^<^'UA); (xii.) and the locus of central axes having a given direction UA relatively to the body is a right circular cylinder whose radius is the reciprocal of the parallel radius of the elliptic cylinder T4>'p=T\ or Sp(j>cf)'p = X^ (xiii.) To each generator of a cylinder (x.) corresponds one of the cylinders (xii.) which is traced out by that generator when the forces are rotated round the vector A. In terms of the vectors a and r of Art. 36, Ex. 4 (rjl A), we may replace (xii.) by TATo-=T<^V, (xiv.) and this equation represents a complex of the second order — the assemblage of lines in the body which become central axes by suitable rotation of the forces. We shall now determine the pitch corresponding to each central axis. Operating by ' on (viii.) we have by (vii.) pct>'k + cl,'YrjX = YKX, (XV.) and operating on this by S^'A or SVA/x, or SVAV^A we deduce pT'X^-SX'Y'qX^TX^SKr}X (XVI.) This equation gives p in terms of the vectors determining the central axes. Again we obtain an equivalent expression by taking the tensor of (xv.)j .a'^d on replacing A by t and YrjX by a- the result is p2T<^'T2-2j0ST<^r,p = ct>p-rjSXp (xix.) because (fip is of the form 2aS/?/o. The function ^rj(/)Tj' for this base-point is 'p = 'p-TX.^.7]S7]p ; (xx.) and supposing u^ to be a latent root and a to be a unit vector along the corresponding axis, it appears on inversion of the function cfxp' — v? that the latent roots {u^, u'^, u"^) of <^r)4>il are parameters of the quadrics of the con- focal system (fixed in the body) S/)(,, to the standard form of Art. 70 gives <^nf> = uaSf3p + tc'aS^'p + u"a"S(3"p (xxii . ) where the unit vectors (3 are likewise mutually perpendicular so that the system of forces may be replaced by A acting at the extremity of 97 and by three couples (such as that due to the unit force +B acting at the extremity of tj + ^ua and — ^ acting at the extremity of r) — ^ua) whose arms (ua, u'a, u"o!') are mutually perpendicular as well as the forces (/?, /?', (3"). The parameters of the confocals (xxi.) touched by an arbitrary line (a, r) are the roots of the quadratic equation (Art. 83, Ex. 2, p. 124). St { f'ylr - u^{M" - <^<^') + ^^t + So-(<^', observing that in general the i/r function of ^^' is V^'i/^ ; or of the equation ^4TT-2_^2(J^/-XT2-T<^'T2-f To-2TA2) + TVrT2 + T'VT^ + ^p^r-^4>^'(T (xxiv.) where J/'( = Tk^TA2) is the second invariant of (^<^' or the first of \/r'i/r. This shows that the central axes touch confocals having the sum of their parameters constant and equal to M" ; and in particular we have Minding's theorem for jo=0 that the lines of action of single force resultants intersect the focal conies of the system (xxi.) since the parameters of the touched confocals are in this case the finite latent roots of ' -Trj'-TX^-7jTX\Srj)T = (xxv.) Moreover (Art. 83 (x.)) this is the reciprocal of the tangent cone to the confocal (xxi.) whose parameter is u^ — Trj^TX^. According as the tangent cone becomes more and more obtuse by variation of the vector rj and finally becomes a tangent plane, the reciprocal cone becomes more and more acute and finally coincides with the normal to the quadric, and the locus of such points is the surface Srj(cf>'-T'n^X^)-h]TX^ = l (xxvi.) This surface is a quartic analogous to Fresnel's wave-surface, and its equation may be reduced to the form . rp TVkt^ TYktj _ Tk ^^~T^'r) Tcf,' . k'Wk7j~Tcj,'U . k-Wkt]' ^xxvii.) remembering that (f)'K=0. In this form it is apparent that the surface consists of a system of circles concentric with the astatic centre, coplanar with the vector k and of radius proportional to that of the elliptic cylinder (xiii.) which is parallel to the radius in the central plane. For points inside this surface the cones of axes are imaginary. The boundary of the region containing the feet of central perpendiculars on the axes has been investigated by Tait {Quaternions, Art. 403). Expressing that Tt; is a maximum when Ur/ is given and when t is subject to the conditions (xxv.) Sr?T=0, St((/)^'-T7/2TA2)t = 0, the equation of the boundary is found to be S7;(X.X-^; (VI.) thus p is the reciprocal of the square of the radius of a quadric and the vector cy terminates on the surface represented by s^+i=«' (-^•) because Vc70'trr || X and therefore Tn = Y(j)Yz;y(f)'zn{Yzjy + 1 = 0; their directions (compare (vi.)) are determined by Sa\ = 0, Sa95>X\-i4-l = 0, (x.) and the corresponding pitches are the roots of Sa(V^~pX+2>^)« = l (XI.) which is the condition that the plane should touch a quadric (ix,). In order to reduce to a canonical form the two-system of wrenches compounded from two given wrenches (/x^, \) and (yU2, X2), we assume in conformity with the foregoing a function (p which satisfies the relations 0Xi = ^tp ^\ = jUL^, ^YW = VeVXiXg (XII.) where e is the spin-vector of cp. The function (0 — Ve)p w^U then be self -conjugate and will have a zero root, VX^Xg being the corresponding axis, and it will be expressible in the form -aiSip-bjSjp. We have (Art. 27, p. 25) (jyp = /X1SX2 ( VX1X2) " V — ^2^^! (^^1^2) " V -f YeVXiX^S ( VX1X2) - V> (Xlll.) and the spin-vector is deducible from the relation 2e = V{(^,X2-Mi)(VXiX2)-n+V6VXiX2(VXiX2)-i. Operating by SVX^Xg we find 2SeyXiX2 = S (^i\ ~ ^^2^1) which gives e = V{(^,X2-/X2Xi)(VXA)-^}-KVXiX2)-^S(^,X2-M2Ai)J Taking the origin at the extremity of the vector e, a wrench of unit intensity compounded from the two wrenches is deter- mined by ^ = (p\=ai cos ui-bj sin u=p{i cos u+j sin u)-\-Yr]{i cos % -hi sinu), \ = i cos u-\-j sin u; (xiv.) whence the vector equation of the cylindroid — the locus of the central axes, and the equation for the pitch are r] = (b — a)k sin u cos u + t{i cos u +j sin u), p = a cos^u -\- b sin% 166 STATICS. [chap. xi. where u is the angle the axis makes with the vector i. The scalar equation of the cylindroid is found on elimination of u ^O be TYW8kr} = (a-h)8ir}Sjr} (XV.) To show that in general a wrench may he resolved in one and only one way into components on six given screws, or to reduce any pair of vectors /j. and X to the forms 6 6 lUL^Xt^lUL^, X = 2^iXp (XVI.) 1 1 where the vectors /m-^... /ulq and Xj . . . Xg are given, we assume in the first place iuin = (pi\n, (^ = 1,2,3); iuL,, = 4 + ^jU2)(Ai + a2)-i.] Ex. 2. Find the locus of intersection of screws of the three-system lJb=(f>X. whose axes are coplanar with the origin. [If ix = (f)X=pX. + Y7]X, ix' = X! =p'X-\-YriX! the axes intersect in 97. Hence ( (the two vectors w and p — r being supposed to have a common origin). In fact the vector w represents in magnitude and direction the angular velocity of the body. Using fluxional notation for the velocities, we may write p = T4-Vto(p-T) = a)Sft)-iT+Vft)(/o~T+Vft,-iT-ira)), ...(ill.) thus analysing the instantaneous motion of the body into a rotation round a line coupled with a proportionate velocity of translation along the line ; or, in Sir Robert Ball's phraseology, we have determined the instantaneous twist-velocity about the instantaneous screw ; the expressions >7 = T — Va)"^t + CCa), _p = Sa)-^T (IV.) being the equation of the line or axis of instantaneous motion and the pitch of the instantaneous screw. (Compare Art. 99.) When the equation of this axis is referred to the moving origin we may write it in the form q~^{ri — T)q=-'^r'^q~'^Tq-\-XL = ri because ft) = gfg~\ ...(v.) for (jo = 2Vqq-'^ = 2Yq(q-^)q-'^ = 2q(Vq-'^q)q-^ = qiq-'^ by (i.). The line rj'= —Yr'^q-'^Tq-\-xi being supposed drawn in the body, the motion of the body brings it into coincidence at the proper instant with the instantaneous axis at the time t. Also the rotation converts i into the angular velocity vector oo at the time t. Thus in dealing with the body itself it is convenient to use the vectors i and zj, and in considering the motion of the body with regard to external objects, the vectors w and p are preferably employed. Let us no longer suppose the vector ray to be constant as in (ii.). Then if the vectors p and CT are still connected by the first equation of the last article, we shall have instead of the first equation of the present article p = T + Yw(p-T) + q^q-'^, /) = T-+g(Yitrr + C7)g-i; ...(vi.) and more particularly when the vector r is constantly zero, p==Ywp + q^q-^, /) = g(t^ + V^C7)g-^ if yo = gC7g-i; ...(vil.) and still more particularly (6 = qLq~^ because (o — qiq~^, Ycocd = 0, Yu = 0. ...(viii.) ART. 106.] EELATIVE MOTION. MOVING AXES. 171 What we really do here is to compare the velocities of a point moving arbitrarily with respect to fixed objects and with respect to the moving body. The vector cr represents the velocity of the point relatively to the body, while p is its velocity relatively to fixed objects. Sometimes a notation such as ?^> = 5tng-i; p = N^p + ^ -, where p = gtrrg-i (ix.) may be employed — but it is not very explicit — to denote the variation of p arising from causes independent of the rotation ; and in this notation we may replace ( VIII.) by "=-ar ^^-^ which expresses that the rate of change of the angular velocity is independent of the rotation. We may for example suppose i, j and k to be fixed relatively to the vectors ^, and a = qiq~^, /3 = qjq~'^, y — qkq''^ to be unit vectors derived from these by the rotation. In this case if p = ax + fiy-\-yz, the derived p takes account of the variations of a, /3 and y as well as of x, y and z, while :^ only refers to the variations of x, y and z and not at all to those of a, /3 and y. These results include the whole theory of fixed and moving axes, there being now no difficulty in writing down deriveds of any order. For example, on difierentiating (vi.) again, we have p = T + Vw (yo — t) + Vft)(/6 — t) + gcJg " ^ + Vcogtng ~ \ and on substituting for p, the general formula of acceleration is jo = r + Vw(p-T)+Vft)Va)(y3-T) + g^g-^ + 2Va)gCTg-\ ...(xi.) which may of course be expressed in terms of i. In the case of a rigid body it is frequently convenient to replace (ill.) by the relation p = (r-\-Ywp, (XII.) where or is the velocity of the point of the body which in- stantaneously coincides with the fixed origin of vectors p. The acceleration of the point at the extremity of the vector p is p = &-i-Yco(T + Yd)p + Y(oVa)p, (XIII.) which follows on substitution for p in the result of difierentiating (XII.). As'in Art. 102, we represent the twist-velocity of the body by the symbol (o-, co), the fixed origin being taken as base-point, and we may replace (iv.) of the present article by cr = (p-{-Yn)w; p = Scrw-'^; rj = Ya(o-^ + Xco (XIV.) 172 FINITE DISPLACEMENTS. [chap. xii. Ex. 1. The instantaneous twist-velocity of a body may be reduced to a pair of simultaneous angular velocities, (^ and /?', round two lines, by means of the relations o- = Ya/? + Va'^', co = /3 + /3', where a and a are vectors to points on the lines. [Compare Art. 102.] I Ex. 2. If p is the pitch of the instantaneous screw and if (o is the angular velocity of a rigid body, the velocity of any point in the body satisfies the relation Sp(o-i=^; I and vectors drawn from a common origin to represent the simultaneous velocities of the points of the body terminate on a common plane. Ex. 3. The locus of points having a velocity of given magnitude is a right circular cylinder Tp = T((T + Y(Tp) or TYco(/)-7?) = (Tp2-^2X(o2)i^ coaxial with the instantaneous axis. Ex. 4. Determine the acceleration centre of a body moving arbitrarily. [In terms of cr and co, if the acceleration of the point at the extremity of the vector a is instantaneously zero, o- + Va)cr + Vtba-l-a)V"a>a = or cr + Vwo" + <^a = 0, where cf)p = Vw/a + coVw/). Hence t/^/o = — wS(i>/o — V . wVcuw . p + w^Sw/) and the third invariant is m = Va)u)2, so that aYwu)^ = (wSw + Y . wYww — w^Sw) . {& + Ywcr).] Ex. 5. The instantaneous acceleration of a point of a rigid body moving in any manner is a linear function of the vector to the point from the acceleration centre, or p = c{i(p — a) where <^/o = Ycb/o + YwYw/) and a = 0. (a) The locus of points having instantaneous accelerations of given magni- tude is one of a system of similar and coaxial ellipsoids Tct>(p-a) = Tp, concentric with the acceleration centre, whose linear dimensions are propor- tional to the acceleration. (6) The function <^ is independent of the velocity of translation, and a change in that velocity merely alters the position of the acceleration centre and of the associated ellipsoids. Ex. 6. The locus of points for which the magnitude of the velocity is momentarily constant is the quadric surface S(o--l-Yw/o)((T-fYt;>/)) = or S{d + Ya)(/)-a)}(^(/)-a) = ; and the locus of points for which the direction of the velocity is momentarily constant is the twisted cubic Y(o-+Va)p)((r-i-Y(oo- + Yw/3-|-(oYa)p) = or Y{d-l-Ya>(p-a)}<^(/o-a) = 0. (a) The equation of the twisted cubic may also be written in the form pYoxJ}^ — {((i) — t(ji) S ((i) — toi) 4- Y . (0 Ycow — w^Sw} . (,dv, d$'=:^d?^+^dw, (i.) and it must be observed that ') ^w, ^T- , , 3(a),) . . and accordingly we may replace (iii.) by this new expression The results of this article have been employed in Art. 94 in connection with the theory of surfaces. Art. 107. In many investigations relating to rotations for- mulae of the type * p=y^a''Sa-''/3-yy-' (l.) present themselves, and it may not be superfluous to make a few remarks about their reduction. It frequently happens that a, j8 and y form a mutually rectangular unit system, and in this case a S = aa-\-h/3+cy we have p = Y/3^yy' .aa + y^^a^'^-yy" .hp + y^a^'^l^y-' . cy, ...(ll.) when we apply the general relation a'^^^^^a-^ if Sa^ = 0, Ta = l (ill.) In order to reduce the coefficient of h^ for instance, it is generally best to start from the central term, a^ in this case, and to replace it by cos irx + a sin ttX, and similarly for successive reductions. Thus we avoid introducing the sines and cosines of the halves of the angles of rotation. It is worth while noticing that ■•■ ■$ da^.a-* = |dir.a-hJda(ai-2^-|-a-i) (iV.) * It may be advisable to refer again to Chap. IV. and its examples. 174 FINITE DISPLACEMENTS. [chap. xii. is expressible in terms of the whole angle using the relation The general relation connecting two quaternions jp and q and two scalars x and y, (^p^qp-^)y=p^qyp-^, (v.) will often be found useful. Ex. 1. A planet rotates about its axis y in the period 2m~^ and a satellite describes a circular orbit round the planet in the period 2n~^ ; show that the motion relative to the planet is represented by p = (^y-mt^ymtyt y-mt^ymt ^(^y-mt^ytntynt^ S5€ = 0, the vectors in this expression being all fixed relatively to the planet ; and reduce the equation to p = y-mt^nt^^-ntytnt^ (a) By taking the epoch when the satellite is in the plane of the equator, the equation may be simplified to p^^y-mtj^ayfitj^ y-ntj^-aymt^ S^7 = where r is the radius of the orbit and where Tra is the angle between the plane of the orbit and the equator. (b) The equation may also be written p = ra(cos Trnt sin 7r7nt - cos Tra sin Trnt cos Trmt) +r(3{cos irnt cos 7rm^+ cos Tra sin Trnt sin Tvmt) + ry sin TT-a sin TTTiif where a = fSy. (c) The condition for a stationary point may be written in the form or 7ia + mVy-"*^-"y^«y'^/3 = 0, and this is equivalent to n = m cos Tra, cos Trnt = 0. Ex. 2. Unit vectors a, f3 and y are directed respectively to the point of upper culmination on the celestial equator, to the east point and to the north celestial pole, while ^, j and k are directed to the south point, the east point and the zenith. Show that the vector directed to a star may be expressed in the forms o- = y -^/?- 2'a/?^y^ = h-'^j-Hj^k'^ where ttz is the hour-angle west, Tvy the declination, ttvj the azimuth west, and t:v the altitude. (a) If 7r6 is the latitude of the place of observation, show that and obtain the quaternion equation yzp2yyz^l^l,^-wp2v-l^w^b^ and hence deduce the formulae of transformation from one set of coordinates- to the other. ART. 107.] ASTRONOMICAL EXAMPLES. 175 Ex. 3. Assuming the effect of refraction to be K times the tangent of the zenith distance, prove that the vector to the apparent place of a star is 0- + CT where (a) Substituting for I' in terms of a and f3 (Ex. 2 («)), verify the successive steps of the transformation (^yky-'^-ya = - (3Y(^--'Y(3y = - ^S^y^^ cos irb + ^^j^+i gi^ ^j, = — sin Trb sin Try - cos 7r6 cos Try cos tt^ + /?(sin irb cos Try - cos ttB sin Try cos tt^) — y cos Tr6 sin ttz. (b) Show that the expression for Z3 reduces to the form _ j8' cos Tr6 sin TT^ + y' (sin Tr6 cos Try - cos Tr6 sin Try cos ttz) sin Trb sin Try + cos Trb cos Try cos ttz where fi' and y' are unit vectors tangential respectively to the parallel of declination and to the circle of declination. (c) If q is the parallactic angle and ^ the zenith distance, show that ^-hrrq^jj^^ ^ = ^o"' " i -3 . U Vy o" . tan f . Ex. 4. An equatorial telescope in imperfect adjustment is directed to a star, and the circle readings are observed to be (j/ + i/')'^ and (z+z')Tr where y and z' are small ; if for zero circle readings the direction of the telescope is a + a', that of the declination axis j3 + 13' and that of the polar axis 7 + 7' where a, f3' and 7' are small vectors perpendicular respectively to a, /3 and 7, show that o- = (7 + yV^'\(i + (3V'^''\o. + a') {(3 + (3y^^{y + yT' ; and neglecting small terms of the second order obtain the relation YI3'{Trz'y + {y-'-^ + y)y'](3-'o. + Y{Try'f3 + {^y-^ + (3)(3'}a = a:. From this and two similar equations corresponding to the results of setting the telescope on two other known stars, deduce the errors in the adjustment which are represented by the small vectors a', ^' and 7'. Ex. 5. The unit of length is taken equal to the focal length of a photo- graphic telescope in perfect adjustment so that were it not for refraction the image of a star would remain fixed on the photographic plate. Assuming the effect of refraction to be K times the tangent of the zenith distance,, show that the image describes on the plate a curve represented by 87^x7 No- where ZTT is the hour angle reckoned towards the west, and where cr, 7 and k are three (coplanar) unit vectors fixed relatively to the plate and directed respectively to the star, to the north celestial pole and, when the telescope is on the star in the meridian, to the zenith. {a) Prove that this curve represents a conic, or a portion of a conic, and that it isj, the intersection of the plane and cone ScTo-=0, S7U(o-^-trr)=S7K, and consider the arrangement of the curves for various values of K and for stars of different declinations. 176 FINITE DISPLACEMENTS. [chap. xii. Ex. 6. The positions of stars are determined by taking transits with a telescope movable about a fixed axis. Show that the hour-angle irz at the time of transit and the declination iry are connected with the reading iru of a circle fixed to the telescope at right angles to the fixed axis by the quaternion equation where 6, b\ c and c' are constants of the instrument, a, jS and y having the same signification as in Ex. 2. (a) If 8 is a unit vector along the axis round which the telescope turns the equation may be written in the form and for an almucantar whose line of collimation makes a constant angle {jra) with the vertical and is in the meridian when u = 0, the equation is| y'(3'-Y = /^' COS 7ra + yS'-^a^"/?" sin rra where irb is the latitude of the place. Ex. 7. If XJo- is the unit vector towards the centre of a planet ; Uct + t the vector towards a marking on the planet in latitude I ; y the unit vector along the planet's axis of rotation ; a the unit vector from the planet's centre towards the point on its equator on the meridian through the marking ; if P is the time of rotation of the planet on its axis and s the angular semi-diameter at the time of observation, show that y sin Z-J-y^-P' a cos 1==ts-^ - Uo-(l -HtV^)^' where t is the time of observation measured from some selected epoch. (a) Denoting the vector on the right by rj, show that y terminates on a fixed circle and verify that y cosec 1= -Y (7)213 + V3ni + Vil2)(^Vin2V3)~^ where rji, 7)2 and 773 are the values of the vector 77 at three times of observation. (c) Show how to deduce the time of rotation. Ex. 8. A polar axis having a fixed direction y carries a declination axis initially parallel to (^ on which is mounted a telescope initially parallel to a. The vectors being all of unit length and the instrument being completely out of adjustment so that no conditions of rectangular! ty are even approxi- mately satisfied, show that when the direction of the telescope is changed to a' by a rotation round the declination axis followed by a rotation round the polar axis, a' = y^^%^-yy-^ while if the rotation is first made round the polar axis and then round the declination axis, ^>^(^Yfiy-yy'ay-\y'(^y-J, and prove the equivalence of these two expressions. (a) If u and v are the tangents of half the angles of rotation round the polar axis and the declination axis respectively, show that the vector equation a-a+uYy{a-\-a!) + vY^{a + a') + uv{{a-a')^(Sy + Y.Yy(^{a + a')] = serves to determine both u and v. (h) Deduce from this the scalar quadratic equation in i^ : S)8(a - a') - 2?*Sy)8a' - ^2s^(a - a')Sy^ - '?*2s^Vy/5(a -f- a') =0. CHAPTER XIII. STEAIN. Art. 108. Homogeneous strain converts vectors (p) in an unstrained body into vectors (o- = (pp) in the manner described in the chapter on the linear vector function (Arts. 63, 64), but the transformation is of less generality. The order of rotation from ^a to 0 (i) must be satisfied. This requires one latent root of (p to be real and positive, and when the roots are all real this is obviously the case. When two of the roots are imaginary, g'-\-sJ — \g" and g' — \l — \g\ the third invariant is {g"^-\'g"'^)g where g is the remaining latent root ; so that here again one root is positive. It follows from this that in every homogeneous strain one direction at least remains unchanged, for we have U^a = Ua if (pa = ga, ^>0 (ll.) If the three latent roots are positive, three lines remain unrotated. In the case of a "pure strain three mutually rectangular directions remain unchanged, and the function (f> is self-conjugate with "positive latent roots. The decomposition of a linear function into a self -conjugate function preceded or followed by a rotation has been considered in Art. 70 ; and by selecting the square root {cfxpr of the function -g=0, {i>-gy=0. [These are excellent examples of the degradation of the symbolic cubic. Art. 66, p. 95.] Ex. 2. If the function p=g{l - f3'Sa)(l - l3Sa)p, Sa/3 = Sa'^' = 0, it is necessary to prove that g is a root of (f>, and that it is equal to the cube root of m. It may be shown that the converse is also true.] Ex. 3. The strain produced by two successive pure strains is generally impure. [Two functions are commutative in order of operation only if they are coaxial (Art. 66, Ex. 2, p. 95).] Art. 110. Lines in the unstrained body whose lengths are altered in a given ratio g are parallel to edges of the quadric cone T'>^ = Sa^'^^ . a^/3^ .(v.) which has a and Va^'^a for double edges. Substituting a + ta\ for /3 in this equation, we get for the edges in the plane SX/o = 0, which passes through a, ,8 = YX(^'cf>a±a(S\-^\lrxlr'Xf), (vi.) after discarding the factor t^. These edges are real for all directions of the vector X, and it easily appears that the upper sign corresponds to SU . a/3= -\- SU . ^ap)q~^' The latent root corresponding to any edge {(i) is (compare (i.)) ±T<^U/5. To determine the rotation which must be superposed on the strain so as to leave unrotated two vectors a and f3 satisfying the condition (iv.) we may utilize Ex. 6, Chapter III., p. 26, and find the rotation which converts U^a and U<^/? into ±Ua and ± U^, having as in (vi.) due regard to the indeterminate sign. It is possible to superpose a rotation on a strain so that all the lines in a plane may be unrotated. It is only necessary to reduce the function <^ to the form given in Art. 109 (iv.), and we have q-Kp.q=gp-f3Sap, (vii.) and the lines in the plane Sap = (or SX.p = 0, compare Art. 109 (x.)) — a cyclic plane of <^'<^ — are unrotated. Art. 111. The displacement at the extremity of the vector p produced by the strain is S = cT-p = {(p-l)p = p(Sp-'iPp-l) + pYp-^cj>p, (I.) which we have resolved along and at right angles to the vector p. When unity is a latent root of the function + -(l>')a = Y .YVa-.a, YV(r = 2e, (iv.) * For example a= + ^SaVfxv . ?^ : SXfiv ; d>'a = SV/tj/Sa?^ : SXfiu. ■ - du du 182 STEAIN. [chap. xiii. where e is the spin-vector of a. Thus for a pure strain at all points, we must have VVo- = 0, or o- = VP (v.) (Art. 56) where P is a scalar function of p. (See p. 74.) Art. 113. For small strains it is convenient to change the notation and to consider the displacement of a point produced by the strain rather than the relation between the vectors to the strained and unstrained positions of the point. We write there- fore for a homogeneous small strain (T = p-\-(pp, (I.) replacing the function (p of earlier articles by l + (p, the function (p being now small, or Tcpp being small in comparison with Tp. Apart from its smallness, however, the new function is of a more general character than the old. We may for example have the order of rotation from (pa to ^/S to (py different from that from a to /5 to y without violating the physical reality of the strain. In fact the ratio of volumes is now U^ S(a + 4.a){p + 4>p)(y + ^y) ^ Sa^y + i:8^afiy ^^ | m", ...(II.) Sa/Sy SajSy and m" is small in comparison with unity. Small strains are superposable (cf. Art. 104 (vi.), p. 169), or because we agree to neglect the terms of the second order (pi(p2p and (p2fpiP- A small strain is resolvable into a pure strain and a small rotation by the relation p + cpp = p-\-(p,p + Y6p = (l-{-Ye){l-^tS A/a, where (^p represents the displacement due to a small pure strain. [The terms may be taken separately. aaVap represents a shrinkage or an expansion to or from one line (a) ; bf^S/3p represents an elongation parallel to another. See Minchin, Treatise on Statics, Art. 379.] CHAPTER XIY. DYNAMICS OF A PAETICLE. Art. 114. The rate of change of the momentum of a particle is equal* to the applied force, or -^^.7np = mp = ^ (I.) where m is the mass ; p the velocity, mp the momentum and ^ the applied force. The moment of momentum of the particle about any point A is Y(p — a)mp = mV(p — a)p; (li.) and if A is a fixed point the rate of change of moment of momentum is equal to the moment of the applied force, for ^^mY{p-a)p = mY(p-a)p = Y(p^a)i, (m.) since Ypp = 0. If the point A is in motion with velocity d, the rate of change of moment of momentum is mY(p-a)p-mYap = Y{p-a)i-mYdp, (iv.) and in this case it depends on the velocity of the point A and on that of the particle P, unless indeed the motion of A is constantly parallel to that of P. Since i . Jm V = - ^iSp/i = - Sp^= - |J S^dp, (v.) . the energy equation is JmTpH { S^dp=^ const =F, (vi.) and for a conservative system of forces (Art. 56 (vii.), p. 74), « 'sidp = P, ^=-VP (VII.) \' ART. 115.] DAMPED VIBRATIONS. 185 Ex. 1. If the applied force is parallel to a fixed plane SAp = 0, deduce the integral SX.p = at + b ; and if it is parallel to a fixed line (/x), show that YiJLp = at + /3 where a, &, a and f3 are constants of integration. Ex. 2. If the force is directed to a fixed centre — the origin of vectors p — show that mYpp = (3 = -d constant vector. Ex. 3. If T is the tangential and N the normal component of the force and V the velocity in any orbit, prove that if C is the curvature of the orbit, [Letting accents denote deriveds of p with respect to the arc, we have p=p'Vf p=p"v^+pv since v = s. Also Tp'' = C and ^=p'T+Vp"N. See Art. 117.] Art. 115. The equation of motion of a particle of unit mass attracted to any number of fixed centres with forces varying as the distance is ^' = Saj(ai — /3) = 2ajai — /oSctj, (l.) the attraction to any centre being proportional to the distance T{a^ — p) and acting along '\J{a^ — p) towards the centre. The scalars a^, a^, etc., define the ratio of the magnitude of the attraction of the centres to the distance, and they are positive for attractive and negative for repulsive forces. If a is the vector to the mean centre of the centres for the multiples tt;^, a^, ... a^, and if a is the sum of the multiples, the equation takes the form p = a{a — p), (a = ^a^, aa^^a^a^); (ll.) and the particle moves as if attracted to the mean centre. The more general equation p-\-2bp-i-cp = 0, (ill.) where h and c are scalar constants, is that of the motion of a particle acted on by a force ( — cp) due to a centre at the origin attracting or repelling (c>'0 or ^y + ^.^y = Q, (ix.) so that the function (j)2 + n(^-^+7i^ has a zero root and y is the corresponding axis. The third invariant of the function must vanish if it has a zero root, and the appropriate values of the scalars n are the roots of the equation ^{<^c^^n^ + n^)X{4>o-\-n^ + n'^)lx{4>2 + n4>^ + 7i'^)v = 0, (x.) where A, jx and v are any vectors. Solving this equation we determine six linear functions with zero latent roots, and the corresponding axes (y^, y^^ etc.), being determined, the solution is ;o = 2iV"^ (^O the arbitrary constants being the tensors of the vectors y. Ex. 1. Show how to determine the constants of integration. [We may have given the initial position and the initial velocity — six constants. For example the solution of (11.) is p = o.-\-y-^ cos 'Jat + y2 sin \Jat, and if /3 = /5 and p = y when t = 0, we have 71 = /? - «., 72*^ = 7-] Art. 116. For a force directed to a fixed centre, the origin of vectors p, ^.^^^ W= ±Up, (i.) and (Art. .114 (ili.)) we deduce at once the integral of moment of momentum Vpp = /3, (il.) where the constant ^ is double the vector area swept out by the radius vector in unit time. Conversely if the vector moment of momentum with respect to any fixed point is constant, that point is a centre to which the force acting on the particle is directed, for = — Ypp or pW^W p. The orbit of the particle lies in the plane Sp/3 = (m.) ART. 116.] CENTRAL ORBIT. 187 In general the vector /3 admits of transformations such as the following (compare Art. 85 (ii.), p. 132) : /3 = V.Ve = V.^.^ = V<„ = VU^.a«, ...