MATHEMATICAL PAPEE8. C. J. CLAY & SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AYE MARIA LANE. DEIGHTON, BELL AND CO. F. A. BROCKHAUS. THE COLLECTED MATHEMATICAL PAPERS OF AETHUE CAYLEY, Sc.D., F.E.S., SADLEKIAN PROFESSOR OF PURE MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE. VOL, I. UNIVERSITY, CALIFORNIA CAMBRIDGE : AT THE UNIVERSITY PRESS. 1889 [All Eights reserved.] CAMBRIDGE : PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. PBEFACE. "DATHER more than a year ago I was requested by the Syndics of the -LlJ University Press to allow my mathematical papers to be reprinted in a collected form : I had great pleasure in acceding to a request so com plimentary to myself, and I willingly undertook the work of superintending the impression of them, and of adding such notes and references as might appear to me desirable. The present volume contains one hundred papers (numbered 1, 2, 3, ...,100) originally published in the years 1841 to 1853. They are here reproduced nearly, but not exactly, in chronological order : and as nearly as may be in their original forms ; but in a few cases where the paper is con troversial, or where it is a translation (into French or English) of an English or French paper, only the title is given : there are in some few cases omissions which are indicated where they occur. The number is printed in the upper inside corner of the page ; it is intended that the numbers shall run consecutively through all the volumes ; and thus a paper can be referred to simply by its number. I have of course corrected obvious typographical errors, and in particular have freely altered punctuation, but I have not attempted to verify formulae. Additions are made in square brackets [ ] ; to avoid confusion with these, square brackets occurring in the original papers have in general been changed into twisted ones { }, but where they occur in a formula it was not always possible to make the alteration. The addition in a square bracket is very frequently that of a date : it appears to me that the proper reference to a serial work is by the number of the volume, accompanied by the date on the title page : the date is always useful, and, in the case of two or more series of a Journal or set of Transactions, we avoid the necessity of a reference to the series; Liouville t. I. (1850) is better than Liouville, Serie 2, t. I. I regret that this rule has not been strictly followed as regards the titles of some of the earlier papers, see the remark at the end of the Contents. A. CAYLEY. January 23, 1889. CONTENTS. [An Asterisk denotes that the Paper is not printed in full.] PAGE 1. On a Theorem in the Geometry of Position .... Camb. Math. Jour. t. n. (1841), pp. 267271 (1841) 2. On the Properties of a certain Symbolical Expression . . 5 Camb. Math. Jour. t. in. (1843), pp. 6271 (1842) 3. On certain Definite Integrals Camb. Math. Jour. t. in. (1843), pp. 138144 (1842) 4. On certain Expansions, in series of Multiple Sines and Cosines 19 Camb. Math. Jour. t. in. (1843), pp. 162167 (1842) 5. On the Intersection of Curves .... Camb. Math. Jour. t. in. (1843), pp. 211213 (1843) 6. On the Motion of Rotation of a Solid Body . . . 28 Camb. Math. Jour. t. in. (1843), pp. 224232 (1843) 7. On a class of Differential Equations, and on the Lines of Curvature of an Ellipsoid .... Camb. Math. Jour. t. in. (1843), pp. 264267 (1843) 8. On Lagrange s Theorem ..... Camb. Math. Jour. t. m. (1843), pp. 283286 (1843) 9. Demonstration of Pascal s Theorem ... Camb. Math. Jour. t. iv. (1845), pp. 1820 (1843) ir> 10. On the Ttieory of Algebraical Curves ... Camb. Math. Jour. t. iv. (1845), pp. 102112 (1844) 11. Chapters in the Analytical Geometry of (n) Dimensions . 55 Camb. Math. Jour. t. iv. (1845), pp. 119127 (1844) /*o 12. On the Theory of Determinants . Camb. Phil. Trans, t. vin. (1849), pp. 116 (1843) CONTENTS. PAGE 13. On the Theory of Linear Transformations .... Camb. Math. Jour. t. iv. (1845), pp. 193209 14. On Linear Transformations . . . . 95 Camb. and Dubl. Math. Jour. t. I. (1846), pp. 104122 15. Note sur deux Formules donnees par MM. Eisenstein et Hesse . 113 Crelle, t. xxix. (1845), pp. 5457 Memoire sur les Hyperdeterminants . . . . 117" Crelle, t. xxx. (1846), pp. 137 *17. Note on Mr Bronwin s paper on Elliptic Integrals . . . 118 Camb. Math. Jour. t. in. (1843), pp. 197, 198 *18. Remarks on the Rev. B. Bronwin s paper on Jacob? s Theory of Elliptic Functions . . . . . ." - 119 Phil. Mag. t. xxn. (1843), pp. 358368 19. Investigation of the Transformation of certain Elliptic Functions 120 Phil. Mag. t. xxv. (1844), pp. 352354 20. On certain results relating to Quaternions . * -~*^ . 123 Phil. Mag. t. xxvi. (1845), pp. 141145 *21. On Jacobis Elliptic Functions, in reply to the Rev. B. Bromvin: and on Quaternions . . . . . - 127 Phil. Mag. t. xxvi. (1845), pp. 208211 22. On Algebraical Couples ........ 128 Phil. Mag. t. xxvn. (1845), pp. 3840 23. On the Transformation of Elliptic Functions . . . . 132 Phil. Mag. t. xxvn. (1845), pp. 424427 24. On the Inverse Elliptic Functions . . . . . . 136 Camb. Math. Jour. t. iv. (1845), pp. 257277 25. Memoire sur les Fonctions doublement periodiques . . . 156 Liouville, t. x. (1845), pp. 385420 26. Memoire sur les Courbes du Troisieme Ordre . . . . 183 Liouville, t. ix. (1844), pp. 285293 27. Nouvelles remarques sur les Courbes du Troisieme Ordre . . 190 Liouville, t. x. (1845), pp. 102109 28. Sur quelques Integrates Multiples ...... 195 Liouville, t. x. (1845), pp. 158168 29. Addition a la Note sur quelques Integrates Multiples . . 204 Liouville, t. x. (1845), pp. 242244 CONTENTS. \ y ix 30. Memoire sur les Courbes a double Courbure et les Surfaces developpables . .. . . . . . . . 207 Liouville, t. x. (1845), pp. 245250 31. Demonstration d un Theoreme de M. Chasles . . . . 212 Liouville, t. x. (1845), pp. 383, 384 32. On some Analytical Formula and their application to the Theory of Spherical Coordinates . . . . . . . 213 Camb. and Dubl. Math. Jour. t. i. (1846), pp. 2233 33. On the Reduction of du^^U, ivhen U is a Function of the Fourth Order ......... 224 Camb. and Dubl. Math. Jour. t. i. (1846), pp. 7073 34. Note on the Maxima and Minima of Functions of Three Vari ables f ........... 228 Camb. and Dubl. Math. Jour. t. i. (1846), pp. 74, 75 35. On Homogeneous Functions of the Third Order with Three Vari ables ........... 230 Camb. and Dubl. Math. Jour. t. i. (1846), pp. 97104 / 36. On the Geometrical Representation of the Motion of a Solid Body ........... 234 Camb. and Dubl. Math. Jour. t. i. (1846), pp. 164167 * 37. On the Rotation of a Solid Body round a Fixed Point . . 237 Camb. and Dubl. Math. Jour. t. i. (1846), pp. 167173 and 264274 38. Note on a Geometrical Theorem contained in a Paper by Sir W. Thomson ......... 253 Camb. and Dubl. Math. Jour. t. I. (1846), pp. 207, 208 39. On the Diametral Planes of a Surface of the Second Order . 255 Camb. and Dubl. Math. Jour. t. i. (1846), pp. 274278 40. On the Theory of Involution in Geometry ..... 259 Camb. and Dubl. Math. Jour. t. n. (1847), pp. 5261 41. On certain Formulce for Differentiation, with applications to the evaluation of Definite Integrals . . . . . 267 Camb. and Dubl. Math. Jour. t. n. (1847), pp. 122128 42. On the Caustic by Rejection at a Circle . . . .273 Camb. and Dubl. Math. Jour. t. n. (1847), pp. 128130 , 43. On the Differential Equations which occur in Dynamical Problems 276 Camb. and Dubl. Math. Jour. t. n. (1847), pp. 210219 c. b x CONTENTS. PAGE 44. On a Multiple Integral connected with the Theory of Attractions 285 Camb. and Dubl. Math. Jour. t. n. (1847), pp. 219223 45. On the Theory of Elliptic Functions ... . . 290 Camb. and Dubl. Math. Jour. t. n. (1847), pp. 256266 46. Note on a System of Imaginaries . . . . 301 Phil. Mag. t. xxx. (1847), pp. 257, 258 47. Sur la Surface des Ondes .... ... 302 Liouville, t. xi. (1846), pp. 291296 48. Note sur les Fonctions de M. Sturm 306 Liouville, t. XL (1846), pp. 297299 49. Sur quelques Formules du Calcul Integral .... 309 Liouville, t. xn. (1847), pp. 231240 50. Sur quelques Tlieor ernes de la Geometric de Position 317 Crelle, t. xxxi. (1846), pp. 213227 51. Probleme de Geometric Analytique .... 329 Crelle, t. xxxi. (1846), pp. 227230 52. Sur quelques Proprietes des Determinants Gauches . 332 Crelle, t. xxxn. (1846), pp. 119123 53. Recherches sur T Elimination, et sur la Theorie des Courses . 337 Crelle, t. xxxiv. (1847), pp. 3045 54. Note sur les Hyperdeterminants . . . . . . . 352 Crelle, t. xxxiv. (1847), pp. 148152 55. Sur quelques Theor ernes de la Geometric de Position . . 356 Crelle, t. xxxiv. (1847), pp. 270275 56. Demonstration of a Geometrical Theorem of Jacobi s 362 Camb. and Dubl. Math. Jour. t. in. (1848), pp. 48, 49 57. On the Theory of Elliptic Functions. . . . . . 364 Camb. and Dubl. Math. Jour. t. m. (1848), pp. 50, 51 58. Notes on the Abelian Integrals Jacobi s System of Differential Equations .......... 366 Camb. and Dubl. Math. Jour. t. in. (1848), pp. 51 54 59. On the Theory of Elimination . . . . . . 370 Camb. and Dubl. Math. Jour. t. in. (1848), pp. 116120 60. On the Expansion of Integral Functions in a series of Laplace s Coefficients .......... 375 Camb. and Dubl. Math. Jour. t. in. (1848), pp. 120, 121 CONTENTS. xi PAGE 61. On Geometrical Reciprocity . . . . . . , 377 Camb. and Dubl. Math. Jour. t. in. (1848), pp. 173179 62. On an Integral Transformation . . . . . . 383 Camb. and Dubl. Math. Jour. t. in. (1848), pp. 286, 287 63. Demonstration d un Theoreme de M. Boole concernant des Integrales Multiples . . . . . . . . 384 Liouville, t. xm. (1848), pp. 245248 64. Sur la generalisation d un Theoreme de M. Jellett qui se rapporte aux Attractions . . . . . . . . , 338 Liouville, t. xm. (1848), pp. 264268 65. Nouvelles Recherches sur les Fonctions de M. Sturm . . 392 Liouville, t. xm. (1848), pp. 269274 66. Sur les Fonctions de Laplace . . . . . . . 397 Liouville, t. xin. (1848), pp. 275280 67. Note sur les Fonctions Elliptiques ...... 402 Crelle, t. xxxvn. (1848), pp. 5860 68. On the application of Quaternions to the Theory of Rotation . 405 Phil. Mag. t. xxxin. (1848), pp. 196200 69. Sur les Determinants Gauches . . . . . . . 410 Crelle, t. xxxvin. (1848), pp. 9396 70. Sur quelques Theoremes de la Geometric de Position . . 414 Crelle, t. xxxvni. (1848), pp. 97104 71. Note sur les Fonctions du Second Ordre . . . . . 421 Crelle, t. xxxvin. (1848), pp. 105, 106 72. Note on the Theory of Permutations ..... 423 Phil. Mag. t. xxxiv. (1849), pp. 527529 73. Abstract of a Memoir by Dr Hesse on the construction of the Surface of the Second Order which passes through nine given points ......... . . 425 Camb. and Dubl. Math. Jour. t. iv. (1849), pp. 4446 74. On the Simultaneous Transformation of Two Homogeneous Functions of the Second Order . . . . . .428 Camb. and Dubl. Math. Jour. t. iv. (1849), pp. 4750 75. On the Attraction of an Ellipsoid . . . . ... 432 Camb. and Dubl. Math. Jour. t. iv. (1849), pp. 5065 62 Xll CONTEXTS. PAGE 76. On the Triple Tangent Planes of Surfaces of the Third Order 445 Camb. and Dubl. Math. Jour. t. iv. (1849), pp. 118132 77. On the order of certain Systems of Algebraical Equations . 457 Camb. and Dubl. Math. Jour. t. iv. (1849), pp. 132137 /78. Note on the Motion of Rotation of a Solid of Revolution . . 462 Camb. and Dubl. Math. Jour. t. iv. (1849), pp. 268270 79. On a System of Equations connected with Malfattis Problem, and on another Algebraical System . . . .. . 465 Camb. and Dubl. Math. Jour. t. iv. (1849), pp. 270275 80. Sur quelques Transmutations des Lignes Courbes . . . 471 Liouville, t. xiv. (1849), pp. 4046 81. Addition au Memoire sur quelques Transmutations des Lignes Courbes . . . . . . . . . . 475 Liouville, t. xv. (1850), pp. 351356 82. On the Triadic Arrangements of Seven and Fifteen Things . 481 Phil. Mag. t. xxxvn. (1850), pp. 5053 *83. On Curves of Double Curvature and Developable Surfaces . 485 Camb. and Dubl. Math. Jour. t. v. (1850), pp. 18 22 84. On the Developable Surfaces which arise from two Surfaces of the Second Order ........ 435 Camb. and Dubl. Math. Jour. .t. v. (1850), pp. 46 57 85. Note on a Family of Curves of the Fourth Order . . 496 Camb. and Dubl. Math. Jour. t. v. (1850), pp. 148 152 86. On the Developable derived from an Equation of the Fifth Order 500 Camb. and Dubl. Math. Jour. t. v. (1850), pp. 152 159 *87. Notes on Elliptic Functions (from Jacobi) . . 597 Camb. and Dubl. Math. Jour. t. v. (1850), pp. 201 204 88. On the Transformation of an Elliptic Integral . . . . 508 Camb. and Dubl. Math. Jour. t. v. (1850), pp. 204 206 89. On the Attraction of Ellipsoids (Jacobi s Method) . . 511 Camb. and Dubl. Math. Jour. t. v. (1850), pp. 217 226 90. Note sur quelques For mules relatives aux Coniques . . 519 Crelle, t. xxxix. (1850), pp. 13 91. Sur le Probleme des Contacts . . . . . 522 Crelle, t. xxxix. (1850), pp. 4 13 CONTENTS. xiii PAGE 92. Note sur un Systeme de certaines Formules . . . . 532 Crelle, t. xxxix. (1850), pp. 14, 15 93. Note sur quelques Formules qui se rapportent a la Multiplication des Fonctions Elliptiques . . . . . . , 534 Crelle, t. xxxix. (1850), pp. 1622 94. Note sur I Addition des Fonctions Elliptiques . . . . 540 Crelle, t. XLI. (1851), pp. 5765 95. Note sur quelques Theor ernes de la Geometric de Position. . 550 Crelle, t. XLI. (1851), pp. 6672 96. Memoir e sur les Coniques inscrites dans une meme Surface du Second Ordre . ......... 557 Crelle, t. XLI. (1851), pp. 7386 97. Note sur la Solution de I Equation x 1=0 . . . . 564 Crelle, t. XLI. (1851), pp. 8183 """98. Note relative a la sixieme section du Memoir e sur quelques Theor ernes de la Geometric de Position . . . . . 567 Crelle, t. XLI. (1851), p. 84 99. Note sur quelques Formules qui se rapportent a la Multiplication des Fonctions Elliptiques . . . . . . . 568 Crelle, t. XLI. (1851), pp. 8592 100. Note sur la Theorie des Hyperdeterminants . . . . 577 Crelle, t. XLII. (1851), pp. 368371 Notes and References . . . . . . . . . . 581 Volumes n, in, and iv of the Cambridge Mathematical Journal have on the title pages the dates 1841, 1843, 1845 respectively, and volume vin of the Cambridge Philosophical Transactions has the date 1849. As each of these volumes extends over more than a single year, I have added, see p. vii, the year of publication for the papers 1, 2, ... 12. In all other cases, the year of publication is shown by the date on the title page of the volume. XV S^&eStU f OF THE (tTNIVJGRsiTT, CLASSIFICATION. GEOMETRY Intersection of curves, 5 Motion of solid body, 6, 36, 37, 68, 78 Theory of algebraical curves, 10 Analytical geometry of n dimensions, 11 Curves and developables, 30, 83 Geometrical involution, 40 Geometry of position, 50, 55, 70, 95, 98 Theory of Curves, 53 Reciprocity, 61 Transmutations of curves, 80, 81 Distances of points, 1 Lines of curvature of ellipsoid, 7 Pascal s theorem, 9 Transformation of coordinates, 20 Cubic curves, 26, 27 Skew Cubics, theorem of Chasles , 31 Spherical coordinates, 32 Theorem of Sir W. Thomson, Equimomental surfaces, 38 Diametral planes of quadric surface, 39 Caustic by reflexion of circle, 42 Wave surface (Tetrahedroid), 47 Centres of similitude of quadric surfaces, 51 Theorem of Jacobi, Confocal surfaces, 56 Theorem of Jellett, Attraction and surface of ellipsoid, 64 Quadric surfaces, 71 Abstract of memoir by Hesse, Quadric surfaces, 73 Cubic surfaces, triple tangent planes, 76 Developable from two quadric surfaces, 84 Quartic curves, 85 Developable from quintic curve, 86 Formulae for two conies, 90 Problem of contacts, 91 Conies inscribed in a quadric surface, 96 XVI CLASSIFICATION. ANALYSIS Determinants, 1, 9, 12, 52, 61, 69 Attractions and multiple integrals, 2, 3, 28, 29, 41, 44, 56, 63, 64, 88 Attraction of Ellipsoids, Legendre, 75 ; Jacobi, 89 Linear transformations, &c. 12, 13, 14, 15, 16, 33, 34, 35, 39, 53, 54, 71, 74, 77, 100 Elliptic functions, 17, 18, 19, 21, 23, 24, 25, 33, 45, 57, 58, 62, 67, 87, 88, 93, 94, 99 Expansions in multiple sines and cosines, 4 Differential equations, 7, 43, 58 Lagrange s theorem, 8 Quaternions, 20, 68 Octaves, 21, 46 Algebraical couples, 22 Maxima and minima, 34 Differentiation, 41 Sturmian functions, 48, 65 Integral Calculus, 49, 62 Elimination, 53, 59 Legendre s coefficients, 60 Laplace s functions, 66 Arrangements, 72, 82 Systems of equations, 32, 77, 79, 90, 92, 96 Equation x 287 - 1 = 0, 97 1] f OF THE (UNIVERSITY \^ C 1. ON A THEOREM IN THE GEOMETRY OF POSITION, [From the Cambridge Mathematical Journal, vol. II. (1841), pp. 267 271.] WE propose to apply the following (new ?) theorem to the solution of two problems in Analytical Geometry. Let the symbols , P , > 7 , P , 7 , &c. ", P", 7 denote the quantities a, a/3 <* P, aft y" aft"y 4- a P"y a fiy" + a"Py a P y, &c. (the law of whose formation is tolerably well known, but may be thus expressed, a, /3 a. ^*j = QJ yg 1 Qj a , /3 , P , 7 a , P, 7 _ ^. P", 7 + 7" * r 7//! 4" /9..T 1 01 > P > 7 ,Q/ / > Oi/ ^ P> 7 i the signs 4- being used when the number of terms in the side of the square is odd, and + and alternately when it is even.) Then the theorem in question is a + a- P + r 7.. , p a! + cr $ + T 7 .., p a." 4- a (3" + T y".. a + a- p + r y.. , p a. + a ft + T y .. , p a." + a /3" + r y".. = r"/3 +r"y .. , p"o." + a" ft" 4- r"y".. \ P , " , r ", P", i -. p , (7 , T .. // // // p , a- , T .. a. , & , 7 .. a , P , 7 .. c. ON A THEOREM IN THE GEOMETRY OF POSITION. [1 (This theorem admits of a generalisation which we shall not have occasion to make use of, and which therefore we may notice at another opportunity.) To find the relation that exists between the distances of five points in space. We have, in general, whatever x lt y lt z lf w lf &c. denote V-, -2x ly -2y 1} -22,, -2iv,, 1 , -2x 2 , -2y 2 , -22!.,, -2w,, 1 ?, -2xs, -2y 5 , -2z 5 , -2w 5 , 1 0,0,0,0,0 multiplied into 1. 1, 2/2, 0, 0, 0, 0, 0, 2 2 -yi- 2 P! ~ J/2 I I *"l ~ 2 ~~ tZ/2 l * * 3 " 2 *^3 "T" > " 2 4 i~ * * > ^2 *^5 * > #1 ~T 1,1, 1,1,0 Putting the ty s equal to 0, each factor of the first side of the equation vanishes, and therefore in this case the second side of the equation becomes equal to zero. Hence x lt y l} z 1} # 2 , y 2 , z 2 , &c. being the coordinates of the points 1, 2, &c. situated 2 2 arbitrarily in space, and 12, 13, &c. denoting the squares of the distances between these points, we have immediately the required relation -2 = 0, o, 12, 13, 14, 15, 1 2 21, 0, 2 23, Q 24, 2 25, 1 2 31, 2 32, 0, 2 34, 2 35, 1 2 2 42, 2 o, o 45, 1 2 51, 52, 5? 54, o, 1 1, 1, 1, 1, 1, 1] ON A THEOREM IN THE GEOMETRY OF POSITION. which is easily expanded, though from the mere number of terms the process is some what long. Precisely the same investigation is applicable to the case of four points in a plane, or three points in a straight line. Thus the former gives The latter gives 2 2 , 12, 13, 14, 1 ax 2 2 2 21 , o, 23, 24, 1 2 2 2 31 , 32, 0, 34, 1 2 2 2 41 , 42, 43, 0, 1 1 , 1, 1, 1, 2 _ 2 0, 12, 13, 1 = 0; 2 2 21, 0, 23, 1 2 2 31, 32, 0, 1 I, 1, 1, or expanding, 4 4 4 2 2 2 2 2 2 12 + 13 + 23 - 2 . 12 13 - 2 . 13 23 - 2 . 12 23 = ; which may be derived immediately from the equation 12 + 13 = + 23, arid is the simplest form under which this equation, cleared of the ambiguous signs, can be put. (The above result may be deduced so elegantly from the general theory of elimi nation, that notwithstanding its simplicity it is perhaps worth mentioning.) Let ,-*,/=?; then n 12 = 23= a", 31=/3 2 , and a from which a, /3, 7 are to be eliminated. Multiplying the last equation by @y, 72, a/3, and reducing by the three first, - 2 - 2 12 . a + . /3 + 23 . 7 + a/3y = 0, - 2 - 2 31 . a + 23 . + 0.7+ a/3 7 = 0, a + /S + 7 + . a/rty = ; 12 4 ON A THEOREM IN THE GEOMETRY OF POSITION. from which, eliminating a, ft 7, afty by the general theory of simple equations, 2 2 o, 12, 13, 1 2 2 21, 0, 23, 1 2 o Us 32, 0, 1 l, 1, 1, = 0. The (additional) equation that exists between the distances of five points on a sphere or four points in a circle, has such a remarkable analogy with the preceding, that they almost require to be noticed at the same time. If a, ft 7, r be the coordinates of the centre, and the radius of the sphere, and _ r 2 > we nave immediately - 20 yi - fy Zl + B = 0, - 2/3y 5 - 2 7 * 5 +8 = 6; whence eliminating a, ft 7, whence, multiplying by we have immediately ~~ . ~" y\ > "" " *"^i * _ O/v _ 9?/ _ 9^ ^^s, ^y 5 > ^^s> L > ^1 > #1 "" ^/l" T = 0; 2/5, 2 2 2 2 o, 12, 13, 14, 15 2 2 2 2 21, 0, 23, 24, 25 2 2 2 2 31, 32, 0, 34, 35 2 2 2 2 41, 42, 43, 0, 45 2 2 2 g 51, 52, 53, 54, = 0. Forming the analogous equation for four points in a circle, and expanding, we readily deduce 84+15* 24 - 2 . T? 34 2 1? 24 - 2 . 14 2 23 13^ 24 - 2 . 14 23 12" 34 = 0, which is the rational, and therefore analytically the most simple form of T2 314+14 23 = 13 24. Euclid, B. vi., last proposition. (It may be remarked that the two factors we have employed in the preceding eliminations, only differ by a numerical factor.) 2] 9 ON THE PROPERTIES OF A CERTAIN SYMBOLICAL EXPRESSION. [From the Cambridge Mathematical Journal, vol. in. (1841), pp. 62 71.] THE series possesses some remarkable properties, which it is the object of the present paper to investigate. We shall prove that the symbolical expression (i/r) is independent of a, b, &c., and equivalent to the definite integral a property which we shall afterwards apply to the investigation of the attractions of an ellipsoid upon an external point, and to some other analogous integrals. The demonstration of this, which is one of considerable complexity, may be effected as follows : Writing the symbol . -=- + T -^ .^r ... under the form 1 + I da 2 1 + m db 2 let the p th power of this quantity be expanded in powers of A. The general term is ( -...- + i * which is to be applied to ON THE PROPERTIES OF A CERTAIN [2 Considering the expression / i * \I+rda*" J if for a moment we write {(1 -*, Ac.; this becomes *;, \i (2i" + 2 - n) . Now it is immediately seen that AJ ^ = PI from which we may deduce 1 _ 2i (2i + 2) ... (2i + 20 - 2) (2i + 2 - n) ... (2i + 2? - n) A V ~ Pi** or, restoring the value of Pl , and forming the expression for the general term of (^), this is (a 2 + &c.) -f 2i (2t + 2 - n) A.- (a . + 6 ... + ^ + + &c. p representing the quantity a 2 + 6 2 + &c. Hence, selecting the terms of the s th order in I, m, &c. the expression for the part of (i/r) which is of the s th order in I, m &c. may be written under the form multiplied by .2.. . U -2(2 + 2-R)( + l) ... (i + ) + P (?} L) 2i (2i + 2) (2i + 2 - n) (2i + 4 - n) ( + 2) . . . (i + s +1) 1 . 2 - &c. [la? + nib- ...= U suppose] which for conciseness we shall represent by U &c. = /S> suppose. 2] SYMBOLICAL EXPRESSION. 7 Now U representing any homogeneous function of the order 2s, it is easily seen that and repeating continually the operation A, observing that A?7, A 2 Z7, &c. are of the orders 2(s-l), 2(s 2), &c. we at length arrive at + 2) (2i + 2? - 4s- n) (2i + 2q - 4s - n - 2) A Changing t into U A O _ pS+i- where in general + i , we have an equation which we may represent by MU A?- 1 Z7 U - .d ., _ 4- 1 /! ., ____ -\- 1 A ; - ~ q> l S+i+i q> * S+i+i +i " ^f* x (2s + 2i + 2i ) (2s + 2t" + 2i + 2) . . . (2s + 2i + 2i + 2r - 2) x (2i+2i / + 2g-2s-n) ... (2i + 2i + 2g-2s-n -2r + 2). Now the value of S, written at full length, is (-1)* 1.2...S u s+i+i _ +&C. and substituting for the several terms of this expansion the values given by the equation (a), we have where in general k x =a t (*A 8 , o& + x - 1 4 8 _i,or-i- + A-x,ok-x) V T ~*~*i ! ~^~ n " x> l bs x+i - 9 i . ^J . . . A being the (x + l) th of the series &... 8 ON THE PROPERTIES OF A CERTAIN [2 Substituting for the quantities involved in this expression, and putting, for simplicity 2i + 2 - n = 2y, we have, without any further reduction, except that of arranging the factors of the different terms, and cancelling those which appear in the numerator and denominator of the same term, 1.2 s 2"* 1 . 1.2. ..*.!. 2... (s-x).1.2...x multiplied by the series (i + s + 1) ... (i + s + x 1) into 1 + 1 x + 7(7 + 1) x (x - 1) lx-7 1.2 (x-7)(x-l-7) (i + s) ...(i + s + x-2) . 1-7 x + . into 7 x(x-l) 7(7+!) x(x-l)(x-2) l -T= -T^-x terms j into to ? = x. Now it may be shown that 1_ (x-r + which reduces the expression for k x to the form 1.2. ..5 2 2 + 1 .1.2...5.1.2...( 5 -x) terms x .. . -|(<+f). v x-1) x-2) + &c. (x + 1) terms ; from which it may be shown, that except for x = 0, k x = 0. The value x = 0, observing that the expression (i+s+l)(i+ s + 2) ...(i + s-1) represents , gives 2] SYMBOLICAL EXPRESSION. or we have simply 2 2s (1.2...s) 2 .(2i+2s) where A = + -^- 2 + ..., U= (la- + mb- ...) . 12.4 Consider the term = ^ \ " 5 a 2A 6 2 x . . . Z A TO** . . . ; 1 . 2 ... X . 1 . 2 ... j* . &c. with respect to this, A s reduces itself to _ 1.2...S /_d V A 1.2...X. l."2.../*.&c! (da) " and the corresponding term of S is 2 - 2x - 1 - 2 - 2 &c - = /rT- S l^ 111} which, omitting the factor a-T^" anc * multiplying by a; 25 , is the general term of the s th order in I, m, ... of 1 The term itself is therefore the general term of / J or taking the sum of all such terms for the complete value of S, and the sum of the different values of 8 for the values 0, 1, 2... of the variable s, we have the required equation x*- 1 dx Another and perhaps more remarkable form of this equation may be deduced by writing --^, ~ -~ , & c . for a 2 , 6 2 , &c., and putting ^ + ^_ + & c . = V 2 , fy 2 = a 2 , mi; 2 = /S 2 , &c. : we readily deduce = ^\ ; : [ gS c. OF THF UNIVERSITY OP 10 ON THE PROPERTIES OF A CERTAIN [2 77 being determined by the equation ^_ b * = 1- *7 2 + a* 77* + & " or, as it may otherwise be written, 2 a 2 frfr ^ ma , + 6M ...._ ._--_- n, it will be recollected, denotes the number of the quantities a, b, &c. Now suppose V= //...</> (a -#, 6-y, ...}dxdy ... (the integral sign being repeated n times) where the limits of the integral are given by the equation l+f + Aa-l; and that it is permitted, throughout the integral to expand the function < (a -#,...) in ascending powers of #, y, &c. (the condition for which is apparently that of </> not becoming infinite for any values of x, y, &c., included within the limits of the integration): then observing that any integral of the form JJ ... aP if ... dxdy &c. ... where any one of the exponents p, q, &c. ... is odd, when taken between the required limits contains equal positive and negative elements and therefore vanishes, the general term of V assumes the form db Also, by a formula quoted in the eleventh No. of the Mathematical Journal, the value of the definite integral // . . . x- r y- s ... dx dy ... is (observing that the value there given referring to positive values only of the variables, must be multiplied by 2 n ): or, as it may be written hence the general term of V takes the form hh...-n** 1 . 2. 3. ..r. 1.2. ...... and putting r + s + &c. =p, and taking the sum of the terms that answer to the same value of p, it is immediately seen that this sum is . 1 . 2 ...p. in (in + l) ... (in 2] SYMBOLICAL EXPRESSION. 11 Or the function <f>(a x, b y...) not becoming infinite within the limits of the integration, we have ff...<j)(a-x, b-y ...)dxdy ... 4) /. 7, _>* S^J r, 1 -..!,* Af J } a+ " * (a " }> the integral on the first side of the equation extending to all real values of x, y> /V2 nj i &c., subject to j- + j-- 2 +...<I. Suppose in the first place < (a, 6...) = . a + b 2 ...y By a preceding formula the second side of the equation reduces itself to r (in) J o V {(n 1 + Kx 2 } (rf + h?a?) . . . (n factors) } n being given by . . -^ + -^ ... - 1. Hence the formula dxdy ... . .. n times + h 2 x 2 ) (n 1 + /i> 2 ) (n factors)} where the integral on the first side of the equation extends to all real values of /y- fit-* x, y, &c. satisfying j- + j- n + &c. . . . < 1 ; if, as we have seen, is determined by / ( IV " 2 - . =1; ,f + h* r rt* + } and finally, the condition of <J>(a x, by...} not becoming infinite within the limits of the integration, reduces itself to -5- + 7- + ... > 1, which must be satisfied by these />* n " quantities. Suppose in the next place that the function (/> (a, 6...) satisfies - o + -.^ + &c. = 0. Ct/Cv \AJ\) The factor ( h- -j-^ + &c. ) may be written under the form V da- ) /To 7 ,,\ ^*~ , /J ., 7o v "" . p . 7 o / C"" ft" \ /To I o\ ^" , P (/I. /I") - + ( - ft 1 ) - 7 h &C. + ft- I -5- + -77- ... I = (/I " ft J -m + &C. wO" ac- \dci~ do - ) do- 22 12 ON THE PROPERTIES OF A CERTAIN SYMBOLICAL EXPRESSION. [2 since, as applied to the function 0> ^ 2 + ^ + &c " is e q uivalent to 5 we have in this case // . . . <f> (a - x, b - y . . .) dx dy. . . (a b V or the first side divided by hh, t ... has the remarkable property of depending on the differences hf-h*, &c. only; this is the generalisation of a well-known property of the function V, in the theory of the attraction of a spheroid upon an external point. If in this equation we put <f> (a, b...)= - ft , which satisfies the required (a 2 +6 2 ...) condition + &c. = 0, then transferring the factor a to the left-hand side of the sign do? S, and putting in a preceding formula, a 2 = 0, j3 2 = h? - h 2 , &c. and 77" + h* for 77 , we obtain 2M,...7r* n a p _ a.-"- 1 ^ _ " Vfo* + ^ 2 ) r (in) J o V [{i? r + ^ 2 + (V - fr 2 ) l i ? 2 + ^ 2 + (V - ^ 2 ) ^1 ( - !) factors] where, as before, the integrations on the first side extend to all real values of x, y, &c., satisfying T^ + f- 2 ...<1; rf is determined by - -, 2 +&c. = 1 ; and a, b,...h, h,, &c. are f\t fl> f) ~l v a 2 b" subject to y- + r- +&c. ... > 1. /r n- For n = 3, this becomes, (a x) dx dy dz ///: + ^) o V[{^ +> + (V - frO ^} I*? 2 + the integrations on the first side extending over the ellipsoid whose semiaxes are A, h lt h tl , and the point whose coordinates are a, 6, c, being exterior to this ellipsoid; also 1: a known thcorem - 13 3. ON CERTAIN DEFINITE INTEGRALS. [From the Cambridge Mathematical Journal, vol. ill. (1841), pp. 138 144.] IN the first place, we shall consider the integral the integration extending to all real values of the variables, subject to the condition * + f +...<-!. and the constants a, b, &c. satisfying the condition a 2 b 2 We have dV _. Tf , .. N (a x)dxdy... ...(n times)- - - 2- / " (n V( + /O r (in) ./ o v [|| + h* + ( V - A 2 being determined by the equation " 2 by a formula [see p. 12] in a paper, [2], " On the Properties of a Certain Symbolical Expression," in the preceding No. of this Journal: having been substituted for the if of the formula. 14 ON CERTAIN DEFINITE INTEGRALS. [3 Let the variable x, on the second side of the equation, be replaced by <f>, where we have without difficulty _ M^. = da where 3> = ( + /r + <) (f + V + </>)... and similarly rfF _ 9\ M /Ill^!*r rf< ^T = r(jn) J (*+V &c ....... From these values it is easy to verify the equation a 2 / ... ~ For this evidently verifies the above values of -^= , n- , &c. if only the term -rr CL (L CL O C I c vanishes ; and we have dV (n-2)/^,...7r* n r df a" \J^ (Jt ~ OF 1 /ln\ I T > rlt\ fc _L 7,2 _L A " ) ./<J> or, observing that 1- and taking the integral from to x , rfF_ (n-2)M / ...7r in / a 2 6 2 \ 1 "df" 2r<in> V f+* l + /v"V\/r(l + ^)^+V)-} in virtue of the equation which determines ^. No constant has been added to the value of V, since the two sides of the equation vanish as they should do for a, b ... infinite, for which values is also infinite and the quantity which is always less than ,,#.;, vanishes. Hence, restoring the values of V and 4>, II... (n times) - - T^ZJ 3] ON CERTAIN DEFINITE INTEGRALS. 15 the limits of the first side of the equation, and the condition to be satisfied by a, b, &c., also the equation for the determination of , being as above. The integral r r fir rlii V = ...(n times)- between the same limits, and with the same condition to be satisfied by the constants, has been obtained [see p. 11] in the paper already quoted. Writing instead of rf, and a? = ^7 , we have ran) where _ Let V = s 9 +-jj- + ... Then by the assistance of a formula, da 2 db 2 J V \a 2 + 6 2 ...y =2l (2t + 2) ---^ + 2 g~ 2 ^ 2t + 2 - n >" ^ +2 g" n) -(a 2 + ^...)^ given in the same paper [see p. 6], in which it is obvious that a, b... may be changed into a x, b y, &c. ...; also putting t = n; we have .. (n times) __^_L^ = *&, ** x fVv* f/ V _i_ 1 Jfl+2 9 2 ? 1 9 _ Ti / J * \ I Now in general, if %% be any function of , (Z But from the equation 2 ^ ,-rr = 1, (C + ) we obtain Ji_ - k ^-i ^ = whence Also _ -4 * (^ + A7rfa { (f + A 2 ) 3 )Ua/ { (| + A a ) 2 )da 2 whence taking the sum S, and observing that * a 2 OF THE UNIVERSITY 16 ON CERTAIN DEFINITE INTEGRALS. [3 22 we and we hence obtain vx= Hence the function ^ "771 i r J ..V (observing that differentiation with respect to is the same as differentiation with respect to 0) becomes integrable, and taking the integral between the proper limits, its value is where We have immediately or whence 16. V Hence restoring the value of V, and of the first side of the equation, II ... (n times) - ^-^ - T- ; a-+6-... in+? with the condition a , + - . . . = 1 2 from which equation the differential coefficients of , which enter into the preceding result, are to be determined. 3] ON CERTAIN DEFINITE INTEGRALS. 17 In general if u be any function of f , a, b ... . Ct> It . ,^ CLil _-\ JL . (t~1l _-\ (Z from which the values of the second side for q = 1, q=%, &c. may be successively calculated. The performance of the operation {- r } [ 3 \ ( ) . upon the integral V. leads in Va/ \dbj \dcj like manner to a very great number of integrals, all of them expressible algebraically, for a single differentiation renders the integration with respect to <f> possible. But this is a subject which need not be further considered at present. We shall consider, lastly, the definite integral U= ...(n times). limits, &c. as before. This is readily deduced from the less general one I . . . (n times) ^ JJ \(a a For representing this quantity by F(h, A,...), it may be seen that /* / n\ ^ T~t /I -i vi but in the value of F(h, h f ...), changing h, h f ... into mh, mh,... also writing m-<f> instead of 0, and m?% for , we have = (| + ^ + 0) (f + ^ + </,)... Hence -L or, observing that ^ is equivalent to ^- , and effecting the integration between the proper limits, d C. ON CERTAIN DEFINITE INTEGRALS. Substituting this value, also /{^> + f + A, + "I f r f^ in the value f U and observing that m = gives =*>> = 1 g ives ? = *> where ^ is determined as before by the equation s a ,,r writing 0-f-| for f, df = d^, the limits of </> are 0, x; or, inverting the limits and omitting the negative sign, _hh ...7r in a S + h + <i> + f+h* + 4 + " r which, in the particular case of n = 3, may easily be made to coincide with known results. The analogous integral is apparently not reducible to a single integral. 19 4. ON CERTAIN EXPANSIONS, IN SERIES OF MULTIPLE SINES AND COSINES. [From the Cambridge Mathematical Journal, vol. in. (1842), pp. 162 167.] IN the following paper we shall suppose e the base of the hyperbolic system of 1 / logarithms ; e a constant, such that its modulus, and also the modulus of - {1 >/l e 2 }, are each of them less than unity; %{e wV(-1) } a function of u, which, as u increases from to TT, passes continuously from the former of these values to the latter, without becoming a maximum in the interval, /{e"^ 1 " 1 } any function of u \vhich remains finite and continuous for values of u included between the above limits. Hence, writing and considering the quantity v / _ 1 gwvi-D^ | e wv (-D j (I e cos u) as a function of m, for values of m or u included between the limits and TT, we have Jl e^f{ ut/( ~ v } cos rm dm 1 e v<-D % ( e V(-Dj (1-ecosu) T o J^Te 1 " 1 x {e u ^~ 1} } (I - e cos it) (Poisson, 3fec. torn. I. p. 650) ; which may also be written l-ecosu and if the first side of the equation be generally expansible in a series of multiple cosines of m, instead of being so in particular cases only, its expanded value will always be the one given by the second side of the preceding equation. 32 20 ON CERTAIN EXPANSIONS, [4 Now, between the limits and TT, the function will always be expansible in a series of multiple cosines of u; and if by any algebraical process the function fp cos TXP can be expanded in the form we have, in a convergent series, /{ *<- 1 >}cosrx{e M ( - 1) }=ao + 22;Xcossu ..................... (6). Again, putting -{l-Jl-e*}=\ .................................... (7), " have j.-H.SrVeo.jm .............................. (8). Multiplying these two series, and effecting the integration, we obtain d ! f _ 2 ( ^ , (9) 1-ecosu and the second side of this equation being obviously derived from the expansion of /Xcosr^X by rejecting negative powers of X and dividing by 2, the term independent of X may conveniently be represented by the notation 2/XcosrxX ....................................... (10); where in general, if FX can be expanded in the form have = % A + 2 A g \ g ................................. (12). (By what has preceded, the expansion of FX in the above form is always possible in a certain sense ; however, in the remainder of the present paper, F\ will always be of a form to satisfy the equation F(-) = FX, except in cases which will afterwards \X/ be considered, where the condition A_ s = A 8 is unnecessary.) Hence, observing the equations (4), (9), (10), WV( 1) D % ( e uV(-i)J (1 _ e cos w =S "ooarmZcoervX/X ......... (13); from which, assuming a system of equations analogous to (1), and representing by II the product ^i^ . . . , it is easy to deduce -j -- (J - 1 e"^- 1 x {e"^ - 1 } (1 - e cos = S_S_" ...ncosrmn(2cosr X X)/(X 1( X,...) (14), 4] IN SERIES OF MULTIPLE SINES AND COSINES. 21 where F (Xj, X-j ...) being expansible in the form r(\, \ a ...) = 2_Z*-Z>-.A, l .*...\ W...[A. l ^ m = A-, lt _+.,.] (15), r(X 1) X 2 ...)=S;S;...^^ 1>Ss ...V>X/ 2 ..., (16), N being the number of exponents which vanish. The equations (13) and (14) may also be written in the forms "*- 1 } =2.; cosrm2cosr x X N j /X (17), VI e 2 _;... n (cosrm) n {2 cos r x \ J -~ 1 * /X ^ < X + ^/(X,, X 2 ...)... (18). \ -^ ^ As examples of these formulae, we may assume x ( e V(-Dj = m = w Hence, putting A, ........................... (20), and observing the equation J~l e V(-D % ( 6 M * (- 1 )} = l-ecosM ........................ (21), the equation (17) becomes IJ -f\ (22). V \* ~~ v J Thus, if 6 ar = cos" 1 ^ - (23), assuming /| e V(-D} = CO3 ^~ e _ / 24) 1 e cos w 1 1 )} {i (X + X" 1 ) - e} A r ...(25), the term corresponding to r = being 7 _^={2X-2e-e(X 2 + l) + 2e 2 X}, =-e .................. (26). ^ v I ~~ 6 Again, assuming f fe^i- 1 ! = = 2 y l J dm l 22 ON CERTAIN EXPANSIONS, [4 and integrating the resulting equation with respect to m, _ a ^ sn rm^ A r = ??i + 22, -A r .................. 28), , f ^ _ y 1 f a formula given in the fifth No. of the Mathematical Journal, and which suggested the present paper. As another example, let /(^}.eos(*-w)-f*S5^ .. ..(29). dm (1 e cos u) 3 Then integrating with respect to m, there is a term - - which it is evident, a priori, must vanish. Equating it to zero, and reducing, we obtain X + X " 5 (31), 1-e 2 l- that is j = * + (X*+l)+ (X* + 3X) + (V + 4** + 8) + ........... (32), a singular formula, which may be verified by substituting for X its value : we then obtain The expansions of sin k (6 - r), cos & (0 -or), are in like manner given by the formulae cos k(6 TV) = ^_^ A,X cos kL cosrm. (34), T / /I \ -* 00 A * i T OA*1 * fits ,_ v smA-(y CT) = 2 _ A r v- cos & - (35), Ar r V where, to abbreviate, we have written Hff!S^te-J (36), Forming the analogous expressions for cos fc (0 - O, 4] IN SERIES OF MUPTIPLE SINES AND COSINES. 23 substituting in cos k (6 - 6 } = cos k(ar- CT ) {cos k (6 - r) cos k (0 - r ) + sin k (0 - vr) sin k (6 - *r )} - sin k(vr- r ) {sin k (0 - r) cos A; (# - w ) - sin & (0 - w ) cos & (0 - r)}, and reducing the whole to multiple cosines, the final result takes the very simple form .. cos k(0-0 )=2_ cos {r m f - rm + k (* - )} A r AV cos kL cos &Z/ \L - jjj ^X - ^j . . .(38). V -------- Again, formulae analogous to (14), (18), may be deduced from the equation F(?/I!, ?w 2 ...) P n din [ 2ir dm 1 cw wio + sin (r l m l + r.m 2 ...) I -5-^ .r ... sin (^m, + r 2 m 2 . . .) F (m 1} r>z 2 . . .) (39), J 27T J o Z7T which holds from w x = to m x = 2?r, &c., but in many cases universally. In this case, writing for F^m^ m. 2 ...) the function O i __ _ 1 _ l-tf-esuLuJ--! ^r Ml v(-i, e 2 v ( -i) i (40) - j ^/ _ i e v (-D ^ | 6 w v (-D j 1 - e cos tt ) J and observing N/l e 2 e sin u v - 1 1 + \e~ MV( ~ 1) rtVoo , / -, . , - , ,.,-. - - - - - =B= 7 - = 1 + 22, [cos su - J - I sin su }\ s ...... (41), l-ecosu 1 -\ e ~ u> ^-v i l an exactly similar analysis, (except that in the expansion the supposition is not made that A Sl< S2 ... = -4_ gl) _s 2 ---) leads to the result T^-esinu/l 1 -1 e^*- 1 ^ {fc"^ - 11 } (1 - e OOB)J cos (lil, + r.,m 2 . . .) 2" cos (r^i^i sin (r- i m l + r z m. 2 . . . ) 2 n sin (r^i^i + r. 2 % 2 \ 2 . . .)/(\i, X 2 ..Q ... (42) (n) being the number of variables M I} w 2 .... Hence also /{e" 1 *"" 1 *, e MaV( ~ 1) ... - ^ ( x + x ~ j i f ^ ......... (43). 24 ON CERTAIN EXPANSIONS, IN SERIES OF MULTIPLE SINES AND COSINES. [4 By choosing for /{e" 1 """" 1 , e^ " 1 ...}, functions expansible without sines, or without cosines, a variety of formulae may be obtained: we may instance l - e 2 - \e (X - X" 1 ) A r having the same meaning as before. Also - -^ ( x A ^r~^~ A " ; -v-r T**~* /A\ where A r = A, e A e "> (^o;. . |Jk 2^ \ i /) \ / ?" Asfain, i + - - 8 e _, r* or, what is the same thing, \ fi _ 1/3 (\ _i_-\-iM j/"\ j.^ n _ il A 2i^ = (49); or, comparing with (44), = Jl - e 2 O I T1 which are all obtained by applying the formula (43) to the expansion of (0 - or), and comparing with the equations (25), (33). 5] 25 5. ON THE INTEESECTION OF CURVES. [From the Cambridge Mathematical Journal, vol. III. (1843), pp. 211213.] THE following theorem is quoted in a note of Chasles Aper9u Historique &c., Memoires de Bruxelles, torn. XL p. 149, where M. Chasles employs it in the demonstration of Pascal s theorem : " If a curve of the third order pass through eight of the points of intersection of two curves of the third order, it passes through the ninth point of intersection." The application in question is so elegant, that it deserves to be generally known. Consider a hexagon inscribed in a conic section. The aggregate of three alternate sides may be looked upon as forming a curve of the third order, and that of the remaining sides, a second curve of the same order. These two intersect in nine points, viz. the six angular points of the hexagon, and the three points which are the intersections of pairs of opposite sides. Suppose a curve of the third order passing through eight of these points, viz. the aggregate of the conic section passing through the angular points of the hexagon, and of the line joining two of the three inter sections of pairs of opposite sides. This passes through the ninth point, by the theorem of Chasles, i.e. the three intersections of pairs of opposite sides lie in the same straight line, (since obviously the third intersection does not lie in the conic section) ; which is Pascal s theorem. The demonstration of the above property of curves of the third order is one of extreme simplicity. Let U = 0, V = 0, be the equations of two curves of the third order, the curve of the same order which passes through eight of their points of intersection (which may be considered as eight perfectly arbitrary points), and a ninth arbitrary point, will be perfectly determinate. Let U , V , be the values of U, V, when the coordinates of this last point are written in place of x, y. Then UV U V=Q, satisfies the above conditions, or it is the equation to the curve required ; but it is an equation which is satisfied by all the nine points of intersection of the two curves, i.e. any curve that passes through eight of these points of inter section, passes also through the ninth. OFTHF 4 UNIVERSITY 26 ON THE INTERSECTION OF CURVES. [5 Consider generally two curves, U m =Q, F n = 0, of the orders m and n respectively, and a curve of the r th order (r not less than m or n) passing through the mn points of intersection. The equation to such a curve will be of the form U = u r - m U m + v r - n V H = 0, u r _ m , v r _, denoting two polynomes of the orders r - m, r - n, with all their coefficients complete. it would at first sight appear that the curve 7 = might be made to pass through as many as {1 + 2 ... + (r -m + 1)} + [I + 2 ... (r- n + 1)} - 1, arbitrary points, i.e. (r - w + 1) (r - wi + 2) + J (r - n + 1 ) (r - n + 2) - 1 ; or, what is the same thing, r ( r + 3) _ mn + (r m n + 1) ( r m ~ n + 2 ) arbitrary points, such being apparently the number of disposable constants. This is in fact the case as long as r is not greater than m + n - 1 ; but when r exceeds this, there arise, between the polynomes which multiply the disposable coefficients, certain linear relations which cause them to group themselves into a smaller number of disposable quantities. Thus, if r be not less than m + n, forming different polynomes of the form afy* V n [a + /8 = or < m], and multiplying by the coefficients of #/ in U m and adding, we obtain a sum U m V n , which might have been obtained by taking the different polynomes of the form aty 8 U m [y + B = or < ?i], multiplying by the coefficients of a?V in V n , and adding: or we have a linear relation between the different polynomes of the forms x*y*V n , and &*fU m . In the case where r is not less than m + n + 1, there are two more such relations, viz. those obtained in the same way from the different polynomes off. xV n , oflif.xU m , and xyt .yV n , &tf .yU m , &c.; and in general, whatever be the- excess of r above m + n- I, the number of these linear relations is 1 + 2 . . . (r - m - n + 1) = $ (r - m - n + 1) (r - m - n + 2). Hence, if r be not less than m+n, the number of points through which a curve of the r th order may be made to pass, in addition to the mn points which are the intersections of U m = Q, V n = Q, is simply $r (r + 3) -win. In the ^ case of r = m + n-l, or r=?n + n-2, the two formulae coincide. Hence we may enunciate the theorem "A curve of the r th order, passing through the mn points of intersection of two curves of the ?w th and w th orders respectively, may be made to pass through r ( r + 3) - mn + %(m + n-r- 1) (m + n-r- 2) arbitrary points, if r be not greater than m + 7i-3 : if r be greater than this value, it may be made to pass through $r(r + 3) mn points only." Suppose r not greater than w + n-3, and a curve of the r th order made to pass through ? (r + 3) - win + (m + n - r - 1) (m + n-r-2) arbitrary points, and mn - $ (m + n - r - 1) (m + n - r - 2) 5] ON THE INTERSECTION OF CURVES. 27 of the mn points of intersection above. Such a curve passes through r(r + 3) given points, and though the mn \ (m + n r 1) (m + n r 2) latter points are not perfectly arbitrary, there appears to be no reason why the relation between the positions of these points should be such as to prevent the curve from being completely determined by these conditions. But if it be so, then the curve must pass through the remaining ^ (m + n r 1) (m + n r 2) points of intersection, or we have the theorem " If a curve of the ?* th order (? not less than m or n, not greater than m + n 3) pass through mn ^ (m + n i 1) (m + n r 2) of the points of intersection of two curves of the m ih and n- th orders respectively, it passes through the remaining (m + n r 1) (m + n r 2) points of intersection." 4 2 28 6. ON THE MOTION OF KOTATION OF A SOLID BODY. [From the Cambridge Mathematical Journal, vol. in. (1843), pp. 224 232.] IN the fifth volume of Liouville s Journal, in a paper "Des lois geometriques qui regissent les de placemens d un systeme solide," M. Olinde Rodrigues has given some very elegant formulae for determining the position of two sets of rectangular axes with respect to each other, employing rational functions of three quantities only. The principal object of the present paper is to apply these to the problem of the rotation of a solid body ; but I shall first demonstrate the formulae in question, and some others connected with the same subject which may be useful on other occasions. Let Ax, Ay, Az\ Ax,, Ay,, Az,, be any two sets of rectangular axes passing through the point A : x, y, z, x t , y t , z t , being taken for the points where these lines intersect the spherical surface described round the centre A with radius unity. Join xx i> yy t > zz <> ky arcs of great circles, and through the central points of these describe great circles cutting them at right angles: these are easily seen to intersect in a certain point P. Let Px =f, Py =g, Pz = h ; then also Px t =f, Py t = g, Pz t = h : and Z#P#, = yPy t = /.zPz,, =0 suppose, being measured from xP towards yP, yP towards zP, or zP towards xP. The cosines of /, g, h, are of course connected by the equation cos 2 /+ cos 2 <7 + cos 2 h = 1. Let a, /3, 7 ; a , /3 , 7 ; a", /3", 7", represent the cosines of x t x, y t x, z t x ; x t y, y t y, z t y ; j; t z, y t z, zz: these quantities are to be determined as functions of/, g, h, 0. Suppose for a moment, Z yPz = x, Z zPx = y, Z xPy = z ; ON THE MOTION OF ROTATION OF A SOLID BODY. 29 a = cos z / + sin 2 / cos 0, a = cos/ cos g + sin /sin g cos (z 0), a" = cos /cos h + sin /sin h cos (y + &}, P = cos # cos/ + sin # sin /cos (z + 0), /3" = cos (JT cos A + sin g sin A cos (x 8), 7 = cos h cos/+ sin A sin/cos (y 0), 7 = cos h cos # -f sin A sin ^ cos (x + 0), y" = cos 2 h + sin 2 h cos ft Also sin g sin A cos x = cos g cos h, sin /i sin /cos y = cos h cos/ sin/ sin g cos z = cos / cos (7, and sin g sin A, sin x = cos f, sin h sin /sin y = cos g, sin/ sin g sin z = cos A. Substituting, a. = cos 2 / + sin 2 /cos0, a = cos/cos # (1 cos 0) + cos A sin 0, OL" = cos/cos h (1 cos 0) - cos g sin 0, /3 = cos # cos/ (1 cos 0} cos h sin 0, /3 = cos 2 g + sin 2 g cos 0, /3" =- cos g cos A (1 cos 0) + cos/ sin 0, 7 =cos Acos/(l cos 0) + cos # sin 0, <y = cos A cos # (1 cos 0) cos/sin 0, 7" = cos 2 A + sin 2 A cos 0. Assume X = tan ^0 cos/ ^ = tan ^0 cos </, v = tan ^0 cos A, and sec 2 = 1 -f X 2 + p? + v ~ = K : then KO. = I + \--fji--v 2 , a = 2 (A/i + i/), a x/ = 2 (v\ -/x), /c/3 = 2 (Ayu - i;), /r/3 = 1 + p? - v* -X 2 , K /3" = 2 (/zy + X), 7 = 2(i/\+^), K7 = 2 (/AI/ - X), ^7" = 1 + i; 2 - X 2 - fj,- ; which are the formulas required, differing only from those in Liouville, by having X, ^ v, instead of ?n, $n, ^p; and a, a , a"; , /S , /3" ; 7, 7 , 7", instead of a, b, c: a , b , c ; a", b", c". It is to be remarked, that , ", ^; 7", 7, 7 , are deduced from ct, a , a", by writing p,, v, \ ; i;, X, ^, for X, //,, i/. Let 1 + a + /3 + 7" = v ; then /cu = 4, and we have Xu = /3" - 7 , /AU = 7 - a", yy = - X 2 u = 1 + a - $ - y", p? v = 1 - a + /3 - 7", 30 ON THE MOTION OF ROTATION OF A SOLID BODY. [f> Suppose that Ax, Ay, Az, are referred to axes Ax, A\, A /., by the quantities /, m, n, k, analogous to X, p, v, K, these latter axes being referred to Ax lt Ay t , Az t , by the quantities l t , m t , n,, k r Let a, b, c; a , b , c ; a", b", c" ; a,, b,, c, ; a/, fe/, c/; a,", 6,", c/ , denote the quantities analogous to a, /9, 7; a , /? , 7 ; a", /3", 7". Then we have., by spherical trigonometry, the formulae a =a,a, + b a + c a", /3 = ab / -\-b bf + c bf, 7 - a C, +6 C/ + e c/ ; a = a a, + 6 a/ + c a,", /3 = a b / + b bf + cbf, 7 = c t + b cf + c cf : a" = a" a, + b"af + c"af, $" = a"b, + b"bf + c"bf, 7" - a" C/ 4- 6"c/ + c"c/ . Then expressing a, b, c; a , 6 , c ; a", 6", c" ; a,, b lt c,\ af, bf , cf ; a/ , &/ , c/ , in terms of I, m, n ; / /t 7?*^ n /; after some reductions we arrive at kk f v = 4 (1 ll t mm, nnf) 2 , = 4IP suppose, kk, (/3" - 7 ) = 4 (I +1, + n t m - nm t ) U, kk t (7 a. ) = 4 (m+m l + l,m lm t ) II, kk, ( - /3") = 4 (71 + n, + m,n - mn,} U ; and hence Il/i = 111 + m / + l t m lm it Hv = n + n t + in t n mn /t wliich are formulae of considerable elegance for exhibiting the combined effect of successive displacements of the axes. The following analogous ones are readily obtained : P 1 + \l -f fj,m + vii , PI, =\ I i>m + fin , p m/ = /ji m \n + vl , Pn / = v n pi + \m : and again, vn / , P t l = \ l, + vm / ^n it f vl, , P t n = v n t + fil t \m t . These formulae will be found useful in the integration of the equations of rotation of a solid body. Next it is required to express the quantities p, q, r, in terms of X, /z, v, where as usual - at at at d* fi , da da" r = f3 Tt + $di +l3 ~dt Differentiating the values of fix, /3 , ft"ic, multiplying by 7, 7 , 7", and adding, up = 2X (7/A - 7 X + 7") + 2/* (7\ - 7> + y"v) + 2i/ ( - 7 - y v + 7 , 31 f>] ON THE MOTION OF ROTATION OF A SOLID BODY. where X , //, v, denote -5-, -~ , -jr. Reducing, we have Kp = 2 (A + vp! v p,} : from which it is easy to derive the system Kp = 2 ( A + vp! v p,\ tcq = 2 ( - i/A + p! + i/A), - or, determining A , p!, v , from these equations, the equivalent system 2A = ( 1 + A 2 ) p + (Ayu,- v ) q + (v\+ p.] r, 2p! = (A/A + v) p 4- ( 1 + p?} q + (p,v X) r, 2i/ = OA - p,}p + (p.v + A ) q + ( 1 + i/O r. The following equation also is immediately obtained, K = K (\p + /J<q + vr). The subsequent part of the problem requires the knowledge of the differential coefficients of p, q, r, with respect to A, p., v; A , p!, v. It will be sufficient to write down the six " 2ft fc ^ from which the others are immediately obtained. Suppose now a solid body acted on by any forces, and revolving round a fixed point. The equations of motion are _ = dt d\ d\ ~ d\ dt dp! dp, dp, d, dT _dT_ dV dt dv dv dv where T = $ (Ap* + Bq* + Cr 2 ) ; V = 2 [j(Xdx + Ydy + Zdz}} dm; or if Xdx + Ydy + Zdz is not an exact differential, -j- , .are independent a\ dp, dv symbols standing for dm , 32 ON THE MOTION OF ROTATION OF A SOLID BODY. [fi see Mecanique Analytique, Avertissement, t. I. p. v. [Ed. 3, p. vn.] : only in this latter case V stands for the disturbing function, the principal forces vanishing. Now, considering the first of the above equations whence, writing p ; 4, /, x , for f, |, |, It rfV = \ (A P ~ " B< ? + Cr "> ~ I E V + ~ Also and hence i f-r -. -- 2 - (^ - vBq Substituting for X , //, v , K, after all reductions, and, forming the analogous quantities in ^, v, and substituting in the equations of motion, these become v (Ap + (C - B] qr] + [Bq + (A - Cf) rp] - X [Cr + (B - A} pq] = H [Ap +(G-B) qr} + X {Bq + (A - C) rp} + {Cr + (B - A] pq] = or eliminating, and replacing /, q, r , by -? -jt. ~, we obtain 6] ON THE MOTION OF ROTATION OF A SOLID BODY. 33 to which are to be joined d\ d/j, dv\ dt dt dtj d\ d/j> dv\ V ~di~* dt + di) _ _ / cZX dfi dv\ where it will be recollected K = 1 + X 2 + fji? + v~ ; and on the integration of these six equations depends the complete determination of the motion. If we neglect the terms depending on V, the first three equations may be integrated in the form 2 _ ,_C-B 2 _ Z _A-_C g _ B-A and considering JD, g, r as functions of ^>, given by these equations, the three latter ones take the form K d\ dp dv 4qr d(f> d<f> d<f> K d\ dfjb dv d$ d<f> d(f) d\ d/ji dv p d^~ d<f> + dj> of which, as is well known, the equations following, equivalent to two independent equations, are integrals, - v) + 2CV (i/X + ^), Kg = 2Ap (\fi + v) + Bq (1 + ^ - \- - v~] + 2Cr (pv - X), Kg" = 2Ap ( v \ - /i) + 2% (fjiv +\) + Cr ( 1 + v- - X 2 - /z-) ; where g, g , g", are arbitrary constants satisfying To obtain another integral, it is apparently necessary, as in the ordinary theory, to revert to the consideration of the invariable plane. Suppose g = 0, g" = 0, g " = V (A*p* + &-q* + CV), = k suppose. 34 ON THE MOTION OF ROTATION OF A SOLID BODY. [6 We easily obtain, where X , /n , v , K O are written for X, p, v, K, to denote this particular supposition, K,Ap = 2 (y X - HQ) k, K Bq = 2 (/v + X ) A;, * O = (l + iv l -V-/O k; whence, and from K O 1 4- X 2 + n<? + v?, K Cr = (2 + 2v 2 r ) k, we obtain */ 2 ) Ap _(I+ V( ?)Bq _ "- k + Cr " /i ~ ^+6V Hence, writing /i = .4^ + Bq^ + C?\ z , the equation ..1-4 reduces itself to f or, intecfratincf, 4tan~ 1 i/ = J . Cr)pqr The integral takes rather a simpler form if p, q, < be considered functions of r, and becomes + Cr V [\k 2 -Bh + (B-C) GV} {Ah -k* + (C-A) GV}] and then, v a being determined, X , /z are given by the equations _ i> Ap + Bq _ v (} Bq - Ap * ~7 \ 7^ /*0 / . /"f ? + C/r A; + Or Hence , m, ?i, denoting arbitrary constants, the general values of X, fj,, v, are given by the equations P = 1 l\ mfjL nv Q , P X = I + Xo + mr 7?/i , P fi = m + /z + wX i/ , In a following paper I propose to develope the formulas for the variations of the arbitrary constants p lt q lt r lt I, m, n, when the terms involving V are taken into account. 6] ON THE MOTION OF ROTATION OF A SOLID BODY. 35 Note. It may be as well to verify independently the analytical conclusion imme diately deducible from the preceding formulae, viz. if X, //,, v, be given by the differential equations, rfX dub dv KP = dt +V dt-^dt d\, du> dv K ^- v dt + cTt + *dt d\ ^ du, dv KT = fA , - X + -J-, at dt at where K = 1 + A- 2 + p? + v z , and p, q, r, are any functions of t. Then if X , /A O , v , be particular values of X, JJL, v, and I, m, n, arbitrary constants, the general integrals are given by the system P =1 X W/A O nv , P X = 1 + X + mv nfi , PO/U, = m + /j, + n\ Iv , P Q v = n -f i/o + l/j, m\ . Assuming these equations, we deduce the equivalent system, (1 + XX + /J,fJ, + WQ) I = X X + V /Ji Vfl , (1 -I- XX + /iyu, + vv ) m = p yu. + \v Xz , (1 + XX + ppo + vv ) n = v v + /j, \ Differentiate the first of these and eliminate I, the result takes the form = + V) (X + vp! - v p) - (i/ - X /i ) ( - i/V + p! + \ v ") + (fio + X i^ ) (ji\ f - \p + v) + K O \ , - v^ + fr + XoiV) - (/* 4- \v) O<X - \^ + ^o ) ~ ^. where X , &c. denote -5- , &c. and K O = 1 + X 2 + ^ 2 + v<?. Reducing by the differential equations in X, //., v; X 0) ^o, V Q , this becomes KojX + lpO 2 + v *)+$q( v -x/i)-^^ +\v )} -K {\ , + %p (/i 2 + ^ 2 ) + %q (> - X /i ) - ^r (/Lt,, + Xv )} = ; or substituting for X , \ , we have the identical equation ^p (K O K - KK O ) = : and similarly may the remaining equations be verified. 52 36 [7 7. ON A CLASS OF DIFFERENTIAL EQUATIONS, AND ON THE LINES OF CURVATURE OF AN ELLIPSOID. [From the Cambridge Mathematical Journal, vol. in. (1843), pp. 264- 267.] CONSIDER the primitive equation =0 ................................. (1), between n variables x, y, z, the constants /, g, h being connected by the equation H(f,ff, h ...... )=0 .................................... (2), H denoting a homogeneous function. Suppose that /, g, h ...... are determined by the conditions fx, + gy, +hz, ...... =0 .............................. (3), (4), fan-* + gy n - 2 + hz n - 2 =0. Then writing X = y with analogous expressions for y, z ; the equations (3) give /, g, h, proportional to x, y, z or eliminating/, g, h by the equation (2), H(X, Y, Z ) = (5). Conversely the equation (5), which contains, in appearance, n (n 2) arbitrary constants, is equivalent to the system (1), (2). And if H be a rational integral function of the order r, the first side of the equation (5) is the product of ; factors, each of them of the form given by the system (1), (2). 7] ON A CLASS OF DIFFERENTIAL EQUATIONS, &C. Now from the equation (1), we have the system fx +gy + hz ...... = .............................. (6), fdx + ffdy 4 hdx ...... =0, fd n ~*x + gd n ~"-y + hd n ~-z ...... = 0, or writing X= y , z , ...................................... (7), dy , dz , ... with analogous expressions for Y, Z ...... ; then from the equations (6), /, g, h ...... art- proportional to X, Y, Z ...... : or, eliminating by (2), H(X, Y, Z ...... )=0 .................................... (8). Conversely the integral of the equation (8) of the order (n 2) is given either bv the system of equations (1), (2), in which it is evident that the number of arbitrary constants is reduced to (w-2); or, by the equation (5), which contains in appearance n(n 2) arbitrary constants, but which we have seen is equivalent in reality to the system (1), (2). Thus, with three variables, the integral of H (ydz zdy, zdx xdz, ocdy ydx) = ..................... (9) may be expressed in the form H(yz l -y l z, zx^ - z,x, asy 1 -ac 1 y)=0 ..................... (10), and also in the form x z = -11} - where H(f, 9, h) = Q .................................... (12). The above principles afford an elegant mode of integrating the differential equation for the lines of curvature of an ellipsoid. The equation in question is (6 2 - c 2 ) xdydz + (c- - a 2 ) ydzdx + (a- - 6 2 ) zdxdy = ............... (13), where x, y, z are connected by x z ?/ 2 z- writi "K ,= , = ". ~ = .............................. (15), these become (6 J - c 2 ) udvdw + (c 2 - a-) vdwdu + (a 2 - b") wdudv = ............... (16), u+ v + w= 1 ....................................... (17). ^IfrORNl^ 38 ON A CLASS OF DIFFERENTIAL EQUATIONS, AND ON THE [7 Multiplying by \(vdu udv) (wdv vdw) (udw wdu)}- 1 , the first of these becomes -a?du -b-dv -c*dw =Q . (vdu - udv)(udw^ wdu) (wdv - vdw) (vdu - udv) (udw - wdu) (wdv - vdw) but writing (17) and its derived equations under the form u+ ( V + W ) = l (19), du + (dv + dw) = 0, we deduce -du (v + w) + u (dv + dw) = -da (20), i.e. du= (vdu udv) -f (udw wdu) (21 ), and similarly _ dv = (wdv - vdw) + (vdu udv), diu = (udw wdu) + wdo vdw). Substituting, b- c 2 c 2 a 2 a- b" I I T~ I wdv vdw udw wdu vdu udv the integral of which may be written in the form where, on account of (17), Ul + Vl + w 1 = l (24); and also in the form fu + gv + hiv = (25), where /, g, h are connected by b- c 2 c 2 a 2 a* + + / 9 9 this last equation is satisfied identically by 6 2 - c 2 c 2 - a 2 . a -b> ~ ~ Restoring x, y, z, x l , y lt z l for u, v, w, u ly v l9 w lt the equations to a Hue of curvature passing through a given point x ly y ly z lt on the ellipsoid, are the equation (14) and a 2 (y*z* - y*zS) b- (zfx* - z*x?) c 2 (ar,y - tfy?) or again, under a known form, they are the equation (14) and (6 2 -c 2 ) a 2 c 2 -a 2 f a 2 -6 2 z* _ ^_ o & + c* - A 2 If- A* - B 2 c- 7] LINES OF CURVATURE OF AN ELLIPSOID. 39 From the equations (14), (2,9) it is easy to prove the well-known form xZ . y 1 4. z * - i ^^ - in fact, multiplying (29) by m, and adding to (14), we have the equation (30), if the equations 6 2 -c 2 1 1 1 c 2 -a 2 1 1 - - i J JY) _ _ 6 2 ^ G* - 4 2 6 2 ~ 6 2 + 1 a 3 -6 2 1 1 -, + in c* A* - B> c 2 (c 2 + 0) are satisfied. But on reduction, these take the form OB 2 -<7 2 ) + (& 2 -c 2 )m0 + ma 2 (6 2 -c 2 ) = 0, .................. (32), (C" 2 - A-) 8 + (c- - a 2 ) m6 + mb 2 (c 2 - a J ) = 0, (A* - B 2 ) 6 + (a 2 - 6 2 ) m& + me 2 (a* - 6 2 ) = 0, and since, by adding, an identical equation is obtained, m and 6 may be determined to satisfy these equations. The values of 0, m are 9= = (a a -6 2 ) (6 2 - c 2 ) (c 2 - a 2 ) (a 2 - 6 2 ) (6 2 - c 2 ) (c 2 - a?) ^ 4*>>v UNIVERSITY-) /v ***^ 40 [8 8. ON LAGRANGE S THEOREM. [From the Cambridge Mathematical Journal, vol. in. (1843), pp. 283 286.] THE value given by Lagrange s theorem for the expansion of any function of the quantity x, determined by the equation x = u + lifx (1 ), admits of being expressed in rather a remarkable symbolical form. The a priori deduction of this, independently of any expansion, presents some difficulties; I shall therefore content myself with showing that the form in question satisfies the equations \JL f TT/ /* 7 ^* /" 7~T/ 7 />\ -r- . f F x fx dx = -yr . / F x dx (2), du J dh Fx = Fu for h = (3), deduced from the equation (1), and which are sufficient to determine the expansion of Fx, considered as a function of u and h in powers of h. Consider generally the symbolical expression 4 (4). ( h j, J involving in general symbols of operation relative to any of the other variables entering into p,h. Then, if H/i be expansible in the form H/i = 2 (Ah m ) (5) , it is obvious that / ^ \ = S {(<f>m A) h m ] (6). dh 8] ON LAGRANGE S THEOREM. 41 For instance, u representing a variable contained in the function H/*, and taking a particular form of (j> (h - \du) From this it is easy to demonstrate du ( \cLuJ J ii \ciu) dh (\duj i ) h\du) * where H A denotes -yy H/i, as usual. Hence also of which a particular case is ^-,-i -, , ,,) d { u = -- <W J Also, V*" 1 ^!! **)- Hi for h = ........................ (12). ) d T ,. u = -- dA * (11). Hence the form in question for Fx is Fx=(^ h ^(F u^} .............................. (13); from which, differentiating with respect to u, and writing F instead of F , Fx - ^ [ Uj a well-known form of Lagrange s theorem, almost equally important with the more usual one. It is easy to deduce (13) from (14). To do this, we have only to form the equation deduced from (14) by writing Fxf x for fa, and adding this to (14), \du) * J f- (Fu e h u ) - hf u Fa e hfu \ du J F u^ u ) (16). \-" / c - 6 42 ON LAGRANGE S THEOREM. In the case of several variables, if x = u + hf (x, X-L . . .), x l = M! + /1,/i (x, a-j . . .), &c (17), writing for shortness then the formula is (a,^,...) ( d\ h dh f^LY dT, I F*hf+hJi+ -\ .(18), {where / (a?) is written to denote j-f(x> x * ), & c -} or the coefficient of h n hf l in the expansion of F ( X > x * ) ..(19) fl " n I fJ \ n \ *.} (.*-} ...Ff*fa 20). is From the formula (18), a formula may be deduced for the expansion of Fix, x, ), in the same way as (13) was deduced from (14), but the result is not expressible in a simple form by this method. An apparently simple form has indeed been given for this expansion by Laplace, Mecanique Celeste, [Ed. 1, 1798] torn. I. p. 176; but the expression there given for the general term, requires first that certain differentiations should be performed, and then that certain changes should be made in the result, quantities Z) z , are to be changed into z n , z^ ; in other words, the general term is not really expressed by known symbols of operation only. The formula (18) is probably known, but I have not met with it anywhere. 43 9. DEMONSTRATION OF PASCAL S THEOREM. [From the Cambridge Mathematical Journal, vol. IV. (1843), pp. 18 20.] LEMMA 1. Let U = Ax + By + Cz = be the equation to a plane passing through a given point taken for the origin, and consider the planes tT 1 = 0, #3 = 0, U 3 = 0, Z7 4 =0, Z7 B = 0, ^ = 0; the condition which expresses that the intersections of the planes (1) and (2), (3) and (4), (5) and (6) lie in the same plane, may be written down under the form A lt A 2 , A 3 , A t . . B! , B z , B 3 , BI C 1 , C 2 , C s , 1/4 A 3 , AI, A 5 , A 6 B s , B 4 , B 5 , B 6 C*s . ^4, C s , C 6 = 0. LEMMA 2. Representing the determinants -2 > 2/2 ) Z 3 y 2/3 ) & by the abbreviated notation 123, &c.; the following equation is identically true: 345 . 126 - 346 . 125 + 356 . 124 - 456 . 123 = 0. 62 44 DEMONSTRATION OF PASCAL S THEOREM. This is an immediate consequence of the equations f\ 2/4, 2/5, 2/6 2/3, 2/4, 2/4, 2/5 Consider now the points 1, 2, 3, 4, 5, 6, the coordinates of these being respectively x lt t/!, z^ ...... # 6 , 2/6, z 9 . I represent, for shortness, the equation to the plane passing through the origin and the points 1, 2, which may be called the plane 12, in the form consequently the symbols 12 X , I2 y , 12, denote respectively y^z^ y^z^, Zja; 2 Z^K I} x\y* x$)\, and similarly for the planes 13, &c. If now the intersections of 12 and 45, 23 and 56, 34 and 01 lie in the same plane, we must have, by Lemma (1), the equation =0. 12 y , 45 y , 23 y , 56 y , . 12,, 45,, 23,, 56,, . 23*, 56*, 34,,, Ql x 23,,, 56 y , 34 y , 61 y 23 Z , 56,, 34,, 61, Multiplying the two sides of this equation by the two sides respectively of the equation = 612.345, 2/6, yi, 2/3 2/4> * , *4 , and observing the equations = 612, 112 = 0, &c. DEMONSTRATION OF PASCALS THEOREM. 45 this becomes 612 .. . 645, 145, 245, 623, 123, . . 423, 523 156, 256, 356, 456, 534 361, 461, 561 = 0, reducible to 612 534 145, 245, 123, . 423 156, 256, 356, 456 361, 461 = 0; or, omitting the factor 612 . 534 and expanding, 145 . 256 . 423 . 361 + 245 . 123 . 456 . 361 - 245 . 123 . 356 . 461 - 245 . 156 . 423 . 361 = 0. Considering for instance x 6 , y 9 , z s as variable, this equation expresses evidently that the point 6 lies in a cone of the second order having the origin for its vertex, and the equation is evidently satisfied by writing o? 6 , y a , z t = x y lt z ly or x 3 , y 3 , z,, or x 4 , y 4 , z 4 , or #5, 2/5, Zt, and thus the cone passes through the points 1, 3, 4, 5. For sc 9 ,y , z 6 = x 2 , y a , z i} the equation becomes, reducing and dividing by 245 . 123, 452 . 321 - 352 . 421 + 152 . 423 = 0, which is deducible from Lemma (2), by writing #, y e , z,= x, y,, z,, and is therefore identically true. Hence the cone passes through the point (2), and therefore the points 1, 2, 3, 4, 5, 6 lie in the same cone of the second order, which is Pascal s Theorem. I have demonstrated it in the cone, for the sake of symmetry; but by writing throughout unity instead of z, the above applies directly to the case of the theorem in the plane. The demonstration of_ Chasles^ form of Pascal s Theorem (viz. that the anharmoiiic relation of the planes 61, 62, 63, 64 is the same with that of 51, 52, 53, "54), is very much simpler; but as it would require some preparatory information with reference to the analytical definition of the similarity of anharmonic relation, I must defer it to another opportunity. 4G 10. ON THE THEORY OF ALGEBRAIC CURVES. [From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 102 112.] SUPPOSE a curve defined by the equation U = 0, V being a rational and integral function of the ?n th order of the coordinates x, y. It may always be assumed, without loss of generality, that the terms involving x m , y m , both of them appear in Z7; and also that the coefficient of y m is equal to unity : for in any particular curve where this was not the case, by transforming the axes, and dividing the new equation by the coefficient of y m , the conditions in question would become satisfied. Let H m denote the terms of U, which are of the order m, and let y - ax, y - ($x , . . y - \x be the factors of H m . If the quantities a, ft ...X are all of them different, the curve is said to have a number of asymptotic directions equal to the degree of its equation. Such curves only will be considered in the present paper, the consideration of the far more complicated theory of those curves, the number of whose asymptotic directions is less than the degree of their equation, being entirely rejected. Assuming, then, that the factors of H m are all of them different, we may deduce from the equation 7=0, by known methods, the series / a / 1 \ y = ax + a H --- h ...... , .............................. [*) x 8" \" y = \x + \ H --- h ...... , X and these being obtained, we have, identically, U =(-*x -a - ...}(y -px-p -...}... (y-\x-\ - ...} ......... (2), 10] ON THE THEORY OF ALGEBRAIC CURVES. 47 the negative powers of x on the second side, in point of fact, destroying each other. Supposing in general that fx containing positive and negative powers of x, Efx denotes the function which is obtained by the rejection of the negative powers, we may write ..... (3), the symbol E being necessary in the present case, because, when the series are continued only to the power x~ m+1 , the negative powers no longer destroy each other. We may henceforward consider U as originally given by the equation (3), the m(m+l) quantities a, a ...a< w >, , /3 ... /3< m >, ... X, X ...X<> satisfying the equations obtained from the supposition that it is possible to determine the following terms a (m+D ) p (m+i) _ x (m+1) , >tt so that the terms containing negative powers of x, on the second side of equation (2), vanish. It is easily seen that a, /3 . . . X, of, /3 . . . X are entirely arbitrary, a", $" . . . X" satisfy a single equation involving only the preceding quantities, a. ", ft " . . . X " two equations involving the quantities which precede them, and so on, until a (m) , /3 (m) ... X (m) , which satisfy (m 1) relations involving the preceding quantities. Thus the m (m + 1) quantities in question satisfy |m(ra-l) equations, or they may be considered as functions of m (m + 1) - |m (m - 1) = \m (m + 3) arbitrary constants. Hence the value of U, given by the equation (3), is the most general expression for a function of the m th order. It is to be remarked also that the quantities a< w+1 >, /3< m+1 > ... X< w+1 >, ...... are all of them completely determinate as functions of a, /3 ... X, ... a< w >, /3< m > ... X (m >. The advantage of the above mode of expressing the function U, is the facility obtained by means of it for the elimination of the variable y from the equation U=Q, and any other analogous one F = 0. In fact, suppose V expressed in the same manner as U, or by the equation > n > n (4), n being the degree of the function V. It is almost unnecessary to remark, that A, B...K,...A^, BW...K are to be considered as functions of |n( + 3) arbitrary constants, and that the subsequent A (n+l) , B (n+l) ...K (nJr ^ ... can be completely determined as functions of these. Determining the values of y from the equation (3), viz. the values given by the equations (2) ; substituting these successively in the equation V=(y-Ax - ...}(y-Bx- ...)... (y-Kx- ...) = (5), analogous to (2), and taking the product of the quantities so obtained, also observing that this product must be independent of negative powers of x, the result of the elimination may be written down under the form E ( ( . n(inn) _ A (mn)\ ( ,, (mn) _ (6X 48 ON THE THEORY OF ALGEBRAIC CURVES. [10 the series in { } being continued only to x~ Jmn ^ > because the terms after this point produce in the whole product only terms involving negative powers of x. It is for the same reason that the series in ( ), in the equations (3) and (4), are only continued to the terms involving x~ m+1 , x~ n+l respectively. The first side of the equation (6) is of the order mn, in x, as it ought to be. But it is easy to see, from the form of the expression, in what case the order of the first side reduces itself to a number less than mn. Thus, if n be not greater than m, and the following equations be satisfied, A = OL, A w = a (1 > . . . A "- 1 * = a**- 1 *, r > n ....................... (7), B =, BV =p w ... B-v =13^, s^r, K=K, RU=K {I > ...K (v -v = K^-v, v >> u, the degree of the equation (6) is evidently mn - r s ... - v, or the curves U = 0, V = intersect in this number only of points. If mn r s ... v 0, the curves U=Q and V=Q do not intersect at all, and if mn rs v be negative, = co suppose, the equation (6) is satisfied identically ; or the functions U, V have a common factor, the number o> expressing the degree of this factor in x, y. Supposing the function V given arbitrarily, it may be required to determine U, so that the curves 17=0, V= intersect in a number mnk points. This may in general be done, and done in a variety of ways, for any value of k from unity to ^m(m + 3). I shall not discuss the question generally at present, nor examine into the meaning of the quantity mn \m (m + 3) { = \m (In m 3)} becoming negative, but confine myself to the simple case of U and V, both of them functions of the second order. It is required, then, to find the equations of all those curves of the second order which intersect a given curve of the second order in a number of points less than four. Assume in general = E(y- Ax- A - \ly-Bx-B -}, \ J x J V x) 7T K then A", B" satisfy A" + B" = Q, and putting B"=-T~^R anc * tnere f re ^ = ~ A~-^B and reducing, we obtain V=(y-Ax-A }(y-Bx-B }+K. Similarly assume U= E ( y ax a -- j ( y fix ft -- ) , \ x J \ x J then a", /3" satisfy oc" + /3" = Q, and putting (3" = -- ^, and therefore a." = -- -^, and reducing, we obtain U = (y - ax - a ) (y $x /3 ) + k. 10] ON THE THEORY OF ALGEBRAIC CURVES. 49 Suppose (1) U=0, V=Q intersect in three points, we must have a = A, or the curve 7=0 must have one of its asymptotes parallel to one of the asymptotes of V= 0. (2) The curves intersect in two points. We must have a. = A, a. = A , or else a=A, fi = B; i.e. U=0 must have one of its asymptotes coincident with one of the asymptotes of the curve V= 0, or else it must have its two asymptotes parallel to those of F=0. The latter case is that of similar and similarly situated curves. (3) Suppose the curves intersect in a single point only. Then either a = A , a? = A , a." = A", which it is easy to see gives U= (y - Ax - A ) (y - fix - ff) + K A ~ A- B or else a=A, a! = A , ft = B, which is the case of one of the asymptotes of the curve U=0, coinciding with one of those of the curve F=0, and the remaining asymptotes parallel. As for the first case, if a, a : are the transverse axes, 6, l the inclinations of the two asymptotes to each other, these four quantities are connected by the equation a 2 tan 0cos 2 ai 2 tan #! cos 2 #! and besides, one of the asymptotes of the first curve is coincident with one of the asymptotes of the second. This is a remarkable case ; it may be as well to verify that U=0, F=0 do intersect in a single point only. Multiplying the first equation by y-Bx- B f , the second by y - fix - @ , and subtracting, the result is (A-p}(y-Bx-B )-(A-B)(y-(3x-p} = Q, reducible to A (B - P) + BP - B /3 y-Ax=- ^_ -, i. e . y-Ax-C=Q. Combining this with V = 0, we have an equation of the form y Bx D = Q. And from this and yAxC = 0, x, y are determined by means of a simple equation. (4) Lastly, when the curves do not intersect at all. Here a = A, a. = A , ft=-B, P = B , or the asymptotes of U=0 coincide with those of F=0; i.e. the curves are similar, similarly situated, and concentric: or else a = A, a! = A , a? = A", ft = B ; here U=(y-Ax-A }(y-Bx-P}+K, or the required curve has one of its asymptotes coincident with one of those of the proposed curve; the remaining two asymptotes are parallel, and the magnitudes of the curves are equal. In general, if two curves of the orders m and n, respectively, are such that r asymptotes of the first are parallel to as many of the second, s out of these asymptotes C. ___ _ 7 OF THE UNIVERSITY 50 ON THE THEORY OF ALGEBRAIC CURVES. [10 being coincident in the two curves, the number of points of intersection is mn-r-s: but the converse of this theorem is not true. In a former paper, On the Intersection of Curves, [5], I investigated the number of arbitrary constants in the equation of a curve of a given order p subjected to ^ pass through the ran points of intersection of two curves of the orders m and n respectively. The reasoning there employed is not applicable to the case where the two curves intersect in a number of points less than raw. In fact, it was assumed that, F = being the equation of the required curve, W was of the form uU+vV; u, v being polynomials of the degrees p - m, p-n respectively. This is, in point of fact, true in the case there considered, viz. that in which the two curves intersect in raw points; but where the number of points of intersection is less than this, u, v may be assumed polynomials of an order higher than p-m, p-n, and yet uU+vV reduce itself to the order p. The preceding investigations enable us to resolve the question for every possible case. Considering then the functions U, V determined as before by the equations (3), (4), suppose, in the first place, we have a system of equations a = A, $ = B e = H (t equations) (8). Assume P= (y - ax - ...)(#-#c- ...) ... (y - 8x - ...).- Q= (y - Ax-. ..) (y -Ex -...}... (y - Hx -...); T = (y-i.x-...} ... (y-KX- ...), whence U=PT, Suppose T = ET + AT, EV . U - ET ,V=EV.PT- ET . = EV.P (ET + AT) -ET.Q (EV + = ET . EV . (P - Q} + EV . P . AT - ET . Q . = E{ET.EV. (P -Q} + EV . PAT -ET.Q = II suppose. In this expression ET, EV are of the degrees m-t, n-t, AT, A^~ of the degree - 1, and P, Q, P- Q of the degrees t, t, t-l respectively. The terms of II are therefore of the degrees m + n-t-l, m-1, n l respectively, and the largest of these is in general m + n - t - 1. Suppose, however, that ra + n - t - I is equal to m - I (it cannot be inferior to it), then t = n ; becomes equal to unity, or A^ vanishes. The remaining two terms of II are ET (P - Q), PAT, which are of the degrees m-1, n - I respectively. II is still of the degree m-1, supposing ra > n. If m = n, the term PAT vanishes. II is still of the degree ra - 1. Hence in every case the degree of II is m + n t l: assuming always that P Q does not reduce itself to a degree lower than t-l, (which is always the case as long as the equations 10] ON THE THEORY OF ALGEBRAIC CURVES. 51 a = A , ft = B ...... 6 = H are not all of them satisfied simultaneously). It will be seen presently that we shall gain in symmetry by wording the theorem thus : the degree of II is equal to the greatest of the two quantities m + n t 1, m l. Suppose next, in addition to the equations (8), we have a = A , = B ...... ?=F, t equations, >... (8 ). Then, taking T , W, P , Q the analogous quantities to T, ^ P, Q, and putting EV. U-ET. V=TL , we have, as before, IT = E {ET . EV . (P - Q } + EV . P AT - ET . The degree of P - Q is if - 2 (unless simultaneously a" = A", J3" = B" ...... " = F" t in which case the degree may be lower). The degrees, therefore, of the terms of II are m+n t 2, n-l, ml. Or we may say that the degree of II is equal to the greatest of the quantities m + n - 1 2, m - 1 ; though to establish this proposition in the case where t = n l would require some additional considerations. Continuing in this manner until we come to the quantity II*- 1 , the degree of this quantity is the greatest of the two numbers m + n t {k ~ 1} k, m l. And we may suppose that none of the equations &fo = A ( *> ...... are satisfied, so that the series II, II ...... n**- 1 is not to be continued beyond this point. Considering now the equation of the curve passing through the mn t t ...t (k ~ l} points of intersection of 17=0, V=Q, we may write W=uU+vV+pU+p U f ...... +p*- n*- 1 = .................. (9), for the required equation; the dimensions of u, v, p, p ...p (k ~ l} being respectively p m, p n; p m n + t+lorp ra+1; p m-n + t + 2 or p m + I ; ...... p m n + t (k ~ l] + k or p m + 1 , the lowest of the two numbers being taken for the dimensions of p, p ... p (k ~ l) . Also, if any of these numbers become negative, the corresponding term is to be rejected. In saying that the degrees of p, p ...... p*-v have these actual values, it is supposed that the degrees of II, II ...... II*" 1 actually ascend to the greatest of the values p- mn + t+l, or m l; m + n t 2, or m-l ; m + n - t tk ~ 1} + k, or ml. The cases of exception to this are when several of the consecutive numbers t, t ...t {k - 1} are equal. In this case the corresponding terms of the series II, II ...... U (k ~ 1} , are also equal. Suppose for instance t, t were equal, II, II would also be equal. A term of p of an order higher by unity than p m n + t + l, or p-m+l, which is the highest term admissible, produces in pll a term identical with one of the terms of p li ; so that nothing is gained in generality by admitting such terms into p. The equation (9), with the preceding values for the dimensions of p, p ...... p (k ~ l) , may be 72 52 ON THE THEORY OF ALGEBRAIC CURVES. [10 employed, therefore, even when several consecutive terms of the series t, t t*~^ are equal. It will be convenient also to assume that p m is not negative, or at least for greater simplicity to examine this case in the first place. u, U, and v, V, contain terms of the form a^y^U, aPy*V, a + /3 ;}> p m, y + S $> p n; pTi contains terms of this form, and in addition terms for which a + /3=p + l in, ry + 8 = p + 1 n. It is useless to repeat the former terms, so that we may assume for p, a homogeneous function of the order p m n + t + l, or p m+1; in which case pH consists only of terms for which a + /3 = p + 1 m, y + S = p + ln. And the general expression of p contains p mn + t + 2, or p ra + 2, arbitrary constants. Similarly p Yl contains terms of the form of those in ull, vV, pH, and also terms for which oL+ft = p + 2m, y+S = p+2n; the latter terms only need be considered, or p may be assumed to be a homogeneous function of the order p m n + t + 2, or p m + 1 , containing therefore p m-n if+3, or p m + 2 arbitrary constants. Similarly p (k ~ l] contains p m n + t (k ~ 1} + k + I or p m+2 arbitrary constants. Assume + k+l where, in forming the value of V the least of the two quantities in ( ) is to be taken; this value also, if negative, being replaced by zero. The number of arbitrary constants in p, p ...p (k ~ l) is consequently equal to V. The numbers of arbitrary constants in u, v, are respectively (1+2 + ( p _ m + i)} and {1+2 + ( p - n +l)} i.e. z(p - m + 1) (p - m + 2), and | (p - n + 1) (p - n + 2) ; thus the whole number of arbitrary constants in W, diminished by unity (since nothing is gained in generality, by leaving the coefficient (for instance of yf} indeterminate, instead of supposing it equal to unity) becomes I (p - m + 1) (p - m + 2) + I (p - n + 1) (p - n + 2) + V - 1, reducible to |p (p + 3) + ^ (p m n + 1) (p m n + 2) mn + V. By the reasonings contained in the paper already referred to, if p -f k in n + 1 be positive, to find the number of really disposable constants in W, we must subtract from this number a number | (p + k - m n+ l)(p + k m - n + 2). Hence, calling (/> the number of disposable constants in W, we have (f) = %p (p + 3) + i (p - m -n + l)(p- m - n + 2) - mn + V - A (11), where A = 0, if p + k m-n + 1 be negative or zero (12), A = | (p + k m n+l)(p + k- m n + 2), if p + k m 71 + 1 be positive ; and y is given by the equation (9). 10] ON THE THEORY OF ALGEBRAIC CURVES. 53 Also, if be the number of points through which the curve W = can be drawn, including the points of intersection of the curves U=0, 1^=0, then Any particular cases may be deduced with the greatest facility from these general formulas. Thus, supposing the curves to intersect in the complete number of points mn, we have < = zP (P + 3) + i (1 H) (p m n + 1) (p m n + 2) mn, 8 being zero or unity according as p<m+nl or p>m+n-l. Reducing, we have, for p ^> m + n 3, zP (p + 3) + i (p m n + 1) (p m n + 2) mn, and for p > m + n 3, (f> = \p (p + 3) mn, Suppose, in the next place, the curves have t parallel pairs of asymptotes, none of these pairs being coincident. Then p 4> m + n t 2, $ = kp (p + 3) + (p mn + l) (p m n+2) mn, e = IP (p + 3) + | (p - m - n + 1) (p - m - n + 2) - t; p>m + n t 2, p ;}> m + n 2, 4> = i/ 3 (p + 8 ) + k (p - m ~ n + 2 ) (p - m - n + 3) - mn + t, e = zP (p + 3 ) + 2 (p ~ *n ~ n + 2 ) (p ~ ~ n + 3), p >m + n 2, In these formulae, if t be equal to 2 or greater than 2, the limiting conditions are more conveniently written p^> m + n 1-2; p $> m + n t 2>m + n 4; p>m + n 4t. Similarly may the solution of the question be explicitly obtained when the curves have t asymptotes parallel, and t out of these coincident, but the number of separate formula? will be greater. In conclusion, I may add the following references to two memoirs on the present subject: the conclusions in one point of view are considerably less general even than those of my former paper, though much more so in another. Jacobi, Theoremata de punctis intersectionis duarum curvarum algebraicarum ; Crelle s Journal, vol. xv. [1836, pp. 285 308]; Pliicker, Theoremes geneVaux concernant les equations a plusieurs variables, d un degre quelconque entre un nombre quelconque d inconnues. D vol. xvi. [1837, pp. 4757]. 