(IV.) where w is the angular velocity of the radius vector, and where w (for a plane orbit) is the angle the radius vector makes with some prime vector or more generally where w is the scalar angular velocity. We may also write ^-^ or^-/5- ^- (y) Tp^~ dt ' Tp^~^ ' dt ' ^^-^ so that for a central force p^-ml3-'^, if i=-mVpT:p-\ (vi.) .In particular when the law of force is that of the inverse square, the scalar m is constant, and (vi.) integrates at once and gives ^= — -m/^-^Up + y where S/3y = by (ii.), (vii.) y being a vector constant of integration. This shows that the hodograph of the motion is a circle whose centre is the extremity of the vector y and whose radius is mT/3~^. Moreover, substituting for p in (ii.), we find the equation of the orbit, ^ ^ im . tt / x l3=^m/3-^Tp + Ypy; (viii.) which is equivalent to the two equations mTp = T/3''-S/3yp, Sfip = 0; (ix.) and which represents a conic referred to a focus as origin. If w is the angle the radius makes w^th the vector y/3 we may replace (ix.) by Tp(l + eco8iv)=p where e = 7n-'^Ty^, p = m"^T^^ ...(x.) and e is the excentricity and p the semi-latus-7'ectum. Taking the tensor of (vii.), utilizing (ix.) and observing that by (x.) Ty'^ = me^p~^, we obtain the energy equation whet&a=p(l—e^)~'^ is the mean distance. Now when w^e resolve the velocity along and perpendicular to p, p = p-'Spp-^p-Wpp=:Vpr + p-''^ if r = Tp', (xii.) 138 DYNAMICS OF A PARTICLE. [chap. xiv. whence on substitution in (xi.) we find = m( 2_£_r r r^ a .(XIII.) which gives on integration the radius vector in terms of the time. Ex. 1. Deduce the usual u and 6 equations for a central orbit by expressing p in the form rk^'i. [Here p = (r + r$k)k^i, f3=r^6k = hk, 0=hu^, r=r'^= -hu' where accents denote differentiation with respect to d and where u = r~^. Thus p= -/i{u' — uk) Ici^ p= - h^u^ (^" ■\-%i)k-' z.] Ex. 2. If a, /5 and y are three unit vectors, a along the radius vector, y perpendicular to the plane of the instantaneous orbit and ^ = ya ; if c is the rate of description of angles by the radius vector in the orbit and if a is the rate at which the plane of the orbit turns round the radius vector, prove that the equation of motion is a (y* — rc'^) + ^(2rc + fc) + ymc = ^. [Here -=V^ = yc, -^ = V . :^/-^ = ad, so that a — Be, y=-/3d and ■■ a p y ^PP r- ^ / I . 0z=ya- ac. Compare Art. 86. By the instantaneous orbit is meant the orbit which a planet would describe round the sun if the disturbing forces were suddenly removed. The equation exhibits the effect of the components of the force along and perpendicular to the radius vector and perpendicular to the plane of the orbit.] Ex. 3. Express the equation of motion in a perturbed orbit in terms of the reciprocal of the radius vector (u), the rate of description of areas (k) and the rate (a') at which the orbit turns round the radius vector per unit description of angle in the orbit ; and show that it is [We have to express everything in terms of A = ^^pp = ^'^^, of u and of a and their differentials with respect to the angle c. Writing thiis p = au~^ we have p = hu^ .^(au~^) = ku^{f3u~^ — au'ir''^), etc.] Ex. 4. Express the equation of motion of a particle in the form a (u" + u) + au'~^-(3u^-y (us" - su") - y -^ (us' - su') = - -^^ where u is the reciprocal of the projection of the radius vector on a fixed plane, a is a unit vector along this projection, y is the unit normal to the plane, /3 = ya, ^is the rate of description of the projection of areas, s is the tangent or the angle between the radius vector and the projection, and the independent variable is the angle in the fixed plane. [Here p = (a + sy)u~^, aa~^ = y, y' = 0, /?'=— a, Zrw''^ = c if c is the angle in the plane. The scalar equations to which the above is equivalent have been much used in the lunar theory.] ART. 117.] PERTUEBED ORBIT. 189 Ex. 5. Prove that the vector curvatures of an orbit and its hodograph are dUd^_ I l_ dUdp_ I J_ dp ~ ^Tp' dp I'Tp' and that for a central orbit they reduce to dVdp _ /3T$ dVdp jB dp T/oTp3' dp T^p2 where f3 = Ypp. (a) Hence the law of nature is the only law for which the hodograph is a circle for all initial conditions. Art. 117. The equation of motion of a particle constrained to move along a curve or on a surface is P=i+i^ (I-) where v is the reaction arising from the constraint. If there is no friction, the reaction is at right angles to the direction of motion or the vector v lies in the normal plane of the constraining curve or is the normal to the constraining surface. The condition Spp = 0, (II.) which is then satisfied, allows us to retain the equations (v.) and (vi.) of Art. 114. In terms of the deriveds with respect to the arc s of the orbit which we now denote by p, p" , etc., we have (compare Art. 85, Ex. 1, p. 133), P = pv, p = p"v^-{-pv, V = S, V = VV, (ill.) or in the notation of Art. 86, p. 134, p = av, p = ^c{d^ + av (iv.) where -y is the velocity ; and the equation of motion is pv^ + pv = $+v (V.) In the case of a constraining curve, the motion must be deter- mined from the energy equation which is alone available for this purpose. For a surface we have, on elimination of the unknown tensor of v, Y(p-i)v = 0, (VI.) and in this equation v is proportional to a known function of p — the result of operating by V on the scalar equation of the constraining surface. (Art. 54, p. 69.) If on the other hand we seek the reaction arising from the curv^.or surface, we have by (ii.) v^p'-^Np'v=p"v^-p-^Ypi= -y\^i^p-p-^^pi. ...(vii.) the energy equation being employed in the last transformation. 190 DYNAMICS OF A PARTICLE. [chap. xiv. For a rough constraint, the equation of motion may be written in the form p = p"v^+pvv' = ^ + v-'npTv^ S/dV = 0, (vm.) where 7i is the coefficient of friction. Resolving along and at right angles to p this equation gives vv' + Sp(= -7iTv, v=p"v^-p'-^Yp'l ; ^. (ix.) whence on elimination of Tv, p"v^-p'-Wp'$=~Vv.7r\vv' + Sp'^); (x.) or again in terms of the vectors p and p, we have nyp(p-$)=U{pv).Sp{p-^\ (XI.) because 8p{p-^) = nT(pv) and Wp(p-() = XJ(pv). Equation (x.) or (xi.) may be employed for a constraining surface. In the case of a curve we must take the tensor of each side to eliminate the unknown Uv. We may remark that it follows from (ix.) that if the curve is a geodesic on the constraining surface V . vp'~Wp'^ = or Svp$=0, (xii.) because (Art. 90) for a geodesic p" \\v. In other words, when the direction of the applied force is coplanar with the normal to the surface and the tangent to the orbit, the curve is a geodesic on the surface, and in particular this is the case when there is no applied force. If the constraining curve or surface is in motion so that. Art. 104, p. 168, the vector p to the particle from a fixed point is connected with the vector CT to the particle from a point moving with the constraint by the equation p-=T + qU5q~'^, (xiii.) in which r and q are supposed to be given functions of t, the equation of motion takes the form (compare Art. 105, p. 171) r + 5'(^ + 2Vttrr + VtCT + YtVt^)g-i = ^ + v, (xiv.) and for a smooth constraint, S^^g-iv = 0, (xv.> qHSq-^ being the velocity with which the particle moves along the curve or surface of the constraint. Ex. 1. A particle moves under gravity on a surface of revolution having its axis vertical. [If k is the unit vector directed vertically downwards, the equation of motion is Y(p — gk)v = 0. Since the surface is of revolution, the vectors v, k and p are coplanar, or Spkv = 0, so that Ykp || Ykv || Yvp. Operating on the equation of motion by S^ or Sp we find the integrable relation Skpp = Oy so that Skpp= —h where h is the constant rate of description of area by the projection of p on the horizontal plane. We have also Svp = and Skp=-z if we write 8kp—-z. From these three equations pSYkpYkv = — hYkv — zYvYkp ; and if the equation of the surface is given in the form Tp=f{z)=f{-Skp) we may put v = Vp-kf'(z) and Ykp=Ykvf{z). Hence pWkp'^ = K^ — z^v^'tp'^ ; and by Art. 114 (vi.) on expressing everything in terms of z we obtain the equation . ,„, z^{f^-^zff+fP) = 2{E^gz){f-2^)-h\ If the surface is spherical fiz) is constant and equal to the radius of the sphere, so that /' is zero. ART. 117.] CONSTKAINED MOTION. 191 Again if iv is the angle the plane of p and k makes with some initial plane^ h = wTYkp^ = w{f^-z'^) from which ic can be found in terms of z by the previous equation. If, on the other hand, the equation of the surface is given in the form Skp=f{Tp), it may be more convenient to obtain an equation in r{=Tp) and r by using Spp + rr = instead of S^p + z = ; and if the equation is of the form Skp = f{TYkp)=f(p) we may use i$VkpVkp+pp = 0.] Ex. 2. A particle slides under gravity within a fine smooth tube which revolves round a vertical axis. [The origin being taken on the axis, the vector to the particle is p = q^q~^ (compare p. 168), and if 7i is the angle through which the tube has been rotated from some initial position, p = q{-UJ + nVh-CS) q-\ •p = q{x^ + 2hYk^ + n^kYkT;5 + vYkm) q'^ ; while the equation of motion is p=gk + v where SvqTDq~^ = 0. Because the axis of q is parallel to k, we find on elimination of the reaction v, S^iiS + n^kNkV5 + n Vkm) =gStyk ; and in this equation n and ii are given functions of t when the law of rotation is known, and CT is a known function of a parameter variable with the time when the form of the tube is known. If the velocity of rotation is uniform, the equation integrates and |(Trj2 + v^ykz:j'^)=gSkT;y + ^C. If for example the curve is a helix with its axis vertical so that ^ = a{i cos ic-hj sin u) + bku we have ?iT^= -{a^ + b'^)u% and Ykuy^= -a% and the equation is u^{d^'+b^)+n^a^ = 2gbn- C : and if the curve is a vertical circle, T:y = a(icosu + ksimt) we have iira^ + nhi^ cos^ u = 2ga sin u — C] Ex. 3. A particle under gravity traverses with uniform velocity a smooth curve which rotates uniformly round a vertical axis. Prove that the curve lies on a paraboloid of revolution. [The equation of the surface on which the curve must lie is n^TVkvy'^ + 2gSkm = coiist.] Ex. 4. Two particles of masses m and m', connected by an inextensible string which remains stretched throughout the motion, are projected from the extremities of the vectors a and a with the velocities /5 and yS' ; prove that the vector to the particle m during the motion is p where p{m + m') = m{a + fit) + m' (a + fi't) + m'T(a - a') . (U(a - a) cos nt + 11 (/3 - /3') sin nt), the scalar n being defined by 7iT(a-a') = T(^-/5'). Ex, 5. If a particle can be made by suitable initial conditions to describe a given curve under the action of a force ^, show that 2/."jSJdp-p'Vp'^=0, p' and p" being the first and second deriveds with respect to the arc and a 192 DYNAMICS OF A PAETICLE. [chap. xiv. suitable constant being included in the integral which is taken along the curve. (a) Hence deduce M. Bonnet's theorem. [We have 7n{p"'V^ + p'vv') = $ which gives mvv'= — S^/a' and mv^= — ^fS^d/a, etc. Conversely if the condition is satisfied it follows that a particle will for suitable initial conditions describe the curve. If ^j, ^2? etc., are forces under which, acting separately, a particle can describe the curve, and if for greater clearness we replace jS^„d/) by (7„ + jS^„d/o (the new integral being taken from any selected point on the curve), we have ^{2p"{Cn + \SiAp) - ^pYpU = 2p{^G,, + jS . 2^,, . &p - p'Yp'^in) ; I or a particle will describe the curve freely under the action of the resultant of the forces provided its mass m and the velocity v satisfy mv^ = llmnVn^ initially.] Ex. 6. Show that the condition of the last example is equivalent to the conditions ^ d * ^ Sp>t = 0, ^^Sp"-'i + 2Sp'i = 0, which assert that the force must be in the osculating plane of the curve, and that the rate of change (as we pass along the curve) of the product of the radius of curvature into the normal component of the force is equal to double the tangential component. Art. 118. Tait has applied the calculus of variations in the following manner in the determination of the curves of quickest descent, or the brachistochrones, for a conservative system of forces. (Quaternions, Arts. 518 and 523.) J .(I.) If the integral ^ = [q. Tdyo= [q. ds is taken along a curve, Q being a given scalar function of p, the variation of the integral corresponding to a variation of the curve is SA = ^SQ . Tdp + Jq . STdp = - ^SSpV .Q.Tdp- ^QSJJdp . ^d^o. The symbols d and S are commutative in order of operation, so that on integrating by parts JQSUdyo . Sdp = JQSUdyo . dSp = [QSVdp . Sp] - ^SSpd (QUdp) where the term in square brackets corresponds to the variation of the limits of the integral. Thus SA=-[QSVdp.Sp]-{-^SSp{d(QVdp)-VQ.Tdp} (ii.) If the integral is stationary, the variation vanishes and the term under the sign of integration in (ii.) must be zero for all vectors Sp. And since Sp may have any direction when the ART. lis.] BRACHISTOCHRONE. 193 curve is not restricted in any manner except at the limits, we must have d(Q\Jdp)-VQ.Tdp = 0, or ^(Q^0-VQ = O (iii.) If on the other hand the curve is constrained to lie on a surface so that SvSp = where p is normal to the surface, the condition is /a \ V.(^(Qp')-VQ) = (IV.) For the brachistochrone the integral A is the time of descrip- tion of the curve or A=t=^{v-Kds, Q = l>-i = (2£'-2P)~* (v.) by Art. 114 (vi.), so that VQ = VP .(^ = VP .Tp-^ The first equation (ill.) now becomes d{Tp-KVdp)-VP.Tp-\dt = or d. p-^ + VF .Tp-^dt = 0, or finally p+p-^.VP.p = (vi.) Tait remarks " It is very instructive to compare this equation with that of the free path (/) + VP = 0); noting how the force — VP is, as it were, reflected on the tangent of the path." Ex. Determine the brachistochrone when gravity is the only force. I [HereVP=— K, a constant vector, and the equation dp~^- KTp~^dt=0 shows that p-^ = a + Kf{t) where a is a constant vector which may without loss of generality be supposed to be perpendicular to k. Substitution gives , df-{Ta:^ + TKY^)dt = 0, and the solution of this is I /=T. K-^atsiXiTaK{t~tQ)^T.K-^atRJin{t-tQ) where 7i = T. ttK. Thus p-^=-Ta(Ua + VKtSLnn{t-tQ)) and p = Ta-^ . cos'^)i{t - ^o)(Ua + U/c tan n{t - ^q)), and on integration p = f3-^n-^Ta-^['Ua{2n(t-tQ) + sm2n{t-tQ)}-VKcos2n(t-to)] which represents a cycloid. (Tait's Quaternions, Art. 524.)] •^- ♦ J.Q. CHAPTER XV. DYNAMICS. Art. 119. Let ni-^, 'tn^, etc., be the masses of particles of any dynamical system which are situated at the extremities of the vectors p^, p^, etc., drawn from a fixed origin. By Newton's second law the equation of motion of the particle wi^ is ^h/>'i = ^i + & + ^i3+etc., (I.) where ^^ is the force external to the system which acts on mi^ and where ^^2 i^ ^^® force due to the interaction of m^ on m^, etc. By Newton's third law action and reaction are equal and opposite, or fl2 + & = 0, Vpi^i2 + V/.2f21 = 0, (II.) these being the conditions that ^^2 ^^^ ^21 should equilibrate. Hence by adding equations such as (i.) for all the particles, and by adding the results of operating on these equations by Vp^, Vp2, etc., we obtain the equations 2mi/,\ = Efp Hm^Yp^p^ = ^^Pi^v ("!•) which are independent of the interactions of the particles. Attending to (ii.) the rate of change of kinetic energy of the system of particles is evidently ^ . i2m,Tft2= _2m,SAft= -2Sft|,-2:S(ft-/5,)^,„ ...(IV.) and because (11.) implies ^^2 Ii Pi~~ P2 ^^ ^^^ ^^^^ ^^^^ ^^ inde- pendent of the interactions provided the relative velocity of every pair of particles is at right angles to the line joining them — or in other words, provided the distance between every pair of particles remains unchanged. Writing M=i:m^, Mp==^m^p^, i=^iv ^ = ^^Piiv = I.m^Yp,p^,(y.) so that M is the total mass of the system, p the vector to the centre of mass, ^ the resultant external force, f] the resultant ART. 1-20.J SYSTEM OF PARTICLES. 195 moment of the external forces with respect to the origin as base point and 6 the resultant moment of momentum with respect ' to the origin, the equations (ill.) become Mp=i, e=r, (VI.) When the external forces are zero, ^ and rj vanish and the integrals of (vi.) are Mp=:at-^fi, e = y, , (VII.) where a, ^ and y are constant vectors ; and when the internal forces are given functions of the distances between the particles, we have also in this case the integral of energy \^m{Tp,^ = 2/. T(p, - p^) where f,^ = U (Pi - A2)/ • T (pi - p^). (viii.) I Art. 120. With reference to a point moving in any arbitrary manner, the extremity of the vector e, the moment of momentum is I e. = l.m^N{p^-e){p^--e) = e-MY{pk + €p-ek)', (l.) and (vi.), Art. 119, may be replaced by M-p = i, e, = rj,-MY{p-eye, (ll.) where r]^ = r]^ Ve^ is the resultant moment of the forces about the extremity of e. In particular when e terminates at the centre of mass, the equations are ^P = i^ Oo = %^ (Ill-) where 6q and tjq refer to the centre of mass. These equations are of the same form as those of the last article. We may note that in general . e,=:O^MVpp=e,-MYp,p,, (IV.) I where p^ = p — €. Ex. 1. Find the locus of points fixed in space about which at any instant the moment of momentum is a minimum. [If the extremity of e terminates at a fixed point Oe = — MYep, and the locus of points for which T^e has a given value is the right circular cylinder T (^ - MYep) = T^e. Writing d = M{pp + Yi^p) we have T(9e2 = if2^2Xp2 + J^2XV(€ - €^) p\ The locus is the line MYep^Ydp . p'^. Compare Art. 99, p. 156.] Ex. 2. A point moves in such a manner that the moment of momentum with respect to it is constant. Determine the particulars of the motion, [If de is constant, the relation (iv.) MYpfpe= -0 + Oe + MYpp gives, on difi"erentiating twice and utilizing the equations of motion (Art. 119 (vi.)), MYp,p, = -rj-{- Ypi MY(p,pe + pe/Se) =-h + Yp$+Ypi because Be is constant. Forming the vectors of the products of right and of left hand members of the first and second of these three relations, and also fdrftiing the scalar of the product of corresponding members of the three relations, we obtain the equation p, =p _ € = ± V((9 - 6'e - MYpp)(7i - Yp^) X {MS{e - 6'e - MYpp){rj - Yp^X-n - V/>^'- V/iJ)rt 196 DYNAMICS. [chap. xv. so that € is expressed in terms of quantities which are known when the motion of the system is given. There are thus two paths corresponding to the double sign symmetrically placed with respect to the path of the centre of mass.] Ex. 3. Refer the equations of motion to variable axes. [See Art. 105 and the formulae of differentiation (vi.) and (xi.), p. 170.] Art. 121. In the case of a rigid body, let e be the vector to any point fixed in it and let co be the angular velocity. Then by Art. 105, p. 170, . • V ^ ^ ^T^ 1 Pi-e=Va)(pi-e), (I.) I because the velocity of the point in the body at the extremity of p^ relatively to that at the extremity of e is due to the angular velocity co. Equation (i.) of the last article may now be replaced ^^ (9e = 2m,V(p,-e)Va)(pi-e) = 0.a), (ll.) so that Of is a linear function of co. The linear function (p^ is fixed relatively to the body because the vectors pi — e, etc., are fixed in the body, but in considering the rate of change of cp^o) we must take account of the change of orientation of the body as well as of the change of co. We have (Art. 105 (ix.)), ^^, and the moment of inertia round a principal axis is maximum, or minimum, or at least stationary in value. If the extremity of e is fixed in space as well as in the body, so that the body moves about a fixed point, it appears from (vi.) that when the body is set rotating under no forces about one of these principal axes, it will rotate permanently round it. For we have Vcocpeco = if w is along a principal axis, and

(/)i-€) or V.a)V(/)i-€)V(u(pi-€).] Ex. 2. If / is a principal moment of inertia at the extremity of the vector €, or in other words a latent root of (f>^, show that P - 27i"/2 + (7i''2 + n')I- {n"n' -n)=0, where n'\ n' and n are three positive scalars, namely, n" = - 2mi(/)i - e)- ; n' — — ^m^m^ (pi - e) {p^ - e)- ; n = ^m^m.^n^ S {p^ - e) {p^ - e) (/O3 - ef. [See EleTTients, Art. 417, and observe that (w - Vtot^) Yii) (w 4- Vwd;) = 0, which is independent of the constants of the body. The corresponding relation gy ^ ^^2^^ _ g VtoD.to) VD.toD.^to = connects ui and its first and second deriveds D! + xY^o. + V (<^i + xf^o,^ (<^i + xi^o. = or i* (^1 + OG)~^a + (^1 -f xy^Ya4)asJmi{x) — 0, ART. 121.] KIGID BODY. 199 where m^{x) is the third invariant of ^i+.r. We may obviously take the first invariant of ^j to be zero without loss of generality, so that the latent cubic of the specified type, and the differential equation for x is reduced to Weierstrass's standard form. The function ^j is of the form ^, =a + 6x + c\/r where v and ^/r are the auxiliary functions for <^, and when the first invariant is taken to be zero, 3a + 26wi" + cm' = 0. The scalars b and c are arbitrary constants of integra- tion. Assuming a=ui + vj + wk where i, j, k are the axes of <^, we see that Au = {B-C)viv, Bv = {C—A)wu, Cw = (A-B)icv, J, B and C being the latent roots of <^ — the principal moments of inertia. Thus _. / BC I CA I AB ''~^^{C-A){A-By^y{A-B){B~Cy''S{B-C){C-AS and the latent roots of (/^j are lh{B-^C-^A)-lc{CA+AB-2BC\ etc. Moreover since by definition of a, we have Sa(^a = 0, Sa(f)^a = and also a'^= + l as may be easily shown, we find Sio(f)(jD = Sa\(f)-i^ + x)a= —bABC, or in the notation of the last example, cABC^ -A2 and bABC=g^.] Ex. 6. Eesolve the vector of angular momentum (fxo, along and at right angles to w, and investigate the relation of the components to the quadric Szj(t>Tn= -1. [Compare Art. Ill, p. 181.] Ex. 7. The motion of a freely moving body is known, and it is required to determine as far as possible its dynamical constants. [The mass cannot be determined, but if we know the particulars of the motion of three points, the extremities of e^, €2 and €3, we can find w from the two equations e^ - €3 = ^w (e^ — Cg)? ^1 - €3 = Vw (e^ — €3). In the next place, to find p, the vector to the centre of mass, we have ei = p + Yo)(€^- p), and ti = p + Y(lj(€i- p) + YwYii)(ei- p), and because p = the second of these relations gives p on solution of a linear equation. To find the function X= —^in^p^SpiX where A is an arbitrary vector, and where the vectors p^, etc., are drawn from the centre of mass of the body to terminate at the particles of mass mj, etc., the problem is solved when we reduce the function ^ to the form cfik= -aaSak-bf3^/3X-cySyX-d8S8X where aa + b/3 + cy-\-d8 = 0, «, 6, c and d being the masses of the four particles and a, ^, y and S being their vectors of position. Now for some scalar x, we have xa = S/3y8, xb= — Say 8, xc = Sa(38, xd= — Sa/?y ; and we also have (Art. 65, Ex. 1, p. 92) VrA=-2a6Vai8Sa^A, m = 2a5cSa/?y2. The second of these serves to determine x^ for it reduces to m=x^abcd2a. SubsiiflUting a for A in the first, we find y(/-a=xbcdV(/3 - 8)(y - 8) ; and when we operate with Sa, S/5 and Sy on this and similar expressions we have Sayf/a= -x^bcd{b + c + d), etc., Sayfr 13 = etc. =x^abcd. It easily appears that the six relations in a, /3 and y imply the remaining six involving 8 when 2aa=0. 200 DYNAMICS. [chap. xv. Assuming first any vector a which satisfies the condition Sayjra = - x%cd(Jb + c + d) — that is any vector which terminates on a certain quadric — we have next the two relations Sa\lr/3=x^abcd, S/Sirl3= -£c^acd{a + c + d) which require the vector f3 to terminate on a conic. Selecting ft there remain the three equations Sa^y = Sft\]ry =j;^abcd, Sy\^y=^ -x^abd{a + h-{-d) which determine y as the vector to a point of intersection of a line and a quadric. Finally, we have S= -(i~Xatt + 6^ + cy).] Art. 122. When an impulse acts on a system of particles, the velocity of the particle m^ is changed from p-^^^ to p^ where ^i(Pi-Pi,o) = ^i + ^i2+^i3 + etc., (I.) where X^ is the external impulse acting on m^ and where \^, \^, etc., are the impulsive actions of the particles m,, m^, etc., on m^. These impulsive interactions satisfy conditions analogous to (ii.) of Art. 119, and we obtain the equations 2^i(A-A.o) = 2Xp i:m,Vpi(^i-/5i,o) = 2VpiXi, (II.) which are independent of the interactions. The work done on the particle m^ by the impulse is (Thomson and Tait, Art. 308) -JS(/Ji+pi,o)(Xi + Xi2 + Ai3+etc.), (III.) and the total work done on the whole system is W= - J2S(/3i+/3i,o)Xi- J2S(/3i-^2 + Pi,o-P2,o)^i2- •••(IV.) For a rigid body it is frequently convenient to define the motion by the velocity (o-) of the point of the body coinciding with the origin and the angular velocity {w). Thus p^ = o- — Yp^w, and if x = 2Xi, 1^, = ^^ p^\, cj>uo = ^Y p^Nc^p^, (v.) so that X is the resultant force and /j. the resultant moment of the impulse with respect to the origin while (j> is the inertia function corresponding to the origin, the equations (ii.) become il/(o--o-o-Vp(a)-a)o)) = X, ifVp(o--o-o) + 0(a)-a)o) = M ; (vi-) and because X^a is parallel to the line joinmg two particles and therefore perpendicular to pi — p2 and to Pi,o — p2,o> ^^^^ expression for the work done is independent of X^g, etc., and reduces to W= -lS((r + o-o)X-iS(a) + a)o)/x, (vil.) because we have 2S (o- + o-o — Vpi ( ft) + ojo) ) Xi = S (o- + o-q) 2Xi + S (o) 4- Wo) 2 VpiXj. When the origin is taken at the centre of mass, (vi.) becomes M(a--(To) = \, 0o(^-^o) = M. (VIII.) where (p^ refers to the centre of mass, and thus we have at once (r = (To+if"^X, ftj = ft)o + ^o"V; (IX.) i ART. 122.] IMPULSES. 201 or, in the language of the theory of screws, when a free body having an instantaneous twist velocity (o-q, Wq) is acted on by an impulsive wrench (/ul, X), the instantaneous twist velocity immediately after the impulse is (o-q + M-'^X, Wq + ^q'VX ^^^ centre of mass being the base-point. (See p. 171.) When the origin is taken at an arbitrary point, we may replace (VI.) by M((T — a-Q — y p(w — Wq)) = \, ^o(ft) — ft)o) = // — VyoX, (x.) where (rr, w), (o-q, coq) and (jul, X) are referred to the origin as base-point and where Q corresponds to the centre of mass. This is easily shown in various ways. The form of the expression (vii.) is independent of the choice of base-point. In particular when the base-point is at the centre of mass, we find from (vii.), (viii.) and (ix.), W= _i(ifo-2 4-Sa)0o«) + K^^o' + Sa)o0o^o) = - J(.^/-iX2-f-SM0o-V)-S((roX + a)oM) (XI.) Ex. 1. Prove that the solution of (vi.) is (w - Wo)(m + MSpcf)-^ + M'^Sp(f)p . p^) = (yfr-\-MxpSp + MYQ) = VaA, (t — Vaa> = d serve to determine the unknowns cr, w and A. We have on elimination of or and A, <^w-i/aVa(o = xa + M^-a^Sa4>a) = My^Ya (a - do) - M^YaYaYa (d - do), where do = 0*0 — Vawo is the initial velocity of the point. Hence in terms of (0 - a>o as given by this equation cr-cro = d- do + Va(a>-(Oo) and A = J/~^(o- — o-q).] Ex. 3. A rigid body is moving in any manner. Suddenly a line in the body is constrained to move in a definite manner. [If a and /? are the vectors from the centre of mass to two points on the line, we may suppose the impulsive wrench to consist of forces A and A' applied at the ex:tremities of a and /3. Hence J/(^-cro) = A + A', (/>(w-a>o) = V(aA-h^A'), o- = d + Vatu = ^ -|- V^cu, where d and ft are the velocities of the extremities of a and ft. From the first and second equation we deduce S(^ — a)^(w — Wo) + i/Sa/3((r-(ro) = 0, which asserts that the moment of momentum about the line is unchanged. 202 DYNAMICS. [chap. xv. We also have (i) = {a — /3 + x){a — 13)~^, where ^ is a scalar to be determined by substituting for w and cr in the equation just found. Solving the linear equation for a; we find to, and hence cr and A and A'.] Ex. 4. A rigid body is moving in any manner. It is required with the least possible expenditure of energy to cause a given point to move in a given manner. [Writing equation (xi.) in the form If = - ^{M(d + Vaw)2 + Swt/)w) + ^ ( J/(ao + Vawo)^ + Soj^cfiOio), we express that this function of to is a minimum. We find cfid) — MaVao) = MY ad, and as in Ex. 2, this gives (u (m - if Sa<^x« + ^^a^Sac^a) = MxIrY ad - M'^YcfiaYaYad, and substituting in cr = a + Vaw, in (f){o) - (Dq) = fx and in i¥(o- — ctq) = A, we determine the impulsive wrench and the instantaneous twist velocity.] Ex. 5. If p and p' are the pitches of the screws of an impulsive wrench and of the instantaneous twist velocity produced by the wrench on a free quiescent rigid body ; if also tTT and th' are the vector perpendiculars from the centres of mass on the axes of these screws, Ma-~M{p' + 77J')(i) = )^, lx = (p + ZJ)X = <^a). (a) Hence in terms of A and o), p' = J/-iSAw-i, CT' = i/-iVAo>-i; p = S<^wA-i, CT=V-^ {p + w)X', and express the moment of inertia about the line through the centre of mass parallel to the instantaneous axis in terms of p, p\ tD' and ct'. {d) The cosine of the angle between the axes of the two screws is ^'(p'^ + TS7'2)~^ ; and if the axes are parallel, that of the instantaneous screw passes through the centre of mass or else the instantaneous motion is a translation. In the former case the pitch and vector perpendicular on the axis of the impulsive screw satisfy the condition CTS(/)AA-i=py<^AA-i. Ex. 6. Determine the dynamical constants of a free body by observing the effects produced by impulsive wrenches in starting the body from a given position. [If p is the vector from a fixed origin to the unknown centre of mass, if an impulsive wrench is (/x. A) and the corresponding twist velocity is (cr, w) for the fixed origin as base-point, the equations are (compare (x.)) J/(o-- Vp(u) = A, (f)(1) = fx-YpX, together with others with accented letters cr', to', fx', A', cr", co", jjl", A" for other impulsive wrenches and the corresponding twist velocities. From these equations M, p and <^ (corresponding to the centre of mass) are to be determined. The mass follows at once from the first equation, and we have i/= S A(o(So-(o)-i = SA'to' (So-'(o')-i = SA V(S(rV')-^ ART. 122.] THEORY OF SCREWS. 203 The vector p is given by V(o-- J/-iA)(cr' - J^-iA')= W/3(oV/)(o'= - pS(o- - Jf -U) w'. And the function ^ can be found from three couple equations. Some rather elegant identities connecting the wrenches and the twist velocities may be deduced from this beautiful problem of Sir Robert Ball's.] Ex. 7. An impulsive wrench of given pitch and intensity is applied to a free quiescent rigid body. The axis of the screw of the wrench passes through a fixed point ; find the direction of the axis so that {a) the kinetic energy, or {h) the angular velocity, generated by the impulse may be as great as possible. [The base-point being taken at the centre of mass, we have if. cr = A, <^(o = (p + Vy)A where TA, p and y are given. The kinetic energy is — ^S(p + Vy)A^"^(/?-f- Vy)A -|J/~^A^, and if this is a maximum subject to the condition that TA is given, we have (jo- Vy)^ is the inertia-function of M^ corresponding to its centre of mass, and where a^ is the vector from the same origin to the point of contact ; cTg, Wg, o-g', w.2', <^2 ^^^ ^2 being in like manner related to the body J/2 and to its centre of mass. (a)' TBhe relative velocity of the points of the bodies in contact is o-/ + V(o/ai - 0-2' - Vwg'ttg = o-j -H Vwittj - cto, — YfJi^d-i + {M{~^ -\- ^^2"^) ■^ + V . (^j-i Vai A . tti -h V . (/)2-^ Va2 A . aa ; or briefly, it is T' = T-f-^A, 204 DYNAMICS. [chap. xv. where ^ is a certain self -con jugate function determined by the circumstances of the impact and where t is the initial relative velocity of the points in contact. (6) For perfectly smooth bodies, VAv = 0, where v is the normal to the bodies at the point of contact, and the value of A corresponding to the end of the "first period" of impact is and the twist- velocity of the body J/j immediately after the impact is where e is the coefficient of restitution. (c) The total loss of kinetic energy is -(1-^2)8x1/2. (Sv4>v)-i. (d) For perfectly rough bodies, YtV = 0. The value of A corresponding to the end of the first period of impact is A= —^~W, and the twist- velocity immediately after the impact is (o-i - (1 + e) Mc^^-W, (Oi - (1 -f e) cfi^-Wa^^-W). (e) For perfectly rough bodies, the loss of kinetic energy is -(l-e2)ST-iT. Art. 123. When a rigid body is not perfectly free but constrained in any manner an impulsive wrench will in general be partially neutralized by the reaction of the constraints. Referred to the centre of mass as base-point, we have for a quiescent body, Mcr = \ — X^, (pCO = JUL — jUL^, (l.) where (/m, \) is the impulsive wrench and (yu,, X,) the wrench on the constraints, or where ( — /ul,, —\) is the reaction of the constraints. In order to determine the instantaneous motion produced by the impulsive wrench (nx, X), it is necessary to know the evoked wrench (jul^, \). We consider the case in which the constraints are smooth, or so that no evoked wrench can generate any motion. In this case the work done by the wrench (/x,, X,) must be zero, or we must have (Art. 122 (vii.)) S(/x,(o + X,o-) = 0, (II.) where (/x,, X,) is any wrench arising from the constraints and where (cr, w) is any possible twist velocity of the body. The screws of (yw,, X,) and of (a-, co) are said to be reciprocal w^hen this condition is satisfied; and for smooth constraints, every possible twist velocity is reciprocal to every possible ivrenck arising from the constraints. A body with one degree of freedom can move only one way from a given position, by a twist about some definite screw (o-i, ft)i). A body with two degrees of freedom can move in a singly infinite variety of ways from a given position ; if (o-^, w-^ and ((72' ^2) ^^^ ^^^ screws about which the body can begin to ART. 123.] CONSTRAINED BODY. 205 twist, it can begin to twist about every screw of the two- system {x^(T^-\-Xc,a-2, x^(jo^-\-X2w^, where x^ and x^ are scalars, as easily appears from the composition of small displacements (o-jd^p Wjd^^) and (o-gd^g' ^2^^2)- Similarly a body with n degrees of freedom can begin to twist about any screw of the Ti-system (Sa^jo-^, ^x^ud^, where ((Ti, cDj) ... (o-„, ft),i) are n independent screws about which the body can begin to twist; and being given n independent screws about which the body can begin to twist, all possible initial motions belong to a given system of twists. Every wrench reciprocal to n independent screws of the freedom is a wrench arising from the constraints, for every such wrench is reciprocal to every possible twist on account of the linear character of the condition of reciprocity (ii.), and no such wrench can generate any motion in the body. By expressing that a wrench (/x^, X,) is reciprocal to n screws of the freedom, the number of its arbitrary constants is reduced from 6 to Q — n since n conditions (ii.) must be satisfied ; and thus the screws of the constraint compose a system of order (6 — ?i). This system can be determined when the system of the freedom is known, and conversely. Again knowing the system of screws of the freedom we can determine what Sir Robert Ball calls the screws of the reduced wrenches. A reduced wrench causes no reaction on the con- straints ; it produces the same initial motion as if the body were perfectly free. In equations (i.) the wrench (yu — /x„ X — A,) is a reduced wrench, or (^co, Ma) is the reduced wrench corresponding to the twist velocity (o-, o)). The system of screws of the reduced wrenches is (02a;^ft)i, M^x-^a-^) when that of the freedom is Suppose now that we select n independent screws of the '71-system of the reduced wrenches and Q — n screws of the (6 — '?^)-system of the constraints, and that (Art. 102) we resolve an impulsive wrench (/x, X) into its components on these six screws, we shall have (compare (xvi.), p. 166), />t = M +M., X = V+X„ (III.) where (yu", X') is the component of (yot, X) belonging to the system of the reduced wrenches and where (/x^, X^) is the component belonging to the system of the wrenches of the constraint. The instantaneous twist velocity is then given by the relations a- = M-^\\ w = (p-y (IV.) Ex.'l. Prove that represent respectively a three-system of screws (fx, A) and the reciprocal three-system (fx', A'), +ct>rk A^iKc^+c^rV, x,\\{4>+rh where i\j and k are three mutually perpendicular unit vectors.] Ex. 3. Determine sextets of co-reciprocal screws. [Take any triad of co-reciprocal screws of a three-system /jl = ^A, and any triad of co-reciprocal screws of the reciprocal system /x= - ^'A.] Ex. 4. Resolve a wrench (or twist) into its components on six co- reciprocal screws. [If {fjLi, Aj) . . . (/xg, Ag) are the six co-reciprocals, we can find a linear function cj) so that (/Xj, A^), (jUg, Ag) and (/X3, A3) belong to the system /x' = <^A'; and then (ju^, A4), (/x., A5) and (/Xg, Ag) will belong to the system /x"= -cfy'X". We assume for the given wrench (/x, A) that /x = ^A'-<^'A" and A = A'-t-A" ; whence we have generally A' = ( -f ()^')~\/x -f c^'A) and A" =-{(f> + (/)')~-'(/x - t^A), and it only remains to resolve A' along A^, A2 and A3, and A" along A4, A5 and Ag in order to obtain the required relations /x = 2^j/Xi, A = 2^iAi.] Ex. 5. Find the {n - 6)-system reciprocal to a given 7i-system. [This has been effected in Ex. 1 for n = 3. Let n = 4, and for any three screws of the system construct the function cf). Resolve any fourth screw (/x„, A,j) as in the last example, so that /x„ = <^A' — ^'A" and A„ = A'-|-A", and take two vectors A5 and Ag which with A" compose a mutually conjugate triad with respect to Sp((f) -f ^')/o = const. Then (-^5(^'A5-^6<^'Ag, ^sA.^ + ^gAg) is the two-system reciprocal to the four-system. To determine the four- system reciprocal to a given two- system, take any function <^ satisfying /Xj = (^Aj, /X2 = <^A2, where (/Xj, Aj) and (/xg, Ag) are two screws of the two-system, and determine the vector A3 conjugate to A^ and A3 with respect to the quadric S/o((^ -!-(j){) (Mcr.f + 80)2^102) : ( J^/So-iO-g + Sbi^cfxo^y, the base-point being coincident with the position of the centre of mass at the instant of the change of motion. [This is the general case of a self-closing gate. By the last example yiMfj^- -I- Sw22) =^( J/So-jcrg + Sw^c/xog) and y{M Swicra + S w^c^cog) = x' {Mg-^ + S(o^(/)0) j), where x : x' is the ratio of the twist velocities just before the change from the screw (o-j, w^) to the screw (o-g, Wo) and just after the change from (0-2, w.,) back to (o-j, Wi). The system of forces being conservative, the magnitude of the twist velocity throughout the partial oscillation during the continuous part of the swing depends solely on the position of the body, and is the same just after the sudden change from (o-g, (02) to (o-j, Wj) as just before the next sudden change from (o-j, w^) to (0-2, Wg). To show that x is greater than x' or that (3/0-1'^ + Sa)i<^w,)(JI/cr2^-|-S(02<^W2)-(3/So-iO-2-|- 8(0^(^0)2)^ is positive, turns on the fact that a^p^ + y^S"'^ — 2Sa/3Sy8 is positive when a, ^, y and 8 are real vectors. The value of this expression lies between the limits (Ta/i^ i TyS)^.] Ex. 9. An impulsive wrench reciprocal to the instantaneous twist velocity of a free body at the moment of its application increases the kinetic energy. [The change of kinetic energy (Art. 122 (xi.) is - ^S/x<^~V - ^i/~^A^, and this is equal to the kinetic energy which the wrench would generate were the body at rest.] Ex. 10. Determine the dynamical constants and the constraints of a rigid body by observing the effects of impulsive wrenches applied to the body when placed in a given position. [Let (/Ai, Aj), (/x,i, A,i) and (o-i, Wj) represent an impulsive wrench, the corresponding opposing wrench arising from the constraints and the twist velocity produced. We know (cr^, Wj) by observation — that is, a screw of the freedom. A second impulsive wrench (/Xg, Ag) being applied, we find a second screw of the freedom (cr2, W2), provided we have not o-g = tcr^^ Wg = ^Wj. In this second case, however, we have a screw of the constraint, for the impulsive wrench (/Xa-if/Xi, A2-^Ai) generates no motion. Administering a third wrench we obtain similarly either a new screw of the freedom or a new screw^ of the constraint ; and from the results of applying six independent wrenches, the screw systems of the freedom and of the constraint become completely known. These systems being known, we can by (m.) resolve an impulsive wrench. (/j^i, Aj) into the reduced wrench (/x/, A/) and the evoked wrench (/^/ij -^/i^J ^^^ '^^ have as many sets of equations J/(o-i — V/3Wi) = A/, (^(Oj = /x/ — V/o A/ as degrees of freedom. For one degree of freedom, the first equation gives the mass i/ = SwiA/ : Sw^o-^ ; and a line locus V(Oi (/jSwi Ai' - Vo-i A/) = 208 DYNAMICS. [chap. xv. for the centre of mass. Eliminating p between this and the second equation, the result is V. Yui^XiXfXi — (^(Oi)SwjAi' = Ai'SVwiAi'Yo-jAi'; or separately, SA/(/)Wi = SA/ju,/ and Swi^w^ = S(a)i/Xi' + crjA/) — A/^So-iWi(SwiA/)~\ The body has therefore a given moment of inertia (Swi~^^(Oi) round Wj, and a given product of inertia ( — SUa>i<^UA/) with respect to Uwi and-UA/ ; but it is otherwise indeterminate. For two degrees of freedom, the two force equations completely determine p, and the couple equations give completely ^cu^ and cfxn^- There remains only one unknown constant, the moment of inertia (SVa>jCU2~-^<^V(OiW2) with respect to the line perpendicular to Wj and Wg. The dynamical constants are completely determinate in the case of three degrees of freedom. Compare generally Art. 122, Ex. 6.] Ex. 11. Two three-systems of screws can be in one way correlated, so that each screw of one system, regarded as an impulsive screw, corresponds to a screw of the other system regarded as an instantaneous screw. (Ball, Treatise, Art. 318.) [This has been virtually proved in the last example. We have to show that if cr = <^ = M{cty,-Yp)(,' + Yp)(2'+yp)K remembering that (f) is self -conjugate, and this holds for all vectors A. Hence (<^2i - <^i'2') ^ - ^Xi^'^ - yX2P^ = Oj where Xi ^^^ X2 ^^'^ Hamilton's auxiliary functions for (f)i and ^3- And because A is perfectly arbitrary, we have (xi + X2')/o = 2^2i i^ ^21 i^ ^^^ spin- vector of ^2^-2y{-x^i + X2)~^^2i)^^K so that to each screw of one system corresponds a definite screw of the other.] Ex. 12. Screws {fx, A) and (cr, co) are connected by the relations A = (^jcr -H (^2^5 /x = <^3a>-l-^4cr, where c^j, cf)^, (f>2 ^^^ ^4 ^^^ ^^^^ given linear vector functions. Find the conditions that {jx', A') should be reciprocal to (cr, w) whenever {fx, A) is reciprocal to (o-', w'). [The general relations of this example establish a homography between screws (/x. A) and (o-, o)) ; and w^hen the conditions of mutual reciprocity are satisfied, the homography is said to be chiastic (Ball). , The conditions are simply S ( Act' -f jaw') = S ( A'cr -f /x'w) or Scr'(<^iCr-h^2w) + ^^'(^3^ + ^4^) = So^(^i<^' + ^2^') + Sw(^3w' + ^40"')) where w, o>', cr and cr' are arbitrary vectors. Putting w and to' both zero, it appears that ^^ must be self -conjugate. In like manner (^3 is self -con jugate, and the condition reduces to So-'(^2"~ ^4')^ = ^o-(*^2 ~" 4^\)^'i which requires (^4 to be the conjugate of ^2- Thus the general chiastic homography is defined by relations of the form A = (^jO- -I- (/)20), IX = <^3(0 + (f)2Cr, where cf)^ and (f>^ are self-conjugate.] ART. 123.] THEOKY OF SCREWS. 209 Ex. 13. The screws of impulsive wrenches applied to a free rigid body at rest in a given position (or the screws of the reduced wrenches applied to a constrained body) are in chiastic homography with the screws of the corresponding instantaneous twist velocities. [Here A = Jf(o- — V/aw), n=(f>o) + MYp{(r—Vp(D) and the conditions are satisfied. This may be seen still more simply by taking the base-point at the centre of mass.] Ex. 14. The united screws of a chiastic homography are co-reciprocal. [For a united screw /i,=^(r, X.=x(i), and for a second united screw /x'=j;V, X' = x'(ii\ and hence ^S (o-a>' -f o-'w) = S (/xw' -f- o-' A) = S (/x-'o) -f crA') = ar'S (cr'to + o-w'), so that the screws are reciprocal or else x=x'. In the latter case every screw of the system (o- + io-', w-f-^w') is easily seen to be a united screw of the homography. The theory is quite analogous to that of the axes of a self- conjugate function. The united screws in the general homography are to be determined by solution of the equations ^w = <^iO--f-<^2*^j ^cr = ^jtu -}- ^^o-. On elimination of o-, we have Compare Art. 115 (x.), p. 186.] Ex. 15. There are n real principal screws for every position of a rigid body having freedom of the nth. order, so that the body will begin to move from rest along one of these screws when a wrench is administered on that screw. [For the centre of mass as base-point, if (/a, A) is on a principal screw, we have fjL=xa', A = :rw and also fx — fx^= (fao and X — X^=M(r. Now if ((Tj, (Uj), (o-g, (02), etc., are screws of the freedom we deduce from these expressions the n conditions xB ((TjO) -1- (TWi) = S(x)(f)0)i + J/So-cTi, etc. ; because the evoked wrench is reciprocal to every screw of the freedom. Also w=2^„(o„ and (r='2tn(rni and on substitution for &> and cr and on elimi- nation of the scalars t, a determinant of the nth order in x is obtained. Putting X equal to one of the roots of this equation, the scalars t can be found from n — loi the conditions. Just as in the case of self conjugate functions, if a root x is imaginary (x' + xJ -Ix"), the corresponding principal screw is imaginary (o-'-hV^o-", A'-t-V^A"): and there is a conjugate principal screw (o-' — v — 10-", A'-v — lA"). By the last example these screws are reciprocal, and we find that S(o'(f)(x>' + M(r'^ + So}"(i)= —r]^f where (*«„ J,) is the evoked wrench. The condition of reciprocity gives Sco<^o>-f-l^So-6-=0 ; and for a permanent screw 6)=x(o, &=xa; and we must have ^=0 because Sw^w + i/o-^ is essentially negative. By means of the equations of constraint we can eliminate ^, and 17^ from the conditions MY (OCT = —^,f Y(aoi= —rj,."] J.Q. O 210 DYNAMICS. [chap. xv. Ex. 17. To find the principal and the permanent screws for freedom of the third order. [Here (r=cf>^it) where (f>i is a given linear function, and the screws of the constraint belong to the reciprocal three-system /x, = — 4>i X^. For a principal screw (i> =j;(r — fx^= xcfi^u) + ^I'A,, J/^^w =X(a — X,; so that CO is an axis and x a root of a determinate linear vector function. For a permanent screw, Y(o(i) = —ri= ii,i MVoxfiiCi) = — ^,; and on elimination of ^, we find and 0) is now an axis of the new linear function , x ^^^ ''^'- The scalar m'' is called the divergence and SVar is the convergence. Again the small strain at the extremity of p due to the dis- placement adt may be resolved into a pure strain, which converts a into a + J(0 + 0')a.d^, and a rotation represented in magnitude and direction by ed^ where e is the spin-vector of (p ; for we have, a + 0a. d^ = (l-fVe.dO(a + 0o«-dO = (l + 0odO(« + ^^«.dO when we neglect dt\ Hence the spin-vector e represents in magnitude and direction the angular velocity of the element at p when 0- denotes the velocity of its points in the field. It remains to exhibit e, m'' and x in terms of a. We have for any three vectors V/3y . 0a + Vya . 0/3 + Va)8 . 0y = - ( V/3ySaV + VyaS^V + Va/3Sy V) . cr = -VaSa/3y = (m''-2e)Sa/3y; and the first quaternion invariant (Art. 67, Ex. 7, p. 97) is m" - 2e = - Vo-, and m" = - S Vo-, e = J Wc (ill.) Further, x<^ = ^^^<^o-> x«= -^•^"^•o', • (iv.) ART. a24.] THE ASSOCIATED LINEAR FUNCTIONS. 213 because, for example, xa = (w" — ^)a = — S Vo- . a + SaV . a-. It is evident that x ^^^ ^ have the same spin-vector. The vector 2e or VVe is the spin-vector of xfr' (Art. 68, p. 98).' Thus m'=-iSVVVTo-(7', ^€ = iY .YVVYcct', (vn.) and this expression for (pe should be verified by operating with (p on the value already obtained for e. It is also a useful exercise to verify that the third invariant is m = iSVV'V''Sa-o-V', (VIII.) .(IX.) but a more familiar form of this invariant is m = du du du dx dy dz dv dv dv dx dy dz ' J* dw dw dw dx dy dz *The device employed here is quite analogous to a transfo|'mation in Aron- hold's symbolic notation. 214 THE OPERATOR V. [chap. XVI. which is obtained by putting p=:ix-{-jy-\-kz, cr = iu-]-jv-\-kw ^=-^^^^^^^=-^9^9^^^ (X.) Ex. 1. Show that in terms of i, j and k, Ex. 2. In terms of three arbitrary differentials of p and of the corre- sponding differentials of o-, Sdo-d'o-dV J 2dorSad'pd"p , 2dpSd'o-d"( Sdpd'pd"p Sdpd'pd"p Sdpd'pd"p Wl"+2€ = 2do-.Yd>d> Sdpd>d'>~' „ , _2dp.YdVdV '^'^'~ Sdpd>d> • (a) If 6.p=cf)^dcr, write down the corresponding functions for ido-=0. Hence the three differentials of cr corresponding to three arbitrary differentials of p are coplanar and Sd(rd'a-d"(r=0. Conversely, if m is identically zero, three differentials of o- corresponding to three arbitrary differentials of p are linearly connected, or Mor-}-Z'd'o-4-rd"o-=0, suppose. Hence cr can receive only two independent variations, or a relation of the form/(o-) = must be satisfied by tr.] Ex. 5. If (J satisfies two scalar relations /i('a = 0, where a is an arbitrary constant vector ; and if it is possible to find a scalar multiplier to render <^dp the differential of a vector function, [Note that (/>'a=-VSa(r if <^dp=d(r.] ART. 125.] INTEGRATION THEOREMS. 215 Ex. 9. If Ci and C^ are the principal curvatures of a surface w = const., show that ^^ ^ ^^_ _ svxj Vw, (7iC2= - ^SYVVVUVi^VV. [See my note, Elements^ Vol. ii., p. 251. If t-^ and Tg are tangents to the two lines of curvature, (7iTi + Sti V . U V?^ = 0, C^r^ + SrgV . U Vw = ; and (Ex. 4), since TUVw= 1, the third invariant of the function — SdpV . UV?« is zero, and Cj, CU and zero are therefore its latent roots.] (ii) Integration Theorems, Art. 125. It has been shown in Arts. 55 and 56 that the form in which the operator V naturally presents itself leads to the two results (pp. 72 and 73). {du.q = Vq.dv, {dp.q = Y(dv.V).q; (l.) the first integral being taken over a small closed surface of which dv is an element of outwardly directed area while dv is the included volume ; and the second integral being taken along a small plane closed curve of directed area dv, where rotation round dv in the direction of the circuiting is positive. In both relations g is a quaternion function of the variable vector p. In order to extend these results to integration over finite regions, we shall first suppose that the quaternion q satisfies certain conditions: — (a) that it is free from discontinuity, (b) that it is single-valued, (c) that it does not become infinite at any point of the region. Further we suppose (d) that the region included in the surface over which we propose to integrate is simply connected, so that any closed circuit drawn in that region can be made evanescent by continuous variation without cutting through the surface. On these suppositions, we divide the region within a closed surface into infinitesimal parallelepipeds, and we apply the theorem of Art. 55 to each. Adding together the integrals I di/ . g over the faces of these parallelepipeds, the sum obtained is equal to the sum of the corresponding elements Vq . dv, but over an interface corresponding to two parallelepipeds the directed elements are opposite, so that if one parallelepiped contributes an element dv . q, the other contributes an equal and opposite element —dv.q; consequently the sum of the integrals \dv . q is the integral over the bounding surface. IVIoreover the sum of the elements Vq . dv is the integral I Vg . dv throughout the volume, and we have \dv . q= IVg . dv, (n.) 2ieS THE OPEEATOR V. [chap. xvr. where the first integral is taken over the surface and the second throughout the volume. Under the same conditions we can fill up any continuous closed curve by a net- work of parallelograms described on any surface terminated by the curve, and if these are all circuited in the same direction the elements contributed by the common sides cancel, and ^dp .q=^V{d..V) .q. (iii.) where dp is a directed element of the curve and di^ a directed element of the surface. Hence it follows because (iii.) has a value independent of any particular surface through the curve that over any closed surface fv(di/.V).g = 0. (IV.) (a) Suppose a surface to exist over which q is discontinuous, and imagine the region of the volume integral to be divided into two regions by the surface of discontinuity. Applying (ii.) to each of these regions and adding, we find \Vqdv = \dv.q-{-ldvi^{qi-q2),.. (v.) an element of the surface of discontinuity furnishing the parts di/12.5'1 and dv^i.q^j or dv^^iq^^-q^). (b) If q is not single-valued, it is not hard to see when infinite values of Vq are excluded from the region that, assuming any one of its values for q at any point of the region, the value of q at every other point of the region is determinate. In fact starting from a point p with a given value of q we can return to p with a different value only if we thread some circuit along which q is indeterminate ; and if q is indeterminate anywhere within the region, its corresponding deriveds must be infinite, which is contrary to supposition. When a curve locus of indeterminate values of q exists in the region, we may enclose it in a tube and so isolate it from the region. The region thus becomes multiply-connected (d), (c) If q becomes infinite at any point, we exclude that point by a small sphere concentric with it and we take account of the surface integral over the sphere, the vectors representing the elements of directed area being drawn outwards from the region, that is, towards the centre of the sphere, and the radius of the sphere being ultimately reduced to zero. Taking the origin at the point, the element of directed area over the surface of the sphere is di/= - Up . r^. dl2 if r is the radius and dI2 an element of solid angle. Then for the sphere ldv.q^-\dn.Vp.r\q (vi.) If over the surface of the sphere q = qo+r-Kq^ + r-^.q2 + r-^.qs + etc., (vii.) the surface integral vanishes unless qo exists, and it generally becomes infinite or indeterminate if q^, etc., exist. Of paramount importance is the case in which q contains the term VTp-i. e= - Up . Tp-'^. e. In this case if no higher negative power of r occurs, the integral becomes jdv.g'= - jdl2.e= -4776, (vm.) or THE UNIVERSITY ART. 125.] SINGULA tolES IN wJI^^feRATION. 217 and we must replace (ii.) by lVq'dv = jdv.q-47rey ....(ix.) the origin being excluded from the volume integral. In general when 5^3, etc., are zero, by a well-known theorem in spherical harmonics (Art. 127) we need only consider the terms in q2 which are linear in U/o and which we may take to be SaJJp + cfiJJp. Writing \Jp = li + mj+nk where I, m and n are the direction cosines of Up, and remembering that jdi2 . ^2 = Att, jdl2 . ^m = 0, etc., we have ldn.Vp(SaUp + Vp) = iTr^i{Sai + cf>i)= -|7r(a+m"-2e), (x.) where m" is the first invariant and where e is the spin vector of . (SaU/3 + Vp) *Th§ interior of a hollow curtain rir.g becomes simply connected when a diaphragm is drawn across one normal section. + The two line integrals are taken in the same sense of rotation round the axis of the small circle. If we choose the minus sign may be placed on the right of the sign of integration, and then we shall have the surface integral equal to the sum of two line integrals taken in opposite directions. 218 THE OPERATOR V. [chap. xvi. suppose where is a linear vector function, the terms not linear in JJp leading to a vanishing integral round the circle. Putting Vp = i cos w+j sinw where i and j are in the plane of the small circle, the integral easily reduces to 7r{jSai — iSaj+j(f>i — i(pj\ and to 7r(Va^ — X'^ + 2Se^) where x' ^-^d € have the same signification as in the chapter on linear vector functions. ^^ Ex. 1. If /(V) is any linear function of the operator V with_ constant ^ coefficients, j/(dv) . g = J/(V) . ^ . d.;. If (dp) . ^ = j/(Vdi^V) . q, and J q .f{dv) = I q ./(V) .dv, j ^ ./{dp) = J q ./(Vd.V). [No step in the proof of the simpler case need be modified. In the second set of relations the operator is placed in front of the operand. See Art. 57, Ex. 11, and M'Aulay's Utility of Qiiaternions in Physics7\ Ex. 2. In general if /(a) is a linear function of an arbitrary vector a while the variable vector p is involved in the constitution of the function, show that l/(dv) = j/(V). d^, j/(dp) = j/(VdvV), where /(V) means that V operates in situ on the variable vector p as involved in the structure of the function. Ex. 3. Prove that j ^^ = - jSdvV . VTp-\ where no infinites occur. [See Tait's Quaternions^ Art. 504. Here the line integral is ^YdpVTp~\ which transforms into jV.Vdvy.VTp-i or j di/V2Tp-i- j Sdi/V . VT^-^.] Ex. 4. Prove that \qdv = llp.Vq.dv-^lpdvq, [This is an example of an extensive class of transformations depending on the invariantal properties of V. Transforming the surface integral, we have ^ pdvq = j p{V)qdVf where V operates both on p and on q. But pS/ = Vp=-3. See Art. 132, p 235.] (iii) Inverse Operations. Art. 126. We shall now establish general solutions for the equations Vp = q, and W = g, (l.) where g is a given quaternion function of /o ; or we shall assign definite interpretations to the functions _p = V-ig and r = V~^q (li.) for all points of an arbitrarily selected region within which infinities do not occur. ART. 126.] INVERSE OPERATIONS. 219 We shall first prove the transformation* \Vu. Vp .dv= \dp. u. Vp— IVi^oVp . dv = {vu.dv.p- j'v(u-Tp-i)V . p^dv-4xp (III.) - in the case in which p does not become infinite within the region, while u tends to the value Tp"^ at the origin which we suppose to be taken within the field of integration, and where 47rp in the third member is 47r times the value of p at the origin. The suffixes are intended to indicate that the affected symbols are free from the operation of V. Surrounding the origin by a small sphere and supposing (V) to operate in situ on u and on p we have \ViL .Vp .dv= l(V)t(, . Vp . dv— Wuq^p . dv = \dv .u .Vp—\ VuqVp . dv for the region between the small sphere and the boundary, the surface integral over the sphere vanishing by the last Article (compare (vii.)). But these integrals may be extended through- out the entire region, for we shall show that the integrals taken through the volume of the small sphere tend to zero when the radius is indefinitely diminished. Within the sphere we may take u = Tp-i and dv = Tp^.dQ . dTp, so that {Vu.Vp,dv=-[Up.Vp.dQ.dTp which vanishes in the limit. A fortiori the integral I VuqVjs . dv= |Tp-i . V^p . dv for the small sphere vanishes. Thus the first part of (ill.) is proved. Again for the field exclusive of the sphere I Vu . Vp . dv= \Vu . (V)2? . dv— \VuV .pQdv = IVu. dv .p — ^TTp— jVuV.p^dv by (viii.) of the last Article because for the surface of the sphere [Vu .dv,p=-\- [Tp-2 . Up . Up . Tp2 . dQ . 29= -[dQp. *It is manifest from the proof of these relations that they are valid when neither p nor u become infinite in the field of integration provided we omit the term in Tp"^ and the term Airp. ^ 220 Al£ tha reg bee wh wh I = f Vi( . d»/ . p - f V2(u - Tp - 1) . jpd'U - 47ri). . . .(ill.)' Changing the origin or replacing /o by p—p in (ill.)', and supposing p to be the current vector in the integrations, we obtain for the particular case in which u = T{p — p)'^ the important identities, ^ = j 4.T(,--,) -j 4.T(,-.:,) +J^ ' 4^T(p-^,) -^- '^ -a^-) '^j47rT(p'-p) ''j4xT(p'-/o)' the second being deduced from (iii.)' by replacing Vu by V'.T(/o'-p)-i or by its equal -VT(/o'-p)-i and taking V out- side the sign of integration. If then Vp = q, we have and in this relation jp' is any function which over the boundary satisfies Vp — q. In like manner - . where r is any function which over the boundary satisfies V^r=q. It may be observed that in these results there is a certain analogy to the solutions of the linear function equations of Art. 65, p. 92. ART. 126.] INVERSE OPERATIONS. 221 If we operate on (vi.) by V and put p = Vr we find on com- parison with the second form of (v.) that V.V-2g = V-ig ....(VII.) because the last integral of (vi.) vanishes under the operation of V (or of — V under the sign of integration operating on V'T(p' — p)~^) provided p does not terminate on the boundary. Ex. 1. Find the potential which produces a given distribution of force in a given field. If f is the force and P the potential, we have to determine a scalar function P from the equation ^= - VP. By (v.) this function is _ ._ _ r svi\dv' f sdvt f Psdvv.Tjp-prn "^ ^- j A7rT(p- p')'^ J 47rT(p- pY J 47r J Ex. 2. A quaternion p which satisfies the equation V^p = throughout a given region is expressible as a surface integral over the boundary ; and a quaternion jD which satisfies Vp = throughout the region is of the form P^-VJ dv' . p' 47rT{p-py Ex. 3. A scalar satisfying the equation VP=0 is constant. A vector satisfying Vo-=0 is expressible in the form (r=VP where P is a scalar function satisfying V^p^o. \ : - ;' . ; • Ex. 4. Construct quaternion functions of p, homogeneous and of the first and second orders, which shall vanish under the operation of V. [For the quadratic function assume p = SpQp - 2H,cf)„pan, and if SVp is identically zero the condition 2^naw=0 must be satisfied. In order that VVp may vanish, we must have ^oP= — 2V<^„/3a„= 4-2<^„V/oan since <^o i^ self -con jugate. Again, because ^^p=0, the first invariants of the functions cf) must vanish. But in general m'Vpa = Yp(f>a + Y<^pa -h cfiYpa, and in the present case 2 V/onp + ^0,n^p4>r>Pi where m„"=0, 2<^„a„=0, vanishes under the operation of V.] Ex. 5. Determine the extent of the arbitrariness in the dissection of a quaternion into the parts V~^SVg' and V~^YVq on the supposition that V~^SV^ is a vector. [The most general expressions for the parts are V~^SVg'+o- and y-iyVg'-o-, where o- is a vector satisfying Vo-=0. See Ex. 2.] Ex. 6. Divide a vector cr into two parts o-i and o-g so that SV(Ti = 0, VVo-2 = 0. [Here (r2=V~^SVo- and a-i = V"iVVo-. We may calculate one of these, say 0-2 by the general formula, and the other is o^ — o-g.] .