54 ON THE THEORY OF ALGEBRAIC CURVES. [10 Addition. As an exemplification of the preceding formulae, and besides as a question interesting in itself, it may be proposed to determine the asymptotic curves of the r th order of a given curve having all its asymptotic directions distinct, r being any number less than the degree of the equation of the given curve. DEFINITION. A curve of the r th order, which intersects a given curve of the ?w th order in a number of points, = mr - \r (r + 3), is said to be an asymptotic curve of the r th order to the curve in question. Suppose, as before, T=0 being the equation to the given curve, \ (w) \ and let 6, <j> . . . &> denote any combination of r terms out of the series a ... X, and , </> ... &/, &c. ... the corresponding terms out of a. ... V, &c. Then, writing (where the quantities 3> (m) , ^i- 1 , ^<>, n ...H (Ml) are entirely determinate, since, by what has preceded, ff, d> ...n satisfy a certain equation, (9", d>", ... H" two equations ...... ^(w)^ ^)(w) ? n (m) (?n 1) equations), we have F = for the required equation of the asymptotic curve. It is obvious that the whole number of asymptotic curves of the order r, is n (n 1) ... (n - r + 1), viz. 1 . 2 ... r curves for each combination of n (n - 1) ... (n ^-^H > as y m pt o tes. Some particular instances of asymptotic curves will be found in a memoir by M. Pliicker, Liouvilles Journal, vol. I. [1836, pp. 229 252], Enumeration des courbes du quatrieme ordre, &c. The general theory does not seem to be one to which much attention has been paid. 55 11. CHAPTERS IN THE ANALYTICAL GEOMETRY OF (n) DIMENSIONS. [From the Cambridge Mathematical Journal, vol. iv. (1843), pp. 119 127.] CHAP. 1. On some preliminary formula. I TAKE for granted all the ordinary formulae relating to determinants. It will be convenient, however, to write down a few, relating to a certain system of determinants, which are of considerable importance in that which follows: they are all of them either known, or immediately deducible from known formulae. Consider the series of terms * i > ^2 ^n (1). AI > A., A n K lt K, K n the number of the quantities A K being equal to q (q < n). Suppose q + l vertical rows selected, aiid the quantities contained in them formed into a determinant, this may be done in -- - =- different ways. The system of determinants -L . H/ ^ Q ~~~ -L so obtained will be represented by the notation (2), A,, A, A H K n and the system of equations, obtained by equating each of these determinants to zero, by the notation (3). A. 2 A ti K,. ..K n 5G ANALYTICAL GEOMETRY OF (n) DIMENSIONS. [11 The n - equations represented by this formula reduce themselves to 1.2 (w q + l) (n q) independent equations. Imagine these expressed by (1) = 0, (2) = (n-q)=0 (4), any one of the determinants of (2) is reducible to the form where 1} 2 ... n _ 3 are coefficients independent of x ly x 2 ...x n . The equations (3) may be replaced by ...\ n x n , /v?! + ..., ...T I .T I + ... =0 (6), and conversely from (6) we may deduce (3), unless X 1; X 2 ,...X n =0 (7). TI , T 2 , . . . T n (The number of the quantities X, //,... r is of course equal to n.) The equations (3) may also be expressed in the form /v* fir .*/ 2 , ... ^ n \ A i 77" "\ A i V /J^ Ai-n.o ~r . . . O)i-f\ 2y . . . /^-i-o-n ~r <&i-l*-n j \)> the number of the quantities X, yu. . . . co being q. And conversely (3) is deducible from (8), unless o, =0.. (9). CHAP. 2. On the determination of linear equations in x 1} x z ,...x n ivhich are satisfied by the values of these quantities derived from given systems of linear equations. It is required to find linear equations in x 1 ,...x n which are satisfied by the values of these quantities derived 1. from the equations gT = 0, 33 = ... <5 = ; 2. from the equations " = 0, W = ... &" = ; 3. from & "=(), W = . . . W = 0, &c. &c., where n x n , .............................. (1), ... + B n x n , and similarly ^", i3",..., &" , 2S ", ..., &c. are linear functions of the coordinates^, x z ,...x n . 11] ANALYTICAL GEOMETRY OF (n) DIMENSIONS. 57 Also r , r"... representing the number of equations in the systems (1), (2)... and k the number of these given systems, (n - r) + (n - r") +...}> n - 1 or (k - 1) n + 1 > r + r" + . . . Assume (2), the latter equations denoting the equations obtained by equating to zero the terms involving x lt those involving a? 2 , &c. ... separately. Suppose, in addition to these, a set of linear equations in V, p! ... \", /j," ... so that, with the preceding ones, there is a sufficient number of equations for the elimination of these quantities. Then, performing the elimination, we thus obtain equations M* = 0, where ^ is a function of #1, a? a ... which vanishes for the values of these quantities derived from the equations (1) or (2) ...&c. The series of equations ^ = may be expressed in the form A , * ,... n",--- On", S, ...o/ , A /", = (3). CHAP. 3. On reciprocal equations. Consider a system of equations ... + A n x n =0, .............................. (1) ? ... + K n x n = 0, (r in number). The reciprocal system with respect to a given function (U) of the second order in #! , x. 2 . . . x n , is said to be d Xl U, A % ,... A n K.,, ... K n = -(2), (n - r in number). It must first be shown that the reciprocal system to (2) is the system (1), or that the systems (1), (2) are reciprocals of each other. C. o 58 ANALYTICAL GEOMETRY OF (n) DIMENSIONS. Consider, in general, the system of equations [11 n U = (3). n U=Q. Suppose 2 U = 2 (a?]x? + 22 (a/3) x a x ft , so that d Xl U=2 (sot) x a (4), (5). The equations (3) may be written x, [a, (I 2 ) + <x 2 (12) . . . + (In)} + . . . + x n [a, (wl) + a 2 (w2) . . . + a n (n 2 )} =0 (6), &c. and forming the reciprocals of these, also replacing d x JJ, d^U ... by their values, we have x, (I 2 ) + a? 2 (12) + . . . x n (In), ...x, (nl) + # 2 (n2) ...+%, (w 2 ) = (7). a, (I 2 ) + a 2 (12) + ...* (Iw), . . . a 1 (nl) + 2 (n2) . . . + a n (7i 2 ) X, (I 2 ) + \, (12) + . . . \ n (In), ... X, (wl) + X 2 (n-2) . . . + \ n ( 2 ) From these, assuming (Pj, (12),... (In) , + (8) (21), (2 2 ),... (2n) (nl), (n2),... (n 2 ) we obtain, for the reciprocal system of (3), ; n =0 (9). Now, suppose the equations (3) represent the system (2) ; their number in this case must be n r. Also if 6 represent any one of the quantities a, /3 . . . X, we have = Q (10), By means of these equations, the system (9) may be reduced to the form A f -L- A IT -i- A f If 1* I V f I If rm <m If T -i-L 1 ** 1 | -i-t- 2 t*> 2 . . | -iJ- )i ^1\ ) ... Ji 1 fcCj [^ J-l- 2 w 2 ... ^^ ^A ^ ^ /i J *AJy> _LJ , 4*. / _^- 2 > . . w , or,,, , <7 r , ., , ... a o which are satisfied by the equations (1). Hence the reciprocal system to (2) is (1), or (1), (2) are reciprocals to each other. 11] ANALYTICAL GEOMETRY OF (n) DIMENSIONS. THEOREM. Consider the equations (& -0, 23 =0... =0) (E"=o, w =o...jr =0), 59 (12), &c. of Chap. 2. The equations d x U , d x JJ , ... d Xn U = 0, AS, AJ,... A n fy i ri A / (r 1 , Lr 2 , ... (JT n \ &C. , d x JJ ,...d XH U 0," " \J% , . . . \J n = 0, ...(13), which are the reciprocals of these systems, represent taken conjointly the reciprocal of the system of equations (3) of the same chapter. Let this system, which contains n - {(n - r] + (n - r ) + . . .} equations, be represented by !#! + flr a fl?a ... + &= (14), r a . . . + j3 n x n = 0. The reciprocal system is [ C* - t* /-v ^ d x lJ,...d Xn l r , =0 b i > Ss > containing (n - r) + (n - r ) + &c. . . . equations. Also, by the formulae in Chap. 2, !#!+ ... +o n x n = X/^ / writing ^ = n - {(n - r) + (n - r ) + . . . }. Also, assuming any arbitrary quantities ^, sets being (/ - 0),) such that (15), , p...<r, r in number). .(16), 7?, n ... 1; 0, ... W (the number of (17), ; 2 r,o ANALYTICAL GEOMETRY OF (?l) DIMENSIONS. [11 From the equations (15) we deduce the (n r) equations i d Xl U, d X2 U,...d Xn U I -0 (18). +... Hence, writing . . . a- r G, and reducing, by the formula (8) of Chap. 1, we have d Xl U, d x lJ,...d XH U = A A A *I > -"-a > -An (19). (20); and similarly may the remaining formulae of (13) be deduced from the equation (15). Hence the required theorem is demonstrated, a theorem which may be more clearly stated as follows : The reciprocals of several systems of equations form together the reciprocal of the equation which is satisfied by the values of the variables which satisfy each of the original systems of equations. (The theorem requires that the number of all the reciprocal equations shall be less than the number of variables.) Conversely, consider several systems of equations, the whole number of the equations being less than the number of variables. These systems, taken conjointly, have for their reciprocal, the equation which is satisfied by the values satisfying the reciprocal system of each of the given systems. CHAP. 4. On some properties of functions of the second order. Suppose, as before, U denotes the general function of the second order, or Also let V denote a function of the second order of the form Y Tf ( r T T r -LJ. *i> * *?; 1, PI, P-2, [22), \) (H being the symbol of a homogeneous function of the second order, and the number r of the quantities a, @...p, being less than n 1). [Observe that a^ ^ 1 ,...p 1> ... <z n , /3 n ,...p n , or say the suffixed quantities a, @,...p (r in number) are used to denote coefficients : a, /3, without suffixes, are any two numbers in the series of suffixes 1, 2, 3,...w.] Then 2t7 - 2&F, k arbitrary, is of the form ]^> ................................. (23). 11] ANALYTICAL GEOMETRY OF (n) DIMENSIONS. Suppose X lt X 2 ,...X n determined by the equations 61 = (24), equations that involve the condition that k satisfies an equation of the order n r, as will be presently proved. Then shall x l =X 1 ... x n = X n satisfy the system of equations, which is the reciprocal of #u x. 2 ,...x n = (25). PI, Pi,---Pn To prove these properties, in the first place we must find the form of V. Consider the quantities % At % B , ,..^ L , (n-r) in number, of the form A = A 1 x 1 + A. 2 x 2 ... +A n x n , (2G), >.B -^1 ^l l" -^2 **"2 -*-*ii &ii j where, if represent any of the quantities 4, 5 ... L, a^ + 0202 ... +o B n =0, (27), B 2 + ... +2 (45) ^,+ ... = 2 (4 2 ) & + 22 (45) &&. Hence, if 2F=2 {a 2 } ^ a 2 + 22 {0/8} a?.*,, (28), we have for the coefficients of this form and consequently the coefficients of 2 U 2k V are [1=] = (P) - /, {p}, [ 12 ] = (12) - k {12}. Hence, representing any of the quantities a, /3 . . . p, 0! {I 2 } +# 2 {12} ... +0 n {In} = (29), I I ^l ) G2 ANALYTICAL GEOMETRY OF (n) DIMENSIONS, whence also 0, [1 s ] + ... n [ln] = 1 (V) + ... n (ln\ 0, [nl] +... 6 n [w 2 ] = 0, (nl) + ... O n (w 2 ). Hence, the equations for determining X l ,...X n may be reduced to X, fa (I 2 ) + ...* (In)] + X 2 fa (21) ...+, (2n)] . . . + X n fa (nl) ... + a n (n 2 )] = C" r^ft a*)*-/ STi[r + l, 1] + ^ [n, 1] + (21) ... +/S, (2w)] . . . + X n 2 )] = 0, + X 2 [ Pl (21) . . . + pn (2n)] ...+X n [p n (nl) ... + f, (n 8 )] - 0. X, [r + 1, 2] ... + X n [r + 1, n] = 0, i, 2]...+ = 0. Eliminating XJ...XH, since the first r equations do not contain fc, the equation in this quantity is of the order n r. Next form the reciprocals of the equations (25). These are x U, d x lJ,...d Xn U A\ > +** > -"-n -0 (31). From which we may deduce a^, U ... + <x n d x ,, U, fadtcL U ... 4- &A Z7, . . . p^ IT. . . p n d Xn U, d Xr+l U, ... d Xtt U > -4,-+i) A n , -L f+l , . .. L n , 6 , which are evidently satisfied by x 1 = X z ...x n = = 0. . . (32), In the case of four variables, the above investigation demonstrates the following properties of surfaces of the second order. I. If a cone intersect a surface of the second order, three different cones may be drawn through the curve of intersection, and the vertices of these lie in the plane which is the polar reciprocal of the vertex of the intersecting cone. II. If two planes intersect a surface of the second order through the curve of intersection, two cones may be drawn, and the vertices of these lie in the line which is the polar reciprocal of the line of intersection of the two planes. Both these theorems are undoubtedly known, though I am not able to refer for them to any given place. 12] 63 12. ON THE THEORY OF DETERMINANTS. [From the Transactions of the Cambridge Philosophical Society, vol. vin. (1843), pp. 1 16.] THE following Memoir is composed of two separate investigations, each of them having a general reference to the Theory of Determinants, but otherwise perfectly unconnected. The name of " Determinants " or " Kesultants " has been given, as is well known, to the functions which equated to zero express the result of the elimination of any number of variables from as many linear equations, without constant terms. But the same functions occur in the resolution of a system of linear equations, in the general problem of elimination between algebraic equations, and particular cases of them in algebraic geometry, in the theory of numbers, and, in short, in almost every part of mathematics. They have accordingly been a subject of very considerable attention with analysts. Occurring, apparently for the first time, in Cramer s Introduction a I Analyse des Lignes Courbes, 1750 : they are afterwards met with in a Memoir On Elimination, by Bezout, Memoires de I Academie, 1764; in two Memoirs by Laplace and Vandermonde in the same collection, 1774; in Bezout s Theorie gtnerale des Equations algebriques [1779] ; in Memoirs by Binet, Journal de I Ecole Poly technique, vol. ix. [1813]; by Cauchy, ditto, vol. x. [1815]; by Jacobi, Crelles Journal, vol. xxn. [1841] ; Lebesgue, Liouville, [vol. n. 1837], &c. The Memoirs of Cauchy and Jacobi contain the greatest part of their known properties, and may be considered as constituting the general theory of the subject. In the first part of the present paper, I consider the properties of certain derivational functions of a quantity U, linear in two separate sets of variables (by the term " Derivational Function," I would propose to denote those functions, the nature of which depends upon the form of the quantity to which they refer, with respect to the variables entering into it, e.g. the differential coefficient of any quantity is a derivational function. The theory of derivational functions is apparently one that would admit of interesting developments). The particular functions of this class which are here considered, are closely connected with the theory of the reciprocal polars of surfaces of the second order, which latter is indeed a particular case of the theory of these functions. In the second part, I consider the notation and properties of certain functions resolvable into a series of determinants, but the nature of which can hardly be explained independently of the notation. ON THE THEORY OF DETERMINANTS. [12 In the first section I have denoted a determinant, by simply writing down in the form of a square the different quantities of which it is made up. This is not concise, but it is clearer than any abridged notation. The ordinary properties of determinants, I have throughout taken for granted ; these may easily be learnt by referring to the Memoirs of Cauchy and Jacobi, quoted above. It may however be convenient to write down the following fundamental property, demonstrated by these authors, and by Binet. a, , /3,... P, pa+ffft..., pa! + a ft ... , p a. + a {3 ..., p a. + pa- ..., (), an equation, particular cases of which are of very frequent occurrence, e.g. in the investigations on the forms of numbers in Gauss Disquisitiones Arithmetic, [1801], in Lagrange s Determination of the Elements of a Comet s Orbit [1780], &c. I have applied it in the Cambridge Mathematical Journal [1] to Carnot s problem, of finding the relation between the distances of five points in space, and to another geometrical problem. With respect to the notation of the second section, this is so fully explained there, as to render it unnecessary to say anything further about it at present. 1. On the properties of certain determinants, considered as Derivational Functions. Consider the function ...)+ (1), *! + ...) + (n lines, and n terms in each line) ; and suppose KU = a, ft, , P, .(2). (The single letter K being employed instead of KU, in cases where the quantity KU, rather than the functional symbol K, is being considered.) And write FU =- nf+S77+..., R + s ?? + . . . , Y U = i A# + K x + ..., Btf + BY + a , ft a! , P R 4- R Y + , & + s Y + (3). (4). The symbols K, F, investigate. possess properties which it is the object of this section to 12] ON THE THEORY OF DETERMINANTS. 65 Let A, B,..., A , B ,...,... be given by the equations: A= B . y , . . . , B = + y , 8 . i > "\ v / ft II // 7 > J e, A = /3", y",... , = y", 8",... (The upper or lower signs according as n is odd or even.) These quantities satisfy the double series of equations, Act + B{3 +...=*, (6). Aa + B(3 + ...=0, &c. AOL + A a. + ... =K, (7) ; Bo. + J B a + ...=0, B/3+B P + ... = *, &c. the second side of each equation being 0, except for the r th equation of the r th set of equations in the systems. Let X, /*, . . . represent the r th , r + 1 th , . . . terms of the series a, /3, . . . ; L, M, . . . the corresponding terms of the series A, B..., where r is any number less than n, and consider the determinant A >-- L (8), which may be expressed as a determinant of the w th order, in the form A ,...L , 0, 0, (9). L^- t 0, 0, , 1, 0, 0,1, c. 66 ON THE THEORY OF DETERMINANTS. Multiplying this by the two sides of the equation [12 *= , /3, (10), , /3 , and reducing the result by the equation (0), and the equations (6), the second side becomes , 0,... (11), 0, ic, K, , u,^ v ( U, ^L6 , V which is equivalent to if r nW v^ n2" K p 1 > " > \ J - "/> or we have the equation A , ... L = * -* ^ , v* , (13), which in the particular case of r = n, becomes A, B,... A , B , =* -> (14), which latter equation is given by M. Cauchy in the memoirs already quoted; the proof in the " Exercises," being nearly the same with the above one of the more general equation (13). The equation (13) itself has been demonstrated by Jacobi somewhat less directly. Consider now the function FU, given by the equation (3). This may be expanded in the form FU = (n%+sr)+...)[A (AZ + AV +...)+ B (B0 + BV+...) + ...]+ (15), ( R f + 8 v + ...) [A (AOS + AV+ ...) + B (Btf + BV +...)+-..] + which may be written = x (A I + BT; +...)+ (16), ON THE THEORY OF DETERMINANTS. 12] by putting A = A (&A + n A + ...) + B (&B + n B +...)+..., (17). (BB ~B = A (sA + s A + ...)+E (sB +s B + ...) + Hence, KFU = A, B, (18), A , B , A, B,... i R, s,... ; A, B, (19), A , B , R , s , A , B , or observing the equation (14), and writing A, B,... =J (20), A , B , (21), R , S , this becomes KFU = Jf. (KU} n - 1 . (921 \ / i ^j^w /. Hence likewise * V / " * *! ***\ *^ / Consider next the equation R# + RV + . . Sac + sV + (24<} f {AT ). ^ + ..., A , B l r) + . . . , A B (25). A , B , . . . R , S, . . . x, A, B, A , B , R , S , x , A, B, Jf > v, (26). x, A, B, x , A , B, 92 67 68 ON THE THEORY OF DETERMINANTS. [12 If the two sides of this equation are multiplied by the two sides of the equation (2), written under the form K = 1, a, /3,... * , 13 , (27), the second side is reduced to -Jf (28), and hence Similarly n-^U n-2_ U also combining these with the equations (22), (23), VFU FVU U (29), (30). .(31) ; (32). KFU KVU KU It may be remarked here that if U, V are functions connected by the equation (33), then in general U=c n ~ l V (34). To prove this, observing that the first of the equations (33) may be written FU-f(e~ l V) (35), i we have V.FU = V .F(c n ^ V) (36), JL i, or Jf.(KU) n -*U=Jr[K(c n -iV}~\ n -*c n -i V (37). If neither /, / nor (KU] vanish, this equation is of the form whence substituting in (33), U = kV. k n ~ l = c . (38), (39), 12] ON THE THEORY OF DETERMINANTS. 69 which demonstrates the equation (34) ; and this equation might be proved in like manner from the second of the equations (33). If however, J = 0, or f = 0, the above proof fails, and if KU = Q, the proof also fails, unless at the same line n = 2. In all these cases probably, certainly in the case of KU=0, n ^ 2, the equation (34) is not a necessary consequence of (33). In fact FU, or ^U may be given, and yet U remain indeterminate. Let U t , a,, fi t ,...A t) B,, &c. ... be analogous to U, a, /?..., A t , B, &c. ... and consider the equation K(KU,.FU+gKU.FU) (40), c t A+gKA t , KsB + gtcB,,... Multiply the two sides by the two sides of the equation (2), the second side becomes, after reduction, ( (). g K (A ft + Bfi + ...), w + gic (A ,< + B, ff +...), Multiplying by the two sides of the analogous equation ,- ,, /, (42), and reducing, the second side becomes (43), (44), whence K (K U, . FU + gK U . FU) = (K U) n ~ l (K U) n ~ l K (U,+gU) (45), and similarly K (KU / . r iIU+gKU. flU,) = (JKLT) n ~ l (KU^} n ~ l K (U / + gU} (46). In a similar manner is the following equation to be demonstrated, (KU t . FU+gKU. FU} = F(KU,. VU + gKU . ^U^= (47), -jr.(KUV-*(KV^x er/t+a/tf..., 0/c + $, V . . . , a / + go. ..., f$ t +g/3 . . . , a/ + aa. .. I 70 ON THE THEORY OF DETERMINANTS. Suppose U = ^t (pg + <ri) + ...) (ax+a x + ...) (48), this expression being the abbreviation of U=(p^ + <rij + ...)(ax + a x +...)+ (49), + [(n 1) lines, or a smaller number], then KU= 2ap, Sao-,... is=0 (50), which follows from the equation (). Conversely, whenever KU=0, U is of the above form. Also FU = - A#+AV+..., EX + E X + ..., (51) Rf;+ 817 + ..., 2a/> , Sacr R ^ + s ?; + . . . , Sft p , which may be transformed into FU= Aaj + AV..., BOJ + BV..., ... R| + S?;..., R ^+S T?..., (52) p , o- , a , a (for shortness, I omit the demonstration of this equation). And similarly, [12 EX + R x P + B 7? + ..., .. ...(53), where it is obvious that if the sum 2 contain fewer than (n 1) terms, FU=0, fU=0. The equations (52), (53) express the theorem, that whenever KU=0, the functions FU, 1U are each of them the product of two determinants. If next u,= u+u, 12] ON THE THEORY OF DETERMINANTS. 71 then in (45) taking <? = -! [the Numbers (56) &c. which follow are as in the original memoir] )} ......... (56), } n ~i . KU, or observing the equation (50), (57). Hence F {(K (U + 7) . VU- KU . V (U + U)} = V {K (U + U) . FU - KU . F(U+ U}\ are each of them the product of two determinants. But this result admits of a further reduction : we have (58) - /3 substituting a, = a + 2/>a, &c. ..., also observing that if the second line be multiplied by x, the third by x , ... and the sum subtracted from the first line, the value of the determinant is not altered, and that the effect of this is simply to change a /5 o/ ... into a, a ... in the first line, and introduce into the corner place a quantity - U, which in the expansion of the determinant is multiplied by zero: this may be written in the form (59), which may be reduced to Jf . (KU) .(K(U+ U)y x .(60), If each of these determinants are multiplied by the quantity (KU} n -\ expressed under the two forms (61), A, B,... A, A , A , B , B, R, 72 ON THE THEORY OF DETERMINANTS. they would become respectively KU. AP+BO-+..., A p+B <r+..., , KU. [12 ..(62), so that finally (63). AP+BO-+..., A t; Aa+A a +. The second side of this may be written under the forms AE + AV + ... , BOJ + BV + ... A(Ap+Ba. .) + A (A p + B a. .) + ..., v(Ap + B<r..)+V f (A p+B <r..)+ KU multiplied into E R(Aa+A a ..)+S(Ba + B a . .) + ..., R (Aa + A a ..) + S (Ba + B a . .)+..., (64). And KU multiplied into Sx+S x <r..)+R (A p+R<r. a + A a ..}+B(Ba + B a . .)+..., A (Aa (Ba+B a ..) (65). And again, by the equations (52), (53), in the new forms x[(Aa + A a ... (KU) 1 (Ba + B a ...)(& +sV ...)]} ...... X ...)...]} ...(67). 12] ON THE THEORY OF DETERMINANTS. 73 Comparing these latter forms with the two equivalent quantities forming the first side of (53), and observing (33), (34), it would appear at first sight that K(U+U, _fK(UU}\n- ~\ KU (2 [(Ap + Bo- ...) (A+ BT; ...) + (A p + 5V...) ( x [(Aa + A a ...)(RX + R O; ...} + (Ba + B a ...}($x + s x ... )...]}, K(U+U).FU-KU.F(U+U) n-2 x [(Aa + A a . . .) ( a + B a . . .) (EX + BV . . .) . . .]}, x [(Aa + A a ... K(U+U}FU-KU.F(U + U) which however are not true, except for n = 2, on account of the equation (57). In the case of n 2, these equations become x; . ..)...] ......... (68), x [(^La 4- A a ...) (A + AV ...) + (Ba + B a ...) (BOD + BV + ...)...] (69), and it is remarkable that these equations ((68), (69)) are true whatever be the value of n, provided 2 contains a single term only. The demonstration of this theorem is somewhat tedious, but it may perhaps be as well to give it at full length. It is obvious that the equation (69) alone need be proved, (68) following immediately when this is done. I premise by noticing the following general property of determinants. The function a+2pa, /3+2o-a, (70), (where "Spa p 1 a 1 + p^a 2 ... + p s a s ), contains no term whose dimension in the quantities a, a ..., or in the other quantities p, a-..., is higher than s. (Of course if the order of the determinant be less than s or equal to it. this number becomes the limit of the dimension of any term in a, a ... or p, a ..., and the theorem is useless.) This is easily proved by means of a well-known theorem, tp a, Sera,... | = (71), c. X? f or THE (UNIVERSITY -^ 10 74 ON THE THEORY OF DETERMINANTS. [12 whenever s is less than the number expressing the order of the determinant. Hence in the formula (70), if S contain a single term only, the first side of the equation is linear in p, a; ... and also in a, a , ..., i.e. it consists of a term independent of all these quantities, and a second term linear in the products pa, pa , ...era, era ,... This is therefore the form of K (U+ U). Consider the several equations K = KU=A a. +B/3 + (72), = &c. it is easy to deduce * / = K(U + U)=KU + Apa + B aa + (73). + A pa + B aa + To find the values of A, B, &c. corresponding to U+ U, we must write = M"/3+N" 7 " + = &C. where M = 7", 8",... , N = B" , e",... i (75), 7 ", 8 ", 8" , e ", r " I" " -KT" " S" Si-rn, [=+ 7, o ,...!, N = o , e,...!, occ. the order of each of these determinants being n 2, and the upper or lower signs being used according as n 1 is odd or even, i.e. as n is even or odd. Hence A t = A+T& aa +N ra + (76), + M" aa" + N" ra" + . . . and therefore K,A-icA t = A-pa +(AB ) aa + (AC )ra + (77), + AA pa + (AB - /CM ) aa + (AC - *N }ra +... + AA"pa" + (AB" - KM") aa" + (AC" - N") ra" + the additional quantities C, r having been introduced for greater clearness. Now the equations AB -KM =A B, AC - K N =A C, (78), AB" - K M" = A"B, AC" - K N" = A"C, 12] ON THE THEORY OF DETERMINANTS. 75 written under the form AB -A B = icM , AC -A C= K N , ..................... (79), AB" - A"B = K M", A C" -A"G = K N", are particular cases of the equation (13), and are therefore identically true. Hence, substituting in (77), KjA-icA^ A 2 pa + A Baa +AGra ... + ..................... (80), + A A pa + A Baa + A Cra . . . + + A"Apa" + A" Baa" + A Cra" ...+ Forming in a similar manner, the combinations K t B tcE,, ... K t A tcAf, K t E tcBf, . . . , multiplying by the products of the different quantities Ax +A a; ..., Bx + B x ....... Rg + Srj..., R % + S t)..., ... and adding so as to form the function K(U+U). FU - KU . F(U+ U), we obtain the required formula, viz. that the value of this quantity is ...}(n %+s <r). ..}+...] ...... (81); x [(Aa + A a . . .) (A# + AV ...}+(Ba + B a . . .) (EX + sV ...) + ...] with this theorem, I conclude the present section, noticing only, as a problem worthy of investigation, the discovery of the forms of the second sides of the equations (68), (69), in the case of 2 containing more than a single term. 2. On the notation and properties of certain functions resolvable into a series of determinants. Let the letters r 1} r 2 ,...r k .................................... n\ represent a permutation of the numbers 1, 2, ... k ....................................... (2). Then in the series (1), if one of the letters succeeds mediately or immediately a letter representing a higher number than its own, for each time that this happens there is said to be a " derangement " or " inversion." It is to be remarked that if any letter succeed s letters representing higher numbers, this is reckoned for the same number s of inversions. Suppose next the symbol r ............................................. (3), denotes the sign + or , according as the number of inversions in the series (1) is even or odd. 102 76 ON THE THEORY OF DETERMINANTS. [12 This being premised, consider the symbol denoting the sum of all the different terms of the form r s--- Ap n <T Sl ...... Ap rk (r Sk ........................... (5), the letters r lt r 2 ...r k ; s 1} s 2 ...s k ; &c ........................ (6), denoting any permutations whatever, the same or different, of the series of numbers (2) [and the several combinations of pa- ... being understood as denoting suffixes of the A s]. The number of terms represented by the symbol (5) is evidently (1.2 ... k) n ....................................... (7). In some cases it will be necessary to leave a certain number of the vertical rows p, o- ... unpermuted. This will be represented by writing the mark (}) immediately above the rows in question. So that for instance t t the number of rows with the (f) being x, denotes the sum of the (1.2 ...ky-* ....................................... (9) terms, of the form r s ...Ap ri a Sl ...6^ l ...Ap rk (T Sk ...6 k $ k ................. (10). Then it is obvious, that if all the rows have the mark (f) the notation (8) denotes a single product only, and if the mark (-J-) be placed over all but one of the rows the notation (8) belongs to a determinant. It is obvious also that we may write t t =2 u v ... where 2 refers to the different permutations, Wj, u 2 ,...u k ; v lt v 2 ,...v k ; &c ............................ (12), which can be formed out of the numbers (2). The equation (11) would still be true, if the mark ((*) were placed over any number of the columns p, cr ... Suppose in this equation a single column only is left without the mark (f) on the second side of the equation ; the first side is then expressed as the sum of a number (1 . 2 . . . k} n -\ or generally (1 . 2 . . . k} n - x ~ l .................. (13), 12] ON THE THEORY OF DETERMINANTS. 77 of determinants, according as we consider the symbol (4) or the more general one (8). And this may be done in n or (n x) different ways respectively. It may be remarked, that the symbol (8) is the same in form as if a single column only had the mark (() over it ; the number n being at the same time reduced from n to (n x + I): for the marked columns of symbols may be replaced by a single marked column of new symbols. Hence, without loss of generality, the theorems which follow may be stated with reference to a single marked column only. Suppose the letters pi, Pa,"-pk , o-i, o-jj, ...o-j.; &c ...................... (14) denote certain permutations of i, a aj ...t; &, &,...&; &c ........ .. ............ (15), in such a manner that Pi = ac ffi> P2 = a 9 2 "- Pk = <Xg k , 0-i=&., o- = @h., ... o-k = /3h k ......... (16). Then the two following theorems may be proved : 1- + = g h ... < 17 > if n be even : but in the contrary case t t +,... (Aa By means of these, and the equation (11), a fundamental property of the symbol (3) may be demonstrated. We have pkfa which when n is even, reduces itself by (17) to (20) = 1.2...* 78 ON THE THEORY OF DETERMINANTS. [12 But when n is odd, from the equation (18), t . (n)*} S( 1/ l) = (21), *, Pk J since the number of negative and positive values of g are equal. From the equation (20), it follows that when n is even, the value of a symbol of the form t !&, (n)} is the same, over whichever of the columns a, @ ... the mark (f ) is placed. To denote this indifference, the preceding quantity is better represented by t !!, A... (71)] ; [ (23), this last form being never employed when n is odd, in which case the same property does not hold. Hence also an ordinary determinant is represented by t (All kk the latter form being obviously equally general with the former one. It is obvious from the equations (17), (18), that the expression (22) vanishes, in the case of n even whenever any two of the symbols a are equivalent, or any two of the symbols /3, &c.; but if n be odd, this property holds for the symbols /?, &c., but not for the marked ones a. In fact, the interchange of the two equal symbols, in each case, changes the sign of the expression (22), but they evidently leave it unaltered, i.e. the quantity in question must be zero. Consider now the symbol t (Al 1 ... (2 (25), kk which, for shortness, may be denoted by t [A.k.Zp] (26). 12] ON THE THEORY OF DETERMINANTS. 79 I proceed to prove a theorem, which may be expressed as follows : t t where AB n ...*u... = S - A rs...i B xy .. (27), (28), the number of the symbols r, s,... being obviously 2p 1, and that of ac, y,... being 2q \. The summatory sign S refers to I, and denotes the sum of the several terms corresponding to values of I from I 1 to I = k. Also the theorem would be equally true if I had been placed in any position whatever in the series r, s . . . I ; and again, in any position whatever in the series x, y ... I, instead of at the end of each of these. With a very slight modification this may be made to suit the case of an odd number instead of one of the two even numbers 2p, 2q; (in fact, it is only necessary to place the mark (f) in [AB\ ...} over the column corresponding to the marked column in [A ...}, {A ...} being the symbol for which the number of columns is odd), but it is inapplicable where the two numbers are odd. Consider the second side of (27) ; this may be expanded in the form ...AB (29), where 2 refers to the different quantities s, ..., x, y,... as in (11). Substituting from (28), this becomes S.S|, ...Sl k (+ S ... X yA 1Sl _ h ...A kSk _ lk ...B Xl y i ^ i ... B^,^ ^ ... (30). Effecting the summation with respect to x, y ... this becomes t f Z?1 1 7 A -Dl -L . . . fri I ^j, ki k + s -a.18,. ..?,- -^fcV"^ ] : )- (31), ^ kk...l k ) S now referring to s, ... only. The quantity under the sign 2 vanishes if any two of the quantities I are equal, and in the contrary case, we have t t 311. ..O =i{B.k.2q] (32), i if l- l \ id a/ ... i k j which reduces the above to t [B.k.2q] .2+ ... ^L SI ..J, ^k.s k ...i k (33), 2 referring to the quantities s ... , and also to the quantities I. And this is evidently equivalent to t t {A.k.2p}{B.k.2q} (34), the theorem to be proved. It is obvious that when p=l, q=l, the equation (27), coincides with the theorem (), quoted in the introduction to this paper. 80 [13 13. ON THE THEORY OF LINEAR TRANSFORMATIONS. [From the Cambridge Mathematical Journal, vol. iv. (1845), pp. 193209.] THE following investigations were suggested to me by a very elegant paper on the same subject, published in the Journal by Mr Boole. The following remarkable theorem is there arrived at. If a rational homogeneous function U, of the n th order, with the m variables x, y . . . , be transformed by linear substitutions into a function V of the new variables, , ??...; if, moreover, 6U expresses the function of the coefficients of U, which, equated to zero, is the result of the elimination of the variables from the series of equations d x U=Q, d y U=0, &c., and of course 6V the analogous function of the coefficients of V: then 0V=E na .0U, where E is the determinant formed by the coefficients of the equations which connect x, y ... with 77..., x and a = (w-l) m-1 . In attempting to demonstrate this very beautiful property, it occurred to me that it might be generalised by considering for the function U, not a homogeneous function of the w th order between m variables, but one of the same order, containing n sets of m variables, and the variables of each set entering linearly. The form which Mr Boole s theorem thus assumes is 0V = Ef . E. 2 a ... E n a . 0U. This it was easy to demonstrate would be true, if 6U satisfied a certain system of partial differential equations. I imagined at first that these would determine the function 6U, (supposed, in analogy with Mr Boole s function, to represent the result of the elimination of the variables from d x JJ=Q, d yi U = 0, ... d x U= 0, &c.) : this I afterwards found was not the case ; and thus I was led to a class of functions, including as a particular case the function 0U, all of them possessed of the same characteristic property. The system of partial differential equations was without difficulty replaced by a more fundamental system of equations, upon which, assumed as definitions, the theory appears to me naturally to depend ; and it is this view of it which I intend partially to develope in the present paper. 1 The value of a was left undetermined, but Mr Boole has since informed me, he was acquainted with it at the time his paper was written; and has given it in a subsequent paper. 13] ON THE THEORY OF LINEAR TRANSFORMATIONS. I have already employed the notation 81 , P , 7 , 8, (1) , P, 7, 8 , _,// /j// // <y/ a i P , 7 > 6 (where the number of horizontal rows is less than that of vertical ones) to denote the series of determinants, 7, (2), , , 7, ", ", 7", which can be formed out of the above quantities by selecting any system of vertical rows; these different determinants not being connected together by the sign +, or in any other manner, but being looked upon as perfectly separate. The fundamental theorem for the multiplication of determinants gives, applied to these, the formula A, B, C, D t ... =E a, , 7, 8, (3), A, B , C , D , a!, p , 7 , 8 , A", B", C", D", a.", p", y>, 8", where A = \a B = \p . A = fj,a + /j, oL + fi"ot" +... [- (4), B = yU,/3 + yt, &C. j (5), meamng of the equation is, that the terms on the first side are equal, each each, to the terms on the second side. This preliminary theorem being explained, consider a set of arbitrarv coefficients represented by the general formula rst (6), assume all integer values, from 1 to TO inclusively. 11 82 Let ON THE THEORY OF LINEAR TRANSFORMATIONS. asftf ... , asf t," ... , [13 (7) represent the whole series, taken in any order, in which the first symbolical letter is or. Similarly, rjmt;..., r/a*/, ................................. (8), the whole series of those in which the second symbolical letter is a, and so on. Imagine a function u of the coefficients, which is simultaneously of the forms u =H P Is/*/ ... , Is/ */ (A), r,/2 * ..., r/2 *,/ ..., &c. ; in which jfiT^ denotes a rational homogeneous function of the order >. The function H is not necessarily supposed to be the same in the above equations, and in point of fact it will not in general be so. The number of equations is of course =n. The function u, whose properties we proceed to investigate, may conveniently be named a " Hyperdeterminant." Any function satisfying any of the equations (A), without satisfying all of them, will be an " Incomplete Hyperdeterminant." But, con sidering in the first place such as are complete Let rst ... be a new set of coefficients connected with the former ones by a system of equations of the form rst...=\ l r lst...+\S2st...+\ n r mst ..................... (9), (where the r in V... is not an exponent, but an affix). Suppose u is the same function of these new coefficients that u was of the former ones. Then consider the first of the equations (A) and the equation (3), and writing r Li (10), we have immediately the equation Consider the new set of coefficients rst ... = rfrlt ... +frr2t ... + ... + fj^" rmt .......... (U). (12), 13] ON THE THEORY OF LINEAR TRANSFORMATIONS. 83 and u the analogous function of these ; then, from the second of the equations (A) and the equation (3), and writing M= n,\ Ai 2 , (13), 2 3 > w In like manner, considering the new coefficients rsi ... , where rsi . . . = zV rsi . . . + v 2 f rs2 + v n * rsm (15), the new function u, and the quantity N given by N = "i 1 . "- (16), v 1 v 2 we have, as before, %*/! or ii = LPMPNPu . . fl 8} V- -W > whence, generally, denoting the last result by u , u = LPMPNP ...u . (B 1} \- Lj > --/ Consider now the function 222... (rst ...x r y s z t ...} (19), where the 2 s refer successively to r, s, t, ... , and denote summations from 1 to m inclusively. If u be looked upon as a derivative from the above function, we may write u = . 222 . . . (rst ... x r y s z t . . .) (20). Assume x r = \ r l x^ + \? x 2 . . . + \ r m x m \ f ^ ^ ) It is easy to obtain 222 . . . (rst. . . x r y s z t ...) = 222 ...(rst... x r y s z t ...),... = 222 . . . (rsi . . . x r y s z t . . .) (22), and the formula for (u) becomes IH) ^,^,^< i^^i" IT // ? i / p /lyJP 1\P (^i / /^ ^^ / T*C/ T 1 ?/ ^ ^ / T-^ 9 \ Proceeding to obtain the expression of the coefficients rsi... in terms of the coefficients rst..., we have rat. ..=222... (\rWrf. ..fgh...) (C), 112 84 ON THE THEORY OF LINEAR TRANSFORMATIONS. [13 where the 2 s refer successively to f, g, h, ... , denoting summations from 1 to m inclusively. Having this equation, it is perhaps as well to retain .U (B, i), instead of (B, 2), that form being principally useful in showing the relation of the function u to the theory of the transformation of functions. It may immediately be seen, that in the equations (B), (C) we may, if we please, omit any number of the marks of variation (), omitting at the same time the corresponding signs 2, and the corresponding factors of the series L, M, N ... Also, if u be such as only to satisfy some of the equations (A) ; then, if in the same formulae we omit the corresponding marks ( ), summatory signs, and terms of the series L, M, N ... , the resulting equations are still true. From the formulas (A) we may obtain the partial differential equations d 22... fort... 22....(Va*.. dpst... d u = 0, or pu, or pu, -J 5 -- \ drpt.. according as a is not equal, or is equal, to ft ; and so on : the summatory signs referring in every case to those of the series r, s, t,..., which are left variable, and extending from 1 to m inclusively. To demonstrate this, consider the general form of u, as given by the first of the equations (A). This is evidently composed of a series of terms, each of the form riiich P= cPQR ...(p factors), */-.-, VC > ( m terms) Q, R, &c. being of the same form ; and we have 22. ..(.. *- V dpst . . . and 22 ...(*... d dfist . . . P = Isftf ..., 1 s"t". ...... (m terms) | = ; */*/-. ,"*"" " asftf ... , as"t"..., ... 13] ON THE THEORY OF LINEAR TRANSFORMATIONS. 85 so that all the terms on the second side of the equation vanish. If, however, /3 = a, 22... \ whence, on the second side, we have cQR ...P 4 cPR . . . Q + &c. =p . cPQR . . . + &c. + &c. or the theorem in question is proved. In the case of an incomplete hyperdeterminant, the corresponding systems of equations are of course to be omitted. In every case it is from these equations that the form of the function u is to be investigated ; they entirely replace the system (A). A very important case of the general theory is, when we suppose the coefficients ,, rst ... to have the property r s t ...= r"s"t" ... , whenever r s t . . . and r"s"t" . . . denote ^ the same combination of letters ; and also that the coefficients X are equal to the coefficients p, v ... , each to each. In this case the coefficients rsi ... have likewise the same property, viz. that r"s"i" ... = r s t ..., whenever r s t . . . and r"s"t" . . . denote the same combination of letters. The equations (B, 1), (B, 2), become in this case u = L n v.u .................................... (B, 3), f ri n ) ( ri n i (S) 2SS J _ t_J _ r^t r r r t ( L n P (S) J _ L J r<tf v v T fR 4>"\ O tft^ ... < -- .. -.p rt>Z ... ZrOCgXt ...Y Lt " . VJ J. TSl ... X^C^X t . . . > ...... [B, 9), where only different combinations of values are to be taken for r, s, t, ... and a, /8, . . . express how often the same number occurs in the series. In the equation (C), /A, v must be replaced by X, the equations (D) are no longer satisfied; the equations (A) reduce themselves to a single one, (so that there can be no question here of incomplete hyperdeterminants) : but this is no longer sufficient to determine the function sought after. For this reason, the particular case, treated separately, would be far more difficult than the general one ; but the formulas of the general case being first established, these apply immediately to the particular one 1 . The case in question may be defined as that of symmetrical hyperdeterminants, (a denomination already adopted for ordinary determinants). It would be easily seen what on the same principle would be meant by partially symmetrical hyperdeterminants. I have not yet succeeded in obtaining the general expression of a hyperdeterminant ; the only cases in which I can do so are the following : I. p = 1, n even, (if n be odd, there only exist incomplete hyperdeterminants). II. p =2, m = 2, n even. III. p = 3. m = 2, ?i = 4. I. The first case is, in fact, that of the functions considered at the termination of a paper in the Cambridge Philosophical Transactions, vol. vm. part I. [12] ; though at that time I was quite unacquainted with the general theory. 1 See concluding paragraph of this paper. 86 ON THE THEORY OF LINEAR TRANSFORMATIONS. [13 Using the notation there employed, we have u= 22 mm ) a complete hyperdeterminant when n is even ; and when n is odd the functions f 1 !...<) }. 22 I 11. ..(n) 22 mm ) ^ mm are each of them incomplete hyperdeterminants. (A) In the case of n = 2, the complete hyperdeterminant is simply the ordinary determinant 11, 12,... Im 21, 22, ... 2m ml, ml, ...mm Stating the general conclusion as applied to this case, which is a very well known one, "If the function U = 22 (rs . x r y s ) be transformed into a similar function 22 (rs.x r y s }, by means of the substitutions _ "X l ^ i "X 2 ,!. i \ in ^ JL r /Vy iA/i T^ /v.j. tx 2 . . . i **f *** in, j y s = PS iii + fv 2/2 + P y ; then ii, 12,... 2l, 22, v, v,... V, V, /v> /^a 1 , /*!*, P-2-, 11, 12,... 21, 22, Also, by what has preceded, so that the theorem is easily seen to amount to the following one " If the terms of a determinant of the m th order be of the form 2 r 2, (rs . x rp y s<r ), r, s extending as before, from 1 to ra inclusively, the determinant itself is the product of three deter minants ; the first formed with the coefficients rs, the second with the quantities x, and the third with the quantities y." 13] ON THE THEORY OF LINEAR TRANSFORMATIONS. 87 In a following number of the Journal I shall prove, and apply to the theories of Maxima and Minima and of Spherical Coordinates, (I may just mention having obtained, in an elegant form, the formulae for transforming from one oblique set of coordinates to another oblique one) the more general theorem, " If k be the order of the determinant formed as above, the determinant itself is a quadratic function, its coefficients being determinants formed with the coefficients rs, its variables being determinants formed respectively with the variables x and the variables y\ and the number of variables in each set being the number of combi nations of k things out of in, ( = 1 if k = m; if k > m the determinant vanishes)." I shall give in the same paper the demonstration of a very beautiful theorem, rather relating, however, to determinants than to quadratic functions, proved by Hesse in a Memoir in Crelle s Journal, vol. xx., " De curvis et superficiebus secundi ordinis;" and from which he has deduced the most interesting geometrical results. Another Memoir, by the same author, Crelle, vol. xxvin., " Ueber die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiter Grade mit zwei Variabeln," though relating in point of fact rather to functions of the third order, contains some most important results. A few theorems on quadratic functions, belonging, however, to a different part of the subject, will be found in my paper already quoted in the Cambridge Philosophical Transactions [12] ; and likewise in a paper in the Journal, Chapters in the Algebraical Geometry of n dimensions [11]. I shall, just before concluding this case, write down the particular formula corre sponding to three variables, and for the symmetrical case. It is, as is well known, the theorem, "If U= Ax* + Ef + Cz* + ZFyz + ZGxz + ZHxy be transformed into by means of (a#y - a/3 Y + a /3 Y - a W + a"/?? - a ^V) 2 (4#tf - ^1F- - BG" - CH- + 2FGH)." (B) Let n = 3, and for greater simplicity m = 2 ; write ^ 6=211, 7=212, (UNIVERSITY^ c = 121, .^ = 122, ^CALJFORN^ d = 221, A = 222, so that U = a x-ty^ + b x^j^ + c x-^y^ + d x.jj.^ + e x-^j^ +fx 2 y l 2 2 -f g x^y^z.^ + h xy^. 88 ON THE THEORY OF LINEAR TRANSFORMATIONS. [13 There is no complete hyperdeterminant (i.e. for p = 1), and the incomplete ones are ah bg cf+de ii / , suppose, ah de bg + cf = u tl , ah cf de + bg = u /fl . Thus, suppose the transforming equations are fl/2 A2 1 "I 2 2 J 2/1 = HI 2/1 + PI 2/2 , 2/2 = /^a 1 2/ 1 + /^ 2 2/2 ; ZT. = V? Z-L + V? 2 - then u ( = (//a 1 p? - /i 2 1 A t i 2 ) (^i 1 v? - v? vf] u , where y, z are changed # = (^i 1 ^ 2 2 - ^ ^i 2 ) (Xi 1 X 2 2 - X 2 X V) w /y , *, * u m = (V X 2 2 - V V) Oi 1 /^2 2 - ^ tf] u,,,, x, y We might also have assumed u t = ad be, or eh gf, u t/ = af be, or ch dg, u ltl = ag ce, or bh df, but these are ordinary determinants. (C) ;i = 4, m = 2. a =1111, i 6 = 2111, j = 2112, c = 1211, A: =1212, d = 2211, Z = 2212, e=1121, m=1122, /=2121, w=2122, ^ = 2211, o=2212, A = 2221, jp=2222. U = a x^iZiWi + b x^y^w-i + c x^^w^ + d a; 2 y 2 2 1 w l + BO;, y^z, w 1 +fx 2 yiZiW 1 + g X 2 y 2 z 1 w 1 + h x^z.w, + i x^y^ z x w 2 +j x^z^Wi + k x^y^w^ + I x^y^w^ + m x^y^Wz + n x^y^w^ + o x. 2 y^w z + p x^y^w.^ we have u = ap - bo en + dm el +fk + gj hi ; 13] ON THE THEORY OF LINEAR TRANSFORMATIONS. 89 so that, with the same sets of transforming equations as above, and the additional one, w n =p 2 1 w l + p<?u>2, we have u = (X^Xa 2 Xa 1 ^ 2 ) (/u*! 1 /^ 2 /u, 2 Vi 2 ) (y^v 2 v^v-f) (p^p/ Pzpf) u \ this is important when viewed in reference to a result which will presently be obtained. If we take the symmetrical case, we have U = a tf + 4/3 a?y + 67 ofy z + 4&ey 8 + e y* ; which is transformed into U = aV 4 + 4/3Vy + 67VY 2 + 4SV?/ 3 + e y \ by means of x = X x + fj, y , u = a e - 4/3 + u=\ t [1111 ...(n) (1222 t then, if we have II. Where p = 2, ra = 2, n is odd. The expression u = \ t fllll. ..(n)] (2222 t fllll ...(w)) (2111 (2222 j (2222 is a complete hyperdeterminant ; and that over whichever row the mark (f) of nonper- mutation is placed. The different expressions so obtained are not, however, all of them independent functions : thus, in the following example, where n = 3, the three functions are absolutely identical. (-4). n = 3, notation as in I. (5). u = a 2 A 2 + b 2 g* + c 2 / 2 + d 2 e z - 2a% - 2ahcf- 2ahde - 2bgcf- 2bgde - 2cfde + 4,adfg + 4>bech, and then }/ /\ i^ 2 ^ i^ 2x2 ( i 2 i 2x2 / i 2 i 2\2 This is in many respects an interesting example. We see that the function w may be expressed in the three following forms: u = (ah- bg - cf+ de)~ +4 (ad- be) (fg - eh) (1), u = (ah - bg - de + c/) 2 + 4 (af- be) (dg - ch) (2), u = (ah cf de + bg) 2 + 4 (ag ce) (df bfi) (3), c - 12 90 ON THE THEORY OF LINEAR TRANSFORMATIONS. [13 which are indeed the direct results of the general form above given, the sign (f) being placed in succession over the different columns : and the three forms, as just remarked, are in this case identical. We see from the first of these that u is of the second or third, from the second that u is of the first or third, from the third that u is of the first or second of the three following forms: u= a, b, c, d > f> 9> which is as it should be. i , u = H z a, b, e, f c, d, g, h a, c, e, g b, d, f, h Let du j- , da L/ n = The following is a singular property of u. du j do then, u being the same function of these new coefficients that u is of the former ones, u = u 3 . To prove this, write p = ah bg cf+de, q = (ad bc), r = ehfg; a, = ap 2qe, e, = - 2ra + pe , b,=bp-2qf, f,=-2rb+pf, c t =cp- 2qg, g, = - 2rc + pg, d, = dp-2qh, h,=-2rd+ph , we have, as a particular case of the general formula just obtained, u / = (p 2 4gr) 2 u = u 2 . u, = u 3 . Also a, = h , e, = d , whence u, = u, that is u = u 3 . There is no difficulty in showing also, that if a", b", ...h" are derived from a , b ...h , as these are from a, b,...h, then a" = tfa, b" = u*b, ...h" = u*h. The particular case of this theorem, which corresponds to symmetrical values of the coefficients, is given by M. Eisenstein, Crelle, vol. xxvu. [1844], as a corollary to his researches on the cubic forms of numbers. Considering this symmetrical case By 3 , 13] ON THE THEORY OF LINEAR TRANSFORMATIONS. 91 so that if V be transformed into U = aV 3 + 3/3V 2 / + 3y V/ 2 + B f, by means of oc=\x y = A./ then if u = a /2 S 2 - 6a S /3y - 3/3 V 2 + 4 /8 8 + 4a y , we have w = (A,/*, X,/*) 6 . M. III. p=3, m = 2, ?i = 4. Notation as in I. \C), where A, B are arbitrary constants, and &, 33, &c. ... p| are functions of the coefficients, given as follows : & = a s p 3 - b s o 3 - c 3 n s + d*m* - e*l 3 +/ 3 & 3 + g s j 3 - h s i 3 . 53 = - a 2 p 2 bo + b 2 o 2 ap + C 2 n 2 dm - d 2 m 2 cn + e*Pfk - f 2 k 2 el - g*j 2 hi + h 2 i 2 gj - a 2 p 2 cn + b 2 o 2 dm + c 2 n 2 ap - d*m*bo + eH-gj - / 2 k 2 hi - g 2 j 2 el + h 2 i 2 fk - ay el + b 2 o 2 fk + cWgj - d 2 m 2 hi + eH 2 ap - / 2 k 2 bo - g*j 2 cn + h&dm - a 2 p 2 hi + b 2 o 2 gj + cWfk - d 2 m 2 el + eWdrn - / 2 k 2 cn - g 2 j 2 bo + h 2 i 2 ap. <& = + a 2 p 2 dm - b 2 o 2 cn - c 2 n 2 bo + d*m?ap - e-Phi + f 2 k 2 gj + ffifk - h 2 i 2 el + a 2 p 2 fk -b 2 o 2 el -c-n 2 hi + d 2 m 2 gj - e 2 l 2 bo + f 2 k 2 ap + g 2 j 2 dm - h 2 i 2 cn + a 2 p 2 gj -b 2 o 2 hi - c 2 n 2 el + d 2 m 2 fk - eH 2 cn + f 2 k 2 dm + g 2 fap -h 2 i 2 bo. B = apbocn - apbodm - apcndm + bocndm - elf kg j + elfkhi + elgjhi - higjfk + apboel - apboj k - cndmgj + dmcnhi - elfkap + elfkbo + gjhicn - higjdm - apbogj + apbohi + cndmel - dmcnkf+ elf ken - elfkdm - gjhiap + higjbo + apcnel - bodmfk - cnapgj + dmbohi - elgj ap +fkhibo + gjelcn - hifkdm -apcnfk +bodmel + cnaphi - dmboel + elgjbo - fkhiap -gjeldm+ hijkcn - apdmel + bocnkf + cmbogj - dmaphi + elhiap -fkgjbo - gjfkcn + hidmel. (B = apdmfk - bocnel - bocnih + dmapgj - elhibo +fkapgj + gjfkdm - hielcn. $ = a 2 phjo - b 2 goip - c 2 nflm+ d 2 emkn - e 2 dlkn +f 2 ckml + g 2 bjpi - h 2 aijo - i 2 phbg + J 2 ogah + k 2 nfde - I 2 mecf + mHdcf - n 2 ckde - o 2 bjah + p 2 aibg + a 2 phkn - b 2 golm - c 2 nfip + d 2 emjo - e*dljo + f 2 ckip + g 2 bjpl - h 2 aink -itphcf +J 2 ogde + tfnfah -Pmebg + mHdbg - n 2 ckah - o 2 bjde + p^aicf + a 2 phlm - b 2 gokn - tfnfjo + d 2 emip - eHdpi + f 2 ckjo + g*bjnk - h 2 aiml -i 2 phde +fgocf + frnfbg - I 2 meah + m 2 dlah - n 2 ckbg - o 2 bjcf + p*aide + tfpdno - b 2 cmpo - c 2 bpmn + d 2 aomn - e 2 hjkl + f-gilk + g 2 flij - h 2 ekji -Mfgh +f-kehg + tfjhef - I 2 igfe + m 2 pbcd - h 2 aodc - o 2 ndab+p 2 mbca + a 2 plng - frhkmo - c 2 ejpn + d 2 fiom - e 2 cpjl +f*doik + g 2 anlj - h 2 bmki -i 2 odfh +J 2 pceg + k 2 mhbf - I 2 nage + m*khbd - nHgac - o 2 ifbd + p^ieca + a 2 plfo -tfekpo - c 2 hjmn + d 2 gimn - e 2 bpkl +/ 2 aolk + g n -dnij - h 2 cmji -i 2 ndgh +J 2 , lic hg + fcpbef - Foafe + itfjhcd - n 2 igcd - o 2 lfab + p 2 keba. 122 92 ON THE THEORY OF LINEAR TRANSFORMATIONS. [13 <& = apbgkn boalhm cndiep + dmcjfo ihjocf + gjidep + kflamh lekbng + apbglm boahkn cndejo + dmcfip ihjode + gijpcf + kflmbg leknah + apcflg bojehk cnjehk + dmilgf ihadno + gjbmcp + kfcbpm eladno + apcflm- bodkne cnahjo + dmbgip ihknde + mdahjo + kfipbg bocfml + apidng bojcmh cnbkpe + dmaflo ihaflo + gjbkpe + fkcjhm elidng + apidfo bojcep nbkhm + cdmalng ihalng + gjkbmh +fkcjpe elidfo |^ = a?honl - tfpgmk - c 2 pfmj + d 2 onie - fipdfg + j 2 oech + k 2 nbeh - I 2 mafg we have, as usual, elfocj + fkpied + gjhmla ignbk + mdnbkg ncmhal obpied + paofjc elcfip + fkedoj + gjhakn hibgml + mdknah ncmlbg obpicf + paojde elmpbc + fkadno + gjadno hipmcb + mdjehk ncilfg obilfg + pahejk elbgip + fkahjo + gjednk hicfml + gjcfml ncpigb leahjo + paknde elmhjc fkidng + gjaflo ihbkpe + mdbkpe cnaflo boidng + pahmcj elmnbk fkalng + gjidfo ihpecj + mdjcep ncidfo obalng + pahmbk. - e 2 dpkj + f 2 ilco + g 2 blni - h?amkj + m 2 bkh - n 2 kadg - o 2 jadf + + 60B - (ap - bo - en + dm - el +fk + gj - hi} 3 , = v suppose. Particular forms of U are A = l, 5 = 0: u = & + 323 + 3< A = l, 5 = 9: u = & + 323 - 6< - 3B + 33 + 9jp - 18C5 - 27f, =0U suppose, where OU=0 is the result of the elimination of the variables from the equations d Xi U=0, d yi U=0, d Zl U=0, d Wl U = 0, d^U^O, d yi U=0, d Zt U=Q, d w JJ=0. In fact, by an investigation similar to Mr Boole s, applied to a function such as U, it is shown that OU has the characteristic property of the function u: also in the present case n is the most general function of its kind, so that 6U is obtained from U by properly determining the constant. This has been effected by comparing the value of u, in the symmetrical case, with the value of OU, in the same case, the expanded expression of which is given by Mr Boole in the Journal, vol. iv. p. 169. Assuming .4=1, the result was B = 9. [Incorrect: the result of the elimination is not 0U = Q, but an equation of a higher degree.] in which a, /3, are indeterminate. We have u = where M = ( V V - W = M (B X V - *, V) 8 (/>> V - 13] ON THE THEORY OF LINEAR TRANSFORMATIONS. 93 and thence v s = Mv s , which coincides with a previous formula, and 0U=M0U: whence, eliminating M, 0u = 0u v v 3 an equation which is remarkable as containing only the constants of U and U : it is an equation of condition which must exist among the constants of U in order that this function may be derivable by linear substitutions from U. In the symmetrical case, or where U = aa? + k$a?y it has been already seen that v is given by v = ae - 4/38 + 37 2 . Proceeding to form 0U, we have 33 = 4 (ae/3 2 8 2 - a 2 e 2 /38 + 37 2 /3 2 8 2 - 3/38T 4 ), <& = 3 (tfey + 2 7 6 + ^T 4 - 4/3 3 8 3 ), B = 6 (ae/3 2 8 2 - 2ae/38 7 2 + 3/3y8 2 - 2/38T 4 ), <& = 3ae7 4 - 4/3 3 8 3 + 7 4 , jp = 6 (aSV + e 2 /3 2 7a - 2/3 3 678 - 28 3 a/37 - 4/3yS 2 . + 4 . /387 4 + 7 3 /3 2 e + 7 3 a8 2 ), CS = 12 (ae/3S7 2 - a/378 3 - erf/3 3 + 7 3 aS 2 + 7 3 e/3 2 + ySSy 4 - 2/3y3 2 ), f^ = (a 2 8 4 + e 2 /S 4 - 4/3 2 e7 3 and these values give 6U= a 3 e 3 - Ga^e - 12a 2 /3Se 2 - 18a 2 7 2 e 2 - 27a 2 8 4 - 27 4 e 2 + SG/SyS 3 + 54 a 2 7S 2 e + 54 a/ - 54/3y e - 64/3 3 S 3 so that this function, divided by (ae 4/3S + 37 2 ) 3 , is invariable for all functions of the fourth order which can be deduced one from the other by linear substitutions. The function ae 4/38 + 37 2 occurs in other investigations : I have met with it in a problem relating to a homogeneous function of two variables, of any order whatever, a, /3, 7, 8, e signifying the fourth differential coefficients of the function. But this is only remotely connected with the present subject. Since writing the above, Mr Boole has pointed out to me that in the transform ation of a function of the fourth order of the form ox 4 + 4iba?y + Qcx^y* + 4<dxy 3 + ey 4 , besides his function 0u, and my quadratic function ae 4<bd + 3c 2 , there exists a function 94 ON THE THEORY OF LINEAR TRANSFORMATIONS. [13 of the third order ace b 2 e ad 2 c 3 + 2bdc, possessing precisely the same characteristic property, and that, moreover, the function 0u may be reduced to the form (ae - 4bd + 3c 2 ) 3 - 27 (ace - ad- - eb 2 - c 3 + 2bdc) 2 ; the latter part of which was verified by trial ; the former he has demonstrated in a manner which, though very elegant, does not appear to be the most direct which the theorem admits of. In fact, it may be obtained by a method just hinted at by Mr Boole, in his earliest paper on the subject, Mathematical Journal, vol. IT. p. 70. The equations d x 2 u = 0, d x d y u = 0, d y -u = 0, imply the corresponding equations for the transformed function : from these equations we might obtain two relations between the coefficients, which, in the case of a function of the fourth order, are of the orders 3 and 4 respectively : these imply the corresponding relations between the coefficients of the transformed function. Let .4=0, .5=0, A =Q, B = 0, represent these equations; then, since A=0, B = 0, imply A = Q, we must have A = AJ/ + M.B, A, M, being functions of X, X , ft, &c. p : but B being of the fourth order, while A, A are only of the third order in the coefficients of u, it is evident that the term M5 must disappear, or that the equation is of the form A = AA. The function A is obviously the function which, equated to zero, would be the result of the elimination of x 2 , xy, y-, considered as independent quantities from the equations ax 2 + 2bxy + cy- = 0, bx 2 + 2cxy + dy 2 = 0, ex 2 + 2dxy + ey 2 = 0, viz. the function given above. Hence the two functions on which the linear transformation of functions of the fourth order ultimately depend are the very simple ones ae - 4>bd + 3c 2 , ace - ad- - eb 2 -c s + 2bdc, the function of the sixth order being merely a derivative from these. The above method may easily be extended : thus for instance, in the transformation of functions of any even order, I am in possession of several of the transforming functions ; that of the fourth order, for functions of the sixth order, I have actually expanded : but it does not appear to contain the complete theory. Again, in the particular case of homogeneous functions of two variables, the transforming functions may be expressed as symmetrical functions of the roots of the equation u = 0, which gives rise to an entirely distinct theory. This, however, I have not as yet developed sufficiently for publication. There does not appear to be anything very directly analogous to the subject of this note, in my general theory : if this be so, it proves the absolute necessity of a distinct investi gation for the present case, that which I have denominated the symmetrical one. 14] 95 14. ON LINEAR TRANSFORMATIONS. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 104 122.] IN continuing my researches on the present subject, I have been led to a new manner of considering the question, which, at the same time that it is much more general, has the advantage of applying directly to the only case which one can possibly hope to develope with any degree of completeness, that of functions of two variables. In fact the question may be proposed, " To find all the derivatives of any number of functions, which have the property of preserving their form unaltered after any linear transformations of the variables." By Derivative I understand a function deduced in any manner whatever from the given functions, and I give the name of Hyperdeterminant Derivative, or simply of Hyperdeterminant, to those derivatives which have the property just enunciated. These derivatives may easily be expressed explicitly, by means of the known method of the separation of symbols. We thus obtain the most general expression of a hyperdeterminant. But there remains a question to be resolved, which appears to present very great difficulties, that of determining the independent derivatives, and the relation between these and the remaining ones. I have only succeeded in treating a very particular case of this question, which shows however in what way the general problem is to be attacked. Imagine p series each of m variables x 1} y 1} ...&c. o; a , 2/ 2 , ...&c. x p , y p ,...&c., where p is at least as great as m. Similarly p* series each of m variables xS, y^, ...&c. x 2 \ y,\ ... , &c. ... x p * , y\~, . . . &c., 96 ON LINEAR TRANSFORMATIONS. [14 j} at least as great as ra\ and so on. Let the analogous variables x, y ... nected with these by the equations be con- x = y = c v = X v x + IJL y" + . . . , where x, y,... stand for x 1} 3/1,... or % 2) y 2 ,... or x p , y p ,...;ac, y\... stand for x , yi,... or a? 2 \ y%,... or a? p) y"^, &c The coefficients X, JM, ..., V, //, ... &c. ; X\ //,\ ..., X \ /A \... remain the same in all these systems. Suppose next, .e. (where 8^, S,/... are the symbols of differentiation relative to x, y, &c.). Then evidently with similar equations for \ ?)\... Suppose fc, &,...& that is to say \\CL\\ is the series of determinants formed by choosing any m vertical columns to compose a determinant, and similarly ||n v ||, &c. Suppose, besides, X, /^,... \/ / f... E" - > - t - / \\ \ , p , ... -. /\ /\ X , p , ... Then, by the known properties of determinants, |(01=^ ||fi v n & c ., i.e. the terms on the one side are respectively equal to the terms on the other. Hence if i.e. n a rational and integral function, homogeneous of the order / in the quantities of the series ||H||, homogeneous of the order / v in the quantities of the series ||fT||, &c., we have immediately \.. Q; 14] ON LINEAR TRANSFORMATIONS. .97 or if U be any function whatever of the variables x, y ... which is transformed by the linear substitutions above into U, then or the function Q U is by the above definition a hyperdeterminant derivative. The symbol Q may be called " symbol of hyperdeterminant derivation," or simply " hyperdeterminant symbol." Let A, B,... represent the different quantities of the series ||n||, A\ B\... those of the series ||fT||, &c. ..., then G may be reduced to a single term, and we may write D = A*B* ...A* B? ... Also U may be supposed of the form where , <l> are functions of the variables of one of the sets x, y,..., of one of the sets US , 2/\ ..., &c., thus is of the form F(ac lt 2/j,...^, y,\ ...), and so on. The functions , <... may be the same or different. It may be supposed after the differentiations that several of the sets x, y,... or of the sets x\ y\ ... become identical : in such cases it will always be assumed that the functions 0, ... into which these sets of variables enter, are similar; so that they become absolutely identical, when the variables they contain are made so. Thus the general expression of a hyperdeterminant is iu which, after the differentiations, any number of the sets of variables are made equal. For instance, if all the sets x, y ... and all the sets x, y ... are made equal, the hyperdeterminant refers to a single function F(x, y ...x\ y ...). In any other case it refers not to a single function but to several. What precedes, is the general theory: it might perhaps have been made clearer by confining it to a particular case: and by doing this from the beginning, it will be seen that it presents no real difficulties. Passing at present to some developments, to do this, I neglect entirely the sets x\ y ... and I assume that the number m of variables in each of the sets x, y ... reduces itself to two; so that I consider functions of two variables x, y only. The functions <s), 3>, &c. reduce themselves to functions PI, F a ... V P of the variables x ly y lt or x 2 , y, ... or x p , y p . Writing also ^2- fli= 12, &c. the symbols A, B ... reduce themselves to 12, 18.... Hence for functions of two variables, there results the following still tolerably general form Q^ = 12 a 13 C. 13 98 ON LINEAR TRANSFORMATIONS. [14 The functions V ly V 2 ... may be the same or different : but they will be supposed the same whenever the corresponding variables are made equal. This equality will be denoted by writing, for instance, to represent the value assumed by QFF FF... when after the differentiations yi =s, 2/3 = #4, #4 = x, y; 2/2 = x ) y j &c. It is easy to determine the general term of Q U. To do this, writing for shortness a + +7 ...=/;, a &c. Ar_ /_yr+s+t.. +/+ ... f.-.L^J IV 17 J [s\ s [t] f " [r Y [sj f [t J or or the general term is ,r+s+t-.. Y ,a-r+s +t ... Y ,/3-s+/3 -8 + . where r, s, ,...& , t ,...t",... extend from to a, & 7...^ , 7 , ...7"... respectively. It would be easy to change this general term in a way similar to that which will be employed presently for the particular case of D^i^^- If several of the functions become identical, and for these some of the letters / are equivalent, it is clear that the derivative Q^ refers to a certain number of functions F, , F 2 . . . the same or different, of the variables x, y; x , y ; . . . and besides that this derivative is homogeneous, of the degrees l} #/, ... with respect to the differential coefficients of the orders f l} //,... &c. of F 1; (consequently homogeneous of the order l + 1 +... with respect to these differential coefficients collectively), homo geneous and of the degrees 2 , 2 ,... with respect to the differential coefficients of the orders / 2 , / 2 ... of F 2 , (consequently of the order 2 + &/... with respect to these collectively), and so on. The degree with respect to all the functions is of course i + Q{ % , _ -f # 2 4. # 2 + ...,=p suppose. In general, only a single function will be considered, 14] ON LINEAR TRANSFORMATIONS. 99 and it will be assumed that \^\U only contains the differential coefficients of the / th order. In this case, the derivative is said to be of the degree p and of the order /. The most convenient classification is by degrees, rather than by orders. Commencing with the simplest case, that of functions of the second order (and writing V, W instead of V 1} F 2 ), we have (where , ^ apply to V and 2 , y* to W). This will be constantly represented in the sequel by the notation B a (V, W). Hence, writing g/F= F-, S x *- 1 S y V = F- 1 ... , we have B a (V, W)= FTF>- F- TT - 1 + ... ; and in particular, according as a is odd or even, B a (V, F)=0, $B a (V, V) = V<V>-^V< 1 V>*- 1 +.. I continued to the term which contains F*F* a , the coefficient of this last term being divided by two. Thus, for the functions \ (ax 2 + 2bxy + cy 2 ), -fa (ax* + 4<bx 3 y + 6cx 2 y 2 + kdocif + eif), &c., if a be made equal to 2, 4, &c. respectively, we have the constant derivatives ac b 2 , ae 4>bd + 3c", ag-6bf+I5ce-Wd 2 , ai - 8bh + 28c^ - 5(]df+ 35e 2 , which have all of them the property of remaining unaltered, a un facteur pres, when the variables are transformed by means of x = \x + py, y = \ x + ^y. Thus, for instance, if these equations give ax 2 + 2bxy + cif = ax- + Zbxy + cy 2 , then ac - b 2 = (Xp - X>) 2 . (ac - b 2 ), and so on. This is the general property, which we call to mind for the case of these constant derivatives. 132 1 00 ON LINEAR TRANSFORMATIONS. [] 4 The above functions may be transformed by means of the identical equation B (V W} 1?" ~ k Bt.(V W} ct \ > / "^ * \ *? / to make use of which, it is only necessary to remark the general formula Thus, if & = !, we obtain for the above series, the new forms ac b 2 , (ae -bd)-3 (bd - c 2 ), (ag - bf) - 5 (bf - ce) + 10 (ce - d 2 ), (ai - bh) - 7 (bh - eg) + 21 (eg - df) - 35 (df- e n -\ &c., the law of which is evident. This shows also that these functions may be linearly expressed by means of the series of determinants ! a, b b, c a, b, c I &c. b , c, d We may also immediately deduce from them the derivatives B which relate to two functions. For example, for functions of the sixth order this is ag + ag~Q (bf + b f) + 15 (ce + c e) - 20dd , which has an obvious connection with ag 66/4- loce - Wd 2 ; and the same is the case for functions of any order. The following theorem is easily verified ; but I am unacquainted with the general theory to which it belongs. "If U, V are any functions of the second order, and W=\U+^V; then B, [B t (W, W), B,(W, (where B relates to X, p) is the same that would be obtained by the elimination of x, y between lf=Q, V = 0." (See Note 1 .) I In fact this becomes 4 (ac - fe 2 ) (a c - b 2 ) - (ac + a c - 2bbJ = 0, which is one of the forms under which the result of the elimination of the variables from two quadratic equations may be written. This is a result for which I am indebted to Mr Boole. 1 Not given with the present paper. 14] ON LINEAR TRANSFORMATIONS. 101 Passing to the third degree, we may consider in particular the derivatives HUVW= < 23 a 3i a 12 a UVW=C a (U, V, W) : writing for shortness A t?-^ 2a - r fi r TT TT r f r~\ r Oy u u , we have the general term C a (U, V, W) = 2{(-y + * +t A r A s A t U>" +t - s V a+ >- t W * + *- r }, where r, s, t extend from to a. By changing the suffixes r, s the following more convenient formula C a (U, V, W) = ^{(-)^U" V^W 3 ^-^[(-) t A p , t A <r+t _ a A t ]}, where t extends from to 2a : p, a, and 3d p a must be positive and not greater than 2or. In particular, according as a is odd or even, C a (U, U, tf) = 0, C m (U, U, U) omitting therein those values of p, a- for which p> or or a- > 3a p a; and dividing by two the terms in which p a- or a- = 3a p a-, and by six the term for which p = a = 3a p a-, = a. In particular, for functions of the fourth or eighth orders we have the constant derivatives ace - ad 1 - b 2 e -c 3 + 2bcd ; aei - Mbd - kafh + Sag 2 + 3tc* + I2beh - 8chd - 8bgf- 22ceg + 24c/ 2 + 24c% - "36def+ loe s ; the first of which is a simple determinant. Thus we have been led to the functions ae 46d + 3c 2 and ace ad 2 eb" c s + 2bcd, which occur in my "Note sur quelques formules &c." (Crelle, vol. xxix. [1845] [15]), and in the forms which M. Eisenstein has given for the solutions of equations of the first four degrees. Let U be a function of the order 4a : the derivative C may be expressed by means of the derivatives B. For, consider the function B[U, B 2a (V, W}]; paying attention to the signification of B, this may be written 102 ON LINEAR TRANSFORMATIONS. [14 where the symbols g g , rj s refer to the two systems # 2 , 7/ 2 : x 3 , y 3 . Thus it is easily seen that we may write , or 10=12 + 13 = 12-31, whence the function becomes of which all the terms vanish except Hence putting = ~ [2a] 2a ~ 2.4...4a we have 5 4o [f7 ; 2a (F, TF)] =#<?( [T, F, W), or in particular U, U, U). Thus for example, neglecting a numerical factor, (ax 2 + 2bxy + cy 2 ) (ex 2 + 2dzy + ey 2 ) - (ba? + Zcxy + dy 2 } 2 = (ac - b 2 ) & + 2 (ad - be) oPy + (ae + 2bd - 3c 2 ) x 2 y 2 + 2 (be- cd) xif + (ce - d 2 ) y 4 , and then e (ac - 6 2 ) - 4d | (ad - be) + Qc fa (ae + 2bd - 3c 2 ) - 46 f (be - cd) + a (ce - d 2 ) = 3 (ace - ad 2 - b*e - c 3 + 2bcd). We have likewise the singular equation 1 / F4al 1 \ where U = Ti - li a ^ a - L ^ r J - a^^y ... +a 4a y 4a ) , &c. LTOCJ \ 1 / If however U= V = W, we must write the reason of which is easily seen. This subject will be resumed in the sequel. 14] ON LINEAR TRANSFORMATIONS. 103 The functions C may be transformed in the same way as the functions B have been. In fact C (U V W} - - a \ / if in particular k = 1, then C,(U, V, W) = , V, W)- - for U , &c. but in general W- l"& nzC l ( U, V, W), where p + p = cr + a- = T + T = 2a - 2, 2o 2 2<x 2 2a 2 2a If ? for J7 P, &C. hence (7 a (C7, 7, W ) = SS {(-)"+" /", p ]/ , o- 1 jy , 3a-p-a2 r a _ i] where J t = L J - ; extends from to a - 1 ; p, a- -I, and 3a-p-<r-2 may have each of them any positive values not greater than 2a - 2. In particular ^(7, U, tT) = 622 {(-)P+ /-,3a-p-<T-l f/-,p+2 yhere p, a need only have such values that p < <r 1, o- 1 < 3a p o- 2. In particular the derivative aei- ... + 15e 3 may be transformed into a, d, g -3 a, e, y -3 6, c, # + 6 6, d, / -15 c, d, e b, e, h r -f i" b > /. ^ c, cZ, A J c, e, g d, e, f c, /, % c, g, h a, e, i > V* "p c ^ - L1BR A^^s/ 9 in which form it is obviously a linear function of the 104 ON LINEAR TRANSFORMATIONS. [14 a, b, c, d, e, f, g b, c, d, e, f, g, h c, d, e, f, g, h, i which is true generally. Omitting for the present the theory of derivatives of the form = 23 a 31* 12 Y UVW, we pass on to the derivatives of the fourth degree, considering those forms in which all the differential coefficients are of the same order. We ma write , V, or if for shortness we have 13.42= D j3>y UVWX. Suppose U=V=W=X, and consider the derivatives which correspond to the same value f of a + /3 + 7. The question is to determine how many of these are independent, and to express the remaining ones in terms of these. Since the functions become equal after the differentiations, we are at liberty before the differentiations to inter change the symbolic numbers 1, 2, 3, 4 in any manner whatever. We have thus but the identical equation multiplied by E a e and applied to the product UVWX, gives Ua+ibc + -^a 6+ie + JJabci whence if a + 6 + c=/-l, we have a set of equations between the derivatives D a ft for which a + 8 + 7 =/ Reducing these by the conditions first found, suppose / is the number of divisions of an integer / into three parts, zero admissible, but permu tations of the same three parts rejected. The number of derivatives is @/, and the number of relations between them is (/- 1). Hence /- (/-!) of these derivatives are independent: only when / is even, one of these is -D/, ,o, i.e. 12 7 34 / . UVWX, i.e. U f UV.W f WX, or B f (U, V] E f (X, W), i.e. [B f (U, u]f ; rejecting this, the number of independent derivatives, when / is even, is H/-@ (/-!)-!. Let E be the greatest integer contained in the fraction shown to be E{ or E S? , 6 6 the number required may be 14] ON LINEAR TRANSFORMATIONS. 105 according as / is even or odd. Giving to / the six forms 6g, Qg + 1, 6g + 2, 6g + 3, 6g + 4, Qg + 5, the corresponding numbers of the independent derivatives are ff> 9 9> 9 + 1, g> g + 1 I thus there is a single derivative for the orders 3, 5, 6, 7, 8, 10,... two for the orders 9, 11, 12, 13, 14, 16, ...&c. When / is even, the terms D/-3, 3 ,o, D/-6,6,o , and when / is odd, the terms .Z)/_ lil>0 , D/_ 4 4 o, Df_ 7 j ]0 , &c. may be taken for independent derivatives : by stopping immediately before that in which the second suffix exceeds the first, the right number of terms is always obtained. Thus, when /=9 the independent derivatives are Aio, A, an d we have the system of equations Aoo + Aio -f Aoi = o, A 21 + D 53l + A 22 = o, Aio + D m + An = o, A + A + A = o, A + A*) + A 2 1 = 0, Asi + ^>441 + ^432 = 0, Aii + A 2 i + Aia = o, D m + D 432 + A23 = o, ABO + Aai + A = 0, ^432 + ^342 + ^333 = 0, which are to be reduced by ^ 900 = ~ -^900 0, Aoi ~ Aio, &C. It is easy to form the table Aoo= A 2 , Aoo=0, Aoo = 0, 110 == 2^ -"2", -^410 -AlO, A2o = ~ Aio, A 2 o = ~ Aioi Aoo = 0, Aii = 0, An = 0, Aw, 1)221 = 0, D^ = Aw, An = o, n =0 . = An= T) _iD _ i 7? 2 ^Sai 3^ -^330 ^ -"6 j A22 = % AsO + ^ -^*6 > c. 14 106 ON LINEAR TRANSFORMATIONS. [14 L KOO = -Og > JJw> v, *TM = ~ 2 -"8 > *TM ) -^530 > -^630 = 2 *4>i ~~ 2 -^540) -^521 ~ ~ J -^530 T^ -^8"> -^621 = 2 -^810 + 2 -^540 1 ^440 = ~ T6 -^530 J^ ->8 > -^540 > -^431 = f-g -^530 ^^ -^8 > -DsSl = 2 "^810 ~ 2 -^540 ) D^= T %A 3 o + T% 8 2 , i? = 0, ^332 ~~ T5 -^530 ~~ TK -"8 > -^441 = ^J -^432 = 2 -^810 + 2 -^540) X333 = 0. Whatever be the value, all the tables except the three first commence thus, according as / is even or odd, -D/.o.o = B/, or D/, 0,0 =0, Df-l, i,0 = 2"-S/> -D/-l,l,0. -D/-2, 2, = I -^/-3, 3, + ^ -S/ 2 . Of-*. 2,0= -0/-1, 1,0, but beyond this I am not acquainted with the law. To give some formulae for the transformation of these derivatives; we have, for example, Zy^,! = ("12. 84) 7 " 1 13. 42 UUUU=13.43B f _ l (U, U] B f ^(U, IT}. 13 . 42 = ffiqtfa^i ~ 1% 2 7 ?3 7 ?4 ViV^aZi + and Zintfa^t -fy-i ( U, U} .B/_i (U, U} = $/_! (^ C = 5/_! (U U- J ) 5/_! ( f- J ?7 ! ), &c. i i (where U- , C/" J stand for tT"- U l , &c.) ; or which reduces itself to according as f is even or odd. 14] ON LINEAR TRANSFORMATIONS. 107 For example, for the orders 3, 5, 7, 9, we have D 210 = - 2 {4 (ac - 6 2 ) (6c2 - c 2 ) - (ad - be) 2 }, D m = - 2 (4 (ae - 4bd + 3c 2 ) (If- 4ce + 3d 2 ) - (af-3be + 2cd) 2 }, D 6 io = - 2 {4 (off - 6bf+ loce - Wd 2 ) (bh - 6cg + lodf- We 2 ) - (ah - obg + 9c/- ode) 2 }, 8io = _ 2 (4 (ai - 8bh + 28cg - 5Gdf+ 35e 2 ) (bj - 8ci + 28dh - oGeg + 35/ 2 ) - (aj - 7bi + 20cA - 28dg The derivatives D will be presently calculated in a completely expanded form up to the ninth order. We have, therefore, still to find the derivatives of the sixth and eighth orders, and a second derivative of the ninth order. For the sixth order, the simplest method is to make use of D^, which is easily seen to be equal to 24 c, d, e, f d, e, /, g For the two others we have the general formula a, b, b, c, d c, d, + B f _,(U>U> 2 )B f _ 2 (U 2 U> ) + 2[B,^(U 1 IT- 0?}, 222 where U , U 1 , U 2 have been written for U* , U l , U> 2 ; a formula which is demon strated in precisely the same way as that for D/_j 1>0 . /_3,3. = - 2 { B f _ z (U^U^)B f _ 3 (U^U^}-QB f _,(U^U^}B f _,(U^U 2 ) (U>* U *) +9 J B / _ 3 (tf> J7- 1 ) B f _ s (U< 2 U^) (in which Z7- , &c. stand for U , &c.). In particular D^ = 2 {4 (ag - Qbf + Uce - Wd 2 ) (ci - Qdh + loeg - 10/ 2 ) - 4 (ah - 5bg + 9c/- 5de) x (bi + 5ch + 9dg - oef) + (ai - Qbh + I6cg - 2Qdf+ loe 2 ) 2 +8(bh- Qcg + I5df- 10e 2 ) 2 }, D^ = - 2 (4 (ag - Qbf+ loce - Wd 2 ) (dj - Qei + Ufh - Wg 2 ) - 6 (ah - 5bg + 9c/- 5de) x (cj- 5di + 9eh - 5fg) + 6 (ai - Qbh + IQcg - 26d/+ I5e 2 ) (bj - Qci + Wdh - 2Qeg + 15/ 2 ) + 36 (6/1 - 6cg + lodf - We 2 ) (ci - 6dh + I5eg - 10/ 2 ; - 9 (bi - 5ch + 9dg - 5ef) 2 - (aj - 6bi + loch- Wdg + 9ef) 2 }. Hence we have all the elements necessary for the calculation of the following table of the independent constant derivatives of the fourth degree, up to the ninth order. [I have arranged the terms alphabetically and in tabular form as in my Memoirs on Qualities, and have corrected some inaccuracies] ; 142 108 ON LINEAR TRANSFORMATIONS. [14 = 2 x DOM = 24 X = 2 x = 2 x a d 2 + 1 ace# + 1 a 2 h 2 + I abed 6 ac/ 2 - 1 abgh- 14 ac 3 + 4 ad 2 # - 1 acfh+ 18 b s d + 4 ade/+ 2 ac# 2 + 24 6 2 c 2 - 3 ae 3 - I adeh 10 b 2 eg - I adfg 60 9 b 2 f 2 + 1 ae 2 # + 40 6cd<7+ 2 b 2 fh + 24 beef - 2 b 2 g 2 + 25 Aw = 2 x bd 2 f - 2 bceh 60 bde 2 + 2 bcfq - 234 a 2 / 2 + 1 J ./ 6d 2 ^ + 40 abef- 10 c 2 d/ + 2 bdeg + 50 acd/+ 4 c 2 ^ 2 + 1 6d/ 2 + 360 ace 2 +16 cd 2 e - 3 be 2 f - 240 ad 2 e 12 d 4 + 1 c 2 e^ + 360 b 2 df +16 c 2 / 2 + 81 6 2 e 2 + 9 12 cd 2 # - 240 6c 2 / - 12 cdef- 990 bcde - 76 ce 3 + 600 fed 3 + 48 d 3 / +600 c 3 e + 48 d 2 e 2 +375 c 2 d 2 - 32 f 2223 f 142 atf + 1 abhi 16 acgi + 36 ach 2 + 20 adfi 52 adgh 60 ae 2 i + 30 aefh + 20 ae# 2 + 60 af 2 g - 40 b 2 gi + 20 b 2 h 2 + 44 bcfi - 60 bcgh 388 bdei + 20 bdfk + 696 bdg 2 + 180 be 2 h - 340 befg - 460 bf 3 + 240 c 2 ei + 60 C 2 fh + 180 c 2 g 2 + 544 cd 2 i 40 cdeh 460 cdfg - 2596 ce?g + 2340 cef 2 - 420 d 3 h + 240 d 2 <?# - 420 d 2 / 2 + 12876 de*f - 3280 e 4 + 1025 + 8632 -^330 ^ -^222 + 2 -"6 #4= equations which give the functions D wo> As3o> -54o b y means of which the others have been expressed. D no = = 2x D l= 4x a 2 ? 2 + 1 j abii 18 j achj + 40 + 2 aci 2 + 32 2 adgj- 56 7 adhi 112 + 7 aefj + 28 + 5 aegi + 224 + 22 aeh 2 _ 27 140 25 afqh + 45 / u aq 3 20 6% + 32 2 b 2 i 2 + 49 + 2 bcgj - 112 + 7 bchi 536 7 WJ + 224 + 22 bdgi + 392 74 bdh- + 896 + 52 be?j - 140 25 befi - 196 + 73 begh 1792 - 23 bf 2 h + 1120 70 bfg 2 + 45 c 2 fj 27 jj c 2 gi + 896 + 52 c 2 A 2 + 400 25 cdei + 45 cdfi 1792 23 cdgh 4256 - 22 ceH + 1120 70 cefh + 560 + 127 ceg 2 + 6272 - 32 cf 2 g 3920 25 d 3 j 20 d 2 fh + 6272 32 d 2 g 2 + 784 + 47 d 2 ei . . + 45 de 2 h- 3920 25 defg- 13328 85 df 3 + 7840 + 50 e 3 g + 7840 + 50 e 2 / 2 - 4704 30 35022 + 698 14] ON LINEAR TRANSFORMATIONS. 109 We may now proceed to demonstrate an important property of the derivatives of the fourth degree, analogous to the one which exists for the third degree. Let U, V, W, X be functions of any order /: then, investigating the value of the expression B 2f _ 2a [B a (U, V), B a (W, X}], this reduces itself in the first place to 0J !/ ~ 2a 12" 84 UVWX, where &, ye refer to U and V, and &, ^ to W and X: this comes to writing ,= + , 7/ 9 = 77 1 +7; 2 , and & = ,+ 4 , ^ = % + ?4; whence 00 = 13+14 + 23 + 24, or the function in question is (13 + 14 + 23 + 24) 2/ ~ 2a F2 a 34" UVWX. But all the terms of this where the sum of the indices of , ^ or ,, 7; 2 or &> ^ o r 174. exceed /, vanish : whence it is only necessary to consider those of the form K r (13 . 42) r (14 . 23/~ a ~ r (T2 . 34)" UVWX, where K r denotes the numerical coefficient [r] r [r] r [/- a - r]/-- [/- a - r]/- or 5 2/ _ 2o [fl a (U, V), B a ( W, X)] = 2 {K r D a , r<f _^ f ( U, V, W, X)}. In particular, if U= V=W=X, writing also B a for B a (U, U), B, f -^(B a , B a ) = 2(K r D a , r , f ^ r ). If a is odd, this becomes = S(^,D a ,,,/_ a _ r ), an equation which must be satisfied identically by the relations that exist between the quantities D: if, on the contrary, a is even, we see that there are as many independent functions of the form B 2 f- 2a (B a , B a ) as there are of the form Z>; and that these two systems may be linearly expressed, either by means of the other. Thus, for the orders 3, 5, 7, the derivatives D are respectively equal, neglecting a numerical factor, to B.(U*. U 2 ), B W (U\ U 2 ), B U (U\ U*); for the sixth order they may be linearly expressed by means of B,,(U\ U>), B* y and so on. All that remains to complete the theory of the fourth degree is to find the general solution of this system of equations, as also of the system connecting the derivatives D. 110 ON LINEAR TRANSFORMATIONS. [14 Passing on to a more general property; let U lf U<,, ... U p be functions of the orders /!> /a ">fp\ a]Q d suppose a function of the degree /i in the variables : suppose that ( U z . . . U p ) contains the differential coefficients of the order r 2 for U 2 , r s for U 3 , &c., so that /j = (/ 2 r<,)+...(f p r p ). Consider the expression B fl {U lt 0(0,. ..#,)}, which reduces itself in the first place to i^... u p , then to K (12^* 13 fs ~ r *. . . lp p where for shortness For if one of the indices were smaller another would be greater, for instance that of _ - f _ y, _ 12 : and the symbols 2 > rj 2 in 12 2 the term would vanish. Hence, writing _ - f _ y, _ ^ 12 : and the symbols 2 > rj 2 in 12 2 Q would rise to an order higher than / 2 , or and ^(U^U,..., U P } = U U 1 U,...U P , we have ^{^i, (U*,.