-~r- 222 THE OPEEATOR V. [chap. xvi. Ex. 7. The general solution of the equation may be written in the form \ m + n n ) [The equation may be transformed into (m + 72.)VSVo-+7iVVVo-=f, and by the last example, VSVo-=(m+7i)-iV-iSVf, VVVo-=7i-iV-iVVf. The solution given above of the equation of equilibrium of an elastic solid may be expressed more simply in the form (r=V~2(7i~^J-m7i~^(m4-?*)~^V~^SV^).] Ex. 8. If V2jo = at all points within a closed surface, and if V^^ =_o at all external points ; if 'p,—'p over the surface and if f, tends to zero at infinity, "d v\vX/-p/) 47rT(/3'-p) • [Integrating throughout external space we find if V2p^=0, see note p. 219, - Jdv' . Vp;.T(p'-p)-^ + jv . T(p'-p)-^ . dv' .p;=0, when p terminates at an internal point so that T(p' — p)~^ does not become infinite. The surface integrals are to be taken over the closed surface and over an indefinitely large surface, but it easily appears that the latter part of the integer vanishes since p^ vanishes at infinity. Putting y^p==0 in (iv.), remembering that p,=p over the closed surface, and subtracting, we have the required result.] Ex. 9. If fnp is a homogeneous function of p of order n satisfying VY„/3=0, show that when Tp where fn-2P» fn-iP, etc., are functions defined by equations such as (il). Any integral of the form P= \pdv .T(p — (t))-^ in which « is the current vector and in which p is independent of p may be expressed in the form P=fV.Tp-\ (IV.) provided Tp is not less than the greatest of the tensors Tw. For (Art. 59 (xi.), p. 79), and we may speak of P as the potential at p due to a distribution of density p although it is not necessary to suppose that p is a scalar. If Q= \qdv. T((a—p)~^ is the potential of a second distribution of density q, the mutual potential is I ^=f^=k-^^^'4«-i^ (-> If the second distribution lies outside a sphere of radius a having its centre at the origin and including the first distribution,, we have by (v.), provided we reduce the temporary vector p to zero after the performance of the operations indicated, and the suffix serves to remind us of this reduction. \i Q—gJ^p) is a solid harmonic of positive order n, and if we suppose the corresponding distribution to be a surface distribu- tion on the sphere, we may replace qdv' by (47r)-i.(2ti + l).a-i.^n(a)').Td,.', 224 THE OPEKATOR V. [chap. xvi. or by (4x)-^(27i + l).a'^+^^n(Uft)).dQ, utilizing Ex. 9, Art. 126, and dropping the accents as being no longer necessary. In this case (vii.) becomes 47r ./( - V) . g,lp\ = {2n + l)a-^^\^P^gn{^w) . dO (viii.) In this expression it is only necessary to take account of terms of order 71 in /( — V), for gnip) vanishes under the operation of terms of higher order, and the results of operation of terms of lower order vanish when p is reduced to zero. If P is a solid harmonic of order — 7i — 1, the form of the function /V is given by (iii.), and ^nd accordingly (viii.) becomes 47r/n(-^)-5'n/> = (-r.l.3 (2n-l)(2^+l).f/,(Ua,).^n(Ur^).dfi;...(x.) while if the order of the harmonic P is — (m + 1) where m is not equal to n, we have J/^(Ua)).^n(Ua)).dQ = (XI.) Again if P = T(p-a)-i = e^'^^.T;o-i = STa"Tp-^-M^(U;o), (xii.) ive find on substitution in (viii.), .47r^„(Ua) = {2n + 1) j^„(Uft))^„(Uco) . dO, = j^^(Ua,)^„(Ua)).dQ because /( - V) . gn{p) = e " ^*^ . grip) = gn{p + a). Hence we can expand any function g(}Jp) in a series of spherical harmonics, the harmonic of order n being £f„(u«)=^?:^^J4„(Uco)^(u«.).dfi (XIV.) Ex. 1. A scalar solid harmonic of order -{n + \) may be expressed in the form Saj V . SagV Sa„V . Tp-\ where aj, a^, ... an are real vectors. [Consider the edge s com mon to the con es Fnp = 0, p^=0. These group into conjugate pairs ^+^1 -1/3' and /3 — *J — l/3\ and each conjugate pair lies in a real plane Sap=0 where a = Y/3f3'. Having determined the vectors tti, ttg, ... a„ we have a relation of the form F„p = p^Fn-2p + 1 . Stti/oSajj/o . . . Sa„p, .(XIII.) ART. 128.] SPHERICAL HARMONICS. 225 where ^ is a scalar and where Fn-op is a homogeneous function of p of order n — 2. If i^nV is the generating operator (see (ix.)) of the harmonic we have, on putting V for p in the above relation, FnV . Tp-^ = t. SaiVSagV ... SttnV . T/o-i because y2Tp-i = 0, and the scalar t can be found by comparing a coefficient.] Ex. 2. If 3' is a quaternion associated with each element of mass of a body, j^dm=/(V) . q^ {T?dm = Vv/(V) . q, where t is the vector from a point in the body to the element dm, where ^-q is the value of q at the origin of vectors t, and where Vv operates on/(V) as if it were a function of a vector V. (a) The first terms of the function /(V) are /(V)= if- J/SToV+i{SV*V - ^(.4 +5+ 0) V2} _ etc., where M is the mass of the body, Tq the vector to the centre of the mass, ^ the inertia function for the origin of vectors r'and A^ B^ C the principal moments of inertia for the same point. [We have j^dm = je - ^^^dm . q^ = \{\ - StV + ^StV^ - etc.)dm . q^ ; and because StV2 = t2V2 + VrV^, j Vt VVrdm = * V and jT2dm= -\{A+B-{-C\ the expansion is justified. Again the differential of /a corresponding to da is d/a= - SdaVa ./a= - {Sdare-^^'dw.] Ex. 3. A heavy body is placed in a field in which the gravitational potential is P. The potential energy of the body ( TF), the resultant force and the resultant couple (A and p) acting on the body and referred to its centre of mass, are Tf=JfP+^SV*V.P, A=ifVP+^SV^V.VP, />t = V^VV.P. (v) Various expressions for V. Art. 128. We shall now examine in greater detail than in Art. 57 the various analytical expressions for the operator V and for V^. In terms of three arbitrary differentials we may write V = Xd+XU+X"d", (I.) where (Art. 54 (vi.), p. 70) sdpd>dv sdpd>d>' ^ ~ sd^d'yody ^''-^ The operator V^ is now ■ *^ V2=2:x2d24-2(AXd'd"+X"VdM')+2VX . d, (iii.) and in the third sum V operates on the vectors X alone and not on the operand of V^. J.Q. p 226 THE OPERATOE V. [chap. xvr. Remembering that V^ is a scalar operator, this breaks up into* the two parts V2 = 2XM-^ + 2SXX(d'd'' + d"d') + 2SV\ . d ; (iv.) = 2VXV.(d'd"-d"d') + 2VV\.d (v.) It is only when the differentials are independent that the order in which the differentiations are performed is indifferent, and it is only in this case that we can generally suppress the terms involving d'd'' — d"d' and similar expressions in (v.). When independent differentials are employed, we use the expression (Art. 57. (iii.), p. 74), V^ _^P2P3^ ^ ^ ^P^Pl A^J[P1P2_ A ; (VI.) ^PlP2p3 ' ^'^ ^PlP2P3 ' ^^ ^PlP2Ps * ^ ^ ' ' or as it may be briefly written ^ = ''>3^+''^3^ + ''«3^' ^^"•> where the vectors v^, v^ and v^ satisfy the relations Sj/ipi + 1 = 0, etc., Sj/2/03 = 0, Sj/g/og = 0, etc. ; (viii.) or again we may put V = V2..^+V^.|- + Vt(;.^, (IX.) ^u dv dw as we see by comparing the results of operation of the forms (vii.) and (ix.) on u, v and lu. Thus ^^ = Vu, i/2 = ^^j p^ = Vw (x.) and VVi/i = 0, VVi/2 = 0, VV,;3 = (xi.) The vectors i/^, u^ ^^^ ^z are the normals at the extremity of p to the three surfaces u = const, v = const, and w = const which pass through that point. The appropriate expressions for V^ are now V^ = 2..^£+22S.,.3.3^+2SV.,.|^; (xii.) or V-2(Vu)^^+2SV.V,«.^+2V%.|^. ...(xni.) Again introducing the operand q for the sake of greater clearness, we may write Vq= , (XIV.) ^PlP2pS because the terms which involve the second deriveds of p, such as ART. 128.] VARIOUS EXPRESSIONS FOR V. 227 ^Pi2p3 • 9' + ^AsPi2 • ^' cancel in pairs. Operating with this form of V on (vi.), we have V2= I ^ [v '^ np2pZ- ^ ^ \ I V ^ pWs • ^PsPl ^ "^ )|,(XY.) SyQi/Og/Ogl 'dvASp^p^p^' 'duJ 9i6\ ^piP2Ps "dvJ )' where the second sum contains six terms, and to this the sign S may be prefixed. Or in terms of the vectors v it easily appears that this reduces to V^= +S,,,{4(3i^^ . ly^l-S^^ • I)}, (xv:.) Whenever the surfaces, u = const., v = const, and w = const., are equipotential surfaces with the corresponding potentials, u, v and w, the operator V^ is a homogeneous quadratic in the ■^ -"i -pi differentiating symbols — , — , — . This property follows directly from (xiii.). The converse is also true. When the surfaces are mutually rectangular, the operator V^ is independent of the products of differentiating symbols. In this case we find from (xv.) the most convenient expression for V2=-^— L— .Z^(T.^^.^) (XVII.) Ex. 1. Determine expressions for V^ where (1) p=u{(i cos 2v +j sin w) sin v + ^ cos v} ; (2) p = u {i cos V +j sin v) + kw ; (3) p=^{{4> + u){(^ + v){(^^-io)].i, as in Art. 84. Ex. 2. If a scalar function P of a scalar function u oi p can be found to satisfy V^P=0, show that 7)2 p 7^p T/2j, {Vuf . ^+ V22^ . g^=0 and VVi^V . 7X^ = 0. ^ ^ ou^ ou (Vw)2 [See (xiii.) for the first condition. The second expresses that VH . (Vw)~^ is a function of «*.] Ex. 3. Given that a family of surfaces w = const, is an equipotential system, show that the potential corresponding to u is f v2w P=jdi^.e"-'(^«)'' "*. [See the last example.] Ex. 4. A family of concentric, similar and coaxial quadrics compose an equip©£p, V^= — ^p, V% = m". The condition of Ex. 2 becomes Y(l>p^p.m" = 0.] 228 THE OPERATOR V. [chap. xvi. Ex. 5. Find the condition that the family of surfaces /(/o, «fc) = should form an equipotential system, and determine the potential when the condition is satisfied. [Imagine u to be expressed as a function of p by solution of the equation f(p,u)=0. On this understanding we may treat /(p, ■w) = as an identity and equate to zero the results of operating on it by v and V^. We find vf+vu . 1=0, vy+2sv«^+ v% |+(v.)^ . ^=o: where V operates on / as if f were a function of p alone, and where consequently V and ^^— are commutative in order of operation on f. ou Utilizing the results of Ex. 2 to eliminate V^u and eliminating Vw we find ?)u ^^ du~?)u' ^°^ \du ' {VfyJ ^ {V/y ' du The condition to be satisfied is that the right-hand member — a function of p and u — should reduce to a function of u alone by aid of the equation f(p, u)=0. If F(p, u) reduces to a function of u alone by aid of the equation /(/o, w) = 0, we must have Vi^+Vw . |^ || V?^ |1 V/ or simply YVJVF=0. Thus the condition required is Ex. 6. Show that the family of confocals Sp((fi + 7i)~^p + l==0 is an equipotential system, and determine the potential. [Here we have y/= -2(<^-|-'^)-V and ^= -(<^ + w)-y = -K^/)' ; also V2/= -22i((t> + u)-H=2i:(a^-\-u)-\ These give §u'^^Tu= -P«-^= -l^log V{(aH.)(6^Hh.)(c^ + .)}, Ex. 7. The condition that the family of surfaces f(p, u) = should compose a system of characteristic surfaces in an optical medium of constant density is r /?)f\-^^ VV/V/(V/)^(|^) }=0. [Hamilton's characteristic functions satisfy the relation TVQ=n, where n is the index of refraction of the medium. If the family of surfaces satisfies the condition we must have Q a function of u, so that \/Q = QVu= - Q'f~^Vf, where the accents denote differentiation with respect to u. Hence when n is constant, TVf.f-^ must reduce to a function of u, or VV/V(TV/./-^)=0.] (vi) Kinematics of a deformable system. Art. 129. It' q is any function of p and t, its total differential may be written in the form dq = qdt-SdpV .q; (l.) ART. 129.] KINEMATICS OF CONTINUOUS MEDIUM. 229 and in particular when we replace dp by crdt we shall write 'Dq = qdt-S(rV .q.dt and J)q = q-Ba-V .q (ll.) When or denotes a velocity, Dtq is the rate of change of the quantity q regarded as associated with the moving point. On the other hand q is the rate in change of ^ at a fixed point, and — Sd/oV , q is the change in the value q from the extremity of p to that of /o + d/o at a given instant. If dyo, dj/ and dv are elements of directed line, directed area and volume respectively, at the extremity of /o in a medium moving with the velocity o-, we have by Art. 124 (ii.), p. 212, Dtiqdv) = (Dtq + m''q) . dv, D,(Strrd,/) = S(D,trT+x^)-d'^ = S^di/, r (iii.) D,(Sc7d^) = S(D,t7 + 0'CT) . dyo = Sgdyo, . where* (Art. 124 (i.) and (ill.)) — ^ ^ \ ( IV ^ because for example we have StrxD^di/ = Scrxdj^ = S^t^di/. In terms of the spin- vector e = JVVo-, the divergence W/"=— SVo- and the self -conjugate part 0^ of we may also write ^ = Dftrr — VeCT -f {m" — 0o)^> § = ^tT^ — ^^'^ + (P ' • • • i^) or explicitly in terms of cr we have t^ = tir - V V Vcrtrr - o-SVct, g = trr - VSo-trr - Vo-VVtjr (vi.) To prove these results observe that ^ = trr — So-qV . CT — trroSVo-+ SctqV . a- = CT-So-(V) . cT-f Ss7(V) . (7- o-oSVct and that ^ = trr — So-qV . trr — VSCTo(r = ^ — So-oV.CT-f VStrro-Q — VSortrr, where (V) operates in situ both on o- and tJT and where o-^ and zs^ are free from the operation of V. In addition we may write (D,-fm")g = g-Sc7(V).g (vii.) because this expression is q — So-qV . q — S Vo- . ^q. We may connect this with previous results by observing that (D<-|-m")Scja) = S(^co + t7w) = S(CTw-|-?7ft)) (viii.) is a consequence of (iv.) where co is any vector function of p and t^ ^Iso SVCT = (D,4-m'0SVt7 (IX.) * See H. A. Lorentz, EncyEopddie der math. Wiss., Vg, p. 75. 230 THE OPERATOR V. [chap. xvi. We may also observe that if w = VVcr, we have by (vi.) since t and p are independent, so that the order of operation by | V and of partial differentiation with respect to t is indifferent. ^ Hence VV^ = a), if ft)=-.VVtrT (x.) From these relations we derive various forms for equations of continuity ; and the voluminal, the areal and the linear equations ; of continuity are respectively ^ (De + m")g = 0, T^ = 0, m-=0 (xi.) The first asserts that qdv does not change for the element of volume ; the second requires Strrdi/ to remain constant for all vector areas dv, and S-nydp remains unchanged if ^ = 0. \ Instead of supposing the quantities q, rn and cr to be functions of p and t, we may take them to be functions of t, u, v and u' where u, v and w are three parameters which individualize the moving point. This is Lagrange's method, and Euler's method is that in which everything is expressed in terms of p and t. The total differential of q we shall now write in the form, ^^=i^*+i^'-+s^^+3>^ (-) ' and following the moving point we have I>eg = ^ (xm.) since u, v and iv remain unchanged. In particular The vectors a and ^ now become ^ = t-^^4^o. t = W-^«|-o. (X.V.) as appears on reference to (iv.). The appropriate form for V in these relations is that given in Art. 128 (vi.) or (xiv.). The element of volume is now —Sp^p2P2dudvdw, and the voluminal equation of continuity is simply (compare (m.)) qSp^p^ps^ const (xv.) Ex. 1. If c is the density of a continuous distribution of matter moving with the velocity cr, Euler's equation of continuity is c = Sy(co-) or D,c = cSVo-; ART. 130.] EQUATIONS OF CONTINUITY. 231 and Lagrange's equation is - cSpip2P3 = 0= const. (a) Hence Drlogc= -^ log 8/31/3.2/03 = SV:^ = SV(r. Ex. 2. Show that (>=6--o-SV(r, rr=(r-Vo-2-Vo-VVo- = D«o--JV.o-2. Ex. 3. Show that ^-P^'^^t Ex. 4. In general VgC7' + VCTg' = t^", V^CT' + Vt;TCT' = m"trT" + g" where Z5" = YTST:5'. [These relations follow most easily from (iv.).] Art. 130. The integral F=-\^r;^dp (I.) taken from one point to another along a curve depends generally on the nature of the curve ; but if VVcr = 0, so that ct = VP, the value of the integral is simply the difference of the values of P at the extremities of the curve. This integral may be called the flow of the vector t7 along the curve. The time rate of change of F as the curve moves with the medium with velocity a- is D,F= - [s^d/o, (II.) and if this integral is independent of the nature of the curve, § = VQ, cT-y(TVVt7 = V(Scrtrr+Q), D,tJ = V(S(7trro+Q) (m.) are different forms of the condition to be satisfied, Q being a scalar function of p and t. Other forms of the condition are VVg = 0, VVc7-VVV(rVVcT = 0, VVD,CT = VV'VS(7C7'; (iv.) or again (Art. 129 (x.)) di/, D,a= - fScodi/, co = YVto (IX.) The circulation is therefore the flux of the vector a)( = VVtrr) through the circuit, and the rate of change of the circulation is the flux of the derived vector w ( = VV^) or the circulation of ^. For any small plane circuit, the circulation — SW^dj/ is the projection of VVct on the normal to the circuit into the area of the circuit. Thus VVct determines the aspect of the unit circuit in which the circulation is a maximum, and it likewise gives the magnitude of the circulation TVVcr in that principal circuit. In like manner w determines the aspect of the circuit in which the rate of change of circulation is a maximum as well as the value of that maximum. The vector D^VVtJ determines the rate of change of the circulation from one principal circuit to another following the motion of the medium. A principal circuit does not generally remain a principal circuit. We note that by (iv.) and by Art. 129 (IV.) to = YVBtTH - VVTSo-ct' = D, VVcT - VVVo-V W; (x.) and in general we have (D,V-VDO.g = V'S(rT.g, (XI.) because D,V . g = V^--So-V . Vg, VD,g = Vg--(V)So-V . g. If a tubular surface, drawn through any circuit, is composed of curves satisfying the differential equation VVCTdyo = 0; (XII.) or, what is equivalent, if SVtrrdi/ = (XIII.) over the tubular surface, the circulation in any evanescible* circuit traced on this surface is zero. In particular if ABC and A'B'C' are two circuits embracing the tube, the circuit ABCAA'B'C'A'A is evanescible and also the circuit AA'A. From this it follows that the circulation in ABCA is equal to that in A'B'C'A', being opposite to that in A'C'B'A'. Hence the circulation * An evanescible circuit may be reduced to zero by continuous variation. ART. 131.] CIRCULATION AND FLUX. 235 is the same in all circuits drawn on the tube so as to embrace it once. The flux of the vector zs through a given surface bounded by a given curve is /> fc i ' / x 6^=- StJTdi/, (xiv.) and the condition that this should depend only on the bounding curve is that the divergence of zs should vanish, or SVtrr = 0, (XV.) as we see by transforming the integral over a closed surface into a volume integral. The rate of change of the flux is T>,G= - [s^d,/, (xvi.) and the condition that this rate of change should depend only on the bounding curve is SVt5 = or SVcj--S(V)(r. SVc7 = 0, or (D, + m")SVt7 = 0. (xvii.) In any case in which SVtrr = 0, if a tube is constructed of the lines Vd/)CT = through a circuit, the fluxes across all sections of the tube are the same, and the value of the flux is the strength of the tube. For a small tube we have, if Tdi/ is the area of a cross section and if dn is the strength, Tdj/Ttrr = d7i, where SV^ = (xvm.) Ex. 1. If VVDiO- = 0, the circulation of the vectors cr in any circuit moving with the medium remains unchanged. [See (ill.) and (iv.). We have D,a- = V(io-2 + ^).] Ex. 2. Show that in Lagrange's method (D,V-VD.)y=|^.9. Art. 131. In Art. 126 we showed that any vector V5 can be expressed in the form (see (iv.), p. 220) trr = Vp, (SV^ = ()), (I.) where |) is a certain quaternion. We shall examine how this quaternion is related to the flow and the flux of the vector trr. In terms of jp, J^=_fScTdyo=-[sVVVp.dp + [Sp], (II.) because — S . VSp . dp = dSp. Hence for a closed circuit, the circuktion depends merely on Vp. If the circulation in every circuit vanishes, the quaternion p reduces to a scalar, as we have already observed. The circulation in general is expressible as C=-js.VVYp.dp=-[s.V2Vp.di. (III.) 234 . THE OPERATOR V. [chap. xvi. We have also I)tF= - fs.(VVp- VSo-VVp-o-V-^Vp)dp + [D,Sp]; .....(iv.) and m and tj are ^^Vp-V^a-Vp-YaV^Np, ^ = Vp-YVY(TVp-crV^Sp. (v.) The flux is G= - {s^du= - isdpV.Sp- {sYpdp, (vi.) since f Sdt'VVp = [sdyoY^. The flux through any closed surface depends merely on Sp. Comparing (ii.) and (vi.) we see that Yp and Sp play a comple- mentary role in these two relations. Various forms may be found for DtG on which we cannot delay. Replacing p by ct in the second form of the identity {Art. 126 (iv.)), we obtain the expression ''=i4:7w^pr^ U.Tip-p) (^"•) applicable throughout a given region, and this exhibits the nature of the quaternion p of the present article. If there is no circulation at the boundary, so that we may put nj = VQ (where Q is a scalar function) in the surface integral, we have on replacing p by VQ in the identity already referred to also putting p^Q in the first form of the same identity and introducing a new scalar function E, f V'Q'dv _ f Sd.-VQ' r du'Q' ^-'^ h^T{p'-p)- }4:^T(p-p)-^^]wr(p'^y ('^•> Substituting for the surface integral from (vill.) in (vii.) and attending to the definition of R in (ix.), we find Moreover R is given by (ix.) as a scalar surface integral depend- ing on the values of Sdi/^ and of Q over the boundary, and V2i2.= throughout the region. In this notation (ii.) and (vi.) become F= -.{sVr,dp + [P + Rl G= - {sdi^V(P+R)--{s^dp. (XI.) If r] = 0, the distribution of the vectors CT is irrotational ; if P is zero there is no divergence and the distribution is solenoidal ; ART. 132.] IRROTATIONAL AND SOLENOIDAL VECTORS. 235 if P and rj both vanish, the distribution is irrotational and solenoid al. If, as in Art. 130 (xviii.), dri is the strength of a tube of vectors VVtn of cross-section Tdo), and if dp is along the tube, we have VVcT . dv = VVtrr . TdwTdp = dyodn because dp || do) 11 NVrs. If the tubes form closed rings and if di/ is the directed element of a surface bounded by a ring, we find (compare (x.)) or again where Q is the solid angle subtended at the extremity of p by the closed ring of strength d^i, because Sd,/'VT(p'-p)-i = Sd/U(yo'-yo) . T(/-p)-2= -dQ. (See Chap. VII., Ex. 22, p. 86.) Hence at any point outside the vortex rings, i.e. at a point at which p does not equal p, we have V^ = ^- V \iUn, CT = V (P + ^ fodTi + i^) (XIII.) This well-known transformation is due to the fact that under the supposed conditions a certain quaternion is reduced to zero by the operation of V. Art. 132. By means of the transformations pS(V)CT = pSVCT-trr, yoV(V)!:T = pVVc7-2cT, pSp(V)CT = pSpVt7 4-VpnT, pVpV(V)cT = pVpVVt^-3Vptrr, ...(i.) which may be verified without difficulty, we obtain the trans- formations, = i [ p VVrrr . di' - ^ L Vdi/trr ; j Vpticr . dt'= — LSpVto . dv-h LSpdi^t^r = ^[pVpVVc7-iLVpVd,.S7 (11.) Another transformation, likewise depending on the invariantal properties of V, is Swtu . dv = I Spa)di/trr — (Spw W -f Spw V'CT)dv ; . . .(iii.) 236 THE OPERATOE V. [chap. xvi. and by introducing p and V into any relation it is generally possible to find a transformation analogous to these. Ex. 1. The momentum and the moment of momentum with respect to the origin of vectors p of a portion of a continuous medium of density c, may be thrown into the forms A = jco-dv = jpS V {car) dv - jcpSdvo- = \ Jp W {ca) . dv - 1 Jc/oVdvcr, )u = JcV/xrdv = - f pSpV(co-)dy + jcpS/adi/o- = ^ JpV/)VV(co-)dv - \\cpYpYAv(T ; and the kinetic energy of the portion may be represented by T= j|cTo-2dv =-\ jcS/ocrdi/a- + jc(S/)o-SVo- + S/oo-VVo-)di; + \ jSpo-Vccrdv. (a) For an incompressible substance of uniform density, if 2€ = Wo-, A = c|o-dv= - cJ/)Sdvcr=cJ/0€d2; - ^cJ/aVdvo", /x = c^YfxrdiV = - 2c J/)S/)€dv + c j/)S/)d vcr = § c J pYped^; - ^ c J/oV/aYdj/o-, T= I c JTo-Mv = - ^ c JS/Do-dvo- + 2c jS/ocredv. Ex. 2. In the notation of Art. 131, the kinetic energy may be expressed by T= -^{c(S7;o-dv + (P+^)So-dv)+i|c(P+i2)dv-i|S7?V(ca-).dv ; and for an incompressible substance of uniform density, T= - ^cJ(S77o-di/ + i?So-dv) - c jSTyedv, and the volume integral is (vii) Equations of motion of a deforinable system. Art. 133. For any system of particles the equations (compare Arts. 119 and 120, p. 194) M.J)tCT = X, DivTf.dm = ^ (I.) are independent of the mutual reactions of the particles com- posing the system, M being the total mass, a- the velocity of the centre of the mass, r the vector from the centre of mass to the particle dm, X the resultant force and jm the resultant couple referred to the centre of mass. Suppose the system of particles to compose a definite portion of a distribution of matter, and let each particle dm be acted on by a force ^dm and a couple ^ydm due to external causes. In addition the portion of matter is subject to the interaction between it and the rest of the matter. The forces of the interaction on the portion may be supposed to be the resultant of a number of forces ^dv acting at each point of the boundary ART. 133.] KINETICS OF CONTINUOUS MEDIUM. 237 of the portion, and $di/ is a linear function of the tensor of dp — the vector element of the surface. Moreover if c is the density, we have dm = cdv, where dv is an element of the volume. The equation (i.) therefore may be replaced by D,o-. [cdv=fcfd?;H-f$di/; (ii.) and D, . [Vtt . cdi;= fc(;?+ VT^)dt;+ [Vr^dv; (m.) and the volume integrals are taken throughout the selected portion while the surface integrals are taken over its boundary. When we take the portion of matter to be small, the volume integrals in (li.) are ultimately of the third order of small quantities and the surface integral is of the second order. Provided therefore D^o- is not excessively large for very small portions and provided ^du is a continuous function of the vector- element of surface dt/, the surface integral must vanish independently of the volume integrals when the dimensions of the portion are greatly reduced ; and if the portion is taken to be a tetrahedron whose vector faces are proportional to a, /3, y and S, we see that the function #di/ at any point must satisfy the condition $(a + ^ + y) = *a + $^ + $7 (iv.) for all vectors a, /3 and y, because we have for the evanescent tetrahedron $a + $^ + $y + *^ = 0, where a + /3 + y + ^ = 0. Thus $ is a linear and vector function. We may therefore apply the integration theorem of Art. 125, Ex. 2, and replace |$di/ in (ii.) by the volume integral I V . dv, in which V operates on $ in situ. Thus we have Dt(T.{cdv={(c^-\-^V).dv', (V.) and when we reduce the portion, we find in the limit D,o- = ^+c-i.#V, (VI.) where Dtor is the acceleration of the centre of mass of a small portion of the matter. Applying the same principles of continuity and of dimensions to (ill.), and taking the portion of matter to be a small parallele- piped whose edges are parallel to a, ^ and y, we find '' -^;ySa/3y + Va#V^y + V/3c&Vya + Vy$a/3 = 0; or simply (Art. 67, Ex. 7, p. 97) C>7-f-2e = 0, (VII.) 238 . THE OPERATOR V. [chap. xvi. where e is the spin-vector of $, as we see more easily by puttino^ i, j and k for a, /8 and y. Provided there is no voluminal distribution of couple, the function $ is self -conjugate. The equation of continuity is c = SV(ccr) or — cSp^p^p^ = C, (viii.) according as we use Euler's or Lagrange's method (Art. 129), and by Art. 128 (vi.) or (xiv.) we may replace (vi.) by W-^+'^-il ■ ^""^^f^ + i ■ *^>3Px + 4 • *V,,,,). ...(IX.) Ex. 1. Find the equation of motion for a perfect fluid. [The force ^dv on the boundary of a portion of the fluid is —pdv, where p is the pressure, remembering that dv is outwardly directed. Hence the equation is D«o- = ^ — c~^Vp.] Ex. 2. Integrating along a stream line, show that ^T(r2 + jS(^ + c-i*V)d/3 is constant for an element of the matter, and find the integral in the case of a fluid acted on by conservative forces. Ex. 3. When the forces acting on a perfect fluid are conservative, the circulation in any circuit moving with the fluid remains unchanged provided the density is a function of the pressure. [We have Dt(T= - V(P+ jc-^;?). See Art. 130, Ex. 1. An independent proof is easily obtained by Lagrange's method, which gives and if this vanishes for all closed circuits VVD«o- = 0.] Ex. 4. If F= - jSo-dp, show that Art. 134. To determine the nature of the stress-function $ for a viscous fluid, we assume as usual that the stress consists of a hydrostatic pressure and of a part linear in the rate of dis- tortion of the fluid, and that the stress-function is coaxial with the strain-function. In the notation of Art. 124, the strain- function is i( + (p'), and the general linear function coaxial with this function and linear in its coefficients is of the form n((f) + (l)')+nm", where n and n' are constants and where m ( — — SV(r) is the first invariant of or (f>' or J (0 H- otherwise if n varies, it must undergo operation by the V which replaces a. In like manner for an isotropic elastic solid, if is the displacement, ^a= -n(SaV .e + VSae)-naS\-e, (v.> assuming that the stress function is coaxial with the strain- function and linear in its constituents. The equation of motion becomes Dt^e = ^-c-''nV^e-c-\n-\-n')V .SVe (vi.) Art. 135. The rate of change of kinetic energy of any finite portion of the matter is Di^cTo-^ . dv = D JiTo-^ . dm ■ • =-[Sa-D,a-.dm=-fScro(c^+^V)dr, (l> and in the last integral V operates on $ but not on a- as indicated by the suffix. Because S(r$V = So-^^V -j- So^^ V, where V operates on the unsuffixed symbols, we may integrate by parts, and we find Dt hcTa-^ .dv=- {cSa-i • 2, (f>2 ^^^ ^3 ^^^ ^^^^ linear vector functions. (b) The work done in altering v and to to v + dv and (o + dco is dTf=-SAdv-S/xd(o; and if the dynamical system is conservative, so that d TT is the differential of a function W of v and w, the functions (f)^ and c^g must be self -con jugate and 2 must be the conjugate of cf>2- (c) In the case of a perfect fluid, the velocity generated in this way must be irrotatiorial, and assuming that o-, as well as A and fi, is a linear function of V and w, we must have (r = V(Sv6' + S(of), where 9 and ^ are vector functions of the vector p. (d) In the case of a solid moving in an infinite liquid of uniform density, or of a solid containing a cavity filled with liquid, the functions 6 and'^ must satisfy V26' = 0, VY=0 throughout the liquid. And at the surface of the solid in contact with the liquid ' ■' S(v + V(op)dv=SdvV.(Sv(9+S(of), so that 6 and f must satisfy the surface conditions dv=SdvV.(9, Vpdj/=SdvV.^. J.Q. Q 242 THE OPERATOR V. [chap. xvi. (e) In this case we may replace the expressions for A and /x by X = Mv + cldv{Svd + ^(oO, /x = <^(o + cjV/3dv(Sv6' + Sa>f) ; and by the aid of the conditions which 6 and f satisfy, it may be shown that jdi/Sa^= JSW . eSaO' . dv, \ypdvSa^= JSVV . ^aC . dv, jVpdvSa6'= JSVV . ^ad' . dv, jdvSaf= JSVV . (9Saf' . dv,^ so that the conditions (b) are satisfied. Also the functions <^j, <^2 ^^^ <^3 depend on the nature of the solid and on the density of the liquid, and they are invariably related to the solid. (/) If the solid is acted on by an applied wrench (17, ^) referred to its centre of mass, the equations of motion, analogous to Euler's equation for a rigid body, are the second equation being obtained by expressing that the rate of change of the moment of momentum (fi + Y-yX) with respect to a fixed point is equal to the moment of the applied forces (17 + Vy^) with respect to that point. (g) When there are no applied forces obtain and interpret the integrals T((/)j V + ^2^) = const., S {(f)iv + (/>2w)( + So)(f)2(i) = const. (h) When the linear momentum is constantly zero, and the angular velocity is that of a certain solid moving round a fixed point under the action of the couple tj. (i) For a steady motion of translation under no forces Yv(f)^v = ; and in general for steady motion when w does not vanish v=- (^i"^ (<^2 + ^) <^j ^to [<^3 - {cf)2 + ^) <^r^ ( = 0, where ^ is a scalar. From this it follows that the axis of the screws of steady motion are parallel to edges of a sextic cone, and in general to each edge of the cone corresponds a single screw. Art. 137. In terms of the displacement 6, the equation for an elastic solid is (compare Art. 134 (vi.)) D,20 = ^+c-i$V, (I.) the velocity or being and $ being a self -conjugate function because there is no voluminal distribution of couple. The displacement is a function of the time and the position vector, and when the strain is small we may neglect the term — S^V , in Di^O. We replace, in fact, Dt^O by the second derived of regarded as a function of t alone, that is by 0. Observe that now V is commutative in order of operation with the result' of differentiating with respect to the time. ART. 137.] ELASTIC SOLID. 243 By Art. 135, the rate at which the forces work in storing and dissipating energy is the integral W -{se^^V.dv (II.) taken throughout the body. By Hooke's law, stress is a linear function of strain. If the strain is multiplied by n, the function $ is likewise multiplied by n. Suppose the strain to be gradually increased from zero so that at any stage the strain is n times the final amount where n is positive and less than unity. In this case (ii.) becomes W= —nn\SO^QV . dv; and integrating between the limits and 1, the total work done in producing the strain in this particular way is seen to be w. ^{se^^v.dv (III.) If the work done is a function of the strain and not of the manner in which it has been produced, the function W is the energy function — a quadratic function of the strain, and the work done in altering the strain in any arbitrary manner is the difference of the values of the energy function corresponding to the final and the initial state. When the energy function exists we see on comparison of (ii.) and (ill.) that in general for any two sets of strain answering to the displacements 6^ and 0^, we have |So-ij) + (jjji) = 0, {iiii) = {jjjj)- For rotation through Itt the functions of u independent of k or involving k twice must reduce to constants. We find in addition to the conditions satisfied for rotation through one right angle that {iiij)=(^jjji)=zQ^ (uu) = (ujj)-\-2{ijij). Expressing that (kaaa), (kaa^) are functions of cos3^t and sinBw, we get -{kiii) = {kjij) = {kijj), -\kjjj) = {kiji) = {kjii). For rotation through an arbitrary angle the symbols linear in k must vanish and the conditions for v='^ir must hold.] Ex. 5. When the energy function exists prove the existence of a self -conjugate function (^ for which the relation 0(a, /3, 7)-e(A a, y) = V. c^Va^. y is identically true. {a) The axes of ^, when determinate, form a natural system of lines of reference, and where a plane of symmetry exists, it is normal to an axis. [The function on the left is obviously a linear function of Va^. Operating by S8 we have (8a/?y)-(8^ay) = (/5y5a)-(^8ya)= -SVy8 we have also (kaaa) = (kin) cos% + S(kuj) cos% sin w + 3 (kijj) cos u sin^w 4- {kjjj) sin^ u because the letters in a symbol involving ^,y and k are completely permutable for this special set of axes. Hence it follows that a plane z = which is a plane of symmetry of the quartic (pppp) and of the quadric Sp^p is a plane of elastic symmetry. The coefficients of the powers of cosu and sin i^ in (kkka) and in (kaaa) must then vanish, and by the special laws of interchange every coefficient of odd order in k vanishes. Suppose now that the plane &jp = or z^ = is a plane of symmetry. The coefficients of the powers of z must be functions of cos u alone. Thus (pppp) =z^a-\- Azhh cos u + Qz'^r'^ (c cos 2u + c') + 4zr^ (d cos 3w + d' cos u) + r^(e cos 4,u + e' cos 2u + e") suppose. If the plane u = v is also a plane of symmetry, this function must be independent of the sign when we put u — v±w, where to is arbitrary. Hence b sin v = c sin 2v = d sin 3v = d' sin v = e sin 4y = e' sin 2v = 0, and unless the quartic is a surface of revolution, the only admissible values of V are ^tt, ^tt and Jtt. Hence planes of elastic symmetry must intersect at angles of 90°, 60° or 45° if every plane through their intersection is not a plane of symmetry. Of course in the second and third cases, the quadric Spp is of revolution. There is no difficulty in writing down the elastic constants for each case. Suppose two roots of (f> to be equal so that there are indeterminate axes in the plane of i and j, and that it is required to find a natural system of lines of reference. We may equate to zero the derived with respect to u of the first of the coefficients (kkka), (kkaa) + 2(kaka), (kaaa), (aaaa) which does not vanish. Determining u from such an equation we take i cos u+j sin u and 7 cos w - 2 sin ^^ along with k as the natural axes of refer- ence. The case in which cf) reduces to a constant will be considered in the next example.] Ex. 6. When the energy function exists, iV2. e(p, p, p) = m(p, z, ^) + 22e(^, i, p) = ,p, is a self -con jugate vector function invariantally related to the elastic structure. [The function is invariantal because V^ is an invariant operator inde- pendent of any particular choice of i, j and k. If a plane of symmetry exists, it is a principal plane of this function, because if k is normal to a plane of symmetry, Sicf>2k and Sjcfi^k both vanish, being of odd order in k. Therefore k is an axis of ^2^ ^^^ 4*2 ^^^ 4* of the last example have a common axis. In terms of the axes ^, ; and k of the last example, it is easy to see that - S i4>2i = 32 (liaa) -|- 2 (eg + eg), - Sicfij = 32 (ijaa), where a stands for i, j and k in the summation. The axes of this function may be used as natural axes of reference when the function <^ of the last example reduces to a constant e. In this case for arbitrary axes, i, j and k are completely permutable in any symbol in which they all occur, and (jkjk)=e + (jjkk), etc.] ART. 188.] ELASTIC SYMMETRY. 247 Art. 138. In the notation of the last Article, the equation of vibrations of an elastic solid, not acted on by voluminal forces, is c9=e(v, v,0), (I.) where, as we have said, is the second partial derived, with respect to the time, of 6, which is a function of t and p. Consider the propagation of a plane wave. If the vector v represents in magnitude and direction the wave-velocity, the equation of a wave-front is u = t-S^, (II.) u for this represents a plane moving at right angles to itself with velocity v. Over a wave-front, the displacement from the mean position is, by definition, the same at every point at any given time. In other words is a function of u and of t. Hence V OIL V OU and generally if /V is a homogeneous function of V of order n, /v»=/©S • <■■■•) In particular (i.) becomes for plane wave motion If the wave is of permanent type, involves t only as involved in u, and if in addition the vibration is harmonic and of frequency p. e=^,= -p^e ^v.) In this case (iv.) becomes e(Uy, Ui;, e)=^ceTv' (vi.) This shows that for a plane wave propagated in the direction Uv, the vibration is parallel to an axis of the linear vector function* G(Ui;, Uu, a), and that the velocity is the square root of the quotient of the corresponding latent root by the density. The solid admits of three plane-polarised waves propagated in the same direction with different velocities. The wave-velocity surface is determined by the equation s{K? \' «)— }{e& \' ^)-W{®G' \' r)-''r}=0- (-'•) which is equivalent to the latent cubic of the function '* e(Ui;, Ui/, a). *The function ©(Uu, Uu, a) is not one of the functions 9 (a, i, i) of the last Article. The second and third vectors may be interchanged in these expressions, not the first and second. 248 THE OPEBATOK V. [chap. xvi. When the energy function exists, the linear function e(Uv,Vv,a) is self-conjugate because we have by the law of interchanges (Art. 137 (VIII.)), S^e(Uu, Vv, a) = Sae(Uy, Vv, /3). In this case the vibrations 0^, 0^, 6^ for any direction of wave propagation are mutually rectangular. Moreover, since the function W is essentially positive, the latent roots of the function O are positive as well as real, and there are therefore three real wave-velocities UfTuj, UfTug and JJvTv^ in any direction. When a linear function has indeterminate axes, the \[r function oi (p — g vanishes where g is the repeated root (Art. 66). The condition for indeterminate directions of vibration is therefore where a and (3 arbitrary vectors. This equation admits of a finite number of solutions (u), which correspond to Hamilton's internal conical refraction. These vectors terminate at double points on the w^ave-velocity surface. The index-surface (MacCullagh) or the surface of wave- sloiuness (Hamilton) is the inverse S{e(M,M,a)-Ca}{e(M,M,/5)-c^}{e(M,M,y)-cy} = 0...(iX.) of the wave- velocity surface (vii.), the vector jm being equal to -v-\ The wave-surface, or the surface of ray-velocity, is the envelope of the plane ^ S^ = l or S/xp=-l, (X.) subject to the condition (vii.) or (ix.). That is, the wave-surface is the reciprocal of the index surface with respect to the unit sphere p"-\-l = 0; or it is the envelope of plane wave-fronts in unit time after passing through the origin ; or it is the wave of the vibration propagated from the origin in unit time; or the vectors p which satisfy its equation represent in magnitude and direction the ray- velocities. When the energy function exists a simple and remarkable expression may be found for the ray-velocity p in terms of /m and 6. The wave-surface may be expressed by elimination between e(^,iJL,e)=ce, de(^,^,e)=cde, Sfjip+i=o, Spd^=o....(xi.) The second equation is in full e(d/x, fx, ^)+e(M, d^, e)+Q(^, p., d^)=cde; ART. 139.] PLANE WAVES IN ELASTIC SOLID. 249 and operating on this by S0 and attending to the law of inter- changes (Art. 137 (viii.)), 2SdfjLe{0, e, M)+sd^e(^, m, o)=csede; and by (xi.) this reduces to ^ SdfxeiO, e, /ul)=--0. Thus every d^ is perpendicular to 0(6, 6, /x) and also to p, so that 9(^, 0, iJL)=^xp where a:; is a scalar. Operating by S/x we find -x = ^^xe{e, 6, ^) = Sae(/>t, m, 0) = ce\ and therefore t e(ue,ve, ^)=cp (xn.) f Further, if we operate on this by Syu and on the first of (xi.) by SO we recover the relation Sp/m + 1 = 0; so that all the relations connecting VO, /m and p are comprised in the two yf^ \fi I ion ^ I e(fji, M, 0) = cO, e(ue, ije,^)=cp (xm.) I (viii) Electro-magnetic Theory. Art. 139. The fundamental circuital laws of the electro- magnetic field are"^ (I.) the circulation ( — I Srjdp) of the magnetic force (tj) in any closed circuit is equal to the flux f ISydi/j of the electric current (y) through the circuit divided by the velocity of light (u) in free space ; (11.) the circulation, with changed sign, (+ Sed/a) of the electric force (e) in any closed circuit is equal to the flux )y^di/j of the magnetic current (y^) through the circuit i-ik divided by u. These laws are symbolized by the relations fs,dp = ySyd., (Sedp=-ySyA; (l-) and because it is implied that the fluxes of the vectors y and y^ through the circuit are independent of any particular surface bounded by the circuit (Art. 130 (xv.)), SVy = 0, SVy^ = (11.) ^We-cf-nnot delay to explain the units employed in this article. Full explana- tion will be found in the article by H. A. Lorentz on Maxwell's Electromagnetische Theorie in Bd. Vg, pp. 63-144, of the Encyhlopddie der mathematischen Wissen- schaften. These units are but slightly modified from Heaviside's rational units. Much use has been made of Lorentz's article and of Heaviside's work in the preparation of the account of the theory given in the text. \ 250 THE OPERATOR V. [chap. xvi. We proceed to define more particularly wliat is meant by the electric and magnetic current fluxes and by the electric and magnetic forces in these laws. The electric current flux through the circuit consists in general of three parts, the flux ( — I Sidv) due to the conduction current (i), the rate of change ( — D^IS^di/) of the electric displacement (S) through the circuit, and the flux (— leSudr) due to the convection current (ev) where e is the density of electrification"^ carried through the circuit with velocity v. In like manner the magnetic current is due to the rate of change ( — D J S/3di/) of the magnetic induction (^) through the circuit, to a conduction current (i^) postulated by Heaviside, but probably non-existent, and to a convection current (e^v^) where e^ is the density of magnetification carried through the circuit with velocity v^. On the whole the integral fluxes are -[Syd»/ = -Djs^d,.- [Stdiz-jeSiydi., -|Sy,d,.= -BSs/3dv-^SiAv-\e^SvAi^ (ni.) In the rate of change of the displacement through the circuit we must take account of the motion of the circuit which we suppose to move with the velocity o; varying from point to point. We have therefore by Art. 129 (iii.), p. 229. | \Sydu = ^S(S + i + ev)di;, jSy,di; = js(^ + ^, + e,Odi/, ...(iv.) where S = S- YVYcrS - crSVS, $ = $- V VVo-^ - crS V/5 (v.) Converting the line integrals in (i.) into surface integrals and expressing that the relations hold for every possible small circuit di/, we arrive at the diflerential equations of circuitation YVr) = -(S + i + ev), VVe=--(/3 + i, + e,0 (vi.) We have not yet explained the meaning of the vectors e and rj. The total electric and magnetic forces at a point consist of impressed forces (e^ and rj;) together with e and rj. Thus if et and rit are the total forces, ei = e + 6^, rit = ri-\-rii\ (VH.) * This is not electrification of the medium. It is due to charges of electricity carried by moving particles, for example. ART. 140.] ELECTRO-MAGNETIC EQUATIONS. 251 and Lorentz further divides the impressed electric force into a part €ic co-operative with e in producing the conduction current and a part ad co-operative with e in producing the displacement. We shall write where the suffix i calls to mind that the force is impressed, c that it relates to conduction current, d to displacement and h to magnetic induction (/3). Expressing that the conduction currents are produced by the forces enumerated, we have f = $(e + e,e), h = ^X^ + mcy, (IX.) and by Ohm's law in the case of isotropic media $ is a scalar — the conductivity — and for anisotropic media is a linear vector function. Similarly we suppose the postulated function $^ corresponding to the postulated magnetic conduction current y^ to be a linear vector function. In like manner, expressing that the displacement (S) and the induction (p) are due to the forces mentioned, S = (p(€-^€ia), ^ = ci)Xrj-\-mby (X.) The phenomena of hysteresis shows that (p and ^^ are not always linear functions of the forces, but we shall only consider the important case in which they are linear functions. For isotropic media, is the (scalar) dielectric constant and cp^ is the magnetic permeability. Some little care is necessary in differentiating these expres- sions when the medium is in motion. Owing to the motion

; -h S>; VVe = S We?;. The electric and magnetic forces evoke mechanical forces, ^ per unit volume, and the stress ^gdv across the directed element 252 THE OPERATOE V. [chap. xvi. dv. If the element moves with velocity cr the activity of these forces on the element is A^dv= -(So-^+S(T$,(V))dt', (III.) the last term, in which (V) operates on $s and on o- in situ, being equal to the surface integral — |S(rsdj/ over the element. The total activity Adv = (A^-i-A^)dv (iv.) is equal to the rate of transfer of energy to the element. The term /= -Sl^-\-Si^^^-\ (v.) is by Joule's law the rate of waste of energy per unit volume owing to the conversion of energy into heat by the resistance. The terms in this expression for the Joulian waste are analogous to the dissipation function of a viscous fluid. The term eSev + e^Sriv^ relates to the convection currents. The work done in increasing the electric displacement by the amount d^ is _Se,.^d^= -s(e-{-€ui)ds= -s^-ms, :....(V1.) where eta is the total electric force operative in producing the displacement. (Compare (viii.) and (x.) of the last article.) Experiments on dielectrics show that an energy function exists, or in other words the work done is the differential of the function • F= - JS(50-i(5= -iS€tciS=-h^eta€td, (VII.) which represents the energy stored in unit volume of the medium and due to the electric force. From the existence of this energy function we infer that is self-conjugate. A similar result holds good for the magnetic induction, and the energy due to this cause is W = - iS/3cp^ -^/3^- i^mlS = - \^m.m ( vm.) The energy stored in unit of volume due to electric and mag- netic forces is the sum of W and W,. - When the medium is at rest the total activity is (ii.) 1 Adv^{J+W+W^-e^.€v-€)$>riv-U^^Neri)dv, (iX.) because in this case S and /3 must be replaced by S and /3. We have accounted for every term except the last. This by a pro- cess of exclusion represents the rate of radiation of energy from the small volume. It may be expressed as a surface integral, J -U^V\€ri.dv= -u\sdvYeriy (X.) and this is the total outward flux of the vector uVerj, the vector area dv being outwardly directed as usual. This vector is the 1 ART. 140.] ENERGY IN ELECTROMAGNETIC FIELD. 253 Poynfcing vector —discovered independently by Professor Poynting and Mr. Oliver Heaviside. It represents in magnitude and ! direction the flux of radiated energy. Granting that the same vector represents the energy flux when the medium is in motion, and there seems to be no adequate reason for doubt, the total activity is Adv = (J-eSev-e,Sr]v,-uSVYer])dv-\-Dt(Wdv-\- W^dv), ...(xi.) the last term being the rate of change of the energy stored in the element dv and due to electric and magnetic causes. Equating this to the sum (A^-\-A.2)dv already obtained, we have BtiWdv-hW^dv) = -(S^0-M+S^0,-i/3).dv-(So-^+S(r*,(V))dv. ....(XII.) By Art. 129 (in.), p. 229, we find D,( Wdv) = (DtW- WSVct) . d^' = -ST>tS.(l>-^Sdv-iSS.J)t(p-KS.dv-WSVa-.dv, where T>t~^ is the result of operating by D« on the function -\ Further, by (i v.) of the same article, S = J)tS- YVYaS = D,^ - 5S Vo- + SSV. a\ and therefore -S^V"'<5=-SD,^.0-M-2TrSV(r-S^V'Sc7>-i^. Hence equation (xii.) becomes + JS/3.D,0,-i./34-iS/3^,-i/3SV(T- S^V'So->,-')5. ...(XIII.) The first term on the right may be written iS^.d,0-i.<5-iSo-VS^o0"%. where V operates on ^~^ alone since we have generally D, = d,-So-V, where d^ refers to the rate of change at a point fixed in space. Consider now the term ^SS . dt,-W. (XVI.) because p^ = p^ if SXp^ = SXp^ for all vectors X ; and again -cl>-^SSSfj,-^,-^^S8iui, ...(xvii.) since ^^ = 02 i^' Si/^^^ = St/02M for all vectors ja and v. Replace p. in this expression by V operating in situ on the various vectors, and we find because +V. VV0r^^.^-<^r^^SV/3~ JVS^o0-^^o>(xviii.) and Y ,YVct>-''S . S = VSSo-''S-SSoV . cp'-'S ART. 141.] STRESS IN ELECTRO-MAGNETIC FIELD. 255 Thus we find for the mechanical force ^, -V. VV0,-i^./3 + ^,-i/3SV^ (XIX.) The stress across any small area is determined by (xvii.). In general the terms in 9 and 9^ are small, and we shall neglect them. The stress across any small area due to the electric displace- ment is when we neglect 9, and ii JUL is parallel to (p~'^6, we have while if jui is perpendicular to 0"^, Thus the stress consists of a tension along the lines JJip'^S and an equal pressure at right angles to these lines, numerically equal to the electric energy per unit volume. Similar results hold for the magnetic stress. Art. 141. When the circuit is at rest, and when there is no convection current the equations of circuitation become S-]-i = uYVr], $= -uVVe, (I.) when we put i, = 0. When moreover the medium is at rest we have (Art. 139 (x.) and (ix.)) ^ = 0(€ + e^^), $ = ^X^-^mb), i = ^(€ + eic); (II.) and from these we obtain the equation i>C6+€id)+^(€ + €ic) + uWVr^YV6 + uYV^i^=0 ....(III.) which is explicit in the vector e. Having determined e from this equation, the impressed forces being known, we obtain S, i and /§ by direct operations on e. The vectors ^ and VV;; are also expressible by direct operations in terms of e. There are two principal types of this equation. For a dielectric non-conductor ^ is zero, and the propagation of the disturbance is by waves. For a conductor incapable of storing electric energy, (p is zero and the propagation is by diffusion. When there are no applied forces the equations (i.) and (ii.) may be^ replaced by ^n + ^^n = u'^^rin, hnCp.rjn + ^,r]n= - uYVcn ( VI.) and the boundary conditions. The scalars b must in general be determined by an equation arising from the boundary conditions. The scalars a depend on the initial state of the disturbance. The particular solutions €„e*«*, ')7ne*«*, are the normal solutions, and for any two normal solutions we have u&We^rf^ + b^Se-^cfie^ + hjSrj^^cfi^rji + Sci^Cg + ^'72*^/'^! = 0, (vn.) iDecause SWeii72 = 87^3 VV'e/ — Se^V V^^g'- Integrating throughout the medium .and converting a volume integral into a surface integral we find, u\SY€i7]r,dv + 62 jS€i<^€2dv + b^\^7)2(t>,rj^dv + ^Se^^e^dv + ^Sri.>^,7]^dv = ; w JS VcgT^idv + 61 jS€2c/)€idv + b2lS7j-^cf)^7j2dv + JSe2€idv + ^Srji^^-q^dv = 0, (viii.) the second equation following by interchange of suffixes from the first. If in either of these equations we replace b^ and 62 by conjugate complex ■expressions b'±xj -lb" , an d at the same time replace e^ and €3 by €'± V- le" .and 7]^ and 172 by rj' ± >J — Irj", the real part of the equations is u\ S (Ve'r]' + V€"r;") d V + 6' j(Se'<^e' + Se"e" + Srj'cfi^r)' + Sr;"<^,7y") dv + j(S€'e' + S€"*e" + S-q'^,7j' + Sr]"^^7j") dv = 0, ... (ix:) remembering in the reduction of this expression that (f> and ^ are self- conjugate (Art. 140 (vii.)). The surface integral is the total inward flux of energy across the boundary due to the disturbances e', 97' and e", rj". If no energy is communicated from outside the boundary, this is zero or negative — zero if no energy from inside escapes, and otherwise negative. The remaining integrals are all negative, the coefficient of b' being minus double the energy stored by the two distributions separately and the remaining integral being minus the energy wasted by conductive friction. Hence in any case b' cannot be positive. If there is no energy radiated and none dissipated, b' must be zero or else e, e", t]' and 17" must vanish so that there is no disturbance. On the whole then, the real parts of the scalars b are zero or negative when the medium receives no external energy ; when in addition there is no dissipation and no radiation of energy across the boundary the real parts are zero, and in this case there are permanent oscillations within the medium, the scalars a being determined once for all by the initial conditions. Art. 142. We shall now give a sketch of the theory of the propagation of light in a crystalline medium adopting Clerk Maxwell's hypothesis. The medium being supposed non-con- ducting the functions $ and $^ disappear, and the equations of a free vibration become ^o= A=^//o=-'^'^^^0' (!•) when (j) and 0^ are two self- conjugate functions which are constant if the properties of the medium are the same for the same directions at all points."* * The suffixes are employed in these equations as we shall have more to deal with the vectors e and rj defined in (11.). ART. 142.] MAXWELL'S THEORY OF LIGHT. 257 Assuming for a plane wave (Art. 138, p. 247) that eo = €sin7iu — S^j, r}Q = r] sin n(t — S-\ (n.) where v is the wave- velocity, we find on substitution in (i.) S = (l)6 = uYv-\ ^ = 0,>/=-uVtf-ie (III.) From these we obtain among other relations — W = S€S = Se; + u-M»;), d/3 = 0,d;7=-uV(di;-^€-|-u-ide); (vn.) operating by Se on the first, or by S;; on the second, and attending to (ill.), we find I Sdv-Wefj = 0, (viii.) because by (iv.) ScdS and S^drj are each equal to — ^dw. As this holds for all values of dv we must have p parallel to Yet], and by (iv.) we find for the ray-velocity uY en I p='^. (IX-) w and this, it should be noticed, is parallel to the Poynting Flux (Art. 140). Again it is easy to deduce from (iii.) and (iv.) the expression for the wave- velocity (v) , Y8S uw , , v"^=^^, or u = ^7-3^ (x.) uw YpS We have now enumerated six vectors depending on the propagation of the wave which are connected by the relations, y^'^YS^, €=-YI3p, v = -^'pS; uw ^ u ^^ u ^ .(XI.) J.Q. 258 THE OPEKATOR V. [chap. xvi. and these vectors when drawn from a common origin pierce a concentric sphere in a pair of supplemental triangles. When some one of the four vectors /3, S, e and rj is given, all the vectors can in general be determined subject to a choice of sign. If € is given, we have S = € = 0, Sr]6(l),€ in w= —St](p^ri. Hence ^, p and v~^ can be found without ambiguity when the sign is selected. The case of exception is when e (or rj) is a solution of the equation Y(/)a,-'^l!v, i8 = ^VU„.^-i5, (XIV.) and from these, when we eliminate (3 and S in turn, and introduce new linear functions ^y have the same latent roots appears from the fact that their latent cubics are equivalent to the equation in Tv^ obtained by eliminating ^ and S from (xiv.). If Tu'^ is the second root of (py and if S' is the corresponding axis, we have Tu^S^^ - M' = u'SYVvcp - ^S' - M = Tv'^SS'^ " M, and therefore (by (xiv.)), since Tu^ is not generally equal to Tu'^, SScp-^S' = 0, S^0,-i/3' = O (XVI.) But these conditions may be written in the form S^' = 8(5^6 = 8/3^7' = S/3',; = 0, (XVII.) ART. 142.] PLANE POLARISED WAVES. 259 where e, 13' t] , etc., correspond to ^'. Thus <5' is perpendicular to e and v, and therefore parallel to /9 by (xi.), and /3' is parallel to (5. In fact we have TO'=+U^, U^^=TXJ(5, (XVIII.) because Uu = UV^^S = UV^'^'. Since ^ and ^' satisfy the relations (compare (xvi.)) S(50-M' = O, S%-i^' = 0, S^Uu = 0, S^'Uu=0, ...(xix.) we easily find on putting Uu = U V(5^' = V<5(5' : TV^«5' in (xv.) that '?^2S^>,-i^'S^0-i^ = Ty2TVdT2, (XX.) and that u(S^>,-M^S^0-^^)^ ,_ u( S^0,-MS^V-^^)^ . . """ W? ' "" ~ Y^ ' ^ -^ This result leads to a simple construction. Let the quadrics SnT0-%= -1 and 8^0,-1^7= -1 (xxii.) be constructed. Then by (xix.) ^ and ^' are parallel to the pair of common conjugate radii in the central plane at right angles to the direction of the wave-velocity. Let vs and cr, be respec- tively the vector radii of the first and second quadrics parallel to (5, and let xs' and trr/ be those parallel to o', then we have u , u , . \ "=-Y^"" = -Y^- <^^™-> and from this construction everything relating to the wave can be determined. For the first set of signs in (xviii.) we have S = Z^Js/w, P = V5'Jw, S' = TS'sfw', j8' = — VS^sfw', \ €=(l)-'^V5jw, n=-'^TS^slw, e=^-'^T^'Jw\ ,;'=-0^-lsT,V^,i;U/o, f(Y:p-^ = U-WJJp.^e\ (XXV.) and as in (xv.) we may write, * Note that 0~^aT has the same direction as the central perpendicular on the tangent plane to the quadric St3'0~^trr= - 1 at the extremity of tD" and that its length is the reciprocal of that of the perpendicular. 260 THE OPERATOR V. [chap. xvi. and if we take e and e' to be the two axes of 0p corresponding to the two finite latent roots Tp-^ and Tp''-^ of the function, we find as before for it appears that Ue"= +U;7, U>?''=+Ue. We can write down results analogous to (xxiii.) and (xxiv.) for the various vectors related to the waves whose ray -velocity- is along a fixed direction JJp. We now return to equation (xii.), which we may write in the form I rj=Ycf>e<{>^€J(~r-y^ ,.\ (XXVII.) where m, is the third invariant of <^, and where * \ • ^ w==-Secf>e, w^^-Secfy^e, io'=-= -Secfycfi-'^cfie, (xxviii.) because we have Expressing p and v~^ in terms of e, by (xi.), _ uY€Y(f>€(f),€ _ u{w,<^€'-wc\>,e) From these equations, on attending to (xxviii.), ^p(w,cf>-wcf>,r^p^o, sp<^rV=-^; (^^^O Sv-\w'-wcl)^)v~^ ;.... (xxxiii.) mX''^w' -iv^~'^)p = u\io^-w4)~^(j))\r^; (xxxiv.) the last relations, which alone are likely to give trouble, being derived from (xxxii.) by operating with {vf — W(^~'^<^){w^~'^(l) — w) on both sides, remember- ing that in this the factors are commutative. From (xxx.) and (xxxi.) the equations of the wave-velocity surface and of the. ray-velocity surface may be written down, and equations (xxxiii.) and (xxxiv.) are suitable for investigating the cases of indeterminations which correspond to external and internal conical refraction. Suppose, for example, that w'=h'^w, where h^ is a latent root of the function 4>r^ and that /3 (not now the magnetic induction) is the corresponding axis while /?' is the axis of the conjugate function <^~^4^ corresponding to the same .(xxix.) * It should be noticed that w' has not here its recent meaning. ART. 142.] ELECTRO-MAGNETIC WAVE-SURFACE. 261 root. The equation (xxxiii.) fails to give a determinate value of p, and operating on it by S/3', we find S/?'