- U p )} = K(U lt U 2 ... U p }: i.e. the first side is a constant derivative of f/i, U^...U P . Suppose t/ j = =-~f (aqxh + ...), or finally, e ( ^ ... ^,) = ,A- - ,/^ + ... (U lf ... U p ). an equation which holds good (changing, however, the numerical factor,) when several of the functions U 1 ... U p become identical. Hence the theorem : if U be a function given by 14] ON LINEAR TRANSFORMATIONS. Ill and be any constant derivative whatever of U, then / , d d \ ~ [/-3 -- xf-^y -j - + ... V dctf * aa/_! / is a derivative of U, and its value, neglecting a numerical factor, may be found by omitting in the symbol D, which corresponds to the derivative @, the factors which contain any one, no matter which, of the symbolic numbers. If, for example, - i Aio = = Gabcd - 4ac 3 - 4>bd 3 + 36 2 c 2 - a*d 2 , or reduces itself, omitting a numerical factor, to 12*I3UUU=-B l {U, B^U, U}}. This may be compared with some formulae of M. Eisenstein s (Crelle, vol. xxvu. [1844, pp. 105, 106] ; adopting his notation, we have $> = aa? + 3bx 2 y + Sexy 2 + dy 3 , F = ^ 5 2 (3>, <) = (ao - b 2 ) x 2 + (ad - be} xy + (bd - c 2 ) f, da 9 do 9 db y da where D is the same as @. Hence to the system of formulae which he has given, we may add the two following : dF_ d dF\ dy dy dx) dx dy dy j d>$> d<&_ d 3 (j) f d 2 d rf 2 <X> d<&\ i - - YTG ~ ~ dx 2 dy ( dxdy dy + df da) d 3 d*3> d$> d 2 <& d$>\ d 3 d 2 dxdy 2 \ dxdy dx dx 2 dy J dy 3 dx 2 dx) the first of which explains most simply the origin of the function <!. It will be sufficient to indicate the reductions which may be applied to derivatives of the form where U, V, W are homogeneous functions. In fact, if the above becomes, neglecting a numerical factor, (S! . 23) a . (B, . 31)" (B, . 12) Y UVW, 112 ON LINEAR TRANSFORMATIONS. [U where the symbols , rj are supposed not to affect the x, y which enter into the expressions H. But we have identically H! 23 + E 2 31 + E 3 12 = 0, an equation which gives rise to reductions similar to those which have been found for the derivatives D a> ^ y , but which require to be performed with care, in order to avoid inacccuracies with respect to the numerical factors. It may, however, be at once inferred, that the number of independent derivatives C a> ^ y is the same with that of the independent derivatives D a> ft y for the same value of a + /3 + 7. From similar reasonings to those by which B {U, B (U, U)} has been found, the following general theorem may be inferred. " The derivative of any number of the derivatives of one or more functions, or even of any number of functions of these derivatives, is itself a derivative of the original functions." For the complete reduction of these double derivatives, it would be sufficient, theoretically, to be able to reduce to the smallest number possible, the derivatives of any given degree whatever. This has been done for the derivatives of the third degree Va,p,y, and for those of the fourth degree, in which all the differentiations rise to the same order (D a ,p, v ): it seems, however, very difficult to extend these methods even to the next simplest cases, extensive researches in the theory of the division of numbers would probably be necessary. Important results might be obtained by con necting the theory of hyperdeterminants with that of elimination, but I have not yet arrived at anything satisfactory upon this subject, I shall conclude with the remark, that it is very easy to find a series, or rather a series of series of hyperdeterminants of all degrees, viz. the determinants a, b, b, c, d, a, 26, c > a, 46, 6c, 4>d, e b, 2c, d b, 4c, Qd, 4e, f a, 26, c, . - c, U, 6e, > 9 b, 2c, d, . a, 46, 6c, 4>d, e, b, 4c, 6d, 4e, f, c, U, 6e, */. 9> 3c , d &c. 3d, , d, e, [I have inserted in these determinants the numerical coefficients which were by mistake omitted.] However, these functions are not all independent; e.g. the last may be linearly- expressed by the square of the second and the cube of (ae - 4-bd + 3c 2 ) ; nor do I know the symbolical form of these hyperdeterminant determinants. a, 6, c, d &c . b, c, d, e c, d, , f d, e, / 9 &c. a, 36, b, 3c, a, 36, 3c, b, 3c, 3d, a, 36, 3c, d, b, 3c, 3d, e, 151 113 15. NOTE SUB DEUX FORMULES DONNEES PAR M. M. EISENSTEIN ET HESSE. [From the Journal fur die reine und angewandte Mathematik, (Crelle) vol. xxix. (1845), pp. 54 57.] MR. EISENSTEIN a donne [Journal, t. xxvu. (1844), pp. 105 106] cette formule assez remarquable : (a*d 2 - 36 2 c 2 + 4ac 3 + 46d 3 - Qabcd) 3 = A*D*- 3&C* + 4.AC 3 + 4D5 3 - QABCD, ou A, B, C, D sont des fonctions donne es de a, b, c, d. Cela peut se gdneraliser comme suit. Soit u = a 2 h? + by + c 2 / 2 + dV - 2ahbg - Zahcf- 2ahde - Zbgcf- Zbgde - Zcfde + 4>adfg + 4<bceh, et de plus , du D , du n 1 du 1 du A. = * -= . Jo = -k -^pf , (j = * -= .... 1 = * -- . 2 da 2 db 2 dc ^ dh Representons par U la meme fonction de A, B, C, ... H, que la fonction u Test des quantite s a, b, c, ... h, Ton a I e quation U = u\ C est un cas particulier d une propri^td generale de la fonction u, que Ton peut enoncer ainsi. Imaginons la fonction transform^e en a!x\y\z\ + b x\y\z + ... + h x\y\z\, c. 15 114 NOTE SUR DEUX FORMULES DONNEES PAR [15 par les substitutions et soit u , la fonction analogue a u, des nouveaux coefficients a , b , c , ... h : alors U, = (\i \a - X, V) 2 (/*i fa - A*a A*i ) 2 (PI Vg - *2 I//) 2 . M. En echangeant seulement les z, ceci donne u = (v^ v 2 Vi) 2 . u, ou a = i/! a + i//e, 6 = Vi 6 + V/, c = ^ c + i//^, d = v l d + y/A, g = v , a + vje, f = v 2 b + Vzf, g = v 2 c + v 2 g, h = v. 2 d + v 2 h. Soit vi =ah-bg-cf-de, Vi = - 2 (ad - be), v 2 = - 2 (eA -/^r), i/ 2 ah bg cf de (= v^, on trouve d abord (z/ji/2 7 ^2^2 ) = ^ 2 , ou w = u 3 , et puis, en ayant egard aux valeurs de A, B, C, ... H : a = H, b = -G, c = -F, d = E, e =D, f = -B, g = -C, h =A, de maniere que u = U, d ou enfin U = u s . La propriete de la fonction u que je viens d enoncer, se rapporte a une the orie assez ge"nerale, d une iiouvelle espece de fonctions alge briques dont je m occupe actuelle- ment, et lesquelles a cause de leur analogie avec les d^terminantes, on pourrait comme je crois nommer " Hyperdeterminantes." Je me propose de poser les premiers fondemens de cette the orie dans un memoire qui va paraitre dans le prochain No. du Cambridge Mathematical Journal (No. xxm.) [13]. A present je passe a une formule de Mr. Hesse [Journal, t. xxvm. (1844), p. 88], qui se rapporte aussi a la meme the orie. Soit V, une fonction homogene du troisieme ordre, et a trois variables x, y, z. Soit W, la d^terminante formee avec les coefficients du second ordre de V, arranges en cette forme : dx* dxdy dxdz dxdy dy 2 dydi dzdx dzdy dz" 15] M. M. EISENSTEIN ET HESSE. 115 soit a une quantite constarite quelconque et soient A et B deux autres constantes de termine s : Mr. Hesse a de montre liquation remarquable mais sans donner la forme des coefficients A et B, ce qui parait etre tres difficile a effectuer. En considerant le cas d une fonction homogene de deux variables seule- ment, mais d ailleurs d un ordre quelconque, on parvient a un theoreme analogue qui m a para interessant. " Soit U une fonction homogene et de 1 ordre y des deux variables x, y, et V U d 2 V H 2 V / fJ 2 V\z kt /, iv y u/ y i Ui v \ .. determmante -5- . -= I -==- , 1 on a das 2 dy 2 \dxdyl ( v -l)(2 v -5yia*} U + {(v - 2) ( v - 3) 3 + ( v -2)( v - 3) (2i/ - 5) Ja 2 } V U. En repre sentant par i, j, k, I, m, les coefficients differentiels du quatrieme ordre de U, on a / = ikm U- jm? k? + 2jkl, J = 4<jl 3& 2 mi, de maniere que /, J sont des fonctions de x, y des ordres 3 (v 4) et 2 (v 4) respectivement." Ce qu il y a de remarquable, c est la forme de ces deux quantites /, /. On voit d abord que la fonction / est la determinante forme e avec les termes i, j, k j, k, I k, I, m. Mais de plus les deux fonctions /, J ont la propriete suivante. Si Ton imagine une fonction du quatrieme ordre transformed en au moyen de en repre sentant par / , / , les memes fonctions de i , j , k , I , m , on a J = (V - X 8 . J, I = (X// - X 4 . 7. 152 116 NOTE SUR DEUX FORMULES DONNEES PAR M. M. EISENSTEIN ET HESSE. [15 L on parviendrait, je crois, a des r&ultats plus simples en considerant une fonction U, homogene en x, y, et aussi en a?,, y 1} et en posant dxdx l dydy l dxdy^ dydx l Par exemple, si U est du second ordre en as, y et aussi en x lt y lt de maniere que U= x? (A on a simplement en repr^sentant par la determinante formee avec les coefficients A , B , G-, A , R, C ; A", B", C"; et multiplied par 2 8 . Mais il faudrait deVelopper tout cela beaucoup plus loin. P.S. Je ne sais pas si Ton a jamais discutd la question suivante, qui doit conduire, il me semble, a des resultats importants. Soient L, M, L , M , U, V des fonctions de x. En e"liminant cette variable entre les equations LU+MV = et L U+M V=0, et en repr&entant 1 dquation ainsi obtenue par [LU + MV, L U + M V] = 0, et de m&me par [17, 7] = liquation obtenue en eliminant x entre U = 0, F=0, il est clair que [LU + MU, L U + M V] contiendra [17, V] comme facteur: quelles sont maintenant les propridtes de 1 autre facteur qu on peut e"crire sous les trois formes: [LU + MV, L U + M V] : [U, V], [LU + MV, LM - L M] : [L, M] et [L U + M V, LM -LM] : [L, M ]) ? 16] 117 16. MEMOIRE SUR LES HYPERDETERMINANTS (TRADUCTION D UN MEMOIRE ANGLAIS INSERE DANS LE "CAMBRIDGE MATHE MATICAL JOURNAL" AVEC QUELQUES ADDITIONS DE L AUTEUR). [From the Journal fur die reine und angewandte Mathematik (Crelle), vol. xxx. (J846), pp. 1-37.] This is a translation of the foregoing papers, [13] and [14], On Linear Transformations, the additions are inconsiderable: in date of publication it was subsequent to [13] but I think preceded [14]. 118 17, NOTE ON MR BRONWIN S PAPER ON ELLIPTIC INTEGRALS. [From the Cambridge Mathematical Journal, vol. in. (1843), pp. 197 198.] This Note was in answer to objections raised by Mr Bronwin in his paper "On Elliptic Functions" Camb. Math. Jour. vol. in. (1843) pp. 123131, to some of the formula; of transformation in the Funda- inenta Nova. There is in it a serious error which was afterwards pointed out by Mr Bronwin. The Note is omitted, as are also the other papers, [18] and part of [21], which refer to the same subject. 18] 119 18. REMARKS ON THE REV. B. BRONWIN S PAPER ON JACOBI S THEORY OF ELLIPTIC FUNCTIONS. [From the Philosophical Magazine, vol. xxn. (1843), pp. 358 368.] This paper is omitted : see [17]. 120 [19 19. INVESTIGATION OF THE TRANSFORMATION OF CERTAIN ELLIPTIC FUNCTIONS. [From the Philosophical Magazine, vol. xxv. (1844), pp. 352 354.] THE function sinamw (</>M for shortness) may be expressed in the form d)U = U II I J + H 77 H 7J7T-} H- II I 1 + ~ 77 r^ 7 -. ) (1) ~ f)nnn IS I Omn I/ ^ / O.-yvi I/^ I / O v-^1 I 1 \ t/ /, / V / where in, m receive any integer, positive or negative, values whatever, omitting only the combination in = 0, m = in the numerator (Abel, CEuvres, t. I. p. 212, [Ed. 2, p. 343] but with modifications to adapt it to Jacobi s notation ; also the positive and negative values of m, mf are not collected together as in Abel s formulas). We deduce from this + l)K i+0J" Suppose now K = aH + a H i, K i = bH + b H i, a, b, a , b integers, and ab a b a positive number v. Also let 6 fH +f H i; f, f integers such that af a f, bf b f, v, have not all three any common factor. Consider the expression ...$ (2 (*- from which u + 2<a) ... <f> (u + 2 (v - 1) <o) ~ ll + * H l + + (2m +1) K i + 2r0J" where r extends from to v l inclusively, the single combination m = 0, m = 0, r = being omitted in the numerator. We may write mK + m K i + r0 = 19] INVESTIGATION OF THE TRANSFORMATION OF CERTAIN ELLIPTIC FUNCTIONS. 121 /JL, fi denoting any integers whatever. Also to given values of p, // there corresponds only a single system of values of m, m , r. To prove this we must show that the equations ma + m b + rf = /JL, ma + m b + rf = p!, can always be satisfied, and satisfied in a single manner only. Observing the value of v, vm + r (b f- bf) = fib - fi b ; then if v and b fbf have no common factor, there is a single value of r less than v, which gives an integer value for m. This being the case, m b and m b are both integers, and therefore, since b, b have no common factor (for such a factor would divide v and b f bf), m is also an integer. If, however, v and b f bf have a common factor c, so that v = ab - a b = C(f>, b f- bf = eft ; then (af - a f) b = c (<f>f - </> /), or since no factor of c divides af - a f, c divides b , and consequently b. The equation for v may therefore be divided by c. Hence, putting - = i/ /? we may find a value of C r, say r t , less than v t , which makes m an integer; and the general value of r less than v which makes m an integer, is r = r t +sv ( , where s is a positive integer less than c. But m being integral, bm , b m , and consequently cm are integral ; we have also cv-jn! + (r, + sv f ) (af a f) = apf a p; and there may be found a single value of s less than c, giving an integer value for m . Hence in every case there is a single system of values of m, m , r, corresponding to any assumed integer values whatever of /j,, //. Hence U = uU(l+ u m , \ -5-11(1 + u , (f>,u being a function similar to $u, or sin am u, but to a different modulus, viz. such that the complete functions are H, H instead of K, K . We have therefore , _ <})U(f) (u+ 2&))... <f> (u + 2 (v 1) &)) Expressing <a in terms of K, K , we have vH = b K - a K i, - vH i = bK - aK i, and therefore va> = (bf-bf )K-(a f-af )K i. Let g, g be any two integer numbers having no common factor, which is also a factor of v, we may always determine a, b, a , b , so that va=gK- g K i. This will be the case if g = b f- bf, g = a f- af. One of the quantities / / may be assumed equal to 0. Suppose f = 0, then g = b f g = a f; whence agbg =vf. Let Tc be the greatest common measure of g, g , so that g = kg t , y = kg /, then, since no factor of k divides v, k must divide /, or f=kf, but 9> = bf> g , = af, and a , b are integers, or / must divide g, g ; whence / = 1, or f=k. Also ag / -bg / =v, where g / and g t are prime to each other, so that integer values may always be found for a and b ; so that in the equation (1) we have gK- g K i ft) = . C7\ v \ /i g, g being any integer numbers such that no common factor of g, g also divides v. a 16 122 INVESTIGATION OF THE TRANSFORMATION OF CERTAIN ELLIPTIC FUNCTIONS. [19 The above supposition, / = 0, is, however, only a particular one; omitting it, the conditions to be satisfied by a, b, a, b , may be written under the form ab a b = v, ag - bg = [mod. 4 > ..... ................. (8) a g - b g = [mod. v], to which we may join the equations before obtained, r ................................... - vH i = bK - aKi, J which contain the theory of the modular equation. This, however, involves some further investigations, which are not sufficiently connected with the present subject to be attempted here. 20] 123 20. ON CERTAIN RESULTS RELATING TO QUATERNIONS. [From the Philosophical Magazine, vol. xxvi. (1845), pp. 141 145.] IN his last paper on Quaternions [Phil. Mag. vol. xxv. (1844), p. 491] Sir William R. Hamilton has alluded to a paper of mine on the Analytical Geometry of (n) Dimensions, in the Cambridge Mathematical Journal [11], as one that might refer to the same subject. It may perhaps be as well to notice that the investigations there contained have no reference whatever to Sir William Hamilton s very beautiful theory ; a more correct title for them would have been, a Generalization of the Analysis which occurs in ordinary Analytical Geometry. I take this opportunity of communicating one or two results relating to quater nions ; the first of them does appear to me rather a curious one. Observing that (1) it is easy to form the equation (A + Bi+Cj + Dk)~ l (a + pi+ y j + 8k) (A + Bi + Cj + Dk) (4+.B i + i [ /3 (A 2 +B*-C 2 - D 2 ) + 2 7 (BC + AD) +j[2/3(BC-AD) + y (A* -& + + 28 (BD -AC) .... (2) + S (A n - - 5 2 - C 2 which I have given with these letters for the sake of reference ; it will be convenient to change the notation and write 162 124 ON CERTAIN RESULTS RELATING TO QUATERNIONS. \i + /xj + i/A;)" 1 (ix +jy + kz] (1 1 [20 1 + X 2 + ^ + v 2 ( i [ x (1 + X 2 - yu, 2 - 1/ 2 ) + 2y (\fj, + v) + 2z (\v - /*) ] -[ +j [2^(XyLt v) + y (1 X 2 + /u, 2 i/ 2 ) + Zz (^v + X) ] (Xv + /i) +2y(/jLv \~) + z (1 X 2 /u, 2 + i/ 2 )] =^ t (oa; + a 7/ + OL"Z) } (3) .(4) suppose. The peculiarity of this formula is, that the coefficients a, /3, ... are precisely such that a system of formulae a. y + OL" z = ,(5) denotes the transformation from one set of rectangular axes to another set, also rectangular. Nor is this all, the quantities X, /*, v may be geometrically interpreted. Suppose the axes Ax, Ay, Az could be made to coincide with the axes Ax,, Ay t , Az, by means of a revolution through an angle round an axis AP inclined to Ax, Ay, Az, at angles f, g, h then X = tan cos/ ^ = tan \Q cos g, v = tan \Q cos h. In fact the formulae are precisely those given for such a transformation by M. Olinde Rodrigues Liouville, t. v., "Des lois ge ometriques qui rdgissent les ddplacemens d un systeme solide " (or Comb. Math. Journal, t. iii. p. 224 [6]). It would be an inte resting question to account, a priori, for the appearance of these coefficients here. The ordinary definition of a determinant naturally leads to that of a quaternion determinant. We have, for instance, or , <p OT . % ^ > y j", ^>", ^ &c., the same as for common determinants, only here the order of the factors on each term of the second side of the equation is essential, and not, as in the other case, arbitrary. Thus, for instance, = (7), Iff, Iff 20] ON CERTAIN RESULTS RELATING TO QUATERNIONS. 125 OT, OT = OTOT OT OT, 4=0 (8), but OT , W that is, a quaternion determinant does not vanish when two vertical rows become identical. One is immediately led to inquire what the value of such determinants is. Suppose OT = x + iy +jz + kw, & =x + iy +jz + kw , &c., it is easy to prove OT OT =-2 i, j, k (9), OT OT x, y, z r ?/ ? * > y , z OT, OT, OT =-2 3, i, j, k (10), OT , OT , OT x , y , z , w OT", OT", OT" of, y , z , w x", y", z", w" w. w =0 (11), OT , OT , OT , OT OT , OT , OT , OT tiJ , OT , OT or a quaternion determinant vanishes when four or more of its vertical rows become identical. Again, it is immediately seen that or , (12) &c. for determinants of any order, whence the theorem, if any four (or more) adjacent vertical columns of a quaternion determinant be transposed in every possible manner, the sum of all these determinants vanishes, which is a much less simple property than the one which exists for the horizontal rows, viz. the same that in ordinary determinants exists for the horizontal or vertical rows indifferently. It is important to remark that the equations OT, </> =0 or OT, OT =0, &c (13) 1 f^ i/ /i f* / / f\ Q 1>c * ^T(p -ZD" Q> U, Of 570 OVST = U 3 OCC. are none of them the result of the elimination of II, 4>, from the two equations OT!! + <<& = 0, (14). OT I! + </> <& = 0, 126 ON CERTAIN RESULTS RELATING TO QUATERNIONS. [20 On the contrary, the result of this elimination is the very different equation OT- 1 .</>- iff - 1 .</> = ................................. (15), equivalent of course to four independent equations, one of which may evidently be replaced by . M<f> - M^ . M(j) = ............................. (16), if M-sr, &c. denotes the modulus of CT, &c. An equation analogous to this last will undoubtedly hold for any number of equations, but it is difficult to say what is the equation analogous to the one immediately preceding this, in the case of a greater number of equations, or rather, it is difficult to give the result in a symmetrical form independent of extraneous factors. I may just, in conclusion, mention what appears to me a possible application of Sir William Hamilton s interesting discovery. In the same way that the circular functions depend on infinite products, such as , &c ................ .. (17), mir [m any integer from oo to GO , omitting m = 0} and the inverse elliptic functions on the doubly infinite products + - - ., &c ....................... (18) mw + nun/ {m and n integers from oo to oo , omitting m= 0, n = 0}, may not the inverse ultra-elliptic functions of the next order of complexity depend on the quadruply infinite products *n(l+- -^T. rr ? (19) V mw + n-cn + ofy+ptykj {m, n, o, p integers from oo to - oo , omitting m = 0, n = 0, o = 0, p = 0}. It seems as if some supposition of this kind would remove a difficulty started by Jacobi (Crelle, t. ix.) with respect to the multiple periodicity of these functions. Of course this must remain a mere suggestion until the theory of quaternions is very much more developed than it is at present ; in particular the theory of quaternion exponentials would have to be developed, for even in a product, such as (18), there is a certain singular exponential factor running through the theory, as appears from some formulae in Jacobi s Fund. Nova (relative to his functions @, H), the occurrence of which may be accounted for, a priori, as I have succeeded in doing in a paper to be published shortly in the Cambridge Mathematical Journal [24]. 21] 127 21. ON JACOBI S ELLIPTIC FUNCTIONS, IN REPLY TO THE KEY. B. BRONWIN; AND ON QUATERNIONS. [From the Philosophical Magazine, vol. xxvi. (1845), pp. 208, 211.] The first part of this Paper is omitted, see [17]: only the Postscript on Quaternions, pp. 210, 211, is printed. IT is possible to form an analogous theory with seven imaginary roots of ( 1) (?with i/ = 2 w 1 roots when v is a prime number). Thus if these be i 1} i 2 , i 3 , i 4 , i 5 , i 6 , i 7 , which group together according to the types 123, 145, 624, 653, 725, 734, 176, i.e. the type 123 denotes the system of equations 2-2*1 = Is > 1$2 = *1 > Ill s = 1"2 , &c. We have the following expression for the product of two factors (X + X l i l +...X 7 i 7 ) (X + X\ i, + ...X 7 i 7 ) X jJi j X 2 X 2 ... X 7 X 7 [23 - h 45 - h 76 - h (oi)K [31 - h 46 - h 57 h (02)] i z [12 H - 65 - h 47 - h (03)] i s [51 - h 62 H - 47 H - (04)] i 4 [14 H h 36 - h 72 - h (05)] t. [24 - h 53 - h 17 - h (06)] i 6 [25 - h 34 - h 61 - h (07)] i, where (01) = X^X\ + X.X , . . . ; 12 = X.X , - X 2 X\ &c. ; and the modulus of this expression is the product of the moduli of the factors. The above system of types requires some care in writing down, and not only with respect to the combinations of the letters, but also to their order; it would be vitiated, e.g. by writing 716 instead of 176. A theorem analogous to that which I gave before, for quaternions, is the following : If A = 1 + X& . . . + \,i,, X = x^ ...+ x 7 i, : it is immediately shown that the possible part of A^JTA vanishes, and that the coefficients of i 1 ,...i 7 are linear functions of x l , ... x 7 . The modulus of the above expression is evidently the modulus of X ; hence " we may determine seven linear functions of x l ... x 7 , the sum of whose squares is equal to cci 2 + ... + xf." The number of arbitrary quantities is however only seven, instead of twenty-one, as it should be. 128 [22 22. ON ALGEBRAICAL COUPLES. [From the Philosophical Magazine, vol. xxvu. (1845), pp. 3840.] IT is worth while, in connection with the theory of quaternions and the researches of Mr Graves (Phil. Mag. [Vol. xxvi. (1845), pp. 315320]), to investigate the pro perties of a couple ix+jy in which i, j are symbols such that tf = ad + j, ij = afi + g j, ji = yi + Bj, If ix +jy iac! +jy^ = iX +j Y, then X = a xx l + a!xy l + yx Y = $ xx l Imagine the constants a, 6 ... so determined that ix +jy may have a modulus of the form K (x + \y} (x + py) ; there results one of the four following essentially inde pendent systems - j- = - \fiBi + (7 + X + = - (7 + p$) (x + fiy) (x 22] ON ALGEBRAICAL COUPLES. 129 The couple may be said to have the two linear moduli, (7 + XS) (x + \y\ - (7 + n% ) (x + fiy) ; A, /a as well as the quadratic one, - (7 + X8) (7 +p$) (x + \y) (oc + py), the product of these, which is the modulus, and the only modulus in the remaining systems. B. i* = - Si + ( 7 + 8 X (X + fj, 7 + X/i) i yj X + XF= 1 ( 7 + = - (y + pS) (x + \y) (x, r l v>. i = X/u X/U, Si + (7 + 8 X + /A) j, A*yL6 X 2 r = r Z+XF=- f Z + /iF= ^ (7 + ya8) (a; + Xy) (^ C. 17 130 ON ALGEBRAICAL COUPLES. [22 The formula- are much simpler and not essentially less general, if yu,= -X. They thus become A . *= Bi +2jj, ij=ji = r yi + fy, J 2 = KSi + jj, X \Y=^(j\8)(x \y) (ar, \y,). A, (Two linear moduli.) B . i 2 = -&i *? => = 7* C . D . + \F = + - (7 + \8) (x A, + X \Y=-(y\S)(x \y) (x, + ~ 7* 7* + X \Y = + (7 + \B ) (* + Xy) (a;, X^). X There is a system more general than (A.) having a single linear modulus q(OX + 7): this is E. i 2 = a (i - 0j) + frqj, or, without real loss of generality, 22] ON ALGEBRAICAL COUPLES. 131 E . i- = a i, ij = o% ji = 7 i, J 2 = 7 i + qj, To complete the theory of this system, one may add the identical equation Y l F-?^"^ 3 ) ( a/ ~ ^7 + e-qM* "e^qS \ x ^ o^fa y where M = * ~ <* ~ ^ a 7 i / _ /3 / _ /) / By determining the constants, so that 3- ,,= ro = ^r~ , the system would gl/ oi da. <x 0j reduce itself to the form A. 172 132 [23 23. ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [From the Philosophical Magazine and Journal of Science, vol. xxvu. (1845), pp. 424 427.] IN a former paper [19] I gave a proof of Jacobi s theorem, which I suggested [21] would lead to the resolution of the very important problem of finding the relation between the complete functions. This is in fact effected by the formulae there given, but there is an apparent indeterminateness in them, the cause of which it is necessary to explain, and which I shall now show to be inherent in the problem. For the sake of supplying an omission, for the detection of which I am indebted to Mr Bronwin, I will first recapitulate the steps of the demonstration. If \ to, \ v ( J ) be the complete functions corresponding to </>, x, then this function is expressible in the form Ax = xU^ X \ - "^ raeo + nv) \ m + Let p be any prime number, p, v integers not divisible by p, arid aw + vv P The function is always reducible to the form mco +nv 1 Analogous to the K, K i of M. Jacobi. 23] ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 133 and thus ^x is an inverse function, the complete functions of which are \ &> , \ v : and to , v are connected with &>, v by the equations &> = - (aw + v), P v = - (a to + g v\ P a, g, a , being any integers subject to the conditions that a, 6" are odd and a , even ; also ag -a g= p, fj, VOL = I p, fj,8 va = Ip, i, I being any integers whatever. In fact, to prove this, we have only to consider the general form of a factor in the numerator of ^-^x. Omitting a constant factor, this is and it is to be shown that we can always satisfy the equation ma) + nv + 2rd = m a> + riv, or the equations pm + 2rfjL m a + n a. , pn + 2rv = m g + rig ; and also that to each set of values of m, n, r, there is a unique set of values of m, ri, and vice versa. This is done in the paper referred to. Moreover, with the suppositions just made as to the numbers a, being odd and a , 8 even, it is obvious that m is odd or even, according as m is, and n according as n is, which shows that we can likewise satisfy and thus the denominator of fax is also reducible to the required form. Now proceeding to the immediate object of this paper, a, , a. , , and consequently &) , v are to a certain extent indeterminate. Let A, B, A , B be a particular set of values of a, , a, , and 0, P the corresponding values of &> , v . We have evidently A, B odd and A , B even. Also AB -A B= p, H>B - vA = L p, pB vA = Lp, =-(Aco + Bv), 134 ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. [23 By eliminating &>, v from these equations and the former system, it is easy to obtain to = a + b U, v = a + b U, where a = l (aB - 6 A \ b = - 1 - (a B - A), p v p a = - (a. B - A ), b = - 1 (a B - S A\ p p^ The coefficients a, b, a , b are integers, as is obvious from the equation //,(a J5 .4) = p(L a lA ), and the others analogous to it; moreover, a, b are odd and a , b are even, and aV - a b = ~ (A B - A B) ( - a ) ; that is ab a b = l. Hence the theorem, "The general values a> , v of the complete functions are linearly connected with the particular system of values 0, U by the equations, w = aO+bU, v =a + b U, in which a, b are odd integers and a , b even ones, satisfying the condition ab a b 1." With this relation between 0, U and co , v , it is easy to show that the function fax is precisely the same, whether 0, U or to , v be taken for the complete functions. In fact, stating the proposition relatively to <j>x, we have, " The inverse function <j>x is not altered by the change of co, v into a/, v , where a> = aa> + v, v = a. w + S v, and a, g, a , satisfy the conditions that a, are odd, a , even, and a a?6 = l." This is immediately shown by writing = m o) + riv , or ra = m a + n V, ftam 3+n ?. It is obvious that to each set of values of m, n there is a unique set of values of m , n , and vice versa : also that odd or even values of m, m or n, n always corre spond to each. It is, in fact, the preceding reasoning applied to the case of p=l. Hence finally the theorem, " The only conditions for determining &> , v are the equations to = - (QUO + v), v = - (ct w + v), p p^ where a, are odd and a. , even, and <* -oi g = p, n&-va! = l f p, p-va=lp, I and I arbitrary integers: and it is absolutely indifferent what system of values is adopted for a , v, the value of fax is precisely the same." 23] ON THE TRANSFORMATION OF ELLIPTIC FUNCTIONS. 135 We derive from the above the somewhat singular conclusion, that the complete functions are not absolutely determinate functions of the modulus; notwithstanding that they are given by the apparently determinate conditions, das I**"" - cW) (1 + eV) dx In fact definite integrals are in many cases really indeterminate, and acquire different values according as we consider the variable to pass through real values, or through imaginary ones. Where the limits are real, it is tacitly supposed that the variable passes through a succession of real values, and thus CD, v may be considered as completely determined by these equations, but only in consequence of this tacit supposition. If c and e are imaginary, there is absolutely no system of values to be selected for CD, v in preference to any other system. The only remaining difficulty is to show from the integral itself, independently of the theory of elliptic functions, that such integrals contain an indeterminateness of two arbitrary integers; and this difficulty is equally great in the simplest cases. Why, a priori, do the functions [ dx [ dx sin" 1 ,*: = . , or log x = \ Jojl -X 2 Jo % contain a single indeterminate integer ? OBS. I am of course aware, that in treating of the properties of such products as (x \ 1 + - , it is absolutely necessary to pay attention to the relations between iitOj j"~ iv U/ the infinite limiting values of ra and n; and that this introduces certain exponential factors, to which no allusion has been made. But these factors always disappear from the quotient of two such products, and to have made mention of them would only have been embarrassing the demonstration without necessity. or THE (UNIVERSITY 136 [24 24. ON THE INVERSE ELLIPTIC FUNCTIONS. [From the Cambridge Mathematical Journal, t. iv. (1845), pp. 257277.] THE properties of the inverse elliptic functions have been the object of the researches of the two illustrious analysts, Abel and Jacobi. Among their most remarkable ones may be reckoned the formulae given by Abel (CEuvres, t. I. p. 212 [Ed. 2, p. 343]), in which the functions </>x, foe, Fa, (corresponding to Jacobi s sin am . a, cos am . a, Aa?/i . a, though not precisely equivalent to these, Abel s radical being [(1 - c 2 ^ 2 ) (1 + e 2 ^ 2 )]*, and Jacobi s, like that of Legendre s [(1 - a?) (1 - k^}\), are expressed in the form of fractions, having a common denominator; and this, together with the three numerators, resolved into a doubly infinite series of factors; i.e. the general factor contains two independent integers. These formulae may conveniently be referred to as "Abel s double factorial expressions" for the functions <f>, f, F. By dividing each of these products into an infinite number of partial products, and expressing these by means of circular or exponential functions, Abel has obtained (pp. 216 218) two other systems of formulae for the same quantities, which may be referred to as "Abel s first and second single factorial systems." The theory of the functions forming the above numerators and denominator, is mentioned by Abel in a letter to Legendre (CEuvres, t. n. p. 259 [Ed. 2, p. 272]), as a subject to which his attention had been directed, but none of his researches upon them have ever been published. Abel s double factorial expressions have nowhere any thing analogous to them in Jacobi s Fund. Nova; but the system of formula? analogous to the first single factorial system is given by Jacobi (p. 86), and the second system is implicitly contained in some of the subsequent formulae. The functions forming the numerator and denominator of sin am . u, Jacobi represents, omitting a constant factor, by H (u), 8 (u~) ; and proceeds to investigate the properties of these new functions. This he principally effects by means of a very remarkable equation of the form I ( M ) = A u 2 + 5/o du . / du sin 2 am u, 24] ON THE INVERSE ELLIPTIC FUNCTIONS. 137 (Fund, Nova, pp. 145, 133), by which @ (u) is made to depend on the known function sin am . u. The other two numerators are easily expressed by means of the two functions H, @. From the omission of Abel s double factorial expressions, which are the only ones which display clearly the real nature of the functions in the numerators and denomi nators ; and besides, from the different form of Jacobi s radical, which complicates the transformation from an impossible to a possible argument, it is difficult to trace the connection between Jacobi s formulae ; and in particular to account for the appearance of an exponential factor which runs through them. It would seem therefore natural to make the whole theory depend upon the definitions of the new transcendental functions to which Abel s double factorial expressions lead one, even if these definitions were not of such a nature, that one only wonders they should never have been assumed a priori from the analogy of the circular functions sin, cos, and quite independently of the theory of elliptic integrals. This is accordingly what I have done in the present paper, in which therefore I assume no single property of elliptic functions, but demonstrate them all, from my fundamental equations. For the sake however of comparison, I retain entirely the notation of Abel. Several of the formulae that will be obtained are new. The infinite product .JL\ ma/ where m receives the integer values +1, +2, ...r, converges, as is well known, as r becomes indefinitely great to a determinate function sin of x ; the theory of which might, if necessary, be investigated from this property assumed as a definition. We are thus naturally led to investigate the properties of the new transcendant 9 , _ r rjTT l 1 4- *** 1 f9\ a ALLLL I 1 -t- - ; (^) . V m<o + nvij m and n are integer numbers, positive or negative ; and it is supposed that whatever positive value is attributed to either of these, the corresponding negative one is also given to it. i = *J ( 1), &> and v are real positive quantities. (At least this is the standard case, and the only one we shall explicitly consider. Many of the formulae obtained are true, with slight modifications, whatever o> and v represent, provided only <o : vi be not a real quantity ; for if it were so, ma> + nvi for some values of m, n would vanish, or at least become indefinitely small, and u would cease to be a determinate function of a?.) 1 Now the value of the above expression, or, as for the sake of shortness it may be written, of the function (3), (m, )J 1 I have examined the case of impossible values of w and u in a paper which I am preparing for Crelle s Journal. [The paper here referred to is [25], actually published in Liouville s Journal]. c. 18 138 ON THE INVERSE ELLIPTIC FUNCTIONS. [24 depends in a remarkable manner on the mode in which the superior limits of m, n are assigned. Imagine m, n to have any positive or negative integer values satisfying the equation <f>(m 2 , n 2 )<T (4). Consider, for greater distinctness, m, n as the coordinates of a point; the equation < (m 2 , n 2 ) = T belongs to a certain curve symmetrical with respect to the two axes. I suppose besides that this is a continuous curve without multiple points, and such that the minimum value of a radius vector through the origin continually increases as T increases, and becomes infinite with T. The curve may be analytically discontinuous, this is of no importance. The condition with respect to the limits is then that m and n must be integer values denoting the coordinates of a point within the above curve, the whole system of such integer values being successively taken for these quantities. Suppose, next, u denotes the same function as u, except that the limiting condition is $ (m 2 , n 2 )<T (5). The curve <j> (m 2 , n 2 ) = T is supposed to possess the same properties with the other limiting curve, and, for greater distinctness, to lie entirely outside of it ; but this last condition is nonessential. These conditions being satisfied, the ratio u : u is very easily determined in the limiting case of T and T infinite. In fact it ( * 1 -=nn i + , (6), u ( (m, ri)\ or Z- = 22JJl + 7 -^j (7), M I (TO, )J the limiting conditions being (f)(m 2 , n 2 )>T (8), </> (m 2 , n 2 ) < T. id % } x x 2 , AN Now M 1 + / -\r = / \~i-7 \ 2 + ( 9 )> ( (TO, n)) (m, n) (m, n) 2 J ** K* 1 ^ * 1 9 \?V I (~\ A A /- =X. 22 7 r fr.*7 r + (10), u (TO, n) (m, n)- or, the alternate terms vanishing on account of the positive and negative values destroying each other, n 1 1 J " i 2 ^r> _ i , ^^ (11^ ~i<~ (TO, n) 2 (TO, w) 4 In general n, n} dmdn + P (12), 24] ON THE INVERSE ELLIPTIC FUNCTIONS. 139 P denoting a series the first term of which is of the form C-^r (m, ri), and the remaining ones depending on the differential coefficients of this quantity with respect to m and n. The limits between which the two sides are to be taken, are identical. In the present case, supposing T and T indefinitely great, it is easy to see that first term of 1 small; and we have u the first term of the expression for I is the only one which is not indefinitely 11 l- = -%Atc\ or u = ue-* At * ........................... (13), \L where A = f f-" W = f I" dmdn n 4\ - J J (m, n) 2 JJ (moj + nvi) 2 the limits of the integration being given by < (m 2 , n 2 )>T .................................... (15), </> (m 2 , n 2 ) < T . Some particular cases are important. Suppose the limits of u are given by m 2 &> 2 < T 2 , n 2 v- <T 2 ................................. (16), and those of u, by m 2 w 2 + wV < T 2 .................................... (17) ; we have A __ [[ dmdn 8 ) (mco + nvi j = -- an coj (T+nvi */ (T 2 - n 2 v 2 ) + nvi - V (T 2 - n 2 v 2 ) + nvi -T+nvi + V 2 (av or, in this case, " = (19). Again, let the limits of u be m 2 a> 2 <R 2 , n 2 v 2 <S 2 (20), and those of u, m 2 a) 2 <R 2 ) n 2 v 2 <S 2 (21). dmdn 1 [j f 1 1 1 11 / ^u j i i eo J \R + nvi R + nvi - R + nvi R + nvi) 182 140 ON THE INVERSE ELLIPTIC FUNCTIONS. [24 where the limits are nV < S 2 , for the terms containing R, nV < $ 2 , for the terms containing R, R-S i R + Si" 4 Sf Sf = _ JL (X - X), if X = tan- 1 ,, X = tan- 1 ^ , the arcs X, X being included between the limits 0, ^TT. Hence . ,. 2(X \) /SIA\ u = ue (24). r . i f S S f Tf ** n ** i *.J ifirs * " $-1 In particular it ^> = -^ , u = u. It = 0, == 1, u = ue * px ; it ^=00, = l, J J\ JK JLI JK Jtt *jr u = ue$&&; where /3 = , for which quantity it will continue to be used. (0V We may now completely define the functions whose properties are to be investi gated. Writing, for shortness, (TO, n) = TOW + nvi (A), (m, n) = (m + ^)co + nvi, (m, n) = TOO) + (ft + ) vi, (m, n) = (m + ^-) w + (n + ^) vi ; we may put ( ^ ^ (B), = nni+ - -, (w, )] (TO, n)J <&x= nnii + 7 ^ 1; ( ( m > n )} the limits being given respectively by the equations mod. (m, n) < T, mod. (m, n) < T, mod. (TO, n) < T, mod. (m, n) < T, T being finally infinite. The system of values m = 0, n = 0, is of course omitted in yx. The functions yx, gx, Gx, <&x, are all of them real finite functions of x, possessing properties analogous to that of u. Thus, representing any one of them by Jx, we have ...(ox 24] ON THE INVERSE ELLIPTIC FUNCTIONS. 141 where J$x is the same as Joe, only for J^x the limits are given by ra 2 o> 2 or 75 (TO + ) 2 <B 2 < R 2 , n?v 2 or (n + ) 2 v 2 < $, (-R, 8, and -~ infinite), and for J_ p ar, by the same 8 formulae, (R, S, and -~ infinite). It is to this equation that the most characteristic properties of the functions Jx are due. The following equations are deduced immediately from the above definitions: y (-ar) = - 7 aj, g(-x)=gx, G(-x) = Gx, <& (-x) = (&x ......... (D), 7(0) = 0, <7(0) = 1, 0(0)=!, ffi(0) = l, 7 (0) = 1. Suppose 7^, <7!#, Cr^, (Six, are the values that would have been obtained for yx, gx, Ox, (&x by interchanging CD and v, then changing x into xi, and interchanging m and w, by which means the limiting equations are the same in the two cases, we obtain the following system of equations : g-i (an) = Gx, G! (on) = gx , j (Xi) = (SlX ; or otherwise, y(an) = vy 1 as ....................................... (F), g (xi) = GjOD, G (xi) = g,x, <& (xi) = (&&, equations which are useful in transforming almost any other property of the functions J. The functions J^x are changed one into another, except as regards a constant multiplier, by the change of x into x + _ . This will be shown in a Note, or it may be seen from some formulae deduced immediately from the definitions of the functions JftX, which will be given in the sequel 1 . Observing the relation between Jx and we have in particular (G), 1 Not given in the present paper. [The Note was given, see p. 154, and the formulas referred to mnst have been the formulas (M) p. 144.] 142 ON THE INVERSE ELLIPTIC FUNCTIONS. [24 where A, B, C, D, are most simply determined by writing x = 0, y Putting at 1TOJ the same time e*" 8 = e T = q^ 1 , /&) whence also /CO <?(!) (!)=.- < 25 > Similarly, the functions J-$x are changed one into the other by the change of x into x + ^vi. We have in the same way Whence , , , vi A =7 where e^ u " = e =g r ~ 1 . It is obvious that the relation between q and ^ is Iq.lq 1 = TT" We obtain from the above 24] ON THE INVERSE ELLIPTIC FUNCTIONS. 143 Also, by making x = -= in the expression for l x(*+s] and # = ~ in that for 2 V */ " / vi\ If I * + "o 1 1 we have ,_. (27) - and the same or an equivalent one would have been obtained from the functions g, G, (&. By combining the above systems, we deduce one of the form g x - D V ; and, observing the equation e^ wvi = 6^= ( 1), with the following values for the coefficients, Collecting the formulae which connect 7(5-), yf-alj these are \2> J \2 I . IVl 2J^~ 17 \2J" V2 144 ON THE INVERSE ELLIPTIC FUNCTIONS. [24 And by the assistance of these B"C" -5- A"D" = B D -f- A C = CD + A B = - 1 . . . (28) , Ul A B + C J) =- A"B" -5- C"D" = - 7 2 ( IT ) - "C" -j- 5"D" = - J/ C + which will be required presently. It is now easy to proceed to the general systems of formulae, (H) = f _ \\rnn ^x(mia-nvi) q\m? q$ni ^ 7 [X + (TO, )} = ( !)+ . ^ {a; + (m, %)} = (_ 1) . (m, w)} = (-!) . (TO, TO)} = 7 {a? + (TO, TO)} = ( - l) m + n . 3>Agac , g{x + (m, n}} = (- l} m . G{x + (m, n)} = (-!)* . , n)}= 7 {# + (m, n)} = (- l) m+w . VA Gae, g{x + (TO, n)} = ( - l) m G {x + (TO, n)} = ( - l) n . 45s {x + (TO, n)} = fl = ( IJwn+im + Jji^Cim + itw-in 7 { + (w, n)} =(-l) m+n . fir {a; + (TO, n)} = ( - I)" 1 . G {cc + (TO, n)} = ( - 1)" . { + (TO, n)} = . 24] ON THE INVERSE ELLIPTIC FUNCTIONS. 145 Suppose # = 0, we have the new systems, Q = (-l} mn q{- m \l-W .......................................... (MUs). 7 (m, ri) = 0, y (m, n) = (- l) m+n , g(m, ri) = (-l) m , G(m, w) = (-l) n , S (w, n) = @ . (w, w) - 0, g (m, n) = (- l) m 3> , G(w, w) = (-l) w ^) (7, (m, n) = D. f_ 1 \m(n + J) ,7 Jm 2 .y Jn^ Jn ^ *j yi y 7 (m, n) = (-l) m+B SP A / , r(m, n) = (-!) ^ ^ G (7??, n) = 0, G (m, n) = ( - l) w ^0^ , (Sf(m, n)= Q _/ 7(m, n)=(-l) m+n H ^ // , flr(w, n) =(-!) Oofi", G(m, n) = (-!) & C", & (m, n) = 0, <& (m, n) = n o D". We obtain immediately, by taking the logarithmic differentials of the functions 7#, gx, Gx, (&x, the equations y x -T- 70; = SS [a? (m, n)}" 1 , ?7i = 0, n = admissible, ...... (iV), i^ ar - gx SS { (m, )}~*j G a; - Ga? = 2S { - (m, ri)}- 1 , G a;^a; = SS{-(i, ri)}- 1 , the limits being the same as in the case of the factorial expressions. Consider an equation gx Gx + 70; fc = S2 [& {^ - (w, ri)}- 1 + 23 {# - (w, w)}" 1 ] ......... (* 29 ) c. 19 OFTHF UNIVERSITY 146 ON THE INVERSE ELLIPTIC FUNCTIONS. [24 we have <&=g(m, n) G (m, w)-5-y (wi, w) (ffir (m, n) = 1 ........................... (30), & = g(m,n)G (m, n) + 7 (m, n) G (in, n) = 5"C" - ^"" = - 1 ... (31). (The application of the ordinary method of decomposition into partial fractions, which is in general exceedingly precarious when applied to transcendental functions, is justified here by a theorem of Cauchy s, which will presently be quoted.) We have thus gxGx -i- yxOfcx = (yx -f- y#) ((Sx ac -=- <?&#), and similarly gxOStx + yx Gx = (y x + jx) - (G x -r- Gx), ............... (0), Gxtfitx -r- <yx gx = ( y x -T- <yx) (g x+ gx\ - b-yxO&x +gx Gx = ( g x -=- gx) (G x -=- Gx), e^x gx +Gx(&x = (G x -r Gx) - (<& x + <&x), tfyxGx -=- (&x gx = (6r aj -f- (&x) -( g x + gx) ; in which we have written e "A ** fW\ 1 , . . .2/1 & Eliminating the derived coefficients, f"- 1 , /"^2 . 7^2 2 Adding these equations, 6 2 = e 2 + c 2 , or 6 = V(e 2 +c 2 ), in which sense it will continue to be used. g*x = &*x - cy#, ................................. (P). G 2 x = <&*x + e^x. Suppose <j>x = ix+<&x, ....................................... (Q), fx = gx 4- C&a;, F<9bf-6r, then f 2 x=l-tf<px, .................................... (R), F 2 x = 1 + e?(j> 2 x, 24] ON THE INVERSE ELLIPTIC FUNCTIONS. 147 and also < # = fx Fx , ($), // _ /AA.n, I? X "~~ C (DX Jj X, ITT/ ^2>-/k /y ^/vi Hence, putting for fx, Fx, their values, or writing <$>x = y, and integrating, dy or y which shows that <j> is an inverse elliptic function. The equations which are the foundation of the theory of the functions </>, /, F, are deduced immediately from the equations (S). (Abel, (Euvres, torn. I. p. 143 [Ed. 2, p. 268.]) These are F (x i v} _ = ^ ...(W) 2 Jo 7(i_ c y)(i +e y) i 1 so that from this point we may take for granted any properties of these functions. We see, for instance, immediately, whence 2 J o 7(1 - c y) (1 + ey) 2 J J(l + c y) (1 - eY) which give the values of (a, v in terms of c, e ; values which may be developed in a variety of ways, in infinite series. We may also express 7(0)1 *^ c -> an d consequently A, B...&C., by means of the quantities c, e. We have only to combine the equations 192 148 ON THE INVERSE ELLIPTIC FUNCTIONS. [24 with the former relations between these quantities, and we have *, A" = (- 1)1 c -* e~^qr l q~ k , B" = - - I* ic* e~* r* ~ which are to be substituted in any formulae into which these quantities enter. The following is Cauchy s Theorem, (Exercises de Math. t. n. p. 289). " If in attributing to the modulus r of the variable z = r {cos^ + V(-l)sinp} ............................. (35), infinitely great values, these can be chosen so that the two functions /*+/(-*) /*-/(-*) 2 .(36), sensibly vanish, whatever be the value of p, or vanish in general, though ceasing to do so and obtaining finite values for certain particular values of p ; then A,_ ~ x-z the integral residue being reduced to its principal value." To understand this, it is only necessary to remark that the integral residue in question is the series of fractions that would be obtained by the ordinary process of decomposition; and by the principal value is meant, that all those roots are to be taken, the modulus of which is not greater than a certain limit, this limit being afterwards made infinite. Suppose now fx is a fraction, the numerator and denominator of which are monomials of the form (jx) 1 (gx} m . . . , I, m... being positive integers, and of course no common factor being left in the numerator and denominator. Let \ be the excess of the degree of the denominator over that of the numerator. Suppose the modulus r of (z) has any value not the same with any of the moduli of (m, n), (m, n), (m, n), (m, n) ....................... (38). ON THE INVERSE ELLIPTIC FUNCTIONS. 149 Then we have r (cosp + i sinp) = raw + nvi + 9 6 being a finite quantity, such that none of the functions JQ vanish, m and n are the greatest integer values which allow the possible part of 6 and the coefficient of its impossible part to remain positive. We have therefore mW + nV = r*-M ................................. (40), M being finite ; or when r is infinite, at least one of the values m, n is infinite. The function fz reduces itself to the form where .F is finite. Hence q t and 5 being always less than unity, fz, and consequently both \ {fe+f( z)} and ^- {/z /( z)} vanish for r=oo, as long as X is positive. ZiZ In the case of A, = 0, the conditions are still satisfied, if we suppose fx to denote an uneven function of x : for when X = 0, the index of exponential in the above expression vanishes, or fz is constantly finite. But fz being an odd function of z, fz+f(z) = Q. And g- {fzf(z}} vanishes for z infinite, on account of the z in - the denominator : hence the expansion is admissible in this case. But it is certainly so also, in a great many cases at least, where fz is an even function of z; for these may be deduced from the others by a simple change in the value of the variable. For instance, from the expansion of <yx -=- goo, which is an odd function, by writing m x+ -~ for x, we obtain that of Gx -=- (&x, which is even. z A case of some importance is when the function is of the above form, multiplied by an exponential e ^^+^. Here writing z = mw + nvi + 6, the admissibility of the formula depends on the evanescence of e Ja (mcu-frtv?) 2 Q i\7, 2 Q^ri* _ _ (42) or, if a = h + ki, this becomes, omitting a finite factor, 6 J m 2 (A/3 ft) <o s - \ ri> (Aj3+A) \P-kmn <ou ^ _ ........... (43), which vanishes if h 2 + k 2 < X 2 /3 2 , i.e. the modulus of a is less than X/3. The limiting case is admissible when the series is convergent. We obtain in this way a very great variety of formulae. For instance, 6 i aa?+bx + y X 2S [( 1 )-""- - 6 i (n , n) 8 +6 (m , n) ^ m? ^ ^\x(m y n)}~ 1 ] ......... (A ), e Jaa; 2 +te ^. qg> = _ ty c ~ J ^]^ I"/ _ (m, n)2+6 (m, n) 2 +& (TO, 150 ON THE INVERSE ELLIPTIC FUNCTIONS. [24 in which the modulus of a must not exceed /3 : in the limiting cases, for a = /3, b must be entirely impossible, and for a = yS, b must be entirely real. The formulae for jx are + yx = ^ (_ i^-n ? n e & (m, n) ^ _ ^ n y-i ............ ^ .=_ 7 # = 22 (- I)" 71 ft e & < m - n > (a - (w, w)}" 1 ; and for 6 = 0, e^ 2 --7tf=22(-l)^ l - w 2 ri2 {as - (m, n}} 1 ..................... (45), e-JP* - 72; = 22 ( - Next the system, {-(m, Gx+gx = - c- 1 S2 (- l) m+w {a; - (m, 71)}" 1 , (Stc + gas ^-ft- 1 2S (- l) m {a; - (m, w)}- 1 ; 7 -r- Ga? = - 6- 1 e~ l SS ( - l) w {x - (m, n)} 1 , gx +Gx= ie- 1 22 ( - l) m+n {x - (m, n)}- 1 , <&x + Gx= ib- 1 22 (- I) 71 {x - (m, n)]- 1 ; 70? +<&x=ic~ l e-i 22 (- l) m+n {x - (m, n)}- 1 , gx +(&x= e- 1 22 (- 1) {x - (m, n)}- 1 , Gx +<&x= ic- 1 22 (- l} n {x - (m, n)}-\ which is partially given by Abel. We may obtain, in like manner, expressions for the functions ^ 7^GV"- (six terms of this form > Gx (twelve) <yx gx 24] ON THE INVERSE ELLIPTIC FUNCTIONS. 151 each of them, except (E ), (the system for which, admitting no exponential, has already been given,) multiplied by an exponential e l ax * +bx , the limits of a being + 2/3, + /3, /3 , 3/3, 2/3, 4/3. For the limiting values, b must be entirely impossible for the superior limit, and entirely possible for the inferior one. Thus the last case is I TJ \ ( * >> in particular _ ^V fg^a (m, n)*+b (m,n) vm? g 2 " 2 [x (m, w)]" 1 ] _ S 2[ e ^<"> 2 +^.n>^<-H> O |t?/ ^~ v^? ^y | ^^ |_^ Q\ g-PH-b fa;-^ fi)}-i] ~^ [eia(m n)2+&lm n> ^ 2(m+il 2 2 2 * ^{^-K *))"*]; + 22 q, imt [x - (m, n) or the analogous formula obtained by" changing @, q ly m into (3, q, n. The function <px, which is even, and for which X = 0, cannot be expanded entirely in a series of partial fractions : but (x a)" 1 $-x may be so expanded. Multiply by (x a), the second side has for its general term (x - a) (Mx + N) {x- (m, n)}~ 2 , equivalent to K + (M x + N ) {x - (m, n)}- 2 . Summing all the K"s, we have an equation of the form (j> 2 x = A + 22[L{x-(m, ri)}- 2 + M {x - (m, n)}- 1 ] ............... (47). To determine the coefficients as simply as possible, change x into x + &> + ^nwi, - e~- c~ 2 ((j>x)-* = A + 22 [L {x - (TO, n)}~* + M {x - (m, n)}~ 1 ] ......... (48), L=-er*(T a [{x-(m, w)} 2 (^)~ 2 ], x = (m, n) ............ (49), M = -e~- c~ 2 d x [{x - (m, n)} 2 (0#)- 2 ], or writing x + (m, n) for x, and therefore x = in the values of L and M, L = - e~ 2 c~ 2 {x 2 (<#)- 2 } =e~ 2 c- 2 .............................. (50), M = - e~ 2 c~ 2 d x {x 2 (<H~1 = 5 whence <$?x = A - e~- c~ 2 22 {x - (TO, n)}~ 2 ........................... (51). 152 ON THE INVERSE ELLIPTIC FUNCTIONS. [24 Integrating this last equation twice, f dxf dx<f> 2 x = %Atf + e- *c- 2 22l [x-(m, n}} .................. (52), or (&x = e-^^ A ^ + ^^dx^<^x ........................... (53), an equation from which it is easy to determine the coefficient A. Suppose for a moment <f> / x=f 9 <pa;da}, $ /l x = h$pdx\ then, since < 2 (x + <w) - $-x = 0, 0/ ( x + <) - fa = $/> $,/ (x + to)- 0/c = </> But similarly $*x </> 2 (co - x) = ; whence whence, writing x = ^ or ^ (a; + o>) - fr/c = ^ (2^ \ I CLI *M# e- ijii# (5o) ; or, comparing these, ( . 2 f ( \\ "P (56), (57), u/ y i/ or writing then .- <> which is the formula corresponding to the one of Jacobi s referred to at the beginning of this paper. Analogous formulas may be deduced from it by writing x + - , or x + , co in . or x + 2 + -fT , instead of x. The following formula, making the necessary changes of notation, are taken from Jacobi. We have 24] ON THE INVERSE ELLIPTIC FUNCTIONS, whence / {< the first side of which is Hence, multiplying by e 2 c 2 , and observing the value of (&x, 153 (60)> (61). (x + a) _ <& (x-a) _ a) 6r(aj-a) If in this case we interchange x, a and add, Ta 5 (x + a) -- Fa <>a ^ e 2 c 2 6a d>x 6 (a + x) .................. (63). [By subtracting, we should have obtained an equation only differing from the above in the sign of a.] Integrating the last equation but one, with respect to a, l& (x + a} + l(& (x-a}~ 2l&x - 2H&a =1(1+ tfc^x^a), the integral being taken from a = 0. Hence r (x + a) (x - a) = (&*x<&*a (1 + e z d><px<pa) ............ (64) ; (x + a) <& (x - a) = C5 2 ^G 2 a + e or whence also 7 (x + a) 7 (x - a) = g(x + a) g (x a) = a) G(x-a)= these equations being obtained from the first by the change of x into x + - , x + , - X "*" 2 "*" "2 ^^ e y ^ orm a most important group of formulae in the present theory. By integrating the same formulae with respect to x, and representing by II (x, a) the if 6 2 c 2 <4a fa Fa &xdx T , . , . integral - , ^ n , -^ - , Jacobi obtains J 1 + e-c^a^x 7^7 - T TjF- Ctr (a; + a) (lira an equation which conducts him almost immediately to the formulae for the addition of the argument or of the parameter in the function H. This, however, is not verv C. 20 154 ON THE INVERSE ELLIPTIC FUNCTIONS. [24 closely connected with the present subject. For some formulae also deduced from (63), by which jar4 < ff . V V/IE / \ * s ex P resse d in terms of the function <, see Jacobi. NOTE. We have 7^ = ^1111 [1 + (ra, n)J (m, n)/ the limits of n being + <?, and those of ra being +jp, in the first case, and p, p l, P in the second case. Also ^- = oc . 3 We deduce immediately ( ^ M / (m, w) ,-, - n .._.. - nn 1+ -, = nn (m, n)J 2 (ra, n) (paying attention to the omission of (ra = 0, n = 0) in y^x, and supposing that this value enters into the numerator of the expression just obtained, but not into its denominator). This is of the form but the limits are not the same in this product and in g$x. In the latter m assumes the value p 1, which it does not in the former; hence and the above product reduces itself to unity in consequence of all the values assumed by n being indefinitely small compared with the quantity (p + ) ; we have therefore Ag ft x ................................. (65), and similar expressions for the remaining functions. To illustrate this further, suppose we had been considering, instead of 7/3 x, the function 7_ J 3#, given by the same formula, but with - = 0, instead of - = oo . We have in this case also q q A different from A on account of the different limits. The divisor of the second side takes the form 24] ON THE INVERSE ELLIPTIC FUNCTIONS. 155 where the extreme values of n are infinite as compared with p. This may be reduced to *TT sin . [x (p + ) w} -4- sin (p + ) w, neglecting the exponentials whose indices are infinitely great and negative. Observing the value of /3 this becomes e ~ <0 3a; , and we have a result of the form of that which would be deduced from the equations 7^ x = e^y^ as, ^la;+ -) = Ag^x. It is scarcely necessary to remark that y-px has the V */ same relations to the change of x into x + -^ as 7^ a; has to that of x into x + -^ . 202 156 [25 25. MEMOIRE SUE LES FONCTIONS DOUBLEMENT PERIODIQUES. [From the Journal des Mathe matiques (Liouville), torn. x. (1845), pp. 385 420.] UN des plus beaux rdsultats des recherches de 1 illustre Abel, dans la thdorie des fonctions elliptiques, consiste dans les expressions qu il obtint pour les fonctions inverses </>a, fa, Fa. (equivalentes a-peu-pres a sin ama, cos ama, Aaraa) en forme de fractions avec un denominateur commun : ce de nominateur et les trois nume rateurs etant chacun le produit d une suite infinie double de facteurs. On ne sait pas a quel point Abel avait pousse 1 investigation des proprietes de ces nouvelles fonctions; on trouve seulement, dans une Lettre a Legendre, imprimee parmi ses (Euvres [t. II. p. 259, Ed. 2, p. 274], qu il s en etait occupd Depuis, les fonctions H, , qui sont essen- tiellement les memes que ces fonctions d Abel, ont e te 1 objet des savantes recherches de M. Jacobi, a qui Ton doit, en particulier, la belle formule log @ (a) - log (0) = Aa 2 ( 1 - -=/) - P | da \ da sin 2 ama, 2 \ KJ Jo Jo qui est vraiment fondamentale, et sur laquelle on peut dire que sa theorie est basee. Mais les expressions qu obtient M. Jacobi pour les fonctions H, , sont sous la forme d un produit d une suite infinie simple de facteurs, ce qui ne met pas a beaucoup pres si bien en evidence la vraie nature de ces fonctions que les expressions d Abel ; celles-ci sont, en outre, si analogues aux formules en produits infinis des fonctions circulates, que Ton est seulement etonne que personne ne se soit avise jusqu ici de les poser, a priori, comme les definitions les plus simples des fonctions doublement periodiques, pour en deduire la thdorie de ces fonctions. C est de cette maniere que je me propose de traiter ici la question. Je prends pour definitions les formules d Abel, en supposant, pour plus de ge ne ralite, que les fonctions completes H, T (K, K de M. Jacobi) sont chacune de la forme A + B J 1 (ce qui donne lieu a quelques integrations assez dedicates) . Et de ces seules Equations, sans me servir en rien de la 25] MEMOIRE SUB LES FONCTIONS DOUBLEMENT PERIODIQUES. 157 thdorie des fonctions elliptiques, je deduis les proprietes fondamentales des fonctions en question, et de la des fonctions elliptiques. On a ainsi quatre fonctions a considerer au lieu des deux H, , dont 1 une est pour ainsi dire analogue a un sinus et les autres a des cosinus. Mais ce qu il y a de remarquable, c est 1 apparition d un facteur exponentiel qui entre presque partout. On le preVoyait d apres les formules de M. Jacobi, mais ces formules n expliquent pas, ce me semble, pourquoi ce facteur s y rencontre : mon analyse le fait voir de la maniere la plus satisfaisante. Ce facteur resulte, en effet, de ce que, pour les produits infinis doubles, il ne suffit pas, pour obtenir un re sultat determine, d attribuer aux deux entiers variables des valeurs quel- conques depuis oo jusqu a oo , meme en supposant I dgalite des valeurs positives et negatives ; il faut, en outre, etablir une relation entre les valeurs infinies que regoivent les deux variables. Mais dans les produits que je considere, on demontre qu en supposant toujours cette e galite des valeurs positives et negatives, quelque liaison que Ton etablisse entre les valeurs infinies, il re sulte toujours la meme valeur du produit, a un facteur exponentiel pres, dont 1 indice est le carrd de x, multiplie par une constante dont la valeur s exprime au moyen d une integrale definie double, et qui depend de la liaison etablie entre les valeurs infinies des variables. C est-a-dire qu en multipliant par un facteur exponentiel de cette forme, convenablement choisi, on peut changer a volonte la relation en question sans affecter la valeur du produit. Voila 1 idee fondamentale du Memoire qui suit. En me servant de quelques formules de M. Cauchy, relatives a la decomposition des fonctions en fractions simples, j e tablis d une maniere rigoureuse des relations entre les trois quotients de mes quatre fonctions, qui sont les memes par lesquelles Abel demontre les propriete s fondamentales des fonctions <f>, f, F. Ces formules une fois obtenues, on peut supposer connue toute la thdorie des fonctions elliptiques. Ces the oremes de M. Cauchy me conduisent, en outre, a un grand nombre de nouvelles formules qui contiennent des suites infinies doubles. Parmi celles-ci, il y en a une qui me fournit la demonstration du theoreme deja cite" de M. Jacobi, thdoreme duquel il d^duit une foule de resultats inteVessants. Je finis en citant ceux qui se rapportent de plus pres aux fonctions dont je parle. J espere reprendre une autre fois la con- siddration d une autre partie de la the orie, dans laquelle j entrevois des conclusions inteVessantes. Soient H, T des quantites finies quelconques, assujetties a la seule condition que la fraction O : T ne soit pas reelle. En repre sentant par m, n des entiers positifs ou negatifs quelconques, mettons, pour abre ger, + wT = (7?i, w) .................................... (1), et considerons une expression de cette forme ........................ (2), ou le symbole II denote, comme a 1 ordinaire, le produit d un nombre infini de facteurs que Ton obtient en donnant a m, n des valeurs entieres quelconques, depuis oo jusqu a 158 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. [25 + x , en excluant seulement la combinaison (m = 0, n = 0). Pour qu une telle expression soit finie, il faut que pour chaque combinaison de valeurs de m, n, il y en ait une autre des memes valeurs avec les signes contraires. Cependant, comme on 1 a deja explique, cela ne suffit pas pour rendre de termine e la valeur de u, Soit < une fonction de m, n qui ne change pas en changeant a la fois les signes de ces deux quantity s ; et imaginons que 1 equation represente une courbe ferme e, dont tous les points s e loignent, au cas limite de T = oo , d une distance infinie de 1 origine. Cela pose , en donnant a m, n des valeurs entieres qui satisfassent a cette condition < < 2 1 , et puis faisant T = oo , on obtient pour u une valeur parfaitement determine e, qui depend de la forme de la fonction <. Soit u ce que devient la fonction u en changeant seuiement 1 equation aux limites dans 1 equation analogue on peut, pour simplifier, supposer que la courbe represented par cette equation soit situee entitlement en dehors de celle que represente 1 dquation mais cela n est pas essentiel. II est facile de trouver une relation tres-simple qui existe entre ces deux expressions u et u. En effet, u :u = Il\l + X (m, n)\ en donnant a m, n des valeurs qui satisfassent a la fois aux deux conditions <j) > T, fj> < T. Done, en considdrant toujours ces valeurs, log u - log u = log II \ 1 + t \ = S log \\ + t (m, n)J ( (m, n)} "* (m, n) m (m, w) Dans cette expression, les termes qui contiennent les puissances impaires x s eVanouissent, a cause des valeurs e gales positives et negatives. Mais puisque, a la limite, m, n ne re9oivent que des valeurs infinies, on peut ne gliger les termes multiplies par #*, &c. Done log u log u = A- ic 2 5] / - c > * (m, n) 3 ou enfm, log u - log u = - | Aa?, u = ue-* Ax ..................... (3), ou j ai reprdsentd par e la base du systeme hyperbolique de logarithmes, et ou A est donnd par 1 dquation 4 = S= ...................................... W- 25] MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. 159 II est facile de voir que Ton peut changer la sommation en integration double, et ^crire dm dn entre les memes limites qu auparavant, c est-a-dire que m, n doivent satisfaire aux deux conditions $ > T, $ < T. Prenons par exemple, pour limite supeVieure, (w 2 = m 2 , n 2 = n 2 ), et pour limite inferieure, (m 2 + n=T 2 ); m, n sont censes contenir T comme facteur, de maniere qu ils deviennent infinis avec cette quantite ; on suppose aussi m > T, n>T, mais cela est seulement pour la clarte. II devient ndcessaire a ce point de de finir de plus pres les valeurs de H, T ; nous ecrirons Q = a> + G>% T = v+vi ................................. (6), ou i = J^I ; o), a) , v, v sont des quantites r^elles, telles que <wi/ a> v ne s evanouit pas ; c est la condition pour que O : T contienne une partie imaginaire. Avec les coordonnees polaires [[ drde f ^(logr-logT) ~JJ r (H -cos 6 + T sin 0) 2 J (II cos + T sin 0) en representant par r ce que devient r a la limite superieure ; 1 integrale doit etre prise depuis 6 = jusqu a 6 2?r. On voit tout de suite que la partie qui contient log T s evanouit; done log r d6 " A. = ("n cos 6 + T sin 0) 2 Soit a un angle positif plus petit que |TT, tel que tan a = , on a evidemment _ + m depuis d = a jusqu a = + a, ou depuis 6= TT a. jusqu a 0=7r + at, et n ** depuis = a jusqu a 6= IT a, ou depuis = 7r + a. jusqu a ^ = 2?r a (le signe ambigu, de maniere que r soit toujours positif). En reunissant les parties opposees de 1 integrale, on obtient A-9 cos 6 + T sin 160 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. [21 ou, en mettant dans la seconde inte grale \TT i =& , et ^TT au lieu de 6, f a (logm-logcos0)d0 [ * (log n - log cos 6) d6 J_ a (11 cos + T sin 0) 2 " j_ a , (T cos + ft sin 0) 2 La premiere intdgrale se reduit a - 2 (log m - log cos 0) T + T ^^ ^ (entre les limites) 2 f tan0c?0 ~r _ 4 (log m log cos a) tan a. f a tan 2 0c0 fl 2 - T 2 tan 2 a J O 2 - T 2 tan 2 = 4 (log m - log cos a) tan a 4 T a I" ft 2 (l + tan 2 0) ft 2 -T 2 tan 2 + li 2 + T 2 J d " IF-T 2 tan 2 L inte grale, dans cette formule ra ( I + tan 2 0) d0 f tan n da; o ft 2 -T 2 tan 2 s exprime tout de suite par les logarithmes, mais il faut apporter une attention particuliere a la maniere de determiner quelle valeur doit etre attribuee a ce logarithme, dans le cas ou la partie reelle en est negative. Je renvoie cette discussion a une Note. Voici le rdsultat auquel j arrive. En representant par Lu la valeur principale du logarithme de u, quand il y a une valeur principale (c est-a-dire quand la partie re elle de u est positive), et e crivant LriU = Lu, ou L(u) TTI, selon qu il y a une valeur principale de log u, ou de log (u), on a tana dx , o 20T " n-T tan a en prenant le signe sup^rieur pour cov ca v positif, et le signe inferieur pour wv w v negatif. La premiere partie de A se rdduit done a 4 (log m log cos a) tan a 4 ( _ ,& T H + T tan a & 2 fV> "*< fl 2 -T 2 tan 2 a ^a 2 + T 2 V 2 T " ft - T tan a) ou, en re tablissant la valeur de a, a 4mn log y/m 2 + n 2 4 / n L O ,. inH + m 2 O 2 n 2 T2 *" 02 _u T2 ( arC n ~ 2 T * : 111 ii 11 1 ii -+- 1 \ 111 1 25] MEMOLRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. 161 Pour avoir la seconde partie de A, il faut changer m, n, H, T en n, m, T, fl. En faisant cela, on change le signe de tot/ co v ; ainsi il faut entire L^n au lieu de L n {. En reunissant les deux parties, et mettant ATT au lieu de arc tan + arc tan . on m n obtient enfin 11 /mfl + nT\ T r /nT + mfl r^^ t ormule que Ton pourrait rendre plus simple en distinguant les deux cas ou le premier ou le second logarithme a une valeur principale ; mais il vaut mieux la laisser sous cette forme. On aurait pu croire que la valeur de A pourrait s obtenir plus facilement en re duisant A a la difference de deux integrates definies, les conditions pour les limites etant donnees, dans la premiere, par m 2 < m 2 , w 2 < n 2 , et, pour la seconde, par m 2 + n 2 < T 2 , en de signant generalement les coordonnees par m, n. Cependant, de cette maniere, on admet dans les deux inte grales les valeurs m = 0, n = 0, qui rendent infinie la fonction a intdgrer. Ainsi la valeur de 1 inte grale dependrait du choix des variables, et Ton obtiendrait des re"sultats inexacts. Autrement dit, on obtiendrait de cette fa^on la valeur de A, a une quantite V pres, qui est la difference de deux integrales de la meme forme, entre des limites m* < fj, 2 , n 2 < v 2 ou m 2 + n~ < r 2 , p, v, T des quantites infiniment petites. On voit tout de suite que V n est pas pour cela infiniment petit (en effet, sa valeur depend du rapport p : v et nullement des valeurs absolues de ces quantites), et, pour en trouver la valeur, il faudrait se servir de 1 analyse pre cedente. Soit, comme exemple, = oo , J.= -* ,.,<- 1)1 1)_ rk n De meme, pour = oo , m 2 ( ^ r U T A = ^" 77 ~ rri ( ~ ^ ~ fli\ _ 2iri _ Ziri *" ~ T (fl + T\ 11T O T TV A V -- | -I- / ^ u i a ~\~ i 6 En representant par A , A" ces deux valeurs de A, on a 4 -^ / *-i^--2 (10), ou j dcris c. 21 162 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIOUIQUES. [25 en prcnant toujours le signe superieur pour cov w v positif, et le signe infe rieur pour wv o) v negatif. Solent u e ce que devient u en prenant = oo , w_ e ce que devient cette meme fonction en prenant = x ; on a evidemment m u_ e : u s = e-* (A ~ A "> x z = e s * 2 . On peut done s imaginer une fonction U donne e par les equations U=-^u_ s =e^ u (12). On verra dans la suite que u g est analogue a la fonction H (u) de M. Jacobi. II convient cependant, pour la symetrie, de conside rer, au lieu de u, la nouvelle fonction U qui vient d etre definie. Quoique suffisamment donnee par ces equations, nous allons en chercher encore une autre definition. Pour cela, il faut trouver la valeur de 1 integrale definie qui determine A, prise depuis la meme litnite inferieure jusqu a celle donnee par 1 equation mod (m, n) = T. Mettons, comme auparavant, m = r cos 6, n = r sin 6 : soient aussi Hi = w w i, T] = v v i. L equation pour la limite superieure se re duit a r 2 (fl cos 6 + T sin 0) (fl, cos + T, sin (9) = T , et celle pour la limite inferieure, a r=T. On trouve tout de suite _ i 2w log (Q cos + T sin (9) (fl, cos + T, sin 0) d0 Olcostf + Tsin 6>) 2 log ^ cos e + T sin 6 ^ ^ cos e + Tl sin ^ (entre e = 0S= 27r} 1 j dd _ i T - n tan T x - fl, tan 0\ 2T J o n+Ttan0 ( O + T tan + Q, + T, tan 0j 1 r *" d0 , T - n tan T, - flj tan 0\ _ T J _ j, H + T tan Vn + T tan H, + T, tan / 25] MEMOIRE SUB LES FONCTIONS DOUBLEMENT PERIODIQUES. 163 D abord ** d6 T! - Q x tan 6 *" M M 1 t ** d6 T! - Q x tan 6 = f J _ ^ fl + T tan flj + Tj tan 6 ~ J l + T tan n x + T x tan 6 dd = 20 - T 2 tan 2 e 2Q 2H On a de meme, en remarquant qu en changeant fi, T en T, fl, on change le signe de wv-tav, iliH- T! tan Oj + D un autre cote, TT. + an, " cela donne . n + n ou enfin f ^ c?# T! Oj tan # _ iri TrT 1 J - iir fl + Ttan0 fij + Tj tan ~ ~ H + Ti ~ cav - a> v et de m^me ( *" d6 (T - O tan (?) _ + wi J _ iff (O + T tan 6>) 2 ~ W$Ti en omettant seulement le dernier terme de 1 autre integrate ; ce que Ton peut verifier, an reste, en differentiant par rapport a H, T 1 dquation de TT + T tan e ~ fi + Ti 212 164 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. [2; On a done, en ajoutant les deux parties qui composent 1 integrale, r\r<r\ /- T <Y> <Y> / / \ IlT il + Ti T (a)v (0 v) II est facile de voir, en ecrivant DTP O X TV ii 1 li + 1 I/ que cette expression ne change pas de valeur en mettant flj, T a , H, T au lieu de H, T, n it TI, pourvu qu on change aussi le signe de i; cela doit eVidemment etre ainsi et peut servir de verification. Soit, pour un moment, MJ ce que devient u en prenant pour limite liquation w 2 + ?i 2 = T 2 , et supposons que dans la fonction u on ait pour limite 1 equation mod (m, n) = T. En re tenant la valeur de A, qui vient d etre trouve e, mais aussi a cause de 1 equation, consequence facile des resultats pr^c^dents, En eliminant t< n on obtient, entre M, Z7, une equation de la forme J/- =t -i2te* M dans laquelle H- 4 , T i-in Tl7r 1 + ~ 12 + Ti~ ftT T r-T-x (tu to v) ou enfin , 7T (t)U + ft) __ 7T O) + tri __ , " HT wu - 6> u * HT mod (wi; - wV) 25] MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. 165 En rassemblant tous ces resultats, on a le systeme de formules (A) 7 o/u) OT mod (ei/ - ft/v) 7T (WV + ft/I/) mod (wv (a v) u, u s , w_ e des fonctions de la forme mais avec des limites diffeVentes, savoir : mod (m, n)=T , T= oo , pour la fonction u ; m m- = m 2 , ft 2 = n 2 , m = oo , n = oo , = oo , pour ? = m 2 , w 2 = n 2 , TO = oo , n = oo , = co , pour M_. v A present, il importe de remarquer que la fonction u g est pe riodique a 1 egard de H, de la maniere d un sinus ou d un cosinus, mais ne Test pas a 1 egard de T ; de meme w_ 8 est periodique a 1 dgard de T, mais non a 1 egard de O. Quant a U, u, elles ne sont simplement periodiques ni a 1 egard de H, ni a 1 ^gard de T. Pour ddmontrer cela, considerons, par exemple, 1 equation = x n ( i + x (m, n (17) [m depuis - m jusqu a m, n depuis - n jusqu a n, m = oo , n = x , = x , le systeme (m = 0, n = 0) toujours exclu] ; en representant par u g ce que devient u g quand on ecrit x + fl au lieu de x, on a evidemment (TO, n)/ 1 \ (m + I, n) (m + 1, n) : n en admettant dans le premier produit la combinaison (w = 0, n = 0), mais excluant (TO + 1 = 0, n = 0), et excluant 1 une et 1 autre du second produit, Cette equation est de la forme x (TO + 1, n)J avec les memes limites que dans u; done u t : u s = (m + 1, ??) : n 1 + (-m, n)J 166 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIOD1QUES. [25 ou II se rapporte a la seule variable n qui doit s e tendre depuis n jusqu a n. Puisqu ainsi n ne re^oit jamais des valeurs qui soient comparables a m, chacun de ces produits se reMuit a 1 unitd, et Ton obtient u t : u t =A , on enfin u e =-u g ....................................... (18), eu posant x = - et remarquant que u est fonction impaire de x, ce qui donne A = -l. Soit de meme u" e ce que devient u s en changeant x en x + T. On a pareillement 1+ * (m, n + I)/ V (m, n)/ ou IT se rapporte a la seule variable m. Mais ici les produits ne se reduisent point a 1 unite; en effet, n f V i + - ou n i - 7- (m, n+1)/ V (m, n) = sin^(# + nT) : sin ^ nT. Mais quand la partie imaginaire de 6 est infinie, on trouve S in0 = l(e<><-,r<)=le > selon que la partie re elle de 6i est positive ou negative. Or la partie reelle de \L est de signe contraire a wv w v ; on a done n (m, de meme, (m, -n) et de la u"t = A"e-** x ut (19), Equation de la meme forme que celle que Ton obtiendrait en posant 25] MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. 167 et remarquant que u_ e est peYiodique a I e gard de T. On voit aussi qu en sommant par rapport a m, il est permis d exprimer u par un produit infini simple, dont chaque facteur est de la forme 7T sin -= (x + nT), mais qu on n arrive pas a un tel re sultat, en sommant par rapport a n. En effet, 1 equation u s = e~ ex 2 u_g fait voir que u s exprime par un tel produit ou chaque terme est de la forme sin ^ (x + wflt), multiplie par le facteur exponentiel e~ Sx \ On etablit de cette fagon 1 identite, a un facteur constant pres, de u s a H(u)\ mais, a present, pour eviter les longueurs, je ne me propose de considdrer aucun de ces produits infinis simples. II faut maintenant, pour le deVeloppement de la thdorie, introduire les trois fonctions qui correspondent au cosinus. Mettant, pour abre ger, .(20), j ecris (B) ^n !i + 1 7 r ( (m, n} (m, n) as (m, n) Zx = -l l + Limites. mod (m, n)=T, T=x mod (m, n) = T. mod (m, n) = T. mod (m, n) = T. (m, n)\ En representant par yx, y~gx, &c., ce que deviennent ces fonctions, et prenant pour limites (m 2 = m 2 , w 2 = n 2 ), (m 2 = m 2 , n" = n-), (m? = m 2 , n 2 = n 2 ), (m 2 = in 2 , n 2 = n 2 ), in a n ,, ou - = oo pour \ s x, &c., = oc pour \-gX, &c,, n m on a, en general, en mettant -/ au lieu de 1 une quelconque des lettres y, g, G, Z, Js = ^J s x = e-^J_ s x (21), ou Jgx est periodique a 1 egard de fl, et J-gX a 1 egard de T. C est cette equation remarquable qui definit la loi de pe riodicite des fonctions J, et de laquelle se deduisent presque toutes les proprietes de ces fonctions. 168 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIOD1QUES. [25 II est clair qu en changeant entre elles les quantites fl, T (quantites que nous designerons comme les fonctions completes), on ne change pas les fonctions yx, Zx, tandis que gx se change en Go;, et Gx en gx. On change de cette maniere en 8 et B en B (par exemple, y e x se change en y~gx, &c.). Cette consideration fait voir que toute propriete des fonctions J est double, et donne le moyen le plus facile pour passer d une propriete quelconque a la proprie te correspondante. On tire immediatement des definitions m ernes les Equations y ( x) = yx, g ( x) gx, G( x} = Gx, Z ( x) = Zx ; (C) gO=l, 0=1, , yo . i. II est facile de demontrer, de la meme maniere dont nous avons prouve la periodicite de J s x a 1 egard de H, que ces fonctions se changent 1 une en 1 autre, a un facteur constant pres, en changeant x en x+^l. Faisant done attention a 1 equation qui lie ensemble Jx et Jx, on obtient le systeme de formules y (x o \ 2t / G(x + l) = Z(x+^L) = Pour determiner A, B, C, D, posons x = ou x = En ecrivant, pour abr^ger, (D) ^ = 6*?- = on trouve Byx, CZx, DZx. fl. (22) d ou Ton deduit cette Equation de condition, On a de meme le systeme (23) (24). Zx, G (x + T) = e-W C yx, .(25). 25] MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. De la, si 169 ce qui donne (E) log q log q l = ?r 2 , il resulte ^ =y(iT) f / = 9 ^ \% ^ )> (<9c\\ . (v) > y (J T), ; ) d ou nous tirons 1 equation de condition 2" (^ T) Z (! T) = Q~* C27) En posant x = ^ T dans 1 equation pour y (a; + \ fl) et a; = \ H dans 1 ^quation pour y (x + ^ T), on obtient aussi y (iQ) g (i T) = ty (T) (^ O) (28), et 1 on deduirait cette meme Equation, ou une equation equivalente, des autres formules, de maniere qu on ne peut pas trouver d autres Equations de condition. Enfin, en rapprochant ces systemes, on obtient y (x + i fl + i T) = e *fl* *"- Y 4 ^, - g (tf + in + iT) = e ^ ^^ gaf, I (29X G (x + 1 II + T) = e*^ <n - Y) C" ^ ^ (a? + 4 H + 1 T) = c** <- Y D >, > avec les equations suivantes pour les coefficients, : V (30). J En rassemblant les Equations entre J(%&), J(^T), on a (31), c. 22 170 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. d ou Ton conclut les formules B"C" : A"D"=B D : A C = CD : AB = -\, A B : C U = -A"B" : C"D" = - f ( T) : Z A"C" : B"D" = ~A C : B D = A D : B C = -A D : B C = [25 (32), dont nous aurons bientot besoin. On peut passer maintenant a ce systeme general de formules, ou j ecris ( ) " au lieu de ( l) w : la, comme toujours, (TO, n) = mfl - nT ; y {x + (TO, n)} = ( - ) m+n yx, g [x+(m, n}}=(-} gx, G {x {- (TO, n)} = ( ) n Z{x + (m, w)} = Cb = I _ \ nm+ln JSx ( m, n) n -\ir&-\m ,, in* . \ / ^1 Z J y {x + (m, ri)} = (- } m+n g {x + (m, n)}=(- ) m (F) Z {ac+ (m, n)} = 3>DGx ; Sx (m, n) Jot* y {x + (m, n)} = (- ) m+n VA Gx, g {x + (TO, n)} = ( - ) w Z{x+(m, n)} = X = ( ) mn+^m+^n x (m, -n) ^ -^ y {a; + (m, n)} = ( - ) m+n XA Zx, g {x + (m, n)} = (- ) XB"Gx, G{x+(m, Ti)} =(-) n XC"gx, Z{x+(m, n)}= XD"yx. 25] MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. Soit x=0: 171 = ( ) n q-l? q-l y (m, n)= 0, g(m, n) = (-)6 , Z (m, n)= ; O, w) = ( - ) m+n > (m, n) = 0, ^ (m, n) = (G) \Tr / \ mn+^n n m 2 n n a in . o \ / !zi i y (m, n) = ( - } m+n V A , g (m, n) = ( ) m M^o-B , G (TO, n) = 0, G (m, n) = ( - } n ^ C", ^ (m, n) = ^o^D ; / "\ mn+$m+ Jn /^ Jm 2 Jm rt Jji 2 i?i . o \ ) "1 T. y (in, n) = ( - ) m+w X ^", g (m, n) = ( - ) m Xo^", On a de suite, en prenant les diffe rentielles des logarithmes des fonctions Jx, (H) g x G x Z x - (m, (B est le coefficient de a,- 2 , dans 1 exponentielle e~^ x - des equations (B) ; il n y a pas a craindre de le confondre avec le B qui entre dans les equations pour y (m, n), &c. : c est par hasard que j ai pris deux fois la meme lettre.) 222 172 MEMOIRS SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. Reduisons en fractions simples la fonction gx Gx : yx Zx. En ecrivant on obtient Done c est-a-dire [25 gxGx : yxZx = ^[L{x- (m, n)}~* + M {x - (m, n)}~ 1 ], L=g(m, ri) G(m, n) : y (m, ri)Z (m, n) = l, M= g (m, n) G (m, n) : y (m, n) Z (m, n) = B"G" - A"D" = 1. gx Gx : yx Zx = 2 [x - (m, n)}~ 1 - 2 {# - (m, n)}- 1 , gx Gx : yxZx = (y x : yx) - (Z x : Zx). (Nous allons bientot justifier, au moyen d un theoreme de M. Cauchy, 1 emploi de cette methode de decomposition en fractions simples qui ne s applique pas toujours aux fonctions transcendantes.) On obtient de meme f gx Zx : yx Gx = (yx : yx)-(G x : Gx), GxZx : yx gx = (y x : y x) - (g x : gx), (I) b 2 yzZx : gx Gx = (g x : gx)(G x : Gx), <?yxgx : GxZx = (G x : Gx)-(Z x : Zx), c 2 yxGx : Zx yx = (Z x : Zx) (y x : y x), en mettant, pour abreger, (J) Done, en eliminant les fonctions d^rivdes, G-x - Z*x = Z 2 x-g 2 x= c?y-x, 1 J ai ^crit - au lieu de - pour me conformer a la notation d Abel ; il est a peine necessaire de remarquer e e que c, e sont, en general, 1 une et 1 autre des quantites imaginaires. 25] MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. ce qui donne, en ajoutant, (K) b- = e 2 + c 2 , ou b = Je* + c 2 , en nous retiendrons desormais la lettre b dans cette signification. Puis 2 X = Z 2 X - C 2 V 2 X, (L) ,- J 173 Soient maintenant (M) x = yx : Zx, fx = gx : Zx, Fx=Gx: Zx. On obtient (N) (O) <f) x =fxFx, fx = - Fx=e?<t>xfx. Ces Equations sont precise ment les e quations fondarnentales d Abel ((Euvres, t, I., p. 143 [Ed. 2, p. 268]), et il en deduit 1 + qui sont les formules connues pour 1 addition des fonctions elliptiques. En effet, on peut e crire 1 dquation fix =fx Fx sous la forme ou, en mettant y = (f>x, (Q) 1 = X = J (I - tf^x} (\ + eWx} dy_ 174 MEMOIRS SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. [25 on peut done de ce point supposer connues toutes les proprie te s des fonctions elliptiques. On a, par exemple, et de la (R) - c y) qui determinent H, T en fonction de c, e. II parait au premier abord que T, H soient des fonctions parfaitement de terminees de c, e. J ai des raisons pour croire que cela n est pas precisement le cas, et que la question admettrait des deVeloppements inter- essants; mais je reserve ce sujet pour une autre occasion. On peut, a 1 aide des Equations entre y(l), &c., et les quantites c, e, exprimer cette suite de fonctions en termes de c, e. On de duit: (S) et de la (T) (*T) = A" = (- B" =-( C" = (-] qu on doit introduire dans toutes les formules ou entrent les quantites A, B, &c. Voici le the oreme de M. Cauchy (Exercices de Mathematiques, tome n. page 289) : " Si, en attribuant au module r de la variable z = r (cos p + i sinp) des valeurs infiniment grandes, on peut les choisir de maniere que les deux fonctions /*+/(-*) /*-/(-*) 2 2^ deviennent sensiblement nulles, quel que soit Tangle p, ou du moins que chacune de ces fonctions reste toujours finie ou infiniment petite, et ne cesse d etre infiniment 25] MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. 175 petite, en demeurant finie, que dans le voisinage de certaines valeurs particulieres de p, on aura pourvu qu on reduise le residu integral a sa valeur principale." On se rappelle que cela veut dire qu en supposant ces conditions satisfaites, la fonction fx peut s exprimer de la maniere ordinaire comme la somme d une suite de fractions simples, mais qu il faut e tendre d abord la sommation aux racines de liquation ^=0, dont les modules sont infe rieurs a une certaine limite qu on fait alors infinie. Soit, par exemple, f ..kax^+bx \ /QQ\ J^ p / x \ k*/> ou Tx, T^x ne contiennent que des facteurs qui sont des puissances entieres des fonctions yx, gx, Gx, Zx. En supposant que r n est d aucune des formes mod (m, n), mod (m, ri), mod (m, n), mod (m, n), mais, d ailleurs, une quantite infinie quelconque, on peut toujours e crire z = r (cosp + i sin p) (m, ri) + (34), ou est une quantite finie telle qu aucune des fonctions y9, g&, G6, Z6 ne s evanouit, m, n sont des entiers dont 1 un au moins est infini, mais qui varient depuis oc jusqu a oo , avec Tangle p. Soit 8 le degre de Tx, 8 a celui de T^x a Tegard de yx, &c. ; soit aussi X = 8j 8 ; on a, en general, une equation de cette forme, f y c irt (m, n) 2 f, Mm 2 /^AAw 2 ^Im+Jn Jf /OS\ JZ 6" (Ji* (* t -f \OO), ou F est fini. En supposant done que la partie reelle de a (m, n) 2 Xfl 2 TO 2 + XT 2 w 2 .. (36) \ * / \ / est negative quelles que soient les valeurs de m, n, on a et de meme Si, au contraire, cette partie reelle est positive, ces trois fonctions sont toujours infinies. II y a cependant un cas particulier a conside rer, savoir, celui ou cette partie reelle se reduit a Pm- ou Qn*, P, Q e"tant des quantites positives. II faut ici que le 176 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. [25 coefficient de n ou de m dans la partie re elle de Im + Jn s evanouisse. Enfin, si les parties re elles s eVanouissent entierement, ce qui ne peut arriver que pour a = 0, 6 = 0, X = 0, fa; est fini. On a done pour fx fonction irapaire, /*+/(- *)_ Q /*-/(-*) 2 2z (la seconde Equation a cause du ddnominateur infmi z). II est cependant certain que, daas plusieurs cas pour lesquels fx est fonction paire, on peut require fx en une suite de fractions simples : par exemple, yx : <gx est une fonction impaire que Ton peut, par ce qui precede, ddvelopper en suite de fractions simples ; en ecrivant x + ^T au lieu de x, on ddduit un pareil de veloppement pour Gx : Zx qui est fonction paire. Remarquons que quand la partie reelle de a (m, nf - XgH 2 ra 2 + XgT-n 2 est negative pour toute valeur de m ou n, la suite pour fx est toujours convergente. En effet, les numerateurs des fractions simples contiennent ce facteur de fz, e ja (m, ?)2 Q JA?, 2 ^jAw2 qui s evanouit pour les valeurs infiniment grandes de m, n. Dans le cas ou la partie reelle est positive, le developpement ne peut jamais etre vrai ; je crois qu en general il est vrai, dans les cas limites, quand la suite que 1 on obtient est convergente. II y a des exceptions cependant ; on en verra une en developpant $*x. Avant de passer aux exemples, developpons la condition pour que la partie re elle en question soit toujours negative. En supposant a = h + ki, on obtient pour cette quantite 1 expression i, / o / 2 \ 7 / (<u 2 a) 2 ) Ztcwo) r- 77 v- + v 2 .(JL/J ^ o/ / XTT mod (cou - w u)) > .................. (37), + n- <h (f 2 v 2 ) ZKVV -- p ^ - -> + 2mn [h (wv oy v } k (a>v + to v)}, , qui doit toujours rester negative. Cela donne, apres quelques reductions, la condition TO . ,o 2\7T(0)V+ to v } .