<^v-i=0, (xxxv.) since (fif3' = b^,f3'. Two other equations for v are obtained by putting w' = b^w in (xxxi.), and these are I Sv-\b^cf>-i-cf)-^)-^v-^ = 0, Sv-^,v-^= -mpH-'^ ; (xxxvi.) and from these three equations we find four values of v~\ say ± v{'^ and ± v{'^. Substituting the value v~^ in (xxxiii.) and replacing w, by its value in terms of p by (xxx.), we get mf}P- - (f> corresponding to the latent roots a^ and c^, the equation is equivalent to the pair mXb^-a^)Sa'p + Sa'cf)Vi-^{m^Sp-'^p + ahc^) = OA ^ ' mX62 - c2) Syp + Sy'(f>v{-^ (m,S/)(/),- V + ^ V) = 0. / (xxxviii.) In order to calculate in the most explicit manner the vectors v{-^, etc., we may by Art. 71, p. 100, reduce the functions ^ and (f) to the trinomial forms <^A=-2aSaA, ^A=-Sa2aSaA, <^r^A= -2a'Sa'A, <^-i A = - 2a- VSa'A, where identically A = - SaSa'A = - 2a'SaA. Putting v~^ — ap + f3'q + y'?', equation (xxxv.) becomes ^'=0, while (xxxvi.) reduces to p^{c~^-b~^) = r^b~^-a~^) and p^ + r^=m^bh(,-^, and we finally get foi' the four vectors v-^=^JKsJ^UeL^^^a^l^\ (x^:x.) I u \ a y a^ — c^ c ^ a^-c^J Again, taking p=cuv + f3;i/ + yz, and substituting in (xxxvii.), we find a simple expression m/Xb^-a^)ouv + (b^-c^)yz} - mXaa^p + yc^r){a;^+f+z^) + {ap + yr)2(P = (xL.) for the equation of the conic traced out by the extremity of p. We notice that m, = SafSy^. In order to obtain more explicit forms for the equations of the wave- surface and the wave-velocity surface, we note that the first equation (xxx.) expands into wj^Spyfrp - w^wSp'^p -\- v^Spyfr^p = 0, where yfr and x/r^ are Hamilton's auxiliary functions and where I By the aid of the second equation (xxx.) this becomes f Spylrp^pylr,p + u^^p'^p + u^ = (xli.) In lijj:e manner ^ Sv-^\jr-^v-^8v-^yJrr^v-^ + u-^Sv-^^-iV-^+u-^ = (xLii.) is the equation of the wave- velocity surface, where ^-iYXp,^Yct>--^XrV + Ycl>r^X-y. 1 262 THE OPERATOR V. [chap. xvi. Other forms may be given to the equation of the wave-surface such as m/2SaV2262c2Sa'p2 _ uHi, 2 (62 + c^) Sa>^ + n^ = 0, derived from (xli.), and V Say ^^ m^a22Sa /o2 _ -^^2 ' derived from (xxx.) by the aid of the trinomial expressions for the functions, but in problems treated by quaternions it is frequently preferable to deal directly with vector expressions rather than with the scalar equations of surfaces obtained by eliminating certain quantities from the vector equations. Ex. Show that the wave-surface may be derived from a Fresnel's wave- surface by a pure strain. [Put Vr/p = p and Sp(w^(f> - tv)~'^p = Sp{w^\lr/ (fiyfr/^ - wm)~^p\ also v^w^ — — wSp^^p = wTp'2j etc.] CHAPTER XVII. PROJECTIVE GEOMETRY. Art. 143. There are several interpretations which may be assigned to a quaternion and which we have not yet explained. We now propose to show that a quaternion is capable of repre- senting a definite point loaded with a definite weight or mass, and throughout this chapter we shall speak rather indifferently of quaternions or of points.* In the identity 9 = Sq.{l+^) = 8q.{l+OQ) if OQ = g, (l.) it is manifest that the point Q at the extremity of the vector OQ drawn from an assumed origin is determined when the qua- ternion q is given, and that Sq is also determined. We regard Sq as a weight or a mass concentrated at the point. We shall sometimes use capital letters concurrently with small letters, q = Q.Sq, Q=1+0Q, (ll.) to denote points of unit weight, or unit points, so that Q.tv denotes the point Q weighted with w. Thus SQ= 1, VQ = 0Q. The difference of two unit points is the vector joining them, Q-P = l + OQ-(l-fOP) = OQ-OP-PQ; (ill.) and the origin is the scalar point 0=1. (IV.) A vector represents the point at infinity along its direction, as appears by allowing Sq to diminish indefinitely in (i.) while Yq remains constant, for OQ will then increase indefinitely in length, so that at last Yq represents the point at infinity in its direction. *See Trails. B. I. A., vol. xxxii., and Phil. Trans., vol. 201, pt. viii. I regret that at the time of publication of these papers I was not acquainted with an able memoir by Dr. James Byrnie Shaw {American Journal of Mathematics, vol. xix., pp. 193-216), in which somewhat similar results are obtained. - 264 PROJECTIVE GEOMETRY. [chap. xvii. The relation = S(p + q + r) + Y(p-^q + r) (v.) contains the principle of the centre of mass. It asserts that the point p + q-h'i' is situated at the centre of mass of p, q and r, and that its weight S{p-\-q-\-T)h the sum of the weights of the three points. In another form, — (m, + mo + mo) I H — ^^^ — , — ^-^ — )• Ex. 1. The middle point of the line ab is ^(a + b). Ex. 2. Interpret the relation regarding ^p + Yq, etc., as representing weighted points. Ex. 3. The centre of mass of equal and opposite weights is at infinity. Ex. 4. The equations of the line a, h and of the plane a, 6, c are q=xa-\-yh^ q=xa-\-yh + zCj where x, y and z are scalars. Ex. 5. Corresponding points of similar divisions on the lines ah and cd are , t and corresponding points of homographic divisions on the same lines are a-\-tb, c+td, t being a variable scalar. [See Art. 37, p. 41.] Ex. 6. The equation q = a+2ht+ct'^ represents a conic. Ex. 7. The equation q^a+tb + u(c + td) ' Represents a ruled quadric, ^ and u being variable scalars. Art. 144. In order to develop this method, it becomes neces-^ sary to employ certain special symbols, and with one exception these are to be found in Art. 365 of Hamilton's Elements of Quaternions, though in quite a diiFerent connection. For any pair of points, we write (a,6) = 6Sa-aS6, [a, 6] = V. VaV6; .(i.) and in particular, for points of unit weight (A = 1 + a, B = 1 +/3), these become ' (A, B) = B-A = /3-a, [A, B] = V. VAVB = Va/3 = Va(^-a). (ll.) ART. 144.] SPECIAL SYMBOLS. 265 Thus (a, b) is the product of the weights into the vector connect- ing the points, and [ab] is the product of the weights into the moment of the vector connecting the points with respect to the scalar point or origin. The two functions (a, b) and [a, b] completely determine the line ab. For any three points we write [a,b, c] = (a,b, c) — [b, c]Sa — [c, a]Sb — [a,b]Sc,) (a, b, c) = S[a, 6, c] = S. VaV6Vc = Sa[6, c], |--("w and for unit points A = l-}-a, B = l+/5, C=l-|-y, these become [A, B, C] = Sa/3y - V^Sy - Vya - Va^, (ABC) = S . a/3y. . . .(iv.) Hence it appears that the quaternion [a, b, c] determines the plane of the points, and regarded as a point symbol [a, b, c] represents the reciprocal of the plane with respect to the unit sphere having its centre at the scalar point. For the vector y [abc] : S [abc] is minus the reciprocal of the vector perpendicular from the origin on the plane SyoV(^y-|-ya + a^) = Sa/3y ; that is, its extremity terminates at the pole of the plane with respect to the unit sphere. The symbol (a, b, c) is the" sextupled volume of the pyramid OABC multiplied by the weights 8aS6Sc. Any quaternion may therefore be regarded as representing at ^ pleasure a plane or a point — reciprocals with respect to the unit ^ sphere. The last special symbol we require at present is {abcd) = ^a[bcd]; (v.) or for unit points, (ABCD) = S/3y^-Say^-fSa/5^-Sa^y (vi.) Thus (ABCD) is the sextupled volume of the tetrahedron ABCD, and {abed) is the same volume multiplied by the product of the weights. It will be observed that the five functions are combinatorial, that is to say, they remain unchanged when to any of the quaternions involved in one of the functions is added a sum of products of the other quaternions multiplied by scalar coefficients. For example, [a + xb + yc, b, c] = [a, b, c]. More generally when the constituent quaternions are replaced by linear functions of themselves with scalar multipliers, the functions are merely multiplied by a scalar. If any linear relation with scalar co- efficients connects the constituents of a function, the value of the function is zero. If any two constituents are transposed the function changes sign, and in fact the laws of combination of the rows or columns of an ordinary scalar determinant are obeyed by the constituents of the functions. 266 PEOJECTIVE GEOMETEY. [chap. xvii. Art. 145. In terms of these functions, the equation of the line ah and of the plane abc are respectively [q, a, 6] = 0, (q, a, h, c) = 0; (l.) the first expressing that q, a and b are linearly connected, or that the plane qah is indeterminate; the second requiring the volume (QABC) to be zero. The equation of the line ab may also be written in the form ' (pqab) = 0, (II.) where ^ is a point wholly arbitrary; and the equation of the plane may be replaced by Sql = 0, where l = [abc], (iii.) the point I being, as we have said, the reciprocal of the plane with respect to the unit sphere* 8.^2^0, (IV.) or S . (1 + 0Q)2 = 0, or OQ^ + 1 = 0. Putting L = 1 + OL, the equation of the plane takes the known vector form S(1+OQ)(1 + OL) = The plane at infinity is S? = 0, (V.) this being the reciprocal of the scalar point (the centre) with respect to the unit sphere ; or otherwise if q represents a point at infinity it is a vector (Art. 143, p. 263), so that Sq = 0. The formulae of reciprocation ([abc]; [abd]) = [ab](abcd); [[abc]; [abd]]= —(ab)(abcd), (vi.) are worthy of notice. They connect two points a and b with two points [abc] and [abd] on the reciprocal of the line ab, and are easily verified by vectors. Formulae, such as these, are often suggested by the forms of the expressions. For example, the left-hand members of the above relations evidently vanish if a, b, c and d are linearly connected. We infer that (abed) is a factor, and the remaining factor must be a combination of (ab) and [ab]. It is often useful to observe that if i, j and k are mutually rectangular unit vectors, (l,i) = i, [ij]=k, [l,i,j]=-k, [i,j,k]=-l, (l,i,j,k)=-l;... (vii.) and relations such as these may be employed to ascertain the numerical factors in expressions such as (vi.). * In ordinary homogeneous coordinates the auxiliary quadric is generally taken to be x^ + y^ + z^ + iv^ = 0. It is more convenient in quaternions to employ the unit sphere as the auxiliary. There is however no loss of generality. (Compare Art. 153 (X.), p. 284.) ART. 145.] PRINCIPLE OF DUALITY. 267 Ex. 1. Two lines, a, 5, and c, c?, intersect if (abcd) = 0. (a) This condition may be also written in the form S(ab)[cd] + S[ab]{cd)=0. Ex. 2. The point of intersection of three planes Slq = Oj Smq = 0, Snq = is q = [l, m, ?i]. Ex. 3. The line of intersection of two planes 8^2-= 0, Smq = is q = [l,m, n], where n is an arbitrary quaternion. Ex. 4. If four planes ?, wi, tj, p have a common point (l, m, n, p) = 0. Ex. 5. The line «, b intersects the plane Slq = in the point aSlb-bSla. Ex. 6. The general equation of a conic is q = at^-\-2bt+Cj where ^ is a scalar parameter. (a) The expression q = a^^^a + ^ (^i + ^2) + <^ represents the pole of the chord joining the points ^, and ^2> or the tangent at ^1 if ^2 is variable. (b) The pole of the line in which the plane Slq = meets that of the conic is q^ ^Slc - 2bSlb + cSla. (c) The centre is q = aSc - 26S6 + cSa. (d) The conic is a parabola if SaSc = (S6)2. (e) What kind of a conic is represented by q = At^ + 2Bt + cl (/) If q, 5'i, q2, qs and q^ are any five points on a conic, and if t, t^, t^^ t^ and ^4 are the corresponding parameters, the anharmonic of the pencil (g - gi^ g - g2) • (g - g3> g - ^4) ^ (^1 - ^2)fe - O (g - g25 g - gs) • (g - g4> g - gl) (2^2 - ^3X^4 - ^1)* Ex. 7. The general twisted cubic is q = {a, 6, c, 4^)3. (a) The equation g = («) ^> c, c^jj^i, ljSy2} 1)^ represents the tangent at the point ^2, h being variable. (6) The osculating plane at a point is q = {a,b,c,d\h,\^t2,\\t^,\\ two of the scalars ^1, ^2» ^3 being variable and the other being fixed. (c) The equation in (a) represents the tangent line developable when ^1 and ti both vary. {d)\i h is given it represents the conic in which the osculating plane at ^1 cuts t¥ie developable. (e) The locus of the poles of a fixed plane 8^5'= with respect to these conies is the conic, q = t^ (aScl - 2bSbl + cSal) + h {aSdl - bScl - cSbl + dScd) + b^dl - 2c^cl + dSbl. 268 PEOJECTIVE GEOMETEY. [chap. xvii. (/) The osculating planes at the points in which the plane Slq = meets the curve intersect in the point q = aSdl-3bScl + 3cSbl-dSal, and this point lies in the plane. (g) The symbol of the osculating plane Spq = at the point t is p—[at + b, bt + c, ct + d]; / ' and this equation also represents the cuspidal edge of the reciprocal developable. (h) The last equation may be written in the form p=t^[abG] + t^[abd] + t[acd] + [bcd]. (^) The symbol of the plane containing three points i^i, ^2? h is p = 3t^t2t3[abc] + 22^2^3 . [abd] + S^j . [acd] + 3[bcd]. (j) The anharmonic of the group of planes joining two variable points on the cubic to four fixed points is constant. Akt. 146. Hamilton has given two relations connecting jSve arbitrary quaternions, a(bcde) + b(cdea) + c(deah) + d(eabc) ■\-e{ahcd) = (i.) ^^^ e{ahcd) = [hcd'\ Sae - [acd] Sbe + [abd] See - [abc] Sde; .. . (ii. ) which are of great importance and which correspond to the vector relations SSa/Sy = aS^yS + ^SyaS + ySa^S = Y/SySaS + YyaS^S + Ya/SSyS. The first has been virtually proved in Art. 39, p. 43, and we may at once verify it by writing xa-\-yb + zc-\-^vd-\-ve = 0, where x,y, z, iv and v are scalar s to be determined. From this, by the combinatorial property, we have = (a, b, c, xa-\-yb + zc+vjd-\-ve) = (a, b, c, wd + ve), which gives the ratio of w to v. This relation enables us to express any point in terms of four given points, so that we may if we choose use an arbitrary tetrahedron of reference, for example abed. The second shows how to refer any point to four given planes Saq = 0, Sbq = 0, Seq = 0, Sdq = 0; and the truth of the formula may be verified by observing that we get consistent results when we operate with Sa, S6, Sc and Sd. It will be observed that the relations (i.) and (ii.) are linear with respect to each of the five quaternions, so that the weights of the points do not enter. In fact, just as in tetrahedral coordinates, geometrical relations depend on homogeneous func- tions of the quaternions. Though it is in general distinctly disadvantageous to employ any system of coordinates in ART. 146.] EELATIONS CONNECTING FIVE QUATERNIONS. 269 quaternion investigations, or even to refer in thought to any tetrahedron or axes of reference until a problem has been reduced to its ultimate simplicity, yet it is worth while observ- ing that if we express a variable quaternion q in terms of four given quaternions a, h, c, d by means of the relation q = xa-{-yh-\-zc + wd, (in.) the scalars x, y, z and w are the anharmonic coordinates of Art. 40, p. 43. ■ Ex. 1. The line de meets the plane ahc in the point d{ahce) — e{ahcd). Ex. 2. Show that {{ahc'\, [def]) = [ef^{ahcd)^[fd]{ahce) + [de'\{ahcf\ [[ahc], [def]]^ - {ef){ahcd) - (fd)(abce) - {de)(abcf). [Compare Art. 145 (vi.). Four points on the line of intersection of the planes abc and def are d{abce) — e(abcd) and d{abcf)—f{abcd), and the functions [a'b'l and —{a'b') for two points on the line are proportional to the right-hand members of the above. The weights are correct, and it on]y remains to determine the numerical factors. Putting d=a and e = 6, we verify the signs by the equations cited.] Ex, 3. The point of intersection of the planes abc, dcf and ghi is I n. b c [[abcl [def], [ghi]]. a {adef) (bdef) (cdef) (aghi) ibghi) (cgki) [Equating the left-hand member to j^a-{-i/b-\-zc, we have x{adef)+y{bdef)+z{cdef) = 0, etc., and to determine the factor we may put «=1, h=i, c==j, [abc]-=-1c, [def] = i, [ghi]=j. The left-hand member becomes -f-1, and the determinant also reduces to -fl.] Ex. 4. Given four triangles a„6„c„, where 7i = l, 2, 3 or 4, show that six times the volume of the tetrahedron determined by their planes is {a^a^b^c^ {bia^2<^^ {c^a^^c^ {a^a^b^c^) {b^a^b^c^ {cia^h<^^) {a^a^b^c^ {h^a^b^c^ {c^a^b^c^) n(a„6„c„) [This follows from the last example.] Ex. 5. Establish the identities Saa' Sa6' Sac' S6a' S66' S6e' Sea' Sc6' Sec' Saa' Sa6' Sac' Sao?' S6a' S66' S6c' ^bd' Sea' Sc6' Sec' Sec^' ^da! ^dh' Scfc' S^' = -S[a6c][a'^)'e']; = —{abcd){a'b'c'd'). 276 PEOJECTIVE GEOMETRY. [chap. xvii. [The first determinant is combinatorial in a, h and c and also in a\ b' and c'. It vanishes if either triangle reduces to a line, and conversely. Hence it must be a scalar function of [abc] and of [a'6V], that is (having regard to the weights) it must be of the form j;SY[abc]Yla'b'c]+^S[abc]S[a'b'c'l where x and y are numerical factors. For a = a' = i, b = b'=j\ c^d = k we get,y= -1, and for a = a'=l, b = b' = i, c = c'—j we find x= —1.] Ex. 6. Prove that S6a' S66' = S(a6)(a'6')-S[a&][a'6']. [This is most easily proved by vectors. Compare Art. 145, Ex. 1.] Ex. 7. Find the equation of the hyperboloid having three given generators a6, a'b' and a"b". [There are various methods of finding this equation, but we shall give a method to illustrate the use of Ex. 3. If p and q are any two points on a generator of the opposite system to the given lines, the conditions of intersection are (pqab) = 0, (pqa'b') = 0, {pqa"b") = 0. Regarding these con- ditions as the equations of planes, p being the variable point, the condition that the planes should intersect in a line is [[5'a6][5'a'6'][^a"6"]] = 0, which becomes {aqab'){bqa"b") — {bqa'b')(aqa"b")=0.] Art. 147. The results of the last article are particular cases of a very general theory applicable not only to quaternions but to any operators or quantities which are associative and commutative in addition."^ If /(a, b) is a function of two quaternions distributive with respect to each, the function f(a,b)-f{b,a) ...(I.) is combinatorial in a and h, for it remains unchanged when we replace a by a + 7/6 or 6 by b-\-xa, because /(a +2/6, b)=f{a, b)+yf(b, b) and/(6, a+yb)=f{b, a)+yf{b, 6). In like manner if f(a, b, c) is distributive with respect to a, b and G the function /(a, b, c)—f(b, a, c) is combinatorial in a and b', the function formed by subtracting from this the result of interchanging a and c is combinatorial in a and b and also in a and c ; and the function of six terms 2±/(a,6,c) (II.) formed by transposing a, b and c in /(a, b, c) in every possible way, by changing the sign after every transposition of a pair of constituents and by adding the results together, is combinatorial in a, b and c. Similarly if f(a, b, c, d) is distributive in a, b, c and d, the sum 2±/(a, b, c, d) (ill.) •^See an interesting paper by Prof. A. S. Hathaway, Proc. Acad, of Science, 1897. ART. 147.] COMBINATORIAL FUNCTIONS. 271 is combinatorial in a, h, c and d ; and finally 2+/(a, h, c, d, e) (iv.) is combinatorial in a, h, c, d, e and vanishes identically because the five quaternions are linearly connected. It is geometrically evident from Art. 144, that every com- binatorial function of two quaternions a and b must be a function of (ab) and [ab] — the two vectors which determine the line ab. Every combinatorial function of a, b and c must be a function of [abc] which determines the plane abc ; and the only combinatorial function of four points is (abed) — the sextupled volume of the tetrahedron determined by them. Hence (ii.) is a linear function of [abc] and (in.) is the, product of a quaternion by the scalar (abed). Now in forming these sums, we may proceed step by step. For example, let us transpose beds in f(a, b, e, d, e), leaving a unchanged. We obtain the sum ^±f(%, h, e, d, e), where the temporary suffix applied to a denotes that it is free from the operation indicated by 2 + . Next interchange a and b and change the sign and permute a, c, d, e, leaving b unchanged. We get -^±f(bo,a,c,d,e). Finally the vanishing combinatorial function (iv.) is expanded in the form 2 ±f(aj)ede) - E ±f{b^aede) + S ±f(c^abde) - 2 ±f(d^abee) + 2+/(6oa6c(Z) = 0, (V.) and this general result includes Art. 146 (i.) as a particular case. Again we may leave two or more quaternions fixed and add together the sums obtained, so that for example 2 ±f(a^b^cd) - 2 ±f(a^c^bd) + etc. = S ±f(abed) ( vi.) These expansions correspond to the expansions of determinants by minors. Ex. Find the sources of the functions (lahc\ d), [[abc], cTj, which are combinatorial in a, b and c, or in other words find linear functions of a, 6, c from which the combinatorial functions niay be derived by- summation and transposition. [Since (abc).Yd=[bc]S.aYd+[ca]S.bYd+[ab]S.cYd and . Y[abc]Sd=:-[bc]SaSd-[ca]SbSd-[ab]BcSd, the first expression is 2± 6cSac?Q. Similarly the second expression is - V . [be] Yd.Sa-Y. [ca] YdSb -Y .[ab] YdSc, and the function may be derived from - V6SVcV^ . Sa or from - b^cd^^^ 272 PROJECTIVE GEOMETRY. [chap. xvii. certain parts of this latter expression vanishing under transposition and summation. As a determinant, the function is a b G Sa S6 Se ^ad Sbd Scd [[abc]d]. and this may be deduced directly as follows. We may assume [[abc]d\ = a^a + i/b + zc since S[a6c][[a6c]or| = ; and we have xSa + i/Sb+zSc = Oj xSad + i/Sbd+zScd=0. The numerical factor of the determinant resulting from this may be determined by substituting special values for a, b, c, d.] Art. 148. We shall now consider the general linear trans- formation of points in space. In analogy with the linear vector function, the linear quaternion function fq is a function which satisfies f(a-\-b)=fa+fb (I.) for all pairs of quaternions a and 6. The relation P=fy (n.) represents the general linear transformation from points q to points p, lines and planes q = a+tb, q = a-{-tb-{-uc, becoming lines and planes p=fa + tfb, q=fa+tfb + ufc, and anharmonic properties being preserved. If four given quaternions, a, b, c and d, are converted by a linear transformation into four others, a\ b\ c and d\ the function which effects this transformation is (compare Art. 62 (IV.), p. 88, and Art. 146 (i.) /g = — {a\bcdq)+b\cdqa)-\-c{dqab)-\-d\qabc)}{abcd)-'^ ; (ill.) and this function is in the quadrinomial form. To reduce a function to the quadrinomial form, we may arbitrarily assume any four quaternions a, 6, c, d and use either of the relations connecting five quaternions. Taking the second, fq = {f[bcd\ Saq -f[acd] Sbq +/[a6c?] Scq -f[abG]Sdq}{abcd)-\ (iv.) and thus a linear quaternion function depends on sixteen constants, four constants being involved in each of the four quaternions f[hcd], etc. In (III.) we supposed the weights given. Let us now determine a function which shall convert five given points A, B, C, D, E into ART. 150.] LINEAR TRANSFORMATION. 273 five others A', B', C, J)\ E', paying no attention to the weights. Such a function is . _ AXBCDg)(BVD'EO B^ACDg) ( A'CTD^EQ •^^~ (BCDA)(BCDE) "*" (ACDB)(ACDE) , CXABDg)(A'B^D'EO DXABCg)(A'B'C'E^) "^ (ABDC)(ABDE) ( ABCD)(ABCE) '' '"^^'^ for replacing g by A we get /A=A'(B'C'D'E')(BCDE)-\ etc., and putting g = E, we have /E = E'(A'B'C'D')(ABCD)-i in virtue of the relation connecting five quaternions. Thus the function (v.) eflfects the required transformation, and it is evidently deter- minate to a scalar factor. (Compare Art. 65, Ex. 5, p. 92.) Art. 149. A linear function f being regarded as producing a transformation of points, the inverse of its conjugate f'~'^ produces ike corresponding tangential transformation. For any quaternions p and q, Spq = Spf'^q' = Sq'f-^p = Sq'p, if q'=fq, p=f-'p (l.) Hence any plane Spq = 0, in which q is the current point and p the symbol of the plane, becomes after the transformation Sp^q' = 0, where q' is the transformed current point and where p^ is the transformed symbol of the plane. In other words when points are transformed by the operation of /, planes are trans- formed by the operation of /'"^. Art. 150. Now the symbol of the plane may be expressed in terms of three points in the plane (Art. 145, p. 266), and therefore for some scalar factor n, nf'--'[abc] = [fa,fb,fc] = r[a, b, c] (l.) since we may either transform the symbol of the plane in one step hy f'~^ or we may transform the points a, h, c which enter into the symbol by /. The function F' is a new linear function analogous to Hamilton's yjr', and it is connected with f'-^ by the relation rf^^j^p'^py (jj ^ The scalar n may be explicitly expressed in terms of four arbitrary points, a, b, c, d, by operating with S.fd on (i.), when we find n(abcd) = (fafbfcfd) = S [abc] Ffd, (iii.) where F is the conjugate of F. Thus in addition to (ii.) we have, n=fF=Ff; (iv.) and we inay also write niabcd) = (fafbfcfd) = (fafbfcfd) F[abc] = [fafbf< J.Q. S F[abc] = [fafbfc] = nf- ' [abc]. j ^""'^ (VII.) 274 PEOJECTIVE GEOMETRY. [chap. xvii. Eep]acmg/by/+^, where t is a scalar, the relations nt = 71 + tn' + tV + tV + ^4 =ftFt = (/+ t){F+ tG + t''H-\- 1^) (vi.) are obtained, where the new scalars n, n'\ n'" and the new linear functions G and B. are defined by n\abcd) = 2(afbfcfd) ; n'\abcd) = ^{cibfcfd) ; n"{abcd) = ^{abcfd) ; G[abc-\ = [a,f%fc\ + [f'^, b,fc]-V[faJ% c]; Hiabc] = [/'a, 6, c] + [a, /6, c] + \a, b, fc\ Moreover, on account of the arbitrariness of t in (vi.), n=fF, n'=fG + F, n"=fH+G, n'''=f+H; ...(viii.) and from the symbolical equations may be deduced the following explicit expressions for the auxiliary functions H=n'"-f; G = n"-n'J+f: F=n' -nJ+n'"P-p] (ix.) and the symbolic quartic n-ny+n"P-n'y'+f = (x.) satisfied by the function /. Art. 151. Let t^y t^, % and t^ be the roots of the scalar quartic t^-n"'t^+nr~-rit-\-n = 0, (i.) so that the symbolic quartic may be expressed in the form {f-k){f-t,)(f-k){f-h)^0 (II.) It follows just as in the case of the vector function that (/-y3i = 0, where (f-t,)(J-t,)(f-t,)q = q, (m.) and that g^ is a fixed point — a united point of the transformation — one of four q^, q^, q^ and q^. The point q is quite arbitrary. The equations P = {f-ti)q, P = if-h){f-t,)q, (IV.) represent respectively a united plane of the transformation and a united line — the plane [q^, q^, gj and the line q^q^. . We have also by the property of the conjugate, Sq;p = &q;(f-t,)q = Q if (/'-yg/ = 0; (V.) and thus the united points (g/, q<^, q^ and q^) of the conjugate (/') are the reciprocals with respect to the unit sphere (Art. 145) of the united planes of /. In other words, the united points of a function and of its conjugate form tetrahedra reciprocal with respect to the unit sphere. Ex. 1. Prove that fq may be reduced to the form y^ = (e + €)S^ + Se'V^ + V^, I and determine its latent quartic in terms of the linear vector function ^, the vector-s e and e' and the scalar e. ART. 152.] UNITED POINTS. 275 [By the distributive principle fq=f^q+fyq, etc. To determine the quartic assume fq = tq = t(Sq + yq), and equate scalar and vector parts. We find (e-t)Sq + SeVq = 0, {-t)-h = 0.] Ex. 2. Construct a function with four zero latent roots. [Assume fa = h, fh = c, fc = d^ /c?=0.] Ex. 3. Examine the nature of the symbolic equation satisfied by the function ^^ = a{bcdq) + b{cdqa) + c'{dqah) + d'{qahc). [Every point a + uh on the line a, 6, is a united point of the function, and the i^ function oi fq — {abcd)q vanishes identically. The quartic degrades into a cubic] Ex. 4. Construct a function satisfying a symbolic quadratic. [This may arise from one of two causes. The function may have two line loci of united points a, b and c,d; or it may have a plane locas of united points a, 6, c. In the first case the latent quartic is a perfect square. In the second it has a triple root. For full details on these matters see Phil. Travis., vol. 201, viii.] Ex. 5. Prove that two real lines remain unaltered by the general real linear transformation. [If the roots are all real of course the six edges of the united tetrahedron remain unaltered. If the roots are all imaginary, th ey o ccur i n co njugate pairs, and the united points must be of the form a ± \/ — 16, c ± /J — Id. The lines ab and cd are real and remain unchanged.] Art. 152. Just as in the case of the vector function, we obtain two new functions /o=i(/+A f.=w-n ('•> on combining a function and its conjugate by addition and subtraction. The function /^ is self -conjugate and the function / is the negative of its conjugate, or /o=/o^ /= -/A ("•)• as we see at once by the property of the conjugate. Since fq is the general linear function of q, ^qfg, or Sqf^q is the general scalar quadratic function, and S?/o? = (m.) represents the general quadric surface, the surface being quite arbitrary both in shape and position, and not now referred to its centre as in Art. 72, p. 106. In like manner Sp/,g = (iv-) is the general equation of a linear complex, or of a family of lines p,'q satisfying a single condition of the first order. For if we replace p hy p-\-tq the equation remains unchanged, for we have generally, by the property of the conjugate (ii.), Sqf,q=-Sqf,q=o. 276 PEOJECTIVE GEOMETEY. [chap. xvii. The equations SqfQa = 0, Sg/6 = (v.) represent respectively the polar plane of the point a with respect to the quadric, and the plane containing the lines of the complex which pass through b. The first equation may be deduced from the result of substituting a-\-tq in the equation of the quadric, when we find Sa/o^ + 2^Sg/oa + i{2s^/og = 0, and if q is on the polar plane, the points in which the line aq meets the quadric must be expressible by a-\-tq, a — tq, because the polar plane is the locus of harmonic means, and the points a, a-\-tq, q, a — tq form a harmonic range. If Slq = is an arbitrary plane we see on comparison with (v.) that the pole of the plane with respect to the quadric is /o~^Z, and that the point of concourse of the lines of the complex which lie in the plane is f,~H. It also appears that Slf^-H = and ^mf-H = (vi.) represent respectively the tangential equation of the quadric, or the equation of the reciprocal quadric; and the tangential equation of the complex (the intersection of the planes S^g = 0, Smg = being a line of the complex), or the equation of the reciprocal complex. A complete account of the nature of the united points of the functions /^ and / is furnished by the theorem of Art. 151. Since /o is its own conjugate, each of its united points is reciprocal to the plane containing the remaining three, or the tetrahedron of united points is self -conjugate to the sphere of reciprocation. We saw in Art. 67, p. 96, that it is impossible for a real self- conjugate linear vector function to have a pair of equal roots without having indeterminate axes, and this because a real line cannot be perpendicular to itself. But a real self -conjugate linear quaternion function may have two of its united points coalesced into a single point provided the point is on the sphere of reciprocation. The argument about real roots does not now apply. For suppose a-\-\/ — \h and a — x/ — 16 to be two united points of a self -conjugate quaternion function, the condition of reciprocity is S(a + >v/^^6)(a-V^6) = Sa2 + S62 = 0, and this condition can be satisfied for real points a and h if one ' point (a) is inside and the other (6) is outside the sphere of reciprocation Sg^ = 0. As regards the function /, the most general form its symbolic quartic can have is //+oP ; /(I +p) = €-Se,p + Y7jp ; e being a scalar, €q, €,, r; being vectors and Q being a self -con jugate linear vector function. Ex. 2. Prove that the latent quartic of the function / is and verify the conclusions respecting the roots and united points of/. 278 PEOJECTIVE GEOMETRY. [chap. xvii. Ex. 3. Prove that ^qf2fi~V29 = ^ is the locus of the poles of tangent planes of the quadric ^qfiq = with respect to the quadric Sqf2q = 0. Ex. 4. The locus of the points of concourse of lines of the complex ^pfn = which lie in the tangent planes of the quadric SqfQq = is the quadric Sqf/o-\f,q = 0. Ex. 5. An arbitrary quadric and an arbitrary linear complex have a common quadrilateral of generators. [Tliis follows by expressing that the point of contact of a plane 8^9 = with the quadric ^qfoq = is the same as the point of concourse of the lines of the linear complex 8pf,q = in the plane. We have /a = ^/^a = 1*^, where t and u are scalars, so thai fQ~\f^a = ta. There are thus four points (a) through which pairs of the common generators pass, and these points are the united points of /o-i/.] Ex. 6. If /i and/2 ^^® ^^y *wo functions, prove that the latent quartics •of /1/2 and of /2/i are identical. (a) Show also that the latent quartic of /o~y, is of the form [The first part follows exactly as in the case of vector functions (Art. 71) ; the second is obtained by combining this principle with the fact that -///o~^ is the conjugate of /o-y,.] Ex. 7. If a, h, a' and h' are the united points of the function f^'^fj •corresponding to the latent roots +t, —t, +t' — 1\ prove that if we take xa + yh za' + wh' _ x'a + y'h z'a! ■\-w'h' the equations of the quadric and the linear complex take the canonical forms ^Sq/aq =xy-\-zw — 0, ^pf,, /KA+y/s)-^^, hUx^^f^'^V- But S/2(/i +.^/2)- V ./2-^ ./2(/i +y/2)-> =Sp(/i+^/2)-y2(/i+y/2)-V =(^-y)-iS^(/i + ^/2)-i[(/i + ^/2)-(/i+3//2)](/i+y/2)-V, and this vanishes since y lies on the three quadrics .r, y, z. This in particular gives the theorem that confocal quadrics cut at right angles.] Ex. 18. The locus of the poles of a plane Sg'a = to the same system of quadrics is the line 5' = (/i+-^/2)«^ or [^/ia/2«] = 0; the locus of the poles of the system of planes Sa(€Spf3 + pSeft) - Y^eSpa + pSea) /? + eYcf>apSpf3 + eYp/3 - eSpfS - pSc^ + epSpfi) = 0. In this transformation we make use of the fact that SY€Yaf^cf>YpYaf3 = SYpYaf^cf^YeYafS in order to have the function in the last expression self -con jugate. If then ek = (f)X- eSAp - pSAe + eS Ap, the condition becomes St^"^t = 0, and putting f=f\-\-xf^^ and therefore ^=^i + ^^2) ^'^ result required is obtained.] Ex. 23. If p is any point ; jo^, ^2* Vz ^^ reciprocals of the tangent planes to the three confocals (parameters t^^ ^2» ^3) which pass through the point ; show that the tangent cone to any other confocal (parameter t) is C 6t V vo V '^ fro where any point q is expressed in the form xp + XiPi-\- X2P2+ x^p^. [The condition that the line p + uq should touch the confocal t is ^. Sq(f+t)-'qSp{f-{-t)-'p-(Sq{f+t)-^py = 0, or Sqkq = if k is me linear function defined by ^ = (/+ ty'q ^p{f+ t)-'p - (/+ tr^p^q (/+ tr^p. Substituting in turn p, p^ { = {f+ti)~^p\ p^-, and JO3 for q, we find hp=0, hpi={t-t,)-^p^Sp{f+t)-^p, etc., 282 PEOJECTIVE GEOMETEY. [chap. xvii. because we have Kf+ h)~'p = (/+ t)-\f+ h)-^p^p{f+ t)-'p - (/+ h)p^p{f+ hY^{f+ ty^p }, which reduces to A(/+ h)-^p = {t- h)-' (/+ h)-'P ^P{f+ t)-'P since Sp(/+ t)-^p - ^p{f+ h)-'p = (h - OSp(/+ ^^1)-^/+ 0>- * The equation Sqhq^O reduces to the required form since SpiP2=^, etc.] Ex. 24. Find the equation of the tangent line developable of the quadrics S .q^ = 0, ^qfq = 0. [If p is the point of contact of a tangent line pq to the common curve, the four conditions S.p'^ = 0, Spq = 0, Spfp = 0, Spfq = 0, show that (P,9,fP,f9) = ^^ or that {f+^)p = (f+y)q, where x and y are two scalars. Substituting for p in the conditions of contact, we find four relations in q, x and y, which are easily seen to be equivalent to three. The second condition gives J. = Sg'(/+.r)~^(/+^)g'=0 ; and because the first and third combine into ^p{f+y)p = 0, they give ^q{f+^v)-\f+y)q=0, or ^S(?(/+^)-i(/4-^)? = 0. Again the second and fourth give S^(/+^)g = 0, or B=^q{f+y)q = Q. To eliminate x and y we have therefore to equate to zero the discriminant of A with respect to x and to employ the condition B=0. On expansion A becomes Sq{F+xG + x^ff+a^){f-i-y)q = 0, and as J5=0, this reduces to the quadratic Sq{F+xG + x^H)(f+y)q = 0, and the discriminant equated to zero gives 4SqII{f+y)qSqF{f+y)q-{SqG(f+y)qy = 0. Putting for y its value in terms of q the required equation is obtained.] Ex. 25. A plane is drawn through the line ab, and through the line cd the plane is drawn which is conjugate to this with respect to the quadric Sqfq — 0. The locus of the intersection of the plane is S[qab]f-^[qcd]=--0. [If g' is a point on the intersection, [qab] and [qcd] are the symbols of the two planes. The equation may be transformed by Ex. 5, Art. 146.] Art. 153. A linear quaternion function has in general sixteen square roots quite analogous to the square roots of a linear vector function. A function and its square roots have the same united points, and the latent roots of the derived functions are the square roots of those of the original, there being sixteen •different sets according to the choice of signs. (Compare p. 99.) In analogy with the reduction of a linear vector function to the product of a conical rotator and of a self -conjugate function, we may write fP=fsftP, fP^ftJsP, where /,=// and ftft^^, ..-(l.) ART. 153.] SQUARE ROOTS OF LINEAR FUNCTIONS. 283 since if we take f^ to be a square root of the product ff we must have /,// = 1 because and thus we have /.=(//')*. A=(ffTV ("•) It appears on counting the constants that /< is not a conical rotator, there being sixteen constants in / and only ten in the self -conjugate function f], so that there must be six in /^. Con- sidered geometrically the function ft converts the unit sphere into itself and leaves unchanged conditions of conjugation with respect to that sphere, because Sfafth = if Sab = 0. Farther, because ff=ff-'^ transformation of symbols of planes effected by the function f is identical with that of points (Art. 149). To study the nature of a function ff. which satisfies the relation f,f:=^=.f;f. or /,=/;-! or /;=/.-^ (III.) we shall endeavour to reduce the function to the form ft=fufr, where /,=//, /, = r( )r-^, (iv.) that is to the product of a self-conjugate function and a rotator. First we notice that if a function/,., which satisfies the condition /,.//=l, converts a scalar into a scalar, it is a conical rotator, affected it may be with a minus sign. For if /Xl)=l =//(!), we have for all vectors p, s/;p=s«/-;(i)=Sp=o. Thus frp is a vector, and the mutual inclinations of vectors and their lengths remain unchanged after operation by/,, because ^frPfrp = ^pp' To effect the reduction (iv.), we notice that we must have //=1- /,(1)=/.(1)> (V.) because /,//=/„/.///„ =A^ and /,(l)=/„/.(l)=A(t). Let us now for the sake of symmetry introduce two quaternions a and b defined by the relations l+/,(l) = a = l+/„(l), 1-/,(1) = 6 = 1-/„(1) (VI.) Theie quaternions are known when the function / is given, and in virtue of (v.), fya = a, fj)=—b, Sa6 = 0, (vii.) so that a and 6 are united points of the function /„. 284 PEOJECTIVE GEOMETRY. [chap. xvii. Take any point c conjugate to the line ah, so that Sac = 0, She = ; and take the point d conjugate to the plane ahc so that Sda = 0, Sdh — 0, Sdc = 0. Then we may assume fuC = c, fj= -d, (VIII.) and it is evident that all conditions (Art. 152, p. 276) are satisfied for the self -conjugation of the function f^, and that fnV—V^ where p is any point whatever. The function j\ is determined by the four conditions (vii.) and (viii.), and the rotator /. is given by /„~yi or by its equivalent /„/. It will be noticed that there is an infinite number of ways in which this reduction may be made, for the point c may be any point whatever on the reciprocal of the line ah. Also the function /„ has two line loci of united pointst — he line ac and the reciprocal line hd. Thus we can in an infinite variety of ways reduce an arbitrary function / to the form • /=/,/„/■. where /. = (//)*, /„^ = 1, /. = r( )_r-i (ix.) As a simple example, consider the transformations which convert one quadric into another, or which change Sg/ig = into 8^/3^ = 0, where _p=/g (x.) We have S.=fM, whence 1=/,'/, \i j=U^f4^\ (xi.) and the function /, is quite arbitrary subject to the condition As another example we propose to show that the intersection of two quadrics is expressible in the form g = (/+OV (xii.) where / is a linear function, t a parameter and a a constant quaternion. If this curve lies on the quadric Sg/^g = 0, the relation S(/+ tfaf,{f+tfa = Sa(/' + oVi(/+ 0*" = must be identically satisfied for all values of t. Now fhf+ oVi"* = (/iV/r* + 0*. /r V' + oVi* = (/rV'/i* + «)*,(xin.) as appears by squaring both members of each equation, so that the condition may be written This becomes rational in t if the square roots involving t are identical, that is if /rV'/i*=/iV/r* or fj\=fj or if f=A-'U (XIV.) where f^ is a self -conjugate function, the condition now becoming S^(/2 + yi)^='^' ^^ Sa/2a = and Sa/i(X = 0. ART. 153.] VARIOUS TRANSFORMATIONS. 285 Finally, q = (fi-% + t)^a, where Saf^a = 0, Saf^a = 0, SafJ'^-^f^a^OixY.) is the curve of intersection of the two quadrics ^qfiq = 0, 8g/2g' = 0, because if we ^^^ fi~^f2=f ^2j ^2 ^^^ ^^^ united points of f^^f{ and if t^^ ^2) h ^^d t^ are the corresponding latent roots we have (see p. 100) /rVi' 'fcii = hf^2, etc. Further if ^, ^, z and w are certain scalars and if a^, 61, c^, d^ are the united points of /i~Vi', ^^ must have fa2=xa^^ /^2=y^n /, where w, v, u' and v' are arbitrary scalars, we establish a homography connecting the points on one cubic with those on the other, and if we equate corresponding powers of t in the relation /. {abcd\t, lf={a'h'c'd'^ut + v, u't + v') we have four relations which determine the function /.] 286 PROJECTIVE GEOMETRY. [chap. xvii. Ex. 5. Prove that q==^/{{f+'V)if+y){f+z)].e, where Se2 = Se/e = Se/2e=0, represents a confocal of a generalized system when two of the parameters x^ ?/, z vary ; the intersection of two confocals when only one parameter varies ; and a point common to the three confocals corresponding to given values of the parameters. (See p. 124.) Ex. 6. The generalized confocals are inscribed to the developable of which /^ \4 q^{f+xYe is the cuspidal edge. [The line of the developable corresponding to x is q={f+u){f-\-x)^e', the osculating plane is q = {f+u){f+v){f-\-x)~^e', the symbol of this plane is [{f+x)-h, f{f+xyh, /2(/+^)-i4 or {f+xf[e, fe, fel or simply p — {f+xye. This plane touches every confocal.] Ex. -7. Eight generators of the circumscribing developable are generators of an arbitrary quadric of the confocal system. [The line (/+w)(/+^Fe is a generator of ^q{f+x)-^q = (), and this is one of eight corresponding to the eight values of e deduced from the conditions of Ex. 5.] Ex. 8. Eight rays of the complex of lines joining points to their correspondents in an arbitrary linear transformation are generators of an arbitrary quadric. [The equation of a ray of the complex is q = {f-{-u)a, where a is arbitrary. This is a generator of the quadric Sqfiq=0 if Sqf-^a = 0, Sa(/'/i+/i/)a = 0, Saff^fa = 0. This is the generalization of Hamilton's theory of the umbilical generatrices.] Ex. 9. The reciprocal of the developable generated by the tangents to the curve q = {f+tTci is p = {f' + tf-^b, where h = [aja,fa\ and where m is a given scalar. Ex. 10. The family of cxxvyq^ q = if+tya includes the right line, the conic, the twisted cubic, the quartic intersection of two quadrics, the ex- cubo quartic and the cuspidal edge of the developable "circumscribed to two quadrics ; the corresponding values of m are 1, 2, - 1 or +3, ^, 4 and f, Ex. 11. The centres of generalized curvature at a point on the quadric ^q{f-\-x)-'^q = Q are c=-^^q and «'=^?, where y and z are the parameters of the confocals which pass through the point q. [The point e = (f+u)(f+x)-^q is situated on the generalized normal at q (Ex. 12, p. 279), and if this point remains stationary, that is if it is the point of intersection of consecutive normals, dc=cdv = (J+u)(f-\-x)-^qdv = {f+u)(f+x)-'^dq + (f+x)-^qdu, since as c is stationary do and c must represent the same point so that dc=cdv, where dv is some small scalar. This condition may be replaced by cl9' = (/+^)~H/+^)2'dy, where w is a scalar, and operating by S(if+x)-^q, we find almost exactly as in Art 82, Ex. 4, p. 122, the required result.] ART. 153.] GENERALIZED CONFOCALS. 287 Ex. 12. The surface of centi-es of the quadric x is represented by Ex. 13. The differential equation of right lines on the surface d?/ dz IS ^ -t = 'Jn{y) '>Jn{z) where n{y) is the fourth invariant of /+?/. [The differential oiq = ^{ (f+ a;) (J+y) (f+ z)\e is ^ dq = h{^^+^)-s/{{f+^)(f+y){f+z)}e; and the differential equation of right lines on the surface is obtained by equating to zero Sd^(/+^)-id^=iS.(^+^)'. (f+y)(f+z)e = i(d,^S.g-% + d.^S.^.) = i(z-y).{dy^Se(f+y)--^e-dz^Se{f+z)-^e}. Now Se(f+y)-h = n(y)-'^Se(F+yG+y^B+y^)e = n{y)-^SeFe in virtue of the conditions satisfied by e.] Ex. 14. Tlie differential equation of generalized geodesies on the surface is ^ n{y){y — w) ^ n{z){z — w) where -m; is a constant of integration. [A generalized geodesic is a curve whose osculating plane contains the pole of the tangent plane with respect to the quadric of reciprocation (S.q^ = 0). Thus {(f+^)~^q, q, dq, d^q) = is the differential equation of a geodesic in terms of q and of its deriveds. Writing this equation in the form {f+ xy^q + tq + udq +vd^q = 0, where t, u and V are scalars, operating by &q, Sdq, S(f+a:)~^q and S(f+x)~^dq, and observing that Sdq{f-{-^)~^dq + Sq(f+x)~^d^q = 0, we deduce Sg(/+^)-% Sdq(f+x)-^d^q S . q^Sdqd^q - SqdqSqd^q Sq (/+ x)-^q "^ Sd^ (/+ x)-'dq S . g^S . d^^ _ §^^^2 - ^• This immediately integrates, and we find Sq(f+x)-^qSdq(f+a;)-^dq = s(S . q^S . dq'- - Sqdq^), where s is a scalar constant. By the last example we have Sdq(f+a;)-^dq = i{z-y){n(^)-^dy^ - n(z)-^dz^)SeFe, and similarly Sq(f,^^)-^q = (y-x)(z-a:)n(a^)-^SeFe; S.q^ = Sefe= -SeFe ; Sqdq^O, S . dq^*=i{x-y){z-y) . n(y)-Kdy^ . SeFe + l(a^-z){y-z) . n(z)-KdzK SeFe. Collecting these results and putting x'+sn(a;) = w, the required equation is obtained.] ^ 288 PROJECTIVE GEOMETRY. [chap. xvii. Art. 154. We shall now give a few examples relating to in- variants of linear transformation and of quadric surfaces, and shall explain their geometrical import.* By Art. 150 (v.), p. 273, the relation {{f'-t)a,{f-t)h,{f-t)c,{f-t)d) = (abcd)(n - n't + n'V - n''V + ^*) . .-. (i.) is an identity for all scalars t and all quaternions a, 6, c and d. In this sense n, Ji , n' and n'" are invariants, and every relation connecting them implies some peculiarity in the nature of the transformation effected by /. But there is a wider sense in which these four scalars are invariants. If n^ and n^ are the fourth invariants of two arbitrary functions /^ ^^^ fv ^^ relation {{fJA-tAh)a, {fJh-tfj,)h, {fJU-tfj,)c, {fJU-ti\U)d)\ = {abcd)n^n^(n-n't-^n''t^-n'''t^ + t% f "^ is evidently true since (fiP, fiq, /{i^, /i<^) = '^i(i59"^'^)> where _p, q, r and s are any quaternions. Thus any relation implying a peculiarity of the function / and depending on its four scalar invariants, implies also a corresponding peculiarity in the mutual relations of the functions /1//2 and f-j^, that is, in the relations of any pair of functions that can be reduced to the forms /1//2 a^nd/i/g. (See p. 98 and Ex. 8, p. 101.) Ex. 1. If the function / transforms any tetrahedron abed into another a'b'c'd' having its vertices on the faces of the original, the invariant n" vanishes and an infinite number of tetrahedra possess the property. The ■converse is also true. [The conditions are {a'hcd) = 0, (ab'cd) = 0, (abc'd) = 0, {abcd') = 0, and because a'=fa, etc., wa find on addition that n"' = 0. Let a, b and c be any arbitrary points, and let d be determined from the first three conditions. Then we have n"'{abcd) = (abcd'\ so that if n"'=0, the point / ^^^-5 undergo the same transformation and may be said to be covariant with the original functions for this type of transformation. {a) The function /^gg, defined as the coefficient of ^^i^g^g in the identity -^1^2^3/l23[«^c] = P^l/l'"'«, ^hfl'^h, '2tJ^-^cl where ^i, t^, t^, etc., are arbitrary scalars, is (to a scalar factor) covariant with the original functions. (b) Examine the nature of the transformations the inverse and the conjugate functions undergo simultaneously with the original functions, and find the condition that self -con jugate properties may be preserved. Art. 155. Several important geometrical and numerical relations may be deduced from the identity + P^(P5PlP2Ps) + P5i2hP2pBP4)='^^ (!•) in which pn is a rational and integi'al homogeneous quaternion function of q of order m^. ART. 155.] NUMERICAL CHARACTERISTICS. 291 The scalar equations iP5PlP2Pz) = ^' (PiP2PzP4) = ^ ("•) represent two surfaces of orders ^^^m^ — 7n^ and S^^m^ — mg respectively, and any point on their intersection satisfies the quaternion equation [PiP2Ps\ = ^^ • 0"-) or else the three scalar equations iP2P3P4P5)==^^ (2hP4P5Pl) = ^^ (P4P5PlP2) = ^- '"(iV.) Hence we see that the curve of intersection of the surfaces (ti.) breaks up into two parts, one of which is represented by (m.), while the other — the complementary curve — is common to the five surfaces (ii.) and (iv.). Now the order of the curve (ill.) must be a symmetric function of m^, rti^ and m^, and that of the complementary curve must be a symmetric function of the five orders m„. The sum of the orders is equal to the product of the orders of the surfaces (ii.), that is, to and accordingly the order of the curve (ill.) and that of the complementary curve are respectively m^<,^ — ^^m^-\'^^m{m^ and mc = Y^^m{nfi^ (v.) Again the points common to the three surfaces (iv.) must either lie on the surfaces (ii.) or else must satisfy the equation (i^4P5) = 0, (VI.) which requires P4 = 'Mi,-2), which reduces by (viii.) to I ^123 = '^123(^/^1 — 2) + ^^m^ + H^^m^^m ^ + ra^m^m^ I = on^m^m^ — Sll-^^m-^'I.^^m^m^ + 2 (E^^m^f -2((E>,)2~2i3m,m2) (ix.) In the next place, to find the number (h-^2s) ^^ apparent double points of the curve (m.), we have (Three Dimensions, Art. 346), ^^123= 4^^23(^123- l)-i'^m (^O The rank (r^) of the complementary curve is determined by Tc = - 1^123 + Wc(22/mi - m^ - m^ - 2), I and this may be reduced to re = ll-^^m^l^j^m{m2 + E^^m^ni^m^ — 211-^^m^m2 , (xi.) and the number of apparent double points is hc= imc(mc-l)- ire. We may denote the complementary curve by the symbol which is intended to denote that the points of the curve satisfy every equation obtained by omitting one symbol. Similarly, i(iPiP2PzP,P5Pe))) = ^ (xni.) 1 ART. 156.] ORDER AND RANK OF CURVES. 293 may be taken to denote the points which satisfy the surfaces obtained by omitting two symbols. These points lie on the curve (xii.) and also on the surface (PiPil^sPe)^^- But the intersection of the curve and the surface includes the points t^^^ on the curve [PiP2P3\ — ^' Omitting these, the number of points is mc(miH-m2+m3+ma)-^i23 = 2i^^i'^2^3 (^i^-) Ex. 1. The curve [q,fq, «] = 0, where / is a linear function, is a cubic ; ita rank is 4 and the number of its apparent double points is 1. Ex. 2. The curve [fiq,f2q,.f'sq] = is a sextic of rank 16 and with 7 apparent double points. It is ibhe locus of points that can be destroyed by functions of the system ^1/1 + ^2/2 + ^3/35 and the locus of united points of functions of the system . /_i_/ /i* -^ ^l./l + ^2./2+*3/3 ^h/l+^*2/2 + ^3/3' where t and u are scalars. Ex. 3. The surface (f^q, f^q, f\q, f^q) = is the locus of united points of a family of linear functions. (a) When the functions are self -con jugate, it is the Jacobian of four quadrics. Ex.4. The curve ((Aq, f 2^^/39, Aq, f 59)) ^^^ is of the tenth order and its rank is 40. (a) The Jacobians of sets of four out of five given quadrics have a common curve, and the Jacobians of sets of four out of six quadrics have twenty common points. Art. 156. If Q is any homogeneous and scalar function of q of order ni, but not necessarily rational or integral, the equation Q=o (I.) represents a surface. We shall write the differential of the function Q in the form dQ = mSpdq, (ii.) where p is a homogeneous function of q of order m — 1. By Euler's theorem concerning homogeneous functions, we see by ("•) ^^^^ Q = Spq = P, ' (III.) where P is the function of p into which Q transforms when q expressed as a fraction of p is substituted in Q, for we may regard g as a function of p since j) is a determinate function of q. Again we shall write generally for the differential of 2^, -'* dp = (m-l)/,dg, (iv.) where /^dg is a linear function of dg and where the constituents of fq involve q in the order m — 2 ; and by Euler's theorem we ^''^^ P=M (V.) 294 PEOJECTIVE GEOMETRY. [chap. xvii. This function /^ is self -conjugate, as we have shown in a more general case (Art. 60 (iv.), p. 79). Now if we differentiate (in.) we have dQ=-Spdq + Sqdp = dP, (vi.) and on comparison with (ii.) we see that dP = nSqdp, where (7i-l)(m-l) = l, .(vii.) and it is easy to verify that n is the order in which p is involved in p. We shall also write generally for the differential of q expressed as a function of ^, d^ = (7i-l)/,dp, (viii.) and the function fp is also self-conjugate and involves p in the order n — 2 in its constitution. Thus for any differential by (iv.) and (viii.) we have dp = (m-l)fr,dq = {m-l)(n-l)fjp.dp^fjp.dp ...(ix.) by (vii.), and accordingly fjp='^=fpf,> (X.) or one function produces on an arbitrary quaternion the same effect as the reciprocal of the other. In particular, applying Euler's theorem to (viii.) as we have already applied it to (iv.), we obtain the relations V=f = 0, Sd>dg = 0, (xvi.) ART. 156.] THE GENERAL SURFACE. 295 remembering that in general Sdpd'^ = Sd'pdq because fg is self- conjugate. Hence to conjugate "*■ tangents (qdq and qd'q) on the surface correspond conjugate tangents on the reciprocal, and the reciprocal of a tangent to the surface is the correspondent of the conjugate tangent, for we have S(p-\-xd'p)(q + ydq) = 0. The differential equation of the asymptotic lines is Sdpdq = 0, (xvil.) these lines being their own conjugates. The differential equation of lines of curv^ature is {pqdpdq) = 0, (XVIII.) for this is the condition that consecutive generalized normals should intersect. If c is a centre of curvature, we have c = q + tp, dc = (l + tfq)dq+pdt = (q-\-tp)du, (xix.) where du and d^ are some small scalars. (Compare Art. 153, Ex. 11.) Hence sls p=fqq we obtain the relation qdu-dq = (fg--' + t)-^qdt; and operating by Sfqq we get Sg/,(/,-' + 0-^? = or S3(/,->+0-'2 = 0, (xx.) since /5(/rH0-' = <-MA-(/r' + 0-'} and Sqf,q = 0. The theory of generalized curvature is thus connected with that of the generalized confocals. The scalar t is the parameter of one of the confocals Sr(/p + ^)~V = which pass through q, r being the current variable. The conf ocal ^ = is SrfqV = 0. The roots of this equation in t determine the centres of cur- vature, and because in terms of ^(=/g"^) it becomes Sq(Fp + tGp + f-Ep-i-t^)q = or Sq(Gp-{-r'Hp-^t^)q = (xxi.) (since Fp = npfp~'^ = npfg and Sqfqq = 0) after discarding the factor f, it reduces to a quadratic and gives two values of t. Ex. 1. The points having common polar planes with respect to two surfaces satisfy the equation {PlP2) = ^'^ the points having collinear polar planes with respect to three surfaces lie on the points having concurrent polar planes with respect to four surfaces generate the Jacobian {p,p,p,p,) = ; the i^oilits having concurrent polar planes with respect to five surfaces lie on * Consecutive tangent planes intersect in the tangent line conjugate to that joining their points ot contact. 296 PROJECTIVE GEOMETRY. [chap. xvii. and the points having concurrent polar planes with respect to six surfaces satisfy the equation ({{p,p,p,p,p,p,))) = 0; provided we write generally dQn = 'in„Spndq, where Qn = is the equation of one of the surfaces. Ex. 2. To find the osculating plane at a point on the curve of intersection of two given surfaces. [The osculating plane must pass through the intersection of the tangent planes at the point q, and its equation must be of the form Spir + tSp2r=0, where Spir=0 and Sj02^=0 are the tangent planes. We have identically Spiq = Sp2q = ^Pidq = Sp2dq = 0, and by (xiv.) the scalar t is determined by the condition Sdpid^' + t Sdp.^dq = 0, so that the osculating plane is SpirSdp2dq - Sp2rSdpidq = 0. This has now to be simplified. Assuming a quaternion a satisfying Sadq = 0, we have d^ = [piP2«J- -^^^o dp^ = (m^ - l)/id5', dpg = (mg - l)f2^q, and accordingly Sdp^dq = (m^ - 1) S[piP24/i l>iP2«] = ('«i " 1) ^ [PiF2«][?^i;P2/r^«]» since /[a6c] = [/-^a/^6/-^c]. By Art. 146, Ex. 5, this becomes Saq SpoF.p^Sp^fr'a SaqSaFiP2 Saf^~hi \SpiqSp^FiP2SpJ,-\ - Sdp^dq = (m^ - 1) Sp2q ^p-2F^p2 ^pjc^ci = (wi - 1) I Sag- SaFiP2 Saf^-^a = -(m,-l)Saq^Sp2F,p2. Hence the osculating plane is (m2 - l)SpirSpiF2Pi - (m^ - 1) Sp2^S;?2^iP2 =0-] Art. 157. If we use the notation d^ to denote that the differential of q is equal to a quaternion a, we shall have for the k^^ polar of a with respect to the surface Q = 0, d/Q = where da = 0, (i.) and if m is the order of the surface, we may write the equation of the k^^ polar in the form d/d/-^Q = (II.) the quaternion r being now the variable point, and ?' being regarded as constant in performing the differentiations indicated. If we write d,Q = Sap, (ill.) we may consider the quaternion p to be derived from the scalar Q by an operator D analogous to Hamilton's operator V, and we shall have generally and symbolically, j^_ lhcd]da-[acd]db + \ahd]dc-[ahc]da . (abed) ' ^' -^ ART. 157.] VARIOUS NOTATIONS. 297 and in particular when q^w+ix+jy+hz, a = l, b = i, c=j and d = k, we have D = - 1- — j~-^k — = - V (v.) diu dx "^ dy dz dw ^ ^ In this notation (i.) and (ii.) become (SaD)^Q = 0, {SaT>y(SrJ)y'-'.Q = ..(vi.) We may also formally identify our notation with Aronhold's symbolic notation by writing the second of these expressions in the forms (Sa6)HSre)— ^- = or e/e,— ^ = 0, (vii.) where e is a symbolic quaternion devoid of meaning unless it enters into a term homogeneous in e and of order in, and where e^ = Ser. There is thus a considerable latitude in the choice of an appropriate notation for the investigation of projective properties of curves and surfaces. Ex. 1. In investigations which involve diflferentials of the third order of the equation of an arbitrary surface of order m, we may write daQ = mSpaj di,daQ = 'm(m-l)Sbfa, dcdi,daQ=m(m — l)(m — 2)Scf2{a,b) with liberty to transpose in any way the quaternions a, b, c, the function /2(a, b) being a bilinear function of a and b (compare Art. 60). (a) In terms of the operator D, p = -'D.Q, /a = -^-i— -.DSaD.$, Ma,b)=—, ^^ ^,DSaDS6D.^. ^ m ^' -^ m(m-l) ^' -^ ^^ ' ' m(m-l)(m-2) ^ {b) We may also write Q = ^eq^, daQ = 'mSeaSeq^, dbdaQ = r)i{m-l)SebSeaSeq, dcdfida^ = m(m-l) (m - 2) SecSebSea, where e is a symbolic quaternion devoid of meaning unless it occurs thrice in a term. (c) We have P =fq =/2 {<1'>9) — eSeq^ ; fa =f<^ (a, q) = e'^ea^eq ; /g (a, 6) = eSeaSeft. ^ And when we differentiate /a totally we find d . fa =f . da + (m - 2)/2 (a, d^). {d) The equation of the Hessian is 71 = or (fajbjcjd) = 0, where n is the fourth invariant of / and where a, 6, c and d are arbitrary points. It may also be expressed in the forms {ee'e'e'") Sea Se'b Se"c Se'"dSeq Se'q Se"q Se"'q = ; ^^ (ee'e"e"ySeqSe'qSe"qSe'"q = ; SaDS6D'ScD"Sc^D'"(D$, D'^', ^"Q% D"'$"') = ; {J)B'B"D'y.QQ'Q"Q"' = 0, where e, e\ e", e"\ etc., are equivalent symbols (compare Art. 147, p. 270). 298 - PROJECTIVE GEOMETRY. [chap. xvii. Ex. 2. If J={p\P2PzPd ^^^ d$n=m„Sjt?„d5', where Qn is homogeneous and of order m„ in the variable 9, q . J= [;?2P3P 4] • ^1 - biPsi^J • Q2 + [PiPiP^ ' Qs - [P1P2P3} • ^4- (a) For an arbitrary differential, and for an arbitrary scalar m, q . dJ= (m - 1) J. dq + ^±Qi. d[p2PsPi] + 2 ± (^i - m,)[p2PsPi]Sp\dq. (b) If four surfaces have a common point, their Jacobian passes through that point. If the orders of the surfaces are all equal the point of common intersection is double on the Jacobian. If the orders of three of the surfaces are equal, the fourth touches the Jacobian. If the orders of two surfaces are equal, the line of intersection of the third and fourth touches the Jacobian. (c) At a point common to the intersection of four surfaces of the same order m, q . dV= - m(m - 1)2 ± [paPapJSd^'/idg', where dpn={m - 1 )fndq ; and hence the equation of the tangent cone at the double point is ^±{ap2Pc^p^Mir=0, where a is an arbitrary constant quaternion. {d) If four surfaces have a common multiple-point of order ^, we find that d^-«.^/=2±[d*-V2, d*-V3, d^-^J.d^^i + So, d^-KJ={d^-'p^, d*-^jt?2, d*-i^3, d'=-V4)+2o', where Hq and 2o' denote sums of terms which vanish when q coincides with the multiple point, and we also have di'Qi = m^^dqd^ ~^pi + vanishing terms. (e) At the multiple point d^-^J and d-**-"* . ^J vanish, and therefore d'^-'^J vanishes (as in (6)), and the Jacobian has a multiple point of order 4^-3 ; and because we may write (as in (a)) d4*-3.gJ=mdg.d**-V+2±(mi-m)[d*-V2. d^^-^s, d^-i/^JSd^d^-'jOi + ^o", it follows when the surfaces are all of the same order that the Jacobian has a multiple point of order 4^-2. Ex. 3. Determine the equation of a surface which meets a given surface at the points of contact of lines which meet it in four consecutive points. [This investigation, though rather long (compare Three Dimensions, pp. 559-567) affords some useful exercise in the manipulation of our formulae. If q is the point of contact and qr the tangent touching at four consecutive points, we have ^ = 0, mSrp = SrD.^ = (), m(m- l)Sr/r=SrD2. ^ = 0, SrD».$ = 0. We may suppose the point r to lie in an arbitrary plane Srl=0, and we have to obtain the resultant of the four equations in r and finally to free it from the arbitrary l. Let Sra = and Sr6 = be the equations of planes through the generators of the quadric (r variable) Sr/r=0 which lie in the tangent plane Srp = 0. Thus we have r = [apr] and r' = [bpl] for the points in which three generators meet the arbitrary plane. One or other of these points must lie on the cubic in r. Hence SrD3 . Q . Sr'D'3 .Q' = 0, or SrD'^ . Q' . Sr'D^ .Q = 0, or . (SrD3.Sr'D'3 + S/D3.SrD'3)^$'=0, ART. 157.] FOUR POINT CONTACT. 