} n- + K + r~z~~> /o\ / 9 : ~/o\ -- j~/ / -- r \ LI*" (i)v)ft + (aiv + oj v) K\ ((o- + CD 2 ) (tn- 2 + m*) mod (cav mv) >, o ^...(38). 25] MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. Les valeurs h = 0, k = 177 satisfont a cette condition, laquelle du reste (en considerant h, k comme les coordonnees d un point) est satisfaite pour tout point situe en dedans d un certain cercle qui inclut 1 origine, et dont on aurait 1 equation en remplaant le signe < par le signe = dans la formule (38). On obtient de cette fa9on une grande varie te de formules particulieres. Par exemple celles-ci : _ c j (m, n)i+b (m, n) Q f&a (m, n) 2 +b (m, n) n L\ / !/ [Y \ mn Jm n %a (m, n)*+b (m,n) Q \ dans lesquelles, pour trouver les limites de a, il faut faire X = 1 dans la formule (38). On a ensuite ce systeme de formules sans exponentielles, gx :jx= S[(-) n {x-(m, n)}- 1 ], Gx : jx = 2 [(-) m {^ - (m, n)}- 1 ], (V) a; : yar = ya; : gx - - Gx : gx =- ^ : gx =- jx : Ga? = - -) m+w {ar - (m, w)}- 1 ] ; [(-) 71 {^ - (w, n)}- 1 ], [(-) m+n (a? - (m, n)}- 1 ], [(-) {a- - (w, w)}- 1 ] ; = -6- 1 e- 1 [(-) 7ft {a? - (m, n)}- 1 ], gtf : Ga; = ie" 1 S [(-) m+n [x - (m, n)}~ 1 ], Zx : Go; = ib~ l S [(~) n { - (w, n)}- 1 ] ; jx : Zx = ic~ l e~ l S [(-) m+n {a? - (m, n)}- 1 ], g.r : Zx = e~ l S [(-) w {# - (m, n)} 1 ], Gas : Zjc= ic~ l S [(-) M { - (w, n)}- 1 ], dont quelques-unes ont 6t6 donn^es par Abel. II faut remarquer que celles de ces equations ou la fonction est impaire sout justifides par le theoreme de M. Cauchy, et que les autres se de duisent de celles-ci. On peut trouver de meme le deVeloppement des fonctions 1 Gx yx gx yx gx ya; gx Zx Gx c - 23 178 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. [2 (celles-ci, qui sont toutes impaires, ont e te deja conside re es ; nous venons de faire voir que le developpement est admissible) ; 1 Zx 1 yx gx Gx yx gx Gx V IT P* 7 1 chacune, except^ celles de la suite ~ , multiplied par un facteur exponentiel Par exemple, ~ (W) v yx gx Gx Zz (m n} . n)-+b (m, n) Mais on est conduit a un resultat beaucoup plus important en considerant, par exemple, le deVeloppement de (f> 2 x. Cette fonction contient non-seulement une suite de fractions simples, mais aussi un terme constant ; nous ecrirons ?-(m, n)}~* + M {x- (m, n)}- 1 ] (39). Pour determiner L, M de la maniere la plus simple, changeons x en x + Jfl + ^T. En faisant attention a la valeur de $>x = yx : Zx, on obtient de la + M {x - (TO, n)}- 1 ] ; L = - e~ 2 c~ 2 {a? - (m, n)} 2 (<H~ 2 > M=- e~* c~ 2 -r- [\x - (m, n)} 2 (<#c)- 2 ], x = (m, n), ou enfin L = - e~ 2 c~ 2 Jf ^ ctoc x ce qui donne (40). En integrant deux fois dx x-(m, n)} (41), 25] MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. 179 ou, a cause de log Zx = - ^Bx z + S log {x (m, n)}, rx rx dx\ tfxdx = %Ix t +e-*c-^\ogZx ........................ (42), Jo Jo en mettant, pour abrdger, (on n a pas ajoute de constante arbitraire, parce que Zx est fonctiori paire de x qui se reduit a I unite pour x = 0). Puis, on a (43). De la il est facile de determiner la valeur de 7. Soit, pour un moment, rx rx (/>/c = I $>-x dx, <f) f/ x = I (j) f x dx. Jo Jo Puisque <f>*(x + l)-(j> *x=0, on a = c) x Mais de liquation on d^duit $ t x + <f) / (O x) ^O, $ lt x (f> t/ (II x) = ou, en faisant x = ^ O, c est-a-dire et de la Mais on a deja Z(ar + fl) = e^ na; qr* Zx = &<#*+& Zx ; done enfin c est-a-dire pV 2 p 2 ^ ri 2 ^^ ............... (44). 232 180 MEMOIRE SUR LES FONCTIONS DOUBLEMENT PERIODIQUES. [25 Soit, pour abre ger, on a cette Equation, (X) Zx = e Jo Jo - qui exprime la fonction Zx au moyen de </>#. C est, en effet, la formule remarquable de M. Jacobi, que nous avons cite e dans 1 introduction de ce Me moire. En changeant seulement la notation, on a, d apres M. Jacobi, (4> a f a Fa <bx fx Fx} (). a ou, ce qui est la meme chose, . (48). 1+ e 2 c 2 fi 2 a (f> 2 x De la, en multipliant par e 2 c 2 , et faisant attention a la valeur de Zx, Z (x + a) _Z (x- a) _ ZZ a ^ 2e 2 c 2 </>q/q Fa <px Z(x+a) Z(x-a) Za ~ 1 + e 2 c 2 <patfx~ ............... ^ Ecrivant dans cette Equation a, x au lieu de x, a, et ajoutant, on obtient Z x Z a Z (x + a) Zx + ~Z^- Z(x + a) =e ~ C( t >a( f )a;( t )(a + ^ ..................... ( 50 >- En inte grant la merne equation par rapport a a, \ogZ(x + a) + logZ(x-a)-2 log Zx - = log (1 + e 2 c 2 </> 2 a fix) c est-a-dire Z (x + a) Z (x - a) = Z*x Z z a (1 + e 2 c 2 <pa <px) .............. ....(52), ou, ce qui est la me me chose, Z(x + a)Z(x-a) = Z 2 xZ 2 a+ eWfxfa ..................... (53), de laquelle on d^duit facilement, en e crivant x + ^fl, x + T, x + ^n + ^T au lieu de x, les equations complementaires (y (x + a)y (x-a) = fx Z 2 a - fa Z*x, g (+a) g (a? -a) =g 2 a;^ 2 a -c* -a = G*x Z a -a 25] MEMOIRE SUE, LES FONCTIONS DOUBLEMENT PERIODIQUES. 181 Quoiqu elle ne soit pas liee tres-dtroitement avec la theorie actuelle, on pent aj outer ici cette autre formule de M. Jacobi, qu il obtient en integrant par rapport a x, au lieu de a, x [*- e 2 c 2 6a fa Fa <tfx dx . . Z(x- a) Z a II (x, a) = I , y ./ - = i fog -= {-{-x-^- ......... (54) Jo 1 + e L c 2 pa <j> 2 x 6 Z (x + a) Za au moyen de laquelle il deduit des formules pour 1 addition des arguments ou des parametres des fonctions de la troisieme espece. On trouve aussi, dans les Fund. Nova, f , 1,1 ., i 1; , ,. //(f ,x . Z(x a) Z (v d) Z (x + y + a} quelques iormules deduites de 1 equation (49) pour exprimer -=-7 - . , - -=-} - 2 - L au moyen de la fonction (/> ; il serait facile de deduire des equations semblables pour les autres fonctions y. g. G. v O Note sur um integrals definie. Soient k^, k des quantit^s reelles dont la premiere est la plus grande; l = T = v + v i des quantite s quelconques, telles que wv w v ne s dvanouisse pas, et dcrivoris u= I -^ _ (55). + Ta; L inte grale u a toujours une valeur finie et determin^e, puisque le de nominateur ne devient jamais ze ro. On a evidemment -f J , r i i X ~ \ (o6) et 1 integrale indefinie est il faut passer de la a 1 integrale definie, entre les limites 0, oo . Soit O + T (as + k) II est facile de voir qu en faisant abstraction d un de nominateur toujours positif, A x est une fonction du second degre en x, et B x se rdduit a la quantitd constante (wv w v}(k l k). Le signe de B x est done toujours le meme que celui de wv w v; quant a celui de A x , puisque evidemment .4^=1, il est clair que si A est positif, A x reste toujours positif, ou change deux fois de signe quand x passe de a oc . Si, au contraire, A est negatif, A x est negatif depuis x = jusqu a une certaine valeur positive de as, x=a, et positif depuis x=& jusqu a x = oo . Considdrons d abord ce dernier cas. En representant par Lx la valeur principale de log x (cela suppose que la partie rdelle de x est positive), on obtient cette valeur de u, 182 MEMOIRS SUR LES FONCTIONS DOUBLEMENT PER1ODIQUES. [25 ou e est suppose une quantite infiniment petite positive. La somme des derniers termes se re duit a i ( B a A a \ = - arc tan \- arc tan ^-- e (59), T ou, comme a 1 ordinaire, arc tan x doit etre situe entre les limites + TT. Dans cette formule, A a _ e est une quantitd infiniment petite et ne gative; A a+f est une quantite infiniment petite et positive ; done, quand B a est negatif, les arcs se reduisent a TT , TT, et si B a est positif, a ^TT , ^TT . On a done 1 T ( H + TA;A 1 . W = T i ~n-urJ v rirl ( 6 )> ou il faut se servir du signe supdrieur ou infe rieur selon que wv w v est positif ou negatif. Dans le cas ou A est positif, si A x reste toujours positif, on obtient tout de suite Si J-a; change deux fois de signe, par exemple pour x = a et x = S, il faut introduire les corrections ^ ( - arc tan -~ + arc tan -~-) +~(- arc tan ^-?- + arc tan -^ } , 1 \ A a _ f A a+ J T V A s _ ( A S+ J qui se de"truisent 1 une 1 autre, en sorte que Ton a le meme rdsultat que si A x etait toujours positif. En se servant de la notation employee dans le Me moire, on a dans tous les cas ou le signe se de termine d apres celui de wv w v. En particulier, k rl/r rk ,!, 1 /O _i_ TT7 _ Q [ ax i r /iz -) i A [ tan a ^ J^ T . /Q + TtanaN Jo O 2 - T^ ~ 2flT 7rt Vn - T tan cj Je ne sais pas si Ton a cherche avant moi la valeur exacte de cette intdgrale ddfinie, qui est cependant la plus simple qu on puisse imaginer, et qui devrait, je crois, trouver place dans les livres eldmentaires. NOTA. Une partie de ces recherches a e te deja imprime e dans le Cambridge Mathematical Journal [24] ; je me suis borne" au cas ou n, T sont de la forme w, vi, ce qui simplifie beaucoup la determination des integrals de"finies doubles; mais la forme ge iieVale des resultats en est tres-peu affectde. 26] 183 26. MEMOIRS SUR LES COUKBES DU TROISIEME ORDRE. [From the Journal de Mathematiques Pures et Appliquees (Liouville), tome ix. (1844), pp. 285293.] CONSID&RONS d abord la surface du troisieme ordre qui passe par les six aretes d un tetraedre quelconque. Cette surface sera touchee selon chaque arete par un seul plan ; je dis que : " les plans tangents selon les aretes opposees se rencontrent en trois droites qui sont dans le meme plan, et chacune de ces droites est situe e entierement sur la surface." En effet, si nous representons par P = 0, Q 0, jR 0, S = 0, les equations des quatre plans du tetraedre, et par a, , 7, B des constantes arbitraires, I e quation de la surface ne peut avoir que la forme aQRS + SPRS + yPQS + BPQR = 0. Soient ^, (01, 1&, jfa ce que deviennent les quantitds P, Q, R, S, quand on change les coordonne es x, y, z en de nouvelles variables , 77, ; 1 equation du plan tangent se reduit facilement a la forme (^ - P) (gRS + ryQS + SQR) + (<a - Q) (<*RS + yPS + BPR) + (3ft- R) (aQS + SPS 4 SPQ) + &-S) (QR + PR + yPQ) = 0. Soient P = 0, Q = ; cette equation devient 1 + a = ; et de meme, si R = 0, 8 = 0, 1 equation devient De ces equations on ddduit les deux suivantes: a<am& + epaa + 7$<&S> + S$<MI = o. 184 MEMOIRE SUR LES COURBES DU TROISIEME ORDRE. [26 On conclut de la premiere, que la droite d intersection des deux plans tangents est situee sur la surface; et de la seconde, que cette droite est dans un meme plan, quelles que soient les deux aretes oppose es par lesquelles on a mene" les deux plans tangents. Ainsi le theoreme esfc de montre . En conside rant une section quelconque de cette surface, ou meme en supposant que P, Q, R, S ne contiennent que deux variables x, y, nous avons ce the orerne : "Etant donne e une courbe du troisieme ordre qui passe par les six points d inter section de quatre droites, les tangentes a la courbe en ces six points se rencontrent deux a deux en trois points qui sont les points d intersection de la courbe par une droite. Les tangentes qui doivent etre prises ensemble sont les tangentes en deux points A, A , tels que A est 1 intersection de deux des quatre lignes donnees, et A 1 intersection des deux autres lignes, points que Ton peut nommer opposes. "De rneme, si une courbe du troisieme ordre passe par cinq de ces points, de telle maniere que 1 intersection des tangentes a la courbe en deux points opposes soit situde sur la courbe, la courbe passe par le sixieme point, et par les points d inter section des tangentes mendes par les deux autres paires de points opposes." A present, posons une courbe quelconque du troisieme ordre et deux points A, A sur la courbe, tels que les tangentes en ces deux points se rencontrent sur la courbe (cela suppose que la courbe est de la sixieme ou quatrieme classe, et non pas de la troisieme). Un autre point B sur la courbe etant pris a volonte , les droites AB, A B rencontrent la courbe en H et h ; et les droites Ah, A H se rencontrent en un point B situe sur la courbe. Les tangentes en B, B , et de meme les tangentes en H, h, se rencontrent sur la courbe en deux points qui sont en ligne droite avec le point d intersection des tangentes en A et A . On peut dire que les deux points B, B , ou les deux points H, h, sont une paire de points correspondante a la paire A, A . Les deux paires B, B et H, h sont evidem- ment correspondantes 1 une a 1 autre, et, de plus, A, A correspond a ces deux paires, de la meme maniere que H, h correspond a A, A , et B, B ; de sorte que Ton peut dire que les trois paires A, A ; B, B ; H, h sont supplementaires 1 une aux deux autres. Soit C, C une autre paire de points correspondante a A, A ; je dis que B, B et C, C sont des paires correspondantes 1 une a I autre; et, de plus, si d un point P quel conque de la courbe, Von mene des lignes aux points A, A , B, B , C, C , ces lignes forment un faisceau en involution. En effet, considerons six points quelconques A, A , B, B , C, C dans le meme plan, et repre sentons par /, g, h les points d intersection de EC et B C, CA et C A, AB et A B, et par F, G, H les points d intersection de CB et C B , CA et C A , AB et A B . Nous aliens faire voir que le lieu d un point P qui se meut de telle maniere que les lignes menses aux points A, B, C, A , B , C forment toujours un faisceau en involution, est une courbe du troisieme ordre qui passe par ces six points, et aussi par les points /, ff , h, F, G, H. 26] MEMOIRE SUR LES COURBES DU TROISIEME ORDRE. 185 Solent a, a , , , 7, 7 les tangentes trigonome triques des inclinaisons des lignes PA, PA , PB, PB , PC, PC , sur une ligne fixe quelconque que Ton peut prendre pour 1 axe des x. Pour que ces lignes forment un faisceau en involution, nous devrions avoir la relation - 7 - 7 ) + ( 7 + 7 - a - a!) + 77 (a + a - - ) = 0, ou, ce qui revient a la meme chose, 1 une quelconque des quatre Equations Soient x p , y p , X A , y A , etc., les coordonnees de P, A, etc. Xp X A Xp Xg (Xp X A ) (Xp X B >) en representant par (PAB }, etc., les quantites telles que (PAR), L equation de la courbe peut done se mettre sous 1 une quelconque des quatre formes PAR . PBC . PGA + PA B . PB C . PC A = 0, PA R . PBC . PC A + PAB . PRC . PC A = 0, PAB . PB C . PGA + PA B . PBC . PC A = 0, PA B . PB C . PGA + PAB . PBC . PC A = 0, qui sont du troisieme ordre. La premiere fait voir que la courbe cherche e passe par les neuf points A, B, C, A , B , C , f, g, h. La troisieme ou la quatrieme fait voir qu elle passe de plus par le point F; la quatrieme ou la seconde, qu elle passe par le point G] la seconde ou la troisieme, qu elle passe par le point H. Ainsi la courbe passe par les douze. points A, B, C, A , B , C , /, g, h, F, G, H. On peut remarquer qu une courbe du troisieme ordre qui passe par dix quelconques de ces points, ou meme par neuf quelconques, pourvu que nous exceptions les combinai- sons de A, B, C, A , B , C , avec /, g, h, ou /, G, H, ou F, g, H, ou F, G, h, ne peut etre que la courbe que nous venons de trouver. On peut aussi remarquer en passant que si A, A , B, B , G, G sont les points d intersection de quatre droites, 1 dquation ci- devant trouve e est satisfaite identiquement, de sorte que la position du point P est absolument arbitraire. Cela e tant connu, je ne m arrete pas pour le dernontrer. Revenons au cas d une courbe donnee, avec trois paires de points A, A , B, B , C, C , comme auparavant, tels que B, B et C, C sont des paires correspondantes a A, A . Les C. 24 186 MEMOIRE SUR LES COURBES DU TROISIEME ORDRE. [26 points A, B, C, A , B , C , G, g, H, h sont situds sur la courbe ; ainsi F, f sont aussi sur la courbe (c est-a-dire que B, B et C, C sont des paires correspondantes), et les lignes menses par un point P quelconque de la courbe et les points A, B, C, A , B , C forment un faisceau en involution : theoreme ci-devant enonce. Conside rons une courbe du troisieme ordre, de la quatrieme classe (c est-a-dire, telle que d un point quelconque on ne peut lui mener que quatre tangentes). D un point sur la courbe, independamment de la tangente en ce point, on ne peut mener que deux tangentes. Prenons les deux points K, L sur la courbe, et menons les tangentes KA, KA , LB, LB touchant la courbe en A, A , B, B . II est clair qu en ce cas A, A et B, B sont des paires correspondantes de points. Mais le cas ge neral ou la courbe est de la sixieme classe est moms simple. Considerons, en effet, pour une telle courbe, les huit points A 1} A. 2 , A 3 , A 4 et B 1} B 2 , B 3 , B t de contact des tangentes menses par les deux points K et L. En choisissant A, A et B de quelque maniere que ce soit, parmi les points A l} A. 2 , A 3 . A 4 et B 1} B 2> B 3 , B 4 respectivement, le point B , qui doit entrer dans cette derniere se rie, ne peut pas etre choisi a volonte, mais est parfaitement deter- mine . En designant convenablement les points B, on peut toujours supposer que les paires correspondantes soient A-^A 2 ou A 3 A 4 avec B& ou B 3 B 4 ; A^A 3 ou A 2 A avec B^ ou B. 2 B t ; A^t ou A 2 A 3 avec BB ou B 2 B 3 . Cela suppose que A&, A 2 B 2 , A 3 B 3 , A,B,; A,B 2 , A 2 B,, A 3 B 4 , A,B 3 , A 3 B l} A 2 B t , A,B 2 ; A.B,, A,B 1} A 2 B 3 , A 3 B 2 , se rencontrent, chaque systeme, dans le meme point. II faut expliquer avec plus d evi- dence la correlation de ces huit points. Imaginons les six points A, A ; B, B ; C, C , tels que A A , BB , CC se rencontrent dans le meme point. Formons, comme auparavant, le systeme de points / g, h, F, G, H. Les propriete s de ce systeme de douze points sont tres-nombreuses. Non-seulement F, G, H ; F, g, h; f, G, h; f, g, H sont en ligne droite ; mais, en outre, les trois droites de chacun des onze systemes que voici se rencontreut dans le meme point: Af , Bg , Ch ; A f, B g , C h ; AA , Gg , Hh ; BB , Hh , Ff CC , Ff , Gg ; A f, BG, CH B g, CH , AF C h, AF , BG ; Af , B G, C H- Bg , C H, A F; Ch , A F, B G. (Cela est connu, je crois; au reste, pour le demontrer, considerons un parallelipipede, tel que gf, GF, AB, A B en soient quatre aretes paralleles, les autres aretes paralleles etant gA, A G, fB, BF; gA, A G, Bf, B F. Soient H le point de rencontre, a 1 infini, du premier systeme d aretes paralleles; C et C les points de rencontre, a 1 infini, des aretes des second et troisieme systemes respectivement; h le point de rencontre des quatre diagonales du parallelipipede; faisons la perspective de ce systeme, designant chaque point de la projection par la meme lettre : Ton voit sans peine que la figure plane, ainsi obtenue, a les proprie tes en question.) 26] MEMOIRE SUR LES COURSES DU TROISIEME ORDRE. 187 A present, en examinant la figure, on voit qu il est permis de prendre pour A 1} A. 2> A 3 , A 4 les points A, A , F, f, et pour B 1} B 2> B 3 , B 4 les points B, B , G, g, pour que les systemes A l} A 2 , A 3 , A^\ B lt B 2> B 3 , B 4 aient la correlation ci-devant trouvee. Le systeme C, C , H, h est supplementaire a ces deux-ci. Toutes les proprie te s que je viens de trouver sont telles qu il y a a chacurie une proprie te correspondante que Ton peut obtenir par la thdorie des polaires rdciproques. Nous avons ainsi des proprie te s non moins mtdressantes des courbes de la troisieme classe et du sixieme ou quatrieme ordre. En prenant pour la courbe du troisieme ordre 1 ensemble d une section conique et d une droite, pour que les tangentes en A, A se rencontrent sur la courbe, il faut que A, A soient situees sur la section conique de telle maniere que A A passe par le p61e de la droite a 1 egard de la conique. Alors B, B ; C, G etant pris de la meme maniere, BC , B G et BC, B C se rencontrent sur la droite. Les lignes tirees d un point P quelconque de la conique, ou de la droite a A, B, C; A , B , C , forment un faisceau en involution. Le premier the oreme et la premiere partie du second sont tres-bien connus ; je ne sais s il en est de meme de la derniere partie de ce the oreme. ADDITION. La droite PP , menee par deux points P, P d une courbe du troisieme ordre, qui correspondent ton jours d une paire correspondante donne e de points A, A , est to u jours tangente d une certaine courbe de la troisieme classe. Pour demontrer ce the oreme, imaginons une paire fixe B, B de points qui corre- sponde a la paire donnee A, A . Soient P = 0, Q = 0, R = 0, S = les Equations des lignes qui joignent A, A avec B, B ; 1 dquation de la courbe donne e du troisieme ordre peut se mettre sous la forme On peut toujours supposer, sans perte de ge ne ralite , que liquation soit identiquement vraie, car chaque Equation de la forme P = peut etre censde cori- tenir une constante arbitraire qui en multiplie tous les termes. Done, en faisant on peut toujours supposer /~1 i O O f~l A Ql Vj/T*J =: ", ty = " *J, 242 188 MEMOIRE SUR LES COURBES DU TROISIEME ORDRE. [26 et 1 dquation de la courbe se transforms en e-R + s ce que Ton peut e crire aussi sous la forme a 78 +a = .............................. (3), _ "*" - - ( + \R) ( + pR) "" ( - + vS) ( - 6 + P S) en determinant convenablement les constantes X,, //,, v, p. f En effet, en reduisant, les deux Equations deviennent identiques au moyen des quatre conditions 8 = k\fi, 7 = kvp, ^ S-S = k\p (vp + v + p) , > .............................. (5), 7 a = kvp ou k est une quantite arbitraire, de maniere que des quantites \ p, v, p, il y en a une seule qui peut etre prise a volontd De 1 equation (4) Ton deduit tout de suite cette autre forme, (\ + i) xQi + i)i i rp( y + i) v (p + i)-]_ -- - ~ (7) et de la nous voyons que les points donnes par les deux systemes d equations (0 + \E = 0, -0 + vS=0), correspondent toujours 1 un a 1 autre et aux points donnas par les deux systemes (0 = 0, = 0), I (8), (0 = 0, S=0)J c est-a-dire aux points A, A . On trouve assez facilement pour 1 Equation de la droite men^e par les points deter mines par les deux systemes (7), points que 1 on peut prendre pour P et P , (fMv-p\)e + \fji(v~p)R + vp(\-^)S = ..................... (9), on C0 + AR + BS = Q (10), en faisant pC av p\ , ) .(11). pB = vp (\ ft). 26] MEMOIRE SUll LES COURBES DU TROISIEME ORDRE. 189 Eliminons des equations (5) et (11) les cinq quantites X, p, v, p, p. Pour operer de la maniere la plus elegante, formons d abord les equations identiques .(12), [fj,v p\ vp (X fi)] (v p) (Xyu, + X + /A) [/j,v - p\ + \/Ji(v p)] (X fj,)(vp + v + p) = (i/X pp) [\p (v p) vp (X fju) + 2 (JJLV p\)], (v p) 2 vp (X fjif = (fJiv p\) (\v ftp), ce qui donne k = - \fjbvp (v\ fip) (2(7 + A B), k = \LbVp (v\ itp) C, p -(13), et de la ou, toute reduction faite, = (14), B- CS) -AG(C-Ba)-BC(C+A) = .(15); et puisque cette Equation est du troisieme ordre a 1 egard des quantites A. B, C, il est evident que la ligne donnee par 1 equation (10) est toujours tangente a une certaine courbe de la troisieme classe. En supposant 7 = 0, 8 = 0, ce qui reduit la courbe du troisieme ordre aux lignes R = 0, 8 = 0, (- a) 6 - 6R-aS= 0, l e qu ation ( 15 ) se partage dans les deux Equations = 0, A(C-B)a + B(C + A) = Q (16), et la courbe de la troisieme classe se rdduit au point (R = 0, 8 = 0) et a une conique. Cela est un the"oreme connu, car Ton sait bien que si A, A sont des points quelconques sur deux droites donnees a, a. , et B, B deux autres points de termine s, sur ces droites, de maniere que I mtersection des lignes AB , A B soit toujours sur une ligne droite donne e, les deux lignes a, a sont divise es homographiquement, B, B dtant deux points corre- spondants; et de plus, que la ligne qui unit les points correspondants de deux lignes divise es homographiquement, est toujours tangente a une certaine conique. Quant au point d intersection des lignes a, a , que nous venons de trouver comme formant avec la conique une courbe de la troisieme classe, on observera que, dans notre the orie, non- seulement les points des lignes a, of se correspondent, mais que le point d intersection des lignes a, a correspond a tous les points de la troisieme droite. La conique touche les deux droites a, a ; cela n a pas, je crois, d analogie dans la the orie ge ne rale. 190 [27 27. NOUVELLES REMARQUES SUR LES COURBES DU TROISIEME ORDRE. [From the Journal de Mathematiques Pures et Appliquees (Liouville), torn. x. (1845), pp. 102109.] JE me propose de developper ici quelques consequences de la theorie que j ai donne e, il y a quelques mois, dans ce Journal 1 , [26], des points correspondants des courbes du troisieme ordre. Rappelons d abord la signification de ce terme et ajoutons-y quelques nouvelles definitions. On dit que les points A, A sont correspondants quand les tangentes a la courbe en ces points se rencontrent sur la courbe. En conside rant les points correspondants A, A et les deux autres points correspondants B, B , on dit que ces paires correspon dent, quand les points d intersection H, h de AB , A B ou AB, A B sont situes sur la courbe. Les trois paires A, A , B, B , H, h sont nommees paires supplementaires ou systeme supplementaire. On dirait de meme que les deux systemes A A , BB , Hh et A^A\, -Bi-B i, HJi : sont des systemes supple mentaires correspondants, si, par exemple, A, A et A!, A\, &c., formaient des paires correspondantes. On peut nommer quadrilatere inscrit le systeme de quatre droites qui passent par les points supplementaires AA , BB , Hh, et conique d involution chaque conique tangente a ces quatre droites, ou, en d autres termes, inscrite dans un quadrilatere inscrit. II est inutile d expliquer ce que veulent dire coniques d involution correspondantes, ou quadrilateres correspondants, ou 1 expression conique correspondante a une paire donne e de points correspondants, &c. Demontrons les thdoremes suivants : THEOREMS I. " Les tangentes mene es par un point P de la courbe a trois coniques d involution correspondantes forment un faisceau en involution." 1 Voyez page 285 du tome ix. 27] NOUVELLES REMARQUES STJR LES COURBES DU TROISIEME ORDRE. 191 Chaque conique peut se reduire a une paire de points qui correspondent aussi aux coniques (ce qui justifie la denomination que nous avons donne e a ces coniques). Considerons un quadrilatere inscrit A A , BB , Hh, la conique d involution tangente aux cote s de ce quadrilatere, et deux paires LL , MM de points correspondants, ces paires etant correspondantes 1 une a 1 autre et a la conique d involution. On salt que les lignes mene es d un point quelconque, et ainsi du point P de la courbe, par les points A, A , B, B , forment avec les tangentes a la conique mene es par ce m6me point, un faisceau en involution. Mais PA, PA , PB, PB , PL, PL , et de meme PB, PB , PL, PL , PM, PM , forment aussi des faisceaux en involution ; done les deux tangentes forment, avec PL, PL et PM, PM , un faisceau en involution. De meme, avec une conique correspondante a la premiere, PL, PL , PM, PM et les deux tangentes forment un faisceau en involution ; done PL, PL et les quatre tangentes forment un faisceau en involution, et de meme en introduisant la troisieme conique. THEOREME II. " On peut circonscrire a une conique d involution donne e une infinite de quadrilateres inscrits correspondants au premier." Soient A, A , B, B , H, h comme auparavant ; et A 1} A\ une paire de points correspondants qui correspondent a ceux-ci; en menant par A lt A\ des tangentes a la conique qui se rencontrent en B l} B\, H lf h 1} on voit d abord que les lignes PA, PA , PA l} PA\ et les tangentes a la conique forment un faisceau en involution; et recipro- quement, chaque point P qui satisfait a cette condition appartient a la courbe. Mais en prenant, par exemple, pour P le point B 1} les lignes PA l} PA\ deviennent iden- tiques avec les deux tangentes, ce qui satisfait a la condition d involution. Done B!, B\, H l , hi appartiennent a la courbe, ou A 1} A\, B 1} B\, H lf h t forment un quad rilatere inscrit correspondant au premier ou a la conique. THEOREMS III. " Les centres d homologie de deux coniques d involution correspon dantes forment un quadrilatere inscrit correspondant aux coniques." Conside rons les deux coniques et une troisieme conique quelconque. Chaque point P pour lequel les six tangentes forment un faisceau en involution appartient a la courbe. En prenant pour P un centre d homologie des deux premieres coniques, les deux paires de tangentes deviennent identiques, ce qui satisfait a la condition d invo lution ; done les six centres d homologie sont sur la courbe, ou ces six points sont les sommets d un quadrilatere inscrit qui correspond aussi aux coniques. Re ciproquement, THEOREME IV. " Le lieu d un point P qui se meut de maniere que les tangentes mene es par ce point a trois coniques donne es quelconques, forment toujours un faisceau en involution, est une courbe du troisieme ordre qui passe par les dix-huit centres d homologie des coniques prises deux a deux, ces centres formant six & six des quadri lateres inscrits correspondants." La demonstration analytique de la premiere partie de ce theoreme, quoique longue, me parait assez intdressante pour trouver place ici. Pour plus de symetrie, prenons j* p , p pour les coordonne es inde finies d un point, et representons par T = 0, T = 0, T" = 0, 192 NOUVELLES REMARQUES SUR LES COURBES DU TROISIEME ORDRE, [27 les Equations des trois coniques, T, &c. etant des fonctions homogenes de la forme , &c. Soient -, ^ les coordonnees du point P; mettons, pour abreger, z z U=Ax 2 + By 2 + Cz 2 + 2Fyz + ZGzx + 2Hxy, (de maniere que W=Q serait liquation de la ligne polaire du point P). Nous avons ur - W 2 = o, pour liquation des deux tangentes mene es par le point P a la conique. Cela est probablement connu. II est clair d abord que cette Equation appartient a une conique qui a un double contact avec la conique donne e, et par la forme a laquelle nous allons require cette equation, on voit ensuite qu elle appartient a un systeme de deux droites qui passent par le point P. En de veloppant et e crivant EG- F*=a, CA- G 2 = b, AB - H* = c, GH-AF=f, HF-BG = g, FG-CH = h, on trouve, en effet, a (yf - zrif + b(z%- xtf + c (xrj - yf et de la, en posant & = bz- + cy 2 - 2fyz, = ayz - gxy - hxz +fa- , 53 = ex 2 + az 2 - 2gxz, (& = bzx - hyz - fxy + gy 2 , (& = ay 2 + bx 2 - 2hxy, |^ = cxy -fxz - gyz + hz\ on obtient + or^ 2 - 2^ - 2<ffif#- 2^77 = o, ou, en transportant 1 origine au point P, ggi +23^ -2^77 =0; et de rn^me, pour 1 equation des tangentes, par ce meme point aux deux autres coniques On obtient done, pour la condition que ces lignes forment un faisceau en involution, = 0, 27] NOUVELLES REMARQTJES SUR LES COURBES DU TROISIEME ORDRE. 193 en repr^sentant de cette maniere le determinant forme avec ces neuf quantites. Formons d abord la fonction ov-mr. Cela se reduit a z [- 2 (fc) x 2 y - 2 (eg) xy* + (ca) y*z - 2 (fa) yz 2 - 2 (bg) xz 2 + (be) x 2 z + 4,(fg) xyz + (ba) z 3 ], , = (/V -/V), &c. Multipliant par p^, formant ensuite les quantitds analogues et ajoutant; ^crivant aussi c est-a-dire (cfg) pour le determinant forme" avec les neuf quantitds c,f,ff, c ,f,g , c",f ,g", on obtient d abord les termes qui se d^truisent ; les autres termes contiennent z 3 comme facteur, et en e cartant cette quantity, Ton obtient en derniere analyse liquation (cbf) a? + (acg) y 3 + (bah) z 3 + 4 (fgh) xyz + [2 (agf) + (cah)] fz + [2 (%) + (abf)] z*x + [ 2 (cfh) + (beg)] x*y + [2 (afh ) + (abg)] yz* + [2 (bgf) + (bch)] zx* + [2 (chg) + (ca/)] xf = (ou Ton pent, si Ton veut, dcrire z = 1). C est liquation d une courbe du troisieme ordre. OBSERVATION. Cette methode peut etre utile dans 1 investigation d autres problemes relatifs aux coniques. Par exemple, la question de determiner les centres d homologie de deux coniques donn^es revient a celle-ci: satisfaire identiquement a liquation r 2 + 2 $ rt + 2& K + 2^ fr) = 0, parce que, en effectuant cela, il est facile de voir que -, - sont les coordonnees du z z centre cherche. Ecrivons a + ka = a, c + kb = b, &c., Ton obtient les six Equations b^ 2 + cy 2 2ft/z = 0, ayz gxy hxz + fee 2 = 0, ex 2 + az 2 2gzx = 0, \>zx hyz iyx + gy* = 0, a?/ 2 + b# 2 2hxy = 0, cocy tzx gzy + hz* = 0, c- 25 194 NOUVELLES REMARQUES SUR LES COURBES DU TROISIEME ORDRE. [27 dont les trois dernieres se ddduisent des autres. On peut, de ces six Equations, eliminer les six quantitds a?, y 2 , z 2 , yz, zx, xy, conside rees comme independantes ; on obtient ainsi, toute reduction faite, (abc - af 2 - bg 2 - ch 2 + 2fgh) 2 = 0, equation qui determine la quantite k ; les trois premieres equations deviennent alors /yi sit e quivalentes a deux, qui suffisent pour determiner les rapports - , - . II serait facile z z de rapprocher cette solution de celle que Ton deduit de la theorie des polaires reci- proques. Par exemple, 1 equation qui vient d etre obtenue entre les quantites a, b, . . . est precise ment celle qui exprime que la fonction se divise en facteurs lineaires. Remarquons encore que, dans la geometric solide. liquation appartient au cone, ayant le point P pour sommet, et circonscrit a une surface du second ordre qui a pour equation T = 0. C est un cas particulier d une autre formule que voici : " En representant par ^ une fonction lineaire des trois variables , 77, (ou de quatre variables sans termes constants), et par P la meme fonction de x, y, z, liquation appartient au cone ayant pour sommet le point dont les coordonnees sont x, y, z, et passant par la courbe d intersection du plan P = 0, et de la surface du second ordre T = 0. Et de meme pour deux variables." Je finirai en citant un theoreme de g^ometrie du a M. Hesse 1 , qui a quelques rapports avec le sujet que je viens de traiter : " Le lieu d un point P, qui se meut de maniere que ses polaires par rapport a trois coniques donnees se rencontrent dans le meme point, est une courbe du troisieme ordre ; et encore cette courbe ne change pas quand on remplace les coniques donndes U=Q, U =0, U" = 0, par trois nouvelles coniques de la forme Cette courbe du troisieme ordre est tout a fait distincte de celle que j ai consideree. 1 Voyez le Journal de M. Crelle, tome xxvni. 28] 195 28. SUE QUELQUES INTEGRALES MULTIPLES. [From the Journal de Mathematiques Pures et Appliquees (Liouville), tome x. (1845), pp. 158168.] J AI donne, il y a trois ans, dans le Cambridge Mathematical Journal, [2], une formule assez singuliere pour I mtdgrale multiple I ... daCi prise entre les limites donne es par 1 equation + + ... = 1, la fonction <f> dtant seulement assujettie a la condition de ne pas devenir infinie entre les limites de 1 inte gration. L expression que j obtiens est une suite infinie, dont le terme ge ne ral est de cette forme, En appliquant ce resultat a un cas particulier, j ai obtenu 1 integrale a n variables analogue a celle qui exprime le potentiel d un ellipsoide homogene, pour un point exte rieur. J ai depuis cherche a ^tendre ces re sultats au cas d une densite variable et e gale a une fonction rationnelle et entiere des coordonnees, et d une loi d attraction selon une puissance quelconque de la distance (toujours a n variables) ; mais quoique j aie r^ussi a effectuer cette gdndralisation, mes formules etaient si confuses et si infe rieures a celles que donne la belle analyse de M. Lejeune-Dirichlet, que je ne les ai jamais publides. Cependant, en revenant il y a quelques jours sur ce sujet, en 252 196 SUE, QUELQUES INTEGRALES MULTIPLES. [28 me fondant sur une integrate plus ge ne rale, j ai trouve que la question e tait a peine plus difficile que dans le cas d une densite constante, et se laissait traiter exactement de la meme maniere. J ai re ussi de cette fa9on a exprimer I inte grale cherche e au moyen d une seule intdgrale ddfinie abe lienne, et de ses coefficients differentiels relatifs aux constantes qui y entrent ; et il m a paru que les formules que j ai ainsi obtenues pourraient n etre pas tout a fait indignes de 1 attention des gdometres. Conside rons 1 inte grale multiple a n variables F, donne e par 1 dquation V = \ dr t . . . ccj 2 " 1 " 1 " 1 . . . (/ termes) #/+i 2a /+i ... (n f termes) < (a x xj, . . .), oil les variables x 1 , ... doivent recevoir des valeurs rdelles quelconques, positives ou negatives, qui satisfassent a la condition et Ton suppose de plus qu il est permis de deVelopper (sous le signe integral) la fonction < suivant les puissances ascendantes des variables ac l ,.... En faisant ce deVeloppement, il est clair que les termes qui contiennent des puissances impaires d une ou de plusieurs des variables se ddtruisent par I inte gration ; done, en ne faisant attention qu aux termes qui contiennent seulement des puissances paires, on a ce terme ge ne ral, . . . da /+1 ou p = T! + . . . ; il est a peine necessaire de remarquer que, dans 1 expression il faut prendre f termes tels que [2^ + I] 2 * 1 4 " 1 , et n f termes de 1 autre forme [2r/ +1 ] 2r /+i, et ainsi dans tous les cas semblables. L inte grale qui entre dans cette formule a etc" trouvee, comme on sait, par M. Dirichlet. Sa valeur est ou 28] SUR QUELQUES INTEGRALES MULTIPLES. 197 Le terme entier devient, apres quelques reductions tres-simples, on Ton dcrit, pour un moment, L, =(2n +3) (2n +5)... (2r : +2a x +1), + l) (2ry +1 + 3)... r /+1 /+1 puis on le transforme en En effet, les symboles -^ et A, 3 -^- dtant conversibles. on a et ainsi de suite. 3 _ ^ j *<ia ^5 j \daj \daj En prenant la somme de tous les termes qui correspondent a une meme valeur de p, on a En posant puis, en observant que ^) doit s ^tendre depuis jusqu a oo , on a, pour 1 integrale cherchde, _ _ \da l "da f \ 1 dh [/ /2P \1 /^YJO = oo / ^7 JL. f \\ \ V ... A /+1 ... ^O p=0 ^2^[p]Pr(p + A+/+in+l) " ; ^| ce qui fait voir que I inte grale V depend de cette seule expression, 198 SUR QUELQUES INTEGRALES MULTIPLES. [28 On peut remarquer, en passant, que cette quantite satisfait a cette equation diffeYentielle, L GJ i CL t dt V dt ^+(2k + 2f+n +!)- dt 2 t at M. G. Boole, de Lincoln, a d^duit une equation semblable de mes formules dans le Mathematical Journal. C est a lui qu on doit 1 introduction dans 1 integrale propose e de cette quantite" t, ce qui, au reste, n est pas d une grande importance ici ; mais j ai cru devoir la conserver, a cause de cette Equation meme, qui pourrait conduire a des resultats inteYessants. A present, soit <f) (i - x^t, ...}= ,. --Ty, \^r s > ou s est un nombre entier ; je pose, pour abreger, \n s = i, k +f+ s = a, ce qui donne r (p + k et ainsi Le cas de cr = 0, qui est le plus simple, est, en effet, celui du Memoire cite, et a ce cas on peut reduire celui de cr entier negatif; il y a meme deux manieres d effectuer cette reduction. Representons, en effet, la valeur de U par cette notation plus complete Ui ; on obtient tout de suite ./I d TT a 9~ -f /2cr 21 _ Ui (t dt a4- JT.V la seconde desquelles equations se deduit de cette formule facile a demontrer, -L ?\n r -iT-^r* 1 1~ r o i \i-a- L" "J L" "J /,, 21 \j k tV -(- . ..) V"l ~r / Nous pouvons done, dans la suite, ne consideYer que le cas de cr entier positif. Commencons par operer la transformation que voici ; en determinant par I e quation 28] SUE, QUELQUES INTEGRALES MULTIPLES. 199 je pose ce qui donne =, +..., ou il faut remarquer que ce , contenu en V, ne doit pas etre affecte des symboles . de V, de maniere qu il faut dcrire fo 8 +...) -_ >o 2 [p + i + <rp+* +1 [p]P U + ^ da* "J {<*> formules qui se pretent mieux aux reductions, quoique plus complique es en apparence. Ecrivons d abord _k __ d^ __ _ _ __ _ _ _ " 1 " " + En de veloppant la p me puissance de ce symbole selon les puissances de A, on a ( / \ a LjW A qui doit s appliquer a <^ j Considerons 1 expression et mettons, pour un moment, ( cette expression se trouve rdduite a d? laquelle, par une formule deja citde, devient 2^-2? [i + p-q_ i]p- [,: +p-q 200 SUR QUELQUES INTEGRALES MULTIPLES. [28 c est-a-dire 2*-* [i +p-q- l]p- [i + p - q - frp-* ^ ^ Le terme ge"ne"ral de U, en faisant abstraction du facteur r / -\ p / g 2 . y se r eduit a ... r t - i a 2 [p + ^ + erfHh-1 [p - ? ]IMI [g] L considerons la partie de ce terme qui est de 1 ordre 6 en l lt ... , elle devient x (i + P - q - Soit, en general, une fonction homogene de 1 ordre 26 des lettres a lt ... : on a et de la, en repetant toujours 1 operation A, et faisant attention a ce que A, A 2 , , &c., on obtie ry i n _ -\ _ i -\q sont des fonctions homogenes des ordres 201, 20 2, &c., on obtient (S) \ri\1 OU tl* , r. x,.^_^ fl = r . nxrl" .-.-> ^ "" L* + ^ + # ~ X - 1]< 1 x [ + p - 6 - i*]^ A^ (W + . . .)* . depuis X = jusqu a X = q. Puis, pour le terme general de U, ~) P q r i ^ i f) _ \ _ i ie -^^ 28] SUR QUELQUES INTEORALES MULTIPLES. 201 le dernier facteur etant independant de q, q doit s e tendre depuis jusqu a p. Mais a cause de [q X]^~ A qui devient infini pour q<\ on pent faire etendre q depuis q = X jusqu a q = p ; ou, en ecrivant q-\ = q f , p-\=p f , i] s etend depuis jusqu a p . Le facteur a sommer est done si, pour un moment, Ton pose C=i + p -%n, M=i+p-0-%n. Cette somme se reduit a c est-a-dire a [0 - et nous avons ainsi le terme general [0 _ ^ [6] e [\] A [6 - p] e -P Ecrivons cela devient ou M=0-X, C = i (l ^~ + ---> p doit s &endre depuis - oo jusqu a 0. Mais a cause de [p Y, qui devient infini, pour p ndgatif, on peut Fetendre seulement depuis jusqu a 0, ou meme seulement depuis jusqu a M, a cause du facteur [M]? . La somme se rdduit a et le terme general devient Ecrivons eiifm C. 202 SUR QUELQUES INTEGRALES MULTIPLES. [28 le terme devient / \A i \.f _ - _ i > A* (I a - 4- Y +K - L/ 2 A II f aut que K soit toujours positif, car autrement le terme s evanouit a cause de A^jC*! 2 + ...) A+(C ; mais pour /c plus grand que a-, le facteur [ cr 1]* s eVanouit, done s etend seulement depuis jusqu a <r. Soit k l + ... = /c, et conside rons les termes de ^ (^ + ...)^ +K qui contiennent ar fc ,...; ces termes seront de la forme ou <?i + = X, c est-a-dire 7 1 JJ / g,+*i a 2*, iq^-l^^^ ou, en re duisaut, [k^ (ar +...)" Soit (X) = (-) A i -" ^--- + &c - ik+ . -X; et de la, de maniere que le terme de U devient Prenant la somme pour X, a 1 aide de (ce qui suppose Z + w > 0), on obtient 28] SUE, QUELQUES INTEGRALES MULTIPLES. 203 2 2 " Or]* [K - a - 1]" a^ . . . 1 /. d \ fcl ._ P (1 - u) u i+K ~ l du " ou, mettant ^ = -^ , . . . , a,*., 1 J c " x -~ en r^tablissant le facteur constant _,... .^. de t^, le terme general de cette quantite est 1 (i) r* ^ l _ ... K+ ,. _ 3 _ - u u^ du PT(z) "[2^...:^ X h 1 ...( ni dh,) " ou &j , &c., sont des entiers positifs quelconques qui satisfont a et K peut s e tendre depuis jusqu a cr. II faut observer qu en diffe rentiant par j-r , &c., on ne doit pas considerer comme fonction de Aj . . . , ce qui est cependant ndcessaire dans liquation entre V et U. 262 204 [29 29. ADDITION A LA NOTE SUR QUELQUES INTEGRALES MULTIPLES. [From the Journal de Math&matiques Pures et Appliquees (Liouville), tome x. (1845), pp. 242244.] ON de montre, au moyen des formules que j ai donnees sur ce sujet, [28], une pro- priete remarquable de 1 inte grale multiple r J prise entre les limites ^i 2 ^_ 2 =i En effet, en supposant toujours que Ton peut developper la fonction <f> selon les puis sances entieres et positives de t, 1 inte grale V peut s exprimer sous la forme ou 7^.1. , f 1 ~v . . . -7 I llrt -7," I I lt>n T7 III ... ^ doifl \ dh^J \ dhj Q P =* { 1P _ \ u = bp-v V2* T (p + 1) P (p + k +/+ ^n + 1) ^ (ai ) Soit 29] ADDITION A LA NOTE SUR QUELQUES INTEGRALES MULTIPLES. 205 entre les mernes limites que V. On a En multipliant par et integrant depuis T=0 jusqu a T= 1, on obtient o /I 7T2\fc+/ puisque en general On a done 1 equation ,. /7 ,. 1 k* .7 / Mettons la valeur de f7 qui en resulte, dans 1 equation entre V et U; faisons aussi t=l, ce qui ne nuit pas a la generalite. On a, en rassemblant les formules, V = I dx 1 . . . dx n . . . # 1 2ai+1 . . . A/ +1 2a /+i ... (f)(a 1 x ly . . .), W = I dx l . . . dx n ... (! x^T, ...}, - 4) ( h sT- (H ce qui ^tablit ce theoreme : La suite des integrates F s exprime au moyen des coefficients differentiels par rapport a ^ . . . h n et a^ ... a/ de la seule integrale f T*^ 1 (1 - T 2 ) fc+ / {dx l ...dx n ...<t> (a, - ^T, . . .). Jo 3 En supposant que les indices dans V soient tous pairs, on a /= 0. Les symboles -T , ... n entrent plus dans 1 expression de V. En changeant la fonction </>, on a ces formules plus simples, 206 AUDITION A LA NOTE SUR QUELQUES INTEGRALES MULTIPLES. [29 V = I da\ . . . dx n #f a i . . . x n - a " (f> (*j , . . . , a rt ), W = fdx, . . . dx n $ (Tx lt ... , Tx n ), ou V s exprime an moyen des coefficients diffeYentiels par rapport a A-,, ... , h n de I integrale P r pl+l (1 _ T y t dxi dXn $(Tx lt ..., Tx n }, Jo J de maniere que nous avons trouve des relations entre des integrales definies qui con- tiennent une fonction indeterminee, assujettie a la seule condition d etre developpable dans les limites de 1 integration selon les puissances entieres et positives des variables 1 , mais dans lesquelles les limites sont donnees par 1 Cette condition est supposes dans la demonstration, mais il me paralt, d apres la forme du resultat, qu elle n est pas essentielle. 30] 207 30. MEMOIRE SUR LES COURBES A DOUBLE COUEBUEE ET LES SURFACES DEVELOPPABLES. [From the Journal de Mathematiques Pures et Appliquees (Liouville), tome X. (1845), pp. 245250. ON trouve, dans la Theorie des courbes algebriques de M. Pliicker, de tres-belles recherches sur le nombre des differentes singularity s (points d inflexion, tangentes doubles, &c.) des courbes planes. Les memes principes peuvent s appliquer au cas des courbes a trois dimensions. Pour cela, considerons une suite continue de points dans 1 espace, les lignes qui passent par deux points consecutifs, et les plans qui passent par trois points consecutifs ; ou, en envisageant autrement la meme figure, une suite de lignes dont chacune rencontre la ligne consecutive, les points d intersection de deux lignes consecutives, et les plans qui contiennent ces lignes ; ou encore de cette maniere : une suite de plans, les lignes d intersection de deux plans consecutifs, les points d intersection de trois plans consecutifs. C est ce que 1 on pent nommer un systeme simple. Ce systeme est eVidemment forme d une courbe a double courbure et d une surface deve- loppable ; la courbe est 1 arete de rebroussement de la surface, la surface est 1 osculatrice developpable de la courbe. Les points du systeme sont des points dans la courbe, les lignes du systeme sont les tangentes a la courbe, les plans du systeme sont les plans osculateurs de la courbe. De meme, les plans sont les plans tangents de la surface, les lignes sont les generatrices de la surface ; pour les points, on peut les nommer les points de rebroussement de la surface. J entendrai dans la suite par les termes ligne par deux points, ligne dans deux plans, une ligne menee par deux points quelconques (non consecutifs en general) du systeme, et la ligne d intersection de deux plans quelconques (non conse cutifs en general) du systeme. Enfin, chaque ligne du systeme est rencontree, en general, par un certain nombre d autres lignes non consecutives du systeme. Je nommerai le point de rencontre d une telle paire de lignes point dans deux lignes, et le plan qui contient une telle paire plan par deux lignes. 208 MEMOIRE SUR LES COURBES A DOUBLE COURBURE [30 Supposons qu un plan donne contient, en general, m points du systeme, qu une ligne donnee rencontre, en general, r lignes du systeme, qu un point donne* est situe , en general, dans n plans du systeme. Le systeme est dit etre de 1 ordre m, du rang r, de la classe n. On voit tout de suite que 1 ordre de la courbe est e*gal a 1 ordre du systeme, ou a m; la classe de la courbe au rang du systeme, ou a r. Et de meme, 1 ordre de la surface au rang du systeme, ou a r; la classe de la surface a la classe du systeme, ou a n. Cela posd, les singularity s proprement dites (ouvrage citd, page 202) sont les deux suivantes, analogues aux points d inflexion et de rebroussement dans les courbes planes: 1. Quand quatre points consecutifs sont situes dans le meme plan, ou, autre- ment dit, quand trois lignes conse cutives sont situdes dans le meme plan, ou quand deux plans consecutifs deviennent identiques; je dirai qu il y a alors un plan station naire, et je repre senterai par la lettre a le nombre de ces plans. 2. Quand quatre plans consecutifs se rencontrent dans le rneme point, ou, autre- ment dit, quand trois lignes se rencontrent au meme point, ou quand deux points conse cutifs deviennent identiques ; je dirai qu il y a alors un point stationnaire, et je repre senterai par le nombre de ces points. Dans le premier cas, il y a un point d inflexion spherique dans la courbe, et une ligne d inflexion dans la surface. On n a pas donne de noms a ce qui arrive dans la courbe et la surface dans le second cas; et puisque la singularite est suffisamment distinguee deja en la nommant point stationnaire, il n est pas necessaire de suppleer a cette omission. On peut dire que ces deux cas sont les singularites simples d un systeme. II y a des singularites d un ordre plus eleve dont on n a pas besoin ici. Ensuite, il y a des singularite s d une autre espece, en quelque sorte analogues aux points et tangentes doubles, mais qui ont rapport a un point ou plan indetermine (hors du systeme) ; savoir : 3. Un plan donne peut contenir, en general, un nombre g de lignes dans deux plans. 4. Un point donne peut etre situe, en general, dans un nombre h de lignes par deux points. 5. Un plan donne peut contenir, en general, un nombre x de points dans deux lignes. 6. Un point donne peut etre situs , en general, dans un nombre y de plans par deux lignes. Ces quatre cas sont les singularites impropres simples du systeme. II faut maintenant chercher les relations qui ont lieu entre les nombres m, r, n, a, , g, h, x, y. Citons d abord les formules de M. Pluck er pour les courbes planes (page 211), en changeant seulement les lettres, pour eViter la confusion. En representant par p 1 ordre d une courbe, par v sa classe, par le nombre de ses points doubles, 77 de ses points 30] ET LES SURFACES DEVELOPPABLES. 209 de rebroussement, a de ses tangentes doubles, b de ses points d inflexion, Ton aura les six equations a = \p . fj, - 2 p?- 9 - (2f + ST?) (p . p - 1 - 6) + 2 . - 1 + f 77 . 77 - 1 + JLI = i/ . ?~i - (2a + 36), = | *. v - 2 *> - 9 - (2a + 36) (> . v - 1 - 6) + 2a . a - 1 + f 6 . 6 - 1 + Gab, dont les trois dernieres se derivent des trois premieres, et vice versa. Considerons ensuite un plan donne quelconque en conjonction avec le systeme. Ce plan coupe la surface suivant une courbe plane. Les points de cette courbe sont les points de rencontre du plan avec les lignes du systeme, les tangentes de cette courbe sont les lignes de rencontre du plan avec les plans du systeme. II est clair que la courbe est de 1 ordre r et de la classe n. Chaque fois que le plan contient un point en deux lignes, la courbe a un point double ; aux points ou le plan rencontre la courbe a double courbure, il y a dans la courbe un rebroussement (car, dans ce cas, il y a deux lignes du systeme qui coupent le plan en un meme point, c est-a-dire qu il y a dans la courbe d intersection un point stationriaire ou de rebroussement). Quand le plan contient une ligne en deux plans, il y a dans la courbe une tangente double ; et enfin, pour chaque plan stationnaire du systeme, il y a dans la courbe une tangente stationnaire ou une inflexion. Done nous avons, dans la courbe, r 1 ordre, n la classe, x le nombre de points doubles, ra de points de rebroussement, g de tangentes doubles, a de points d inflexion, ce qui donne les six equations (equivalentes a trois) : n = r . r 1 (2# + 3m), a = 3r.r-2-(&c + 8m), = ir.r-2r 2 -9-(2# + 3m) (r . r - 1 " 6) + 2x . x - 1 + f m . m - 1 + = n.nl (2g + 3a), m = 3n . n 2 (6g + 8a), x = \n. n-2 -ft- - 9 - (2g + 3a) (n . n-l - 6) + 2g . g-l + f a . a-1 + (ou Ton peut remarquer la syme trie a 1 dgard des combinaisons n, a, g, et r, m, x}. De meme, en consideVant un point donne quelconque en conjonction avec le systeme, ce point determine, avec la courbe a double courbure, une surface conique, et, par des raisonnements pareils, on fait voir que, dans cette surface conique, 1 ordre est tit, c. 27 210 MEMOIRE STIR LES COURBES A DOUBLE COURBURE [30 la classe r, le nombre des lignes doubles n, des lignes de rebroussement , des plans tangents doubles y, des lignes d inflexiori h, ce qui donne les equations (dont trois seulement sont inde"pendantes) : r=m. rn^l - ( 2h + 36), = 3m. ra-2- (Qh + 8), -2 m 2 - 9 - (2h + 3 > (m.m-1 -6)+ 2h . h - h = \ r . r - 2 r 2 - 9 - (2y + 3n) (r . r -"l - 6) + 2y . y - 1 + f n . n - 1 + 6 (dans lesquelles on remarque la correspondance r, n, y\ m, , h: et, en les comparant avec les autres six equations, la correspondance m, r, n, a, , g, h, x, y\ n, r, m, , a, h, #, y. ) En corisiderant une courbe a double courbure d un ordre donnd m, on peut attribuer a h, des valeurs quelconques (entre certaines limites), et Ton a alors, pour determiner les autres quantit^s, les equations suivantes: y = i m. m - 2 m 2 - 9 - (2h + SS) (m. ^1 - 6) = r.r-l-(2x; + 3m), g = |- r . r - 2 r 2 - 9 - (2x + 3m) (r.r-l-6) + 2.*-l-f| w . m - 1 + Au cas d une courbe plane les trois premieres equations continuent d etre vraies, mais les trois autres n ont pas de sens. Le cas le plus simple est celui d une courbe du troisieme ordre dans 1 espace. On a, dans ce cas, m = 3, n = l, =(). et de la le systeme m, n, r, a, g, g, h, x, y ; 3, 3, 4, 0, 0, 1, 1, 0, 0: c est-a-dire 1 osculatrice developpable d une courbe du troisieme ordre est une surface du quatrieme ordre, &c. On obtient aussi un systeme tres-simple, en dcrivant m = 4, h = 2, ^" = 1, ce qui donne m, n, r, a, g, g, h, a, y ; 4, 4, 5, 1, 1, 2, 2, 2, 2. 30] ET LES SURFACES DEVELOPPABLES. 211 Mais cela n appartient pas, a ce que je crois, aux courbes les plus ge ne rales du quatrieme ordre. Le probleme de classifier les courbes a double courbure au moyen des surfaces que Ton peut faire passer par ces courbes, ou de trouver la nature d une courbe qui est 1 intersection de deux surfaces donnees, parait appartenir plutot a la theorie des surfaces qu a celle des courbes. Je n ai rien de complet a offrir sur cela. Seulement je crois pouvoir dire que quand la courbe d intersection de deux surfaces des ordres /JL, v est de 1 ordre pv (ce qui est le cas general), on a toujours de maniere que la classe de la courbe est au plus mn (m +n 2). Mais j espere revenir une autre fois sur cette question. II est presque inutile de remarquer que pour un systeme qui est reciproque polaire d un systeme donne , il faut seulement changer m, r, n, a, <?, g, h, x, y en n, r, m, , cr, h, g, y, x. Par exemple, les deux systemes qui viennent d etre considers ont des reciproques de la m^me forme. 27- 2 212 [31 31. DEMONSTRATION D UN THEOREMS DE M. CHASLES. [From the Journal de Mathematiques Pures et Appliquees (Liouville), tome x. (1845), pp. 383384.] " Soient P, P des points correspondants de deux figures homographiques ; si la droite PP passe toujours par un point fixe 0, les points P sont situes sur une courbe du troisieme degre, qui passe par ce meme point." T 7/ ? IT ?/ Z Soient - . . les coordonne es de P ; , -. , . celles de P . En supposant WWW WWW que x , y , z , w sont des fonctions lineaires (sans terme constant) de x, y, z, w, les deux figures seront homographiques. f) -* Soient, de meme, ^ , ^ , K les coordonne es de ; , , - - les coordonnees d un O O O t7 37 37 point quelconque T. Puisque P, P , sont sur la meme droite, on peut faire passer un plan par les quatre points P, P , 0, T. Cela donne tout de suite 1 equation ( \, fJb , V , 37 ^ OL , & , y, 8 x , y , z, w ^x , y , z , w (en representant de cette maniere le determinant forme avec les quantites X, //,, v, &c.), equation qui doit etre satisfaite quels que soient X, /-i, v, or, et qui equivaut ainsi aux deux conditions 8 } , y , z } = 0, < x , y, w > 0. y I I , y , z } (x , y , w } 00 *U Z Ces deux dernieres equations sont du second degre par rapport aux quantite s - , - , - . w w w Le point P est done situe a 1 intersection de deux surfaces du second ordre. Mais ces surfaces ont en commun la droite representee par les equations ay - x = 0, ax - y = 0. Done elles se coupent de plus suivant une courbe du troisieme degre qui passe evidemment par le point 0, parce qu on satisfait aux Equations en e crivant x : a = y : = z : <y=tv : S. 32] 213 32. ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION TO THE THEORY OF SPHERICAL COORDINATES. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 22 33.] SECTION 1. THE formulae in question are only very particular cases of some relating to the theory of the transformation of functions of the second order, which will be given in a following paper. But the case of three variables, here as elsewhere, admits of a symmetrical notation so much simpler than in any other case (on the principle that with three quantities a, b, c, functions of b, c; of c, a; and of a, b, may symmetrically be denoted by A, B, C, which is not possible with a greater number of variables) that it will be convenient to employ here a notation entirely different from that made use of in the general case, and by means of which the results will be exhibited in a more compact form. There is no difficulty in verifying by actual multiplication, any of the equations here obtained. It will be expedient to employ the abbreviation of making a single letter stand for a system of quantities. Thus for instance, if 8 = 6, <, ^r, this merely means that <() is to stand for 4> (0, <, i/r), k& for kff, k<f>, k^r, &c. Suppose then =, v, , ....................................... (1), Q = A, B, C, F, G, H .......................... (2). W (, a/, Q) = A& + Brjr, + C& + F + 7/) + G (& + ?& + H (ft + fr) ... (3), the function W satisfies a remarkable equation, as follows : 214 ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION [32 write &=BC-F S , .................................... (4), 3$ = CA -G 2 , y=FG -CH. = a, as, or, $, <s, we have W (a)!, co.-,, Q) W (&> 3 , ft> 4 , Q) W((a 1 , <w 3 , Q) W(w 2 , &> 4 , Q) = W(fi) 1 (o i , &) 2 ft) 3 , ) ... (7); of which we may notice also the particular cases W(to lt <o s , Q)W(o> 3 , o) 3 , Q)- W(a) 1 , ft) 3 , Q)W(a> 2 , ft> 3 , Q)=W(<Wz, co^ 3 , ) ... (8), 1 , o> 2 , Q)} 2 =Tf(o>, w, <S) ... (9). To these we may join the following formulae, for the transformation of the function W. Suppose oh = aajj + ay + a"^ , b^ + b ^/i + b"^ , ca;, + C T/J + c"^ ......... (10), u 2 = a# 2 + a 2/2 + a"^ 2) b^ 2 + b y 2 + b"^ 2 , ca? 2 + c y 2 + c"^ 2 , then, writing ^ =a , b , c .................................... (11), # =a , b , c x , ^" = a", b", c", Pi = *i> yi, ^i ................................... (12), P 2 = X 2 , 2/ 2! ^ 2 , =W(g,g,Q), W(g ,g ,Q), W(g",g",Q\ W(g ,g",Q\ W(g",g,Q), W (g,g } Q) (13), we have W(<o lt o> 2 , Q)=F(p lt p 2 , ) ........................... (14). Similarly, writing W ( gg , g"g, ), ^ (^, fl^, Q), TT( g"g, g g", ), (15), we have F ( w^, <p 2 &) 3 , ) = F (yip 4 > ^2^3, ^) ........................ (16), in which equations may obviously be changed into Q. 32] TO THE THEORY OF SPHERICAL COORDINATES. 215 SECTION 2. Geometrical Applications. Consider any three axes Ax, Ay, Az, and let X, p, v be the cosines of the inclinations of these lines to each other. Let A, M, N be the inclinations of the coordinate planes to each other ; I, m, n the inclination of the axes to the coordinate planes. Suppose, besides, a = l -x 2 ....................................... (17), b = i - /A 3 , c = 1 - i/ 2 , f = fjuv X , g = i/X fji . f) = Xytt V , k = l -\ -^-v- + 2\fjLv ................................. (18); we have the following systems of equations : V(bt)cosA = -f, V (to) sin A = V (&), V (a) sin Z = V (&) ......... (19). V (ca) cos M = - g, V (ca) sin M = V (&), V (^) sin m = V (&) V (ab) cos N = - 1), V (at)) sin N = V (&), V (C) sin w = V (&). 3+ vb + p$=k, ................................. (20). v&+ J) + xg = 0, Ata + xb + g = 0. f)+^b+ /*f = 0, ................................. (21). i/f) + b + xf = &, f=0. 9+ rf+ M = 0, ................. . ............... (22). z/g + f + XC = 0. yU,Q + Xf + C = &. be- P = ka ................................ (28). ca- tf = kb, ab - t 2 = kc , gb - af = ^/, bf - bg = fy, fg - eft = M, abc - af- - bg- - cb- + 2fgb = k- ................................... (24). 21 G ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION Imagine now a line AO, and let a, /3, 7 be the cosines of its inclinations to the three axes. Suppose also, 6, <, ^ being its inclinations to the coordinate planes, we write sin 6 , sin d> sin v = If we consider a point P on the line AO, at a distance unity from the origin, we see immediately, by considering the projections in the directions perpendicular to the coordinate planes, that the coordinates of this point are a, b, c. By projecting on the three axes and on the line AO, we then obtain the equations a a+vb+/j,c .................................... (26), 13 = va + b +\c, <y = fj,a + \b + c, l=aa+/3b + 70 .................................... (27), from which we obtain &a = aa + f)/3 + 97 ................................... (28), kb = fja + 1)/3 + f7, kc = Qa + f/3 + 7, I=aa+j3b + yc .................................... (29), and hence 1 = a 2 + b- + c 2 + 2\bc + 2/mc + 2vab ........................... (30), k = aa 2 + 1/3- + C 7 2 + 2f^7 + 2ga 7 + 2f)a/3 .................... (31). Hence writing a, b, c = t ....................................... (32), a, & 7 = T ....................................... (33), 1, 1, 1, X, /K, i/ = q ................................. (34), H, b, C, f, Q, i) = (J .................................. (35), we have the equations l = W(t, t, q) .................................... (36), k=W(r, r, q) .................................... (37). Let ^.0 be any other line, and 8 its inclination to AO : of, ft , 7 , a , b , c, the quantities corresponding to a, /3, 7, a, b, c, and similarly , r to , r. We have of course 1- W(t, t , q) .................................... (38), =TT(T , T , q) .................................... (39). 32] TO THE THEORY OF SPHERICAL COORDINATES. 217 We have besides, by projecting on the line AO , the equation cosS = a a+/3 / 6+7 / c ................................. (40), or the analogous one (41). From either of which we deduce cos 8 = aa + bb + cc + X (be + b c) + p (GO! + c a) + v (ab + a b) ............ (42), k cos B = aaa + W + 77 + f (J3y + ffy) + g (y f + y<xf) + f) (a/3 + a ) ...... (43) ; which may otherwise be written cosS=W(t, t , q) .................................... (44), kcoaS=W(T, r, q) .................................... (45); or again, observing the equations which connect the quantities t, r, forms which, though more complicated, have certain advantages ; for instance, we derive immediately from them the new equations , tf, q)} S ........................... (48) smd= ( T ,. ur/ / ? \i V { F (r, T, q) F (T, T , q)} written more simply thus sinS= W (<?, ft" , q) ....................................... (50), V^sinS^VfFCrr 7 , ^?, q)} ................................... (51); to these we may join .. , , q)} / ..(53) , rr , q)} SECTION 3. 0?i Spherical Coordinates. Consider the points JT, Y, Z, on the surface of a sphere, as the intersections of the three axes of the preceding section, with a sphere having its centre in the origin. It is evident that X, /j,, v are the cosines of the sides of the spherical triangle XYZ, c. 28 218 ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION [32 A, M, N are its sides, I, m, n are the perpendiculars from the angles upon the opposite sides. Let P be the point where the line A intersects the sphere : the position of the point P may be determined by means of the ratios : 77 : , supposing , ?/, denote quantities proportional to the a, ft, 7 of the preceding section, i.e. : rj : = cos PX : cos PY : cos PZ (54) ; or again, by means of the ratios x : y : 2, supposing x, y, z denote quantities proportional to the a, b, c of the preceding section, i.e. sin Px sin Py sin Pz x : y : z = ^- : -, : . = ( O 5). sin X sin Y sin Z (Px, Py, Pz are the perpendiculars from P on the sides of the spherical triangle XYZ). These last equations may be otherwise written, x sin X sin PZY y sin Y sin PZX ysin Y = siuPXZ zBinZ "sinPZF zsinZ smPYX (56). The ratios : 77 : or x : y : z, are termed the spherical coordinate ratios of the point P. The two together may be termed conjoint systems : the first may be termed the cosine system, and the second the sine system. The coordinates of the two systems are evidently connected by : 77 : % = x + vy + /ji2 : vx + y +\z : px + \y + z ............ (57), or x : y : 2 = ^ + ^rj + ^ : ^f + ^+f^ : Q + irj + c ............ (58). The systems may conveniently be represented by the single letters w = *l, ? ....................................... (59), P = x, y, z ....................................... (60). Fundamental formula of spherical, coordinates ; distance of two points. Let P, P be the points, 8 their distance, &>, p the conjoint coordinate systems of the first point, a/, p of the second ; we have obviously rn t \P> p > q) /r . 1N ViTFfop, q)W(p ,p , q)} - V { W ( pp , pp , q)} sin o = ,p, q) W(p , p , q)} 32] TO THE THEORY OF SPHERICAL COORDINATES. 219 * TF (,_ , $)_ Or COS O , / -m- , _ x -rrr , / / ^\i ft, 1 . V{^( , , q)} -IT sm 6 = f . s cot o = , <, q) Tf ( , w , W(co, ft) , g) V{F(ft>ft/, W, q)} Equation of a great Circle. Let the conjoint coordinate systems of the pole be e = a, b , c ....................................... (63), e = , A 7 ....................................... (64), then, expressing that the distance of any point P in the locus from the pole is equal to 90, we have immediately the equations W(p, e, q) = .................................... (65), W (a), e, <1) = .................................... (66), which may otherwise be written in the forms af + &77+c = ....................................... (67), 7=0 ....................................... (68), or the equation of a great circle is linear in either coordinate system. Conversely, any linear equation belongs to a great circle. Suppose the equation given in the form A% + Bri + CS=Q .................................... (69); or by an equation between cosine coordinate ratios : the sine system for the pole is given by e = A, B, C ....................................... (70), and the cosine system by .................. (71). Suppose the circle given by an equation between sine coordinates, or in the form y + C^ = .................................... (72), the cosine system of coordinates for the pole is given by c = A, B, C ....................................... (73), arid the sine system by fC, gA + fB + cC ............... (74). 282 220 ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION [32 It is hardly necessary to observe, that if A% + Bri+CS=0 .................................... (75), AX + By + = .................................... (76), represent the same great circle, A : B : C= A+vS+fiC : vA+ B + \C : pA + \B + C ...... (77), A : B : C = aA + f)B + gC : f)A + bB+ fC : gA +fB +cC ...... (78). Inclination of two great Circles. Let the equations of these be ( AZ + Bv + C^O .............................. (79), ( or A x + B y + Cz = .............................. (80), f A + B v + C t;=0 .............................. (81), \or A ar + B y + C z = .............................. (82), and let e, e, have the same values as above, and e , e, corresponding ones. To obtain the inclination of the two circles, we have only, in the formulae given above for the distance of two points, to change p, p , w, a/, into e, e , e, e. The distance of a point from a given circle may be found with equal facility ; for this is evidently the complement of the distance of the point from the pole of the circle. In like manner we may find the condition that two great circles intersect at right angles, &c. There are evidently a whole class of formulae, not by any means peculiar to the present system of coordinates, such as C z} ........................ (83), for the equation of a great circle subjected to pass through the points of inter section of Ax + By + Cz = 0, A x + B y + C z = 0. Again, oo, y, z =0 (84), a, b, c a , b , c for the equation of the great circle which passes through the points given by the sine systems a : b : c and a : b : c , &c., and which are obtained so easily that it is not worth while writing down any more of them. Transformation of Coordinates. Let X 1} Fj, Z lt be the new points of reference, and suppose X l is given by the conjoint systems e = a, b, c. e = ot, ft, 7; and similarly F : , Z l , by the analogous systems e, e ; e" , e". 32" TO THE THEORY OF SPHERICAL COORDINATES. 221 Suppose, as before, P is given by one of the systems w, p; and let w,, p l be the new systems which determine the position of P with reference to X lt F 1; Z-,. In the first place, X 1; ^, v,, are given by the formulae W(e , e", q) W (J, e", q) _, l ~JW(e , e , q) W(e", e", q)] V{F(e / , e , q) F(e", e", q)] " , e, q e, 6, 5", e", q) Tf (e, e, q)} V{F(", e /7 , q) If (e, e, F(e, e , q) 1/1 ~ V {F (e, e, q) TF (e , e , q){ V{^ (e, 6, q) F (e , e , q) The system w is evidently given immediately by W(e, p, q) W(e , p, q) Tf (^ p, q) , , q)T </{W(J, e , q)} *J{W(e", e", q)} TT(e. to, q) W(4, . q) ^(^ o>, q) e, q)} </{W(J, ^ q)} x/iF(e-", T> q) and from these we may obtain the system p-^ , by means of the formula} fik : &?i + fi^i + Ci^i ...... ( 88 >- This requires some further development however. We must in the first place form the system HI, ti, C 1; f 1; Qi, !h: this is done immediately from the formulae of Sect. 2, and we have W(7e", 77, q) feTr(g, g. g) , e , q) ^(e", e", q)~ F(e , e , qf)"TT(e", e", q) " , e^e, q) kW (gg ge, q) Cl = tr(fl", e"", q) W(e", e, q) = Tf (e", e", q) Tf (e, e, q) W (j,~e2, q) feF(g. g, q) Tf (e, e, q)~ TF (e , e , q) e7, q) l = W(e, e, q) V{Tf (e , e , q) TF(e", e", q)} " W(e, e, q)V(F(e , TTq) F(e", e", q)} Tf(^?, ?? , q) _ A?TT(g, g, g) __ 7^e r , q) V { TT (e 7 ; eVq) "Tf STTq)! " ^(e , e , q)V{^(e"> e", q) If (e, e, q)} W(e 7 e", Te, q) A;F(ey , ^, q) , e, q) F(e , e , q)} Tf(e", e", q) V[Tf (e, 6, q) W (e 7 , e , 222 ON SOME ANALYTICAL FORMULA, AND THEIR APPLICATION [32 x, : yi : z^^{W(e, e, q)) x ............................................................ ... { W(e, p, q) W ( e^, 7J\ q) + W(e\ p, q) W( 7?, e^e, q) + W(e", p, q) W ( eV 7 , e?, q)} : 4{W(e , e , q)} x {Tf (e, p, q) TF(e5 77, q)+ Tf (e , p, q) W (e^e, e^e, q)+W(e", p, q) W (Se, eJ, q)} : J{W(e", e", q)} x {JT(e, p, q) Tf (e7, eV 7 , q) + TF(e , ^ q) ^(ee 7 , ^e, qf) + Tf (e", p, q) TF ( e7, e^ q)} ; these may be reduced to the very simple form , , q)} > ^, q)} . , q> , q- and in like manner we obtain , e, q)| W(5 fJ ),q) ....................... (02), > . q)}Tf(?6, ^, q), e", e", (j)] W (^, p, q). It will be as well to indicate the steps of this reduction. Consider the quantity in { } in the first line of the equation which gives the ratios x l : y : z l ; and suppose for a moment eV 7 = I, m, n, &c. : then, selecting the portion of the expression which is multiplied by a, this is or, snce la + l a + l"a" = ~ele" , Ib + I V + l"b" = 0, Ic + I c + l"c" = 0, this reduces itself to ee e .alt;, which is a term of ee e" W( eV 7 , &), q) ; and by comparing the remaining terms in the same manner, it would be seen that the whole reduces itself to eeV 7 W ( eV 7 , &>, q) ; whence the formulae in question. The formulae (86), (87), (91), (92), completely resolve the problem of the transfor mation of coordinates; they determine respectively pi from p or w, coj from p or w. To complete the present part of the subject we may add the following formulae. Suppose a?! : y 1 : z 1 = &x l + a ?/! + a"z l (93), : IM?! + b y! + b"y, , : cx l + c y l + c" z l , 32] TO THE THEORY OF SPHERICAL COORDINATES. 223 which we see from the preceding formulae is the form of the relation between the systems p l and p. And suppose, as before, \ 1} /AJ, v l are the cosines of the distances of the new points of reference X 1? Y 1? Z x . We can immediately determine the relations that must exist between these coefficients, in order that they may actually denote such a transformation. For this purpose write a , b , c =j .................................... (94), a , b , c =j , a", b", c"=j". Then the distance between the point P and any other point P is given by the formula , P , q ) W( Pl , K, e> . - ~ , p, q) W (p t p , q)} V { W ( Pl , Pi , ) W ( Pl , fr , where . } But we must evidently have W(p l , p^, qO COS 6 = , -rT-/ - or the quantities must be proportional to the quantities q, i.e. W(j,j, q) : W(j ,j f , q) : W (/ ,/ , q) : F (/,/ , q) : If (/, j, q) : F(j", j, q) = 1:1 : 1 : Xj : Mi : "i ...(98). And in precisely the same manner, if instead of x, y, z, a? l5 y l , z lt in the above formulae, we had had , 77, : 1( T/J, ^i, the result would have been ^0 , j, 1) : ^ (/, /, : ^0 ". f, *) : ^(/- / > <l) = W(j", j, q) : W (j", j, q) -a:t):C : f : : i - It is hardly necessary to remark, that throughout the preceding formulas an expression, such as W (p, p , q), is proportional to either of the quantities or and may be changed into one of these multiplied by an arbitrary constant, which may be always made to disappear by a corresponding change in another quantity of the same form : thus, for instance, W Cp, ,_q) _ gSZ + yi + SS . W(p~p, q) xt + y<n + zS but these forms being unsymmetrical, it is better in general not to introduce them. All the preceding expressions simplify exceedingly, reducing themselves in fact to the ordinary formulae for the transformation of rectangular coordinates in Geometry of three dimensions, for the case where the triangle XYZ has its sides and angles right angles. As this presents no difficulty, I shall not enter upon it at present. 224 [33 33. ON THE REDUCTION OF ~ WHEN U IS A FUNCTION OF v U THE FOURTH ORDER [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 7073.] IT is well known that the transformation of this differential expression into a similar one, in which the function in the denominator contains only even powers of the corre sponding variable, is the first step in the process of reducing [~ to elliptic integrals. J V ^ And, accordingly, the different modes of effecting this have been examined, more or less, by most of those who have written on the subject. The simplest supposition, that adopted by Legendre, and likewise discussed in some detail by Gudermann, is that u is a fraction, the numerator and denominator of which are linear functions of the new variable. But the theory of this transformation admits of being developed further than it has yet been done, as regards the equation which determines the modulus of the elliptic function. This may be effected most easily as follows. U=a P =aa? + kba?y + 6cx*y + 4<dxy* + ey*. Also let P = a V 4 + ^b x -y + GcV 2 / 2 + 4dVy 3 + e y * be what P becomes after writing X = \x + fiy , +6cV- +4dV 3 +e u \ 33] ON THE REDUCTION OF du + jU, &C 225 Suppose, moreover, k = A./A, \ / fi, I =ae -4bd + 3c 2 , I =a e -4& d we have evidently or J = ace ad 2 be" c 3 + 2bcd , J = a c e - a d * - Ve ~ - c 3 + We d ; xdy ydx = k (x dy y dx ), xdy ydx _ , x dy y dx ~ Hence writing and therefore we obtain y i y u = *- , u =,; x x xdy ydx du x dy y dx du ~ ~ du , du the equation between u and u being Next, to determine the relations between the coefficients of U and U . Since P, P are obtained from each other by linear transformations (Math. Journal, vol. iv. p. 208), [13, p. 94], we have between the coefficients of these functions and of the transforming equations, the relations whence also Suppose now or whence also C. / = _ I 3 ~ I 3 U = a (I + pu *) (1 +qu ~}, d =Q, 6c = a (p + q), e = a pq- = ifa a 3 (P 2.9 226 ON THE REDUCTION OF du + jU, WHEN [33 (p + gf (B4<pq - p* - g 2 ) 2 = 27 J 2 7 3 therefore whence also lOSwY ( <r\ /V = 1- /3 > which determines the relation between p and q. Also *Ja so that du 121 j V{(1 + PU *) (1 + ^w /2 )] If in particular p = - 1, writing also q for g, 127 where Suppose, for shortness, / 3 ; then ( ? 2 + 14g + I) 3 - 16% (q - I) 4 = 0. / 1 , A 3 i.e. fg-| 1-14 Let aJ a then 3 which determines 0. And then + 4(3 q ~ 0-1 33] U IS A FUNCTION OF THE FOURTH ORDER. 227 Suppose q at. is one of the values of q ; the equation becomes s + 14/3* + I) 3 -I) 4 a (a -I) 4 /3 4 (/3 4 -l) NOW a .Hi-rf) - 16(/S 8 + 14/3 4 +l) then (q 2 + Uq + 1) = ^ + ^ q ~ ~ { which values satisfy the above equation: hence also, identically, ,, / i\t /1-/3YH /i+0Y) ( A - (f - (f - jgi) { - (nJ) } {? - (yzg) } jf - (I or the values of </ take the form + * 1 (Comp. Abel (Euv. torn. i. p. 310 [Ed. 2, p. 459].) The equation 6 S has its three roots real if 27 4>M is negative, and only a single real root if 27 41/ is positive. Writing the equation under the form (0 + 3)3 _ 9 (0 + 3)2 + (27 - Jf ) (0 + 3) - (27 - 4Jf ) = 0, we see that in the former case 6 has two values greater than 3, and a single value less than 3. Writing the equation under the form (6 - I) 3 + 3 (6 - I) 2 + (3 - M) (6 - 1) + 1 = 0, (3 - M is negative) the positive roots are both greater than 1. Hence, in this case, q has four positive values and two imaginary ones. In the second case 6 has a single real value, which is greater than 3 and less than 1. Hence q has two negative values and four imaginary ones. In the former case, I s 27/ a is positive, and the function U has either foul- imaginary factors or four real ones. In the second case, I s Z7J- is negative, or the function U has two real and two imaginary factors. 292 228 [34 34. NOTE ON THE MAXIMA AND MINIMA OF FUNCTIONS OF THREE VARIABLES. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 74, 75.] IF A, B, C, F, G, H, be any real quantities, such that (A + B + C} (ABC - AF 2 - BG- - CH* + 2FGH) are positive ; the six quantities BC-F 2 , CA-G 2 , AB-H 2 , AK, BK, CK, (where K = ABC - AF 2 - BG 2 - CH 2 + 2FGH) are all of them positive. It is unnecessary to point out the connection of this property with the theory of maxima and minima. To demonstrate this, writing as usual EG -F-=A , GH~AF=F , CA -G- =B , HF-BG = G , -H* = C , FG - and K as above: then if A", B", C", F", G", H", K be formed from A , B , C , F , G , H , as these and K are from A, B, G, F, G, H, we have the well-known formulae A" = KA, F" =KF, K = K\ B" = KB, G" = KG , C" =KC, H" = KH, 34] NOTE ON THE MAXIMA AND MINIMA OF FUNCTIONS OF THREE VARIABLES. 229 It is required to show that if A + R + C and A" + B" + C" are positive, A , B , C , A", B", C" are so likewise. Consider the cubic equation (A - k) (B f - k) (C - k) - (A - k) F n - - (B - k) G 2 - (C - k) H 2 + ZF G H = 0, the roots of which are all real. By the formulae just given this may be written and the terms of this equation are alternately positive and negative ; i. e. the roots are all positive. Hence the roots of the limiting equation (B -k)(C -k)-F 2 = are positive, i. e. B + C and B C are positive : but from the second condition B , C are of the same sign : consequently they are of the same sign with B + C , or positive. Also A" = B C F - is positive. Similarly, considering the other limiting equations, A , B , C", A", B", C" are all of them positive. In connection with the above I may notice the following theorem. The roots of the equation (A - ka) (B - kb) (C - ck) -(A- ka) (F - kf) 2 - (B - kb) (G - kg) 2 - (C - kc) (H - kh) 2 + 2(F- kf) (G - kg) (H - kh) = 0, are all of them real, if either of the functions Ax 2 + By 2 + Cz 2 + 2Fyz + ZGxz + ZHxy, ax 2 + by 2 + cz 2 + %fyz + 2gxz + %hxy, preserve constantly the same sign. The above form parts of a general system of proper ties of functions of the second order. 230 [35 35. ON HOMOGENEOUS FUNCTIONS OF THE THIRD ORDER WITH THREE VARIABLES. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 97 104.] THE following problem corresponds to the geometrical question of determining the polar reciprocal of a plane curve of the third order: the solution of it is also important, with reference to the linear transformations of homogeneous functions of three variables of the third order; reasons for which it has appeared to me worth while to obtain the completely developed result. Let o U = ace + bif + c.2 y + %iy-z + %jz*x + 3ka?y + Si^z 2 + Sj^z? + Skixy* + Slxyz (1 ). It is required to eliminate x, y, z, \ from the equations U=0 (2), dU . d-* + ^ = f-^-O f +XC-0, dz ) From the equations (2), (3), we obtain immediately = c + 77^ + ^=0 (4); and thence @# = 0, % = 0, 02 = (5); 35] ON HOMOGENEOUS FUNCTIONS OF THE THIRD ORDER &C. so that a single equation more., such as 231 (6), where <E> is homogeneous and of the second order in x, y, z, would, in conjunction with the equations (3) and (5), enable us to eliminate linearly the seven quantities x 2 , y"-, z 2 , yz, zx, xy, \. Such an equation may be thus obtained. Let L, M, N, R, S, T, be the second differential coefficients of U, each of them divided by two. The equations (3) may be written Lx+Ty +Sz + X = 0, ................................. (7). Tx + My + Rz + \7j = 0, And joining to these the equation (4), = 0, we have, by the elimination of x, y, z, in so far as they explicitly appear, and X, an equation <J> = of the required form. Hence we may write Z, T, 8, | T, M, R, T/ 8, R, N, (8); or substituting for L, M, N, R, S, T, and expanding, <l> = A x 2 + Bf + Cz n - + 2Fyz + 2 Gzx + 2Hxy (9), where the values of A, B, C, F, G, H are (10). [I omit these values (10), and the values in the subsequent equations (13), (14), (20) : the values (10) and (13) serve for the calculation of (14), FU, the expression for which with the letters (a, b, c, f, g, h, i, j, k, I) in place of (a, b, c, i, j, k, i 1} j lt k lt 1} is reproduced in my "Third Memoir on Qualities," Phil. Trans, vol. CXLVI. (1856) pp. 627647: the values (20) serve for finding that of (21), K (U), but the developed expression of this has not been calculated.] Performing the elimination indicated, the result may be represented by k, j, I, JL k, |-0 (11). k , b , i,, i , I , k l} 77 j, I, ? h, c, hi 277 . r f A B C F G H Partially expanding, .(12). 232 ON HOMOGENEOUS FUNCTIONS OF THE THIRD [35 The values of the coefficients a, b, c, f, g, h may be useful on other occasions: they are as follows (13)- Substituting these values the result after all reductions becomes = F(U)= (14). It would be desirable, in conjunction with the above, to obtain the equation which results from the elimination of x, y, z from the equations dU dU dU -z = 0. -r- = 0, -T =0 (15) dx dy dz (i.e. the condition of a curve of the third order having a multiple point), but to effect this would be exceedingly laborious. The following is the process of the elimination as given by Dr Hesse, Crelle, t. xxvui. (and which applies also to the case of any three equations of the second order). Forming the function VU, of the third order in x, y, z, by means of the equation ~^= L, T, 8 T, M, R 8, R, N (L, M, N, R, 8, T, the same as before). Then, in consequence of the equations (15), we have not only (16), which is very easily proved to be the case, but also (17), ~VU=0, ^- ax dy , ~ dz .(18), as will be shown in a subsequent paper "On Points of Inflection." [I think never written.] And from the six equations (15), (18), the six quantities x\ y\ z*, yz, zx, xy, may be linearly eliminated : we have V U= Aa? + By 3 + Cz 3 where the values of the coefficients A, B, ... A are and the result of the elimination is k, , j , I , j lt k =0 .(20), .(21). a , k , b , ii , i , I , &i ji , . i c , ii , j , I A, K lt J, L, J,, K K, B ,/!,/, L, K, J lt I , C, I lt J, L {K (IT) is consequently, as is well known, a function of the twelfth order in a, 6, c, 35] ORDER WITH THREE VARIABLES. 233 The equation VU=0, combined with that of the curve, determine, as Dr Hesse has demonstrated in the paper quoted, the points of inflection of the curve. It may be inferred from this, that if U reduce itself to the form ...(22), U = (ax* + fiy 2 + yz* + Ztyz + ZKXZ + Z\xy) P=VP. P a linear function of x, y, z : then V U takes the form .(23), where p is of the second order in the coefficients of P, and also in the coefficients a, /?, 7, i, K, X : and cr is equal to the determinant a, X, K (24), -, i , 7 multiplied by a numerical factor. If U is of the form U=PQR (25), then VU=pPQR = pU (26), and this equation is consequently the condition of the function U being resolvable into linear factors. The equation in question resolves itself into a b c i j k i-, j-, k I J > J L a system which must contain three independent equations only. It would be interesting to verify this a posteriori. c. 30 234 [30 36. ON THE GEOMETEICAL REPRESENTATION OF THE MOTION OF A SOLID BODY. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 164 167.] LET P, Q, R, be consecutive generating lines of a skew surface, and on these take points p , p\ q , q; r , r such that pq , qr are the shortest distances between P and Q, Q and R, &c. Then for the generating line P, the ratio of the inclination of the lines P, Q to the distance pq" is said to be "the torsion," the angle q pq is said to be the deviation, and the ratio of the inclination of the planes Qpq and Qqr to the inclination of P and Q is said to be the "skew curvature." And similarly for any other generating line; so that the torsion and deviation depend on the position of the consecutive line, and the skew curvature on the position of the two consecutive lines. The curve pqr is said to be the minimum distance curve [or curve of striction]. {When the skew surface degenerates into a developable surface, the torsion is infinite, the deviation a right angle, the skew curvature proportional to the curva ture of the principal section, i.e. it is the distance of a point from the edge of re gression, multiplied into the reciprocal of the radius of curvature, a product which is evidently constant along a generating line. Also the curve of minimum distance becomes the edge of regression.} A skew surface, considered independently of its position in space, is determined when for each generating line we know the torsion, deviation, and skew curvature. For, assuming arbitrarily the line P and the point p, also the plane in which pq lies, the position of Q is completely determined from the given torsion and deviation; and then Q being known, the position of R is completely de termined from the skew curvature for P, and the torsion and deviation for Q; and similarly the consecutive generating lines are to be determined. Two skew surfaces are said to be " deformations " of each other, when for correspond ing generating lines the torsion is always the same. Thus a surface will be deformed if considering the elements between the successive generating lines P, Q as rigid, these 36] ON THE GEOMETRICAL REPRESENTATION OF THE MOTION OF A SOLID BODY. 235 elements be made to revolve round the successive generating lines P, Q and to slide along them. {They are " transformations", when not only the torsions but also the deviations are equal at corresponding generating lines: thus, if the sliding of the elements along P, Q be omitted, the new surface will be, not a deformation, but a transformation of the other.} No two skew surfaces can be made to roll and slide one upon the other, so that their successive generating lines coincide, unless one of them is a deformation of the other : and when this is the case, the rolling and sliding motions are completely determined. In fact the angular velocity of the generating line is the angular velocity round this line, into the difference of the skew curvatures of the two surfaces; the velocity of translation of the generating line in its own direction is to the angular velocity of the generating line, as the difference of the deviations is to the torsion. {This includes also the case in which one surface is a transforma tion of the other, where the motion is evidently a rolling one.} A skew surface moving in this manner upon another of which it is the deformation, may be said to " glide " upon it. We may now state the kinematical theorem : " Any motion whatever of a solid body in space may be represented as the gliding motion of one skew surface upon another fixed in space, and of which it is the defor mation." a theorem which is to be considered as the generalization of the well-known one "Any motion of a solid body round a fixed point may be represented as the rolling motion of a conical surface upon a second conical surface fixed in space." and of the supplementary theorem " The angular velocity round the line of contact (the instantaneous axis) is to the angular velocity of this line as the difference of curvatures of the two cones at any point in the same line, to the reciprocal of the distance of the point from the vertex." The analytical demonstration of this last theorem is rather interesting: it depends on the following formulae. Forming two determinants, the first with the angular velo cities round three axes fixed in space, and the first and second derived coefficients of these velocities with respect to the time ; the other in the same way with the angular velocities round axes fixed in the body; the difference of these determinants is equal to the fourth power of the angular velocity into the square of the angular velocity of the instantaneous axis. To show this, let p, q, r be the angular velocities round the axes fixed in the body ; u, v, w those round axes fixed in space ; &> the angular velocity round the instantaneous axis ; V, fl the two determinants : the theorem comes to V - n = M , where M = <w 2 (p 2 + q 2 + r 2 - w 2 ), or or (u 2 + v 2 + w - - w -). Here u = a p + (3 q + 7 r , v = a p +fi q + yr, w = a"p + (3"q + y"r ; 302 236 ON THE GEOMETRICAL REPRESENTATION OF THE MOTION OF A SOLID BODY. [36 whence u = a. p + ft q + 7 r , (the remaining terms vanishing as is well known); and therefore vw v w = a (qr 1 q r) + ft (rp r p) + 7 (pq p q), wu w u = of (qr q r) + ft (rp r p) + 7 ( pq p q), uv u v = a." (qr q r) + ft" (rp r p) + 7" (pq p q). Hence vw" v"w = a. (qr" q"r) + ft (rp" r"p) + 7 (pq" p"q) + u a? wu" - w"u = a (qr" - q"r) + ft (rp" - r"p) + 7 (pq" - p"q) + v o? - vwa> , uv" - u"v = a" (qr" - q"r) + ft" (rp" - r"p) + 7" (pq" - p"q) + w ar - and multiplying these by u , v , w , and adding, the required equation is immediately obtained. In fact, if r be the distance of a point in the instantaneous axis from the vertex, and p, o- the radii of curvature of the two cones at that point, then as may be shown without difficulty : and the angular velocity of the instantaneous axis Jfi is given by the equation OT = - ; hence the relation between the two angular veloci ties is = --- - p cr r 37] 237 37. ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 167 173 and 264274.] THE difficulty of completing elegantly the solution of this problem, in the case where no forces act upon the body, arises from the complexity and want of symmetry of the ordinary formulse for determining the position of one set of rectangular axes with respect to another set ; in consequence of which it has hitherto been considered necessary to make a particular supposition relative to the position of the fixed axes in space, viz. that one of them shall be perpendicular to the "invariable plane" of the rotating body. But some formulse for the above purpose, given also by Euler, are entirely free from these objections. Imagine two sets of axes Ax, Ay, Az, Ax t , Ay,, Az r The former set can be made to coincide with the second set, by a rotation 6 round a certain axis AR, inclined to Ax, Ay, Az at angles f, g, h. (As usual f, g, h are the angles RAx, RAy, RAz considered as positive, and the rotation is in the same direction as a rotation round Az from x towards y.) This axis may be termed the resultant axis, and the angle 6 the resultant rotation. The formulae of Euler express the coefficients of the transformation in terms of the resultant rotation and of the position of the resultant axis, i.e. in terms of 6 and of the angles f, g, h, whose cosines are connected by the equation cos 2 / + cos 2 # + cos 2 h = l. This idea was improved upon by M. Rodrigues (Liouv. torn. v. p. 404), who intro duced the quantities tan 9 cos/, tan 6 cos g, tan 9 cos h, (quantities which will be represented by A, ft, v) by means of which he expressed the 238 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 coefficients as fractions, the numerators of which are very simple rational functions of the second order of \ p, v, and which have the common denominator (1 + X 2 + /r + i> 2 ). These quantities may conveniently be termed the "coordinates of the resultant rotation," and the denominator or the square of the secant of the semi-angle of resultant rotation will be the " modulus " of the rotation. The elegance of these results led me to apply them to the mechanical question, and I gave in the Journal (vol. in. p. 224), [6], the diffe rential equations of motion obtained in terms of \, ji, v: which I integrated as in the common theory, by supposing one of the fixed axes to be perpendicular to the invariable plane. Though my attention was again called to the subject, by the connexion of some of these formula? with Sir William Hamilton s theory of quaternions, no other way of performing the integration occurred to me. The grand discovery however of Jacobi, of the possibility of reducing to quadratures the two final differential equations of any mechanical problem, when the remaining integrals are known, induced me to resume the problem, and at least attempt to bring it so far as to obtain a differential equation of the first order between two variables only, the multiplier of which could be obtained theoretically by Jacobi s discovery. The choice of two new variables to which the equa tions of the problem led me, enabled me to effect this with the greatest simplicity; and the differential equation which I finally obtained, turned out to be integrable per se, so that the laborious process of finding the multiplier became unnecessary. The new variables fl, v have the following geometrical interpretations, fl = k tan ^ 6 cos /, where k is the principal moment, 6 as before the angle of resultant rotation, and / is the incli nation of the resultant axis to the perpendicular upon the invariable plane, and v = k" cos 2 J ; where, if we imagine a line A Q having the same position relatively to the axes in fixed space that the perpendicular upon the invariable plane has to the principal axes of the rotating body, then J is the inclination of this line to the above perpendicular. To the choice of these variables I was led by the analysis only. It will be seen that p, q, r are functions of v only, while X, ft, v contain besides the variable fi. In obtaining these relations a singular equation O 2 = KV k- occurs (equation 13), which may also be written 1 + tan 2 ^ Q cos 2 / = sec 2 \ 6 cos 2 \ J, in which form the inter pretation of the quantities /, J has just been given. The equation (17), it may be remarked, is self-evident : it expresses that the inclination of the resultant axis to the normal of the invariable plane, is equal to the inclination of the same axis to the line AQ. Now the resultant axis having the same inclination to the axes fixed in space as it has to the principal axes, and the line AQ the same inclinations to these fixed axes that the normal to the invariable plane has to the principal axes, the truth of the proposition becomes manifest. The correspondence in form between the systems (10) and (14) is also worth remarking. The final results at which I arrive are, that the time and the arc whose tangent is fl -r- k, are each of them expressible as the integrals of certain algebraical functions of v. The notation throughout is the same as that made use of in the paper already quoted. The equations of rotatory motion are , _ dp _dq _ dr _ d\ _ d/j, _ dv ~"~~~~~ ( ] 37] ON THE ROTATION OF A SOLID BODY BOUND A FIXED POINT. 239 where A = l {( 1 + X 2 ) p + (X/* - v ) q + (\v + ^} r], } M = l[(>X + z/) p + ( 1 +/Jt?)q + (f*v-\)r], )- ..................... (3). N ={(*\ -/*) p + Oiv+X) 2 + (l + *)f},, [whence also XA + fiM + vN = % K (\p + fiq + vr) .............................. (3 bis)]. In the case where the forces vanish, the first three equations become simply P = ^(B Q=l(C R = (A and here the usual four integrals of the system are Ap* + Btf + Ci* = h ................................... (5), Ap (1 + A. 2 - p? - v*) + 2Bq (Xyu, - i/) + 20?- ( v\ + fi) = a (1 + X 2 + ^ + v~), ^ 2Ap (V + v) + Bq (1 + p? - v- - X 2 ) + 2O (pv - X) = b (1 + X 2 + /r + v 2 ), - ...... (6), 2Ap (v\ - /*) + 2% (jju- + X) + Cr (1+ i/ 2 - X 2 - ^ 2 ) - c (1 + X 2 + /r + K 2 ), , or as they may also be written, a (1 + \- - p - v") + 2b(X/t + v) +2c (i;\ -fi) = Ap(l+ X- + fj? + v 2 ), "| 2a (\/A - v) + b (1 + v? - 1>* - X 2 ) + 2c (pv + \) = Bq(l+\-+ ^ + v-), j- ...... (6 bis) ; 2a (i/X + /a) + 2b (/*!/ - X) + c (1 + v* - X 2 - /*-) = Cr(l+ X- + ^ + ir), j to which we may add, ^ 2 > 2 + 5 2 2 2 + CV = P ................................. (7): where /t 2 = a- + b 2 + c 2 ....................................... (8). Introducing the quantities K, fl, (the former of which has been already made use of) given by the equations -i +*+*+". 1 _ (9) fl = \Ap + pBq + vCr, ) 240 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 The equations (6) may be written under the form + ZfiCr - 2vBq = K (Ap + a) - 2Ap, } +2/in + 2vAp = K (Bq +ty-2Bq, .................... (10), 2\Bq - ZfjuAp + 2vl = K (Cr + c) - 2CV , j whence also, multiplying by Ap, Bq, Cr, and adding, 2fl 8 = K {k 2 + (Apa + Bq}) + Ore)} - 2& 2 ....................... (11), or writing fc + (Apa, + Bqb + CVc) = 2y ............................. (12), this becomes & = Kv-k 2 ....................................... (13): an equation, the geometrical interpretation of which has already been given. From the equations (10) we deduce the inverse system ali - bCV + cBq = 2\u - ZlAp, > aCV + bfl - cAp = 2pv - tlBq , v ....................... (14), + l>Ap + cH = 2vv - flCr , } which are easily verified by multiplying by O, Cr, Bq; or by Cr, fl, Ap ; or Bq, Ap, fl : adding and reducing, by which means the equations (10) are re-obtained. Hence also if for shortness bo + cr, /-j f-\ we have, multiplying by p, q, r, and adding, Q,-V = 2v(\p + fjq + vr)-nh ........................... (16). To these may be added the equation fl = aX + b/i + cv .................................... (17), which follows immediately from either of the systems (10) or (14). We may also put the equations (10) under this other form, 2XO - 2/xc + 2z^b = K (Ap + a) - 2a, ^ 2Xc + 2yuft - 2i/a = K, (Bq + b) - 2b, > ............... (10 bis). -2Xb +2yLta +2vl= K (Cr +c)-2c )y It may be remarked now, that p, q, r are functions of v ; since we have to deter mine these quantities, the three equations Ap 2 + Bf + Cr 2 = h, AY + &q* + C 2 r* = fc, ........................... (18). Apa, + Bqb + CVc = 2v - k", j 37] ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. 241 Also \, //,, v are given by the equations (14) as functions of p, q, r, O, i.e. of v, fl; so that every thing is prepared for the investigation of the differential equation between v, fl. To find this we have immediately (19), from the equations (4) and (15). V is of course to be considered as a given function of v. Again, fldn = $(Kdv + vdtc) ................................. (20), where die = 2 (\d\ + ^dp, + vdv) .............................. (21) ; or from the equations (1), (3), [and (3 bis}], dic = K (\p + /m,q + vr)dt ................................. (22). Hence, from (16), 2vdK = K {l(h + 3>)-V}dt .............................. (23); or 2 (vdx + tcdv) = xfl (h + 3>) dt ............................. (24), whence dfl = ^tc(h + <) dt, O2 I J.2 V (25), and therefore, from (19), nrr&=^w-* v (26), ii -)- A uv the required differential equation, in which <l>, V are given functions of v, i.e. they are functions of p, q, r by the equations (15), and these quantities are functions of v by (18). The variables in (26) are therefore separated, and we have the integral equation where 8 is the constant of integration. The equation (19) gives also and thus the solution of the problem is completely effected. The integrals may be taken from any particular value v of v. The variable H may be exhibited as the integral of an explicit algebraical function, by recurring to the variable < of the paper quoted. Thus if Ap? + % 2 + Cr? = A, Bq lo + Gr u c = 2v -k 2 ; 31 242 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 then the values of p, q, r are respectively where . d6 2dv dv -. d<b and then 4 tan" 1 -j- = 28 + in which form it is exactly analogous to the equation there obtained, p. 230, [6, p. 34] 4 tan " 1 *-/j (Jc + Cr) pqr 0?i tfie Variation of the Constants, when the body is acted upon by Forces. The dynamical equations of a problem being expressed in the form ddTdTdV ^_ = dt dp d/jb dp ddT_dT = dV dt dv dv dv suppose the equations obtained from these by neglecting the function V, are integrated; each of the six integrals may be expressed in the form a=f(\, n, v, V, /, i/, t\ where a denotes any one of the arbitrary constants. Assume dT dT dT d* = u W = v Tv = w > then \ , fi, v may be expressed in terms of X, p, v, u, v, w, and the integrals may be reduced to the form , p, v, u, v, w, t). These equations may be considered as the integrals of the proposed system, taking into account the terms involving V, provided the constants [say a, b, c, d, e, t] be sup- 37] ON THE ROTATION OF A SOLID BODY BOUND A FIXED POINT. 243 posed to become variable. We have, in this case, by Lagrange s theory of the variation of the arbitrary constants, the formulae da dV , dV , dV dV dV , . , _ . _ da db\ /da db da db\ \du d\ d\ duj \dv dp dp dvj \div dv dv dw) and in which F is supposed to be expressed as a function of a, b, c, d, e, f, t. Thus the solution of the problem requires the calculation of thirty coefficients (a, b), or rather of fifteen only, since evidently (a, b) = - (b, a). It is known that these coeffi cients are functions of a, b, c, d, e, f, without t ; so that, in calculating them, any assumed arbitrary value, e. g. t = 0, may be given to the time. In practice, it often happens that one of the arbitrary constants, e.g. a, may be expressed in the form a = F(\ p, v, u, v, w, t, b, c, d, e, /), where b, c, d, e, f are given functions of A, p, v, u, v, w, t. In this case, it is easily seen that we may write (1\ f/ 7 \1 . / 7 \ ^^ / 7 7 \ CtCt / -, >. CLCt/ , ~ - . Qj(JL a, b) = {(a, &)} + (c, 6) -j- + (d, b)-r-j + (e, b) -y- + (/, b) , where, in the calculation of {(a, b)}, the differentiations upon a are performed without taking into account the variability of b, c In the particular problem in question, the following are the values of the new variables u, v, w (Math. Journal, memoir already quoted, [6]), 2 u = -( Ap vBq + pCr), ... (W} K ^ 2 v = - ( vAp + Bq - \Cr), 2 w = - ( pAp + \Bq + Or) ; fC equations which may also be expressed in the form 2 A p = ( 1 + X. 2 ) u + (\p + v ) v + (v\ p) w (30), 2iBq = (\p y)M + (l+/A 2 ) / y-(- (pv + X) w , 2Cr = (v\ + p)u + (pv X) v + ( 1 +v*)w, or putting for shortness \u + pv + vw = is (31), 312 244 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 these become 2Ap = \^ + u+vv-fji,w, .............................. (32). 2Bq = pin vu + v + \w, 2O = VW + fJM \V + W, whence also 2fl = /ccr .......................................... (33). Substituting the values of Ap, Bq, Or, given by (30) in the equations (6), we deduce 2a = X-cr + u vv + /j,w ................................. (34), 2b = /itjr + vu+ v \w, 2c = VST fj,u + \v + w, whence also 2 (aX + byu, + cv) = KTX ................................. (35), which in fact follows from (33) and (17). And likewise the inverse system, 2 u = -( a + t/b /AC) .............................. (36;. b + Xc), 2 w = - ( /*a Xb + c). /c It is easy to deduce k 2 = | K [w 2 + v 2 + w 2 + OT 2 ] .............................. (37), *] ..................... (38). Again, from the equations (10 bis), K (bCV - cBq) = - 2X (a 2 + b 2 + c 2 ) + 2a (Xa + fib + i/c) + 2 (bi; - = - 2XA; 2 + 2 (a + bv - C/A) H and, forming also the similar expressions for K (cAp aCV), and K (&Bq b Ap), we thus obtain nw--PX = bCr-c5 ? ................................. (39), 2 flv /C 2 fC = aBq cAp : to which many others might probably be joined. The constants of the problem are a, b, c, h, e, &. Of these a, b, c are given as functions of X, p, v, u, v, w, by the equations (34) ; in which -cr is to be considered as 37] ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. 245 standing for \u + /JLV + vw. {These determine k 2 , which is however given immediately by (37).] As for h, we have ~ A \r*-JtrJ D \ J - 1 1J f1\ v * t where Ap, Bq, Cr are given as functions of X, p, v, u, v, w by (32), in which also or stands for \u + n*v + vw. Again, [dv (h + <&)dv uV in each of which V, <> are functions of v, and of a, b, c, h, partly as entering explicitly into these functions, partly as contained implicitly in p, q, r, which enter into V, <&, and are functions of v, h, k given by (18). After the integration v is to be considered a function of X, p, v, u, v, w given by (38). Both of the integrals may be supposed taken from a certain value u of v, which may be considered as an absolutely invariable arbi trary constant, since without it we have the right number, six, of arbitrary constants. First to find (a, b), (b, c), and (c, a). From (34) we have (a, b) = { ( 1 + X 2 ) (fm w) (\u + OT) (X//, + v) + (X/A v)([AV + Iff) (\V + W) ( 1 + yur) + (v\ + /*) ( u + /j,w) (\w v) (/ji,v X)} whence the system (b, c) = -a, (c, a) = -b, (a, b) = -c (40). Also we may add (k, a) = y (a, a) + 7 (b, a) + 7 (c, a) = 0, or (k, a) = 0, (k, b) = 0, (k, c) = (41 ), which will be useful in calculating some of the following coefficients. Proceeding to calculate (a, h), (b, h), (c, h). It is seen immediately that (a, h) =2{p (a, Ap) + q (a, Bq) + r (a, Cr)}, where Ap, Bq, Cr, are given by the equations (32), so that (a,Ap) = ${ (l+\ 2 )(\u+ er)-(l +X 2 )(Xw+ BT) v) (Xv w) (X/u, + v ) (Xv + w) v + \w) (v\ /u,) ( v + \w)} (42). f OF THE MDTHIVl RSITT ^V. <Z> 246 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 Similarly (a, Bq) = { ( 1 + X 2 ) (pu + w) ( \u + w) (X/i v } + (\fi - v ) GUI; + BT) - ( \v + w](l + p-) + (v\ + p, ) (JJLW u} ( v + \w) (/j,v + X )} i.e. (a, Bq} = 0, (43), and similarly (a, Or) = ; whence (a, A ) = 0, and therefore (b, A) = 0, (c, A) = (44) ; also (k, A)=0, (45). Next we have to determine (a, e), (b, e), (c, e). Here e being a function of a, v, w, X, p, v, a, b, c, A, we must write (a, e) = {(a, e)} + (a, b) ~ + (a, c) j- + (a, A) , , M , -u de de i. e. (a, e) = (a, e) + b ^ c -jr . * ft C* ff n But e=t 2 1^7 ; and thence {(a, e)} = = (a, v), and v is given immediately as a function of X, p, v, u, v, w, by the equation (38). Hence (a, v) = I [ ( 1 + X 2 ) {(1 + K) uv + X^ 2 } - ( \u + r) {u + X (1 + *) tsrj + (XyU, V } {(1 + K) V-ST + yLtOT 2 } ( \V + W) {v + fJL (1 + K) OTJ + (i/X + fi) {(1 + K)WW+ I/OT-)} - (- v + Xw) {w +v (1 + K) <&}] = ^ {(1 + K) sni X (1 + K) TS- + \K X (w 2 + v- + w 2 ) ?<CT| XCT - X (u- + V 2 7,2-\ / 9/^ 2 X \ = iur---=i(Ou-- -) (by (37) and (33)}, K \ K J ............................................................ (46); whence {(a, e)} = _^ (bCr cBq), The terms b -= c -^- are evidently of the form F (v) F (v ). (ZC (XD If therefore we suppose v = v , we have (a, e) = -(b(7r -c%) ................................ (47), 37] ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. 247 if p , q , r , V n refer to the value v of v, i.e. if * + Cr* = h ................................. (48), {This implies evidently Ap a si + Bqjo + Cr c = 2u - k\ h * ChC 1 7? ^ 1 (W dc db V V</ an equation which it is interesting to verify. In fact, from the value of e ,de de f . / d d \ 1 f , 1 / dV dV\ dc db J v \ dc db) V " J v V 2 \ c?c~ C db/ or we have to show that if for shortness g=b-- dc db Now V containing a, b, c explicitly, and also as involved in p, q, r, we have ~ suppose. The equation to be verified becomes du - - = 2 Now, observing that 8k = 0, we have A 2 p8p + B 2 qSq + C 2 r8r = 0, A a8p + B b8q + C cSr = - (bCr - cBq). Also, dp do dr -- ^ + Cr- r - = 0, dv dv - dv * dv dv dp m da n dr -/^ + J Bb- 7 ^+C Cj- av dv dv 248 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 Avhence evidently dp_ -2 . dq_ -2 , d/r_ = - 2 gr dv~\)Cr-cBq P> dv~\>Cr-cBq q dv bCr - cBq dV_ -2 , ^-~ or the equation to be verified is simply which follows immediately from the three equations just given for the determination of dp dq dr , dv dv du * From these values also (k, e)=0 Next, to calculate (A, e), but the three last terms being evidently such as to vanish for v=v , we may neglect them, and consider (h, e) as the value which {(h, e)} assumes for this value of v. Now \(h, e)} = 2p {(Ap, e)} + 2 9 {(J? ?> e)} + 2r {(Or, e)}, where {(Ap, e)} = -^(Ap, v), and (Ap, t>) = i[ ( 1 + ^ 2 ) {(1 + ) " OT + x ^ 2 ] - ( Xw + CT ) 1" + X (1 + *) r] + (\fji + v) {(1 + ) w + Atw 2 } - (\w - w) [v + n (1 + K) BT} + (i;\ - /i) {(1 + ) WOT + OTST 2 } - ( V + XW) {?/; +!/(!+*) OTJ = 1 {(1 + K ) vu + \K^ - X (1 + *) *r 2 - X (w 2 + v 2 + iv 2 ) - vu} = (a, v) whence {(Ap, e)} = - ~ (bCr - cBq) . . ..... (51), and therefore {(% e)] = - 37] ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. 249 whence {(h, e)} = 2, and therefore (h, e) = - Next, to find (a, 8), (b, 8), (c, 8), (h, 8), (52). (observing where (a, = 8 + 8" suppose, (a, 8) = (a, 8 ) + (a, 8"), (a, ) = -(a, .) + /e 2 <nr 2 + 4& 2 = 4 (fl 2 + ^) = , = (a, KV + X 2 ) (*u H- 2XCT) -(\u+ -BJ) X v} (KV + 2yu,t*7) ( Xw + w) /c/i v + \w KV = $ K (u + XOT) = Ap- vBq + fjtCr + XH = ^ (a + Ap) K by equations (29), (33), and (10) ; that is (a, 8 ) = - (a + Ap). (53), Also (a, 8") ft (a, v) + (a, b) ~ + &c. whence or putting v = v , and therefore also c. (a, 8) = {a + Ap - - (b(7r - cBq~)} + Fv - Fv , (a, 8) = x { I- (b,8)=A (c, S) = {c (54), 32 250 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 Again, ( h, 8) = 2p (Ap, 8) + 2q (Bq, B) + 2r (Cr, 8), (Ap, B) = (Ap, xvr) = \ { ( 1 + X 2 ) (KU + 2Xw) - (\u + VT) K\ + (Xp + v ) (KV + 2/Lt-sr) (\v w) KJJL + (v\ + /* ) (KW + 2vm ) ( v + \w) KV } = |- K (u + X-OT) = ^ ; (a + J._p) .................................... (55) ; k therefore (Ap, 8 ) = (a + Ap) ..................................................... (56), 25lt Fv - Fv , (Ap, B)=~{ a -Ap- and similarly for (Bq, 8), (Cr, S). Substituting, and neglecting the terms which vanish for v = v 0> fi ^ h (h, $)=- (h, 8) = ..................................................................... (57). Lastly, to find (e, 8), where, in {(e, 8)}, the differentiations upon e are supposed not to affect the constants a, b, c. Neglecting the terms which vanish for v = v , (e, 8) ={(e, 8)}, {(e, 8)} = {(e, 80} + {(e, 8")}, {(e, 80} = [{(, B )}]+(e,k)^=[{(e, 8 )}]; where, in [{(e, 8 )}], the differentiations upon e and 8 do not affect the constants. 37] ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. 251 neglecting the terms which vanish for v = v , therefore (e, B) = [{(e, 8 )}] + [{(e, 8")}] since Hence (u, K) K) v CT} (2i/ + K (if + tf + w 2 ) + 2 (1 + K ) (K - 1) CT 2 + K (1 + ) - /C ( - 1) -CT 2 - ( + 1) CT 2 } therefore = - a, (c, a) = - b, (a, b) = - c, Hence, recapitulating, (b, c) (a, A)=0, (b,A) = 0, (a, e) =-=-(bC7r -c%), V (b, e) = - ^ (c4p - a(7r ), (c, e) = - =- (b, S) = ^ V (c, S) = * (A, a) = o, (59), .(60). (61), 322 252 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 and therefore da dV .dV 1 ., ,dF k , h + ,. f1 K xl dV + 2 a ^ dV dV 1 dF k , "~ k dV - db dF dF + {b + % - -- i fc I Cr " ~raff -b^L )l^n+ * F V dc j V de to which we may join dk - dV (Q*\ Tt llE We have thus the complete system of formulae. 38] 253 38. NOTE ON A GEOMETRICAL THEOREM CONTAINED IN A PAPER BY SIR W. THOMSON. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846;, pp. 207, 208.] IT is easily shown that if three confocal surfaces of the second order pass through a point P, then the square of the distance of this point from the origin is equal to the sum of the squares of three of the axes, no two of which are parallel or belong to the same surface (the squares of one or two of the axes of the hyperboloids being considered negative) ; i. e. if - a? y 2 z 2 r + : + r = then In fact these equations give (a 2 -6 2 )(a 2 -c 2 ) *_(b* + h)(b* + k)(b 2 +l) (& 2 -a 2 )(6 2 -c 2 ) (c 2 - a 2 ) (c 2 - 6 2 ) and adding these and reducing, we have the relation in question; which is also imme diately obtained by forming the cubic whose roots are h, k, I, 254 NOTE ON A GEOMETRICAL THEOREM &C. [38 From this property may be deduced the theorem given by Mr Thomson in the preceding memoir ["On the Principal Axes of a Solid Body," pp. 127 133 and 195 206, see p. 205]. In fact, writing r- = a? + y 1 + z", and k = -a 2 -b 2 -c 2 + h, we see that in consequence of these relations the equations -. + + -, = 1, a- b 2 c 2 a? if- z- i ___/ i i 1 I o . J n 7 I . o i _o 7. r 2 + a 2 -h r 2 + b 2 -h r 2 + c 2 - a? y 2 z 2 ~~2 T, + f*~l + ~ l* = are equivalent to two independent equations, i.e. the third can be deduced from the two first. Now the first equation is that of an ellipsoid (or generally a surface of the second order, since a, b, c are not necessarily real). The second is that of what may be called a conjugate equimomental surface, defining the term as follows: "The conjugate QL 77 Z" equimomental surfaces of an ellipsoid (or surface of the second order) i + r- 2 + 1 = ^> are the equimomental surfaces derived in the usual manner from any surface of the second x 2 y 2 z 2 y? y~ ^ order 7 + , y , + -,- - = 1, which is confocal with the conjugate surface + j-<, + -* = 1 h a 2 h b 2 h c- a- b- c- of the given ellipsoid," viz. by measuring along any line through the centre distances equal to the axes of the section by a plane through the centre perpendicular to this line, and taking the locus of the points so determined for the equimomental surface. The third equation is that of a surface confocal with the given ellipsoid; hence the theorem, "The curves of curvature of a given ellipsoid lie upon a system of conjugate equimomental surfaces." But since the first and second equations are evidently satisfied by the combination of the first equation with the relation r 2 = h, which is that of a sphere, we have also, "The curve of intersection of the ellipsoid with any one of the conjugate equimomental surfaces, is composed of the line of curvature and a spherical conic." And these two curves being each of them of the fourth order make up the complete curve of intersec tion, which should obviously be of the eighth order. It would be an interesting question to determine the relations existing between the curve of curvature and the spherical conic, which have been thus brought into connec tion by means of the conjugate equimomental surfaces; i.e. between the two curves obtained by combining the equation of the ellipsoid with i I 7.0 . 7_ r 2 = a 2 + b 2 + c 2 + k, respectively: but it will be sufficient at present to have suggested the problem. 39] 255 39. ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER. [From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 274 278.] LET U = Ax 2 + By 2 + Cz 2 +2Fyz + 2Gx;z+2Ha;y = 0, be the equation of a surface of the second order referred to its centre, and let ax + a y + o! z be the equation of one of its diametral planes ; then, as usual (A -u)a + Ha + Go! = 0, Ha+(B- u) a! + Fa." = 0, Ga + Fa + (0 - u) a" = 0, which are equivalent to two independent equations, and consequently capable of deter mining the ratios a : a : a.", provided that u satisfy the cubic equation that is obtained by eliminating ct, a. , a! from the three equations. We have from the second and third, from the third and first, and from the first and second equations respectively, : : " = , : , : <&, = &, : to, : $, = <&, : $, : t ; where, if = BC -F* , T& = FG -Off, ,, to,, (>,, jF /; CGr,, yfy, are what these become when A, B, C are changed into 256 ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER. [39 A a, B u, u, so that &, = &-(# + G )t* + w 2 , t = <& - (A + B) u + u 3 , Hence the equation ax + a!y + a"z = may be written in the three forms or, what comes to the same thing, as follows, (Ax Gz) + vx = 0, <&z + u(Gx + Fy + Gz] in which for shortness v has been written instead of The elimination of -u, v from these equations gives a result = 0, where is a homogeneous function of the third order in x, y, z\ and this equation, it is evident, must belong to the three diametral planes jointly, i.e. % must be the product of three linear factors, each of which equated to zero would correspond to a diametral plane. Thus the system of diametral planes is given by + (&z, Ax + Hy + Gz, x l = 0, + $z, Hx + By +Fz, y + <&z, Gx + Fi/ +Cz, z or developing the determinant, as follows, + { G (QD - + { H(8L- + { F (&- + {-H(< - + {-F(8L- + - G i3 - - (C - B) - (H$ - -$(B- A] -(Gift- + ^ (C - B) A - -f OT xyz; 39] ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER. 257 or reducing 6 = [F (G 2 - H 2 ) -GH(G- B}} a? + {G(H 2 -F 2 )-HF(A-C)}y 3 + {H (F 2 - G 2 ) - FG (B-A)} z 3 + {G(A-B)(B-C) + FH(A+B-2C) + G(F 2 +G 2 - -H(( In the case of curves of the second order, the result is much more simple ; we have Ax + Hy, x = 0, Hx+By, y i.e. =H(f-tf) + (A-B)xy = 0, for the equation of the two diameters. The above formulae may be applied to the question of finding the diametral planes of the cone circumscribed about a given surface of the second order, (or of the lines bisecting the angles made by two tangents of a curve of the second order). Considering the latter question first : if be the equation of the curve, and a, ft the coordinates of the point of intersection of the two tangents, the equation of the pair of tangents is or making the point of intersection the origin, ^ + P-l\-( + Py} 2 = o i. e. (fix - ay) 2 - (b 2 x 2 + a 2 y 2 ) = ; whence A = ft 2 - b \ B = a. 2 -a 2 , H=- a/3, and the equation to the lines bisecting the angles formed by the tangents is aft (x- - y 2 ) -{a. 2 - ft 2 - (a 2 - b 2 )} xy = 0, which is the same for all confocal ellipses; whence the known theorem, "If there be two confocal ellipses, and tangents be drawn to the second from any point P of the first, the tangent and normal of the first conic at the point P, bisect the angles formed by the two tangents in question." c. 33 258 ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER. [39 In the case of surfaces, the equation of the circumscribing cone referred to its vertex as origin, is whence B = 7 2 a 2 + ofc 2 - a 2 c 2 , C = a?b 2 + /3 2 a 2 -6 2 a 2 , Hence, omitting the factor 6 2 c 2 a 2 + c 2 a 2 /3 2 + a 2 6 2 7 2 a 2 6 2 c 2 , we have & = a 2 - a 2 , and the equation of the system of diametral planes becomes = 0- a 2 #7 (c 2 - b 2 ) a? + /3 2 7 (a 2 - c 2 ) # 3 + 7^/3 (b 2 - a 2 ) ^ + 7 a {a 2 (c 2 - 6 2 ) + /S 2 (6 2 + c 2 - 2a 2 ) - 7 2 (b 2 - a 2 ) -f (6 2 - a") (c 2 - 6-)) //^ + a^8 {- a 2 (c 2 - 6 2 ) + /3 2 (a 2 - c 2 ) + 7 2 (c 2 + a 2 - 26 2 ) + (c 2 - fr) (a 2 - c 2 . ] ^^ + 73 {a 2 (a 2 + b 2 - 2c 2 ) - /S 2 (a 2 - c 2 ) + 7 2 (6 2 - a 2 ) + ^a 2 - c 2 ) (6 2 - a 2 ;] xf 2^9 (a 2 (c 2 - 6 2 ) - /8 2 (a 2 - c 2 ) - 7 2 (6 2 + c 2 - 2a 2 ) - (a 2 - c 2 ) (c 2 - 6 2 )} ?/-^ - 7 { - a 2 (c 2 + a 2 - 26 2 ) + yS 2 (a 2 - c 2 ) - 7 2 (b- - a 2 ) - (6 2 - a 2 ) < a 2 - c 2 )} z\r - 7 { - a 2 (c 2 - 6 2 ) - /S 2 (a 2 + 6 2 - 2c 2 ) + 7 2 (6 2 - a 2 ) - (c 2 - 6 2 ) (6 2 - a 2 )} tfy -f ;(a 2 - 6 2 ) (6 2 - c 2 ) (c 2 - a 2 ) + (a 4 + /3y) (6 2 - c 2 ) - (/3 4 + 7 2 a 2 ) (c 2 - a 2 ) - ( 7 4 + a 2 ^) (a 2 - b-) + a 2 (b 2 - c 2 ) (2a 2 - 6 2 - c 2 ) + /S 2 (c 2 - a 2 ) (2b 2 - c 2 - a 2 ) + 7 2 (a 2 - fr) (2c 2 - a 2 - 6 2 )} and since this is a function of a 2 b 2 , 6 2 c 2 , and cr a 2 , the equation is the same for all confocal ellipsoids ; whence the known theorem, " The axes of the circumscribing cone having its vertex in a given point P, are tangents to the curves of intersection of the three surfaces, confocal with the given surface, which pass through the point P." 40] 259 40. ON THE THEORY OF INVOLUTION IN GEOMETRY. [From the Cambridge and Dublin Mathematical Journal, vol. n. (1847), pp. 52 61.] WHEN three conies have the same points of intersection, any transversal intersects the system in six points, which are said to be in involution. It appears natural to apply the term to the conies themselves ; and then it is easy to generalize the notion of involution so as to apply it to functions of any number of variables. Thus, if U, V... be homogeneous functions of the same order of any number of variables x. y ... , a function <"), which is a linear function of U, V ... , is said to be in involution with these functions. More generally may be said to be in involution with any system of factors of these functions : or if U, V . . . be given functions of x, y, z ... , homogeneous of the degrees m, n ... , and u, v, ... arbitrary homogeneous functions of the degrees r m, r n...; then, if % = uU+vV+..., is a function of the degree r, which is in involution with U, V... . The question which immediately arises, is to find the degree of generality of , or the number of arbitrary constants which it contains. And this is a question which, from the variety and interest of its geometrical inter pretations, has very frequently been treated of by geometers, though never, I believe, in quite so general a form, (the number r has almost always had particular values given to it, except in a short paper of my own, on the particular case of two curves, in the Journal, vol. in. p. 211 [5]). 1 There is also an analytical application of 1 The first suggestion of the problem is contained in a memoir of Euler s " Sur une contradiction apparente dans la doctrine des lignes courbes," Mem. de Berlin, t. iv. [1748] p. 219. It is noticed also in Cramer s Introduction a V analyse des lignes courbes [1750]. The following memoirs also have been published on the subject : Plucker, " Eecherches sur les courbes algebriques de tous les degres," Gerg. Ann. t. xix. [182829] p. 97 ; " Recherches sur les surfaces algebriques de tous les degres," p. 129 ; (a great number of memoirs on particular applications of the theory are contained in Gergonne ;) Jacobi, "De relationibus quae locum habere debent inter puncta intersectionis duarum curvarum vel trium superficierum dati ordinis, simul cum enodatione paradoxi algebraici," Crelle, t. xv. [1836]; Plucker, " Theoremes g6neraux concernant les equations d un degre quelconque entre un nombre quelconque d inconnues," Crelle, t. xvi. [1837], (but this last must be read with caution, as several of the theorems are incorrect, or at least stated without the proper limitations); and the Einleitende Betrachtungen, in Plucker s " Theorie der algebraischen Curven" [1839]. The following memoirs of Hesse, con taining developments relative to the case of three surfaces of the second order, may likewise be mentioned, "De curvis et superficiebus secundi gradus," Crelle, t. xx. [1840] p. 285; and " Ueber die lineare Construction des achten Schnitt-punctes dreier Oberflachen zweiter Ordnung, wenn sieben Schnitt-puncte derselben gegeben sind," Crelle, t. xxvi. [1843] p. 147. 332 260 ON THE THEORY OF INVOLUTION IN GEOMETRY. [40 the theory, of considerable interest, to the problem of elimination between any number of equations containing the same number of variables. Suppose, for instance, two equa tions, U = 0, V0, when U, V are homogeneous functions of x, y of the degrees m, n respectively. To eliminate the variables it is sufficient to multiply the first equation by x n ~ l , x n ~ 2 y..., y n ~ l , and the second by x m ~ 1 ..., y m ~ l , and from the equations so obtained to eliminate linearly the (m + n) quantities x" 1 ^- 1 , x m+n - 2 y..., y m + n ~\ But in the case of a greater number of equations it is not at first obvious how many new equations should be obtained ; and when a number apparently sufficiently great have been found, it may happen that the equations so obtained are not independent, and that the elimi nation cannot be performed. But in showing the connexion that exists between these different equations, the theory of involution explains in what manner a system is to be formed, which includes all the really independent equations, and gives the means of detecting the extraneous factors which appear in the result of the linear elimination of the different terms ; but I do not see at present any mode of obtaining the final result at once in its reduced form free from any extraneous factors. Let X, Y, ... be given homogeneous functions of the same degree of any number of variables, and suppose a, /3 . . . being constants, and the number of terms in the series being g ; contains therefore g arbitrary constants. If however, by giving to a, /3 ... particular values !, /3i ..., or or 2 , & ... , and representing by lt @ 2 ... the corresponding values of , we have identically j = 0, 2 = 0, . . . (h equations) ; then the constants in group themselves together into a smaller number g h of arbitrary constants. This supposes, however, that the last mentioned equations are linearly independent; if there are a certain number k of equations fy = 0, 3> 2 = . . . , (where 3> l} ^> 2) ... are linear functions of lf 2 ...) which are identically satisfied, inde pendently of the h equations, then the equations in question are equivalent to h k equations, and the function contains g (h k) or g h + k, arbitrary constants. Similarly if the functions <X> are not independent; so that the number of arbitrary constants really contained in is always Consider now the case of a function @, homogeneous of the r th degree in the variables x, y...{(0+l) in number}. Let U, V... be functions of the degrees m, n..., and suppose where u, v ... are arbitrary functions of the degrees r in, r n, ... [r is supposed throughout greater than m, n ...}. Suppose for shortness that the number of terms in the complete function of 6 variables, and of the order p, i.e. the quotient r/ng , is 40] ON THE THEORY OF INVOLUTION IN GEOMETRY. 261 represented by [p, 6} ; then the function @ contains apparently a number ([r-m, 6} + [r-n, 0] + ...) of arbitrary constants. But since we should have identically = by assuming u = LV, v = - LU, w = 0, &c.... (L the general function of the order r m-ri), or u = MW, v = Q, w = - MU (M the general function of the order r-m-p) &c., the number N must be diminished by [r mn, 0] + [rmp, 0] + [r - n p, 0] + . . . ; but the equations just obtained are themselves not linearly independent, and in conse quence of this the number of arbitrary constants has to be increased by [9 m n p, 0] + . . . ; and so on. Hence finally the whole number of arbitrary constants in the function 6 is N=[r- m, 6] + [r - n, 6} + [r -p, 6] + ... [r m n, 0] [r mp, 0] [r n p, 6] ... + [r m np, 0] + . . . + &c. &c (A). This however supposes that all the numbers r m, r n..., r m n..., are positive: whenever this is not the case for any one of them, the corresponding term is obviously to be omitted. With this convention the equation (A) gives always the correct number of arbitrary constants in : it will be convenient to represent it in the abbreviated form N ={r : m, n, p, ... : 0}. An expression analogous to this, for the particular case of r = m, but incorrect on account of the omission of all the terms after the second line, has been given by M. Pllicker (Crelle, torn. xvi. p. 55), and even some of his particular formulae are incorrect. But proceeding to examine some particular cases: if r >m+n+p+ ... 1, then in the expression (A) either no terms are to be omitted, or else the terms to be omitted reduce themselves to zero, so that N is given by this formula continued to its last term. It will be subsequently shown that in this case {r : m, n, p ... : 6} = [r, 0] mnp ...; or in the case of two or three variables, we have the theorem, " If a curve or surface of the order r be determined to pass through the mn points of intersection of two curves of the orders m and n, or the mnp points of intersection of three surfaces of the orders m, n, p; then if r>m + n 3, or r>m + n+p 4<, the curve or surface contains precisely the same number of arbitrary constants as if the mn or mnp points were perfectly arbitrary." This is natural enough ; the peculiarity is in the case where r }> m + n 3, or 4. For instance, for two curves, r ^ m + n 3, we have {r : m, n : 2} = [r m, 2] + [r n, 2] = [r, 2] mn + [r m n, 2], 262 ON THE THEORY OF INVOLUTION IN GEOMETRY. [40 or the new curve contains ^ [m + n r I] 2 more arbitrary constants than it would do if the mn points, through which it was made to pass, had been perfectly arbitrary ; a result given before in the Journal, [5], In the case of surfaces, if r ^ m + n+p A. Then assuming r>m + n 4, m+p 4, or n+p 4, we have {r : m, n, p:3} = [r- m, 3] + [r - n, 3] + [r -p, 3] [r m n, 3] [r mp, 3] [r np, 3] = [ r , 3] ~ ninp [r m np, 3] ; or the surface contains [m + n+p r I] 3 more arbitrary constants than it would do if the mnp points, through which it was made to pass, had been perfectly arbitrary. Similarly, in the case where r is not greater than one or more of the quantities m + n 4, m+p 4<, n+p 4. Thus in particular, if r be not greater than the least of these quantities {r : m, n, p : 3} = [r, 3] mnp + [r n p, 3] + [r m p, 3] + [>* m n, 3] [? m n p, 3] ; or the surface contains & [m + n +p - r - I] 3 - \n +p - r - I] 3 - % [m +p - r - 1] -[m + n-r- I] 3 more arbitrary constants than it would otherwise have done. Again, for a surface of the r th order, subjected to pass through the curve of intersection of two surfaces of the orders m, n, {r : m, n, 3} = [r - m, 3] + [r - n, 3] - [r - m - n, 3] ; in which the last term, or ^ [m + n r I] 3 , is to be omitted when r$>m + n 4. The function of the ?- th order, which is satisfied by the systems of values which satisfy the equations of the orders m, n... contains, we have seen, [r, m, n, p ... 0] arbitrary constants ; hence it may be determined so as to pass through this number, diminished by unity, of arbitrary points. But the equation being determined in general by the condition of being satisfied by [r, 6] I systems of variables, it will be com pletely determined if, in addition to the above number of arbitrary systems, we suppose it to be satisfied by a number N=[r, 0] {r; m, n, p... : 6} of systems satisfying the equations above. Hence the theorem " The equation of the r th order which is satisfied by a number N=[r, ff]-{r\ m, n, p... : 0} of systems satisfying the equations of the orders m, n, p... is satisfied by any systems whatever which satisfy these equations." In particular " The surface of the r th order which passes through a number [r, 0]-{r :m, n : 40] ON THE THEORY OF INVOLUTION IN GEOMETRY. 263 of points in the curve of intersection of two surfaces of the orders m, n, or through [r, 6] {r : m, n, p : 6} of the mnp points of intersection of three surfaces of the orders m, n, p, passes through the curve of intersection, or through the mnp points of inter section." Thus a surface of the second order which passes through eight points of the curve of intersection of two surfaces of the second order passes through this curve ; and any surface of the second order which passes through seven of the points of intersection of three surfaces of the second order passes through the eighth point. (The first theorem obviously fails if the eight points have the relation in question, i.e. if they are the eight points of intersection of three surfaces of the second order.) Again " The curve of the r th order which passes through [r, 6] [r : m, n : 6} of the points of intersection of two curves of the orders mn, passes through the remaining points of intersection." e.g. "Any curve of the third order which passes through eight of the points of intersection of two curves of the third order, passes also through the ninth point." Consider next the following question, which [as regards particular cases] has been treated of by Jacobi in the memoir already quoted (Crelle, torn. xv.). "To find the number of relations which must exist between K(0 + l) variables, forming K systems, each of which satisfies simultaneously equations of the orders m, n, p. . . respectively ; the number <f> of these equations being anything less than ; or <j> being equal to 0, provided at the same time K=mnp ...." Suppose m <j; n, n<p... and write [m, 6] [in : m, n, p . . . : 6} = N , [n, 0]-{n :n, p ........ : 0} = N , fee. Imagine the equations of the orders n, ... given. Any function of the m th order which is satisfied by N of the systems of values which satisfy the given equations, and any particular equation of the m th order, is satisfied by the remaining K N systems of values. Hence assuming N systems, satisfying the equations of the orders n, p ... but otherwise arbitrary, the remaining systems must satisfy these equations, and a completely determinate equation of the m th order; i.e. there must be < rela tions between the variables of each system, and consequently $ (K N) relations in all. Similarly, if the equations of the orders p ... were given, N systems of variables might be assumed satisfying these equations, but otherwise arbitrary ; the remaining N N systems satisfy (< 1) determinate equations, or the number of relations between the variables is (</> - 1) (N - N )... ; continuing in the same manner the total number of relations between the variables is in which however any term (<- 1) (N - N ) or (< - 2) (N-N ) ... &c., which becomes negative, must be omitted. It is obvious that we may write more simply N = [m, 0]-l-{m- n, p...0] , N =[n, 6]-I-[n ; p...0], &c. 264 ON THE THEORY OF INVOLUTION IN GEOMETRY. [40 In particular, to find the relations which must exist between the coordinates of mn points in order that they may be the points of intersection of two curves of the orders m, n respectively: here K=mn, N= \ [m + 2] 2 - [m-n + 2] 2 = (Zmn - n 2 + 3n), W = %(n* + 3n + 2), so that NN = m(m ri)l which becomes negative when m = n; hence in general the required number of conditions is mn 3w + 1, but when m = n, the number in question becomes (n I) (n 2). Passing to the case of surfaces; to determine the number of relations which must exist between the coordinates of mnp points, in order that they may be the points of intersection of surfaces of the orders m, n, p respectively. The number required is 3 (mnp -N) + 2(N-N ) + (N - N"}, where N=[m, 3]-l-[ra-n, 3] - [m -p, 3] + [w-n-p, 3] (this last term to be omitted when m < n +p 3), N =[ n , 3]-l-[n-p, 3], jr-foq-i. If, for instance, m>n+p 3, so as to retain the term [m np, 3], and n>p, so as to retain the term N - N", the number becomes, after all reductions, 2mnp + np 2 - 4>np -Zp* -\(p- 1) (p - 2) (p - 3), a formula given by Jacobi. If, however, n=p, this number must be augmented by unity. Again, for m < n + p 3, the required number is 2mnp + np 2 - 4>np - 2p 2 - J (p - 1) (p - 2) (p - 3) I (n+p m 1) (n + p m 2) (n+p m S), which however must be augmented by unity if ra = ?i or n=p, and by 3 if m n=p. But without entering into further details about this part of the subject, which has been sufficiently illustrated by the examples that have been given, I pass on to notice the application of the above theory to the problem of elimination. Imagine (0+1) equations between the (6+1) variables, the first sides of these being, as before, rational and integral homogeneous functions of the variables of the orders m, n, p ... respectively. Writing m + n + p ... 6 = r, and multiplying the first equation by all the terms of the form x a y&... of the degree r m, the second equation by all the terms of the same form, of the degree r n, and so on, there result a certain number of equations, containing all the terms af-yt... of the degree r. But these equations are not independent; and the reasoning in the former part of the present paper shows that the number of inde pendent equations is given by the symbol {r : m, n, p... : 6}; the number of terms x?yP... is evidently [r, 6] ; and it will be shown immediately that for the actual value of r, [r, 0]-{r; m, n, p ... : 6}=Q (B) ; so that the number of quantities to be linearly eliminated is precisely equal to the number of equations, or the elimination is always possible. I may mention also that, 40] ON THE THEORY OF INVOLUTION IN GEOMETRY. 265 supposing the coefficients of all the equations to be of the order unity, the order of the result, free from extraneous factors, may be shown to be [r-m, 6} + ... -2 {[r m n, 0] + ...} +3 {[r-m- n-p, 0] + ...}-&c. = mn... + mp... +np... +&c (C), (the equality of which will be presently proved) a result which agrees with that deduced from the theory of symmetrical functions; but I am not in possession of any mode of directly obtaining the final result in this its most simplified form. My method, which it is not necessary to explain here more particularly, leads me to the formation of a set of functions P, Q, X, Y, Z, 6 in number, such that Z divides Y, this quotient divides X, and so on until we have a certain quotient which divides P, and this quotient equated to zero is the result of the elimination freed from extraneous factors. It only remains to demonstrate the formulae (A), (B), and (C). Suppose in general that (K) denotes the sum of all the terms of the form m a n b . . . , which can be formed with a given combination of k letters out of the <f> letters m, n, p ... , and let 2 (k) denote the sum of all the series (k) obtained by taking all the possible different combinations of k letters. It is evident that 2 (k) is a multiple of (</>), {(<) denoting of course the sum of all the terms m a n b ..., m, n... being any letters whatever out of the series m, n, p...}. Let g be the number of exponents a, b, ... , then (<) contains [0J 7 terms, also (k) contains [k] ff terms, and the number of terms such as (k) in the sum 2 (k) is [<]*"* -r- [< k\^~ k . Hence evidently or, what comes to the same thing, Let A be an indeterminate coefficient, a- a summatory sign referring to different systems of exponents ; then or, giving to k the values 1, 2 ... 0, multiplying each equation by an arbitrary coeffi cient, and adding, putting also for shortness <rA (<j> k) Z7$_i, we have + $_! fj-r + A whence in particular, 2 U^ - 22 1^_, + . . . = a {(</> - g) W-^ c. 34 266 ON THE THEORY OF INVOLUTION IN GEOMETRY. [40 which are still equations of considerable generality. If now <f> = and U 6 is a function of m _|_ n + p + . . . of the order 6, the quantity a- (O 9 ^ A (6)} reduces itself to the single term of U g which contains the product mnp.... Hence, if U =[a + m + n+p ... , 0] in which afterwards a = r -m-n- p- ... we have the formula (A). Again, if </> = and U 9+1 is a function of m + n+p... of the order d, the sum a- (O t 1 -^ A (<)} vanishes; whence writing U e+l = [m + n+p ... - 6, 0], we have the formula (B). Similarly, if in the second formula <J> = 0+I, and U B +\ is a function of m + n+p ... of the degree 0, then reduces itself to the term which contains inn ... + np ... + mp ... + &c. ; whence, if U 0+1 = [m + n + p + . . . - B, 6], we have the formula (C). 41] 267 41. ON CERTAIN FORMULAE FOR DIFFERENTIATION WITH APPLI CATIONS TO THE EVALUATION OF DEFINITE INTEGRALS. [From the Cambridge and Dublin Mathematical Journal, vol. n. (1847), pp. 122128.] IN attempting to investigate a formula in the theory of multiple definite integrals (which will be noticed in the sequel), I was led to the question of determining the (t + l) th differential coefficient of the 2t th power of V(# + X) - \A> + /*) ; the only way that occurred for effecting this was to find the successive differential coefficients of this quantity, which may be effected as follows. Assume <i = {(x + X) (x + /*)}** { V(a? + X) - _L_ 77 = ~ = _ (x + X) (x +7) ~ VK^+ X) (x (a; [*J(x + X) - V(a? + p)Y or, attending to the signification of U^i, fa U *,i = P ( x ~ ^) 2 ^fc-2,1-! - (k Hence _^^,= ^ M 5 ^ ^ M = *<*-- ^) 2 ^-1 + ( &c. 342 268 ON CERTAIN FORMULAE FOR DIFFERENTIATION WITH APPLICATIONS [41 from which the law is easily seen to be of the form (where the extreme values of are and (r-1) respectively) and K r9 is determined by K r+1 , 0+1 = (r - 1 - 10) A ne+1 + (i - g r + 2 + 2(9) 7f r . This equation is satisfied by = r(r ri for in the first place this gives - - r(2r-3-20)r(i-r+l) and hence the second side of the equation reduces itself to F(r - 1 - 0) T(2r - 1 - 0) T(i - r + + 1) where the quantity within brackets reduces itself to (i - r) (2r - 1 - 0), so that the above value reduces itself to -fiT r+1 , e+1 , which verifies the equation in question. Also by com paring the first few terms, it is immediately seen that the above is the correct value of K r ^ e , so that - i \dx) U >*- > r(^)r(0 + l)r(2r-l-0)r(t-r+l) ( X ~ri U_ 2r+l+ ^_ r+e+l ...(1), 8 extending as before from to (r-1). In particular if i be integer and r = i+l, (since the factor F (i- r + 0+ 1) - T (i - r + 1) vanishes except for = on account of r (r - r + 1) = w ). Thus also, if r be greater than (i + 1), = i + 1 + s suppose, then >a ^~r; ., , : . _ " / f\ ..\*s afl TJ /\ 41] TO THE EVALUATION OF DEFINITE INTEGRALS. 269 where 6 extends only from 6 = to 6 = s, on account of the factor F (9 s) -r F ( 5), which vanishes for greater values of 6: a rather better form is obtained by replacing this factor by , rq + g) The above formulae have been deduced on the supposition of i being an integer; assuming that they hold generally, the equation (2) gives, by writing (i ) for i, +i TV 1 or integrating (t + 1) times by means of the formula this gives (5); whence also / T*^ a/V T -i- I ^ (ft _[__ 1 i w W/w JL ^ J. It* ^^ o J X Jo ( + \) i+1 (a? + ^)* r (i + 1) (VX + \V) 2 * and from these, by simple transformations, (rtf ff* I^ 2 f /T 1 - - A?|^ 2" /f T* I ^ I -1 I 1 ^ { ff - -- ** / \ r" i w/w A o -- \ t t^ 2 / v (a - a;)*-* (a? - /8) f -f efo T | T (z - ^) (a - ft)* p (a - a; 7,3 (a - {(a - ) + m (* - /S)}* ~~Ti (Vm + l)^- 1 These last two formulae are connected also by the following general property: it Tf / i -\ If (a. b, i)= then (a, b, i} = JT^ (a - 0) (a + b - i, i, ft)" ............... (8), which I have proved by means of a multiple 2 integral. From (6) we may obtain for <y < 1 a -*)- dg r(j)r(i + a) < This is immediately transformed into Jo (ax* + bx + c] i Ti which is a particular case of a formula which will be demonstrated in a subsequent paper. [I am not sure to what this refers.] [The triple integral I j (M -I ^--1^-1 e -(x+my)u-x- 270 ON CERTAIN FORMULAE FOR DIFFERENTIATION WITH APPLICATIONS [41 which however is only a particular case of rda(l-aty+ (1 -2 7 a; + 7 s )- i ^ h8* (l - Z^x + ^Y I * * I T~l 1 T"l / 1 \ = r fr- + n- ft*" 1 (* -)"** - 10 ) which supposes 7 and - each less than unity. This formula was obtained in the case of (i + ) an integer, from a theorem, Leg. Gal. Int., torn. II. p. 258, but there is no doubt that it is generally true. From (9), by writing x = cos 0, we have Jo (1 - 2 7 cos + 7 2 / = r (i + 1) ( n ) which may also be demonstrated by the common equation in the theory of elliptic functions sin ($ - 6) = 7 sin $, as was pointed out to me by Mr [Sir W.] Thomson. It may be compared with the following formula of Jacobi s, Crelle, torn. xv. [1836] p. 7, sin 2 *- 1 dd 1 f" cos(i-)0d0 Consider the multiple integral w _ f dxdy ... -] (x -af + ... the number of variables being (2i + l) (not necessarily odd), and the equation of the limits being then, as will presently be shown, W may be expanded in the form where A = a- + b" + ...... and \ extends from to x . Suppose next y = [ _ dxdy ... _ J {(x - a) 2 . . . + u 2 } 1 (x 2 + ... v*) i+1 the number of variables as before, and the limits for each variable being oc , x . We have immediately v= r __L _ dW Jo (Z + vJ+i d% ^ W as before, i.e. z* r (x + 1) r ( + \ + TO THE EVALUATION OF DEFINITE INTEGRALS. 271 But writing u-, v" for X, /* in the formula (5) (u and v being supposed positive), the integral in this formula is T (i + 1) " v(u + v) 2i hence, after a slight reduction, V= -r- -8^- or a remarkable formula, the discovery of which is due to Mr Thomson. It only remains to prove the formula for W. Out of the variety of ways in which this may be accom plished, the following is a tolerably simple one. In the first place, by a linear trans formation corresponding to that between two sets of rectangular axes, we have W= or expanding in powers of A , and putting for shortness R = a? + y* . . . + u 2 , the general term of W is /_y A A - (t + X + r | 2(r 75-j-x-o. j j J y " the limits being as before x- + y 2 + . . . = %. To effect the integrations, write V^Vy. &c. for x, y ... so that the equation of the limits becomes x + y + ... = 1. Also restricting the integral to positive values, we must multiply it by 2 2i+1 : the integral thus becomes f gr+i+i I x a-k y^ ...{(+ y ...)+ w 2 }- -*-" dxdy ... equivalent to Hence, after a slight reduction, the general term of W is where cr may be considered as extending from to X inclusively, and then X from to oo . But by a formula easily proved - A -i - 22X * ( ~ )<r - 272 ON CERTAIN FORMULA FOR DIFFERENTIATION WITH APPLICATIONS, &C. [41 where a extends from to X. Hence, substituting and prefixing the summatory sign, W = TT^ S = where \ extends from to oo , the formula required. [I annex the following Note added in MS. in my copy of the Journal, and referring to the formula, ante p. 267; a is written to denote X-/x. N.B. It would be worth while to find the general differential coefficient of U k i . Wk,i=-( k + V U k-l,i + ^ U k-2,i-l> from which it is easy to see that % U k.i = (~r K r,0 U k-r,i \r-6 K- 29 TT I r,t k-r-0,i-i TC ri TT r,r k 2r,i-r The general term of 9 +1 U k i is which must be equal to I \r-Q v ,,20+2 TT II A r+l,0+l a u k-r-e-2,i-8-l therefore ^r+l 9+1 = (^ + *-^- 20 -2) K r 6 . In particular whence K r =[k + i A , which appears to indicate a complicated general law. Even the verification of K f l is long, thus the equation becomes r+1 [k + i - 2] r ~ l {fc 2 + (t - r- 1) ft- 4rt} - (ft +i - r- 2) r {ft +(i- r )k-$(r-l)i}[k + i- 2] r ~ 2 = (ft- r)[ft + which is identical, as may be most easily seen by taking first the coefficient of k s , and then writing k = r, k=-i, k- -i-1.] 42] 273 42. ON THE CAUSTIC BY EEFLECTION AT A CIECLE. [From the Cambridge and Dublin Mathematical Journal, vol. II. (1847), pp. 128 130.] THE following solution of the problem is that given by M. de St-Laurent (Annales de Gergonne, t. xvn. [1826] pp. 128 134) ; the process of elimination is somewhat different. The centre of the circle being taken for the origin, let k be its radius; , 6 the coordinates of the Aluminous point; , 77 those of the point at which the reflection takes place ; x, y those of any point in the reflected ray : we have in the first place F + ? 2 = & 2 ....................................... (1). There is no difficulty in finding the equation of the reflected ray 1 ; this is ^ = ................. (2), 1 To do this in the simplest way, write pMf-aO + fo-y) 8 , <r=(-a) 2 +07-6), then, by the condition of reflection, /) + <r = min., p, ff being considered as functions of the variables f, ??, which are connected by the equation (1). Hence or, eliminating X, P whence (^x - ly) 2 [( - a) 2 + (ij -&)] = faa - #) 2 [( - x)* + fa - y)"-]. This may be written {(nx-Hy) (f-o) -(na-&!) (-x)} fax-ty) ( - a) + (r,a - &) ( the factors in { } reduce themselves respectively to P and ??P, where P=f(&-7/) -T, (a-x) +ay - bx ; omitting the factor P, (which equated to zero, is the equation of the line through (a, b) and (, 77), ) and replacing (-) + ? fa -&) and (-*) + ?? fa- 2/) by fc 2 -c|-6ij and k*-x-riy, respectively, we have the equation given above. c. 35 274 ON THE CAUSTIC BY REFLECTION AT A CIRCLE. [42 or, arranging the terms in a more convenient order, (bx + ay)(?-r) 2 ) + <2(by-ax)Zr)-k 2 (b + y) + k 2 (a + x)i) = () ............ (2 ). Hence, considering 77 as indeterminate parameters connected by the equation (1), the locus of the curve generated by the continued intersections of the lines (2) will be found by eliminating f, 77, X from these equations and the system k^(b + y) = .................. (3), ^(a + x) = .................. (4), and from these, multiplying by , 77, adding and reducing by (2), we have = .............................. (o), which replaces the equation (2) or (2 ). Thus the equations from which 77, X are to be eliminated are (1), (3), (4), (5). From (3), (4), (5), by the elimination of , 77, we have - X [X 2 - 4 (bx + ay) 2 } - 4& 2 (by -ax)(a + x) (b + y) - k 2 (a + x) 2 [X + 2 (bx + ay)] Q ............................................. (G), or, reducing, - X 3 + X (4 (a? + b 2 ) (x 2 + y 2 ) - k 2 [(a + x) 2 + (b + y) 2 ]} -2k 2 (bx-ay)(x 2 + y 2 -a 2 -b 2 ) = .............................. (7); Avhich may be represented by 2E = .................................... (7 ). Again, from the equations (4), (3), transposing the last terms and adding the squares, also reducing by (1), fr [(a + x) 2 + (b + y) 2 ] = & 2 X 2 + 4& 2 (a 2 + b 2 ) (x 2 + y 2 ) a.x)] .......... . ....... (8); but from the same equations, multiplying by , 77 and adding, also reducing by (1), k 2 \ + 2(bx + ay)(% 2 - rf) + 4fij (by - ax) + k 2 [- (6 + . /) + *7 ( + )] = ............ (9), or reducing by (5) and dividing by two, & 2 X + (bas + ay) (f 2 - 77") + 2^ (by - ax) = ........................ (10). 42] ON THE CAUSTIC BY REFLECTION AT A CIRCLE. 275 Using this to reduce (8), k 2 [(a + z) 2 + (b + y) 2 ] = 4 ! (a 2 + b 2 )(x 2 + y 2 ) + 3\ 2 ..................... (11), or, from the value of P, -3\ 2 +Q = ................................... (12), which singularly enough is the derived equation of (7 ) with respect to \ : so that the equation of the curve is obtained by expressing that two of the roots of the equation (7 ) are equal. Multiplying (12) by X and reducing by (7 ), - \Q + 3R = 0, or, combining this with (12), whence, replacing R, Q by their values, we find 27 Ar (bx - ay) 2 (x 2 + f -a 2 - b 2 ) 2 - {4 (a 2 + b 2 ) (x z + f) - k 2 [(a + x) 2 + (b + yY\} = 0, the equation of M. de St-Laurent. 352 276 43. ON THE DIFFEEENTIAL EQUATIONS WHICH OCCUR IN DYNAMICAL PROBLEMS. [From the Cambridge and Dublin Mathematical Journal, vol. II. (1847), pp. 210 219.] JACOBI, in a very elaborate memoir, " Theoria novi multiplicatoris systemati sequa- tionum differentialium vulgarium applicandi " (*), has demonstrated a remarkable property of an extensive class of differential equations, namely, that when all the integrals of the system except a single one are known, the remaining integral can always be deter mined by a quadrature. Included in the class in question are, as Jacobi proceeds to show, the differential equations corresponding to any dynamical problem in which neither the forces nor the equations of condition involve the velocities ; i. e. in all ordinary dynamical problems, when all the integrals but one are known, the remaining integral can be determined by quadratures. In the case where the forces and equations of condition are likewise independent of the time, it is immediately seen that the system may be transformed into a system in which the number of equations is less by unity than in the original one, and which does not involve the time, which may afterwards be determined by a quadrature 2 ; and, Jacobi s theorem applying to this new system, he arrives at the proposition "In any dynamical problem where the forces and equations of condition contain only the coordinates of the different points of the system, when all the integrals but two are determined, the remaining integrals may be found by quadratures only." In the following paper, which contains the demonstra tions of these propositions, the analysis employed by Jacobi has been considerably varied in the details, but the leading features of it are preserved. 1 Crelle, t. xxvn. [1844], pp. 199268 and t. xxix. [1845], pp. 213279 and 333376. Compare also the memoir in Liouville, t. x. [1845], pp. 337346. " For, representing the velocities hy x , y ... the dynamical system takes the form dt : dx : dy ... : dx : dy ... =1 : x : y ... : X : Y ... , and the system in question is simply dx : dy ... : dx : dy ... = x : y ... : X : Y ... . 43] ON THE DIFFERENTIAL EQUATIONS &C. 277 1. Let the variables x, y, z, ... &c. be connected with the variables u, v, w, ... by the same number of equations, so that the variables of each set may be considered as functions of those of the other set. And assume dx dy ... Vdudv... ; if from the functions which equated to zero express the relations between the two sets of variables we form two determinants, the former with the differential coefficients of these functions with respect to u, v, ... and the latter with the differential coeffi cients of the same functions with respect to x, y, ... the quotient with its sign changed obtained by dividing the first of these determinants by the second is, as is well known, the value of the function V. Putting for shortness ^ = a dy = o . dx = a! d_y_ du du * dv & * " du du dv ., dv . and te- A > dy = B - d-x = A Ty = ff <- V is the reciprocal of the determinant formed with A, B, ... ; A , B , ... , &c.; or it is the determinant formed with a, (3, . . . a. , ft , ... , &c. From the first of these forms, i.e. considering V as a function of A, B, ... ^-_V, dV =-V<3 dV --Va> -- V/3 dA dB dA a dB ~ V/ *"" where the quantities a, /3, ... a , , ... and A, B,...A , B ,... may be interchanged pro vided - V be substituted for V. (Demonstrations of these formulas or of some equivalent to them will be found in Jacobi s memoir "De determinantibus functionalibus," Crelle. t. xxii. [(1841) pp. 319359].) ^ dV + ad A + ftdB + ...+ a!dA + jS dB + . . . = 0, or reducing by dA dB dA dB ; = -j , . . . ; -y = -j , ... &c. dy dx ay dx this becomes 1 ,_ (dA , , dB . \ . (dA dB . dv + a [-y- dx+ -j- dy+ ... ) + ft -=- dx+-=- dy + ... ... V \dx dx J \dy dy n fdA dB \ v /dA . dB . + a -T- dx + -j- dy + . . . ) + ft (--=- d + ^- dy + \dx dx ) \dy dy y 278 ON THE DIFFERENTIAL EQUATIONS [43 or, reducing, 1 ._ fdA dA \ 7 fdB dB \ . ^dV + _+-.- + ... }dx+ -+-- + ...)dy+ ... = 0; V \du dv J \du dv J whence separating the differentials and replacing A, A ,...; B, B , ...; by their values 1 dV d du d dv _ V dx du dx dv dx 1 dV d du d dv V dy du dy dv dy (in which V, u, v ...; x, y ... may be substituted for V, x, y ... ; u, v ...). 2. Let X, Y ... be any functions of the variables x, y, ... and assume n _ ydu du u A -j |- y -j- + ... ax dy V=X + Y + dx dy U, V, ... being expressed in terms of u, v, ____ Then "" Y ( d du d dv \ v I d du d dv du T dv \du dx dv dx \du dy + dv dy + /dX" du dX dv \ idY du dY dv \du dx dv dx ") \du dy dv dy Also, whatever be the value of M, dMV du ~dv~ "=*-&r ~dy~^ " and from these two properties, dMVU dMVV _ fdMX dMY du dv \ dx ~dy~ f ) 8. Consider the system of differential equations dx : dy : dz ... =X : Y : Z ... (where, for greater clearness, an additional letter z has been introduced). From these we deduce the equivalent system du : dv : dw ... = U : V : W .. 43] WHICH OCCUR IN DYNAMICAL PROBLEMS. 279 Suppose that u and v continue to represent arbitrary functions of x, y, z, ... but that the remaining functions w, ... are such as to satisfy W = 0, . . . (so that w, ... may be considered as the constants introduced by obtaining all the integrals but one of the system of differential equations in x, y, z, ...), we have dMVU dMVV - idMX dMY dMZ 1 = V j 1 -j 1 -j h . . . du dv \ dx dy dz Also the only one of the transformed equations which remains to be integrated is du : dv = U : V, or Vdu - Udv = 0, (in which it is supposed that U and F are expressed by means of the other integrals in terms of u and v). Suppose M can be so determined that dMX dMY dMZ 7 T 7 T 7 I U, dx dy dz (M is what Jacobi terms the multiplier of the proposed system of differential equations) : then dMVJf dMVV = Q du dv or i/V is the multiplier of Vdu Udv = Q, so that fiV ( Vdu - Udv) = const. Hence the theorem : " Given a multiplier of the system of equations dx : dy : dz, ... =X : Y : Z ... (the meaning of the term being defined as above), then if all the integrals but one of this system are known, the remaining integral depends upon a quadrature." Jacobi proceeds to discuss a variety of different systems of equations in which it is possible to determine the multiplier M. Among the most important of these may be considered the system corresponding to the* general problem of Dynamics, which may be discussed under three different forms. 4. Lagrange s first form 1 . Let the whole series of coordinates, each of them multiplied by the square root of the corresponding mass, be represented by x, y, ... and in the same way the whole series of forces, each of them multiplied by the square root of the corresponding mass, by P, Q, ... ; then the equations of motion are d?x = fry _ d? d? 1 I have slightly modified the form so as to avoid the introduction of the masses, and to allow x (for instance) to stand for any one of the coordinates of any of the points, instead of standing for a coordinate parallel to a particular axis. 280 ON THE DIFFERENTIAL EQUATIONS [43 where X-p + X d 4- u.** + A r + A, j 1- /i -j h . . . , da; ax v n^ d = +X where = 0, <I> = 0, . . . are the equations of condition connecting the variables, and d 2 x \, fi, ... coefficients to be determined by substituting the values of -7 , &c. in the equations =-^ = 0, ^ = 0, &c. It is supposed that as well P, Q, ... as , <>,... are inde- Ctt/~ QJv pendent of the velocities. In order to reduce these to an analogous form to that previously employed, we have only to write dt dt which gives dt : dx : dy : dz ... : dx : dy : dz ... = 1 : x : y : z ... : X : F : Z ... Supposing that M is independent of x , y , z, ... the equation on which it depends becomes immediately ,,, , r fdX dY dZ \ SM + M (-=-; + -=-, + -y-, + . . . ) = 0, \da; dy dz J where for shortness s_d_,tjl_ > d , d dt dx " dy dz To reduce this we must first determine the values of X, /z . . . , and for this we have dt 2 dx dt 2 ~+... 43] WHICH OCCUR IN DYNAMICAL PROBLEMS. 281 where for greater clearness an additional letter of the series , < . . . has been introduced, and where & -(IHw + -- , _ (d d<& d c<E>\ \dx dx dy dy) Hence differentiating with respect to x , _ ~ d d\ , du, dv + h + b^-+f = dx dx dx J dx , dv or representing by K the determinant formed with the quantities a, h, g, ... h, b, f, ... g, f, c, ... and by A, H, G, ... H, B, F, ... G, F, C, ... the inverse system of coefficients, we have 9AX + TTXj,rX ^Jf f) 2 .4.6-5 + -" -j 1~ Go -j ...}+ K -j-, = 0, dx dx dx / dx n?( ^.m ^ vz ^K n \-Ht>-j -- 1- Job -j- + 1C -3- ... I + K -y , = 0, \ dx dx dx J dx .j^^ ^K n 2 trd) -j -- h fo j -- h Co -y- . . . + K -J-, = 0, dx dx dx J dx u . , . , d d<& <W , , ,. whence, multiplying by -7- , -=- , -j- , ... and adding, } 9 + BB(\... +2HS ...+K d = dx) \dx) dx dx " dx and, forming the similar equations with the remaining variables and adding, ASa + BSb + CSc ... + 2F8f+ 2GSg + 2HSh + ... + K + / + .. = ; \dx dy dz / , T r(dX dY dZ \ i.e. SK+K , + -T-J+ -j-,+ ... =0. \dx dy dz J c. 36 er THE GAI , 282 ON THE DIFFERENTIAL EQUATIONS [43 Thus the equation in M reduces itself to KBM-MtK-0, which is satisfied by M K. It may be remarked that K reduces itself to the sum of the squares of the different functional determinants formed with the differential coefficients of , <> ... with respect to the different combinations of as many variables out of the series x, y ... - 5. Lagrange s second form. Here the equations of motion are assumed to be d dT_dT_ p = dt dx dx __ dt dy dy ddT_dT_ R==Q dt dz dz where 2T represents the vis viva of the system, x, y, z, ... are the independent variables on which the solution of the problem depends, and x , y , z ,... their differential coeffi cients with respect to the time. It is assumed as before P, Q, R ... do not contain x, ij, z, ... . Suppose these equations give dt : dx : dy : dz ... : dx : dy : dz ... = 1 : x : y : z ... : X : Y : Z ...; then the equation which determines the multiplier M takes as before the form d7 dZ .,, vr/dX dY dZ \ . m+M(=5 + j-, + 7r> + ... =0 - \dx dy dz ) \dx dy To reduce this equation, substituting for T its value which is of the form T = $ (ax 1 * + by 2 + c/ 2 ...4 2/yV + Zgz x + Zhx y ...), and putting for shortness L = ax + hy +gz ... , M=hx + by +fz ... , N = gx +fy +cz ... 43] WHICH OCCUR IN DYNAMICAL PROBLEMS. 283 the equations which determine X, Y, Z ... are (JT aX + hY+gZ... + SL -~-P = 0, dy dT dz gX+fY+cZ...+SN-~-R=Q. Hence, differentiating with respect to oc , dX , dY dZ dX^ ,dY f dZ ^ ^_^ = dx dx J dx " dx dy dX ,dY dZ dN dL or representing by K the determinant formed with a, h, g, ... h, b, /, ... g, f, c, ... and by A, H, G, ... H, B, F, ... G, F, C ... the inverse system of coefficients, we have v dX f dM dL\ n (dN dL\ K -T-, + ASa + Hek + GSg ... + + H [-5 r- I + 1 -a ?- I - - 0. dx \dx dy ) \dx dz) and similarly HXh. -4- 7?M 4. FXf a- 77 (^ _ &M\ + * i dx J Hence, adding, and thus we have as before, though with symbols bearing an entirely different signification, and thence K8M-M8K = (), and M = K. 362 284 ON THE DIFFERENTIAL EQUATIONS, &C. [43 (The value of K in this section may I think be conveniently termed "the deter minant of the vis viva," with respect to the variables x, y, z, . . . . It may be remarked that "the determinant of the vis viva" with respect to any other system of variables u, v, w, ... is =V*K, V as before.) 6. Third form of the equations of motion. [Hamiltonian form.] Here writing dT > dy and taking t, x, y,...%, y, for the variables of the problem [and considering T to be expressed as a function of these variables: to denote this change it would have been proper to use instead of T a new letter H] the equations of motion reduce them selves to dj = _dT dt dt dx + dt d? dt or putting for shortness they become dT_ dT_ ~~ A ~ dT dT U TJ = Y , -j = rl, * drj dr) dt . dx \ dy : dz ... : d% : dy : d% ... = 1 : H : H : ft ... : X : Y : Z ... . Hence writing the equation in M under the form + =0 , * * I V where S- + S + H +*+ F V dt dx dy d% dy we see immediately that (P, Q ... being as before independent of the velocities, and con sequently of , 77, , ...), f + < 0,^ + ^= dx d% dy drj Hence SJf=0, which is satisfied by M=l. 285 44. ON A MULTIPLE INTEGRAL CONNECTED WITH THE THEORY OF ATTRACTIONS. [From the Cambridge and Dublin Mathematical Journal, vol. II. (1847), pp. 219 223.] MR BOOLE [in the Memoir " On a Certain Multiple definite Integral " Irish Acad. Trans, vol. xxi. (1848), pp. 40 150] has given for the integral with n variables yti r r \ fa ^ i w* war tr= iZ 2 1 n ^ J [(a - xj + (b - 2/) 2 . . . + <p + * limits 7^+ 2 + <!> the following formula, or one which may readily be reduced to that form 1 , , r _fg..,^ f Ss-^ds I (2), *- V2 " r f/ ^ >? V IV" i y /V ^^/-") where in which a 2 6 2 ti 2 "7 and 77 is determined by i* v * =71^ + ^1 +7 (4 a? 6 2 w 2 1 = 7^-- + ^--... + -- / " + V 9 + V y 1 See note at the end of this paper. 286 ON A MULTIPLE INTEGRAL [44 Suppose f = g=... = <x>; also assume then the integral becomes IT _ dandy... _ 2 2 ** 1 ^ (x - of + (y - b 2 ) . . . + u^ n+< J the limits for each variable being oo , oo . Now, writing f-s for s and f-v) for t], the new value of 77 reduces itself to zero, and ~ Also cr = ; but where r- = a 2 + & 2 + . . . whence also putting -^ for , <b {<r + t(l cr)] becomes i.e. (t if for a moment A = -- + - + v-. 1 + s s Hence Sf- or substituting in U, and replacing A by its value, or, what comes to the same, IT = w ~ ++ where 44] CONNECTED WITH THE THEORY OF ATTRACTIONS. 287 The only practicable case is that of q = q, for which 9r r ^n + ?) Consider the more general expression r jUMt-i <k - ) J o (w 2 s 2 + * + w 2 )i" P , fv 2 s 2 + is + 7 e- r*-** -- - & .............................. (9); by writing 2u*/s = \7(s + 4>uv) + yV, the upper sign from 5 = GO to s = - , and the lower one from s = - to s = 0, it is easv to v v J derive Now, by a formula which will presently be demonstrated, J o {\/(s + 4tuv) + *Js}-w + (V(g +-4,uv) *Js V(s + 4>uv) _ es fr+i Jo J o V s V( s TQ- whence ., JJS . Thus, by merely changing the function, by means of the formula and hence in the particular case in question 223+1 V 2q ^J (n+i) /" - s ~^( s + 4 ^)~ i " 7 ( s +^ + 2w )~ in+9 ^ ......... (14), But as there may be some doubt about this formula, which is not exactly equivalent either to Liouville s or Peacock s expression for the general differential coefficient of a 288 ON A MULTIPLE INTEGRAL [44 power, it is worth while to remark that, by first transforming the ^n th power into an exponential, and then reducing as above (thus avoiding the general differentiation), we should have obtained 229+1 3,25 Tj-i (n+i) - f d6 t ds e^-i 1 e >-q) Jo Jo which reduces itself to the equation (14) by simply performing the integration with respect to 6; thus establishing the formula beyond doubt 1 . The integral may evidently be effected in finite terms when either q or q - % is integral. Thus for instance in the simplest case of all, or when q = , Tj-id-D 1 f dxdy ... _ U = f"(nTi) ( + 2t0* <"- = J _ (rf + f + v^ < w+1 > {(as - a) 2 + ... u*}* "^ a formula of which several demonstrations have already been given in the Journal. The following is a demonstration, though an indirect one, of the formula (11): in the first place r { V(s + 4mQ + Vs} Jo V F /I / a -* (16), (where as usual i = V 1) : to prove this, we have or, putting -kw; V^ = V(* + 4?<v) V (which is a transformation already employed in the present paper), the formula required follows immediately. Now, by a formula due to M. Catalan, but first rigorously demonstrated by M. Serret, *m ( Z (Liouville, t. VIIL [1843] p. 1), and by a slight modification in the form of this equation which, compared with (16), gives the required equation. 1 A paper by M. Schlomilch "Note sur la variation des constantes arbitraires d une Integrate definie," Crellc, t. xxxin. [1846], pp. 268280, will be found to contain formulae analogous to some of the preceding ones. 44] CONNECTED WITH THE THEORY OF ATTRACTIONS. 289 NOTE. One of the intermediate formulas of Mr Boole [in the Memoir referred to] rnay be written as follows : i p /"* S = - I da. I dvv q cos [(a cr) v + qir] <fxz, IT J o Jo or what conies to the same thing, putting i=^j 1, and rejecting the impossible part of the integral, r 1 f dtx dv vi e w <"-" 62 Jo Jo 7T r 1 1 r 00 i.e. S = I 1 (f>ot. da, I = e? q7n I dvv q e iv (a ~ <T) . Jo 7T J Now (a a-} being positive, 1 n 7T i.e. I=-e+^ i r(q + l)(a- ( r)-<i-\ 7T or, retaining the real part only, / = sin qir T (q + 1) (a - 7T But (a cr) being negative, !_ 7T I= l ~e-^T(q+l)(<r or, retaining the real part only, / = 0. Hence or putting a = ^ + t (l - a-), or a-a = t(l - a), the expression in the text. Mr Boole s final value is dq which, though simpler, appears to me to be in some respects less convenient. C. 290 [45 45. ON THE THEOBY OF ELLIPTIC FUNCTIONS. [From the Cambridge and Dublin Mathematical Journal, vol. n. (1847), pp. 25G 266.] ADOPTING the notation of the Fund. Nova, except that for shortness sn u, en u, dn u are written instead of sinam u, cosam u, V am u, let the functions (u), H (u) be denned by the equations \ Aw* (l- "!)- \ K \ TT J ir(K -Ziu) /-, ,. (i), (2), it is required from these equations to express sn u in terms of the functions H (u}, (w). To accomplish this we have d* 1 d* lid Y - log sn u = -- r , sn M --- T- sn u) du* * sn u du* sn 2 u \du J |sn j 1 = A; 3 sn 2 1 whence also -f 2 log sn u = A; 2 sn 2 u - k* sn 2 (u + iK ). <w? If for a moment & u = I du sn 2 u, >lr // u= I du I du sn 2 u, J o J J then log sn w = A^i|r /y w ^ 2 ^^ (w + i^T ) + -4w + B ; 45] ON THE THEORY OF ELLIPTIC FUNCTIONS. 291 or writing u for u and subtracting, ty lt u being an even function, 2 Au = iri - &Y,, (iK - u) + k 2 ^ (iK + u), or putting u = K, 2AK = Tri - & 2 f (iK -K) + k^,, (iK + K). Now sn 2 (u + K) - sn 2 (u -K) = 0, and therefore ^ (u + K) - ^ ( u -K) = 2 or Also E (u) = u - Hence log sn u = k^ //u - k*^ H (u + iK ) + uiK 1 - + + B xi. / iKJ - u i. e. log sn u log (M + z.K ) log tt + + B , ZxL or, changing the constant, /, ^ (w + ^0 sn M = (7e ^-^r - ^ . @w Now, to determine (7, write u iK for u ; this gives -1- -c e ^ (u ~ iK ] u k&nu (u- iK ) and again changing u into u, whence, multiplying these last two equations, 2A" 372 292 ON THE THEORY OF ELLIPTIC FUNCTIONS. [45 Or U = r-ry 6 l\/K whence sn u -. rr e w U(u) and the equations (1), (2) and (3) may be considered as comprehending the theory of the functions H (w), (%). The preceding process is, in fact, the converse of that made use of in the Fund. Nova; Jacobi having obtained for sn u an expression in the form of a fraction, takes the numerator of it for H (u) and the denominator for (u), and thence deduces the equations (1), (2), the intermediate steps of the demonstration being conducted by means of infinite series; the necessity of which is avoided by the preceding investigation. I proceed to investigate certain results relating to these functions, and to the theory of elliptic functions which have been given by Jacobi in two papers, " Suite des notices sur les fonctions elliptiques," Crelle, t. ill. [1828] p. 306, and t. iv. [1829] p. 185, but without demonstration. In the first place, the equation = (4) d& is satisfied by 2 = (u) or 2 = H (u). It will be sufficient to prove this for 2 = (u), since a similar demonstration may easily be found for the other value. The following preliminary formulae will be required: dE *HE~ + T- **---- which are all of them known. Now, writing (u) under the slightly more convenient form Bu = v (^rr we have d, 7/ / 7/7 \ ^ )w = < u \k 2 ^> I + k~ L du cn 2 du ^ ^ \ 1 1 /. ^ \ = dn 2 M - ^ + \ u (k 2 - -^ } + k 2 /o du cn- du* ~L ^^("v r d u r i <w^ i 2 ^^ _^ j , jz, l w 3E "p-"^ Jo w* Jo du -jj dn- 45] ON THE THEORY OF ELLIPTIC FUNCTIONS. 293 The success of the process depends upon a transformation of the double integral Jo du Jo du -jj- dn 2 u : CtrC to effect this we have d , c t , d -n- dn 2 u 2k sn u ( sn u + k -77 sn u } ; die \ dk J but, by a known formula, whence k 2 -jj sn u = k en w dn u / en 2 udu + k en 2 M sn ?t ; GiA? -77 sn u = 77, sn u dn 2 u k 2 en u dn it f du en- it, a A; k 2 d 2k or jT dtf u ~ ~ 7/i ( sn2 w dn 2 M & 2 sn it en u dn ?t / du en 2 2k ( t d \ } = Y^ \ sn 2 u dn 2 w + |^ 2 ( -7- en 2 w / du en 2 M [ ; K (^ \du j J whence d 2k / dw Jo dw -77 dn 2 it = 77^ {/ duf Q du sn 2 it dn 2 v + -|& 2 J dit (en 2 u fdu en 2 - fdu en 4 u)} ^ ~ ~~ 772 (Jo du f du (2 sn 2 u dn 2 u & en 4 u) + ^k 2 (/ dw en 2 it) 2 }. A/ But - 7 sn 2 u = 2 (en 2 u dn 2 w sn 2 it dn 2 u k 2 sn 2 it en 2 it) = 2 (k 2 2 sn 2 u dn 2 it + & 2 en 4 M) ; ait 2 or, integrating, sn 2 u = k 2 u 2 2 Jo duf du (2 sn 2 it dn 2 u k 2 en 4 it) ; whence at length / 7 / 7 (X ln 17rt \ K A/ / /. 7 > \ > ait Jo ait -77 dn 2 % = ^A-it 2 + $ rr 2 sn 2 + --^ (J era en 2 ) - Also so that d it 1 (E dk ~\ ( T? / Jf\" ^1 = 5 7 -,>. {= - dn 2 M - M 2 ( k 2 - ~] - k 4 (fdu en 2 u) 2 [ ZiKK " (^jt \ jl. / OF THE UN I VERSITT, 294 ON THE THEORY OF ELLIPTIC FUNCTIONS. [45 Substituting these values of , M, , &u and -rr u in the equation (4) in the place du du" d/c of the corresponding differential coefficients of 2, all the terms vanish, or the equation is satisfied by S = (it), and similarly it would be satisfied by 2 = H (u). Assume now irK _ iru then observing the equation we have d2 = Tj^d du ~ 2K dv _ __ ,, _ E 7T ~ * dk ~ kk * V K) dv iJW dv whence, substituting in the equation (4), this becomes T^-4^ = - (5), air da) which is of course satisfied as before by 2 = (), or 2 = H (u), an equation demon strated in a different manner (by means of expansions) by Jacobi in the Memoirs referred to. Consider next the equation ~T r Znu I K -fr -j T An/cK -jj- =U (o), du 2 \ K) du dk (n being any positive integer number). Then, by assuming 7TJK 11TTU K K we should be led as before to the equation (5). Hence, considering 0w or Hw as K functions of u and -~ , the equation (6) is satisfied by assuming for S a corresponding nK function of nu and -^ . Let \ be the modulus corresponding to a transformation of the >i th order; then A, A being the complete functions corresponding to this modulus, A K T- = n ^ , so that the equation (6) will be satisfied by assuming 2 = 0,0^0 or S = H 7 (nu), where y , H^ correspond to the new modulus X. 45] ON THE THEORY OF ELLIPTIC FUNCTIONS. 295 Assume now in the equation (6), Hence, substituting, J^ 2 (0M . *) - 2>m (V 2 -5) ^ ( n ^ *) + 2wM- 2 (# )* (n ~ 1) 4 [("& )"* ( M) " " *] = ; but or effecting the differentiation, and eliminating ^- by means of the equation obtained rt/c from (4) by writing 2 = @w, _. dk 2kk *u { d T X du j 2A^" V " K Substituting in (6) and reducing, - -F> U . 7 , , JI/0 j -- u (k 2 --js)\ :j - + 2n kk 2 -= 7 rtw V jK/ du dk + n du But d log u (,. E\ 7 , 7 -? = u (k jfl = K Jo du en 2 whence / dw en 2 u) ~ + 2nkk 2 -^ + n (n - 1) A 3 sn 2 u . z = (7) which is therefore satisfied by _ z ~ 296 ON THE THEORY OF ELLIPTIC FUNCTIONS. [45 and each of these values is an algebraical function of sn u, (viz. either a rational function or a rational function multiplied by cnwdnw)- Also, in the transformation of the ?i th order, H,(nu) J\ sn, u = -TT^T ( ; (2), (nu) so that it is clear that the above values of z may be taken for the denominator and numerator respectively of ^Xs^u; i.e. these quantities each of them satisfy the equa tion (7). By assuming x = \/& sn u, a = k + j , K this becomes -4)=0 ......... (8); which is therefore satisfied by assuming for z either the numerator or the denominator of V^sn.w (the transformation of the n th order), which is the form in which the property is given by Jacobi. In the case where n is odd, the denominator is of the form and then the numerator is where vM 1 and all the remaining coefficients may be determined from these, the modular equation being supposed known. But the principal use of the formula is for the multiplication of elliptic functions, which it is well known corresponds to the case where n is a square number. Writing n = v~, when v is odd, the denominator is (the sign according as i/ = (4p + l) or (4p-l)); and the numerator is obtained from this by multiplying by x and reversing the order of the coefficients. When v is even the denominator is ... (+ or , according as v = 4>p or v = 4p + 2), so that there are only half as many co efficients to be determined ; but then the numerator must be separately investigated. 