299 where the accents applied to D and Q are temporary marks connecting operator and operand. Now this may be written in the form (4:B^-3ABC)QQ'=0, where .4 = SrDSr'D, 25 = SrD'Sr'D + Sr'DSrD', (7=SrD'Sr'D', and it is easy to express the operators A, B and C in terras of the function/. In virtue of the definition of the planes Sra = 0, Sr6 = 0, we have identically Srfr = SraSrb + SrpSrc, where Src = is some plane. Hence we find on replacing r and r' in A^ B and C by [apl] and [bpl] that A=S[plD]f[plDl B = S[plI)]f[plD'l C=S[plI)']f[plT>']. Remembering that p=fq and that Spq = 0, we have by Art. 146, Ex. 5, Sql SqB' B=-n Sql Slf-H S//-1D' =S{DSql-lSq'D)F(D'^ql-lSqI)'), SqT> Slf-^D SDf-^I)' with similar expressions for A and C, where F=nf-^ is Hamilton's auxiliary function. Writing for the moment e = Y>^ql — l'^qD and remembering that D and D' operate on Q and Q' solely and not on q as involved in the structure of the operators, we proceed to expand and to operate on ^. We have B^q^ {^eFJy . S^^ - ^eFl^qD'f . q = Sei^D'3. q. SqP - 3m(m - l)(m - 2) . n(SeFeSeFlSql^ - SeFl'-Seq^ql\ because by the identities at the beginning of this example SqT>' . Sei^D'2 . Q'=7n{m - l)(m - 2)SFefFe = m{m - l)(m -2).n. SeFe, SqI)'^.SeFD'.Q'=m{m-l)(m-2)SFep = m{m-l)(m-2).n.Seq=0, since Seq=0 and SqJ)'^ . q = 7n{m-l){m-2) . Q=0. We retain for a purpose the term in Seq. In like manner BC.q = SD'i^D' . SeFD' . q . S^^^ _ SgD'(SD'i^D' . q^eFl + 2^lFJy^eFT>' . q)^qP + S^D'2(Se/^D' . q^lFl + 2^lFiy . q . ^eFl) . S^^. The term Sg-D' . SD'i^D' . q may be reduced by writing for the moment D' = 2a'SaD', where as is easily seen 2S«a' = 4. This term becomes m{m — l)(wi — 2)2Sa/7^a' = 4m(w - l)(m — 2) . ?i, and hence we find ABC. q^^eFe . SeFD' . SD'i^D' . ^'S^^^ - m{m - l)(m - 2)n{\'^eFe'^eFl'^qP- - ^eFeSlFlSeq^ql), From these two relations we get, if e' = D'Sql — ISqD', {4B^-SABC)q==(4S€Fe'^-3SeFeSeFe'Se'Fe').Q' = (4Se/^D'3 - ^SeFeSeFUSB'FB') . Q . SqP - Sm{m - l)(m -2).7i. (SeFeSlFl - 4SeFl^)SeqSql, and the last term vanishes because 8^9' = 0. Now it will be observed that the operator in the first term is precisely the same as the original operator with D' substituted for D'Sql - lSqT>'. This remark allows us to write down the result of operating on QQ' in the form {4Bf^ -3ABC)Qq ={4SI)FD'^ -SSBFDSBFD'SD'FD') . Qq .Sql^ - 3m{m -l){m-2).n. (SB'FB'SIFI - 4^J)' FP)SqJ)' . q . Sql*, 300 PKOJECTIVE GEOMETRY. [chap. xvii. the object of the retention of the term in Sqe being now apparent. But the term Ave have retained vanishes by the reduction we have already made use of. Thus Sql^ comes off as a factor, and the equation of the surface is (4SDi^D'3 - SSD/'DSDi^D'SD'i^D') . QQ' = 0.] EXAMPLES TO CHAPTER XVII. Ex. 1. A right line meets three fixed lines aa\ hb' and cc'. The locus of the harmonic conjugate of the point of intersection on the third line with respect to the points on the other two is the intersection of the planes {hh'cq){aa'cc') + {aa cq){bb'cc') = ; {bb'c'q){aa'cc') + {aa'c'q){bb'cc') = 0. Ex. 2. The general equation of a quadric through the conic Sqfq=0, Slq = is Sqfq-SlqSrq = 0. Find the value of I' in order that the quadric may be a cone having its vertex at a and show that the equation of the cone may be written in the ^^^"^ S { ^S^a - aSlq }/{ qSla - aSlq } = 0. Ex. 3. A plane aa'p is drawn through a fixed line aa', and the lines in which it meets the planes 8^5- = and Srq = are joined to the points b and b' respectively. The equations of the joining planes are (qaa'p)Slb-(baa'p)Slq = and (qaa'p)Srb' -{b'aa'p)Srq = 0, respectively, and when p varies the locus of their intersection is the quadric surface ^^g^^ _ ^g^^^ ^g^,^, _ ^,g^,^^ ^^ ^,^ _ ^ Ex. 4. The four faces of a tetrahedron pass each through a fixed point, a, b, c and d respectively. The three edges in the face jt? which contains the point d lie in the planes, I, m and ?i respectively. The vertex q opposite the face p is the intersection of the planes SqlSap - SqpSal = 0, SqmSbp - SqpSbm = 0, ^qnScp - SqpScn = 0, and the vertex q describes the cubic surface (aSql-qSal, bSq7i~qSbm, cSqn-qScn, d) = 0, having the intersection of the fixed planes as a double point. Ex. 5. Find the locus of the vertex of a tetrahedron, if the three edges which pass through that vertex pass each through a fixed point, if the opposite face also passes through a fixed point and the three remaining vertices move in fixed planes. Ex. 6. A plane passes through a fixed point d, and the points in which it meets three fixed lines a^ag* ^1^2 '^^^ ^1^2 ^^'^ joined by planes to three other fixed lines a^a^, b^b^, and C3C4. The locus of intersection of the planes is the surface («l(«2«3«4?)-«2(«^l«3«4^)5 h{hh^i ^^^ ^^^ tangent plane to the cone along this edge touches at the point p the quadric of which the edge is a generator. (6) When p lies on the Jacobian curve [fiP^ Ap^ Ap] = % the cubic cone breaks up into a plane and a quadric cone. The cone is a member of the family of quadrics, and the plane touches at p all the quadrics of the family which pass through jt?. (c) The locus of points of contact of a plane 8^5' = with quadrics of the family is the cubic curve in which the plane cuts the surface {h M^ M^ M) = ^ '^ and the locus of points of contact of pairs of the quadrics is Ex. 12. The integral of the differential equation (dg', /^) = 0, or dq=fq.dt, where /is a linear function, may be written in the form q = e^^ . a, where a is a quaternion constant of integration. {a) This integral represents a doubly infinite family of curves, and a determinate curve of the family passes through an arbitrary point provided it is not a united point of the function/ (6) The equation p = e~^-^' .h angent line develops S6a=0, S6/'a = 0, S6/2a=0 is the reciprocal of the tangent line developable of the curve determined by a if the conditions are satisfied. (c) An arbitrary plane which does not pass through a united point of /is osculated by a single and determinate curve of the family. ■^For another form see Art. 14G, Ex. 5, 302 PROJECTIVE GEOMETRY. [chap. xvii. {d) An arbitrary tangent line to an arbitrary curve of the family is cut in a constant anharmonic ratio at the point of contact and at the points of intersection with three of the united planes of/ (e) A right line which cuts the faces of the tetrahedron in points having a certain anharmonic ratio touches a definite curve of the family, and if p and q are two points on the line {P^ 7i-space, we find that |m " 11^ — - y^mOLiU^ . • • am = 2 + ypaia2 • • • apVm -pap+iap+2 • • • «m- (XI.) \_P_ l^~-P This includes a number of relations such as SY^a/3y = a V^iSy - /^V.^ay + y V,a^ ; aYsPyS-i3Y^ayS + yY^al3S-SY^al3y = if \>^yd^ = 0. Again when the m vectors lie in a space of 171 — 1 dimensions so that they are linearly connected, we have relations of the form A- ^r 2» m - 2«oa3 . . . a»n _ 1 > oaia-,rt /vtt \ V 7. ' V-^A^/ V„i-iaia2«3 •.. «w-i which may be verified by operating with Y^a^ , etc. In particular for two and three dimensions, we recover the formulae, Art. 2T (III.) with Spa/3 = and Art. 26 (ii.). The theory explained in this article may be compared with Grassmann's Ausdehnungslehre* Grassmann's inner ]?voduct of two quantities is the function —Y^afi, and his outer product of a^, ag, . . . a^ is Yma-^a^ • • • am- These so-called products are thus only parts of a complete associative product. Aht. 159. There is a remarkable difference between this general theory and the theory of quaternions which may be illustrated by a special example. The sum of a number of vector areas is not an area vector, or the homogeneous quadratic function of the units ^ = V2aa' + y2/3/3' + V2yy + etc (i.) cannot generally be expressed in the form Ypp. The geometrical reason for this is that two planes, for example p = x{i^ + ^2'^2 ^^^ p = oo^i^-\-xJ.^, have not necessarily a common line although they may have a common point — the origin of the vectors p in the example. To discover a canonical system of vector units in terms of which a homogeneous function (q) of order m may be expressed, observe that qp = Ym+iqp-^Y,n-iqp, and that the line vector ^iqYm-iqp is not generally parallel to p but that it is a linear and distributive function of p. We are thus led to consider the linear and distributive function \ = a^^Y^Ai^, and Y^Ai^ is also an axis and it is perpendicular to i^ and parallel to i^ suppose. This shows that in terms of the canonical units A = ^12^2 + ^uhh + • • • + Ct.2m - 1, 2mi2m - 1^2.1, (HI.) SO that a quadratic in 2m + 1 or in 2m units may be reduced to Til terms involving pairs of units, or to the sum of m area vectors. There is obviously indeterminateness in the units to the extent that ij may be any unit in a definite plane — that of i-^ and i^, and ig may be any unit in another definite plane, and so on. An expression such as A corresponds to an angular velocity in the space of three dimensions. Consider the transformation which converts line vectors (p) into line vectors ((r = (pp) and which preserves unchanged lengths and mutual inclinations, so Y^a-a = Y()i, (v.) wliere cr is the component of the vector a^ which is perpendicular to every line in the flat, or in other words, where trr is the vector perpendicular from the origin to the flat. But when we know oT and the product of the vectors i we know the flat,* and we have V r 1 V ni - 1 Let J«t where U has its quaternion signification. We notice also that the point V r -i P™ = l+-^f^^ = l---^ (VII.) is the reciprocal with respect to the auxiliary quadric Y^g- = of every point in the flat — in other words, this point is the point in the 77i-flat of the origin and of the m points a which is reciprocal to the (m — l)-flat of the points a. In point symbols the equation of the flat is [gai«2...aj = 0, (VIII.) the vanishing of this equation being equivalent to (ii.). Other general expressions admit easily of interpretation on the principles laid down in this article. *The vector equation of the flat is p = uT + 2xi»i. INDEX. The Numbers refer to pages. When a chapter is quoted, the Table of Gontents may he consulted with advantage. Aberration, 85. Academy, Royal Irish, 101, 152, 163, 164, 263, 306, 308. Acceleration, trajectory of point under uniform, 64; relative, 171 ; angular, 171 ; centre, 172 ; of a particle, 184 et seq. ; of a rigid body, 194 et seq. Activity of forces on element of strained medium, 240 ; of electric and mag- netic forces, 251. Addition of vectors, 1 ; of quaternions, 9 ; of vector-arcs, 16 ; of weighted points, 264 ; of vector areas in hyper- space, 306. Algebra, vectorial, 11. Algebraic sign - , use of, 2. Algorithm of «, j, k, 11 ; for hyper- space, 303. Almucantar, example on, 176. Amplitude of versor, 27. Analytical expressions for V, 74, 225. Anchor-ting, 59. Angle of quaternion, 13; differential of, 69 ; directed, 30 ; Eulerian, 33 ; of intersection of spheres, 50 ; element of solid, 86 ; of contact, 134 ; of tor- sion, 134 ; subtended by vortex ring, solid, 235. Angular acceleration, 171 ; momentum, 184, 195; velocity, 170, 187; of emanant, 132 ; of strained element, 212 ; in hyperspace, 307. Anharmonic coordinates, 43 ; equation of sphere in, 54 ; in relation to weighted points, 269 ; ratio of collin- ear points, 41, 45; of four points in space, 56 ; of points on a conic, 267 ; generation of hyperboloid, 65 ; pro- perties of ruled surface, 139 ; of twisted cubic, 268 ; unaltered by linear transformation, 272 ; complex of lines cutting faces of tetrahedron in constant, ratio, 302. Anisotropic medium, 243, 251. Apparent double points, 292. Appendix to new edition of Elements of Quaternions, 99, 135, 211. Arc, vector-, 17; cyclic, 118; of curve, 134. Area, directed, 23 ; of spherical triangle, ;^3 ; -vector in hyperspace, 303. Areal coordinates, 48 ; velocity, 186 ; equation of continuity, 230. Aronhold's notation, 213, 297. Aspect of plane, 23. Associative addition of vectors, 2 ; multiplication of i,j, k, 10; of quat- ernions, 11, 119 ; of vectors in hyper- space, 303. A statics, 160. Astronomy, examples from, 84, 85, 130, 174, 188. " Asymptote of conic, 64 ; of curve, 152. Asymptotic cone, 107 ; lines on surface, 295. Attraction to fixed centre, particle moving under, 185, 186 ; Green's theorem, 218; spherical harmonics, 222. A usdehnungslehre, 306. Auxiliary fuiictions, % and ^, 90, 91 ; F, G and B, 274 ; quadric, 266 ; for hyperspace, 309. Axes of linear vector function, 94 ; of self -con jugate function, 96 ; of quad- ric, 111 ; of section of. Ill ; of screw- systems, 163; moving, 171 ; for curve, 134; for surface, 146; for orbit, 188; for body, 196; of inertia, 197; of elastic symmetry, 245. Axis of quaternion, 13 ; radical, 51 ; Poinsot's central, 156, 169; instan- taneous, 170. Ball, Sir R. S.. theory of screws, 156, 163, 170, 203, 205. INDEX. 311 Base-point, 171, 19o. Bilinear function, 297. Binet's theorem on axis of inertia, 197. Binomial, 134, Biquadratic equation of linear quater- nion function, 274, Biquaternions, 20, 58, Bisecting sides of spherical triangle, triangle, 31, Body, rigid, dynamics of, 196 et seq. ; under no applied forces, 198 ; dyna- mically equivalent to four particles, 199 ; dynamical constants of, 199, 202, 207 ; impact of, 203 ; con- strained, 2<>4 ; resultant force and couple on gravitating, 22.5 ; moving in fluid, 241. Bonnet's theorem, 192. Brachistochrone, 192, Bright curves on surface, 87. Bulkiness of fluid, 240, Burnside, theory of groups, 104. Calculus, icosian, 104 ; of variations, 192. Canonical, form of V, 7o ; of two linear functions, 100 ; of screw-sj'stems, 164 ; equations of quadric and linear complex, 278 ; vectors for rotation in hyperspice, 307, Cavity filled with liquid, motion of body containing, 241, 'Central, sections of quadric. 111 ; sur- faces, uon-, 117 ; axis of forces, 1.56, . 163 ; of displacement, 169 ; orbit, 186. Centre, mean, of tetrahedron, 5 ; of mass, 5, 264 ; of circle inscribed to triangle, 48 ; radical, .52 ; of quadric, 117; of curvature of curve, 1.34; of spherical curvature, 136 ; locus of mean, of corresponding points, 152 ; of furces, Hamilton's, 157 ; astatic, 160 ; of three-system of screws, 164 ; particle attracted to, 185, 186. Centres, of curvature of quadric, 122 ; surface of, 125 ; of surface, 144 ; of generalized curvature, 286, 295. Chain on surface, equilibrium of, 166. Characteristic surfaces in optics, 228. Characteristics of curves and surfaces, numerical, 290. Charpit's differential equations, 151. Chiastio homography, 208. Circle, inverse of line, 53; at infinity, ima¥:inary, .54 ; monomial equation of, o5 ; quaternion equation of, 58 ; vector equation of, 82 ; ellipse pro- jected into, 83 ; osculating, 134, 136, 152 ; surface generated by, 154 ; excluding point from integration, I Circuit, integration round, 73, 215 ; circulation and flux, 232; moving in perfect fluid, 238 ; electro- magnetic, 249. Circuitation equations for electro-mag- netic field, 250. Circular, points at infinity, 96, 126 ; sections of quadric, 113 ; of cone, 118; in relation to strain, 178; tangent cylinder, 115; point at in- finity in hyperspace, 307. Circulation of vector, 232. Circumscribed developable of confo- cals, 126 ; generalized, 286. Clifford, biquaternions, 21. Coaxial, spheres, 51, 53 ; linear vector . functions, 95, 97 ; stress and strain functions, 238. Co-efficient, differential, 63, 67 ; of .friction, 190; of restitution, 204; virtual, of screws, 206 ; of viscosity, 239. Coelostat, example on, 1.30. Coincidence, of axes of function, 94 ; of united points, 275, CoUinearity, of three points, 5, 37, 266 ; of three planes, 39. Collision of two bodies, 203, Combinatorial functions, 265, 270, 304. Commutative, addition of vectors, 1 ; multiplication, 17 ; order of diff"er- entiation, 79 ; linear functions, 95 ; small displacements, 169 ; strains, 182. Complementary curve, 291. Complex, or imaginary, 3, 20, 58 ; n^ roots of quaternion, 28 ; of right lines, 40 ; surfaces formed by lines of, 153 ; related to astatics, 161 ; of axes of inertia, 197 ; linear, '2~'y€t seq. ; of lines con necting corresponding points, 278 ; of generators of systems of quadrics, 301 ; tetrahedral, 302. Composition, of wrenches, 164, 204; of displacements, 168. Concurrence of four planes, 39, 267. Concyclic quadrics, 121, Conductivity, electrical, 251, Cone, tangent to sphere, 49 ; to quadric 108 ; to confocal, 124 ; standing on curve, 65 ; of axes of system, 0^ -f- <02» 101 ; asymptotic,. 107 ; edges of, in plane, ilO; and sphero-conic, 118; through five lines, 121 ; of revolu- tion through three lines, 126 ; differ- ential equation of, 149; tangeut to generalized confocal, 281. Confocal, quadrics, 121 ; tangent cones, 124 ; vector equation of, 124 ; re- lated to astatics, 162 ; related to axes of inertia, 197 ; equipotential system, 228 ; generalized confocals, 279 ; quaternion equation of, 286. 312 INDEX. Congruency of lines, 41 ; surfaces generated by lines of, 153 ; focal and extreme points, 153 ; of axes of three-system of screws, 164, 103. Conic, related to triangle, 48 ; vector equations of, 63; focal, 114, 126; sphero-, 118 ; and Pascal hexagon, 121 ; orbit, 187 ; on wave-surface, 261 ; in point symbols, 264, 267 ; anharmonic property of, 267. Conical refraction, elastic solid, 248 ; dielectric, 260, Conical rotation, represented by q { )q~'^, 18 ; related to spherical triangle, 32 ; in terms of Euler's angles, 33 ; inscribed polygon, 55 ; examples, 60; differential oi qaq~^, 69, 169 ; and linear function, 100 ; and astatics, 160 ; finite displace- ments, 168 ; examples, 173 ; strain, 178 ; and linear quaternion function, 283 ; in hyperspace, 307. Conicoid, see Quadric. Conjugate, of quaternion, 12 ; of pro- duct, 15 ; radii of conic, 63; of linear function, 89 ; axes of function and of its, 94 ; quadric, 107 ; radii of qua- dric, 110, 112; of quaternion function, 273, 275 ; tangents, 295. See Self-conjugate. Connected region, 217. Conservative system of forces, 184 ; acting on perfect fluid, 238. Constant, curve having ratio of curva- ture to torsion, 137. Constants of linear function, 88, 178 ; vector, of integration, 137, 186; dy- namical, of rigid body, 199, 202, 207; elastic, 239, 244 ; dielectric, 251 ; of quaternion function, 272. Constrained motion of particle, 189; of rigid body, 204. Construction of product of two quater- nions, 15 ; fourth proportional to three vectors, 31 ; ellipsoid, 114 ; vectors related to wave in dielectric, 259. Contact, of line and sphere, 49 ; and quadric, 107 ; and confocals, 124 ; four point, of tangent, 298. Continuity, equation of, 72, 230, 238 ; areal and linear, 230. Convention respecting rotation, 7 ; nota- tion, 20. Convergence of vector, 72, 212. Co-ordinates, six, of a line, 40 ; anhar- monic, 43, 48, 54, 269 ; curvilinear, 66, 74, 227 ; Cartesian, 75 ; elliptic, 124, 286; homogeneous or tetrahedral, 268* Coplanar versors, 27. Coplanarity of four points, 5, 38 ; in point symbols, 266. Co-reciprocal screws, 206. Co-residuals on cubic, 101. Correspondence, see nomographic, Transformation. Cov^ariant linear functions, 101, 290. Cremona transformation, 101. Cross ratio, see anharmonic. Crystalline medium, damped oscilla- tions in, 186 ; propagation of light in, 256. Cubic, of linear vector function, 93, 100 j twisted, 93, 104; cone, 101; twisted, locus of feet of normals, 109 ; of points of contact with confocals, 123; tangent line and osculating plane, 133 ; related to moving body, 172 ; developable generated by, 267 ; locus of points in perspective with corre- spondents, 278 ; transformation of, 285 ; characteristics of, 293. Curl of vector, Wo-, 73, 213. Current, electric and magnetic, 250. Curvature, of curve, 132 et seq. ; of surfaces, 141 ef seq., 215 ; of quadric, 122, 125 ; of orbit, 189 ; generalized, 286, 295. Curve, in terms of parameter, 62 ; of intersection of confocals, 125 ; me- trical properties of, 131 et seq. ; uni- cursal, 152 ; intersection of quadrics, 285 ; complementary, 291 ; character- istics of, 292. Curves, family of, 148 ; ^ = {/+ tf'a, 286; q = e*f.a, 301. Curvilinear coordinates, 66, 74, 124, 226. Cusp, condition for, 63, 83. Cuspidal edge, 126, 136, 268, 286. Cyclic planes of quadric, 113, 178; arcs of sphero-conic, 118. Cyclical transposition under sign S, 16» Cycloid, 83, 193. Cylinder, right circular, 45 ; standing on curve, 65 ; circular tangent, to quadric, 115; case of general quadric, 117 ; geodesic on, 137; torsal tangent planes of, 140 ; differential equation of, 149 ; related to astatics, 161. Cylindroid, 84, 165. D symbol of diflferentiation, 229 ; of operator analogous to V, 296. Damped oscillations, 186. Deformation of surfaces, 145, Degraded, cases of quaternions, 9, 19 ; symbolic equations, 95, 275. Degree, see Order. Degrees of freedom, 204. Delta, Hamilton's operator V, 70, 21 U See Operator. INDEX. 313 De Moivre's theorem, 27. Derivative, 63. Determinants and combinatorial func- ^ tions, 270, 305. Developable surface, 65 ; circumscribing confocals, 126 ; related to curve, 135, 139 ; generated by tangent planes along curve on surface, 142; of twisted cubic, 267 ; circumscribing quadrics, 280 ; tangent-line, of two quadrics, 282; circumscribing generalized con- focals, 286. Development of quaternion function, 79, 85 ; of vector of curve in terms of arc, 134. Deviation from osculating curve, 152. Diaphragm, 217. Dielectric, 251 et seq. DifiFerence of two points, 263. Diflt'erential, 63, 66 ; condition for per- fect, 74, 86, 214 ; indeterminate, 87 ; of equation of surface, 142 ; equation of geodesic, 141, 152; of lines of curvature, 144, 147 ; of family of surfaces, 149 ; of curves traced on surfaces, 287- Differentiation, chap, vii., 62; general formula, 66 ; successive, 79 ; with respect to moving axes, 167 et seq. ; of deformable elements, 212 ; follow- ing moving point, 229. Diffusion of electromagnetic disturb- ance, 255. Dilatation in strain, 178. Direct and inverse similitude, 14. Directed area, 23 ; angle, 31 ; curva- ture, 132, 141 ; volume in hyperspace, 304. Discontinuity in integration, 216. Displacement, of a body, 18, chap, xii., 168 ; in strain, 180 ; electric, 250. Dissipation function, 240, 252. Dissociative multiplication, 11. Distortion of elements, 212, 229 ; of vnscous fluid, 238. Distributive, multiplication of vector by scalar, 4 ; by vector, 8 ; property of scalar of product, 6 ; of product, 9 ; of diflferential, 66 ; of linear func- tion, 88 ; multiplication for hyper- space, 303. Disturbance in electromagnetic field propagated by waves or by diffusion, 255. Divergence of vector, 212. Divisiou, of vectors reduced to multi- plication, 11 : homographic, 41, 65, 1.52, 264. Dodecahedron, 104. Double points, on wave surface, 248, 261 ; apparent, 292; on Jacobian, 298. Duality for point symbol, 265. Dynamical constants of a body, 199, 202, 207. Dynamics, of a particle, chap, xiv., 184; of system and rigid body, chap. XV., 194; of continuous medium, 236; electro-, 249. Eight square roots of linear function, 99; umbilical generators, 125; gen- eralization of, 279, 286 ; generators which are also normals, 279. Elastic solid, isotropic; 222, 239; aniso- tropic, 242 et seq. ; symmetry, 245. Electro-magnetic theory, 249 et seq. ; of light, 256. Element, rate of change of, 212, 229. Elements of Quaternions referred to, I, 3, 7, 29, 31, 34, 45, 53, 55, 56, 59, 82, 85, 114, 118, 120, 121, 132, 156, 157, 197, 211, 264; appendix to, 99, 135, 211, Elimination of a vector, 39, 105. Ellipse, vector equation of, 63, 82 ; pro- jected into circle, 83 ; parallactic, 85 ; aberrational, 85 ; differential equation of surface generated by, 149 ; related to astatics, 163 ; locus of feet of per- pendiculars on generators of cylin- droid, 166 ; in conical refraction, 261. Ellipsoid, -^ Hamilton's construction for, 114 ; vector equation of, 152 ; strain, 177. Ellipsoidal linear function, 178. Elliptic, logarithmic spiral, 82 ; co- ordinates, 124 ; functions, 198 ; gen- eralized, co-ordinates, 286. Elongation, 181. Emanant, 131, 138. Energy equation, for particle, 184, 187 r system of particles, 194 ; rigid body, 197 ; for impulses, 200 ; for contin- uous medium, 239 ; in electro-mag- netic theory, 251 ; function, for elastic solid, 243 ; for dielectric, 252. Envelope, examples, 128, 129 ; differen- tial equation of, 149, 151 ; wave- surface as, 248, 257. Epicycloid, 83. Equality of vectors, 1 ; vector-arcs, 17 ; points, 263. Equilibrium, static, 156 ; astatic, 16(». Equipotential surfaces, 227. Euler's angles, 33 ; four square identity, 16 ; exponential formulae, 28 ; theorem on curvature, 143; equations of motion of rigid body, 196 ; of fluid, 230, 238. Evoked wrench, 201. E volutes on polar developable, 139. * See Linear vector function, the use of an eUii>8oid being to a great extent superseded. 314 INDEX. Exact differential, 74, 86, 214. Excentricity of orbit, 187. Excess, spherical, .S3. Expansion, of quaternion function, 79, 85 ; of vector of curve in terms of arc, 134 ; in series of spherical harmonics, 223, 224. Exponential of quaternion, 28, 34 ; differential of, 86 ; for hyperspace, .308. Extreme points on line of congruency, 154. Families of curves and surfaces, 148. Family of equipotential surfaces, 227 ; of curves, q-{f+ty''a, 2S6 ; q=e^''a, 301. Five vectors, 43, 44, 54 ; quaternions, 43, 269 ; points linearly transformed into five, 272 ; surfaces, 291. Flat space, 303 ; symbol of, 308. Flow of a vector, 231. Fluid, motion, 72, 229, 236 ; viscous, 238, 240 ; motion of solid in, 241. Flux through circuit, 233 ; strength of tube of, 233, 235 ; in electro-magnetic theory, 249; of radiated energy, Poynting, 252, 257. Focal, property of quadrics, Salmon's, 114; form of equation, 116; for sphero-conic, 120 ; conies on develop- able, 126 ; points on line of con- gruency, 153 ; conies related to astatics, 162. Foci of central sections of quadric, 129. Force, moment of, 23 ; in statics, 156 ; in dynamics, 184, 194 ; central, 186 ; impulsive, 200 ; electric and magnetic, . 251 ; in electro-magnetic field, me- chanical, 255. Forces, reduction to two, 158 ; con- servative, 184, 238 ; of interaction, 194 ; system of forces, see Wrench. Formula, a, 11 ; b, 8; of differentiation, 66. Formulae, depending on products of vectors, chap, in., 23; of trigono- metry, 25, 30. Four numbers involved in quaternion, 9 ; squares, identity connecting, 16 ; vectors, identities connecting, 24 ; symmetrical relations for, 42 ; linear function rendering four vectors par- allel to, 92 ; particles equivalent to rigid body, 199 ; -system of screws, 201) ; consecutive points on tangents to surface, surface through, 298. Fourth proportional to three vectors, 31. Fractions, relations reduced by partial, 122. Freedom, degrees of, 204. Fresnel, 163, 262. Frictional constraint, 190. Function, anharmonic, of collinear points, 41, 45, of points in space, 56, on a conic, 267 ; linear vector, 88 ; elliptic, 198; dissipation, 240; energy, for elastic solid, 243, for dielectric, 252 ; combinatorial, 270, .304 ; linear quaternion, 272. See Linear function. J Fundamental formulae of trigonometry, ] plane, 25 ; spherical, 30. Gate, self-clos'ng, 207. Gauss, operator, 104 ; measure of cur- vature. 144, 147 ; integration theorem, 215. Generalised, normal, 279 ; curvature, 286, 295 ; geodesic, 287. Generation of ruled quadric, 65 ; of ellipsoid, 114 ; of ruled surface, 137. Generators of quadric, 103, 116 ; um- bilical, 125 ; common, and of linear complex, 278 ; generalized umbilical, 279, 286 ; eight, are also normals, 279 ; complex of, of doubly infinite family of quadrics, 301. Geodesic on cylinder, 137 ; differential equation of, 141, 152; curvature, 141, 148 ; Joachimstal's theorem, 152 ; motion of particle along, 190 ; gen- eralized, 287. Geometrical meaning of invariants, 98, 288. Geometry of Three Dimensions, Salmon's, 291, 292, 298. Geometry, projective, chap, xvii., 263, 308. Gilbert's theorem on confocals, 124. Grassmann, 306. Graves, R. P., Life of Hamilton re- ferred to, 16, 211. Gravitating body in field of force, 225. Green's theorem adapted to quaternions, 219. Groups, theory of, examples relating to, 80 ; referred to, 104. Half-line, half-cone, 45. Harmonic, mean of two vectors, 41, 50, 56, 109 ; properties of triangle, 45, of polar and quadric, 50, 109 ; spherical, 70, 76, 222 et seq. Hathaway, A. S., 270. Heaviside, Oliver, 11, 249, 250, 253. Helix, v^ector equation of, 64, 82 ; vector twist of, 133 ; constant curva- ture and torsion, 137 ; osculating, 152; particle moving on, 191. Herpolhode, 198. Hessian of surface, 297. Hexagon, Pascal, 121. f INDEX. 315 Hiyhtr Plane Curves, Salmon's, referred to, 101, 105. Hodograph, 83, 187, 189. Homographic, ranges, 41, 42, 264 ; . locus of line joining corresponding points, 65 ; locus of mean centre of points on, 15*2 ; screw-systems, 208 ; correspondence of points on twisted cubics, 285. Homography, chiastic, 208. Hooke's law, 243. Hydrodynamics, 72, 228 et seq. Hyperbola, 64 ; section of quadric, rectangular, 111 ; focal, 114. Hyperboloid, homographic generation, 65, 264 ; locus of transversals, 103, 270 ; generators of, 1 16 ; Une of striction of, 140 ; equilibrating forces on generators of, 158. Hyperspace, chap, xviii., 303. Hypocycloid, 83. Hysteresis, 251. Icosian calculus, 104. Identity, Euler's four square, 16 ; con- necting four vectors, 24 ; live quater- nions, 269. Ikosahedron, 104. Imaginary, of algebra, 3, 20, 58 ; n^^^ roots of quaternions, 28 ; roots and t '• axes of linear function, 95, 96, 177 ; conjugate, vectors, 95, 224, 307 ; united points of linear transforma- tion, 275, 276. Impact of two bodies, 203. Impulse, 200. Impulsive wrench, 201, 204 ; genera- ting motion of solid in fluid, 241. Indeterminatenpss of versor of null i quaternion, 19; of tensor of bi- 5 ' quaternion, 21 ; of a differential, 87 ; • in solution of equations, 92 ; of axes - of linear function, 95, 96 ; of square roots of function, 99 ; in value of function, 216 ; related to conical refraction, 248, 260 ; of normal to plane in hyperspace, 303. Index-surface, 248, 261. Induction, magnetic, 250. Inertia function for rigid body, 196 ; Binet's theorem on axes of, 197 ; deduced from observed motion, 199, 202, 207 ; related to jqchii, 225. Infinites in field of integration, 216, 219. Jutinitv, anharmonic equation of plane at, 44, of circle at, 54 ; vector to circular points at, 96, 126, 307 ; vector representing point at, 263 ; equation of plane at, 266. Inflexion on curve, 83. Initial positions in astatics, 160. Inscription of polygon to sphere, 55, 56. Instantaneous twist-velocity, 170, 201 ; orbit, 188. Integrability, condition of, 74, 86, 214. Integrals, line, 73, 215, 219, 231 ; sur- face, 72, 215, 219, 233 ; variation of, 192, 231, 233. Intensity of wrench, 163. Interaction of particles, 194, 200, 236. Interpretations and formulae, chap iii., 23 ; for projective geometry, 263 et seq. Intersection of, line and plane, 35, 267, 269 ; planes, 39, 267, 269, 306 ; two lines, 39, 267 ; line and sphere, 49 ; spheres, 50, 54 ; confocals, 121, 123, 125 ; quadrics, 285 ; generalized con- focals, 286 ; curve and complemen- tary, 292 ; of two surfaces, osculating plane to curve of, 296. Invariants, of linear vector functions, 91, 97 ; geometrical meaning of, 98 ; of two functions, 100; derived by operation of V, 102 ; depending on V, 211 ; of linear quaternion function 274 ; of quadrics and linear trans- formations, 288. Inverse, or reciprocal of vector. 1 1 ; of product 12 ; similitude, 14 ; trans- formation, 90 ; operations of V, 218. Inversion, geometrical, 52, correspond- ing elements in, 69 ; of linear func- tions, 90; of