45] ON THE THEORY OF ELLIPTIC FUNCTIONS. 297 In general, by leaving n indeterminate, and integrating in the form of a series arranged according to ascending powers of x* ; then, whenever n is a, square number, the series terminates and gives the denominator of the corresponding formula of multi plication ; but the general form of the coefficients has not hitherto been discovered. X . u By writing r- instead of x, and then making n infinite, the equation (8) takes the form and it is worth while, before attempting the solution of the general case, to discuss this more simple one 1 . Assume then it is easy to obtain C r+ , = -(2r + 1) (2r + 2)C r - (2r + 2) aC r+1 + 2 (a 2 - 4) ^g? . The general form may be seen to be and then C r+1 P -pC r p = -r (2r - 1) C r _f~ l + 16 (r + 2 - 2p) C^ 1 . The complete value of C r p (assuming C r = 0) is given by an equation of the form where C r p , 1 C r p , ...... are algebraical functions of r of the degrees 2p 2, 2p 4, &c. respectively; but as I am not able completely to effect the integration, and my only object being to give an idea of the law of the successive terms, it will be sufficient to consider the first or algebraical term C r P, which is determined by the same equation as C r P, and is moreover completely determined by this equation and the single additional 1 Writing (/3 + 2) for a, and putting z-c^ p, this becomes and if p = 2Z n p n , ^- { Sn + l } Z n from which the successive values of Z , Z lt &c. might be calculated. 38 298 ON THE THEORY OF ELLIPTIC FUNCTIONS. [45 relation (7,. 1 = 1, since the arbitrary constants of the integration affect only the terms multiplied by 2 r , S r , &c. Assume C r p = __ - _ (2P-> LP \r - 21 2 *>- 2 + 2*- 2 MP \r - 3] 2 ^- 3 -f ... 2~P +1 XP [r - [p ~ ] and substituting this value, (l-p)LP =(l-p){LP-* (1 -p)MP- 2p (2 - 2p) LP = (1 - p) {MP- 1 (1 - p) NP - 2p (3 - 2p) MP = (l-p) {NP~ I - IMP 1 + (1 -p) QP - 2p (4 - 2p) NP =(l-p) {OP- 1 - 3NP- 1 + 30MP- 1 } , the law of which is obvious, the coefficients on the second side in the <?th line being 1, 42-19, and (2q - 3) (2q - 2) respectively. By successive integrations and substitutions LP - LP~ I =0, LP = 1, MP - MP- 1 = p - 11, MP = (p - 1) (2p - 7), (the constants determined by I/ 1 = 0, ^ J = 0, O 2 = 0, P 2 = 0, . . . so as to make Gj> con tain positive powers only of r). The following are a few of the complete values of C r p , the constants determined so as to satisfy C P +? = Q (except CJ = 1), and the factorials being partially developed in powers of r, viz. C r = \ (r - 4) (r - 5) (4r 2 - 24r +51), O r 4 = ^ {( r - 5) (r - 6) (r - 7) (Sr 3 - 60r 2 + 286r + 63) + 384 (9r 2 - 93/- + 242 - 2 . 4 -% &c. (it is curious that (7 5 4 , 6 6 4 , (7 7 4 , all three of them vanish). It seems hopeless to con tinue this investigation any further. Returning to the equation (8), and assuming for z an expression of the same form as before, we have, corresponding to the equations before found for the co efficients C r , dC C r+ * = - (2r + 1) (2r + 2) (n - 2r) (n - 2r - 1) C, - (2r + 2) (n - 2r - 2) a(7 r+1 + 2w (a 2 - 4) -^ . 45] ON THE THEORY OF ELLIPTIC FUNCTIONS. 299 The case corresponding to the denominator in the multiplication of elliptic func tions is that of C = 1, C l 0. It is easy to form the table Cz= 8 TO (TO -1) (TO -4) a, C 4 = - 4 re (n - 1) (TO - 4) [TO + 75] - 32n (n - 1) (n - 4) (n - 9) a 2 , <7 5 = 96 TO (TO -1) (TO -4) (TO -9) [TO + 44] a + 128 n (n - 1) (n - 4) (TO - 9) (n - 16) a 3 , C 6 = - 24 ?i (n - 1) (n - 4) (TO - 9) [17n 2 + 403w + 9000] - 960 n (n - 1) (n - 4) (TO - 9) (n - 16) [n + 41] a 2 - 512 n (n - 1) (n - 4) (n - 9) (n - 16) (w - 25) a 4 , G; = + 96 n (TO - 1) (n - 4) (n - 9) (n - 16) [79n 2 + 2825w + 36180] a + 7168 TO (n - 1) (TO - 4) (TO - 9) (TO - 16) (TO - 25) [n + 42] a 3 + 2048 TO (TO - 1) (TO - 4) (TO - 9) (n - 16) (TO - 25) (TO - 36) a 5 , C 8 = - 48 TO (TO - 1) (TO - 4) (n - 9) [283n 4 - 26978n 3 + 277827TO 3 - 5491932w + 1 27764000] - 3840 TO (TO - 1) (TO - 4) (TO - 9) (TO - 16) (n - 25) [23w s + 1069/2 + 23436] a 2 - 15360 TO (TO - 1) (TO - 4) (TO - 9) (TO - 16) (TO - 25) (TO - 36) [3n + 133] a 4 - 8192 TO (n - 1) (n - 4) (TO - 9) (TO - 16) (TO - 25) (TO - 36) (w - 49) a", &c. in which of course the coefficient of the highest power of TO, in the successive co efficients C r , is the value of C r obtained from the equation (8). With regard to the law of these coefficients I have found that C r = (-y +1 2 2r - s TO (TO - I 2 ) ... {n - (r - I) 2 } 6V a*-* + 2 2r - 6 TO (TO - I 2 ) ... [TO - (r - 2) 2 } GV a r - 4 + 2 2| - TO (TO - I 2 ) ... {TO - (r - 3) 2 } C r s of- 6 + &c. (where however the next term does not contain, as would at first sight be supposed, the factor n (TO - I 2 ) ... (TO - (r - 4) 2 }). And then cy = i, GV = (r - 3) [TO (2r - 7) + (r - 1) (8r - 7)], C r 3 = ( r _4)( r _5)[ TO 2 (4r 2 -24r + 51) + TO (32r - 220r 2 + 41 2r - 255) + 2 (r - 1) (r - 2) (32r 2 - 88r + 51)]. 382 300 ON THE THEORY OF ELLIPTIC FUNCTIONS. [45 In conclusion may be given the following results, in which, recapitulating the notation x = \/k sn u, a = k + , A# _ # (3 - 2o^ + af) ill? GTl 4<-?/ - - V - - - (26 + 16a 2 ) af + sn 5u = x {5 - 20a 8 + (62 + 16a 2 ) ar* - 80a^ - lOoai 8 + 360a# 10 - (300 _ + (368a + 64a 3 ) a; 14 - (125 + 1 60a 2 ) a; 16 + {1 - 5Qa? + 140aw; 6 - (125 + 160a 2 ) a? + (368a + 64a 3 ) x 10 - (300 + 240a 2 ) a (62 + 16a 2 ) of" - 2 &c. Thus, writing # 2 for a 2 ; A; = 1, and therefore a =2, (3 + 8^ + 6^-a! 6^-8^-3^ (1 - 3 2 ) (1 + x 2 ) 3 1 - 3a? where = tan u. (And in general in reducing tan nu the extraneous factor in the numerator and denominator is (1 + x 2 )^ nln ~ l} .) 46] 301 46. NOTE ON A SYSTEM OF IMAGINARIES. [From the Philosophical Magazine, vol. xxx. (1847), pp. 257 258.] THE octuple system of imaginary quantities i lf i 2 , i s , i 4 , i 5 , i 6 , i 7 , which I mentioned in a former paper [21], (and the conditions for the combination of which are contained in the symbols 123, 246, 374, 145, 275, 365, 167, i.e. in the formulae with corresponding formulae for the other triplets i^ie, &c.,) possesses the following property ; namely, if i a , ip, i y be any three of the seven quantities which do not form a triplet, then Thus, for instance, V*3*4J but i 3 . (i 4 and similarly for any other such combination. When i a , ip, i y form a triplet, the two products are equal, and reduce themselves each to 1, or each to + 1, according to the order of the three quantities forming the triplet. Hence in the octuple system in question neither the commutative nor the distributive law holds, which is a still wider departure from the laws of ordinary algebra than that which is presented by Sir W. Hamilton s quaternions. I may mention, that a system of coefficients, which I have obtained for the rectangular transformation of coordinates in n dimensions (Crelle, t. XXXII. [1846] " Sur quelques proprietes des Determinans gauches " [52]), does not appear to be at all con nected with any system of imaginary quantities, though coinciding in the case of n = 3 with those mentioned in my paper " On Certain Results relating to Quaternions," Phil. Mag. Feb. 1845, [20]. 302 [47 47. SUE LA SURFACE DES ONDES. [From the Journal de MatMmatiques Pures et Appliquees (Liouville), torn. XL (1846), pp. 291296.] LA surface des ondes est un cas particulier d une autre surface a laquelle on peut donner le nom de tetraedro ide, et dont voici la proprie te fondamentale : " Le tetrae droide est une surface du quatrieme ordre, qui est coupee par les plans d un certain te"traedre suivant des paires de coniques par rapport auxquelles les trois sommets du te traedre dans ce plan sont des points conjugues. De plus : les seize points d intersection des quatre paires de coniques sont des points singuliers de la surface, c est-a-dire des points ou, an lieu d un plan tangent, il y a un cone tangent du second ordre." On verra plus loin que la surface eut e"te suffisamment definie en disant qu un seul de ces points etait un point singulier ; 1 existence des quinze autres points singuliers est done une propriete assez remarquable. Mais, avant d aller plus loin, il convient de rappeler la signification des points conjugues par rapport a une conique : cela veut dire que la polaire de chacun des trois points passe par les deux autres. Par mi les proprie tes d un tel systeme, on peut citer celle-ci : "Les points d intersection de deux coniques, par rapport auxquelles les memes trois points soiit des points conjugues, sont situe s deux a deux sur six droites, lesquelles passent deux a deux par ces trois points." De la: "Les quatre points singuliers compris dans chaque face du te traedre sont situes deux a deux sur trois paires de droites, lesquelles passent par les trois sommets dans cette face." Autrement dit, les seize points singuliers sont situes deux a deux sur vingt-quatre droites, lesquelles passent six a six par les sommets et sont situdes six a six dans les plans du tetraedre. Done, en conside rant les six droites qui passent par le meme SUE LA SURFACE DES ONDES. 303 sommet, par ces droites combiners deux a deux, il passe quinze plans, y compris trois plans du tetraedre ; en excluant ceux-ci, il y a pour chaque sommet douze plans, chacun desquels passe par quatre points singuliers; done la courbe d intersection d un de ces plans avec la surface a quatre points doubles, ce qui ne peut pas arriver pour une courbe du quatrieme ordre, a moins qu elle ne se re duise a deux coniques. Done : "II y a, outre les plans du tdtraedre, quarante-huit plans, douze par chaque sommet, lesquels rencontrent la surface suivant des paires de coniques." On pourrait de meme chercher combien il y a de plans qui rencontrent la surface suivant des courbes avec trois points doubles, &c. Passons aux cones circonscrits ; on demontre facilernent par 1 analyse cette proprie te : " Les cones circonscrits a la surface ayant pour sommets les sommets du tetraedre, se reduisent a des paires de c6nes du second ordre, lesquelles la touchent suivant les deux coniques de la face oppose e." De plus : " Les seize plans qui touchent quatre a quatre ces paires de cones sont des plans tangents singuliers, dont chacun touche la surface suivant une conique." II y a de plus, entre ces plans, des relations analogues a celles qui existent entre les points singuliers, de maniere que Ton deduit de meme le the oreme : "II y a quarante-huit points, douze a douze dans les quatre plans da tetraedre, pour chacun desquels les cones circonscrits se re duisent a des paires de cdnes du second ordre." Ajoutons que ces quarante-huit points correspondent d une maniere particuliere aux quarante-huit plans, et que les cones circonscrits touchent la surface suivant les coniques situe es dans les plans correspondants. La rdciproque d une surface du quatrieme ordre est, en general, de 1 ordre trerite- six; mais ici, a cause des seize points singuliers, cet ordre se reduit de trente-deux, savoir, a quatre. Et les proprie tes qui viennent d etre enoncdes par rapport aux cdnes circonscrits montrent que la reciproque e tant de cet ordre est ne cessairement une surface de la meme espece ; c est-a-dire : "La reciproque d un te trae droide est aussi un te trae droide." Done le te traedroide est surface de la quatrieme classe. Puisque cette surface ri a pas de lignes doubles ou de lignes de rebroussement, il n y a pas de reduction dans le nombre qui exprime le rang de la surface, lequel est ainsi e gal a douze, c est-a-dire le cone circonscrit est ordinairement de Fordre douze. Mais nous venons de voir que ce cone est seulement de la quatrieme classe (en effet, la classe du cone est la meme chose que celle de la surface) ; done il y a reduction de cent vingt-huit dans la classe du cone. Les seize points singuliers de la surface donnent lieu a autant de lignes doubles dans le c6ne, ce qui effectue une reduction de trente-deux; il y a encore une reduction a effectuer de quatre-vingt-seize, qui doit avoir lieu a cause des lignes doubles ou des lignes de rebroussement du cone. En supposant qu il y a y de celles-ci et x de celles-la (outre les seize lignes doubles dont on a fait mention), il faut que Ton ait 2x + 3y = 96, 304 SUR LA SURFACE DES ONDES. [47 ou x + y ne doit pas etre plus grand que trente-neuf; mais cela ne suffit pas pour determiner ces deux nombres. Nous pouvons encore ajouter que les seize c6nes qui touchent la surface aux points singuliers sont circonscrits, quatre a quatre, a quatre surfaces du second ordre, et que les seize courbes de contact des plans singuliers sont situdes quatre a quatre sur quatre surfaces du second ordre. Je vais donner maintenant une idee de la the"orie analytique. En representant par x = 0, y = 0, z = 0, w = les equations des quatre faces du te traedre, et par 7=0 Fequation de la surface, U sera une fonction homogene du quatrieme degre de x, y, z, w, laquelle, en faisant eVanouir une quelconque des variables, doit se diviser en deux facteurs du second degre", fonctions paires des trois autres variables; de plus, a cause de la condition par rapport aux points singuliers, U ne peut pas contenir de terme xyzw. On doit done avoir U = Atf + % 4 + Cz* -H 2Fy 2 z 2 + 2Gz 2 x 2 + 2Hx 2 y 2 + %Lx 2 iv 2 + 2My 2 w 2 + 2Nz 2 w 2 ou il faut que les coefficients A, B, C, P, F, G, H, L, M, N aient un systeme inverse > de la forme 0, 0, 0, 0, /, g, h, I, m, n. Done, en formant 1 inverse de ce systeme, on obtient, toute reduction accomplie, U=mnfx 4 + nlgy* + Imhz 4 +fghw* + ( lf-mg-nh)( Iy 2 z 2 +fx 2 w 2 ) + ( If + mg nh) (mz 2 x 2 + gy~u> 2 ) + (_ If - m g + n h) ( nx 2 y 2 + hz*w*) = 0. En effet, en ecrivant X = If mg nh, p = lf+ mg - nh, v = lfmg + nh, to = If+mg + nh, V = I 2 / 2 + m?g* + n*ti> - 2mngh - Znlhf- Zlmfg, 1 equation de la courbe pourra s ecrire sous les quatre formes (2mn/a? + nvy* + mpz* + /Xw 2 ) 2 = V (ri>y* + m?z* +/W - ZmfzW - 2fnw 2 y 2 - 2nmy 2 z 2 ), ( nvx 2 + Znlgy 2 + l\z 2 + gpw 2 ) 2 = V (I 2 z 4 + g 2 w 4 + n*& - 2gnw z x 2 - 2nl x 2 z 2 - 2lg z 2 w 2 ), l\y 2 + Zlmhz 2 + hvw 2 ) 2 = V (h 2 w 2 + m?x* + l*y 4 - 2ml x 2 y 2 - 2lh y 2 w 2 - 2hmw 2 x 2 ), + gpy 2 + hvz 2 + 2fghw 2 ) 2 = V (/ 2 ^ + g 2 y* + h 2 z* - 2gh y*z 2 - 2hf z 2 x 2 - 2fg x 2 y 2 ) ; 1 En general, quand deux systemes de quantites sont exprimes line"airenient les uns au moyen des ^autres, on dit que les deux systemes de coefficients sont des systemes inverses ; on passe facilement de la a 1 idee du systeme inverse des coefficients d une fonction du second ordre, et de tels systemes se rencontrent si souvent, que Ton doit avoir un terme pour exprimer sans circonlocution cette relation. 47] SUR LA SURFACE DES ONDES. 305 ce qui met en evidence les e quations des sections de la surface par les quatre plans du te traedre, et aussi celles des points singuliers, lesquelles sont, en effet, ny mz fw = 0, Iz gw nx = , hw mx ly 0, fx gy hz = ; et ces plans touchent la surface suivant les courbes d intersection avec les surfaces = 0, &c., &c., ce qui demontre le th^oreme e nonce par rapport aux seize courbes de contact des plans singuliers, et de la celui pour les seize cones tangents aux points singuliers. Pour de duire de la la forme ordinaire de 1 e quation de la surface des ondes, e crivons I = a/37 (by c/3), m = afiy (ca ay), n = a@y (a/3 &a), /= katx. (by - c/3), g = kb/3 (cy. - ay), h = key (a/3 - ba), equations qui suffisent pour determiner les rapports a : b : c : a : /3 : y : k au moyen de I, in, n, f, g, h. De cette maniere, 1 e quation de la surface se reduit a a/3y (ax 2 + by 2 + cz-} (ax 2 + fty 2 + yz 2 ) - kaz (by + c/3) x 2 w 2 - kb/3 (ca + ay) y 2 w 2 - key (a/3 + 62) z 2 w 2 + k 2 abcw 4 = 0, laquelle se re duit a la surface des ondes en ecrivant |, 77, etant des coordonne es rectangulaires, c est-a-dire en faisant la transformation homographique du tdtraedro ide, de maniere que Fun des plans du te traedre passe a 1 infini, et que les trois autres deviennent rectangulaires. De plus, en particularisant la transformation de maniere que trois des coniques d intersection se re duisent a des cercles, cette surface rentre dans la surface des ondes. II va sans dire que, dans le cas general, plusieurs des points ou des plans dont nous avons parle sont ne cessaire- ment imaginaires ; 1 enumeration de tous les cas differents aurait ete d une longueur effrayante. II y aurait beaucoup a dire sur les cas particuliers ou quelques-unes des coniques se re duisent a des paires de droites (reelles ou imaginaires). Je me contente d ^noncer cette proprie td, tres-facile a ddmontrer, de la surface ordinaire des ondes: au cas ou c = 0, cette surface peut etre engendr^e par un cercle (ayant le centre de la surface pour centre, et dans un plan passant par 1 axe des z\ lequel se meut de maniere a passer par la conique a 2 x 2 + c 2 if = a 2 b 2 . On trouve, dans le Cambridge and Dublin Mathematical Journal, t. I. [1846], p. 208, [38] la demonstration d une autre proprie te de la surface des ondes par rapport aux lignes de courbure des surfaces du second ordre. c - 39 306 [48 48. NOTE SUR LES FONCTIONS DE M. STURM. [From the Journal de Mathematiques Pures et Appliquees (Liouville), torn. XL (1846), pp. 297299.] ON sait que la suite des fonctions fx = (x - a) (x - b) (x - c) (oc-d) ... , jfyc = S (x b) (x c) (x d) ... , f?x = 2(a b) *(x c)(x d) . . . , est de la plus grande utilite dans la theorie de la resolution numerique des equations. En effet, on obtient tout de suite a leur moyen le nombre de racines reelles comprises entre deux limites quelconques. II etait done interessant de chercher la maniere d exprimer ces fonctions par les coefficients de fx. Soit, pour cela, in un nombre quelconque, pas plus grand que le degre n de cette fonction. En prenant k pour tn^ me racine de la suite a, b, c, ... , et mettant, pour abreger, P = (_)*> <<- (a - b) (a - c) ... (a - k) (b - c) ... (b - k) ... (j - k), dans laquelle expression I, a,..., a" 1 - 1 , 1, b,..., b m ~ l , 1, k,..., k m ~\ 48] et, de plus, NOTE SUR LES FONCTIONS DE M. STURM. 307 - a) (x - b) . . . (x - k) dans laquelle le coefficient de x~ r est e"gal a (\^m,(mi) T n ri /J, r \ /A _ Z \ ) [_(* \y vj . . . {U fi c est-a-dire a 1, a, ... , a m ^ 2 , 1, b, ... , b m - 2 , (a; -a) Done enfin le coefficient de x~ r dans f m x : fx est egal a 1, a,..., a m ~ l x 1, a,..., a m ~ 2 , a r ~ l I , K, . . . , K ", K ou, an moyen d une proprie te connue des determinants, en repre sentant comme a 1 ordinaire par 8 q la somme des q iima> puissances de toutes les racines, ce coefficient devient egal a Sf Sf Sf Sf Ml _i, f m , ... , ^ 2m _ 3 , ^,. +M De la, en mettant ^ = V] } tX/ *~~" Ct <S Sf <3 T o a a r/r I > Av_)o , . ^^>1^1 -^ 1 Sf Sf Sf Y 7 en multipliant par /c, et mettant ou Ton suppose on obtient >0 , !,..., O m _ 2 , (j/ M , i r jSf jSf SO *-"! > ^2 > i *Jm 1 ; HJW+1 , i > ^m > > ^aw 3 > ty-zmi , 392 308 NOTE SUR LES FONCT1ONS DE M. STURM. [48 ou r peut ne s e tendre que depuis jusqu a n - m, puisque f m x est fonction entiere. Au moyen des relations coimues qui existent entre les quantity s S q , on a Qm+s,r = (-Y Pr+i Sm+s-2 ... + H p n Q m+S .r = (-Ypr + > +_ + (-Y +m+g - S n + (-Y +m+s ~ 2 p r+m+g -i (r + m + 8-1) ............ r + m + 5 et de la, en posant Q m+ s , r = (-X+" 1 - 1 p r+m &_i . . . + (-)"- 1 p n Sr+m+s-n-i ... [r + J + *>], Q m+ s,r - (-Y +m - l p r+m -SU . . . + ( _ 1 ( r + m + s - n - 1) . . on peut, par les propridtds des determinants, reduire Q m+St r a Q m+s, r dans 1 expression de f m x. Nous avons done exprimd cette fonction au moyen des coefficients p 1} p 2 , ... et des sommes 8 lt S 2 , ..., <Sm-i> lesquelles s expriment facilement par ces memes coefficients (et pour calculer fa, /.&,..., f n x, on aurait seulement besoin de calculer ces sommes une fois pour toutes jusqu a ^_ 3 ) ; il serait done facile de former des tables de ces fonctions, pour les Equations d un degre quelconque, ce qui pourrait a peine s effectuer d aucune autre maniere. 49] 309 49. SUR QUELQUES FORMULES DU CALCUL INTEGRAL. [From the Journal de Mathematiques Pures et Appliquees (Liouville), torn. xn. (1847), pp. 231240.] SoiT x + y V 1 ou x + iy une quantite imaginaire quelconque ; faisons p=<Jx 2 + f, 6 = arc tan y - (1), Ou p etant une quantite positive, et 6 un arc compris entre les limites J TT, -^TT. Cela pose, ecrivons (x + iy) m = p m e im6 (x positif ) , | (x + iy} m = p e im <*"> (x negatif) ; j (dans la seconde de ces formules, il faut prendre le signe supe rieur ou inferieur, selon que y est positif ou negatif.) Au cas de x positif, la valeur du second membre sera ce que M. Cauchy a appele valeur principale de (x + iy} m . Au cas de x negatif, on peut aussi, a ce qu il me semble, nommer cette valeur valeur principale. Cela parait contraire a la the orie de M. Cauchy (Exercices de Mathematiques, t. I. [1826] p. 2); mais la demonstration que Ton y trouve de I impossibilite d une valeur principale pour x negatif ne s applique qu au cas ou Ton suppose que le signe + est toujours le meme sans avoir egard au signe de y. Seulement, selon nos definitions, il importe de remarquer qu il n y a pas de valeur principale pour x ne gatif, au cas particulier on y = ; ou plutot dans ce cas, et dans ce cas seulement, la valeur principale devient inde termme e. Soit, en particulier, x ; ]es deux formules conduisent au meme resultat, savoir = ( yyn^m*; (3^ 310 SUR QUELQUES FORMULES DU CALCUL INTEGRAL. [49 le signe comme auparavant. Car en conside rant x comme infiniment petit positif, on obtient 6 = \ir , et, en conside rant x comme infiniment petit ne gatif, = + ITT , et de la 6 TT = |TT . Ainsi cette formule est toujours vraie, sans qu il soit necessaire de considerer iy comme limite de x + iy, x positif ou x negatif. Remarquons encore que cette fonction (iy) m reste continue quand y passe par zero, ce qui a lieu aussi pour (x + iy) m , x positif, mais non pas pour (x + iy} m , x ne gatif. Les memes remarques s appliquent aux valeurs principales des logarithmes, lesquelles doivent se definir d une maniere analogue par les Equations log (x + iy) = log p + id (a; positif), ^ ! (4), log (x + iy) = log p + i (6 TT) (x negatif), J le signe ambigu, comme auparavant. On demontre sans difficulte que ces valeurs principales satisfont en tout cas aux equations (x + iy) m (x + iy ) m = [(x + iy) (x + iy )] m , \ > (o); log (ao + iy) log (x + iy ) - log [( + iy) (x + iy )] ) seulement ces equations deviennent indeterminees au cas oil, 1 une des quantites x, x, xx yy dtant negative, la quantite correspondante y, y , xy + x y s evanouit. Au moyen de cette definition de la valeur principale d un logarithme, on obtient dx I A + Ba^ ou a, /3 sont reels, et A, B sont assujettis a la seule condition qu au cas ou a, yS ^ seraient de signes contraires, la partie imaginaire de ^ ne s dvanouisse pas. En effet, dans ce cas, 1 int^grale et la valeur principale du logarithme deviennent toutes les deux indeterminees. Sans doute il y a une valeur que Ton peut appeler principale de I integrale, mais cette valeur n est ^gale a aucun des logarithmes de A , DO> et ^ es notions des valeurs principales d une inte grale et d un logarithme n ont pas de rapport ensemble. Ce resultat s accorde avec celui que j ai trouve dans mon Mdmoire " Sur les /auctions doublement periodiques," [25]. 49] SUR QUELQUES FORMULES DU CALCUL INTEGRAL. 311 Je passe a quelques autres applications de ces principes, qui ont rapport a la theorie des fonctions F. Soit d abord r un nombre positif plus petit que 1 unite, et ecrivons r "> U**\ (ix) r - 1 e ix dx ; J -00 on obtient ,,00 ,,00 U = / (ix) r ~ l e ix dx + (- ix) r ~ l e~ !x dx, Jo Jo on enfin, puisque x est positif dans ces deux integrates, au moyen de la formule connue, ,00 of- 1 e ix dx = e*% riti Tr, on en conclut U = 2 cos (r - ) TT . Tr = 2 sin rir . Tr, savoir , 00 sin nr . Tr = l I (ix) r ~ l e ix dx . J 00 On obtiendrait de meme (- ix r ~ l e ix dx . 30 Au premier coup d ceil, ces Equations pourraient paraitre en contradiction 1 une avec 1 autre : mais cela n est pas ainsi, parce qu il n est pas vrai que (- ic) r ~ l soit e gal a (-iy-*(ix) r -\ (-I)*- 1 etant facteur invariable; mais au contraire (-ix) r ~ l = e (r - l]ni (ix} r - 1 , selon que x est positif ou negatif. Dans la premiere de ces Equations, on pout remplacer ix par c+ix, et dans la seconde, -ix par c-ix, c etant positif; mais cela n est pas permis pour c negatif, a cause de la discontinuity de valeur de (c + ix}^ ou (c-ix) r - 1 dans ce cas, quand x passe par zero. En multipliant la premiere Equation par e~ c , et diffe rentiant un nombre quelconque de fois, en admettant, pour les valeurs negatives de r, 1 e quation la formule ne change pas de forme, et Ton obtient r sm nrTr . e~ c = \ I (c + ix} r ~ l e ix dx ........................... (7) J -oo et de meme, au moyen de la seconde Equation, / 00 = 4 | (c - ix)*- 1 e !x dx .............................. (8), 312 SUR QUELQUES FORMULES DTJ CALCUL INTEGRAL. [49 dans lesquelles formules c est positif, et r est un nombre quelconque entre 1, x , sans exclure la limite supe rieure ; seulement, pour r = 0, le facteur sin rirTr se re duit a TT. Au cas ou r est plus grand que ze ro, on peut, si Ton vent, ecrire aussi c = 0. Dans tous les cas, on peut remplacer sin rTrlY par . _ . II importe de remarquer que ces memes inte grales sont absolument inexprimables au cas ou c est negatif: en effet, en e crivant c au lieu de c (c positif), on obtiendrait r r x c o (_ c + ixy-i e dx = e (r -^ (c - ix) r ~ l e ix dx + er (r ~ l} vi (c - ix} r ~ l e ix dx. J 30 JO J X Mais, par la seconde des Equations dont il s agit, / r o 0=J (c - ix) r - 1 e ix dx+ (c- %x) r ~ l e dx JO J ao done r * f x I (- c + ix) r ~ l e ix dx = 2i sin r?r I (c ix) r ~ l e ix dx. J -x JO Or 1 integrale au second membre ne peut pas s exprimer par les transcendantes connues, a moins que r ne soit entier. Ecrivons encore, c etant toujours positif, / =: ( c + ixf~ l e dx, J I l = l (c- ix) r ~ l e ix dx, Jo . 7 2 = (c + ixy- 1 e~ ix dx, Jo r ^3 = I ( c ~~ v&f e~ ix dx. (9). Toutes les fonctions I ( c ix) r ~ l e !x dx, entre les limites 0, x , ou oo , 0, ou oo , oo , s expriment facilement au moyen de I, /!, 7 2 , / 3 . Mais ces quatre fonctions ne sont pas connues ; seulement, au moyen des equations qui viennent d etre trouvees, on obtient 2 sin r-jr Tr.e~ c = D- On deduit encore de ces memes formules : sin rir Fr . e~ c (10). (c + ix) r ~ l cos xdx / oo = i I ( J 00 c + ix) r ~ l sin xdx, f* 00 / 00 = 1 (c t) r ~ 1 cos xd= i I (c ix) r ~ l sin J -00 J X (11). 49] SUR QTJELQUES FORMULES DU CALCUL INTEGRAL. 313 En supposant que r 1 est entier negatif, I inte grale (c 2 + 2 ) r ~ 1 cos xdx se decompose facilement dans une suite d integrales de la forme / oo I (c + ix) p ~ l cos xdx ; J 00 et, en prenant la somme de celles-ci, on obtient la formule (c 2 + x*}^ cos xdx = ; J e~ r (9 + 2c)~ e- 6 d6 (12), l (1 r) J laquelle est due a M. Catalan. Cependant, ni cette demonstration ni celle de M. Catalan ne s appliquent au cas ou r n est pas entier ; la formule subsiste encore dans ce cas, comme M. Serret 1 a demontre rigoureusement. En essayant de la verifier, jo suis tombe sur cette autre formule : (c 2 + i2 ) r ~ 1 cos xdx \ (13), + Ac v i Q e e dv ce qui suppose, comme auparavant, que rl soit ne gatif; en comparant les deux valeurs, on est conduit au resultat singulier (en dcrivant a au lieu de 2c, et 4- (1 p] pour ?) : Y0 + CE (14), (6 + a)^- 1 ) e~ e d0 lequel peut se demontrer sans difficult^, quand p est entier positif impair, en developpant les deux membres suivant les puissances de a; cela se fait au moyen de (15)t ou r s etend depuis jusqu a i(p-l). J ai deduit de cette formule (14) des formules assez remarquables qui se rapportent aux attractions, lesquelles paraitront dans un numero prochain du Cambridge and Dublin Mathematical Journal, [41]. On peut encore demon trer cette formule singuliere : 40 314 SUR QUELQUES FORMULES DU CALCUL INTEGRAL. [49 en effet, pour la verifier, il suffit de reduire I inte grale, d abord k /oo /.oo I (ixf- 1 (- ix) a ~ l e ix da>+ I (- ix) r - 1 (ix) a ~ l e~ ix dx, Jo Jo puis a r*> r 30 (r-a) iri I x r+a-z & ix fa + (a-r) ni x r+a-2 & -ix ^ Jo Jo dont les deux parties s obtiennent au moyen d une formule donne e ci-dessus. Si, an lieu de ( ix) a ~ l , Ton avait (ix) a ~ l , 1 integrale se rdduirait a zero ; mais cela rentre dans une formule plus simple. J ajouterai encore ces deux formules-ci, /_ i r \a ~\ { ^-e ix dx , I (17), r = J _ . C + IX dont je supprime la demonstration. En e crivant dans la derniere x au lieu de x, puis ajoutant, on a <i \a (18); -oo C- + X- d ou Ton deduit tout de suite cette formule de M. Cauchy (19). La formule (7) peut etre considered comme une definition de la fonction Fr au cas de r ndgatif; et, a ce point de vue, elle a, ce me semble, quelques avantages sur celle que M. Cauchy a donnee au moyen des intdgrales extraordinaires. Je passe a la de finition, au moyen d une inte grale ddfinie, de la seconde inte grale eul^rienne, ~ . . Ym Tn (20), quand m ou n, ou tous les deux, sont negatifs. Soit d abord m et n tous les deux positifs, mais n plus petit que 1 unite ; on obtient au moyen de la formule (7) et de la definition ordinaire des fonctions F, 1 f* f * Ym Yn = ^. dx dy x m ~ l (iy) n ~ l e~ x+iy , 2sinn7rJ )_ et de la, en mettant y = ax, dy = xdz, et integrant par rapport a x, J_ x (l-w} m+n> 49] SUE, QTJELQUES FORMULES DU CALCUL INTEGRAL. 315 ou en e crivant k + ai au lieu de ai, k etant positif, ~, 1 [ (k + ai)*- 1 da ~-- ..... formule qui est vraie, meme pour les valeurs negatives de n. En effet, si cette formule est vraie pour une valeur quelconque particuliere de n, elle sera aussi vraie pour la valeur n+p, p dtant entier positif quelconque; ce qui se demontre au moyen des formules de reduction : done il ne s agit que de la demontrer au cas ou n est positif et plus petit que 1 unite, et dans ce cas, puisque la fonction a intdgrer est toujours finie, on peut e crire sans crainte k + ai au lieu de ai, ce qui la reduit a la formule qui vient d etre de montre e ; done cette formule (21) est la definition cherchee, au cas ou n est ne gatif, ou positif et plus petit que 1 unite. Si, de plus, m est ne gatif, ou positif et plus petit que 1 unite , I m sera positif, et Ton de duira ~ , . _ Tm Tn _ F (1 m n) Tn sin (m + n) tr ~ T(m+n) ~ T(l-m) sinrnvr c est-a-dire ~ , x sin (m + n)Tr . $ (m, n) = - V- g (1 - m - n, n) ; sin mir ou enfin, par 1 equation (21), g ( m> n ) = s in(m + n)ff [ - ^^ _ ^ _ t 2 sin WTT sin nir] _ laquelle suppose seulement que m + n soit negatif, ou positif et plus petit que 1 unite. Elle pre sente une analogic assez frappante avec I e quation ordinaire g (m, n) = I a 1 " 1 (1 - a) "- 1 da J qui correspond aux valeurs positives de m, n; de meme que 1 equation (21) est analogue a cette autre forme Q n ~ l da ^ Wf* qui correspond aussi aux valeurs positives de m et n. On peut se proposer de ve rifier I e quation (m et n positifs et plus petits que 1 unite) i r * r * Tin Tn = -r-. - : - dx I du (ix)- 1 (iy) n - 1 e i(x+ ^, 4 sm nnr sin mr J _ , J _, l en transformant le second membre au moyen de x = ay. Pour cela, on distinguera quatre cas, selon que x et y sont tous les deux positifs ou n^gatifs, ou 1 un positif et 1 autre ne gatif. En mettant dans les deux premiers x = ay, dx = yda, 402 316 SUR QUELQUES FORMULES DU CALCUL INTEGRAL. [49 et en integrant par rapport a y, on trouvera que les deux portions correspondantes de 1 integrale double se rduniront en ^ a.- 1 dot F (TO + n) . 2 cos (m + n) TT I Q , a \m+n > c est-a-dire a 2 cos (m + n) TT Tm Tn. Pour les deux autres portions de 1 int^grale double, en ecrivant ae= -ay, on verra qu il faut encore distinguer les deux cas a < 1 et a > 1 . Les quatre integrates ainsi obtenues se reuniront cependant dans les deux n a m ~ l da. [ x a* 1 dot 2 cos 7i7T T (m + n)j^ l _ a)m+n , 2 cos mjr j^ (g _ 1)m+n , savoir, apres quelques reductions faciles, dans celles-ci, 2 cos mr sin mir _, 2 cos nnr sin WITT _ : 7 r Tm Tn, = 7 r 1 m 1 n, sin (w + w) TT sm (m + n) lesquelles se reuniront en cos (m n)7r Tm Tn. On obtient done enfin 1 equation identique 4 sin WTT sin HTT = 2 cos (m n) TT 2 cos (TO + n} TT ; ce qui suffit pour la verification dont il s agit. 50] 317 50. [From the Journal fur die reine und angewandte Mathematik (Crelle), tome xxxi. (1846), pp. 213227.] 1- EN prenant pour donne un systeme quelconque de points et de droites, on peut mener par deux points donnes des nouvelles droites, ou trouver des points nouveaux, savoir les points d intersection de deux des droites donnees ; et ainsi de suite. On obtient de cette maniere un nouveau systeme de points et de droites, qui peut avoir la proprie te que plusieurs des points sont situes dans une meme droite, ou que plusieurs des droites passent par le meme point ; ce qui donne lieu a autant de theoremes de ge ometrie de position. On a deja etudie la theorie de plusieurs de ces systemes; par exemple de celui de quatre points; de six points, situe s deux a deux sur trois droites qui se rencontrent dans un meme point; de six points trois a trois sur deux droites, ou plus ge neralement, de six points sur une conique (ce dernier cas, celui de l hexagramme mystique de Pascal, n est pas encore epuise; nous y reviendrons dans la suite), et meme de quelques systemes dans 1 espace. Cependant il existe des systemes plus generaux que ceux qui ont ete examines, et dont les proprietes peuvent etre apercues d une maniere presque intuitive, et qui, a ce que je crois, sont nouveaux. Commengons par le cas le plus simple. Imaginons un nombre n de points situe s d une maniere quel conque dans 1 espace, et que nous designerons par 1, 2, 3, ... n. Qu on fasse passer par toutes les combinaisons de deux points des droites, et par toutes les combinaisons de trois points des plans ; puis coupons ces droites et ces plans par un plan quelconque, les droites selon des points, et les plans selon des droites. Soit a/3 le point qui cor respond a la droite mene e par les deux points a, /3; soit de meme (3y le point qui correspond a celle mene e par les points j3, y, et ainsi de suite. Soit de plus a@y la droite qui correspond au plan passant par les trois points a, (3, y, etc. II est clair que les trois points a/3, ay, (3y seront situes dans la droite u(3y. Done en representant par 318 SUE, QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [50 N z , N 3 , ... les nombres des combinaisons de n lettres prises deux a deux, trois a trois etc. a la fois, on a le theoreme suivant : THEOREMS I. On pent former un sysieme de N z points stints trois d trois sur N 3 droites: savoir en representant les points par 12, 13, 23, etc. et les droites par 123, etc., les points 12, 13, 23 seront situes sur la droite 123, et ainsi de suite. Pour n = 3, ou n = 4, cola est tout simple ; on aura trois points sur une droite, ou six points trois a trois sur quatre droites; il n en re sulte aucune proprie te ge ome trique. Pour n = 5 on a dix points, trois a trois sur autant de droites, savoir les points 12, 13, 14, 15, 23, 24, 25, 34, 35, 45 et les droites 123, 124, 125, 134, 135, 145, 234, 235, 245, 345. Les points 12, 13, 14, 23, 24, 34 sont les angles d un quadrilatere quelconque 1 , le point 15 est tout a fait arbitraire, le point 25 est situe sur la droite passant par les points 12 et 15, mais sa position sur cette droite est arbitraire. On determinera depuis les points 35, 45; 35 comme point d intersection des droites passant par 13 et 15 et par 23 et 25, c est-a-dire des droites 135 et 235, et de meme 45 comme point d inter section des lignes 145 et 245. Les points 35 et 45 auront la propriete geome trique d etre en ligne droite avec 34, ou bien tous les trois seront dans une meme droite 345. Etudions de plus pres la figure que nous venous de former. En prenant le cinq numeros dans un ordre de terraine, par exemple dans 1 ordre naturel 1, 2, 3, 4, 5, les cinq points 12, 23, 34, 45, 51 pourront etre considered comme form ant un pentagone que nous representerons par la notation (12345). Les cotes de ce pentagone sont eVidem- ment 123, 234, 345, 451, 512. De meme les points 13, 35, 52, 24, 41 peuvent etre considered comme formant le pentagone (13524) dont les cotes sont 135, 352, 524, 241, 413. Ce pentagone est circonscrit au premier, car ses cote s passent evidemment par les angles 15, 23, 45, 12, 34 du premier: mais il est de meme inscrit a celui-ci, car ses angles sont situes respectivement dans les cotes 123, 345, 512, 234, 451 de ce meme pentagone. Done les pentagones (12345), (13524) sont a la fois circonscrits et inscrits I un a 1 autre, done: THEOREMS II. La figure composee de dix points, trois a trois dans dix droites, pent etre consideree (meme de six manieres differentes} sous la forme de deux pentagones, inscrits et circonscrits I un a 1 autre. Ou encore THEOREME III. Etant donne" un pentagone quelconque, on pent tou jours trouver un autre pentagone qui y est d la fois circonscrit et inscrit. Ce second pentagone peut satis- faire d une seule condition donnee quelconque. 1 II faut avoir egard toujours a la difference entre quadrilatere et quadrangle ; chaque quadrilatere a quatre cotes et six angles, chaque quadrangle a quatre angles et six cotes. 50] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 319 Si par exemple le second pentagone doit avoir un de ses angles sur un point donnd d un cote du premier, la construction se deduit tout de suite de ce qui precede. Ces paires correspondantes de pentagones forment une figure connue. On en trouve la construction dans une note de M. [J. T.] Graves dans le Philosophical Magazine [vol. xv. 1839], mais la meme figure est encore mieux connue sous un autre point de .vue. En effet, considerons le point 12, et les droites 123, 124, 125 qui passent par ce point; puis les triangles dont les angles sont 13, 14, 15 et 23, 24, 25. Les cdtds de ces memes triangles sont 134, 335, 145 et 234, 235, 245, et les cotes correspondants se rencontrent dans les points 34, 35, 45 qui sont en ligne droite. Done le the oreme sur les penta gones est le suivant: "Si les angles de deux triangles sont situes deux a deux dans trois droites qui se rencontrent dans un point, leurs cotes homologues se coupent dans trois points en ligne droite" Remarquons aussi que ce the oreme particulier (en n empruntant rien des trois dimensions de 1 espace) reproduit le the oreme general relatif au nombre n. II n y a pour cela qu a considerer n droites passant par le meme point, et qui peuvent etre designe es par 1, 2, 3, ... n. En choisissant d abord les points 12, 13, tout triangle dont les trois angles sont situes dans les droites 1, 2, 3, pendant que deux de ses cotes passent par 12, 13, a la proprie te que le troisieme cotd passe par un point determine 23 situe dans la droite passant par 12, 13. En prenant arbitrairement le point 14, on obtient avec les droites 1, 3, 4 ou 1, 2, 4 les nouveaux points 34, 24 qui sont en ligne droite avec 23, et ainsi de suite. Passons au cas n = 6. II existe ici quinze points situes trois a trois sur vingt droites, ou bien vingt droites qui se coupent quatre a quatre en quinze points. II n y a point ici des systemes d hexagones, mais il existe un systeme de neuf points qui est assez remarquable. Divisons d uiie maniere quelconque les nume ros 1, 2, 3, 4, 5, 6 en deux suites par trois, par exemple en 1, 3, 5 et 2, 4, 6, et considerons les neuf points 12, 14, 16, 32, 34, 3G, 52, 54, 56. Les droites qui passent par 12 et 32, 14 et 34, 16 et 36, savoir 132, 134, 136, se rencontrent dans le meme point 13. De meme les droites qui passent par 32 et 52, 34 et 54, 36 et 56 se rencontrent dans 35, et les droites qui passent par ]2 et 52, 14 et 54, 16 et 56 se rencontrent dans 15. Les points 13, 15 et 35 sont sur la meme droite 135. En considerant les points 12, 14, 16 comme formant un triangle, et de meme les points 32, 34, 36 et 52, 54, 56, cela revient a dire que les droites menees par les angles homo logues des triangles prises deux a deux, se rencontrent trois a trois dans trois points situe s dans la meme droite. Ou bien, ce que Ton savait deja par le theoreme 3 : les cotes homologues des triangles se rencontrent trois a trois dans trois points situe s eri ligne droite. En effet, les cotes des triangles sont 124, 126, 146 pour la premiere, et 324, 326, 346 et 524, 526, 546 pour les deux autres. Les trois premiers cotes se 320 SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [50 rencontrent dans 24, les autres dans 26 et 46, et ces trois points sont dans la droite 246. Maintenant tout cela arrive e galement en combinant les colonnes verticales, ou en conside rant les neuf points comme formant les trois autres triangles dont les angles sont 12, 32, 52; 14, 34, 54; 16, 36, 56. Cela donne lieu au theoreme suivant: THEOREME IV. Le systeme de quinze points, situes trois a trois sur vingt droites, contient (et cela meme de dix manures differentes) un systeme de neuf points qui out la propriete de former de deux manieres differentes trois triangles, tels, que les droites qui passent par leurs angles homologues, prises deux a deux, se rencontrent dans trois points qui sont en ligne droite, tandis que les cotes homologues des triangles se coupent trois a trois en trois autres points qui sont aussi en ligne droite. Dans la seconde maniere de former les triangles, ces deux systemes de trois points en ligne droite sont seulement echanges. II ne reste qu a savoir combien il y en a d arbitraires dans le systeme de quinze points situes trois a trois sur vingt droites. En supposant le systeme forme pour le nombre cinq, on peut prendre arbitrairement 16 et 26 sur la droite 126 qui est determine e par les points 12 et 16. Done 12, 13, 14, 15 et 16 sont arbitrages et 23, 24, 25, 26 sont arbitrairement situes sur des droites donne es. L existence des droites 345, 346, 356, 456 constitue autant de theoremes geometriques ; c est-a-dire, chacune de ces droites est de termine e par trois points. En essayant d approfondir la theorie de six points sur la meme conique, on ren- contrera un systeme de neuf points, tel que ceux que nous venons d examiner; mais il est moins general. II existe des relations entre les points qui n ont pas lieu dans le systeme general. Je renvoie cette discussion a une section separe e de ce me moire, et je passe au cas de n = 7. Pour ce cas on a tout de suite le theoreme suivant: THEOREME V. Le syst&me de vingt et un points situes trois a trois sur trente-cinq droites, peut tire considere (mme de cent vingt manieres differentes} comme compose de trois heptagones, le premier circonscrit au second, le second au troisieme et le troisieme au premier. Les heptagones par exemple peuvent tire (1234567), (1357246), (1526374). Dans ce systeme 12, 13, 14, 15, 16, 17 sont arbitrages, et 23, 24, 25, 26, 27 le sont sur des droites donne es ; les droites 345, 346, 347, 356, 357, 367, 456, 457, 467, 567 sont de termine es chacune par trois points. Dans le cas ge neral 12, 13 ... In sont arbitraires, et 23 ... 2?i le sont sur des droites donnees. II existe % ( n ~ 2 ) ( n ~ 3 ) ( n ~ 4 ) droites dont chacune est de terrnine e par trois points. Un theoreme analogue a celui-ci a lieu quand n est un nombre premier : savoir le suivant : THEOREME VI. Le systeme de N 2 points, situes trois a trois sur N 3 droites, peut tire considere (meme de ^^ manieres} comme compose de (n-l) n-gones, le premier \ n I / circonscrit au second, le second au troisieme, etc., et le dernier a a premier. 50] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 321 Je ne connais pas d autres cas oil 1 idee des nombres premiers se presente dans la geometric. II sera peut-etre possible do trouver des theoremes analogues a III, IV, V, pour toutes les formes du nombre n, mais je n ai pas encore examine cela. Le theoreme general I, peut etre considere comme 1 expression d un fait analytique, qui doit e galement avoir lieu en considerant quatre coordonndes au lieu de trois. Ici une interpretation geometrique a lieu, qui s applique aux points dans Vespace. On peut en effet, sans recourir a aucune notion me taphysique a I egard de la possibilite de Vespace a quatre dimensions, raisonner comme suit (tout cela pourra aussi etre traduit facilement en langue purement analytique) : En supposant quatre dimensions de 1 espace, il faudra considerer des lignes de termine es par deux points, des demi-plans determines par trois points, et des plans determines par quatre points; (deux plans se coupent alors suivant un demi-plan, etc.). L espace ordinaire doit etre consider^ comme plan, et il coupera un plan selon un plan ordinaire, un demi-plan selon une ligne ordinaire, et une ligne selon un point ordinaire. Tout cela pos^: en considerant un nombre n de points, et les combinant deux a deux, trois a trois, et quatre a quatre par des lignes, des demi-plans et des plans, puis coupant le systeme par 1 espace con sidere comme plan, on obtient le theoreme suivant de geometric a trois dimensions : THEOREMS VII. On peut former un systeme de N 2 points, situes trois a trois dans N 3 droites qui elles-memes sont situees quatre a quatre dans N 4 plans. En representant les points par 12, 13, etc., les points situes dans la meme droite sont 12, 13, 23 ; et les droites e tant representees par 123 etc. comme auparavant, les droites 123, 124, 134, 234 sont situees dans le meme plan 1234. En coupant cette figure par un plan, on obtient le theoreme suivant de geometric plane : THEOREME VIII. On peut former un systeme de N 3 points situes quatre a quatre dans NZ droites. Les points doivent dtre representees par la notation 123, etc. et les droites par 1234, etc. Alors 123, 124, 134, 234 sont dans la meme droite designee par 1234. De meme, en considerant un espace a p + 2 dimensions, on obtient la proposition suivante, encore plus generale : THEOREME IX. On peut former dans Vespace un systeme de N p points, qui passent p + 1 d p + 1 par N p+1 droites, situees p + 2 d p + 2 dans N p+2 plans, ou bien pour la ge ometrie plane, un systeme de N p points, situes p + 1 a p + 1 dans N~ p+1 droites. Des the oremes analogues a IV et V seraient probablement tres nombreux et tres compliques. Les reciproques polaires auront e videmment lieu pour tous ces theoremes ; on pour- rait aussi les ddmontrer directement d une maniere analogue. C. 41 322 SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [50 SUR LE THEOREME DE PASCAL. En conside rant six points sur la meme conique, et les prenant dans un ordre determin^, pour en former un hexagone, on sait que les cote s opposes se rencontrent dans trois points situe s en ligne droite. En prenant les points dans un ordre quel- conque, on en peut former soixante hexagones, a chacun desquels correspond une droite ; il s agit maintenant de trouver les relations entre ces droites. M. Steiner a prouve dans son ouvrage Systematische Entwickelutigen u. s. w. [1832], que ces soixante droites passent trois a trois par vingt points, et il ajoute que ces vingt points sont situe s quatre a quatre sur quinze droites. La premiere partie de ce theoreme peut etre demontrde assez facilement, comme nous le verrons : mais pour la seconde partie, je n ai pas reussi a trouver les combinaisons de quatre points qui doivent etre situe s en ligne droite, et il me parait meme qu il est impossible de les trouver 1 . Cherchons les combinaisons des droites qui doivent passer trois a trois par le meme point. Soient 1, 2, 3, 4, 5, 6 les six points situe s sur la meme conique. Considerons d abord 1 hexagone 123456 que Ton obtient en prenant les points dans un ordre deter mine. Suivant le theoreme de Pascal les trois points 12.45, 23.56, 34.61 (ou 12.45 designe le point d intersection des lignes passant par les points 1, 2 et 4, 5) sont situes en ligne droite. Considerons les six hexagones 123456 143652 163254 143256 123654 163452 qu oii tire du premier en permutant les nombres 2, 4, 6 correspondants aux sommets alternes de 1 hexagone. Pour les trois premiers on fait les permutations cycliques de ces nombres (savoir 246, 462, 624), pour les trois autres on fait d abord une inversion 426, 1 Je ne sais pas s il existe une demonstration de la seconde partie du theoreme; je n ai pu la trouver nulle part. Au cas que cette partie du theoreme n 6tait pas correcte, il parait que Ton devra peut-e~tre lui substituer la proposition suivante : " Les vingt points de"terminent deux a deux dix lignes qui passent trois a trois par dix points." On verra dans ce qui suit, de quelle maniere il faudrait combiner ces points. fVoir 55.] 50] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 323 puis les permutations cycliques (426, 264, 642). En ecrivant les combinaisons des points qui doivent etre situes en ligne droite, on a 12.45 23.56 34.61 36.12 56.14 52.34 45.36 14.23 16.52 14.25 43.56 32.61 36.14 56.12 54.42 25.36 12.34 61.54 Suivant cette table les points sur la meme horizontale sont en ligne droite. On remarquera d abord que les trois premieres droites passent par les angles des triangles dorit les cotes sont 36, 45, 12 et 14, 23, 56. Les cotes homologues de ces triangles se rencontrent en 36.14, 45.23, 12.56 qui sont en ligne droite, c est-a-dire, par un theoreme deja cite: les trois ligncs passent par un meme point. On aurait e te conduit au meme re sultat en observant que les trois premieres droites passent par les triangles dont les cotes sont 14, 23, 56 et 52, 16, 34 ou enfin 52, 16, 34 et 36, 45, 12. De meme les trois dernieres droites passent par le meme point. Done il a ete demontre ce qui suit : THEOREM E X. En consider ant les trois hexagones qu on obtient en permutant cydique- ment les angles alternes du premier, les trois droites qui y correspondent se rencontrent dans un meme point. Les soixante lignes passent done trois a trois par vingt points. Ajoutons qu aux trois hexagones de ce theoreme correspondent d une maniere particuliere trois autres hexagones, ou que les vingt points doivent se combiner deux a deux d une maniere particuliere. Mais on se formera une idee plus claire du systeme en remarquant que les neuf droites 36, 45, 12 14, 23, 56 25, 61, 34 out entre elles une relation qui est polaire reciproque de celle entre les neuf points du theoreme IV. Pour faciliter cette comparaison, je prendrai d abord le theoreme analogue pour les tangentes d une conique. THEOREME XI. Soient 1, 3, 5 et 2, 4, 6 des tangentes a une meme conique et 12, etc. les points d intersection de ces droites : les neuf points 36, 45, 12 14, 23, 56 25, 61, 34 412 324 SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [50 peuvent etre determines an moyen de six points de 1 espace A, B, C, a, (3, y, de maniere que A a, etc. repre sente le point d intersection de la droite passant par A, a avec le plan de la figure. Les points sont correspondants entre eux de cette maniere : Act, A/3, Ay BZ, B(3, By COL, C/3, Cy seulement les points 36, 23, 34 etc. sont en ligne droite, ce qui n aurait pas lieu pour les points Act, Bft, Cy, si la position de A, B, C, OL, ft, y dtait arbitraire. On est done conduit a ce probleme : Trouver six points A, B, C, a, ft, y dans 1 espace, tels, qu en representant par Act, etc. 1 intersection de la droite mene e par AOL avec un plan donne", les combinaisons des points (A a, B(3, Cy) (Aft, By, COL) (Ay, BOL, (7/8) (AoL, By, Cft) (Aft, BOL, Cy) (Ay, Bft, Ca) soient en ligne droite. Pour le theoreme de Pascal, cela donne : THEOREME XII. Soient 1, 3, 5 et 2, 4, 6 des points d une conique, les neuf liynes 36, 45, 12 14, 23, 56 25, 61, 34 peuvent etre considerees comme les projections des lignes AOL, Aft, Ay BOL, Bft, By COL, Cft, Cy HUT le plan de la figure, ou A, B, C, OL, ft, y sont six plans, dont la relation reste encore d determiner. En effectuant la solution du probleme que j ai indiquee on aurait, a ce qu il me semble, un point de vue tout a fait nouvea u d envisager les coniques. 50] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 325 Je vais ajouter encore quelques reflexions sur la maniere de chercher les relations qui existent entre les vingt points. En ecrivant settlement les angles alternes des hexagones, on a cette table : 1.2.3 1.2.4 1.2.5 1.2.6 1.3.4 1.3.5 1.3.6 1.4.5 1.4.6 1.5.6 A chaque symbole correspondent six hexagones, qui, a ce que nous avons vu, se part-a gent en deux paires de trois hexagones, et a chaque combinaison de trois, il correspond un point. II y a done deux points qui correspondent au symbole 1.3.5, deux qui correspondent au symbole 1.3.6, deux au symbole 1.5.6 etc. En repre sentant done par 35, 36, 56, les droites passant par ces paires de points, il me parait probable que ces droites aient ensemble les relations du theoreme I, (savoir que 35, 3(5, 56 se ren- contrent dans un point etc.), ce qui donnerait lieu au theoreme hypothetique que j ai enonce dans une note. Voila, a ce que je puis apercevoir, la seule maniere syme trique de combiner les droites. Mais au moins les symboles 1.3.5 1.3.6 1.5.6 ont entre eux des rapports singuliers. En effet, e crivons pour chacun les neuf points du theoreme XII, on a ce tableau : 36, 45, 12 14, 23, 56 25, 61, 34 35, 64, 12 14, 23, 56 26, 15, 34 35, 46, 12 14, 25, 36 26, 13, 54 qui ne contient que quatorze points. Cela me rite des recherches ulte rieures. 326 SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [50 Demonstration analytique du -theoreme de Pascal, et de la premiere partie de celui de M. Steiner. Formules relatives au meme sujet. Solent P = 0, Q 0, R = les equations des lignes 12, 34, 56. On dernontrera assez facilement que les equations des lignes 45, 61, 23 peuvent etre repre sentees par Q + ^R = 0, En effet les six points 1, 2, 3, 4, 5, 6 seront situes dans la conique car en faisant dans cette equation P = 0, 1 equation se re duit a \ (Q + XE) (\Q + R) = ; A, c est-a-dire, la conique contient les points determines par ( P = 0, vP + Q + \R = ), ( P = 0, fjiP + \Q + R = ), ou bieh les points 1, 2 ; et de meme elle contient les autres points 3, 4, 5, 6. Les fonctions P, Q, R sont cense es contenir chacune deux constantes arbitraires ; done on a neuf constantes arbitraires dans ce systeme, qui par consequent est tout-a-fait general. On pent former le systeme suiyant d equations : 12. P =0, 13. \/*P+ Q+ \R = 0, 14. XP+ nQ + \/*R = 0, 15. P+ vQ+ v\R = 0, 16. vP+ Q+ \R = 0, 23. pP+ \Q+ R = 0, 24. P + /AQ+ (*R = Q, 25. XP + v\Q + V R = 0, 26. v\P+ \Q+ R = Q, 34. Q =0, 35. 36. 45. P+ V Q+ fiR = Q, 46. vP + Q + vpR = 0, 56. R = 0. 50] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 327 Ecrivons les equations des lignes comprises dans la table de neuf points ci-dessus donnds. On a d abord + pQ + vR = 0, f* P + vQ + nR = 0, " P = 0, , XP + fiQ+-\fjuR = 0, -..* pP + \Q+ R = 0, *t 12 = 0, \P + v\Q + vR = 0, >* vP+ Q + \R = 0, 3 Q = 0. En combinant la seconde et la troisieme colonne verticale du tableau, on - obtient pour les trois points d intersection des cotes opposes de 1 bexagone 123456, les equations (P = 0, /!*Q + 1/12 = (R = 0, \P + fj,Q = (Q=0, \P + vR = 0\ qui appartiennent a trois points situes sur la droite \P + pQ + vR = 0, ce qui suffit pour demontrer le thdoreme de Pascal. On obtient de merae, en combinant les autres paires de colonnes verticales, deux systemes de trois points, respectivement situes dans les droites + + ^=0 et vX /i v , lesquelles, avec la droite qu on vient de trouver, se rencontrent evidemment dans un meme point, determine par les deux equations XP + fj^Q + vR = et Pi~\ ~D \J i X fjb v Voila une de monstration de la premiere partie du theoreme de M. Steiner. Les equations que nous venons de trouver appartiennent au point d intersection des trois droites qui correspondent au premier des trois hexagones du symbole 1.3.5. -Pour trouver 1 autre point correspondant de la meme maniere a ce symbole, il faut combiner les colonnes horizontales, ce qui donne pour les coordonne es de ce point : (X //,!/) P = (/i v\) Q = (v X/x) R. En cherchant de meme les expressions des points qui correspondent aux symboles 1.3.6 et 1.5.6, on obtient des re sultats moins e le gants, mais qui valent peut-etre la peine d etre enonce s ici. Je forme cette table complete : 328 Pour 1.3.5 I. SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. Systemes de trois lignes, qui se rencontrent dans un point. [50 \P vR II. Pour 1.3.G I. II. Pour 1.5.6 I. (X - A*") P = (V - V) R , (v XA*) R = (f4 V\) Q , (ft- V \)Q=(\- A^)P; = 0; + \Q + \fjLvR) + (i/XP + fivQ + R) = ; (x - = 0; (fiv - \IJL-V~ - X + Xyu, 2 ) P + Oz; - X/xV - X + Xi/ 2 ) P + - - Xi; 2 ) P + - X/i 2 ^) Q + i/ 2 + \(*?v) Q + (f* 2 v (fi~v \/AV-} R = 0, (v XA*) R = o, ,v - v + \u) R = ; n. - (\i/ 2 - XAAV-AW/ + x^ 2 ) P + (^ - /W 2 ) Q + (X - X/^V) P + (i/X - /Ltv 2 + A 4 - X/i 2 ^) Q + (V - ^ 2j/ + y - V"") ^ = - Note sur le theoreme de M. Brian chon. On pent donner une demonstration semblable de ce theoreme, en prenant pour les equations des six tangentes celles-ci : 1. P=0, 3. Q = 0, 5. R - 0, 4. a P + /3 <2 + 7 ^ = 6. a P + /3 Q + 7 ^ = > 2. a"P + ^ // Q + 7 // P = 0, et en cherchant la relation entre les coefficients qui est necessaire pour que ces six equations appartiennent aux tangentes d une meme conique. On obtient facilement (07 - a 7 ) (/3V - /3V) ( 7 "/3 - 7/3") - (aV - V) (/3" - /3a") ( 7 /3 - 7 /3), ce qui exprime aussi la condition pour que les trois diagonales se rencontrent dans un meme point. 51] 329 51. PROBLEMS DE GEOMETRIE ANALYTIQUE. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn. xxxi. (1846), pp. 227230.] Trouver explicitement les coordonne es des centres de similitude de deux surfaces du second ordre, dont chacune est circonscrite d une meme surface de cet ordre. LEMME. Soit U = Ax 2 + By 1 + Cz- + 2Fyz + 2Gzx + 2Hxy + 2Lxw + 2Myw + 2Nzw + Pw- (1), 1 expression generale d une fonction homogene du second ordre a quatre variables, et conside rons les de rivees KU = A, H, G, L H, B , F, M G, F, G, N L, M, N, P OL, A, H, G, L /3, H, B, F, M 7, G, F, C, N S, L, M, N, P (2), ou dans F^ U les lettres o, p ecrites en bas servent a indiquer les variables a, & 7 , B et ^, 77, p, w qui doivent entrer dans 1 expression de cette fonction; par exemple FffU est ce que devient F po U, en ecrivant f 77, p, a> an lieu de a, , 7 , 8. Cela posd, soit U = Ax 2 + By* + Cz 2 + ZFyz 4 ZGzx + ZHxy + 2Lxw + 2Myw + 2Nzw + 2Pw 2 , . . . (3) . V = ax + fty + y Z + Sw, /^ C> 42 330 PROBLEME DE GEOMETRIE ANALYTIQUE. [51 on aura cette Equation identique : qui subsiste meme pour un nombre quelconque de variables. L expression analytique du theoreme consiste en effet en ce que les reciproques de deux surfaces du second ordre, circonscrites 1 une a 1 autre, sont deux surfaces du OC II Z second ordre qui ont cette meme relation. Car en prenant , , pour les coordon- w w w nees d un point, les equations de deux surfaces circonscrites 1 une a 1 autre sont ^ F! = O } (6). . Les equations de leurs reciproques polaires (par rapport a x 2 + y- + z 2 + w 2 = 0) sont c est-a-dire, et vertu du theoreme qui vient d etre pose : ou JK(U+ F 2 ) est constant ; c est-a-dire les premieres parties des Equations ne different entre elles que par le carre* de la fonction lineaire ( F op IT) ; ce qui prouve le theoreme en question. SOLUTION. Soient C r +F 1 2 = 0et U + Vf = (8), les equations des deux surfaces, dont chacune est circonscrite a U = 0. Les expressions de Fj et F 2 sont et F 2 = civX + /3 2 y + j 2 z + S. 2 w, et les lettres o 1( o 2 ecrites en bas se rapporteront a a 1} fr, y^ Sj et a 2 , /S 2 , 7,,, So respectivement. Mettons de plus, pour abreger, Les polaires des deux surfaces ont pour Equations et ces surfaces polaires se rencontrent eVidemment selon les courbes situe es dans les plans exprirnes par les Equations ^K z F 0ip U^K^ p U=Q (12); equations qu on peut dcrire sous cette forme tres simple : (13), 51] en mettant PROBLEME DE GEOMETRIE ANALYTIQUE. c^ 331 7 = ^ 72, (14), et supposant que o se rapporte a a , ft , 7 , 8 . On a enfin, en se servant de la notation complete des determinants, (15), /3 , H, B, F, M 7 , G, F, C, N 8 , L, M, N, P equation qui est double, a cause des doubles valeurs de a, (3 , 7 , 8 . Les poles de ces plans sont les centres de similitude des deux surfaces donnees. Soit done identiquement ..-(16), a) = on a 13 , H, B, F, M 7 , G, F, C, N 8 , L, M, N, P x : y : z : w =^ : 23 : pour les coordonnees - , , - des deux centres de similitude. donnees par I e quation (16), savoir par ou gl = a. , H, G, L , B, F, M 7 , F, C, N 8 , M, N, P (17), 3, <, 5B sont &c. (18), a , A, H, G, L P, H, B, F, M 7 , G, F, C, N 8 , L, M, N, P OL, P, 7 , 8 sont donnds par (14), et K lt K 2 repre sentent ce que devient le deter minant A, H, G, L H, B, F, M G, F, C, N L, M, N, P en ecrivant A + af, B + (3-?, C + 7^, P + 8^, F + fi^i , N + J&, ou A+xf, &c. au lieu de A, B, C, P, F, G, H, L, M, N. 422 (19), 332 [52 52. SUR QUELQUES PROPRIETES DES DETERMINANTS GAUCHES. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn. xxxn. (1846), pp. 119123.] I. JE donne le nom de determinant gauche, a un determinant forme par un systeme de quantity s \. s qui satisfont aux conditions V8 = -Vrlr=M] .................................... (1). J appelle aussi un tel systeme, systeme gauche. On obtiendra des formules plus simples (quoique cette supposition ne soit pas essentielle), en considerant seulement les systemes pour lesquels on a aussi Vr=l ............................................. (2). Je suppose dans tout ce qui va suivre, que le determinant est de 1 ordre n, et que par consequent les suffixes variables r, s, &c., s etendent toujours depuis Funite jusqu a n. En posant les equations 2 tr \ r _ s X r =P g , %s\-.s%s~Qr .............................. (3), j exprime les systemes inverses qui determinent les P, Q par les x, de la maniere suivante : Kx r =^ e K r _ s P s , et Kx s = 2 r A r iS Q r ........................... (4), ou K de signe le determinant forme avec les quantites \. s , et A r s le coefficient differentiel de K par rapport a \r. g ; bien entendu, que la differentiation doit etre 52] SUB QUELQUES PROPRIETES DES DETERMINANTS GAUCHES. 333 effectuee avant d avoir particularism ces quantites par les Equations (1), (2). On sait que ces fonctions A satisfont aux conditions A, r-g A r s = K, \ r s A r > g =0 , ^T.S A r s = K. (5). Je tire des equations (4), en dchangeant r et s dans la derniere de ces Equations : 2 g A r . 8 P 8 =2A.rQ (6), et de la, en multipliant respectivement les diffe rentes equations de ce systeme par \,, ,., et prenant la somme de ces produits : 2 s (S r X s -. r A r . s )P g = S s (SA .rA S .r)Q S (7). On a d abord par les Equations (5) 2 8 (SA .r A s-r ) Q s = KQg (8) ; puis par les Equations (1) et (2) 2 r A, S . r A r s = 2A S . t , SrV.s A,. s ^9^ c est-a-dire par les equations (5) : 2 r \s .rA r .. s =2A S -S ; S ^FS} et SrVrA^^aA^-JST / ce qui donne S,(2 r VrA r ..)P t = 2(2,A^. t P,)-JTP^ (11). Substituant les e quations (8) et (11) dans la formule (7), on obtiendra, en ecrivant r au lieu de s : (12), et egalement KP S = 2 (S r A r s Q r ) KQ S (13) Posant maintenant Kci r s = 2A r g ; et les formules (12), (13) se changeront en Ps et P s = ^ r a r _ s Q r (15): 334 SUE QUELQUES PROPEIETES DES DETERMINANTS GAUCHES. [52 equations qui sont necessairement dquivalentes. On a done identiquement = == S r r .s a r * = 1, (16), f ^. n n , = O v t* (2) c est-a-dire, on a trouve un systeme de w 2 quantites a r s , fonctions explicites et rationnelles d un nombre ^n(n l) de variables independantes, qui satisfont identi quement aux formules (16, 1) et (16, 2). On sait qu en gdometrie cela veut dire que pour n = 2 ou n3 de tels systemes donnent les coefficients propres a effectuer la transformation de deux systemes de coordonne es rectangulaires ; nous dirons par analogic, que des systemes qui satisfont aux equations (16) pour une valeur quelconque de n, sont propres a effectuer la transformation entre deux systemes de coordonnees rectangu laires. On a done le theoreme suivant : Les coefficients propres a la transformation de coordonnees rectangulaires, peuvent etre exprimes rationnellement au moyen de quantitds arbitrages \ r , soumises aux conditions ^s.r = ^g.r [ r =M] ; \ rr =l. Pour developper les formules, il faut d abord former le determinant K de ce systeme, puis le systeme inverse A r _ s ,... et ecrire Kdr s = 2A r s [r$s]; Kat r r = 2A r r -K ; ce qui donne le systeme cherche. Soit par exemple n = 3. Ecrivons pour le systeme des quantites \. s (1 7 ) ; ce qui donne K = 1 + X 2 + p? + v 2 , et pour le systeme des fonctions A r 1 + A, 2 , X/A + v , v\ fj,, "] \fJb-V, 1+/A 2 , JJ<V + \, r ........................... (18). v\ + ft, /j,v X , 1 + v 2 . ) De la on obtient pour le systeme de coefficients a, /S, 7; a , /3 , 7 ; a", ft", 7": ce qui se rapporte a la transformation x = a. x l + /3 y-L + 7 z l , x l = ax + a!y + a"z , (20), z = af x, + ^"y, + y Zi , z, = yx + j y + y"z, 52] SUR QUELQUES PROPRIETES DES DETERMINANTS GATJCHES. 335 de deux systemes de coordonnees rectangulaires. En" effet, les coefficients X, /A, v out une signification ge ome trique : Les axes Ax, Ay 1 , Az l vont coi ncider avec les axes Ax, Ay, Az, par la rotation 6 autour d un certain axe AR (qu on peut nommer " Axe resultant "). En prenant /, g, h pour les inclinaisons de cet axe a Ax, Ay, Az, on a X = tang |-0 cos^ //,= tang \Q cos g, v = tang ^0 cos h. Cette expression de 1 axe AR est due a Euler; les quantite s X, //,, v ont e te introduites pour la premiere fois, par M. Olinde Rodrigues, dans un memoire "Sur les lois ge ome triques qui regissent les deplacements d un systeme solide " (Liouville, torn. v. [1840]), ou il donne [des expres sions semblables a celles] qu on vient de trouver ici, pour les coefficients de la trans formation, en termes de X, fji, v. Ces memes quantites X, //-, v (il y a a remarquer cela en passant) sont lie es de la maniere la plus etroite avec celles de la belle the orie de Sir W. Hamilton sur les Quaternions. Je les ai applique es a la the orie de la rotation d un corps solide. Avant de donner une idee des resultats auxquels je suis parvenu, je passe aux formules de transformation qui se rapportent au cas de n = 4. Je prends ici pour le systeme des quantite s X: -a, -b, a, 1, - 1. - ce qui donne, en mettant pour abreger, af+ bg + ch= 6, K = 1 + a 2 + b 2 + c- +f 2 + g 2 + h" -f et puis pour les quantites A,. s le systeme 1 +/ 2 + g 2 + A 2 , fd + a + bh eg, gd + b + cf ah, -fO -a + bh-cg , 1 +/ 2 + b 2 + c 2 , -cd-h +fg - ab, g0b+cf-ah, c0+h+fgab, l+g 2 +c 2 +a 2 , - hd - c + ag - bf , - b6 - g + hf - ca, a0+f+gh-bc : de maniere que pour Ka, Ka. , Ka," , Ka!" , K{3, Kj3 , K0", Kff", (21), (22), hO + c + ag - bf, b6 + g + hf ca, a0f+gh- be, ... (23), (24), KB, KB , KB", KB ",; on obtient le systeme suivant : 1 +f 2 +g-+h 2 -a*-b--c*, 2(f6+a+bh-cg) , 2 (g0 + b + cf-ah) 2 (-fd -a+bh-cg) , l+f* + b 2 +c 2 -# 2 -/i 2 -a 2 , 2 (-c<9 -h +fg - ab) 2(-h6-c + ag-bf) , 2 (- b0 -g + hf- ca) , 2 (a0 +f + gh - be) et ainsi de suite pour des valeurs quelconques de n. -/ 2 -/i 2 -i-, 2(-a0-f+gh-bc) 336 STIR QUELQUES PROPR1ETES DES DETERMINANTS GAUCHES. [52 II. Maintenant je vais citer les formules que j ai presentees dans le Journal de Cambridge, t. ill. (1843) p. 225, [6], pour la rotation d un corps solide autour d un point fixe. Mettant, comme a 1 ordinaire, les vitesses angulaires autour des axes principaux p, q, r, les moments du corps pour ces memes axes =A, B, C, et la fonction des forces = V: les equations cities pourront etre ecrites sous la forme dp do dr d\ dy, dv -* _ , P~ Q~ R A ~ M ~ N (26), dV A = 1 {( 1 + X 2 )^ + (\fi - v ) q + (\v + fi) r}, ... (27). N = 2 En effet, pour obtenir ces formules, il n y a qu a chercher au moyen de X, /n, v, et de leurs ddrivees par rapport aux temps X , //, v, 1 expression de la fonction T = ^ (Ap 2 + Bq 2 + Cr 2 ) (qui exprime la demi-somme des forces vives} : cela fait, les formules generales que Lagrange a donne es pour la solution des problemes de dynamique, conduisent imme diatement aux equations en question. Dans le memoire cite j ai integre ces equations pour le cas ou la fonction V est zeVo, et en prenant, comme dans la theorie ordinaire, le plan invariable pour plan des deux axes. Ce n est que dernierement que j ai trouve la maniere convenable de traiter ce systeme d dquations ; je le fais au moyen de deux nouvelles variables O, v, entre lesquelles je trouve une equation differentielle dont les variables sont separees, et j exprime en termes de celles-ci les autres variables du probleme, y compris le temps ; et cela sans aucune supposition particuliere, relative a la position des axes des coordonne es par rapport au plan invariable. Le deVeloppement de cette thdorie paraitra dans le prochain No. du Cambridge and Dublin Mathematical Journal, [37]. Je m occupe aussi de la recherche des formules pour les variations des constantes arbitraires relatives aux perturbatrices. II serait bien inteVessant (comme probleme d analyse pure) d e"tendre ces recherches au cas d une valeur quelconque de n ; il faudrait pour cela, chercher les valeurs des quantite s analogues a p, q, r, former une fonction T, en prenant la somme des Carre s de chacune de ces quantite s, chaque carre multiplie par un coefficient constant, et puis former les d dT dT equations -r..^r, ~T.=^> &c., analogues aux equations de Dynamique. Mais je nai at a\ a\. encore rien trouve sur ce sujet. 53] 337 53. RECHEECHES SUE L ELIMINATION, ET SUE LA THEORIE DES COURBES. [From the Journal fur die reine und angewandte Mathematik (Crelle), tome xxxiv. (1847), pp. 3045.] EN designant par U, V, W, ... des fonctions homogenes des ordres w, n, p, &c. et d un nombre egal de variables respectivement, et en supposant que ces fonctions soient les plus g&ierales possibles, c est-a-dire que le coefficient de chaque terme soit une lettre indetermin^e : on sait que les e quations U=0, V=Q, W = 0, ... offrent une rela tion 6 = 0, dans laquelle les variables n entrent plus, et ou la fonctiori , que Ton pent nommer Resultant complet des equations, est homogene et de 1 ordre np ... par rapport aux coefficients de U, de 1 ordre mp ... par rapport a ceux de V, et ainsi de suite, tandis qu elle n est pas decomposable en facteurs. Cela pose, supposons que les coefficients de U, V, ... , au lieu d etre tous indetermine s, soient des fonctions quel- conques d un certain nombre de quantites arbitraires. Substituant ces valeurs dans la fonction , cette fonction sera toujours le Resultant complet des e quations. Mais peut quelquefois etre decomposable en facteurs, dont quelques-uns doivent etre elimine s. En effet les coefficients de U, V, ... peuvent contenir des quantitds 97, ... censees comme variables, et d autres quantites a., jB, ... qui sont constantes, et il peut s agir de la relation entre , 77, ... qui est necessaire pour que les e quations U =0, V= 0, &c. puissent subsister conjointement. Dans ce cas tout facteur A du resultant complet C H ), qui ne contient pas les coefficients variables , 77, ... , doit etre rejete. En supprimant ces facteurs, et exprimant par <1> le facteur qui reste, cette fonction est alors ce que nous nommerons Resultant reduit. Cependant les facteurs A, proprement dits, ne sont jamais des facteurs etrangers, et ce n est qu a cause du point de vue particulier, auquel on a envisage" la question, qu ils ont e te rejete s ; d un autre point de vue ces facteurs auraient pu devenir le Resultant reduit. Nous les nommerons done, a 1 exemple de M. Sylvester, Facteurs speciaux. C- 43 338 KECHERCHES SUR ^ELIMINATION, ET SUR LA THEORIE DES COURBES. [53 Pour faire voir tout cela avec plus de clarte", prenons pour exemple le probleme suivant de Geometric analytique: " Trouver les equations da systeme des lignes tirees par un point donne aux points d intersection de deux courbes donne es." Soient U=Q, F=0 les equations des deux courbes, U, V etant des fonctions homogenes des variables x, y, z, des ordres m et n respectivement, ce qui revient a prendre x : z et y : z pour coordonnees d un point. De meme il faut exprimer par (OL, /3, 7) le point dont les coordonnees sont : 7 et /3 : 7; et ainsi pour tous cas semblables. II s entend, qu on suppose partout que les coefficients de U, V, ou de U, restent absolument inde termine s. Representons par (a, ft, 7) le point donne", et par ( f), un Pi nt quelconque d une des lignes dont il s agit. En eliminant x, y, z entre les Equations 7=0, F=0, et a? (- 717) + y(7f -) + *(?- = ............ (1), on obtient liquation cherchee = 0. Ici est une fonction homogene de 1 ordre mn par rapport a /3 777, 7^ et a? 7 ~~ fi > de l or( ire n par rapport aux coefficients de U; et de 1 ordre n par rapport a ceux de F; de plus cette fonction est decomposable en mn facteurs lineaires par rapport a , 77, , dont les coefficients sont des fonctions irrationnelles de a, fi, 7 et des coefficients de U et F (en effet toute fonction homo- gene de j3yn, 7^ a , ay fit; est dou^e de cette propriete", qui subsiste encore en echangeant entre eux 77, f et a, /S, 7). Chaque facteur lin^aire, ^gal^ a zero, appar- tient a une des lignes en question. Voila pourquoi 0=0 est consideree comme equation du systeme des lignes. Soit maintenant propose le probleme : "Trouver 1 equation du systeme des tangentes tirees d un point fixe a une courbe donnee." II y a ici a eliminer x, y, z entre les equations 7=0, dU n dU . dU ft - J ---j-=0 et dx dy dz .(2). Le resultant complet est une fonction homogene de 1 ordre n (n - 1) par rapport a fit-yri, 7^ -af et ^-/Sf {n repr^sente ici 1 ordre de la fonction U}, de 1 ordre n par rapport a a, /3, 7, et de 1 ordre 2n - 1 par rapport aux coefficients de U. Mais il existe dans ce cas un facteur special L\ qui est ce que devient U en e crivant a, /3, 7 a la place de x, y, z. En le mettant de cote, le resultant reduit < est fonction de 1 ordre w(n-l) par rapport a PS-vn, y-Z et ay - /3f, et de 1 ordre 2(n-l) par rapport aux coefficients de U, et liquation <1> = correspond au systeme de tangentes. 53] RECHERCHES SUE I/ELIMINATION, ET SUB LA THEORIE DES COURBES. 339 En mettant x, y, z a la place de /3 7*77, 7^ a%, ay (3g, <& devient une fonction de 1 ordre n(n 1) par rapport a x, y, z, et de 1 ordre 2(n 1), comme elle 1 etait ci-dessus, par rapport aux coefficients de U. Nous de signerons cette nouvelle valeur de <l> par FIT; c est-a-dire nous reprdsenterons par FU le resultant r^duit des Equations -(3), adU+QdU d _U = Q t xx + yy + zz 0, FU dtant une fonction des ordres n(n 1) et 2(w 1) par rapport a x, y, z, et aux coefficients de U. On sait que I e quation FU=Q est celle de la polaire reciproque de la courbe, par rapport a la conique auxiliaire x 2 + y 2 + z* 0. Or les equations (2) peuvent etre ecrites aussi sous la forme dU dU = dx dy dz .(4). + + dx dy dz j Ici le resultant complet est de 1 ordre n(n 1) par rapport a a, ft, 7, ou a ^, 17, (car ce resultant complet doit etre comme ci-dessus fonction de ce meme ordre de /3 777, 7^ a^ et a?/ /3) et de I ordre (w 1) (3w 1) par rapport aux coefficients de U. Le facteur special dans ce cas est done une fonction de I ordre 3 (n I) 2 des coefficients de U, et il est facile de trouver sa forme ; car on satisferait aux equations (4) en posant dU =Q dU =Q dU^ Q dx dy dz le resultant de ces Equations, que nous designerons toujours par KU, doit done se presenter comme facteur spe cial du rdsultant complet du systeme. Mais KU etant une fonction des coefficients de I ordre 3 (n I) 2 , elle est precisement le facteur special dont il s agit. (II est clair que liquation KU=Q serait la condition ne cessaire, pour que la courbe put avoir un point multiple.) Reprenons le premier systeme. On satisfait a la derniere equation en ecrivant x cd + ^m, y = ftl + rjm, z = yl+m ..................... (6). Soient [U], [V] ce que deviennent U. V par cette substitution. En e liminant I, m entre les deux Equations [17] =0, [V]=Q, on obtiendra le meme resultant que ci-dessus. En effet, le re sultant de ces deux equations est des ordres n et m par rapport aux coefficients de U et V, et de I ordre 2mn par rapport a a, ft, 7, tj, f: done il faut qu il soit e gal a , a un facteur numerique pres. On a done le theoreme suivant: 432 340 RECHERCHES SUR ^ELIMINATION, ET SUR LA THEORIE DES COURBES. [53 THE OREME I. Liquation du systeme des droites mendes par un point donne (a, ft, 7) aux points d intersection des deux courbes 17=0, F = 0, se trouve en eliminant les nouvelles variables I, m entre les deux Equations [U] = Q et [F] = 0, ou [U], [F] sont ce que deviennent U et V par les substitutions En ope rant egalement sur le second systeme d dquations, on obtient directement le re sultant reduit, sans que I ope ration soit embarrassee par aucun facteur special. En effet, on peut remplacer le systeme par dU dU dU a +/3- r +7- r =0, dx dy dz f ^ + ,!*? +f */=o, * dx dy . dz et en faisant les substitutions (6) dans la derniere equation, les deux autres equations se changent en -0 et jj u et j dl dm ou [V] est ce que devient U par cette substitution. Le re sultant de ces Equations est de 1 ordre 2ro(w-l) par rapport a a, ft, 7, f, 17, {c est-a-dire une fonction de ft^ ^ t ylf-af, arjft%, de 1 ordre w(w 1)}, et de 1 ordre 2 (w 1) par rapport aux coefficients de U ; done il n y a plus de facteur special. De la suit : II. L e quation du systeme de tangentes mendes du point donne (a, ft, 7) a la courbe U=0, se trouve en e liminant I, m entre les Equations dl dl _ [ U] etant ce que devient U par les substitutions x = ctl + %m, y = {3l + r)m et z yl + w. En representant liquation par < = 0, ^> est une fonction de /3 - 777, 7^ af, ar) ft%, et en remplac,ant ces fonctions par x, y, z, on obtient 1 equation de la polaire rdciproque (par rapport a 2 + y 2 + z n - 0) de la courbe donne e. Ce beau theoreme est du a M. Joachimsthal, qui me 1 a communique I e te passe pendant mon sejour a Berlin, avec une demonstration. On de duit de la, comme cas tres particulier, une forme du resultant des deux Equations ax 2 + 2bxy + cy* = et a! a? + Wxy + c y 2 = 0, cite e dans mon mdmoire sur les hyperde terminants (t. xxx. de ce journal, [16]). En effet, soit # 2 = z, acy = -y, 2/ 2 = x, et f7=xz y 2 , on aura evidemment U=0, dU 7 dU dU ,dU , ,,dU , ,dU a- r +6- r -+c-r- = 0, et a ~r +b -j- + c -j- =0, dx dy dz ax dy dz 53] RECHERCHES SUR ^ELIMINATION, ET SUR LA THEORIE DES COURBES. 341 done en cherchant le resultant de ces equations de la maniere indique e dans le the oreme, on obtient la formule en question 4 (ac - b 2 ) (a c - b 2 ) - (ac r + a c - 2 bb ) 2 = 0. Cependant la veritable generalisation de cette formule, a ce que je crois, reste encore a trouver. II suit des principes developpes dans le me moire cite, que le re sultant cles deux equations L = 0, M=0 (ou L, M sont des fonctions homogenes des deux variables I, m) peut toujours etre prdsentd sous la forme @ = VLL ... MM ... , ou V est une composition d expressions symboliques, telles que (12) a (13)* 3 ... , dans lesquelles (12) =d h 9, 2 -9 Wl d L , &c. Par exemple pour L = al 2 + 2blm + cm 2 , M = a l 2 + 2b lm + c m 2 , on a = {(12) 2 (34) 2 - (13) 2 (24) 2 } LL MM, {c est-a-dire, comme ci-dessus, @ = 4 (ac - b 2 ) (a c - b 2 ) - (ac + a c - 2bbJ\. En appliquant cette the orie a 1 elimination des inconnues entre les Equations [U] = 0, [ F] = 0, on obtient ydi, et d m = et en faisant cela donne equation qui peut etre presented sous la forme abr^ge e On obtient le resultant cherche en faisant cette substitution dans tous les symboles que contient V, et en introduisant U, V au lieu de [V], [F]. Les memes remarques peuvent etre appliquees au cas d une elimination entre les , ,. dL dL deux equations ^ = 0, ^ = 0, car le re sultant sera exprime ici sous la mme forme @ = VLL .... Cherchons par exemple 1 dquation de la polaire re ciproque d une courbe du second ordre : en observant que le re sultant des equations ~ = = (ou L dl dm est une fonction du second degre") sera exprime sous la forme = (12) 2 ZZ, on obtient imme diatement pour la reciproque de la courbe du second ordre U=0, 1 e quation laquelle se reduit en effet a la forme connue. Passons au cas d une courbe du troisieme ordre. Comme pour une fonctiou de deux variables, le re sultant des Equations dL_ dL ~FJ "> ~7 dl dm 342 RECHERCHES STIR ^ELIMINATION, ET SUR LA THEORIE DBS COURBES. [53 aura la forme 8 = (12) 2 (34) 2 (13) (42) LLLL, on a pour la polaire de U 0, - 2F 7 = (P12) 2 (P34) 2 (P13) (P42) UUUU = 0, et il serait facile de calculer par la le coefficient d une puissance ou d un produit quelconque des variables. Par exemple le coefficient de z 8 se re duit a {(12) "} 2 {(34) /// } 2 {(13) "} 2 {(42) "} UUUU, ou, toute reduction faite, et en supposant 6 U = aa? + by s + cz 3 + 3iy 2 2 + 3jz*x + 3kx 2 y + S^yz 2 + fyzx 2 + Sk^y 2 + Qlxyz a : 2 (Gbc&i 4t 3 c 4t 1 3 6 + 3z\ 2 6 2 c 2 ). Liquation completement developpee, que j ai donne e pour cette polaire dans le Cambridge and Dublin Mathematical Journal, t. I. [1846], p. 97 [35], et que j ai obtenue par une elimination directe, pourra ainsi etre verifie e. Nous allons passer maintenant a la theorie des points d inflexion et des tangentes doubles de la courbe U = 0. Ces singularity s peuvent etre traitdes par une analyse semblable, en remarquant que parmi les n (n 1) tangentes, menees a la courbe d un point P situd sur la courbe, la tangente en ce point se presente generalement deux fois, et trois fois, selon que le point P est un point d inflexion, ou un point de contact d une tangente double. De signons comme ci-dessus par ( U) ce que devient U en ecrivant Ix + m, ly + mrj, Iz + m a la place de x, y et z. En mettant %d x + tjd y + %d z d, on a evidemment [U] = l n U + l n ~> m dU+ l n ~*m"- X Z et en eliminant I, m entre = 0, ~ - = 0, on obtient un resultant = 0, ou @ est dl dm une fonction de U, dU, d 2 U, ... d n U, et qui a, comme le remarque M. Joachimsthal, la propriete dont il s agit. En ecrivant U = 0, contient le facteur (3f7) 2 , et en mettant de cotd ce facteur et dcrivant dU=Q, @ contient le facteur (3 2 f/) 2 ; et ainsi de suite. Nous avons suppose que le point P, auquel appartiennent les coordonne es x, y et z, est un point de la courbe ; de maniere que Ton a actuellement U 0. En faisant done cette supposition et en e liminant le facteur (dU) 2 , liquation =0 prend la forme XdU+Y(d 2 U) 2 =Q (puisqu en mettant dU=0, liquation contiendra a gauche le facteur (d 2 U) 2 }. Dans le cas ou P est un point d inflexion, ou un point de contact d une tangente double, cette equation contiendra dU comme facteur: done il faut que F(3 2 t/") 2 contienne ce facteur, c est-a-dire : ou d 2 U, ou Y, contiendra le facteur dU. Dans le premier cas il s agit d un point d inflexion, dans le second cas d un point de contact d une tangente double. Considerons d abord les points d inflexion. Comme d"U contient dU comme facteur, il faut que cette fonction devienne zero pour toutes les valeurs de , 77, qui font 53] RECHERCHES SUE I/ELIMINATION, ET SUE LA THEORIE DES COURBES. 343 eVanouir dU. De signons par L, M, N les coefficients differentiels du premier ordre de U, de maniere que dU= LI; + Mij +N Cette quantity s eVanouit identiquement en faisant f : = ftN - yM, < t j = yL-otN, S=aM-ftL: done il faut que d*U sevanouisse par la substitution de ces valeurs, quelles que soierit les quantites a, ft, y. En designant par D ce que devient le symbole D par cette substitution, c est-a-dire en faisant D = a (Md x - Ndy) + 13 (Nd x - Ld z ) + y (Ld y - Mb x \ la condition d un point d inflexion se re duit tout simplement a equation dans laquelle les symboles d x , d y , d z ne doivent pas contenir L, M, N. Nous reviendrons sur cette Equation dans une note; pour le moment il suffit de remarquer, qu en vertu de relations qui existent entre L, M, N et les de rive es a, b, c, f, g, h du second ordre, on a identiquement (n - I) 2 D 2 U= n (n-l)V.U-(ax + fty + yz) 2 (V U), ou ^ est une fonction de a, ft, y et des de rive es a, b, c, /, g, h, dont la forme sera donnee dans la suite, et VU=abc- of 2 - bg* - ch* + 2fgh. Done a cause de 7=0, la seule condition pour determiner les points d inflexion est 1 equation V U = 0, qui est de 1 ordre 3 (n - 2) par rapport aux variables, et de 1 ordre 3 par rapport aux coefficients de U. Cela est deja connu par les recherches de M. Hesse. J ai donnd ici cette e quation pour faire voir la liaison qui existe entre cette question et celle de trouver les tangentes doubles, a laquelle je vais passer maintenant. On obtient 1 e quation qui determine les points de contact de ces tangentes, en faisant les memes substitutions = /3N-yM, &c. dans la fonction F, et en e galant a zero les coefficients des differentes puissances ou produits de a, ft, y. Remarquons que cette fonction F s obtient par une fonction de 1 ordre w 2 -w par rapport a x, y, z, u a 77, et de 1 ordre 2(w-l) par rapport aux coefficients, en eliminant les facteurs (3CT) 2 et (d 2 Z7) 2 , qui ensemble montent au degrd 4w - 6 par rapport a x, y, z, a 6 par rapport a f, 77, et a 4 par rapport aux coefficients. Done F est des degres w 2 -n-6, w 2 -5w-6 et 2n - 6 par rapport a x, y, z, a %, r 1 , et aux coefficients de U, respectivement. En substituant done 77, cette fonction devient de 1 ordre n 2 -n-6 par rapport a a, ft, y, de 1 ordre n 3 - 2n 2 - Wn + 12 par rapport a x, y, z {savoir (n*-5n + 6) + (n-l)(n 2 -n-6)}, et de 1 ordre n* + n-l2 par rapport aux coefficients {savoir (2n-Q) + (n*-n-6)}. Mais on sait que les conditions de levanouissement de F doivent se require a une seule e quation; et cela ne pent arriver que de la meme maniere dont la reduction analogue a eu lieu pour les points d inflexion. En ecrivant 344 RECHERCHES SUR L/ELIMINATION, ET SUR LA. THEORIE DES COURBES. [53 done [FJ a la place de ce que devient F apres la substitution, il faut que Ton ait identiquement : N etant fonction de a, /?, 7 du degre n 2 n 6. II parait de plus probable que cette fonction aura la meme forme que celle pour les points d inflexion, savoir N = {ax + {3y + yz) n ~~ n ~ 6 . (II 7, comme ci-dessus V/7, est cense exprimer non pas un produit, mais une derive e de V: de meme plus bas PU et QU.) Cela etant, HU sera du degrd (n 2)(w 2 9) par rapport a x, y, z {c est-a-dire n 3 2n 2 10% + 12 _( W 2__ w _6)j ) e t du degrd n? + n 12 par rapport aux coefficients. On a done le theoreme suivant : THE OREME. On trouve les points de contact de tangentes doubles, en combinant avec liquation de la courbe une equation HU=0 de 1 ordre (n 2)(w 2 9) par rapport aux variables, et de 1 ordre n 2 + n 12 par rapport aux coefficients; c est-a-dire, puisqu il correspond deux points de contact a une tangente double, le nombre de ces tangentes est dgal a \n (n 2) (n 2 9) : theoreme demontre indirectement par M. Pliicker. Cherchons maintenant les equations du systeme des tangentes aux points d inflexion et du systeme des tangentes doubles. Pour trouver la premiere equation, il faut eliminer x, y, z entre les trois equations dU dU dU U = 0, VU=0, x-y-+y^- + z =0. ax ay dz Le resultant complet sera du degre 3ra (n - 2) par rapport a x, y et z, et de 1 ordre 9w 2 18n + 6 {savoir 3(n 2)(ra JL)+ 3n(n l) + 3(n 2)} par rapport aux coefficients; mais il existe ici le facteur spe cial (KU) Z , et ce facteur etant elimine, on obtient uri rdsultant reduit Q U = 0, du degre 3n (n 2) par rapport aux variables, et du meme degre 3n (n 2) par rapport aux coefficients. De meme on aura 1 equation du systeme des tangentes doubles, en eliminant x, y, z, entre les trois equations dU dU dU U=0, UU=Q, et x-T-+y- r +z :7 - = 0. (Lx dy dz Le resultant complet est ici du degre n (n 2) (w 2 9) par rapport a x, y, z, et du degre" 3?i 4 - 5n 3 - 29n 2 + oln - 18 {savoir (n - 1) (n - 2) (n 2 - 9) + n (n - 2) (?i 2 - 9) + n (n - 1) (?i 2 + n 12)} par rapport aux coefficients. Mais il existe de meme dans ce cas un facteur special (KUY"- 11 -* {du degre" 3 (n- 1) 2 J> 2 -n- 6) = 3w 4 - 9n 8 - 9n 8 + 33- 18}, et en 1 e liminant, le resultant reduit sera du degre 4<n (n 2) (n 3) par rapport aux coefficients. Mais le terme de cette Equation a gauche sera evidemment un carre ; on aura done pour 1 equation du systeme des tangentes doubles, la relation PU=Q, oil PU est une fonction du degre |?i (n 2) (n 2 9) par rapport aux variables, et du degre 2n (n 2) (n 3) par rapport aux coefficients. Done on pourra former le tableau suivant des degres des differentes Equations obtenues : 53] RECHERCHES SUR I/ELIMINATION, ET SUR LA THEORIE DES COURBES. 345 Degre"s par rapport Equation de la courbe U=0, aux variables. n n (n 1) 3n (n - 2) (n - 2) (n 2 - 9) 3w (n - 2) Aw (n - 2} (n 2 - 9^ s aux coefficients. 1 3 (n - I) 2 3 (n + 4) (n - 3) 3w (n - 2) 2n (n. VMn X\ Condition d un point multiple KU=0, Polaire re ciproque Ft r =0, . Courbe des inflexions V U = 0, Courbes des contacts des tangentes doubles II U = 0, Systemes des tangentes aux points d inflexion QU=0, Systeme des tangentes doubles P U = 0, . La polaire de la polaire re ciproque FFU sera evidemment du degre (n 2 -n) (n 2 - n - 1) par rapport aux variables, et du degre 4(w- 1) (n*-n - 1) par rapport aux coefficients. Cette polaire de la polaire contiendra, comme on le sait par la theorie geome trique deVeloppee par M. Plucker, les facteurs U, (PU) 2 et (QU) 3 ; il faut y ajouter encore le facteur constant KU, et Ton aura ddfinitivement 1 equation . U, dans laquelle il sera facile a verifier que les deux cote s sont des memes degres par rapport aux variables et aux constantes. En effet on a 4 (n - 1) (n 2 -n-I) = 3(n-I) 2 + 4n (n - 2) (n - 3) + On (n - 2) + 1, et n(n-l)(n*-n-l) = n(n-2)(ri 2 - 9) + 9w (n - 2) + n. Mon but a ete ici de donner une idee precise des theoremes a d^montrer, pour former une theorie toute analytique des polaires reciproques; je n ai fait qu avancer ces the oremes (sans chercher a les d^montrer), pour faire voir leur liaison avec la the orie de Felimination et avec celle des hyperdeterminants ; c est a cette derniere en par- ticulier qu il faut, je crois, recourir pour ddmontrer la formule donne e ci-dessus [F] = A. U + (ax + /3y + vz)^-"- 6 (UU), et pour trouver definitivement la forme de la de rivee lit/", au moyen de laquelle on determinera les points de contact des tangentes doubles. Je serais bien aise que ces recherches puissent de quelque maniere faciliter la solution du probleme des reciproques des surfaces: objet, qui est reste* encore dans une complete obscurite. Note sur les points din flexion. Je vais d abord rassembler plusieurs formules qui se rapportent au systeme des coefficients dans le developpement de D 2 U. On a D 2 U = aa 2 + b/3 2 + c 7 2 + 2fy3 7 + 2g 7 a + 2ha/3, ou a = M 2 c - 2MN/+ N 2 b , c =L-b - 2LMh + M 2 a, C. 44 346 RECHERCHES SUR I/ELIMINATION, ET SUR LA THEORIE DBS COURBES. [53 f = - MNa + NLh + LMg - L 2 f, g = + MNh-NLb+LMf-M 2 g, h = + MNg + NLf - LMc - N 2 h. Nous ecrirons de plus, pour abr^ger, bc-f 2 = , gh-af = $, ca-g 2 =W, hf-bg=&, ab-h 2 = <&, fg -ch=ffi, bc-f 2 =(gt), &c. &c. et enfin < = a# + 33 Jf 2 On obtient identiquement La, + M h + Ng = , Lh + M\y + Nf = 0, Lg + Mf + Nc = Q; /f -MN$=(Nh-Lf)(Lf -Mg), gg - NL& = (Lf - %) (% - m) , Ah - LMffi = (Mg - Nh) (Nh - Lf} ; 4>L*MNff + b (L* b + M*a) + c (N 2 a + i 2 c) - be - 2Z 2 aa = (L*b- M*a) (N*a + D c) c (if 2 c + N%} + a (i 2 b + M 2 a) - ca - 2M 2 6b = (M*c - N 2 6) (# 6 - ^ 2 ) a (^ 2 a + D c) + b (i/ 2 c + N 2 b) - ab - 2^ 2 cc = (N 2 a - D c) (M 2 c - N* b) - a 2 + 2a (l/ 2 c + J^ 2 6) = - (Jlf 2 c - N 2 b) 2 , - b 2 + 2b (^ 2 a + L 2 c) = - (N 2 a - L 2 c) 2 , - c 2 + 2c (Z 2 6 + M 2 a) = -(L 2 b- M 2 a) 2 ; L 2 MN& + Ng (+ Lf+Mg- Nh) + Mh (Lf - Mg + Nh) -gh = -MN (Nh - Mgf , + Lh (- Lf+ Mg + Nh) + Nf (Lf+ Mg - Nh) - hf = - NL (Lf - Nh) 2 , + M (+ Lf-Mg + Nh) + Lg (Lf+ Mg + Nh)-fg =- LM (Mg - Lf) 2 ; 2VZ 2 = - a 2 a + (ab - 2A 2 ) b + (ca - 2g 2 ) c + 2 (of- 2gh) i-1ag g - 2ah h , 2Vl/ 2 = + (ab-2h 2 ) a -6 2 b+ (be - 2/ 2 ) c- 26/f + 2 (bg - 2hf) g-<2bhh, = + ( Ca - 2^ 2 ) a + (be - 2/ 2 ) b - c 2 c - 2c/f - 2c^rg + 2 (ch - 2/gr) h ; 53] RECHERCHES SUR L ELIMINATTON, ET SUR LA THEORIE DES COURBES. 347 2V MN = + (af- 2gh) a - 6/b - c/c - 26cf - 2c% - 2fyh, = - a^ra + (&# - 2A/) b - c#c - 2cAf - 2cag - 2a/h, =~aha- bhb + (ch - 2fg) c - 2bgf - 2a/g - 2ab h. Uans ees equations, si Lx + My + Nz etait facteur de ax 2 + by 2 + cz 2 + 2fyz + 2gzx + 2hxy, on aurait evidemment a = 0, b = 0, c = 0, f=0, g = 0, h = 0, et de la V = 0, < = un carre, ce qui donnerait des formules plus simples, et auxquelles on pourrait encore donner plusieurs autres formes. Par exemple on tire d un de ces systemes, 1 -~ MN$ = (Mg Nh) -, &c., ou S = (Mg-Nh)(Nh-Lf)(Lf-Mg\ et de la les expressions L M N_ " " _ JF " ft L 3 / 2 M s g 2 N 3 h 3 ~j + l + -m auxquelles on pourrait aj outer encore plusieurs systemes analogues, ce qui se ferait sans la moindre difficulte. Jusqu ici L, M, N, a, b, c, f, g, h ont e te des quantity s quelconques. En supposant qu elles soient les derivees du premier et du second ordre d une fonction U, on a (n 1) L =ax + hy + gz, (n - 1) M = hx + by +fz, (n l)N = gx +fy + cz, n (n-l)U=ax 2 +by 2 + cz 2 + 2fyz + 2gxz + 2hxy, et la substitution de ces valeurs donne la formule du texte: (n - I) 2 &U= n(n-l)V.U-(aa; + @y + yz) 2 (V U), (V U} dtant ici = V, et M* dtant donnee par 1 equation Cela pent etre verifie facilement par la substitution actuelle : mais nous allons le demontrer par la theorie des hyperdeterminants. En effet, soit (123) le determinant symbolique formd avec d Xl , d Vi> d Zl ; 8^, d Vt , d Zi ; d Xt , d ys , d Zl , (^23) le determinant sym- bolique forme avec a, 0, 7; 9^, d ys , 9^; d X3 , d yi , d, 3 , et ainsi de suite; puis soient 2\, T,, ...les fonctions xd Xi + yd Vl + zd Z[ , xd Xl + ydy 3 + zd Zt , &c. et p = ouc + @y + yz, on aura identi- quement : p (123) = - T, (423) - T z (431) - T a (AI2), 442 348 RECHERCHES SUR L ELIMINATION, ET SUR LA THEORIE DES COURBES. [53 et en elevant au carre les deux membres de cette equation, ajoutant le facteur U-JJ^Uz, et reduisant les variables apres avoir differentie suivant so, y, z, puis en tenant oorapte des equations (123) 2 U&U T* A (23) 2 U,U 2 U 3 = 2n (n-l)U. V, T,T,A (23) A (31) U, U 2 U 3 = - (n - I) 2 D*U, on obtiendra le resultat dont il s agit. II sera aussi facile de de duire de cette formule quelques propriete s de la courbe V [/"=(), dans les cas d un point double ou d un point de rebroussement de la courbe 7=0. En effet, pour un point double, les derive es L, M, N de U, et par suite D*U, et ses de rivees du premier ordre, s evanouissent. De plus, pour un point de rebrousse ment on a M* = 0. Done, pour un point double on a V U 0, ou la courbe exprimee par cette Equation passe par chaque point double (y compris les points de rebrousse ment) de la courbe t/"=0. Prenons la derive e de liquation en question, en affectant du symbole 3 = gd x + rjd y + d z ses deux membres. Supprimant les termes qui se re dui- sent a zeVo aux points doubles, cela donne c est-a-dire qu il y aura aussi a chacun de ces points un point double de la courbe V U = 0. Passant aux derivees du second ordre, on aura Or ici et 9 2 a, &c. se reduisent a 2 (cdM* - 2fdMdN+ bdN*), &c. savoir (en mettant a^ + hrj+g^, hg + brj + ft;, g%+fy + c% a la place de dL, dM, dN, et en ayant ^gard a la condition Vf7=0) a 2d 2 U.&, &c. ; on aura done d*Du = 2? 2 U . W, ou enfin (n - 1) (n - 2) &U . V = - (OLX + $y + yz) 2 3 2 V T, c est-a-dire: les deux courbes U=0, V[7=0, se toucheront dans les points doubles. Enfin pour un point de rebroussement d*VU s eVanouit, c est-a-dire, il existe un point triple dans la courbe V [/"=(). Mais il peut etre demontre que deux branches de la courbe se touchent en ce point, et qu elles touchent aussi la courbe U = ; c est-a-dire qu il y aura dans la courbe VU=0 un point de rebroussement, et une autre branche de la courbe qui passe par ce point. Pour cela il faudra passer aux derivees du troisieme ordre. Cela donne, en supprimant les termes qui s evanouissent : (n - I) 2 c s &U = 3w (n - 1) 8^ . d n ~U- (cue + $y 53] RECHERCHES SUR I/ELIMINATTON, ET SUR LA THEORIE DES COURBES. 349 Ici on a 9 3 D 2 Z7=a 2 8 3 a + &c., 9 3 a - 6 (dcdM 2 - ZdfdMdN + dbdN*) + 6 {dM&M -f(dM&N + dN&M ) + bdN&N], &c., et les deux lignes de cette expression se reduiserit a Qd 2 U.dM, et a zero, respective- ment. En effet, en rempla^ant dM, dN par leurs valeurs, le coefficient de 2 dans la premiere ligne devient 6 (h*dc-2ghdf+g 2 db) =6a(bdc + cdb -2fdf) (et a cause de 33 = 0, = 0, jp = 0) = 6a3^l ; et egalement pour les autres termes. T>f meme, le coefficient de | 2 dans la seconde ligne devient 6 {chdh f(gdh + hdg) + bgdg}. En y ajoutant 3(h-dc-2ghdf+g 2 db ), savoir 3a3.^, la somme se re duit a 39 (ch? - 2fgh + bg 2 ) = 33 (a& - V) = 3a3^t ; done le coefficient en question s evanouit. En cherchant de la meme maniere les autres coefficients, on trouvera les valeurs dont il s agit, et ainsi 9 3 a = 6? 2 6 r . 9^, &c. et de la &D 2 U= 68 2 tW done enfin 3 (n - 1 ) (n - 2) 9 2 UV9 = - (ax + j3y + ^zf d & V U ; ce qui suffit pour demontrer le theoreme, qui peut e tre e nonce comme suit : "II existe un point double de la courbe VU=0, pour chaque point double de la courbe U = 0, et les deux courbes ont des tangentes communes dans ces points. De plus, il existe un point triple de la courbe VU=0, pour chaque point de rebroussement de la courbe U=0, savoir tin point de rebroussement dont les deux branches touchent la tangente de la courbe U=0, et encore une troisieme branche, qui passe par le point de rebroussement." II suit de la que dans le cas d un point double, ce point doit etre considere comme la reunion de six points d intersection, et dans celui d un point de rebrousse ment, de huit points d intersection. C est de cette maniere que Ton se rend compte du theoreme de M. Pliicker qui dit que 1 effet de ces deux singularites est de faire disparaitre respectivement six ou huit points d inflexion de la courbe donnee. Examinons, en finissant, la thdorie des points d osculation. II est facile de voir que la condition d un tel point (savoir d un point dans lequel la tangente coupe la courbe en quatre points consecutifs) consiste en ce que d 3 U contient dU comme facteur, ou, autrement dit, que I e quation D 3 U=0 est identiquement vraie. On obtient ainsi dix conditions, qui se reduisent assez facilement a six; mais pour les re duire a une seule condition, il faut prendre la derivee, avec le symbole D de la valeur donne e ci-dessus de D-U. On doit cependant ne pas oublier que D.D-U, outre le terme D 3 U, contient aussi des termes que Ton obtient en faisant ope rer les symboles d x , d y , d z sur les quantitds L, M, N qui entrent dans D 2 U. Car il est convenu que les symboles 9*> d y , d z dans D 3 U, ne doivent pas affecter les lettres L, M, N. Cependant il est remarquable que dans le cas actuel, ces termes se detruisent entierement. En effet, en les de signant par fl, on obtient fl = W [(My - N/3) (ad L + kd x + gd N ) + (N* - L/3) (hd L + bd +/8 A .) + (L0 - My) (gd L +fd x + ca. v )] . DU, 350 RECHERCHES SUR L ELIMINATION, ET SUR LA THEORIE DBS COURBES. [53 a 2 -u, ou il faut d abord effectuer les operations d L , 9^ 9^ qui se rapportent a D, et ensuite rap porter les d x , d y , d z a la function U. Cela donne O = 2 [a (Ndy -Md z )+/3 (Ld z - Nd y ) + 7 (Md x - Ld y )] x (fdy ~ bd z ) ~ M(cdy ~fd z )] [L (gd z -cd x )-N(ad z -gd x )] [M (hd x ady) - L (bd x hd y )] [- L (hd z + gdy - 2/9^) + M (ad z - gd x ) + N (dd y - hd x )] + 7 a [- H (fd x + hd z - 2gd y ) + N (bd x -hd y ) + L (bd z - fd y )] + *0[-N (gd y +fd x -<2hd z ) + L (cd y -fd z ) + M(cd x - gd z )] ou d x , d y , d z se rapportent seulement a U. Or tous les termes de cette expression s evanouissent. Par exemple le coefficient de a 3 devient (Ndy - Md z ) [N (fd y - bd z ~) - M (cd y -fd z }] U = [-N 2 (fdy 2 - bdy) + M* (cdyd z -fd/) - MN (cd y 2 -fd^ z +fdyd z - 69/) U = 0, et ainsi pour les autres termes ; done on a H = 0. Done, en transportant a 1 autre cote de 1 equation les termes qui contiennent U, DU ou VU, on obtient cette formule tres simple : (n - I) 2 &U = - (OLX + &y + 7 z ou la seule condition d un point d osculation (en ayant dgarcl a 1 equation V?7=0) se reduit a BVU = Q. Savoir les derivees d x VU, d y W, d z VU doivent etre proportionnelles a d x U, dyll, d z U (ce qui equivaut a une seule condition, en vertu de U=0, VU=Q: comme on le voit facilement). Cela donne le theoreme suivant. THEOEEME. "Dans le cas d un point d osculation, les deux courbes /" = (), VU=0 se touchent." II n y a presque pas de doute que la derive e WU ne se reduise toujours a la forme R . U + S . V If. En effet, M. Hesse 1 a de montre pour les fonctions de trois variables et du troisieme degre, et moi, je 1 ai ve rifie pour les fonctions de deux variables d un degre quelconque. Cela e tant, les points d inflexion de la courbe VU=Q sont situes aux points d intersection avec 7=0, et au cas ou les deux courbes se touchent, ce point de contact doit etre conside re comme la reunion de trois points d intersection : done, "Tout point d osculation peut etre envisage comme point de reunion de trois points d inflexion." Nous demontrerons encore, d une maniere conforme a celle dont nous avons trouve 1 expression de L*U, 1 expression qui vient d etre donne e pour IPU a, moyen de la formule (123) 2 = {T, T 53] RECHERCHES SUR I/ELIMINATION, ET SUR LA THEORIE DES COURBES. 351 En multipliant les deux membres par (A14<) + (J.24) + (J.34), le terme a gauche peut etre presente sous la forme p 2 (A6ty (123) 2 , ou d Xf) , d y0> d ze se rapportent a tous les systemes de variables. En y appliquant le produit U-JJJJ^U^ (les variables identiques apres les differentiations), on obtiendra Qp 2 (A04t)U 4t VU = 6 DVU. p 2 . Pour la droite de 1 equation on a d abord trois termes comme T-c (A14>) (J.23) 2 UJJJJJiJ^ lesquels se detruisent evidemment, a cause de (J.14) U-JJt = ; puis six termes de la forme (Alty (J.23) (-431) UiU z U s Ut, qui se detruisent aussi, puisqu en dchangeant 1,3 et 2,4, le terme change de signe ; puis trois termes comme Tf {(/124) + (^34)} (^423) 2 U^UzU^U^, &c. savoir T* U L {(A24) + (A34,)} (A23)* U,U 3 U 4 = - 2n (n - 1) U. DV ; c est-a-dire tous ces termes sont 6w (n- 1) U D^ ; enfin trois termes 27 7 1 2 T 2 (^L23) (^31) (^134) UJJJJJJi = 2(n-l)* IPU, ou, tous pris ensemble, 6 (n - 1)*D 3 U. Done, en supprimant le terme (5n( n I)UD y, a cause de U=0, on obtient la me me Equation que ci-dessus, savoir : On pourrait croire qu il y a une equation analogue (n pour les derivees du quatrieme degre, mais cela n est pas. En effet, il est facile de voir que pour un point d osculation, D 2 VZ7 se rdduit a WU, a un facteur pres, c est-a-dire Ton aurait D-VU=0, puisque VV?7 s eVanouit aux points d osculation: done on aurait generalement pour un point d osculation D*U=Q; mais cela est seulement le caractere des points d osculation d un plus haut degre, savoir de ceux ou la tangente rencontre la courbe en cinq points conse cutifs. Pour le quatrieme degre le probleme devient trop complique pour etre traite de cette maniere. 352 [54 54. NOTE SUE LES HYPEKDETERMINANTS. [From the Journal fur die reine und angewandte Mathematik (Crelle), tome xxxiv. (1847), pp. 148152.] I. SOIT V une fonction homogene de x, y de 2p i6me ordre. En egalant a zero les coefficients differentiels du p i&me ordre de cette fonction, pris par rapport a x, y, et en eliminant ces variables, on obtiendra entre les coefficients de la fonction un nombre p d equations. Or parmi ces Equations il y aura toujours une seule du second ordre, savoir (U, U}=0 (suivant la notation dans mon memoire sur les hyperde terminants, t. xxx. [(1846), 16]. Par exemple, en e"crivant t au lieu de x : y on a identiquement c (at + b) -b(bt + c) = (ac - 6 2 ) t, ( et - (4idt + 2e) (bt 2 + 2ct+d) + (3ct + 2d)(c 2 + 2dt + e) = (ae - 4bd + 3c 2 ) V, ( gt" ) (at 3 + 3b? + Set +d) - ( 6/4 2 + 3# ) (bt 3 + 3ci 2 + 3dt + e ) + (loot* + 18ft + Qg ) (ct 3 + 3dt* + Set +/) loet + 6/) (dt 3 + 3e? +3ft+g) = (ag- 66/+ loec - NOTE SUR LES HYPERDETE11MINANTS. 353 et ainsi^de suite: il ne reste qu a determiner la loi des coefficients numeriques des teurs a gauche. Pour cela, reprewmtons par A, B, C, ... les coefficients de (1+,)* et par A, B, C , ... les coefficients de (!-)-*. On aura pour ces nombres le systeme suivant : J + A A, - A B, - B A, + A C, + B B, +C A, Les nombres IK, L ... de la derniere ligne seront a determiner au moyen d une autre regie : il faut faire evanouir les sommes des nombres dans la mSme ligne verticals ces .ombres etant pris avec leurs signes actuels. Je suis parvenu par induction a ces es, mais il ne serait pas, je crois, tres difficile de les demontrer directement. II me parait possible que tous les hyperdeterminants qui se rapportent a la fonction V, pmssent etre trouves en e liminant entre les equations (en nombre de p) dont il s agit b cea ckns le cas ou U est de 1 ordre 2p ou 2p + l; au moins cette regie se vjfie pour les fonc hons de deuxieme, troisieme et quatrieme ordres, et cela paraissait (a priori) moms probable pour les derivees dun degre plus eleve que pour celles du second degre pour lesquelles, comme on vient de le voir, il est effectivement vrai. Cela etant il y aura seulement un nombre p de de rive es inde pendantes pour les fonctions du 2/> Ifime et du (2p + iy & ordre: conclusion que je ne d . Soit V = 6a&oZ + 3& 2 c 2 -a 2 d 2 -4ac 3 -46H et repre sentons par V x le determinant forme avec les coefficients differentiels du second ordre de V p rapport a a, Tc d < )ll B.T1 1 cl (propriete qui a un rapport singulier avec celle qu a demontre e M. Eisenstein par rapport aux coefficients du premier ordre de la meme fonction V). La demonstration que je puis donner de , theoreme est a la vdrite assez conroliouA, mais je ne vois pas d aut, on obtierit a ~ > ab , ac 3r, ad + 9q ba , b" + 2r, be - q , bd-3p ca - 3r, cb -q , c 2 + 2/>, c d da+9q, db-3p, dc , d 2 >u le determinant ne contient que les termes di V, . 81 (9 (pr - g-). - 2a y - 2<fr - 1 2y (abf + cd*) - 18? (Kp + <* r ) c - C (pr + g) (acp + bdr) - 2adq (Spr - f) - ISbcq (pr + 45 354 NOTE SUE LES HYPERDETERMINANTS. d ou, en reduisant au moyen des expressions ap = - (2bq + cr), dr = - (2cq + bp), ad = be - 3g, on tire V x = 81 {9 (pr - q*f + 4 (b 2 p + 2bcq + c 2 r) (pr + q-) + Qtf (3/>r - f)} , on enfin, au moyen de 2 (b 2 p + 2bcq + c-r) = 3j)r : V, = 243 (pr - q-)- = 3V 2 ; voila liquation qu il s agissait a demontrer. III. En considerant a : d, b : d, c : d comme repre*sentants les trois coordonnees d uu point, ou, si Ton veut, des fonctions lineaires de ces coordonnees, 1 equation V = appartient evidemment a une surface developpable (de quatrieme ordre). Mais la condi tion pour que 1 equation U=0 (V etant une fonction homogene de quatre variables) appartienne a une surface developpable, est, que le determinant forme avec les coefficients differentials du second ordre de la fonction, s evanouisse (theoreme de M. Hesse, t. xxvm. [(1844) pp. 97 107, "Ueber die Wendepunkte der Curven dritter Ordnung"]). Done il faut que Vj s eVanouisse au moyen de V = 0, c est-a-dire, il faut que V l contienne le facteur V ; ce qui s accorde parfaitement avec 1 e quation qui vient d etre presented. Mais il ne pent etre prouve de cette maniere que 1 autre facteur doit aussi se reduire a V, et meme cela n est pas vrai si V x vient d une fonction d un plus haut degre que le quatrieme. II suit de cela qu en supposant toujours que les coefficients soient des fonctions lineaires des coordonnees, le rtfeultat = de 1 elimination de x, y entre ^ = 0, ^- = appartient toujours a une surface developpable. De meme I ^limination de x, y entre les trois equations - = - 7 r- = 0, -T-; = conduit aux Equations de 1 arete de rebrousse- dx 2 dxdy dif d*U d s U ment de la surface, et de plus, en eliminant entre les equations " = d-W _Q rf " = o, on obtient les points de rebroussement de 1 arete de rebroussement. dx dy- dy 3 Cela conduit h quelques resultats remarquables. Par example, la surface developpable dont 1 equation est V - Qabcd + 36 2 c 2 - 4ac 3 - 4b*d - aW, a pour arete de rebroussement la courbe dont les equations (equivalentes a deux equations seulement) sont bd-c" = 0, ad-bc = 0, ac - b- = 0, ce qui est une courbe du troisieme ordre seulement. Car en considerant deux quelconques de ces trois equations, par exemple celles-ci : bd - c" = 0, ad-bc = 0, ces equations appartiennent a deux surfaces du second ordre qui ont en commun la droite d = 0, c = 0: cela s accorde avec un re"sultat que j ai doune dans mon memoire sur les surfaces developpables dans le journal de M. Liouville [t. x. (1845), 30]. Egalement, en considerant uiie Equation de quatrieme degre en t, on obtient une surface developpable. (ae - 4>bd + 3c 2 ) 3 - 27 (ace + 2bcd - ad- - b 2 e - c 3 ) 2 = 0, qui a pour arete de rebroussement la courbe du sixieme ordre exprimee par les equations ae - l)d _)- 3 C 2 = 0, ace + 2bcd - ad- b 2 e - c 3 = 0. Je n ai pas completement rdussi a expliquer 54] NOTE SUE, LES HYPERDETERMINANTS. 355 pourquoi cette courbe du sixieme ordre a une osculatrice developpable, seulement du sixieme ordre, mais cette reduction s opere en partie au moyen des points de rebrousse- ment de la courbe, qui se trouvent au moyen des Equations at + b = 0, bt + c = 0, ct + d = 0, dt + e = 0. Pour savoir a combien de points ces equations correspondent, il faut re- marquer que, a, b, c, d, e etant des fonctions line aires des coordonnees, on aura toujours entre ces quantites une equation line aire telle que ou A, B, ... sont des constantes. Done, en eliminant a, b, c, d, e, on obtient A Bt + Ctf Dt 3 + Etf = 0, Equation du quatrieme ordre, et a chaque valeur de t il correspond un des points dont il s agit ; done la courbe du sixieme ordre a quatre points de rebroussement. Egalement la surface developpable qui correspond a une equation du ??i icme ordre, est de 1 ordre *2(m 1); 1 arSte de rebroussement est de 1 ordre 3 (m 2), et il y a dans cette courbe un nombre 4 (m 3) de points de rebroussement. II faut toujours se rappeler que ces surfaces developpables ne sont pas les surfaces deVeloppables les plus generales qui existent de 1 ordre 2 (m 1), excepte dans le cas des surfaces developpables du quatrieme ordre. IV. II vaut peut-etre la peine de donrier en passant une demonstration de ce thdoreme de M. Chasles : " Le plan qui passe par trois points qui se meuvent avec des vitesses uniformes dans trois droites quelconques, enveloppe une surface developpable du quatrieme degre." En effet, en supposant que a : 8, /3 : 8, 7:8; of : 8 , ft : 8 , 7 : 8 : a" : 8", @" : 8", 7" : 8", soient des fonctions line aires du temps ((8, 8 , 8") peuvent etre constants, ou, si Ton veut, des fonctions lineaires du temps, ce qui correspond a un cas un peu plus general que celui de M. Chasles), on peut prendre ces valeurs pour coordonnees des trois points mobiles. Done, en prenant x : w, y : w, z : w pour coordonnees d un point quelconque du plan, on obtient 1 e quation de ce plan, en egalant a zero le determinant forme avec les valeurs x, y, z, w ; a, /3, 7, 8 ; a. , ft , 7 , 8 ; a", (3", y", 8" ; ce qui donne une equation de la forme a + Sbt + 3c 2 + dt? = 0, a, b, c, d etant des fonc tions lineaires de x, y, z, w; et cela suffit pour demontrer le theoreme dont il s agit. V. En finissaiit j indiquerai un principe de classification des courbes a double courbure qui me parait etre de quelque importance ; savoir, on pourra distinguer les courbes qui ne peuvent pas etre 1 intersection complete de deux surfaces, de celles qui peuvent 1 etre. Par exemple, en faisant passer par une courbe donnee du troisieme ordre deux surfaces du second ordre, la courbe n est pas 1 intersection complete des deux surfaces ; celles-ci se coupent dans cette courbe et dans une certaine droite. Quel est le theoreme analogue pour les courbes du n k me ordre ? Peut-on, par exemple, toujours combiner avec une courbe donnee d un ordre quelconque, une autre courbe qui est 1 intersection complete de deux surfaces, de maniere que 1 ensemble des deux courbes soit une intersection complete de deux surfaces ? Et si non : de quelle maniere trouvera- t-on les equations generales d une courbe de ?i i0me ordre ? Quel est le degre de gene- ralite de ces equations ? II y a une foule d autres questions qu on pourrait ici proposer. J ai propose une question analogue dans le point de vue analytique, mais elle est restee sans reponse. 452 356 [55 55. SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [From the Journal fur die reine und angewandte Mathematik (Crelle), tome xxxiv. (1847), pp. 270275. Continued from t. xxxi. p. 227, 50.] III. LORSQUE j avais sous la plume la premiere partie de ce me moire, je ne savais pas que la derniere partie du theoreme de M. Steiner sur 1 hexagramme de Pascal (savoir que les vingt points d intersection des soixante droites sont situes, par quatre, dans quinze droites) avait deja e te demontree d une maniere aussi simple qu ele gante par M. Pliicker dans son memoire, "Uber ein neues Princip der Geometric" (t. v. [1828] p. 269). En supposant maintenant cette demonstration connue, je veux examiner de plus pres la correlation de ces vingt points, en adoptant une notation plus commode. Soit a/itySe^ une permutation quelconque des nombres 1, 2, 3, 4, 5, 6 : cette permu tation peut etre nommee directe ou inverse, selon qu elle est formee par un nombre pair ou impair d inversions. Des six permutations a/3ySe%, a&y%e/3, ay/3e ; aSy/3e, a/3y%e&, a/y8e/3, les trois premieres ou les trois dernieres sont directes. Nous representerons les trois permutations directes par (c^ye). Les trois droites que donne le the oreme de Pascal, applique aux hexagones correspondants a ce symbole, se coupent dans un des vingt points dont il s agit: point qui peut etre represente par la meme notation (aye). En supposant que a@y8e est une permutation directe, le point aye correspond au point ~ j de M. Pliicker. Partout dans cette section on pourra changer les mots " directe " et "inverse. 55] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 357 Les six permutations des lettres a, 7, e ne donnent qu un seul point ; de maniere que les vingt points, dont il s agit, sont 123, 124, 125, 126, 134, 135, 136, 145, 146, 156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456. Or, pour trouver comment il faudra combiner ces points, j e cris cfrye et je tire de la le systeme Les quatre points seront situes sur la meme droite, que Ton peut representer par afi.yS.e. Les quinze combinaisons, quatre a quatre, des vingt points, seront 135, 146, 236, 245 dans la droite 12.34.56 A. 136, 154, 234, 256 134, 165, 235, 264 146, 152, 342, 356 142, 165, 345, 362 145, 126, 346, 325 152, 163, 453, 462 153, 126, 456, 423 156, 132, 452, 436 163, 124, 564, 523 164, 132, 562, 534 162, 143, 563, 542 124, 135, 625, 634 125, 143, 623, 645 123, 154, 624, 653 12.35.64 12.36.45 13.45.62 13.46.25 13.42.56 14.56.23 14.52.36 14.53.62 15.62.34 15.63.42 15.64.23 16.23.45 16.24.53 16.25.34, ou les droites s obtiennent en permutant dans 12 . 34 . 56 d abord les derniers trois nume ros et puis dans ces trois permutations les derniers cinq nume ros. Par la la maniere de trouver les droites est claire. Cette dnonciation des points et des droites, dont il s agit, en meme temps qu elle est parfaitement symetrique, est la seule qui se presente naturelleigg|r[^|jeftdant la (UNIVERSITT, OF CALIFORNIA- 358 SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [55 symetrie en est si complique e et si peu manifests qu il sera bon d adopter ime autre notation. Pour cela, je forme le tableau auxiliaire suivant, dont 1 arrangement est assez clair : 351 135 ace cea 513 eac 246 bdf 246 fbd 246 dfb 146 235 acb edf 346 251 cef abd 546 213 ead cfb 236 145 acd bef 256 341 ceb fad 216 543 eaf deb 245 136 acf bde 241 356 ced fba 243 516 eab dfc I)e la, en ecrivant B. 123 = bcf , 234 = abe , 124 = cde , 235 = def, 125 = abd, 236 =.acd, 126 = aef , 245 = acf, 134 = adf, 246 = bdf, 135 = ace , 256 = bee , 136 = bde , 345 = bed , 145 = bef , 346 = cef, 146 = abc , 356 = dbf, 156 = cdf, 456 = ade; et de plus C. 12.34.56 = 12.36.45 = 13 . 45 . 62 = ab , 13 . 46 . 25 = cd , 13 . 42 . 56 = ef , 14 . 56 . 23 = Id , 14.52.36 = ae , 14. 53. 62 = cf , 15.62.34 = rfe, 15.G3.42 = 6c , 15. 64. 23 = ef , 16.23.45 = ce , 16. 24. 53 = ad, 16. 25. 34 = bf savoir (en representant les points 123, 124, &c., par bcf, cde, &c., et les droites 12.34.56, 12.35.64, &c., par ac, be, &c.) on verra dans le tableau (A) que les points situ^s dans la droite ac sont ace, acb, acd, acf, que les points dans la droite be sont bed, bef, bea, bee, et ainsi de suite, de maniere que ce systeme des vingt points et des quinze droites est pre cise inent le systeme reciproque de celui des quinze points et des vingt 55] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 359 droites que nous avons considdrd dans la premiere section de ce memoire ; ou, autre- ment dit, que les vingt points et les quinze droites sent les projections sur un plan, des points et des droites d intersection de six plans dans 1 espace. Seulement la figure plane, ainsi formee, contient quatorze quantites arbitraires, tandis que le systeme de six points sur une conique n en contient que onze, de sorte qu il doit y avoir des relations entre ces six plans. On obtient done la forme suivante plus complete du theoreme de M. Steiner (theoreme qui en meme temps est le complement du thdoreme XII. 1 de ce memoire). THOREME XIV. " Les soixante droites correspondantes aux hexagones formes par six points d une conique se coupent trois a trois dans vingt points qui peuvent etre consideres comme les projections des points d intersection d un systeme de six plans (dont d ailleurs la liaison reste encore a chercher)." Egalement THEOREME XIII. " Les soixante points correspondants aux hexagones formes par six tangentes d une conique sont situ^s trois a trois sur vingt droites de terminees par des plans qui passent par trois points quelconques d un systeme de six points dans 1 espace (la liaison de ce systeme de six points etant encore a chercher)." Soient a, /; b, g ; c, h les points correspondants d un systeme de points situes sur la meme droite, et en involution. Nommons "faisceau" les trois cotes d un quadrilatere qui se coupent dans un meme point et " triangle " les trois cotes qui ne se coupent pas dans un meme point (de sorte que dans tout quadrilatere il y aura quatre faisceaux et quatre triangles). Les quadrilateres dont les cotes passent par les six points en involu tion, peuvent etre classes en deux systemes : Dans le premier les faisceaux passeront par f, g, h ; f, b, c ; g, c, a; h, a, b, et les triangles par a, b, c; a, g, h; b, h, f; c, f, y ; dans 1 autre il en sera le contraire. Deux quadrilateres qui appartiennent a ces deux systemes respectivement, peuvent etre dits " en rapport inverse 1 un a 1 autre." Soient A BCD, A B C D deux quadrilateres non situes dans le meme plan, et soumis a la condi tion que les cCtes DA, B C - DB, C A - DC, A B EC, D A - CA, D B - AB, D C coupent la droite dans les points f, g, h, a, b, c respectivement. Les deux tetraedres A BCD, AB C D , et egalement les te*traedres AB CD et A BC D ; ABC D et A B CD ; ABCD et A B C D seront respectivement inscrits et circonscrits 1 un a 1 autre. Car en considerant par exemple ces deux-ci: A BCD, AB C D : A est dans le plan B C D , B est dans le plan C D A (parce que les droites AB, C D se rencontrent) ; C est dans le plan D AB (parce que AC et B D se rencontrent), et D est dans le plan A B C (parce que AD et B C se rencontrent). Egalement A, B , C", D 360 SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [55 sont situ^s dans les plans BCD, CDA , DA B et A BG respectivement ; et cela ve rifie la relation dont il s agit. Ce the oreme est du a M. Mobius qui 1 a obtenu par son Calcul Barycentrique (ce Journal, t. III. [1828], p. 273), et en considerant un systeme polaire dans lequel le plan reciproque d un point quelconque passe toujours par le point meme (Statik, c. VI. 86, et ce Journal t. X. [1833], p. 317). On trouve aussi quelques remarques sur ce sujet dans 1 ouvrage " Systematische Entwickelunyen u. s. w." de M. Steiner, p. 247. Je ne croyais pas inutile d en faire voir la relation avec la theorie de 1 involution. Remarquons aussi que non seulement les quadrilateres ABGD et A B C D , mais aussi ceux-ci ABC D et A B CD, ACB D et A C BD, ADB C et A D BC sont en rapport inverse entre eux. Par cela la syme trie de la figure est complete e ; mais on n en tire pas de nouveaux systemes de tetraedres inscrits et circonscrits. M. Mobius a demontre qu il n existe pas des quadrilateres reels simples, a quatre cotes et quatre angles, inscrits et circonscrits. Mais en considdrant les points imaginaires, 1 existence en est possible, et on trouve des systemes de cette sorte parmi les neuf points d inflexion d une courbe de troisieme ordre. Je renvoie cette discussion a une autre occasion, V. Je me bornerais, sans examiner de plus pres la figure qui en resulte, a demontrer le theoreme suivant : " Si un point et n droites sont donnas, les points d intersection de chaque droite avec la polaire du point, relative aux autres n 1 droites, sont situes sur une meme droite polaire du point, relative au systeme des droites." Je prends un systeme de droites, considere comme formant une courbe, et j entends par polaire ou droite polaire, la derniere des polaires successives du point, relative a la courbe. En representant analytiquement la courbe par V = 0, V est une fonction homogene d un ordre quelconque en x, y, z\ et si a : ft : 7 sont les valeurs de x : y : z relatives au point, 1 e quation (adx + fidy + yd^U^O est celle de 1 une quelconque des polaires successives. Soit p = 0, q = 0, r = 0, ... les equations des droites, p, q, r, ... seront des fonctions lineaires de x, y, z. Soient comme plus haut x : y : z = a : /3 : 7 les Equations qui determinent le point: 1 e quation de la droite polaire du point, relative aux n droites, est ... =0. Soient a, b, c, ... ce que deviennent p, q, r, ... en ecrivant a, /3, 7, ... au lieu de x, y, z, ... , on obtient aisement oid x + fidy + yd z = adp + bdg + . . . ; et de la on tire (ad p + bd q + cd r + . ..) n ~ l pqr ... - 0, pour la polaire cherche e. Les differentiations etant effectuees selon p, q, r, ... , comme variables inddpendantes, on obtient pbc ...+ aqc ... + ... =0 55] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 361 ou, ce qui est la meme chose : f + 1 + n + ...-o. a b c De meme la polaire du point, relative aux droites q = 0, r = 0, ... , est et par cet-te raison Fintersection de cette droite avec p = est eVidemment situe e sur la droite polaire du point, relative a toutes les droites ; ce qui prouve la proposition dont il s agit. Par exemple, en considerant les droites EG, CA, AB qui passent par trois points A, B, C: la polaire d un point 0, relative aux droites AB, AC, est une droite Act telle que AB, AC; AO, Aa. ferment un faisceau harmonique. Soit a le point d intersection de AOL et EG, et supposons le meme pour les points ft, 7: les points a, ft, 7 seront situes (comme on le salt) sur une mme droite, qui est celle que je nomme polaire de 0, relative aux cote s du triangle, et que M. Plucker a nomme " harmonicale." On sait de plus que les droites A a, Bft, Cy peuvent etre construites comme suit : en prenant a, b, c pour les points d intersection de OA, OB, OC avec EG, CA, AB respectivement, les droites be, EG ; ca, CA; ab, AB se rencontreront dans les points a, ft, 7; ce qui offre une regie facile pour construire la polaire d un point, relative a un n ombre quel- conque de droites. Remarquons en finissant que la conique qui passe par les points A, B, C, et qui touche deux des trois droites Aa, Bft, Cy, touche aussi la troisieme et est effectivement la polaire conique du point, relative aux trois cote s du triangle. En combinant cela avec la proprietd connue que la r ifime polaire de , relative a la meme courbe, passe par 0, si la (n r) leme polaire d un point 0, relative a une courbe du ?i it>me ordre, passe par un point , on obtiendra le the oreme 22 de M. Steiner (ce Journal, t. iv. [1828] p. 209). C. 46 362 [56 56. DEMONSTRATION OF A GEOMETRICAL THEOREM OF JACOBI S. [From the Cambridge and Dublin Mathematical Journal, vol. in. (1848), pp. 48 49.] THE theorem in question (Crelle, t. xn. [1834] p. 137) may be thus stated : "If a cone be circumscribed about a surface of the second order, the focal lines of the cone are generating lines of a surface of the second order confocal to the given surface and which passes through the vertex of the cone." Let (a, /3, 7) be the coordinates of the given point, the equation to the given surface. The equation of the circumscribed cone referred to its vertex is y 2 z-\ /a 2 /3 2 7 2 \ fax $y jz\ 2 _ + + ~ 1 ~ + " + = whence it is easily seen that the equation of the supplementary cone (i.e. the cone generated by lines through the vertex at right angles to the tangent planes of the cone in question) is (ax + {3y + 7-z) 2 a?oP fry 2 c~z 2 = 0. ( J ) Suppose we have identically (ax + /3y + jz) 2 - a-x 2 - b 2 y 2 - cV - h (x 2 + y- + z 2 ) = (Ix + my + nz) (I x + m y + n z) ; Ix + my + nz = Q will determine the direction of one of the cyclic planes of the supple mentary cone, and hence taking the centre for the origin the equations of the focal lines of the circumscribed cone are x <* _y /3 7 I m n = Q being the equation of the first cone, that of the supplementary + C^ 2 + 2 jfyz + 1(&zx + 2^xy = 0, these letters [denoting the inverse coefficients BC-F 2 , &c.]. 56] DEMONSTRATION OF A GEOMETRICAL THEOREM OF JACOBl s. 363 It only remains therefore to determine the values of I, m, n from the last equation but one. The condition which expresses that the first side of this equation divides itself into factors is easily reduced to 2 ry 2 i _ / 1 / \\ ^ 2 ~ Next, since the equation is identical, write ^ ft y x = - we deduce = Again, putting I m ~a + A the whole equation divides by IV mm nri , ~7T "*" /TZT + *a _i_ ;, "*"*) + h P + h c 2 + h a factor whose value is easily seen to be = 1. And rejecting this, we have I 2 m 2 n 2 + + ~~ Thus of the three equations (A), (B), and (C), the first determines h, and the remaining two give the ratios I : m : n. It is obvious that T 2 is the equation of the surface confocal with the given surface which passes through the point (a, ft, 7). The generating lines at this point are found by combining this equation with that of the tangent plane at the same point, viz. and since these two equations are satisfied by x = a + Ir, y = ft + mr, 2 = y + nr, if I, m, n are determined by the equations above, it follows that the focal lines of the cone are the generating lines of the surface, the theorem which was to be demonstrated. It is needless to remark that of the three confocal surfaces, the hyperboloid of one sheet has alone real generating lines; this is as it should be, since a cone has six focal lines, of which four are always imaginary. 462 364 [57 57. ON THE THEORY OF ELLIPTIC FUNCTIONS. [From the Cambridge and Dublin Mathematical Journal, vol. in. (1848), pp. 50 51.] WE have seen [45] that the equation n (n - 1) a*z + (n - 1) (ox - 2x s ) ^ + (1 - ax* + at) ^ - 2n (of - 4) ^ = is integrable, in the case of n an odd number, in the form z = B + B^ ... + j5 i(n _ 1) a; n ~ 1 ; and the coefficients at the beginning of the series have already been determined ; to find those at the end of it, the most convenient mode of writing the series will be and then the coefficients D r are determined by A +2 = (2r + 3) (n - 2r -- 3) D r+l a-*2n (a 2 - 4) d ^ +l ct& - (2r + 3) (2r + 2) (n - 2r - 2) (n - 2>- - The first coefficients then are A=l, D, = (n - 1) a, A = 2 (w - 1) (71 + 6) + (w - 1) (n - 9) a 2 , A = 6 (n - 1) (n - 9) (n + 10) a + (w - 1) (n -9)(n- 25) a , A = - 36 (n - 1) (n 3 - I3n 2 + 36rc + 420.) + 12 (r i -l)(n-9)(n-25)(w + 14)a 2 + (n - 1) (w - 9) (n - 25) (n - 49) a 4 , 57] ON THE THEORY OF ELLIPTIC FUNCTIONS. 365 A = - 1 2 (n - 1) (n - 9) (47w s - 355n s + 3188n + 31500) a + 20 (w - 1) (n - 9) (n - 25) (n - 49) (n + 18) a :! + (TO - 1) - 9) (/i - 25) (n - 49) (n - 81) a 5 , X>, ; = - 24 (n - 1) (n - 9) ( 23ro 4 + 2375w 3 - 14638w s + 116100/1 + 693000) - 12 (n - 1) (n - 9) (493n 4 - 8882w 3 + 70317w 2 - 361641w - 7276500) tf + 30 (n - 1) (n -9)(n- 25) (n - 49) (n - 81) (n + 22) a 4 + (n - 1) (ro - 9) (n - 25) (TO - 36) (n - 49) (TO - 81) (n - 121) a 6 , &c. And, in general, A- = (n - 1) (n - 9) . . . {n - (2r - I)-} of + r (r - 1) (TO - 1) (TO - 9) ... {/i - (2r - 3) 2 } (n + 4r - 2) a -- , &c. (where however the next term does not contain the factor (w 1) (n 9) ... [n ("2r- 5) 2 ]). In the case when TO = v-, then in order that the constant term may reduce itself to unity, we must assume this is evident from what has preceded. 366 [58 58. NOTES ON THE ABELIAN INTEGRALS. JACOBI S SYSTEM OF DIFFERENTIAL EQUATIONS. [From the Cambridge and Lublin Mathematical Journal, vol. III. (1848), pp. 51 54.] THE theory of elliptic functions depends, it is well known, on the differential fi IT (J ?/ equation , , < + -TT-^T-V = 0, ( fx denoting a rational and integral function of the fourth order), the integral of which was discovered by Euler, though first regularly derived from the differential equation by Lagrange. The theory of the Abelian integrals depends in like manner, as is proved by Jacobi, in the memoir " Considerationes generates de transcendentibus Abelianis" (Crelle, t. ix. [1832] p. 394) to depend, upon the system of equations _ n ~~ "i ...... where fx is a rational and integral function of the order 2w 1 or 2n, and the sums S contain n terms. The integration of this system of equations is of course virtually comprehended in Abel s theorem; the problem was to obtain (n 1) integrals each of them containing a single independent arbitrary constant. One such integral was first obtained by Richelot (Crelle, t. xxm. [1842] p. 354), "Ueber die Integration eines merkwiirdigen Sys tems Differentialgleichungen," by a method founded on that of Lagrange for the solution of Euler s equation; and a second integral very ingeniously deduced from it. A complete system of integrals in the required form is afterwards obtained, not by direct integration, but by means of Abel s theorem : there is this objection to them, however, that any one of them contains two roots of the equation fx = 0. The next paper on the subject is one by Jacobi, " Demonstratio Nova theorematis Abeliani" (Crelle, t. xxiv. [1842] p. 28), in which a complete system of equations is deduced by direct integration, 58] NOTES ON THE ABELIAN INTEGRALS, &C. 367 each of which contains only a single root of the equation fa = 0. But in Richelot s second memoir " Einige neue Integralgleichungen des Jacobischen Systems Differential- gleichungen" (Crelle, t. xxv. [1843] p. 97), the equations are obtained by direct inte gration in a form not involving any of the roots of this equation ; the method employed in obtaining them being in a great measure founded upon the memoir just quoted of Jacobi s. The following is the process of integration. Denoting the variables by x lt x 2 ... x n , and writing Fa =(a-a ?1 )(a-a? 2 )...(a-a? n ), so that F X! = (asj, - a? 2 ) . . . (x l - x n ), &c. then the system of differential equations is satisfied by assuming that x l , x...x n are functions of a new variable t, determined by the equations dt F fy (In fact these equations give 5 .^ N = dt 2 fs- = 0, &c.) v(fat) Fx From these we deduce, by differentiation, = __ . fl ^ V/ * 2 * F x, - (a?, - a?) F x (where 2 refers to all the roots except x,) and a set of analogous equations for Dividing this by a.x l , where a. is arbitrary, and reducing by \ 1 /, _ x + x, 2 (d-x 1 )(x 1 -x) 2 (a. - x) (a. - x,) ( Xl -x we have 1 cfcgj _ 1 d fxj a a?! dt 2 ~2 (a. - x,} dx, (F x^ + * (a - Xl } ^ x, ^ (a - x) F x ~ ^ that is _ a - x l dt 2 2 (a- x,) dx, (F x .1 v VIA) A T (a - #0 ^ (a - ) F x (a- ^) 3 (^X) - 2a) _ iv i F x F x, (oL-x)(tt-x l )(x l -~xY 368 NOTES ON THE ABELIAN INTEGRALS. [58 and taking the sum of all the equations of this form, the last term disappears on account of the factor x^x in the denominator, and the result is fx f *J(fx) I 2 iv fa * - a- x d? ~ * a -x dx (F x)* (a - x} F x) (a - x This being premised, assume which, by differentiation, gives d y - eft- and thence *2/__i^/v J_ ^ + i v2 _JL. ^ ^ 2 ~ 2 dt*a-x dt + 2y *(a-x)* \dt , v 2 that 1S = l S Substituting the preceding value of 1 a - a; _ fx v 1 rf A ] _ * + y 2 2 2 ~ Now the fractional part of ," , 2 is equal to \-^ / 4. "*" . _ _. (a - ^) 2 (^ a;) 2 "" " a - dx (F x)* Also if L be the coefficient of x 2 1 in ya, the integral part is simply equal to L, (since (Fa) 2 is a function of the order 2w, in which the coefficient of a 271 is unity). Hence the coefficient of y in the last equation is simply (Fa? 58] JACOBl s SYSTEM OF DIFFERENTIAL EQUATIONS. 369 dii viz. multiplying by the factor 2 ~j- , and integrating, Hence replacing y and ~ by their values we have , > (a - a;) for one of the integrals of the proposed system of equations : and since a. is arbitrary, the complete system is obtained by giving any (n l) particular values to a, and changing the value of the constant of integration C ; or by expanding the first side of the equation in terms of or, and equating the different coefficients to arbitrary constants. The a posteriori demonstration that all the results so obtained are equivalent to (n l} independent equations would probably be of considerable interest. c. 47 370 [59 59. ON THE THEORY OF ELIMINATION. [From the Cambridge and Dublin Mathematical Journal, vol. in. (1848), pp. 116 120.] SUPPOSE the variables X lt X, ... , g in number, are connected by the h linear equations ! = a 1 X 1 + a. 2 X z . . . = 0, these equations not being all independent, but connected by the k linear equations these last equations not being independent, but connected by the I linear equations ^ = a/ <, + .; <&, + ... = 0, !>, + ...=0, and so on for any number of systems of equations. Suppose also that g -h + k-l + ... = Q; in which case the number of quantities X lf X,,... will be equal to the number of really independent equations connecting them, and we may obtain by the elimination of these quantities a result V = 0. 59] ON THE THEORY OF ELIMINATION. To explain the formation of this final result, write 371 V = !, & or, .; ft", # which for shortness may be thus represented, V = where H, O , fl", fl" , fl"", ... contain respectively /<, A, /, /, ?;, n, ... vertical rows, and g, k, k, m, m, p, ... horizontal rows. It is obvious, from the form in which these systems have been arranged, what is meant by speaking of a certain number of the vertical rows of fl and the supplementary vertical rows of fl ; or of a certain number of the horizontal rows of H" and the supplementary horizontal rows of II , &c. Suppose that there is only one set of equations, or y = h : we have here only a single system H, which contains h vertical and h horizontal rows, and V is simply the determinant formed with the system of quantities fi. We may write in this case V = Q. Suppose that there are two sets of equations, or g = h k : we have here two systems H, IT, of which fl contains h vertical and h k horizontal rows, fT contains h vertical and k horizontal rows. From any k of the h vertical rows of H form a determinant, and call this Q ; from the supplementary h k vertical rows of H form a determinant, and call this Q : then Q divides Q, and we have V = Q -~ Q . 472 372 ON THE THEORY OF ELIMINATION. [59 Suppose that there are three sets of equations, or y = h-k + l: we have here three systems, fl, fl , fl", of which fl contains h vertical and h-k + l horizontal rows, fl contains h vertical and k horizontal rows, fl" contains I vertical and k horizontal rows. From any I of the k horizontal rows of fl" form a determinant, and call this Q" ; from the k - 1 supplementary horizontal rows of fl , choosing the vertical rows at pleasure, form a determinant, and call this Q ; from the h-k + l supplementary vertical rows of fl form a determinant, and call this Q: then Q" divides Q , this quotient divides Q, and we have V = Q -=- (Q -=- Q"). Suppose that there are four sets of equations, or g = h-k + l- m: we have here four systems, fl, fl , fl", and II" , of which fl contains h vertical and h k + l m horizontal rows, fl contains h vertical and k horizontal rows, fl" contains I vertical and k horizontal rows, and fl " contains I vertical and m horizontal rows. From any m of the I vertical rows of fl " form a determinant, and call this Q "; from the lm supplementary vertical rows of fl", choosing the horizontal rows at pleasure, form a determinant, and call this Q" ; from the k - 1 + m supplementary horizontal rows of fl , choosing the vertical rows at pleasure, form a determinant, and call this Q ; from the h k + l m supplementary vertical rows of fl form a determinant, and call this Q : then Q " divides Q", this quotient divides Q , this quotient divides Q, and V = Q -H [Q + (Q"+ Q "}}. The mode of proceeding is obvious. It is clear, that if all the coefficients cr, /3, . . . be considered of the order unity, V is of the order h 2k + 31 &c. What has preceded constitutes the theory of elimination alluded to in my memoir " On the Theory of Involution in Geometry," Journal, vol. n. p. 52 61, [40]. And thus the problem of eliminating any number of variables x, y ... from the same number of equations 7=0, F = 0, ... (where U, F, ... are homogeneous functions of any orders whatever) is completely solved; though, as before remarked, I am not in possession of any method of arriving at once at the final result in its most simplified form ; my process, on the contrary, leads me to a result encumbered by an extraneous factor, which is only got rid of by a number of successive divisions less by two than the number of variables to be eliminated. To illustrate the preceding method, consider the three equations of the second order, U = a x- + b if + c z 1 + I yz + m zx + n xy = 0, V= a a? + b y* + c z- + 1 yz + m zx + n xy = 0, W = a"x 2 + b"f + c"z 2 + l"yz + m"zx + n"xy = 0. Here, to eliminate the fifteen quantities x*, y\ z*, y z z, z 3 x, a?y, yz 3 , zx 3 , xy 3 , y 2 z*, zW, x*y*, x*yz, y 2 zx, z 2 xy, we have the eighteen equations x*U = 0, y-U = Q, z 2 U =0, yzU=0, zxU = 0, xylf = 0, a*V = 0, y~V = 0, z n -V = 0, yzV = 0, zxV = 0, xyV = 0, **W=0, y*W = 0, z n -W=0, yzW=(\ zxW = 0, 59] ON THE THEORY OF ELIMINATION. 373 equations, however, which are not independent, but are connected by a"a?V+ b"y* V + c"z* W + l"yz V + m"zx V + ri xy V - (a x 2 W + b f W + c z 2 W + I yz W + m zx W + rixy W) = 0, ax*W + by~W + cz^W + lyzW + mzxW + nxyW - (a"x U + b"y 2 U + c"z*- U + l"yz U + m"zx U + ri xy U) = 0, a V U + b f U + c z* U + I yz U + m zx U + rixy U (ax-V + by*V + czV +lyzV + mzxV + nxyV} = 0. Arranging these coefficients in the required form, we have the following value of V. a a a" b b b" c c c" I b I b I" b" m c m c n a ri a n" a" I c I c I" c" m a m a m" a" n b n b n" b" c b I c b I c" b" I" c a m c a in c" a" m" b a n b a n b" a" n" I a n m I a n m I" a" n" m" m n b I m n b I m" n" b" I" n in I c n m I c ri m" I" c" a" b" c" I" m" ri -a -V -c -I -m -ri a" b" c" I" in" n" a b c I m n a b c I m ri a b c I in n 374 ON THE THEORY OF ELIMINATION, which may be represented as before by [59 V = Thus, for instance, selecting the firsr, second, and sixth lines of O to form the determinant Q , we have Q a" (afb" a"b ) ; and then Q must be formed from the third, fourth, fifth, seventh, &c. ... eighteenth lines of H. (It is obvious that if Q had been formed from the first, second, and third lines of H , we should have had Q = ; the corresponding value of Q would also have vanished, and an illusory result be obtained; and similarly for several other combinations of lines.) 60] 375 60. ON THE EXPANSION OF INTEGRAL FUNCTIONS IN A SERIES OF LAPLACE S COEFFICIENTS. [From the Cambridge and Dublin Mathematical Journal, vol. in. (1848), pp. 120, 121.] SUPPOSE S = AIJ. + A I IJ! I - I + ..., = & +a 1 Q s _ 1 + .................................... (1), where, as usual, Q , Q 1} &c. are the coefficients of the successive powers of p in (1 2/j,p + p~)~-. Assume A 2 ) where A refers to 0; then expanding this expression, first in powers of p, and then in a series of terms of the form \/(l + A 2 ) . Q, and comparing these with the preceding values of S, 1.3...2g-l / 2A y s ~~ (3). 2A Assume - r. o. that is. 1 + A 2 A^^ii-va-a 2 )}* ............... (4). Then ^-rrr^i^- ^-p-Va-W* 1 ^ ........................ (5); 376 ON THE EXPANSION OF INTEGRAL FUNCTIONS IN A SERIES &C. [GO or, expanding this last equation in powers of S, (2r + lK/ 1 1 fr (2r + 7) & (2r + 9)(2r + ll) S* \ ^ _ <W = -r- *2FT + 22 + ~27r * + 2.4.6 ~ 2^ + or, replacing the successive terms of the form 8 . C by their respective values, (24 2r 4. 6 ... (2r + 4) s _ r = (2r + 1) | 3>5- / 2r+1)2 ^- + .T.5...(2r + 3)2^ ^ m 2) (2k + 4)... (2r + 4&) , } ^ - _ i-(2r+ 2^ + l)2^ m * } Thus, if S = fjf, so that A s _ r = 0, except in the particular case A = 1, <**-! = 0, or ZS TO -f 1 ^/t T ^^ V^ /v T */ ^"5 (8) ""* 2 s 3 . 5 . . . (2s - 2& + ^f/CI ^7 T\ \^*^ I ^/ \i 1 ^T ) . . . ^0 i* > / y o 4/? 4- 1 1 ^ ^ = 1] ...C9). which of course includes the preceding case. By substituting the expanded values of the coefficients Q, or again, by determining the value of (1 yu,) tS in terms of these coefficients, and equating it with that given in Murphy s Electricity, [8. Cambridge, 1833], p. 10, or in a variety of other ways, a series of identical equations involving sums of factorials may readily be obtained. The mode of employing the general theory of the separation of symbols made use of in the preceding example, may easily be applied to the solution of analogous questions. 61] 377 61. ON GEOMETRICAL RECIPROCITY. [From the Cambridge and Dublin Mathematical Journal, vol. in. (1848), pp. 173179.] THE fundamental theorem of reciprocity in plane geometry may be thus stated. "The points and lines of a plane P may be considered as corresponding to the lines and points of a plane P in such a manner that to a set of points in a line in the first figure, there corresponds a set of lines through a point in the second figure, (namely through the point corresponding to the line); and to a set of lines through a point in the first figure,, there corresponds a set of points in a line in the second figure, (namely in the line corresponding to the point)." ^ from this theorem, without its being in any respect necessary further to particularize the nature of the correspondence, or to consider in any manner the relative position of the two planes, an endless variety of propositions and theories may be deduced, as, for instance, the duality of all theorems which relate to the purely descriptive properties of figures, the theory of the singular points and tangents Jf curves, &c. ^ Suppose, however, that the two planes coincide, so that a point may be considered indifferently as belonging to the first or to the second figure : an entirely independent series of propositions (which, properly speaking, form no part of the general theory of reciprocity) result from this particularization. In general, the line in the second figure which corresponds to a point considered as belonging to the first figure, and the line in the first figure which corresponds to the same point considered as belonging to the second figure, will not be identical ; neither will the point in the second figure which corresponds with a line considered as belonging to the first figure, and the point in the first figure which corresponds to the same line considered as belonging to the second figure, be identical. C 48 378 ON GEOMETRICAL RECIPROCITY. [61 In the particular case where these lines and points are respectively identical (the identity of the lines implies that of the points and vice versd} we have the theory of " reciprocal polars." Here, where it is unnecessary to define whether the points or lines belong to the first or second figures, the line corresponding to a point and the point corresponding to a line are spoken of as the polar of the point and the pole of the line, or as reciprocal polars. " The points which lie in their respective polars are situated in a conic, to which the polars are tangents." Or, stating the same theorem conversely, " The lines which pass through their respective poles are tangents to a conic, the points of contact being the poles." To determine the polar of a point, let two tangents be drawn through this point to the conic, the points of contact are the poles of the tangents ; hence the line joining them is the polar of the point of intersection of the tangents, that is, " The polar of a point is the line joining the points of contact of the tangents which pass through the point." Conversely, and by the same reasoning, "The pole of a line is the intersection of the tangents at the points where the line meets the conic." The actual geometrical constructions in the several cases where the point is within or without the conic, or the line does or does not intersect the conic, do not enter into the plan of the present memoir. Passing to the general case where the lines and points in question are not identical, which I should propose to term the theory of "Skew Polars" (Polaires Gauches), we have the theorem, . " Considering the points in the first figure which are situated in their respective corresponding lines in the second figure, or the points in the second figure which are situated in their respective corresponding lines in the first figure, in either case the points are situated in the same conic (which will be spoken of as the pole conic ), and the lines are tangents to the same conic (which will be spoken of as the polar conic ), and these two conies have a double contact." This theorem is evidently identical with the converse theorem. The corresponding lines to a point in the pole conic are the tangents through this point to the polar conic ; viz. one of these tangents is the corresponding line when the point is considered as belonging to the first figure, and the other tangent is the corresponding line when the point is considered as belonging to the second figure. The corresponding points to a tangent of the polar conic are the points where this line intersects the pole conic ; viz. one of these points is the corresponding point when the line is considered as belonging to the first figure, and the other is the corresponding point when the line is considered as belonging to the second figure. 61] ON GEOMETRICAL RECIPROCITY. 379 Let i be a point in the pole conic, and when i is considered as belonging to the first figure, let il^ be considered as the corresponding line in the second figure (/! being the point of contact on the polar conic). Then if / be another point in the pole conic, in order to determine which of the tangents is the line in the second figure which corresponds to j considered as a point of the first figure, let il> 2 be the other tangent through /: the points of contact of the tangents through j may be marked with the letters J^, J 2 , in such order that /jJs, /oJj meet in the line of contact of the two conies, and then jf Jj is the required corresponding line. Again, / and i, as before, if B be a tangent to the polar conic, then, marking the point of contact as J^ let J 2 be so determined that 1^, I<*Ti meet in the line of contact of the conies : the tangent to the polar conic at J 2 will meet the pole conic in one of the points where it is met by the line B, and calling this point j, B considered as belonging to the second figure will have j for its corresponding point in the first figure. Similarly, if the point of contact had been marked J 2 , J^ would be determined by an analogous construction, and the tangent at J l would meet the pole conic in one of the points where it is met by the line B (viz. the other point of intersection) ; and representing this by j , B considered as belonging to the first figure would have f for its corresponding point in the second figure, that is, considered as belonging to the second figure, it would have j for its corresponding point in the first figure (the same as before). Similar considerations apply in the case where a tangent A of the polar conic, considered as belonging to one of the figures, has for its corresponding point in the other figure one of its points of intersection with the polar conic; in fact, if A represents the line il l , then A, considered as belonging to the second figure has i for its corresponding point in the first figure, which shows that this question is identical with the former one. To appreciate these constructions it is necessary to bear in mind the following system of theorems, the third and fourth of which are the polar reciprocals of the first and second. If there be two conies having a double contact, such that K is the line joining the points of contact, and k the point of intersection of the tangents at the points of contact : 1. If two tangents to one of the conies meet the other in i, ^ and j, j 1 respectively, then, properly selecting the points j, j 1} the lines ij lt i^j meet in K. And 2. The line joining the points of intersection of the tangents at i, j lf and of the tangents at i lt j passes through k. Also 3. If from two points of one of the conies, tangents be drawn touching the other in the points /, I I and J, J 1} then, properly selecting the points J, J 1} the lines meet in K. And 4. The line joining the points of intersection of the tangents at /,/j and of the tangents at 7 1} J passes through k. 482 380 ON GEOMETRICAL RECIPROCITY. [61 These theorems are in fact particular cases of two theorems relating to two conies having a double contact with a given conic. It may be remarked also that the corresponding points to a tangent of the pole conic are the points of contact of the tangents to the polar conic which pass through the point of contact of the given tangent, and the corresponding lines to a point of the polar conic are the tangents to the pole conic at the points where it is intersected by the tangent at the point in question. We have now to determine the corresponding lines to a given point and the corresponding points to a given line, which is immediately effected by means of the preceding results. Thus, if the point be given, " Through the point draw tangents to the polar conic, meeting the pole conic in A 1} A 2 and B lt B 2 (so that J.j5 2 and A 2 B 1} intersect on the line joining the points of contact of the conies), then A 2 B 2 and A^ are the required lines." In fact A l} B l and A 2 , B. 2 are pairs of points corresponding to the two tangents, so that -di-Bj and A 2 B 2 are the lines which correspond to their point of intersection, that is, to the given point, and similarly for the remaining constructions. Again, " Through the point draw tangents to the pole conic, and from the points of contact draw tangents to the polar conic, touching it in a 1} a 2 and /3j, /3 2 (so that a^, and Oo/Sj intersect on the line joining the points of contact of the conic), then aj/3, and o^8 2 are the required lines." So that A l , B l , !, & are situated in the same line, and also A 2 , B 2 , a.*, /3 2 . Again, if the line be given, " Through the points where the line meets the pole conic draw tangents to the polar conic C 1} C 2 and D 1} D 2 (so that the points C^D* and (7 2 A lie on a line passing through the intersection of the tangents at the points of contact of the tangents), then C^A and 2 D 2 are the required points." Again, " At the points where the line meets the polar conic draw tangents meeting the pole conic, and let 7!, y 2 and B 1} S 2 be the tangents to the pole conic at these points (so that the points y^ and 72^ lie on a line through the intersection of the tangents at the points of contact of the conies), then ^ , S 1 and <y. 2 , B 2 are the required points " ; so that C 1} D 1} 7^ Sj pass through the same point and also (7 2 , D.,, 7 2 , 8 2 . " The preceding constructions have been almost entirely taken from Plucker s " System der Analytischen Geometric," 3, Allgemeine Betrachtungen iiber Coordinaten- bestimmung. I subjoin analytical demonstrations of some of the theorems in question. Using x, y, z to determine the position of a variable point, and putting for shortness = ax + a y + a"z, 77 = bx + Vy + V z, % = ex + c y + c"z. 61] ON GEOMETRICAL RECIPROCITY. 381 then if the position of a point be determined by the coordinates a, /3, 7, the equation of one of the corresponding lines is (that of the other is obtainable from this by writing a, b, c ; a , b , c ; a", b", c" , for a, a , a" ; b, & , 6" ; c, c , c"). Hence if the point lies in the corresponding line, this equation must be satisfied by putting <x, /3, 7 for x, y, z; or, substituting x, y, z in the place of a, /3, 7, the point must lie in the conic U= ax 2 + b y 2 + cV + (b" + c ) yz+(c + a") zx + (a + b)xy = 0, (which equation is evidently not altered by interchanging the coefficients, as above). Again, determining the curve traced out by the line f + /&; + 7t= 0, when a, fj, 7 are connected by the equation into which U=0 is transformed by the substitution of these letters for x, y, z ; we obtain 2a , a! +b, a" + c rj, a + b, 2b , b"+ c a" + c, b" + c , 2c" which is also a conic. It only remains to be seen that the conies U = 0, V = have a double contact. Writing for shortness V = a , b , c a, b , c a", b", c" it may be seen by expansion that the following equation is identically true, V = 4V U - [x (ab" - a"b + a c - ac ) + y (b c - be + V a - b a") + z (c"a r - c a" + cb" - c"b)J, which proves the property in question. Suppose the equations of the two conies to be given, and let it be required to determine the corresponding lines to the point defined by the coordinates a, /3, 7. Writing, to abbreviate, U=Ax*+By- + U = Ao? + B/3* + W = Aax + B$y + P = Ix + my + nz, P = la + m@ + ny, K = ABC - AF* - BG* - CH 2 + 2FGH, ZHxy, 2Ha/3, H (ay + #r), 382 ON GEOMETRICAL RECIPROCITY. [61 & = BC -F* , 33 = CA-G 2 , & = AB - H* , = GH-AF, <& = gtP + 33m 2 + OTw 2 + Zffmn + 2feZ + 2 + P? + <M (7y - #*) + 33m + Jfw) (OLZ - 7<e) suppose U = Q represents the equation of the polar conic, UP 2 = Q that of the pole conic. The two tangents drawn to the polar conic are represented by UU TF 2 = 0, and by determining k in such a way that UU W 2 k (U P 2 ) may divide into factors the equation represents the lines passing through the points of intersection of the tangents with the pole conic. Thus if k=U , the equation reduces itself to U P 2 TP = 0, or W=\/(U )P, the equation of two straight lines each of which passes through the point of inter section of the lines P = 0, W = 0, (that is, of the line of contact of the conies, and the ordinary polar of the point with respect to the polar conic) ; these are in fact the lines A^BZ, AJ}! intersecting in the line of contact. The remaining value of k is not easily determined, but by a somewhat tedious process I have found it to be In fact, substituting the value, it may be shown that which is an equation of the required form. To verify this, we have, by a simple reduction, or, writing for shortness <yy ftz = g, az (&l 2 + 38m 9 + arw 2 + 2$mn + 2&nl + ffi - \m% + 33W77 + ODw^+ JP nrj = K{A (mS - nrff + B(n%- Itf + C(lrj- m) 2 + 2F (n - l) (Irj -m%)+2G (Irj - m& (ml; - n^) + 2H (m? - nrj) (ng - 1$}, which is easily seen to be identically true. 62] 383 62. ON AN INTEGRAL TRANSFOEMATION. [From the Cambridge and Dublin Mathematical Journal, vol. in. (1848), pp. 286 287.] THE following transformation, given for elliptic functions by Gudermann (Crelle, t. XXIIL [1842], p. 330) is useful for some other integrals. -rp _ dbc dba dca + abc (be ad) z (be -ad) + (d-b- then, putting K = (be ad) + (d b c + a)z, we have, supposing a < b < c < d, so that (b a), (c a), (d b), (d - c) are positive, K (y a) = (b - a) (c a) (d z), K(y-b} = (b-a}(d-b)( C -z\ K(y-c} = (c-a}(d-c)(b- z\ K(y-d) = (d-b)(d-e)(a-z\ K*dy = -(b-a)(c-a)(d-b)(d-c}dz. In particular, if a = where M = (b- a) a+ ^ +1 (c - a) a+y+1 (d - b)P +s+l (d - c)y +s+l . Thus, ifa = /8 = 7 = 8 = -, __ dy _ _ _ dz In any case when y = a, y = b, the corresponding values of z are z = d, z = c; the last formula becomes by this means dy c {-(z-a)(z-b}(z-c}(z -d}}^ 384 [63 63. DEMONSTRATION D UN THEOREMS DE M. BOOLE CONCERNANT DES INTEGRALES MULTIPLES. [From the Journal de Mathematiques Pures et Appliquees (Liouville), torn. xm. (1848), pp. 245248.] " Soient P, Q des fonctions de n variables x, y, ... ; lesquelles fonctions satisfont a la condition r ^n p \niti pG(v,u )i \ 7 7 I 7>i)4_Ol/il i /~1 \ fllnfltl [iV^^ll] I : . . . IJLJiiVuU . . . V i ...., \ x /> J-oo \/H(v, w) ou, comme a 1 ordinaire, i = V - 1 ; G (v, w), H (v, w) sont des fonctions homogenes de v, w des ordres 1 et n respectivement (on verra, dans la suite, qu il y a plusieurs fonctions P, Q qui satisfont a une Equation de cette forme). Cela e tant, posons (2), les limites de I inte gration etant donn^es par la condition P = 1 ; et soient On aura pour 1 integrale V cette formule, ^ rss-^ds T(^n + q)J V0 dans laquelle (5)." 63] DEMONSTRATION D UN THEOREME, &C. 385 Ce the oreme remarquable est du a M. Boole, qui me 1 a communique sous une forme un peu diffe rente 1 , en me priant d y supplier la demonstration et d en faire part aux gdometres; il ne m a fallu, pour le prouver, que modifier un peu le procede dont s est servi M. Boole meme, dans son Mdmoire : " On a certain Multiple Definite Integral," Irish Transactions, t. xxi. [1848]. Je vais done reproduire cette demonstration en 1 appliquant au probleme dont il s agit. On de montre par une analyse semblable a peu pres a celle par laquelle se demontre le theorem e de Fourier, que 1 expression (6), 1 p f 00 r 00 ^-\ da I dv I dwe - (a ~ p>v ~ <iw+ ^^ n+ i }n ^ l ...w^ nJrq ~ l fa se re duit (en n y faisant attention qu a la partie reelle) a ^ n+ ou a ze ro, selori que la quantit^ P se trouve ou ne se trouve pas comprise entre les limites 0, 1. Done, en substituant cette integrate triple dans 1 expression de V, on peut etendre depuis - oo jusqu a GO les integrations par rapport aux variables x, y,...\ de cette maniere, et en rdduisant par I e quation (1), on obtient tout de suite [-<? {v , w) ] i w ^n +( ,~i dw- . - fa. ..(7) * J Done, en ecrivant v , vds - , dw = s s 2 (ce qui donne, par les equations (3), G (v, w) = va, H (v, w) = v n s~ n (j>], les limites par rapport a la nouvelle variable s seront x , 0, et Ton obtiendra, en changeant 1 ordre des integrations, TT*" r ds Ss-i-i dans laquelle expression 1 f 1 f 00 S= - fta* I da l dv vfi e (a -* }vi fa IT " JO Jo 1 M. Boole ecrit S= -T 1 \ "^y expression a la verite plus simple, mais qui donne lieu, ce me semble, a quelques difficultes. 49 38G DEMONSTRATION D UN THEOREME DE [63 et il ne s agit plus que de faire voir 1 identite de cette valeur avec celle qui est donnee par 1 dquation (5). Pour cela, je remarque que Ton aura (10), -80 dv ifl e (a -*> =r(q+l) e i?+i> ( a _ a-)-i J cm =T(q + l) e-1 + >< (a - a)-?" 1 , i r x selon que (a <r) est positif ou ne gatif; les valeurs correspondantes de e^ qjri I dvifl e (a ~* )vi if J o sont +1) (a -o-)-?- 1 et ~ e~^ T (q + I) (a - a}~^ ......... (11); or, en ne faisant attention qu aux parties reelles, et en reduisant par une propriete connue des fonctions F, ces valeurs se reduisent a =. - - (a o-)"^" 1 et zero respective- ment ; d apres cela, 1 equation (9) se reduit a et enfin, en ecrivant on obtient pour 8 la valeur donne e par liquation (5), de maniere que la formule dont il s agit se trouve comple tement demontrde. II parait difficile de trouver les formes gene rales de P, Q {rien n etant, je crois, connu stir la solution des equations telles que liquation (1)} ; mais des formes parti- culieres se prdsentent assez facilement. Ainsi, en ne conside rant que les exemples que m a donnes M. Boole (lesquels j ai depuis v^rifie s), soit (13). En substituant ces valeurs dans liquation (1), les integrations s effectuent sans difficult^ et sous la forme necessaire, et Ton obtient G (V, W) = WV* - ( fi + Wt J--)^ f ff(v> w) = w n y ou enfin v- ...), (j> = l ........................... (14). Soit encore (ce qui comprend, comme cas particulier, le probleme des attractions) P= ? + i + --- Q=v* + (a-xY + (b-y)*+ ............... (15); 63] M. BOOLE CONCERN ANT DES INTEGRALES MULTIPLES. on obtient sans plus de difficulte 387 Soit encore, pour dernier exemple, on aura, pour ce cas, (17); (18); X 2 ) (i 2 s + A 2 ) les formules qui se rapportent a cet exemple aussi bien qu au premier sont, je crois, entierement nouvelles. 492 388 [64 64. SUR LA GENERALISATION D UN THEOREMS DE M. JELLETT, QUI SE RAPPORTE AUX ATTRACTIONS. [From the Journal de Mathematiques Pures et Appliquees (Liouville), torn. xui. (1848). pp. 264268.] LES formules qu a donnees M. Jellett pour exprimer les attractions d un ellipsoide au moyen de 1 expression de la surface de 1 ellipsoide re ciproque (t. XL de ce Journal, [1846], page 92) peuvent s e tendre au cas d un nombre quelconque de variables. Pour demontrer cela, je pars de cette formule (1), I ij -I |~/V _ -^ ~i ^ a-ti*,-L,> i mm tea -^ H ... + q) , + s dans laquelle n est le nombre des variables x, y, ... , et ou (2), u 2 s+f" s + g 2 s a 2 b 2 it 2 I ; : H . formule due a M. Boole qui 1 a de montree sous une forme un peu differente (Irish Transactions, t. xxi.). La modification que j y ai introduite se trouve demontree dans le Cambridge and Dublin Mathematical Journal, t. II. p. 223 (^ [44]. 1 Cette formule peut d ailleurs se deduire comme cas particulier de la formule tres-generale de M. Boole que M. Cayley a de montree dans le cahier precedent. (J. L.) ; [63]. 64] SUR LA GENERALISATION D UN THEOREMS DE M. JELLETT, &C. 389 On ddduit de la, en ecrivant fx, gy, . . . au lieu de x, y, ... , en reduisant a zero les quantite s a, b, ... , u (ce qui donne aussi 77 = 0), et en donnant une forme convenable a la fonction <, r(tf + y*...)i-l n dxdy ... = 2^ p s^ n ~ s ds J (/v+#y...) 2 ~r(i w -2)J V S + 2 s + < 2 ... (les limites de 1 integrale au premier membre de cette Equation e"tant donnees par x* + y 2 ... = 1). Done, en e crivant * = fiff on aura ^ = f r 7i ^T I // -* =v=~ ( 4 ) 1 \%" ~~*) J o V(s +y 2 ) (s + (/ - ) . . . Soit S ce que devient S en ecrivant ^ , - , . . . au lieu de y, </, . . . : en ecrivant en meme y t/ temps - au lieu de s, on obtient s 2 = p /i^_ 2 s ) ^ J A// ^ 5 ^ ; et de la, en ecrivant i! 2 y =J , l^ g ^ G) (6) on d^duit .P = Cela etant, remarquons que 1 integrale Jo V est fonction homogene de 1 ordre (2 n) par rapport aux quantitds f, g, ... (en effet, cela se voit tout de suite en faisant s =f~d}. Done ds ou, en e crivant - -1, &c., au lieu de -- ^^, &c., " s ff_JL. ^_ ^ & _ 2 f cfe JoVs+/ 2 s + g* ) V( 5 +/ 2 )(s + ^ 2 )... J o V(s+/ 2 )(s+^ 2 )... " 300 SUR LA GENERALISATION D UN THEOREME DE M. JELLETT, [64 et de la aussi ds + s + } as - = 2/ 2 ^-- , &c. (10). Done enfin ds , (11). ds I a* ( s s -s c est-a-dire " a 2 F+G + ... " 7~2\ En particularisant d une maniere convenable la formule (1), on obtient, pour le cas n 2 If. de T-+ ..<>!, cette formule connue J t/ f dxdy... ^ "~ J F(a ) 2 + (b y) 2 . . .] in-1 [ (13) as (/.- ,^y- i 1 equation des limites ^tant, comme auparavant, ^ + + ... = 1J ; / y et de la, vu la formule (12), resulte L expression de S, en ecrivant r cos a, rcos/3, ... au lieu de x t y,..., remplacant dxdy... par r n ~ l dr dS, et integrant depuis r = jusqu a r = l, se re duit a _ n . -J 9 de sorte qu au cas de w=3 cette fonction se reduit a 1 expression qu a donnde M. Jellett pour la surface d un ellipsoide. Done, en se rappelant que les attractions 64] QUI SE RAPPORTE AUX ATTRACTIONS. 391 . ^ dV dV dV . ,. . sent representees par --^- , -TV , -j , on voit que 1 equation (14) equivaut, pour ce cas, CtCL CLu CtC aux formules de M. Jellett. Remarquons, qu en transformant I inte grale (4) de la meme maniere dont nous avons transforme I inte grale (8), on obtient TT*W... r f 1 I \ s* n - 2 ds ~ ce qui donne pour n 3 cette expression tres-simple de la surface de 1 ellipsoide aux demi-axes f, g, h, formule qui se ve rifie tout de suite au cas de f=g = h. L expression encore plus simple que donne 1 equation (4), savoir, V = _ nrfWK \ _ "__^ .( 18 ) ? n est pas exempte de difficult^ a cause de la valeur apparemment infinie du second membre de I e quation. Au cas d une sphere, cela se re duit a (1+*) ce qui serait, en effet, exact si la formule m ~ l ds Tw T?i (1 + s) m+n r (m + n) subsistait pour les valeurs negatives de m. Cela nous apprend que les integrates de la forme >~ g ~ ldS (W) ne sont pas a rejeter au cas des valeurs positives de q; il est meme facile, en repetant continuellement le proce de* de reduction que nous venons d employer, de pre senter ces inte grales sous une forme ou il n y a plus de terme infini. En m aidant de 1 analogie de quelques formules qui se trouvent dans mon Memoire Sur quelques formules du cakid integral (t. XL de ce Journal, p. 231 [49]), je crois meme pouvoir avancer que cette inte grale doit se remplacer par ou, comme a 1 ordinaire, i = V - 1, et ou k denote une quantite quelconque dont la partie re elle ne s evanouit pas. Mais je renvoie cette discussion a une autre occasion. 392 [65 65. NOUVELLES RECHERCHES SUE LES FONCTIONS DE M. STURM. [From the Journal de Mathematiques Pures et Appliquees (Liouville), torn. xui. (1848), pp. 269274.] EN ddveloppant une remarque faite par M. Sylvester dans un Mdmoire publie il y a huit ou neuf ans dans le Philosophical Magazine, j ai trouve des expressions assez simples des fonctions de M. Sturm, composees au moyen des coefficients memes ; ce qui convient mieux que d exprimer ces fonctions, comme je 1 ai deja fait dans le t. xi. de ce Journal, p. 297, [48], par les sommes des puissances. D ailleurs il ne m est plus ndces- saire de parler des expressions de M. Sylvester, ou meme des divisions successives de M. Sturm ; mais ma methode fait voir directement que les fonctions que je vais definir sont doue es de la proprie te fondamentale sur laquelle se repose la theorie de M. Sturm, a savoir que, en considerant trois fonctions successives, la premiere et la derniere fonc- tion sont de signe contraire pour toute valeur de la variable qui fait evanouir la fonc- tion intermediaire ; cependant je n ai pas encore reussi a de montrer dans toute sa ge ne ralite 1 equation identique d ou depend cette proprietd. Soient d abord V, V des fonctions du meme degrd n, V = ax n + l>x n ~ l + ... , V = a x n + b x n ~ l + ... , et ecrivons v, a xV, a b c V a V, a b xV, a b c V a b 65] NOUVELLES RECHERCHES SUB LES FONCTIONS DE M. STURM. 393 x*V, xV, V, a?V, xV, V a a b a b a c b a c b a d c b d c b e d c e d c , &c. (ce qui suffit pour faire voir la loi de ces fonctions successives F 1} P 2 , ...). II requite des proprie te s eldmentaires des determinants que ces fonctions sont des ordres n 1, n 2, n S, &c., respectivement, par rapport a la variable x. En effet, dans F l le co efficient de x n se reMuit a a, a , savoir, a zdro ; de meme, dans F 2 , les coefficients a a de x n et x n ~ l se reduisent chacun a zero ; et ainsi de suite. Soient encore p,= a, a b b a, . a , . b a b a c b c b d c d c P, - a, . b a c b a c a, . . b a . b a b d d f d b d &c., P f a, a j c c 1 a, a, . b a b a c b c b i e d e d a, . a , b a . b c b a c d c b d e 9 d f c e e q b a c b d c * &c. (ce qui suffit pour indiquer la loi). On aura entre ces differentes fonctions F, P, P cette suite remarquable d equations identiques, PJ\ + P/P 2 ) F 2 P 2 P 3 + P 2 P 3 ) F 3 &c., lesquelles equations, dans ce M^moire, seront prises pour vraies. Cela etaiit, il est evident que F et F 3 seront de signe contraire pour touie valeur de x qui fait evanouir F 2 ; F 2 et F t seront de signe contraire pour toute valeur de x qui fait eVanouir F 3 ; et ainsi de suite. C. 50 394 NOUVELLES RECHERCHES SUE LES FONCTIONS DE M. STUHM. [65 Ces formules renferment le cas ou les deux ibnctions V, V ne sont pas du meme degre (en effet, pour les y adapter, on n a besoin que de faire evanouir quelques-uns des premiers coefficients de V ou de V). II est done permis de supposer que V soit la derive e de V. Dans ce cas, F^aV, et on verra dans un moment que les fonctions F t> F 3 , ... contiennent chacune le facteur a 2 , de maniere qu il convient decrire F 2 = a 2 F 2 , p =c fV a , >>>( Ce facteur a 2 peut etre eVidemment e carte , et ce sera de meme avec le facteur a de F, pourvu, ce que je supposerai dans la suite, que a soit positif. On aura de cette maniere les fonctions V, V, F 2 , F,, ... douses des proprie^s des fonctions de M. Sturm. En effet, elles seront pre cise ment les fonctions fas, fa, f&, ... du Memoire deja cite", ce qui cependarit pourrait etre difficile a de montrer a priori. On deduit tout de suite des expressions de F 2 , F 3 , ... , F, a b xV, V na xV, a b c d V, a b c afV, na xV, na V na n Ib na n-lb n 2c n-lb n-3d n- 2c n-lb &c. Ces formules se simplified au moyen des propriety s connues des determinants, et en xV-nV=- U, eel a donne aV t = V, U, V , aV 3 = xV, V, xU, U, a a b b na b a b c b 2c b d c 3d 3c ou enfin, et en ecrivant un autre terme de la suite, afin d F 2 =- U, V , F 3 = - F, xU, U, b na a b b 2c b c 3d 2c F 4 = - xV, V, x*U, xU, V a . b . b a 2c b c b 3d 2c na F | na , &c.; faire voir la loi, n-lb 3 mieux F n a n-lb , &c., d c 4e 3d n Ib e d 5f 4e n- 2c 65] NOUVELLES RECHERCHES SUR LES FONCTIONS DE M. STURM. 395 formules dans lesquelles V = ax n V = nax n ~ l + n 1 bx n ~ 2 + ... , U = bx n ~ l + 2cx n ~ 2 + ... , et ou, en substituant ces valeurs, on peut commencer pour V 2 avec le terme qui contient .c n ~~, pour V 3 avec le terme qui contient x n ~ 3 , et ainsi de suite, puisque les termes des ordres plus hauts s evanouissent identiquement. Voila, je crois, les expressions les plus simples des fonctions de M. Sturm. Je donnerai en conclusion ces formes developpees des fonctions jusqu a F 4 . F = ax 11 -4- bx n ~ l -\- cx n ~ 2 + V rtri ^9rr n ~ 2 f 9 flLv i&LtJj + b {n-l bx n ~ 2 + n-2 cx n ~ s + . . . ; F, = [2nabc - n~-l b 3 ] + [ - 2n 2 ac + n-l ab- ] {^ex n ~ 9 + 5/^ n ~ 4 + . . . + [3na-d - (3>i - 2) abc + (n - 1) b 3 ] {3dx n ~ 3 + 4ex n ~* + ... + [ - 3abd + 4ac 2 - b-c ] {(?? - 2) cx n ~ 3 + (n - 3) dx*-* + . . . ; F 4 = A [fx n ~* + gx n ~ 5 + ... + B {ex n -* + C {Ggx n + D {5fx n + E 4ex n + F {n 3dx n ~* + n 4>ex n ~ a + ...; dans cette derniere expression j ai mis, pour abre ger, A = 9na-bd 2 - 8na?bce + (4w - 4) ab 3 e - (lO/i - 12) ab"cd + (4n - 8) abc 3 -(n-2) i 3 c 2 + (2n - 2) b*d, B = Wntfbcf- I2na 2 bde - !Qna 2 c-e + 18w,*cd 2 - (on - 5) ab 3 f + (18/i - 16) ab 2 ce + (3n - 9) atfd- - (24n - 36) abtfd + (8w - 16) ac 4 - (3?i - 3) e + (5n - 7) b 3 cd - (2n - 4) 6 2 c s , C = 8na 3 ce - 9na 3 d 2 - (4w - 4) a 2 b 2 e + (10% - 1 2) tfbcd - (4w - 8) a 2 c 3 - (2n - 2) ab 3 d + (n-2) a&V, D = - I0na s cf+ I2na 3 de + (on - 5) a 2 i 2 /- (2n - 8) a?bce - (I2n - 9) aW + (4n - 12) a?c~d -(n-l) ab s e + (Qn - 9) ab*cd - (4n - 8) abe 3 - (2n - 2) b*d + (n - 2) b 3 c~, 502 396 NOUVELLES RECHERCHES SUE LES FONCTIONS DE M. STURM. bo E = Iona 3 df- I6na s e* - (Ion - 10) a?bcf+ (I7n - 12) o?Ue + (16/i - 16) a*c?e - (I5n - 18) a 2 cd 2 + (on - 5) ab 3 f - (I7n - 18) atfce + (Un - 24) abc 2 d -(n- 3) atfd 2 _ (^ _ 8) ac 4 + (3w - 3) 6 4 e - (3w - 5) Z> 3 cd + (w - 2) 6 2 c 3 , ^ = - 15a a 6d/+ 16a 2 6e 2 + 20a 2 c 2 /- 27a 2 d 3 - 48a 2 cc?e - 5ab 2 cf + 7ab 2 de - 4>abc*e - ISabcd 2 II serait evidemment inutile de vouloir pousser plus loin ces calculs. [MS. addition in my copy of Liouville. Quoique ces expressions soient ce qu il y a de plus simple pour le calcul numerique des fonctions de M. Sturm, cependant sous le point de vue analytique il convient de modifier un peu la forme de ces expressions. En effet en e*crivant mettons P = Q = v= n\. ... , = ax n ~ l + 0! bx n ~ 2 + ... , cx r Ton aura V = nP, U = nQ, V = Px + Q ; et cela donne apres une reduction legere P, Q a b V - rP P rO V 3 X , -L , <<, ^J a . 6 0J> a 00 b xP, P, x*Q, xQ, Q b a Q^c b f/ib a 0-4 0iC I) 0oC 0-J) 0%6 04 01^ 04 0,c ej 3 e 04 et ainsi de suite.] 66] 397 66. SUR LES FONCTIONS DE LAPLACE. [From the Journal de Mathematiques Pures et Appliquees (Liouville), torn. xm. (1848), pp. 275280]. J AI rt^ussi a e"tendre a un nombre quelconque de variables la theorie des fonctions de Laplace, en me fondant sur le the oreme que voici: "Les coefficients I, m, ... , I, m , ... etant assujettis aux conditions I 2 + m? + . . . = 0, et les limites de 1 integration etant donnees par ^ + ^+... = 1 ....................................... (2), Ton aura pour toutes valeurs entieres et positives de s, s , except^ pour s = s , I (Ix + my + . . . ) s (I x + m y + . . ./ dxdy . . . = ................................. (3), et pour s = s , j (Ix + my + ...) (l x + m y+...ydxdy...=N s (ll + mm +...) s ............ (4), en faisant, pour abreger, _ TT^ r (s + 1) (ou n denote le nombre des variables)." En admettant d abord comme vrai ce the oreme qui sera demontre plus bas, je remarque qu il est permis d ecrire ^ , ^ , ... , ^ , J-, , ... , a u lieu de I, m, ... , V, m ,... , ou ces nouveaux symboles se rapportent a de certaines fonctions / / de a, b, . . . et de a , b , ... respectivement. De la ce nouveau the oreme: 398 SUR LES FONCTIONS DE LAPLACE. [66 " Les fonctions /, / e tant assujetties aux conditions d*f d*f 1 . J _| J I- =0 <* * ....(6), #/ #/ _ df + dP 4 > on aura (entre les memes limites qu auparavant), except^ pour s = s , et pour .9 = s , \ rln. "rib " I \ da * db I (8). d d d d II est facile de voir que les expressions d d V, / d d satisfont a 1 equation et, de plus, qu elles sont les fonctions entieres et homogenes, des degres s et s respective- ment, les plus ge ne rales qui puissent satisfaire a cette Equation. On a done ce theoreme : "Soient V s , W# les fonctions entieres et homogenes des degres s et s respective- ment, les plus ge ne rales qui satisfassent a 1 e quation (9); on aura toujours, excepte an cas de s = s , sWdxd ...=0 .................... (10) (les limites etant les memes qu auparavant)." Ecrivons a present valeur qui satisfait a la premiere des Equations (8), et nous obtiendrons par la differen tiation successive, en faisant attention a la seconde de ces memes equations, ? + + -) (a! - (11) -S + l ; (56] SUR LES FONCTIONS DE LAPLACE. 399 En representant, cotnrae auparavant, par W s la fonction (x Y f soit Wg ce que devient W g en dcri^ant a, b, ... au lieu de x, y, ... , c est-a-dire ecrivons On deduit de la, et au moyen de 1 equation (IT), en substituant dans 1 equation (8), la formule F(+l) J V da *dfc^ (12), d \ * / S en faisant, pour abreger, ou bien Mt= rfo-I)(n + 28)(n + 28-2) (13) " Soit maintenant 1 Qo Q s _ _ [(a - xf + . . .J^- 1 ~ (a 2 + 6 2 + . . .^^ (a 2 ou, autrement dit, soit 1 equation (12) devient . = M s W s ................................. (16) {ou la valeur de M s est donnee par 1 dquation (13)}. Les deux e quations (10) et (16) contiennent la thdorie des fonctions W s , Q s , lesquelles comprennent evidernment, comme cas particulier, les fonctions de Laplace. Pour de montrer le theoreme exprime par les e quations (1), (2), (3), (4), (5), je fais d abord abstraction des equations (1), et j e cris x= l+ l r)+ 1"$..., y = m% + m tj + m"%. . . , ou les coefficients sont tels que liquation 400 SUR LES FONCTIONS DE LAPLACE. [66 soit identiquement vraie. Cela suppose que les valeurs de p, p , q soient respectivement l 2 + m z + ... , l 2 + m 2 + ... , et II + mm + ... , et que les sommes de produits telles que II" + mm" + ... , I l" + m m" + ... se re duisent chacune a zero. De la dandy ... = *Jpp - q 2 ^p 7 ... lac +my + . . . = p% + qrj, I x + m y + ... =q% + p f ij. En repre sentant par / I inte grale au premier membre de liquation (3), cela donne / = */pp - q* A/p" ... 1 equation des limites etant Cette integrale se simplifie en ^crivant f^ + i=f" car alors ~ r Vp et de la 1 equation des limites etant f/ + ^;+...-i. En supposant s > s , 1 integrale s ^vanouit pour p = 0, et de meme quand s > s, elle s ^vanouit pour p = Q. Done, en ^crivant p = 0, p =0, on aura toujours, excepte pour s = s , 1 equation 7 = ; ce qui revient a 1 ^quation (3). Au cas de s = s , en ecrivant de meme p = 0, p = 0, on trouve (ou, com me a 1 ordinaire, i = V 1). En faisant attention a la valeur de q, et en com- parant avec 1 equation (4), cette derniere Equation sera demontree en verifiant la formule Soient pour cela , = p cos 0, t) / p sin 6. 66] SUR LES FONCTIONS DE LAPLACE. 401 Cela donne (en omettant la partie imaginaire qui s evanouit evidemrnent) +1 cos 8 6 cos sd dp d0d / .... En effectuant d abord 1 intdgration par rapport a ..., les limites de ces variables sont donnees par et Ton trouve tout de suite 1 / pM+i (j _ ^in-i C os s cos s0 dp d6. i) J Tj-^l-l Cette formule doit etre inte gre e depuis p = a p = 1, et depuis 6 = a 6 = 2?r ; mais en multipliant par quatre, on peut n e tendre I inte gratioii par rapport a que depuis = jusqu a = |TT. De la cos s cos s0 de ; et enfin, au moyen des formules connues cos 8 cos s0 d0 on retrouve la formule (5), laquelle il s agissait de demontrer. Ainsi le thdoreme fonda- mental est completement etabli. c. 51 402 [67 67. NOTE SUH LES FONCTIONS ELLIPTIQUES. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn, xxxvu. (1848), pp. 5860.] SOIT x = \Jk sin am u et a. = k + j : la fonction \Jk sin am nu (ou n est un entier) , K peut etre exprimee sous forme d une fraction dont le denominateur est une fonction rationnelle et entiere par rapport a a; et a. En exprimant ce denominateur par z, on aura n*(tf-l)x*2 + (n*-l)(ax-2a?)^ + (l-cuK* + a?) d ~-2n*(tf-4} C ^=0... (1). dx dx* da Cette equation est due a M. Jacobi, (voyez les deux me moires "Suite des notices sur les fonctions elliptiques," t. in. [1828] p. 306 et t. iv. [182.9] p. 185.) En essayant d integrer cette equation a moyen d une suite ordonne e suivant les puissances de x, et en considerant en particulier les cas n = 2, 3, 4 et 5, j ai trouve que les differentes puissances de a se pre sentent et disparaissent d une maniere assez bizarre : (voyez mon me moire " On the theory of elliptic functions," Cambridge and Dublin Math. Journal, t. n. [1847], p. 256, [45].) J ai reconnu depuis que cela vient de ce que la valeur de z est compose e de plusieurs sdries inde pendantes ; une quelconque de ces series ordonnee selon les puissances descendantes de a va a I infmi ; mais, en combinant les differentes series, les termes qui contiennent les puissances negatives de a se detruisent. Par rapport a x chacune de ces series ne contient que des puissances paires et positives, car les puissances negatives qui y entrent apparemment, se reduisent toujours a zdro. En effet, on satisfait a liquation (1) en supposant pour z une expression de la forme z s = 2 2 *< n - 2s af(n-w) 2,^ ... -\-( I) 1 ? 2 2s < n - 2s >- 2 9 afcn-w-q Z q (2) 67] NOTE SUR LES FONCTIONS ELLIPTIQUES. 403 (ou s est arbitraire). Cela donne pour Z q 1 dquation aux differences mele es: + ZK S (n ~ 2s) ~ - I28n 2 [s (n - 2s) -q+2] Z fJ _ 2 = 0. Pour integrer cette equation, supposons ou la semination se rapporte a o- et s e tend depuis a = jusqu a <r = q. Toute reduction faite, et ayant mis pour plus de simplicity n 2 2ns = X, 2ns = /A, on obtient pour Zq" 1 expression 4. [- (q - a) \ - <rp + (q - + (q - <T) (\ - 2q + 4<r + 2) (\ - 2q + 4<r + a- (fi + 2q - 4o- + 2) (ji + 2q - + IGcr (q - <r) [V - 2 (9 - 2) (\ En supposant la valeur de Z^ egale a Funite 7 , les valeurs de Zg" se trouvent comple tement d^terminees ; malheureusement la loi des coefficients n est pas en evidence, excepte dans le cas de cr = 0, ou a- = q. En calculant les termes successifs, on obtient z g = r i + (4a) ,S(7l 28) 2 1 1.2 1 /U> yfji O^ X~ "X (\ 4A (\ ^\ [ 1.2 ( 1 1.2.3 1 1.2.1 X 1 i^ / r>\ ZUA,tt\ IX I a -< i 1 a,(\ 1 1.1.2^ 1 1^ A.+ /J yti (yU, 4) (/Jt, 5) + 1.2.3 &C. 512 404 NOTE SUB LES FONCTIONS ELLIPTIQUES. [67 On aurait une valeur assez ge nerale de z en multipliant les diffe rentes fonctions z s chacune par une constante arbitraire, et en sommant les produits ; mais dans le cas actuel ou z denote le ddnominateur de V&sinamrm, la valeur convenable de z se reduit a ou s est un nombre entier et positif, entre zero et \n ou \(n \\ On aura par exemple dans le cas n = 5 (les sigues e tant tous positifs si n est impair, et alternative- ment positifs et ndgatifs si n est pair), en supprimant les puissances negatives de a (lesquelles s entredetruisent) : - a 125^ + SOOtf 12 + 275a; 16 ), et de la : 2 = l- 50^ + 140o^ - (125 + 160a 2 ) a* + (368a + 64a 3 ) x w - (300 + 240a 2 ) x + 3600KC 14 - 105 16 - 80a^ 18 + (62 + 16a 2 ) ^ - ce qui est effectivement la valeur de z que j ai trouve e dans le m^moire cite pour ce cas particulier. 68] 405 68. ON THE APPLICATION OF QUATERNIONS TO THE THEORY OF ROTATION. [From the Philosophical Magazine, vol. xxxiil. (1848), pp. 196 200.] IN a paper published in the Philosophical Magazine, February 1845, [20], I showed how some formulae of M. Olinde Rodrigues relating to the rotation of a solid body might be expressed in a very simple form by means of Sir W. Hamilton s theory of quaternions. The property in question may be thus stated. Suppose a solid body which revolves through an angle round an axis passing through the origin and inclined to the axes of coordinates at angles a, b, c. Let A. =; tan \Q cos a, p = tan \Q cos b, v = tan W cos c, and write A = 1 + i\ +jfji + kv ; let x, y, z be the coordinates of a point in the body previous to the rotation, x-i, y l , , those of the same point after the rotation, and suppose II =ix +jy + kz , then the coordinates after the rotation may be determined by the formula n x = AHA- 1 ; viz., developing the second side of this equation, U 1 = i (a x + @ y + yz ) 406 ON THE APPLICATION OF QUATERNIONS TO THE THEORY OF ROTATION. [68 where, putting to abbreviate K = 1 + X 2 + y"- 2 + v 2 , we have KOL = 1 + X 2 - fjf - v\ tea = 2(\fi + v) , tea." = 2 (Xi; - /A) these values satisfying identically the well-known system of equations connecting the quantities a, /3, 7, a , /3 , 7, a", /3", 7". The quantities a, 6, c, 6 being immediately known when X, /u,, p are known, these last quantities, completely determine the direction and magnitude of the rotation, and may therefore be termed the coordinates of the rotation ; A will be the quaternion of the rotation. I propose here to develope a few of the consequences which may be deduced from the preceding formulas. Suppose, in the first place, II = A 1, then IIj = A 1, which evidently implies that the point is on the axis of rotation. The equation IL, = II gives the identical equations X(cc-l) + p (3 + z/7 = 0, X a! + / u(/3 / -l) + i 7 / =0, X a" + yu/3" + i/(7 // - 1 ) = 0; from which, by changing the signs of X, fi, v, we derive X (a - 1) + fj, a! + v a" = 0, X +/i(/3 -l)+i> (3" -0, X 7 + p 7 +i/(7"-l)=0. Hence evidently, whatever be the value of II, AHA- 1 - II = 0, if after the multiplication i, j, k are changed into X, /JL, v, a property which will be required in the sequel. By changing the signs of X, /A, v, we also deduce =i ( ax + a y + a! z] + k (ysc + yy + i"z\ where a, /3, 7, a , /3 , 7 , a", /3", 7" are the same as before. Let the question be proposed to compound two rotations (both axes of rotation being supposed to pass through the origin). Let L be the first axis, A the quaternion of rotation, L the second axis, which is supposed to be fixed in space, so as not to alter its direction by reason of the first rotation, A the corresponding quaternion of rotation. The combined effect is given at once by Hi = A (AHA- 1 ) A - 1 , that is, n x = A ALT 68] ON THE APPLICATION OF QUATERNIONS TO THE THEORY OF ROTATION. 407 or since (if Aj be the quaternion for the combined rotation) IIj = AjIIAj" 1 , we have clearly M l denoting the reciprocal of the real part of A A, so that Mr 1 = 1 - XX - w - vv. Retaining this value, the coefficients of the combined rotation are given by Xj = M! (X + X + jjfv jjiv ) , fr = M! (jJL + p + Z/X - V\ } , v l = M 1 (v+v + X> - X/A ) ; to which may be joined [if ^ = 1 + Xj 2 + ^ + vf\, . = KK K, K , *! as before. A or A may be determined with equal facility in terms of A , A],, or A, Aj. These formulas are given in my paper on the rotation of a solid body (Cambridge Mathematical Journal, vol. ill. p. 226, [6]). If the axis L be fixed in the body and moveable with it, its position after the first rotation is obtained from the formula 11! = AH A" 1 by writing n=A 1. Repre senting by A"-l the corresponding value of tt 1} we have A ^AA A" 1 , which is the value to be used instead of A in the preceding formula for the combined rotation, thus the quaternion of rotation is proportional to AA A^A, that is to AA . Hence here A! = M! AA , which only differs from the preceding in the order of the quaternion factors. If the fixed and moveable axes be mixed together in any order whatever, the fixed axes taken in order being L, L ,... and the moveable axes taken in order being Z , L .. then the combined effect of the rotations is given by A t Jf...A"A AA f A. ..., M being the reciprocal of the real term of the product of all the quaternions. Suppose next the axes do not pass through the same point. If a, , 7 be the coordinates of a point in L, and r = ai+ j + yk, then the formula for the rotation is or j = AHA- 1 - (AFA- 1 - r), where the first term indicates a rotation round a parallel axis through the origin, and the second term a translation. 408 ON THE APPLICATION OF QUATERNIONS TO THE THEORY OF ROTATION. [68 For two axes L, L fixed in space, n x = A AII (A A)- 1 - (AT A - 1 - P) - A ( ATA~ l - T) A - ; and so on for any number, the last terms being always a translation. If the two axes are parallel, and the rotations equal and opposite, A-A - 1 , whence u, = n + A (r - P) A ^(F - P) ; or there is only a translation. The constant term vanishes if i, j, k are changed into V, p , i/, which proves that the translation is in a plane perpendicular to the axes. Any motion of a solid body being represented by a rotation and a translation, it may be required to resolve this into two rotations. We have u, = A^Ar 1 + T, where T is a given quaternion whose constant term vanishes. Hence, comparing this with the general formula just given for the combination of two rotations, T = - (A PA " 1 - P) - A (AFA- 1 - F) A - 1 , the second of which equations may be simplified by putting A ~ 1 TA = S, by which it may be reduced to S = (A -T A - P) - (AFA- 1 - F), which, with the preceding equation A 1 = M 1 A A, contains the solution of the problem. Thus if A or A be given, the other is immediately known ; hence also S is known. If in the last equation, after the multiplication is completely effected, we change i, j, k into A, p, v, or V, p , v, we have respectively, S = A -T A - P, S= - (AFA- 1 - F), which are equations which must be satisfied by the coefficients of P and F respectively. Thus if the direction of one axis is given, that of the other is known, and the axes must lie in certain known planes. If the position of one of the axes in its plane be assumed, the equation containing 8 divides itself into three others (equivalent to two independent equations) for the determination of the position in its plane of the other axis. If the axes are parallel, A, p, v are proportional to A , p, v ; or changing i, j, k into A, p, v, or A , p , v, we have 8=0; or what is the same thing, T=0, which shows that the translation must be perpendicular to the plane of the two axes. If p, q, r have their ordinary signification in the theory of rotation, then from the values in the paper in the Cambridge Mathematical Journal already quoted, dA . dK K (ip +jq + kr) = 2-A + -; 08] ON THE APPLICATION OF QUATERNIONS TO THE THEOEY OF ROTATION. 409 but I have not ascertained whether this formula leads to any results of importance. It may, however, be made use of to deduce the following property of quaternions, viz. if A! = JfjA A, M l as before, then in which the coefficients of A are considered constant. To verify this a posteriori, if in the first place we substitute for /q its value M^KK , we have and thence Also dfCi _ , , , die 2 -i dt~ l ~ K ~di + W 1 ~dt Kl CA! . 1 dM 1 ,, n ,rtA . j- 1 A! + jf- - - 1 KI = M 2 K -j- A, at M l at dt dAi /I dM, . A /^A\ . 1 t . ,, n ., A 75 A - (g, -S A + M A 7ft) A - , TIT A + Jtf A * A A which reduces the equation to Hence observing that and omitting the factor A from the resulting equation, 2 dM, , ^A , A +AA=K or since -TJ = 1 XX yLtyLt I/Z/, substituting and dividing by A , we obtain or finally, 9 v d\ ,dfji ,dv\ ,., dA dA ., I X -ji+p -fr + v JT = * A -i _ _ A dt dt dt dt dt .d\ .djt, , dv which is obviously true. C. 52 410 [69 69. SUE LES DETERMINANTS GAUCHES. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn, xxxvui. (1848), pp. 9396: continued from t. xxxn. p. 123, 52.] J AI iiomme "Determinant gauclie" un determinant forme par un systeme de quantites \ r s , qui satisfont aux conditions X,. r = - V( r * S ) .................................... (1), (ou les valeurs de r, s s e tendent depuis 1 unit^ jusqu a n). Or ces determinants peuvent facilement etre exprimes par un systeme de deter minants pareils, dont les termes satisfont a ces conditions meme dans le cas ou les valeurs de r et s deviennent egales, ou pour lesquels on a ces determinants peuvent etre nomme s " gauches et symetriques." En effet, soit H le determinant gauche dont il s agit, cette fonction peut etre presentee sous la forme 2.2 + n 18 x ltl x 2 . 2 ........................ (3), ou H. est ce que devient O si X^, X,. a , &c. sont reduits a ze ro ; Oj est ce que devient le coefficient de \ ^ sous la meme condition, et ainsi de suite; c est-a-dire : H est le determinant formd par les quantite s \ r s en supposant que ces quantites satisfassent aux conditions (2), et en donnant a r les valeurs 1, 2, 3,...n; ^ est le determinant forme pareillement en donnant a r, s les valeurs 2, 3, . . . n ; H 2 s obtient en donnant a r, s les valeurs 1, 3, ...n, et ainsi de suite; cela est aise de voir si Ton range les quantites XT ., en forme de carrd. 69] SUB LES DETERMINANTS GAUCHES. 411 Or les determinants gauches et symetriques (savoir les determinants dont les termes satisfont aux conditions (2)) se reduisent a zeVo pour un n impair, et pour un n pair aux carres des fonctions que M. Jacobi a traitdes dans son memoire " Ueber die Pfaff - sche Integrations-Methode " (t. II. [1827], p. 354, de ce journal) et dans le mdmoire "Theoria novi multiplicatoris sequationum differentialium " t. XXIX. [1845], p. 236, &c. En effet, on voit d abord (par ce que dit M. Jacobi) que le determinant s evanouit pour un n impair, et que pour un n pair il aura pour facteur la fonction dont il s agit ; mais je ne sais pas si Ton a deja remarque que 1 autre facteur se rdduit a la meme fonction. On obtient ces fonctions (dont je reprends ici la theorie), par les propri^tes gendrales d un determinant, ddfini de la maniere que voici : en exprimant par (12 ... n} urie fonction quelconque dans laquelle entrent les nombres symboliques 1, 2, ...n, et par + , le signe correspondant a une permutation quelconque de ces nombres, la fonction S(12...n) (ou 2 designe la somme de tons les termes qu on obtient en permutant ces nombres d une maniere quelconque) est ce qu on nomme Determinant. On pourrait encore generaliser cette definition en admettant plusieurs systemes de nombres 1, 2, ...n; 1 , 2 , ... n ; ... qui alors devraient etre permutes independamment les uns des autres ; on obtiendrait de cette maniere une infinite d autres fonctions, meritionne es (t. XXX. [1846] p. 7). Dans le cas des determinants ordinaires, auquel je ne m arreterai pas ici, on aura (12 ... n) = X a-1 Xp 2 XK.TI- Pour les cas des fonctions dont il s agit (les fonctions de M. Jacobi), on supposera n pair, et Ton ecrira (12 ... TO) = \L2\s.4... \n_i.n, ou X r . g sont des quantites quelconques qui satisfont aux equations (1). La fonction sera compose e d un nombre 1.2 ... n de termes; mais parmi eux il n y aura que 1.3 ...(n 1) termes differents qui se trouveront re pe tes 2* ra (1 . 2 ... ^n) fois, et qu on obtiendra en permutant cycliquement d abord les n 1 derniers nombres, puis les n 3 derniers nombres de chaque permutation, et ainsi de suite ; le signe etant toujours + . II pourra etre ddmontre, comme pour les determinants, que ces fonctions changent de signe en permutant deux quelconques des nombres symboliques, et qu elles s eVanouissent si deux de ces nombres deviennent identiques. De plus, en exprimant par [12...w] la fonction dont il s agit, la regie qui vient d etre e nonce e, donnera pour la formation de ces fonctions : [12...n]=\ 1 . 2 [34...TO] + X 1 . 3 [4...?i-2] ... +X litl [23 ... (n- 1)]. Cela pose", revenons aux determinants gauches et symetriques; et soit d abord n un nombre impair. Alors le determinant sera compost de plusieurs termes, chacun multiplie par le produit de 1 une des quantites \ 2 , X 1-3 , ... X 1>n par une des quantites X 2 j, X 3-1 , ... \ n .i- II est facile de voir que pour chaque terme, multiplie par Xj . a Xp . l (a. ^p /3), il existera un terme egal et de signe contraire, multiplie par \i.p\ a .i- Or Xj.aX/s.j = X aa Xj _$: done KO O vl 412 SUR LES DETERMINANTS GAUCHES. [69 ces deux termes se detruiront. Restent les termes multiplies par X 1-a A j: le coefficient d un terme de cette forme sera un determinant gauche de 1 ordre n 2, mais precise- ment de la forme du determinant meme de 1 ordre n dont il s agit. Or n dtant impair, n 2 le sera egalement : done tout determinant gauche et symetrique s evanouira, si cela a lieu pour les determinants pareils d un ordre inferieur de deux unites. Or pour n . = 3 on a e videmment X 1-2 A 2-3 X 3il + X2.iX 3 . 2 A 1-3 = 0, done: "Tout determinant gauche et symetrique d un ordre impair est zero." Soit maintenant n un nombre pair. Considerons le determinant gauche et syme tri- que plus general qu on obtient en donnant a r les valeurs a, 2, 3, ...n, et a s les valeurs (3, 2, 3, ... n. En developpant cette fonction comme on vient de le faire dans le cas d un n impair, il se presentera d abord un terme multiplie par X^. Mais ce terme sera un determinant gauche et symetrique de 1 ordre n 1 (savoir celui que Ton obtiendrait en donnant a r, s les valeurs 2, 3, ... n), et comme n 1 est impair, ce terme s evanouira. Puis le coefficient de X a . a \3 ./s (ou de X a . a X^.^) sera le determi nant gauche et symetrique qu on obtient en donnant a r les valeurs 2, 3, . . . n (a. excepte), et a s les valeurs 2, 3, ...n (@ excepte ). En admettant que ce determinant gauche se reduit a [( + 1) ... w2 ... (a - 1)] [(/S + 1) ... n2 ... ( - 1)], on aura X a . , [(a! + 1) . . . (a! - 1)] . X^ . [(& + 1) ... (& - 1)] pour le terme general, et la somme de tous ces termes se re duira a ou enfin a [23 ... n] [23 ... n]. Or si le theoreme en question a lieu pour ri 2, il aura lieu aussi pour n. Pour n . = 2 le determinant est X a-j3 X 2 .2 Xo -jS X a . 2) c est-a-dire Xa.a^/3.2 ou [a2] . [/32] : done la meme chose a toujours lieu et on obtient le thdoreme que voici : " Le determinant gauche et symetrique qu on obtient en donnant a r les valeurx a, 2, 3, ... n, et a s les valeurs /3, 2, 3, . . . n (ou n est pair), se reduit a et en particulier, en donnant a r, s les valeurs 1, 2, ...n, ce determinant se reduit a [12...71] 2 ." Appliquons ces theoremes a la reduction de 1 equation (3). En supposant pour plus de simplicity X r . r =l, on aura pour un n pair: O = [123 . . . ] + [34 . . . n] 2 + [56 . . . nj + . . . + 1 ; + [24 ... w] 2 09] SUR LES DETERMINANTS GAUCHES. 413 et pour un n impair: H = [23 ... ?i] 2 + [45 . . . /i] 2 + ... + 1. Particulierement pour it = 4 on obtient : n = [1234p + [12] 2 + [13] 2 + [14] 2 + [23p+ [34]*+ [42] 2 + = (^i.-A. 4 + A 1 . 3 X,. 4 +X 1 . 4 Ao. 3 ) 2 + X 1 . 2 2 + X 1 . 3 2 +X 1 . 4 :! +X 3 . 4 "-f X 4 ../ + et pour = 3 : resultats qui s accordent parfaitement avec ceux que j ai donnes dans mon premier memoire sur ce sujet. II serait possible de trouver pour les quantites A,. g (memoire citd) des expressions analogues a celles que nous venons de donner pour fi : mais elles seraient beaucoup plus compliquees. 414 [70 70. SUB, QUELQUES THEOKEMES DE LA GEOMETRIE DE POSITION. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn, xxxviu. (1848), pp. 97 104 : continued from t. xxxiv. p. 275, 55.] v - LE theoreme de Pascal applique au cas ou une conique se reduit a deux droites, donne lieu a un systeme de neuf points situes trois a trois dans neuf droites qui passent reciproquement trois a trois par les neuf points. Je representerai ces points par 1, 4, 7 ; 2, 5, 8 ; 3, 6, 9, et les droites par 135,426,789; 129,483,756; 186,459,723. On peut prendre pour la conique dont il s agit, deux quelconques de ces droites qui ne se rencontrent pas dans un des neuf points, ou ce qui revient au meme, deux quelconques des droites qui appartiennent a un des trois systemes dans lesquels les droites viennent d etre divisees ; alors la troisieme droite sera celle qui contient les points de rencontre des cotes opposes de 1 hexagone. Par exemple, en prenant pour conique les deux droites 135, 246, 1 hexagone sera 123456, et les points de rencontre des cdtds opposes, savoir de 12 et 45, de 23 et 56, et de 34 et 61, seront respectivement 9, 7, 8 : c est-a-dire des points situe s dans la droite 789 ; de maniere quo : THEOREME XV. "La figure forrnec par neuf points situe s 3 a 3 dans 9 droites peut etre conside ree, de neuf manieres differentes, comme resultante du the oreme de Pascal applique au cas ou la conique se reduit a deux droites." II y a, comme 1 a remarque M. Graves dans le memoire cite du Philosophical Magazine, une autre maniere assez singuliere d envisager la figure. Considerons pour cela les trois triangles 126, 489, 735, qui peuvent etre derives assez simplement de 70] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 415 1 arrangement 135, 426, 789. Le premier de ces triangles est circonscrit au second; car les cote s 26, 61, 12 contiennent respectivement les points 4, 8, 9 ; dgalement le second est circonscrit au troisieme ; et le troisieme au premier. On tire encore du meme arrangement 135, 426, 789 un autre systeme pareil de triangles ; savoir 189, 435, 726. Pareillement les arrangements 129, 483, 756 et 186, 459, 723 donnent lieu chacun a deux systemes analogues. Done : THEOREMS XVI. La figure formee par neuf points situes 3 a 3 dans 9 droites peut etre conside re e, de six manieres diffeVentes, comme composee de trois triangles circonscrits cycliquement 1 un a Fautre. Repre sentons maintenant par 1, 2, 3, 4, 5, 6, 7, 8, 9 les neuf points d inflexion d une courbe du troisieme ordre (six de ces points seront necessairement imaginaires). Ces points sont, comme on sait, situes 3 a 3 dans douze droites qui passent recipro- quement quatre a quatre par les neuf points, et qui peuvent etre repre sentees par 135, 426, 789 ; 129, 483, 756 ; 186, 459, 723 ; 147, 258, 369 ; de sorte que ce systeme est un cas particulier de celui que nous venons de considerer. En conside rant deux quelconques des droites qui ont un point en commun, par exemple 147, 186, et les deux autres droites qui passent par ce meme point 135, 129, on obtient par la deux quadrilateres 4876 et 3952 qui ont pour centre commun le point 7 et dont chacun est circonscrit a 1 autre. On aurait pu obtenir par ces memes droites 147, 186, 135, 129 des autres systemes pareils ; done THEOREME XVII. Huit points quelconques, parmi neuf points d inflexion d une courbe du troisieme ordre, peuvent etre considered de trois manieres differentes comme formant deux quadrilateres circonscrits 1 un a 1 autre. Les diagonales de ces six quadrilateres se reduisent a quatre droites qui passent par le neuvieme point d inflexion. VI. Sur les figures reciproques. Soient : &>, 77 : w, p : &> les coordonnees d un point. Les deux plans definis par les equations (a %+b rj + c p + d w) x^ (a % + b r) + c p + d &>) y z !t] + a"p + "&>) x >- = 0, et y z seront les plans re ciproques du point donne. II peut arriver que le plan rdciproque d un point quelconque passe par ce point meme ; ce qui implique aussi I identite des deux plans re ciproques. Ce cas particulier a 4t6 1 objet des recherches de M. Mobius. Je parlerai dans la suite des reciproques de cette espece en les appelant Re ciproques gauches. Mais geneValement, pour que les plans re ciproques d un point passent par ce point meme, il faut que le point soit situs sur une certaine surface du second ordre ; et 416 SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [70 alors tous les reciproques des points situ^s sur cette surface seront des plans tangents d une autre surface du second ordre. Ces deux surfaces pourront etre identiques ; ce qui implique aussi 1 identite des deux plans reciproques. Ce cas particulier constitue en effet la thdorie connue des Polaires reciproques. Je me propose ici d examiner la theorie du cas ge neral, ou les deux surfaces ne sont point identiques. On pourra demontrer que dans ce cas les deux surfaces ont necessairement en commun quatre droites: ces droites se rencontrent de maniere a former un quadrilatere gauche que je repre senterai par ABCD. II est eVident que les deux surfaces se touchent aux points A, B, C, D. En effet, le plan DAB est le plan tangent de 1 une et de 1 autre de ces surfaces au point A ; et de meme ABC, BCD et CDA sont respectivement les plans tangents aux points B, C et D. Les deux reciproques du point A se reduisent a ce meme plan DAB, et il en est dgalement pour les points B, C, D: il suit de la que les droites AC et BD sont reciproques 1 une a 1 autre, tandis que les droites AB, BC, CD, DA sont respectivement reciproques chacune a elle-meme. Les reciproques d un point quelconque passent par la droite qui est 1 intersection du plan polaire du point par rapport a la premiere surface, et de la reciproque gauche du point determine d une maniere qui sera explique e tout a 1 heure. Done les recipro ques d un point quelconque de la premiere surface passent par la droite qui est 1 intersection du plan tangent de la premiere surface au point dont il s agit, et de la reciproque gauche du meme point ; de maniere que si cette droite d intersection etait connue, les reciproques d un point de la premiere surface seraient les plans tangents de la seconde surface menes par cette droite d intersection ; ou, pour trouver les reciproques d un point de la premiere surface, on n a qu a chercher la reciproque gauche dorit je viens de parler. Dans ce systeme de reciproques gauches, les reciproques gauches des points A, B, C, D sont (comme dans le systeme des reciproques que nous considerons) les plans DAB, ABC, BCD et CAD, et de la les droites AC et DB sont reciproques gauches 1 une a 1 autre, tandis que les droites AB, BC, CD, DA sont reciproques gauches chacune a elle-meme. Mais cela ne suffit pas pour determiner le systeme des reciproques gauches. En etfet, le systeme des reciproques que nous considerons n est pas completement determine au moyen des deux surfaces ; il contient encore une quantite arbitraire (on peut facilement se satisfaire de cela). Or remarquons que la reciproque gauche d un plan quelconque, passant par la droite AB (ou par 1 une quelconque des droites AB, BC, CD, DA), est situe e dans cette meme droite. Considarons un plan doune quelconque pAB passant par cette droite AB\ la reciproque gauche de ce plan sera un point P de la droite AB, dont la position pourra etre prise a volonte. Au moyen de ce plan pAB et de sa re ciproque gauche P, sur la droite AB, on pourra facilement construire le systeme complete des reciproques gauches. Car soit q un point quelconque, et representons par Q la reciproque gauche du plan qAB (Q sera aussi un point de la droite AB) ; considerons les quatre plans DAB, CAB, pAB, qAB qui se rencontrent selon la droite AB, et qui ont respectivement pour reciproques gauches les points A, B, P, Q (situes sur cette meme droite AB) : le rapport anharmonique des quatre 70] SUB QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 417 plans sera egal au rapport anharmonique des quatre points ; ce qui suffit pour determiner le point Q, puisque les quatres plans et les trois autres points sont donnes, et la construction graphique pour determiner ce point Q est parfaitement connue. II est d ailleurs evident que la droite rnenee par un point donnd, de sorte qu elle rencontre des droites reciproques gauches 1 une a 1 autre, est situee dans la reciproque gauche de ce point. Done, en menant par le point q la droite qui rencontre les deux droites AC et BD, le plan passant par cette droite et par le point Q, est la reciproque gauche du point q qu il s agissait de trouver 1 . Egalement, on pourrait construire la reciproque gauche d un plan donnd Done enfin, pour trouver les re ciproques d un point de la premiere surface : (A) " Construisez le plan tangent a ce point de la premiere surface, et construisez de la maniere explique e ci-dessus, la reciproque gauche du point. Par la droite d inter- section de ces deux plans menez deux plans tangents a la seconde surface : ces deux plans seront les reciproques qu il s agissait de trouver." On pourra construire d une maniere analogue les re ciproques d un plan tangent de la premiere surface. En effet : (B) " En construisant les reciproques du point de contact avec la premiere surface du plan dont il s agit, les points de contact avec la seconde surface, de ces deux plans, seront les reciproques dont il s agissait." Egalement, pour trouver les re ciproques d un plan tangent donne de la seconde surface : (C) " Construisez le point de contact de ce plan avec la seconde surface, et con struisez la re ciproque gauche de ce meme plan: la droite mende par ces deux points rencontrera la premiere surface dans deux points qui seront les reciproques que Ton desirait." Et pour trouver les re ciproques d un point de la seconde surface : (D) "Construisez les reciproques du plan tangent passant par le point donne de la seconde surface : les plans tangents a la premiere surface passant par ces deux points, seront les reciproques qu il s agissait de trouver." En effet les theoremes (C, D) ne sont que des transformations des the oremes (A, B) au moyen de la theorie des polaires re ciproques. Enfin, pour trouver les re ciproques d un point quelconque, menez par ce point trois plans tangents ou a la premiere ou a la seconde surface, et construisez les reciproques de ces plans au moyen du the oreme (B ou C) : les deux plans menes par ces points reciproques, pris trois et trois ensemble, et combine s de maniere que 1 intersection des deux plans coincide avec 1 intersection de la polaire par rapport a la premiere surface ut de la re ciproque gauche du point donne, selon la rernarque ci-dessus, seront les reciproques cherche es ; et de la meme maniere pour les reciproques d un plan quelconque. 1 II y a un cas tres simple qui merite d etre consider^ ; savoir celui ou le point P coincide avec A. Hans ce cas Q coincide aussi avec A, et la reciproque gauche de q se reduit au plan qAC. De meme, si les points P, B coincident, la reciproque gauche de q se reduit au plan qBD. c - 53 418 SUR QUELQTJES THEOREMES DE LA GEOMETRIE DE POSITION. [70 Je n essaierai pas d enume rer ici le grand nombre de relations descriptives qui pourraient etre tirees des constructions qu on vient d expliquer. L on remarquera sans peine 1 analogie parfaite qui existe pour toute cette thdorie et la thdorie correspondante de la geometric a deux dimensions, telle que M. Pliicker 1 a expose e, "System der analytischen Geometric," [4, Berlin, 1835], pp. 78 83. On pourra aussi consulter sur ce point un memoire [61] que je viens de composer pour le Cambridge and Dublin Mathe matical Journal, et qui paraitra prochainement. La verification analytique de ces theoremes donne lieu a des developpements assez interessants. Pour obtenir 1 equation de la premiere des surfaces du second ordre, il suffit d ecrire f, 77, p, w au lieu de x, y, z, w, dans 1 dquation de 1 un ou de 1 autre des plans re ciproques. En supposant que x = Q, y 0, z 0, w = soient des plans conjugues par rapport a la surface, et en remarquant que chacune de ces coordonne es peut etre censee contenir un facteur constant, I dquation de la premiere surface peut etre ecrite sous la forme x 2 + y- + z* + w 2 = 0. On aura alors pour a, b, c, d ; a , b , c , d" ; a", b", c", d" ; a ", b" , c", d " un systeme de la forme I, h, g, a; h, I, f, b; g, f, I, c; a, b, c, I, et les equations des plans reciproques deviendront (%x + Tjy + pz + ww) ( x( . hrj+gp ao))^ =0; + y( ^. -fp-b* Op . )) ce qui prouve le theoreme e nonce ci-dessus ; savoir que les deux reciproques passent par la droite d intersection de la polaire et de la rdciproque gauche du point. On obtient sans difnculte, pour 1 equation de la seconde surface : (x hy + gz aw) 2 + (hx + y -fz bw) 2 + ( gx +fy + z cw) 2 + (ax + by + cz + w)- = 0, ou sous une autre forme plus commode : (x 2 + y 2 + z 2 + w 2 ) + (. hy + gz aw) 2 + (hx . fz - bw) 2 + ( gx +fy . - cw) 2 + (ax +by + cz. ) 2 = 0. Les coordonne es des points A, B, C, D seront determine es au moyen des expressions . hy + gz aw _ hx . fz bw gx +fy . cw _ ax + by + cz . x y z w ou, en introduisant la quantitd inde termine e s : sx hy + gz aw = 0, hx + sy fz bw 0, gx +fy +sz cw Q, ax + by +cz +sw = 0. 70] SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 419 En effet, on demontrera aise ment, non seulement que les points ddfinis par ces equations sont situes dans Fintersection des deux surfaces, mais aussi que les deux surfaces se touchent dans ces points- ci ; ce qui fait voir qu en combinant convenablement les quatre points, les deux surfaces doivent se couper (comme nous 1 avons deja avance), suivant les droites AB, BC, CD, DA. Je vais examiner encore de plus pres le systeme d e quations qui determinerit ces points. On en tire tout de suite, pour determiner s, 1 equation s 4 + jus 2 + @ 2 = 0, ou Ton a fait pour abrdger : ^ = a 2 + b 2 + c- +/ 3 +g 2 + h\ = a f+ bg + ch. Supposons encore et s 2 a+f=F, s 1 dquation qui determine s, pourra etre ecrite sous cette autre forme : AF+BG + CH=0. J ai trouve que les equations des droites AC et BD peuvent etre presentees sous les formes assez elegantes Cx . - Az - Gw = 0, -Bx + Ay . -Hw=0, Fx + Gy + Hz . = 0, , Hx . - Fz - Bw = 0, -Gx + Fy . - Cw = 0, Ax + By + Gz . =0, ou chacuu de ces systemes d equations n est equivalent qu a un systeme de deux equations. En entrechangeant les racines de liquation en s 2 , c est-a-dire en ecrivant 2 au lieu de s 2 , les deux systemes ne font que s entrechanger ; comme en effet cela doit etre. Enfin, les Equations des plans ACD, BAG, ou des plans ABD, BDC peuvent etre exprime es sous les formes PQ = ( . - Hy + Gz - A w)- + (Hx. - Fz - 5w) 2 + (- Gx + Fy . - Cw} 2 + (Ax RS=(.- Cy +Bz-Fw) 2 +(Cx.-A2-Gwy + (-Bx + Ay.-Htv) 2 equations desquelles on tire les relations identiques PQ+ RS=-s- O 2 - 4@ 2 ) (x 2 + y- + z 2 + w 2 ), B 2 PQ + s*RS = s 4 (p? - 4 2 ) x [(-% + gz - aw) 2 + (hx . . ,fz - bw) 2 + (- gx -fy . - civ) 2 + (ax + by + cz. ) 2 ], qui mettent en evidence ce que Ton savait deja, savoir, que les deux surfaces contiennent les droites AB, BC, CD, DA. 532 420 SUR QUELQTJES THEOREMES DE LA GEOMETRIE DE POSITION. [70 Mettons pour un moment, pour abre ger: . hrj + gp (id> = I , hg . fp b(o= m, a!~ + brj + cp . = p , de sorte que 1 equation de la reciproque gauche du point (f, 77, p, &>) devient Ix + tny + nz + pw = ; supposons de plus que, en combinant cette equation avec les equations des droites AC et ED respectivement, on obtient x : y : z : w = X : Y : Z : W et x : y : z : w = X t : Y / : Z / : W, respectivement. De la on tire X = . - Cm + Bn - Fp ,} X t = . - Hm + Gn - Ap, } Y = Cl -An-Gp,\ Y,= HI . + Fn-Bp, Z =-Bl+Am . -Hp, W= Fl+Gm + Hn Z / = Gl + Fm . Cp , W,= Al+Bm +Cn . ) et de la, identiquement : 2 -^)^ =07-^7, , - s 2 z, , ce qui correspond en effet au theoreme dnonce ci-dessus, savoir que la reciproque gauche d un point rencontre les droites AC, BD en deux points tels que la droite passant par ces deux points passe par le point meme. Les demonstrations des differents theoremes d analyse dont je me sers, n ont point de difficulte s. 71] 421 71. NOTE SUE LES FONCTIONS DU SECOND ORDRE. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn, xxxvm. (1848), pp. 105106.] SOIENT x, y, ... des variables dont le nombre est 2n ou 2?i + 1 ; representons par , TI, ... un nombre egal de fonctions lineaires des variables x, y, ... et soit ZHxy et V= H B J ai trouve que les deux fonctions U et V peuvent etre reduites a la forme u = \ n + fi , ou X, p, \ , p sont des coefficients constants, fi est line fonction du second ordre des n variables (fonctions lindaires de x, y,...}, et est une fonction de n ou de n + 1 variables (fonctions lineaires de x, ?/,..,); selon que le nombre des variables x, y, ... est 2n ou 2n + 1. Par exemple pour trois variables x, y, , on a et les coniques represente es par les Equations U=Q, V = Q ont entre elles un contact double. Cela se rapporte a la theorie des reciproques dans la geometric plane, telle que M. Pliicker 1 a prdsentde dans son " System der analytischen Geometrie " [4, Berlin, 1835]. 422 NOTE SUR LES FONCTIONS DU SECOND ORDRE. [71 De menie pour quatre variables x, y, z, w on a fj, zw, V = \ xy + /jfzw, et les surfaces representees par les equations 7=0, V = se coupent dans quatre droites, ou bien se touchent aux quatre sommets d un quadrilatere gauche. Cela se rapporte egalement a la theorie des re ciproques dans 1 espace : the orie dont j ai parle dans mon Me moire " Sur quelques the oremes de la ge ometrie de position," VI, [70]. II y a a remarquer que dans le cas ou les coefficients de x, y, ... dans les functions , 77, ... forment un systeme syme trique, c est-a-dire, ou % = Aac+Hy+... , ri = Hx + By + ... , on a, \ : fjL = \ : /JL , et de la V KU, K etant donne par l e<j[uation R= A, H,... H, B, ce qui est connu. Remarquons enfin que dans le cas general ou = ax + b y + ... , 77 = a a: + Hy + ..., la function V ne change pas de valeur en ecrivant au lieu de ces expressions = ax + a y + ... , 77 = bx + b y + . . . , propriete qui se rapporte aussi a la theorie des reciproques. 72] 423 72. [From the Philosophical Magazine, vol. xxxiv. (1849), pp. 527 529.] IT seems worth inquiring whether the distinction made use of in the theory of determinants, of the permutations of a series of things all of them different, into positive and negative permutations, can be made in the case of a series of things not all of them different. The ordinary rule is well known, viz. permutations are con sidered as positive or negative according as they are derived from the primitive arrangement by an even or an odd number of inversions (that is, interchanges of two things) ; and it is obvious that this rule fails when two or more of the series of things become identical, since in this case any given permutation can be derived indifferently by means of an even or an odd number of inversions. To state the rule in a different form, it will be convenient to enter into some preliminary explanations. Consider a series of n things, all of them different, and let abc ... be the primitive arrangement ; imagine a symbol such as (xyz) (u) (vw) . . . where #, y, &c., are the entire series of n things, and which symbol is to be considered as furnishing a rule by which a permutation is to be derived from the primitive arrangement abc ... as follows, viz. the (xyz) of the symbol denotes that the letters x, y, z in the primitive arrange ment abc ... are to be interchanged x into y, y into z t z into x. The (u) of the symbol denotes that the letter u in the primitive arrangement abc ... is to remain unaltered. The (vw) of the symbol denotes that the letters v, w in the primitive arrangement are to be interchanged v into w and w into v, and so on. It is easily seen that any permutation whatever can be derived (and derived in one manner only) from the primitive arrangement by means of a rule such as is furnished by the symbol in question 1 ; and moreover that the number of inversions requisite in order to obtain the permutation by means of the rule in question, is always the smallest number of 1 See on this subject Cauchy s " Memoire sur les Arrangemens &c.", Exercises d Analyse et de Physique Mathematique, t. in. [1844], p. 151. 424 NOTE ON THE THEORY OF PERMUTATIONS. [72 inversions by which the permutation can be derived. Let a, /3 ... be the number of letters in the components (xyz), (u) (vw), &c., X the number of these components. The number of inversions in question is evidently a - 1 + /3 - 1 + &c., or what comes to the same thing, this number is (n - X). It will be convenient to term this number \ the exponent of irregularity of the permutation, and then (n-\) may be termed the supplement of the exponent of irregularity. The rule in the case of a series of things, all of them different, may consequently be stated as follows: "a permutation is positive or negative according as the supplement of the exponent of irregularity is even or odd." Consider now a series of things, not all of them different, and suppose that this is derived from the system of the same number of things abc ... all of them originally different, by supposing for instance a = b = &c., /=# = &c. A given permuta tion of the system of things not all of them different, is of course derivable under the supposition in question from several different permutations of the series abc .... Considering the supplements of the exponents of irregularity of these last-mentioned several permutations, we may consider the given permutation as positive or negative according as the least of these numbers is even or odd. Hence we obtain the rule, "a permutation of a series of things not all of them different, is positive or negative according as the minimum supplement of irregularity of the permutation is even or odd, the system being considered as a particular case of a system of the same number of things all of them different, and the given permutation being successively considered as derived from the different permutations which upon this supposition reduce them selves to the given permutation." This only differs from the rule, "a permutation of a series of things, not all of them different, is positive or negative according as the minimum number of inversions by which it can be obtained is even or odd, the system being considered &c.," inasmuch as the former enunciation is based upon and indicates a direct method of determining the minimum number of inversions requisite in order to obtain a given permutation; but the latter is, in simple cases, of the easier application. As a very simple example, treated by the former rule, we may consider the permutation 1212 derived from the primitive arrangement 1122. Considering this primitive arrangement as a particular case of abed, there are four permutations which, on the suppositions a = b = l, c = d = 2, reduce themselves to 1212, viz. acbd, bead, adbc, bdac, which are obtained by means of the respective symbols (a)(bc)(d); (abc)(d): (a)(bdc); (abdc), the supplements of the exponents of irregularity being therefore 1, 2, 2, 3, or the permutation being negative; in fact it is obviously derivable by means of an inversion of the two mean terms. 73] 425 73. ABSTRACT OF A MEMOIR BY DR HESSE ON THE CONSTRUCTION OF THE SURFACE OF THE SECOND ORDER WHICH PASSES THROUGH NINE GIVEN POINTS. [From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 44 46.] THE construction to be presently given of the surface of the second order which passes through nine given points, is taken from a memoir by Dr Hesse (Crelle, t. xxiv. [1842], p. 36). It depends upon the following lemma, which is there demonstrated. LEMMA. The polar plane of a fixed point P with respect to any surface of the second order passing through seven given points, passes through a fixed point Q (which may be termed the harmonic pole of the point P with respect to the system of surfaces of the second order). PROBLEM. Given the seven points 1, 2, 3, 4, 5, 6, 7, and a point P, to construct the harmonic pole Q of the point P with respect to the system of surfaces of the second order passing through the seven points. The required point Q may be considered as the intersection of the polar planes of the point P with respect to any three hyperboloids, each of which passes through the seven given points; any such hyperboloid may be considered as determined by means of three of its generating lines. These considerations lead to the construction following. 1. Connecting the points 1 and 2, and also the points 3 and 4, by two straight lines, and determining the three lines, each of which passes through one of the points 5, 6, 7, and intersects both of the first-mentioned lines, the three lines so determined are generating lines of a hyperboloid passing through the seven points c 54 426 ABSTRACT OF A MEMOIR BY DR HESSE ON THE [73 Two other systems of generating lines (belonging to two new hyperboloids) are determined by the like construction, interchanging the points 1, 2, 3, 4. And by interchanging all the seven points we obtain 105 systems of generating lines (belonging to as many different hyperboloids, unless some of these hyperboloids are identical). 2. It remains to be shown how the polar plane of the point P with respect to one of the 105 hyperboloids may be constructed. Drawing through the point P three lines, each of which passes through two of the three given generating lines of the hyperboloid in question, the points of intersection of the lines so determined with the generating lines which they respectively intersect, are points of the hyperboloid. Hence, constructing upon each of the three lines in question the harmonic pole of the point P with respect to the two points of intersection, the plane passing through the three harmonic poles is the polar plane of P with respect to the hyperboloid. Hence, constructing the polar planes of P with respect to any three of the 105 hyperboloids, the point of intersection of these three polar planes is the required point Q. PROBLEM. To construct the polar plane of a point P with respect to the surface of the second order which passes through nine given points 1, 2, 3, 4, 5, 6, 7, 8, 9. Consider any seven of the nine points, e.g. the points 1, 2, 3, 4, 5, 6, 7, and construct the harmonic pole of the point P with respect to the system of surfaces of the second order passing through these seven points. By permuting the different points we obtain 36 different points Q, all of which lie in the same plane. This plane (which is of course determined by any three of the thirty-six points) is the required polar plane. Hence we obtain the solution of PROBLEM. To construct the surface of the second order which passes through nine given points 1, 2, 3, 4, 5, 6, 7, 8, 9. Assuming the point P arbitrarily, construct the polar plane of this point with respect to the surface of the second order passing through the nine points. Join the point P with any one of the nine points, e.g. the point 1, and on the line so formed determine the harmonic pole R of the point 1 with respect to the point P, and the point where the line PI is intersected by the polar plane. R is a point of the required surface of the second order, which surface is therefore determined by giving every possible position to the point P. This construction is the complete analogue of Pascal s theorem considered as a construction for describing the conic section which passes through Jive given points. And it would appear that the principles by means of which the construction is obtained ought to lead to the analogue of Pascal s theorem considered in its ordinary form, that is, as a relation between six points of a conic, or in other words to the solution of the problem to determine the relation between ten points of a surface of the second order; but this problem, one of the most interesting in the theory of surfaces of the second order, remains as yet unsolved. The problem last mentioned was proposed as a prize question by the Brussels Academy, which subsequently proposed the more general 73] CONSTRUCTION OF THE SURFACE OF THE SECOND ORDER &C. 427 question to determine the analogue of Pascal s theorem for surfaces of the second order. This of course admitted of being answered in a variety of different ways, according to the different ways of viewing the theorem of Pascal. Thus, M. Chasles, considering Pascal s theorem as a property of a conic intersected by the three sides of a triangle, discovered the following very elegant analogous theorem for surfaces of the second order. " The six edges of a tetrahedron may be considered as intersecting a surface of the second order in twelve points lying three and three upon four planes, each one of which contains three points lying on edges which pass through the same angle of the tetrahedron ; these planes meet the faces opposite to these angles in four straight lines which are generating lines (of the same species) of a certain hyperboloid." It is hardly necessary to remark that all the properties involved in the present memoir are such as to admit of being transformed by the theory of reciprocal polars. 542 428 74. ON THE SIMULTANEOUS TRANSFORMATION OF TWO HOMOGE NEOUS FUNCTIONS OF THE SECOND ORDER [From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 47 50.] THE theory of the simultaneous transformation by linear substitutions of two homogeneous functions of the second order has been developed by Jacobi in the memoir " De binis quibuslibet functionibus &c., Crelle, t. xn. [1834], p. 1 ; but the simplest method of treating the problem is the one derived from Mr Boole s Theory of Linear Transformations, combined with the remark in his " Notes on Linear Transformations," in the Cambridge Mathematical Journal, vol. IV. [1845], p. 167. As I shall have occasion to refer to the results of this theory in the second part of my paper " On the Attraction of Ellipsoids," in the present number of the Journal [75], I take this opportunity of developing the formula in question ; considering for greater convenience the case of three variables only; i Suppose that by a linear transformation, x = a. x l + /3 2/1 we have identically, ax- + by- + cz* + 2fyz + 2gxz + Zhxy - a&i* Aoc" + By* +- Cz 2 + 2Fyz + ZGzx + ZHccy = A,x Of course, whatever be the values of a, b, c, f, g, h, the same transformation gives ar 2 + lay 2 + cz* + Zfyz 74] TRANSFORMATION OF TWO HOMOGENEOUS FUNCTIONS, &C. 429 provided that we have a* = aa 2 + ba 2 + ca" 2 + 2foV + 2ga"a + 2haa , bj = a/3 2 + b 2 + c/3" 2 + 2f/3 /3" + 2g"/3 + 2h/3/3 , Cl = a 7 2 + b 7 2 + C7 //2 + 2f7 / 7 // + 2g 7 "7 + 21177 > fi = a/3 7 + b/3 7 + c/3"y" + f 09 y" + /3V) + g ("7 + 7") + h W + /3V), g! = aya + byV + cy"a" + f ( 7 a" + 7" a ) + g (y"a + 7 a") + h (ya + 7 a), v h x = aa/3 + bo /Q 7 + ca"/3" + f (a /S" + a"/3 ) + g (a /x /3 + a/3") + h (a/3 + a /3). Representing for a moment the equations between the pairs of functions of the second order by u= u lt U = U 1} v = Vn we have, whatever be the value of \, Hence, if *! ~i ^L \ ~i~ 9<i 3 A- \ + H,+\, X 71+^1 +gi, Vi + ^i+fi, Hence, since a, b, c, f, g, h, are arbitrary, LH XAj + fTj, Xflri+ G! Ct 8 7 = H ; then , 13 , i + 0i + gi =n. X + ^ + a, XA + tf + h, X, + G+ g + F, + f x XA + ^T + h, X& + .5+b, xy+j^+f + (7, + c, X# -|- G + g, \f + F+f, Xc +C+c + G, , X/j + ^ , Xd + (7 2 -n 2 \a + A, \h+H, H, G , \c which determine the relations which must subsist between the coefficients of the functions of the second order. We derive a, A, # ^ 6, / / y> / c Oi, AI, ^i ^i, &i, /i / and by comparing the coefficients of a, &c., if we write for shortness, &c., then ?, \f+F , X/+.P, Xc+(7 430 ON THE SIMULTANEOUS TRANSFORMATION OF TWO HOMOGENEOUS [74 a /3 7 each of which virtually contains three equations on account of the indeterminate quantity X. A somewhat more elegant form may be given to these equations ; thus the first of them is a, 13, \fh + H lt 7, X^ + ft , , , X Cl + ^ 1, , \f+F , Xc + C from which the form of the whole system is sufficiently obvious. The actual values of the coefficients a, /3, &c. can only be obtained in the particular case where y x = g 1 = Aj = F! G! H = 0. If we suppose besides (which is no additional loss of generality) that a 1 = & 1 = Ci=l, then the whole system of formulae becomes \a + A , H, + G ; l=II 2 j a, h, g , b, f > f> c or II 2 = tc~ l suppose ; and then 7 X + X) (A 1 + X) /3 2 L 1 + X)(5 1 + X) 7 2 --, L! + X) (^ + X) 7 /2 =-23, (B, + X) (C, + X) a a" + (^ + X) (A, + ?i + X) (1 + X) o"a + (C x + X) (^ + X) 0" J3 + (A, + X) (B, + X) 7 " 7 = - , ^ + X) (C x + X) a a + (C/j + X) (A^ + \)/3 /3 + (A, + X) (B, + X) 7 7 = X 74] FUNCTIONS OF THE SECOND ORDER. 431 where, writing down the expanded values of &, 23, G, jp, (ffi, f^, (\b+ B)(\c+C)-(\f+F)* = 8, (Xc+ (Xa + J. ) (X& + B) - (\h + NY (XA, + H) (\f+ F) - (\6 + 5) (X(7 + G) = G)- (\c + C) (\h + H) = By writing successively X=-^ 1; X = -17 1 , X = -C,, we see in the first place that J-i, B^, C l are the roots of the same cubic equation, and we obtain next the values of a 2 , /3 2 > 7 2 > & c in terms of these quantities A lt B^, C l9 and of the coefficients a, b, &c., A, B, &c. It is easy to see how the above formulas would have been modified if a lt b 1} d, instead of being equal to unity, had one or more of them been equal to unity with a negative sign. It is obvious that every step of the preceding process is equally applicable whatever be the number of variables. 432 [75 75. ON THE ATTRACTION OF AN ELLIPSOID. [From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 50 65.] PART I. ON LEGENDRE S SOLUTION OF THE PROBLEM OF THE ATTRACTION OF AN ELLIPSOID ON AN EXTERNAL POINT. I PROPOSE in the following paper to give an outline of Legendre s investigation of the attraction of an ellipsoid upon an exterior point, [" Mdmoire sur les Integrates Doubles," Paris, Mem. Acad. Sc. for 1788, published 17.91, pp. 454 486], one of the earliest and (notwithstanding its complexity) most elegant solutions of the problem. It will be convenient to begin by considering some of the geometrical properties of a system of cones made use of in the investigation. 1. The equation of the ellipsoid referred to axes parallel to the principal axes, and passing through the attracted point, may be written under the form I (x - of + m (y - b) 2 + n (z - c) 2 - k = 0, (where A/T, \/ - , \/ ~ are the semiaxes, and a, b, c are the coordinates of the attracted point referred to the principal axes). Or putting la? + mb z + nc 2 k = 8, this equation becomes la? + my 2 + nz 2 - 2 (lax + mby + ncz) + 8 = 0. The cones in question are those which have the same axes and directions of circular section as the cone having its vertex in the attracted point and circumscribed about the ellipsoid. The equation of the system of cones (containing the arbitrary para meter &>) is (la? + my* + nz 2 ) B (lax + mby + ncz} 2 + or (x 2 + y 2 + z-} = ; or as it may also be written, (or -t- 18 I 2 a 2 ) x 2 + (w 2 + mS ra 2 6 2 ) y 2 + (o> 2 + n8 n 2 c 2 ) z 2 2mnbcyz 2nlcazx 2lmabxy = 0. 75] ON THE ATTRACTION OF AN ELLIPSOID. 433 For <w = 0, the cone coincides with the circumscribed cone ; as <w increases, the aperture of the cone gradually diminishes, until for a certain value, = O, the cone reduces itself to a straight line (the normal of the confocal ellipsoid through the attracted point). It is easily seen that ft 2 is the positive root of the equation l*a? _m*b 2 tftf n 2 + is 2 + ^ = 7"5v a different form of which may be obtained by writing fi 2 = ^ , being then determined by means of the equation la? ml 2 nc 2 = are the semiaxes of the confocal ellipsoid through the attracted point. In the case where to remains indeterminate, it is obvious that the cone intersects the ellipsoid in the curve in which the ellipsoid is intersected by a certain hyperboloid of revolution of two sheets, having the attracted point for a focus, and the plane of contact of ^ the ellipsoid with the circumscribed cone (that is the polar plane of the attracted point) for the corresponding directrix plane : also the excentricity of the hyperboloid is - *J(l 2 a 2 + m 2 b 2 + n-c 2 ), which suffices for its complete determination. For o> = 0, the hyperboloid reduces itself to the plane of contact of the ellipsoid with the circum scribed cone, and for <w = fl, the hyperboloid and the ellipsoid have a double contact, viz. at the points where the ellipsoid is intersected by the normal to the confocal ellipsoid through the attracted point. If o> remains constant while k is supposed to vary, that is, if the ellipsoid vary in magnitude (the position and proportion of its axes remaining unaltered), the locus of the intersection of the cone and the ellipsoid is a surface of the fourth order defined by the equation (lx 2 + my 2 + nz- - lax - mby - ncz) 2 = o> 2 (a? + y+ z"), and consisting of an exterior and an interior sheet meeting at the attracted point, which is a conical point on the surface, viz. a point where the tangent plane is replaced by a tangent cone. The general form of this surface is easily seen from the figure, in which the ellipsoid has been replaced by a sphere, and the surface in question 55 434 ON THE ATTRACTION OF AN ELLIPSOID. [75 is that generated by the revolution of the curve round the line CM. The surface of the fourth order being once described for any particular value of co, the cone corre sponding to any one of the series of similar, similarly situated, and concentric ellipsoids is at once determined by means of the intersection of the ellipsoid in question with the surface of the fourth order. It is clear too that there is always one of these ellipsoids which has a double contact with the surface of the fourth order, viz. at the points where this ellipsoid is intersected by the normal to the confocal ellipsoid through the attracted point ; thus there is always an ellipsoid for which the cone corresponding to a given value of w reduces itself to a straight line. Consider the attracting ellipsoid, which for distinction may be termed the ellipsoid S, and the two cones C, C , which correspond to the values to, w dw of the variable parameter. Legendre shows that the attraction of the portion of the ellipsoid S included between the two cones C, C is independent of the quantity k, which determines the magnitude of the ellipsoid : that is, if there be any other ellipsoid T similarly situated and concentric to and with the ellipsoid 8, and two cones D, D , which for the ellipsoid T correspond to the same values o>, w dw of the variable parameter ; then the attraction of the portion of the ellipsoid S, included between the two cones C, C , is equal to the attraction of the portion of the ellipsoid T included between the two cones D and D . By taking for the ellipsoid T the ellipsoid for which the cone D reduces itself to a straight line, the aperture of the cone D is indefinitely small, and the attraction of the portion of the ellipsoid T included within the cone D is at once determined ; and thus the attraction of the portion of the ellipsoid S included between the cones C, C is obtained in a finite form. Hence the attraction of the portion of the ellipsoid 8 included between any two cones C t , G lt corresponding to the values &>,, w /t of the variable parameter, is expressed by means of a single integral, and by extending the integration from w = to &>=./(), the attraction of the whole ellipsoid is obtained in the form of a single integral readily reducible to that given by the ordinary solutions. It is clear too that the attraction of the portion of the ellipsoid S included between any two cones C / , C e/ , is equal to that of the portion of the ellipsoid T included between the corresponding cones D, and D u . Hence also, assuming for the ellipsoid T, that for which the cone D t/ reduces itself to a straight line, and supposing that the cones C, and D, coincide with the circumscribing cones, the attraction of the portion of the ellipsoid S exterior to the cone C lt is equal to the attraction of the entire ellipsoid T. More generally, the attraction of the portion of the ellipsoid 8 included between the cones C t and C /t is equal to the attraction of the shell included between the surfaces of the two ellipsoids, for which the cones D t and D tl respectively reduce themselves to straight lines. 2. Proceeding to the analytical solution, and resuming the equation of the ellipsoid Ix* + my z + nz- 2 (lax + mby + ncz) + 8 = 0, and that of the cone (la? + my z + nz 2 ) 8 (lax + mby + ncz}- + <o- (x 2 + y 2 + z-} = ; 75] ON THE ATTRACTION OF AN ELLIPSOID. 435 consider a radius vector on the conical surface such that the cosines of its incli nations to the axes are * P, Q, R and being functions of the parameter o>, and of another variable <, which determines the position of the radius vector upon the conical surface. Also let p be the length of the portion of the radius vector which lies within the ellipsoid ; then representing by dS the spherical angle corresponding to the variations of <w and 0, the attraction in the direction of the axis of x is given by the formula Also by a known formula 2 dR _ dR dQ\ Q fdR dP _ dR dP\ R fdJP dQ_dP dQ\] b da) d<f) da)J \d<j> da> da) d<j>) \d(f> da) &> d<f)J} and it is easy to obtain p = IP 2 + mQ 2 + nR> The quantities P, Q, R have now to be expressed as functions of &), </>, so that their values substituted for x, y, z, may satisfy identically the equation of the cone. This may be done by assuming Q = mb (or + nS) (la + jj. cos < J -- ~ nc sin = nc (&) 2 + mS) la + .cos <f>] H -- r where p = (co 2 + raS) (to 2 + nS) m~b (a) 2 + n8) n-c~ ((o- + mS), If 2 = m z b 2 (to 2 + nS) + n~c (&> 2 + m8), ( --_ . _ (&) 2 + te to 2 -I- wo &) 2 + no j a system of values which, in point of fact, depend upon the following geometrical con siderations : by treating a; as a constant in the equation of the cone, that is, in effect by considering the sections of the cone by planes parallel to that of yz, the equation of the cone becomes that of an ellipse ; transforming first to a set of axes through the centre and then to a set of conjugate axes, one of which passes through the point where the plane of the ellipse is intersected by the axis of x, then the equation takes the form -n + 15l| an( ^ ^ s satisfied by = J.cos0, r) = Bsm<f>, and ?, " being ./I J) X X of course linear functions of these values, the preceding expressions may be obtained. 552 436 ON THE ATTRACTION OF AN ELLIPSOID. [7 5 The substitution of the above values of P, Q, R (a somewhat tedious one which does not occur in the process actually made use of by Legendre) gives the very simple result, Pla)d(0d<f> and the formula for the attraction becomes P*&) 2 da d<f> IP 2 + mQ 2 + nR 2 which is of the form A = 2 I /&> 2 c?&>, where IP 2 + mQ* + nR 2 which last integral, taken between the limits <f> = and <f> = 2?r, and multiplied by 2ft) 3 dw, expresses the attraction of the portion of the ellipsoid included between two consecutive cones. The integration is evidently possible, but the actual performance of it is the great difficulty of Legendre s process. The result, as before mentioned, is independent of the quantity k, or, what comes to the same thing, of the quantity 8: assuming this property (an assumption which in fact resolves itself into the consideration of the ellipsoid for which the cone reduces itself to a straight line, as before explained), the integral is at once obtained by writing S A where A represents the positive root of the equation This gives p _ Va? (of + mA) (&) 2 + rcA) Q = Imab (w 2 + rcA), R = Inac (ft) 2 + mA), values independent of <, or the value of / is found by multiplying the quantity under the integral sign by 2?r : and hence we have A = 4<7rla J L 3 a 2 (&) 2 + ?iA) dco (&> 2 + wA) 2 (&) 2 + wA) 2 + w 3 6 2 (w 2 + wA) 2 (w 2 + /A) 2 + w 3 c 2 (&) 2 + ?iA) 2 (co where of course A is to be considered as a function of &>. By integrating from w = w t to to = w u , we have the attraction of the portion of the ellipsoid included between any two of the series of cones, and to obtain the attraction of the whole ellipsoid we must integrate from to = to &> = . /f J, where is determined as before by the equation la? mb* nc 2 _ / i if T r > i k + 1% k + raf k + nj; 75] ON THE ATTRACTION OF AN ELLIPSOID. 437 and it is obvious that for this value of co we have A = 8. The expression for the attraction is easily reduced to a known form by writing y = - ; this gives A = 4,-rrla I J I 3 a? ( (k + my) 2 (k + nyf + m s b 2 (k + nyf (k + ly) 2 + n 3 c 2 (k + ly) 2 (k + Also , ( l-a- m 2 b 2 n 2 c 2 \ to 2 = k( j 7 + -, h -, ; \K + ly k + my K + nyj whence = _ k [Pa 2 (k + my) 2 (k + nyf + m s fr (k + nyf (k + ly) 2 + n s c 2 (k + ly)* (k + my) 2 ] 2 (k + ly) 2 (k + my) 2 (k + nyf 7 x/7 . . (k + lyY (k + myY (k + and thus A where for the entire ellipsoid the integral is to be taken from y = to y = oo . A better known form is readily obtained by writing a; 2 = . + * , in which case the limits k + ly for the entire ellipsoid are x = 0, x = 1. It may be as well to indicate the first step of the reduction of the integral /, viz. the method of resolving the denominator into two factors. We have identically, (A - 8) (IP 2 + mQ 2 + nR 2 ) = o> 2 (P 2 + Q 2 + R 2 ) + A (ZP 2 + mQ 2 + nR 2 ) - (laP -f- mbQ + ncR) 2 , and the second side of this equation is resolvable into two factors independently of the particular values of P, Q, R. Representing this second side for a moment in the notation of a general quadratic function, or under the form ; .[ AP 2 + BQ 2 + CR 2 + 2FQR + 2GRP + 2HPQ, we have the required solution, IP 2 + mQ 2 + nR 2 = where, as usual, 33 = CA - G 2 , <& = AB - H 2 , and the roots must be so taken that V(- 23) V(- <B) = JF { J = (GH - AF)}. I have purposely restricted myself so far to the problem considered by Legendre : the general transformation, of which the preceding is a particular case, and also a simpler mode of effecting the integration, are given in the next part of this paper. 438 ON THE ATTRACTION OF AN ELLIPSOID. [7! PART II. ON A FORMULA FOR THE TRANSFORMATION OF CERTAIN MULTIPLE INTEGRALS. Consider the integral = lF(x, y,...}dxdy ... , where the number of variables oc, y,... is equal to n, and F (x, y,...} is a homogeneous function of the order p. Suppose that x, y, ... are connected by a homogeneous equation -fy- (x, ?/,...) = containing a variable parameter &> (so that o is a homogeneous function of the order zero in the variables x, y, ...). Then, writing r 2 = x 2 + if + . . . , x = ra, y = r/3, . . . the quantities a, /3, ... are connected by the equations and we may therefore consider them as functions of co and of (n 2) independent variables 0, <f>, &c. ; whence dxdy ... = r n ~ l V dr dw dO ... , where V= , ft, ... da. d(3 da) da) da. cZ3 F(x, y,...)=rF(aL, ...), V = [ r^"- 1 F (a, /3,...)V drda>d0 ..., or, integrating with respect to r, r I J which, taken between the proper limits, is a function of a, /3, . . . , equal / (a, /3, . . . ) suppose; this gives V = If (a, p, . . .) F (a, /3, . . .) V da> d0. . . , in which I shall assume that the limits of o> are constant. If, in order to get rid of the condition a 2 + /3 2 . . . = 1, we assume 75] ON THE ATTRACTION OF AN ELLIPSOID, the preceding expression for V becomes 439 in which D = Assume p , q , dp dq da) da) dp dq dO dd where the number of variables , 77, ... (functions in general of a>, 6, &c.) is n, and where the coefficients P, Q, &c. are supposed to be functions of eo only. We have &P - P df JL P *? 4. P" d % 4. d$~ M* dd^ dd^ and, substituting these values as well as those of jj, q, &c., but retaining the terms P , -58- . &c. in their original form, the determinant D resolves itself into the sum , aw do) of a series of products, dp dq i dw da " >* 77 , t, , P , Q , dr) d P" , Q" , Td dd Let "^ be the function to which -v/r (p, q, ...) is changed by the substitution of the above values of p, q, ... so that ^ is a homogeneous function of , 77, f , . . . and we have the relation ^F = 0. (It will be convenient to consider |, 77, , . . . as functions of to, #, &c., such as to satisfy identically this last equation.) We deduce 1 1, . 1 X . 17 , r , " F dr) # dd dd 1, = &c. = $ suppose, dd r , d0 440 ON THE ATTRACTION OF AN ELLIPSOID. [75 dW dW where for shortness X= ,,. , Y= -5, &c. The substitution of these values gives D = dp da) day " x, P , Q , Y, P , Q , z, P", Q" , where it will be remarked that the successive horizontal lines (after the first) of the determinant are the differential coefficients of M/ 1 , p, q, ... with respect to , with respect to ?;, &c. In general, if i/r denote any function of p, q, ... these quantities being themselves functions of o>, , ?,..., and , 77, ... containing ; also if \P be what ty becomes when for p, q, ... we substitute their values in at, , 77, ... ; then we have identically dm da) dai X , P , Q , y , p f , Q , Z : P", Q", In the present case however, writing for shortness ^ (p, q, ...) =-^r, this function -vjr contains w explicitly as well as implicitly through p, q, &c. The formula is still true if for -- we substitute , ^- . -* on the second side denoting a partial differential da) day da> da> coefficient taken only so far as <w is explicitly contained in yjr. And considering p, q,... as functions of o>, , rj,... (, 77,... themselves functions of to and of other variables which need not here be considered), T/T or W vanishes identically, and we have \^- = 0. Hence, in the last formula, we have to write -3^- instead of ~- , and we thus derive dco da) dm P, Q,.., = dp dq P , Q , da) da) : x, P , Q , Y, P , Q , z, P", Q", 1 This formula, or one equivalent to it, is given in Jacobi s memoir "De Determinautibus Functionalibus," Crelle, t. xxn. [1841] p. 31 J. 75] ON THE ATTRACTION OF AN ELLIPSOID. whence D becomes p ~ d^jr -L , (g , ... -j O, aco which is the value to be made use of in the equation 441 The principal use of the formula is where -fy is a homogeneous function of the second order of p, q, .... Thus, suppose also, for the sake of conformity to the usual notation in the theory of transformation of quadratic functions, writing a, a , ... /3, ft , ... instead of P, P , ... Q, Q , ... and putting after the differentiations =1, we have p= <z+a r] + a" +..., values which we may assume to give rise to the equation (where 77, ... are taken to be functions of 6, &c. such as to satisfy identically the equation 1 = rj 2 + (? +...) Hence, by a well-known property, if A, H,... H, B, K > we have - /{fc^l- vt * r so that, observing that in the present case X=l, and therefore we have C. 56 OF THE UNIVERSITT 442 ON THE ATTRACTION OF AN ELLIPSOID. [75 The remainder of the process of integration may in many cases be effected by the method made use of by Jacobi in the memoir "De binis quibuslibet functionibus &c." Crelle, t. xn. [1834] p. 1, viz. the coefficients a, a , &c., 0, &c., may in addition to the conditions which they are already supposed to satisfy, be so determined as to reduce any homogeneous function of p, q, r, ... entering into the integral to a form containing the squares only of the variables. This method is applied in the memoir in question to the integrals of n variables, analogous to those which give the attraction of an ellipsoid ; and that directly without effecting an integration with respect to the radius vector. I proceed to show how the preceding investigations lead to Legendre s integral, and how the method in question effects with the utmost simplicity the integration which Legendre accomplished by means of what Poisson has spoken of as inextricable calculations. Consider in particular the formula the number of variables being a,s before n, and (x, y,.--) h denoting a homogeneous function of the order h. The equation for the limits is assumed to be I (x - of + in (y - by + . . . = k. Assume ^(x, y , ...) = o> 2 (> 2 + 2/ 2 + ...) + 8(lz i + my 2 + ...)-(lax + mby + ...) 2 , (where 8, = la 2 + mb 2 ... k, is taken to be positive) ; or more simply, ^ (x, y . . .) = (o> 2 + IB I 2 a 2 ) x 2 + (&r + m8 7H.-6 2 ) y* + ... 2Zra abxy Here ^ = h + 2i 3. Also, putting for shortness lap + mbq + . . . = A, lp 2 + mq- +... = <, it is easy to obtain (p q \_ 1 p h+ti+n- s [ ( A J (PJ Qf-^ d\!r _ Also F ( p. g. ...)=- 5 =r - > 3 = 2<wp 2 , p- 5 - da) K = (o) 2 + 18) (<w 2 + mS) ...11 - IS ^- s - &c V \ to- + to to" + ??io / values which give 2 [ (-)*<"-!> cop 21 - 1 [(A + up) A+2i+n - 3 - (A - &>p) ;t+2f+n ~ 3 ] (p, q,...) h Sdca dO ... V = , ^~; 75] ON THE ATTRACTION OF AN ELLIPSOID. where it will be remembered that p, q,... are linear functions (with constant terms) of (n 1.) variables 77, , ..., these last mentioned quantities being themselves functions of (n - 2) variables 6, &c. such that 1 if g* . . . = identically. If besides we suppose 1 rf that <& = lp 2 + mq 2 + ... reduces itself to the form p - Q & c -> we have, by the formula of the paper " On the Simultaneous Transformation of two Homogeneous Equations of the Second Order," [74], PJV QJ" ^K+^-^K + mS-mX)...^-^^^^-^^^^^-...), which is true, whatever be the value of X. It seems difficult to proceed further with the general formula, and I shall suppose n = 3, i=0, h=l, (x, y...} h =x, or write f xdxdy dz the equation of the limits being I (x a) 2 + m(y b) 2 + n(z c) 2 = k. Here we may assume 77 = cos 0, =sin0, (values which give $ = 1). And we have r&) 2 d&) f (a + a! cos + a" sin 0) d6 J ** J 1 cos 2 sin 2 from = to = 2?r ; or, what comes to the same thing, F=8 * J j. _ cos 2 _ sin 2 P~~Q~ J2 from = to = |TT. Hence F=47T we have from the formulae of the paper before quoted, _QR (B-mP)(C-nP)-F 2 ~ ( P -Q)(P-R) B, C, F, being the coefficients of y 2 , z 2 , yz in i/r (x, y, z), viz. and consequently PQR \(B-mP)(C-nP)-F 2 }* ^~ (p _ Q) (P _ .R) 562 444 ON THE ATTRACTION OF AN ELLIPSOID. [75 Also from the equation -- , -- -- -- w + to tX eo 2 + mo ra\ ft) 2 + differentiating with respect to X, and writing \ = P, PV QJV R) (a) 2 + 18- IP) (ft) 2 + m8 - mP) (ft) 2 + n8 - nP) _ 1 f l s a 2 m 3 b z n 3 c 2 ) K [(ft) 2 +18- IP) 2 (ft) 2 + m8 - raP) 2 + (&) 2 + n8 - nP) 2 } or, as this may be written, PQR K (P-Q)(P-R) and from the values first written down, for E, C, F, we obtain (5 mP) (C nP) F 2 = (a) 2 +raS)(ft) 2 +wS)-m 2 & 2 (&) 2 +??S)-M 2 c 2 (ft^ -(a) 2 + m8 - mP) (ft) 2 + n8 - nP) - m 2 6 2 (w 2 + nS - nP) - n 2 c 2 (w 2 + m8 - mP) Pa? (ft) 2 + m8 - mP) (&> 2 + 118 - nP) ^ - 2 , ;<v_7P , the last reduction being effected by means of the equation _ 2 a 2 __ m 2 b 2 ?i 2 c 2 ~ to 2 + 18 - IP ~ ft) 2 + m8 - mP ~ a 2 + n8 - nP = Hence PQR {(B-mP)(C-nP)-F 2 }l K (P-Q)(P- R) la n 3 c 2 -IP) 2 T (ft) 2 +wS-mP) 2 T (ft) 2 +wS-nP) 2 Substituting this value, and multiplying out the fractions in the denominator, V = 4fjrla x to 2 (a) 2 + 18- IP)^ (&) 2 + m8 - ?nP) f (ft) 2 + nS - nPf da> 3 a 2 (ft) 2 + mS - mP) 2 (co 2 +n8- nP) 2 + m 3 b 2 (to 2 +n8- nP) 2 (a) 2 +l8- IP) 2 + n s c 2 (a 2 +18- IP) 2 (a> 2 +m8- mP) 2 the reduction of which integral has been already treated of in the former part of this present memoir. 76] 445 76. ON THE TRIPLE TANGENT PLANES OF SURFACES OF THE THIRD ORDER. [From the Cambridge and Dublin Mathematical Journal, vol. IV. (1849), pp. 118 132.] A SURFACE of the third order contains in general a certain number of straight lines. Any plane through one of these lines intersects the surface in the line and in a conic, that is in a curve or system of the third order having two double points. Such a plane is therefore a double tangent plane of the surface, the double points (or points where the line and conic intersect) being the points of contact. By properly determining the plane, the conic will reduce itself to a pair of straight lines. Here the plane intersects the surface in three straight lines, that is in a curve or system of the third order having three double points, and the plane is therefore a triple tangent plane, the three double points or points of intersection of the lines taken two and two together being the points of contact. The number of lines and triple tangent planes is determined by means of a theorem very easily demonstrated, viz. that through each line there may be drawn five (and only five) triple tangent planes. Thus, considering any triple tangent plane, through each of the three lines in this plane there may be drawn (in addition to the plane in question) four triple tangent planes: these twelve new planes give rise to twenty-four new lines upon the surface, making up with the former three lines, twenty-seven lines upon the surface. It is clear that there can be no lines upon the surface besides these twenty-seven ; for since the three lines upon the triple tangent plane are the complete intersection of this plane with the surface, every other line upon the surface must meet the triple tangent plane in a point upon one of the three lines, and must therefore lie in a plane passing through one of these lines, such plane (since it meets the surface in two lines and therefore in a third line) being obviously a triple tangent plane. Hence the whole number of lines upon the surface is twenty-seven ; and it immediately follows that the number of triple tangent planes is forty-five. The number of lines upon the surface may also be obtained by the following method, which has the advantage of not assuming 446 ON THE TRIPLE TANGENT PLANES OF [76 a priori the existence of a line upon the surface. Imagine the cone having for its vertex a given point riot upon the surface and circumscribed about the surface, every double tangent plane of the cone is also a double tangent plane of the surface, and therefore intersects the surface in a straight line (and a conic). And, conversely, if there be any line upon the surface, the plane through this line and the vertex of the cone will be a double tangent plane of the cone. Hence the number of double tangent planes of the cones is precisely that of the lines upon the surface. By the theorems in Mr [Dr] Salmon s paper " On the degree of a surface reciprocal to a given one," Journal, vol. II. [1847] p. 65, the cone is of the sixth order and has no double lines and six cuspidal lines : hence by the formula in Pliicker s " Theorie der algebraischen Curven," [1839] p. 211, stated so as to apply to cones instead of plane curves, viz. n being the order, x the number of double lines, y that of the cuspidal lines, u that of the double tangent planes, then u = $n (n - 2) O 2 - 9) - (2x + 3?/) (n 2 - n - 6) + 2x (x - 1) + Qxy + y(y- 1), the number of double tangent planes is twenty-seven, which is therefore also the number of lines upon the surface. Suppose the equation of one of the triple tangent planes to be w = 0, and let x = 0, y=0, be the equation of any two triple tangent planes intersecting the plane w = in two of the lines in which it meets the surface. Let 2 = be the equation of a triple tangent plane meeting w = in the remaining line in which it intersects the surface. The equation of the surface of the third order is in every case of the form wP + kxyz = Q, P being a function of the second order, but of the four different planes which the equation 2 = may be supposed to represent, one of them such that the function P resolves itself into the product of a pair of factors, and for the remaining three this resolution into factors does not take place. This will be obvious from the sequel: at present I shall suppose that the plane 2 = is of the latter class, or that P = represents a proper surface of the second order. Since x = 0, y = 0, 2 = 0, are treble tangent planes of the surface, each of these planes must be a tangent plane of the surface of the second order P = 0, and this will be the case if we assume P = a? + * + z* + w" 2 !\ / 1 m H + zw ( n + - m) \ n and considering x, y, z and to as each of them implicitly containing an arbitrary constant, this is the most general function which satisfies the conditions in question. We are thus led to the equation of the surface of the third order: U = w \a? + f + 2* + w- 4- l + j)+yw(m + ~} + zw (n + - H +kxu2 = 0, >/ V mj V nj) 76] SURFACES OF THE THIRD ORDER. 447 I have found that by expressing the parameter k in the particular form / 1 V 2 ( Imn ,- , \ Imn] ~7~ ~TT 2 (p Imn -j V Imn) or, as this equation may be more conveniently written, p2 _ ^32 I ^ k = -, r ; a = Imn + , , 8 = Imn ~-, -- r r , - - , 2 (p a.) Imn Imn ^ the equations of all the planes are expressible in a rational form. These equations are in fact the following : [I have added, here and in the table p. 450, the reference numbers 12 , 23 , &c. constituting a different notation for the lines and planes.] (w) w=0 ...................................................................... 12 (6) lx + m y + nz + w\l + (l-\(m-(n-\] = 0, ............ 23 Imn (a?) = ............................................................. 12.34.56 (y) 2/ = 0, ....................................................................... 42 ^ s=0, ....................................................................... 14 31 (f) lx + ^ +- +w = 0, ...41 m n (g) | +my + - +w=Q, 34 (h) ^ +$- +nz + w = 0, ..13.24.56 1 A somewhat more elegant form is obtained by writing p = 2q + a ; this gives 2 / 1 \ -. _. -. q \ hnnj / ( q + -. ) , &C. \ 448 ON THE TRIPLE TANGENT PLANES OF [76 (0 (g) (E) (x) (y) (z) (x) (y) (i) (m) (n) (1) (m) (5) 0) (m,) (n) ON THE TRIPLE TANGENT PLANES OF a: y + my -\-nz-\-w 24 , y ...14.23.56 m fy lx + mv + \-w 43 n x , l(p-a)+ 2mn w _ Q 12 35 46 v -\- - w 52 n (p - a) + 2lm 15 P + & 1 . 2 y (P 0.) -\ x \ w 12.36.45 62 <j/ i . _ fin o 1 2 z -f- w . 16 2n 1 x -\ y -f- fiz -f- w 56 21 1 IT - if -4- & -\- in (1 45 n (p a) n 1 2m x | mil z | w 4 6 2m 1 15.26.34 16 . 24 . 35 14.25.36 65 n (p a) n 1 21 ST ^~ 77 " 1 7? ^ 1 ?/J 1 / 1 21 I T 1 ?/ " 1 WJ (l m u m ( p a) n ( a) w _ r 1 -I- n^ 4- ID 2m m lx l(p ~ a) y \ l z \w-0 . ... 46 1 m(p a) , # 4- my ^-^ z + w 0. . 4 5 76] SURFACES OF THE THIRD ORDER. 449 (I) - m( ^~^-x + my + -z + w = 0, ....................................... 16.25.34 (m,) j x- n ^~ 0) y + nz + w = 0, ........................................ 15.24.36 (S,) lx + - y - l (% ~ ^ z + w = 0, ................................... 14.26.35 ra 2m V/> /Q/l + iim-a-Kl-m*-iiF- = ............ 51 (q) m-- + ^+[>/(p-a)-2m(l-^ - ?)]- = o, ............ 35 (r) mac + ly--^- + [lm (p - a) - 2n(l-l 2 - w 2 )] -^= = 0, 13.25.46 (p) _ +y + , + -( p - ) -l- -, . ....... 26 7 u 2 * p-ft (q) . __. + , | n p a I \_nl ^ m a _ m t a p a. \_ Im L n\ I 2 m 2 / J p /3 (p,) _. + + _| mB l._ i _ ( p_ a )_ .0, ..25 2 n m y+ _ J^i- - (p-)_ =0 ,.. .15.23.46 2 i w V w 2 Z 2 / VJ nl + 3 x y p a. FI/I 1 M/ \ 2 1 - + - z-bn>n -1 --- -a- p + /3 =0, ......... 53 m /n (7 /j/i (P,) - -^ + ^ + ^-[^(l-^ 2 -^)(p-)-2m W ]=0, ......... 61 t~) OL 4J1 ~ " l-n^-Z 2 )- a -2n~ = 0, ......... 36 = 0. ...13.26.45 In fact, representing the several functions on the left-hand side of these equations respectively by the letters placed opposite to them respectively, the function U is expressible in the sixteen forms following U = = wgg + krjzx, = whh + k^xy, = wee + kfrt, c. 57 450 ON THE TRIPLE TANGENT PLANES OF [76 = w ll, + kyzx, = wi~nm f + kzxy, = w nn / + kxyz, = w 1,1 + kyzx, = wmjm + kzxy, = w n,n + kxyz, = w pp, + kgyz, = w qq, + krjzx, = w rr, + Arxy, w PP/ + ^y z> = w qq, + &?7zx, = w rr, + &xy, (being the forms containing w, out of a complete system of one hundred and twenty different forms). The forty-five planes pass five and five through the twenty-seven lines in the following manner : (a,) (w, x, f x, x)...12 (60 (w, y, rj, y, y)... 2 (d) (w, 0, f, z, z)... 1 (a,) (|, f, ^, p, p,) ... 2 (6 2 ) (n, g, d, q, q/ ) ... 23, (c 2 ) (?, h, ^, r, r,)... 3 (a s ) (|, f , 0, P, P,) 1 (6 3 ) 0/> g> ^, q> q/) 3, (c 8 ) (f, h, B, r, r/; ... 13, (c 6 ) (z, 1 , m y , p, q,)... 15, (c,) (z, 1 , m y , p , q 7 ) ... 6 , where each line may be represented by the letter placed oppositive to the system of planes passing through it. The twenty-seven lines lie three and three upon the forty-five planes in the following manner : (w) oAd, (f) a 3 b 6 c 4 , (1) u 5 & 7 c 9 , (p) a 3 6 7 c 6 > (0) a 2 6 2 c 2 , (g) 6 3 c 8 a 4 , (m) b & c 7 a 9 , (q) 6 3 c 7 a 6 , (0) a 3 b 3 c 3 , (h) c 3 a 6 & 4 , (n) 050769, (r) (a 4 ) (x, > h , 1 , I ) ...34, (Of) (x, m,, n, q/. r) ...46, ft) (y, h, f , m, m,) ...24, (67) (y. ^ . 1 , r /. P) ... 5, (c*) (*> f , g > n , *,) ...14, M ( z , I , m, p/ q) ... 5 , (a s ) (ar, g> h , 1 , I ) ...56, (O (x, m,, n , q/. r) ...36, (&.) (y. h, f , m, "0 ... 4, ft) (y> 5 / , I , ? /^ P) ...26, (c,) (, f , g > n, *,) ... 4 , (*) (z, I , m, p/ q) ...16, (a.) (x, m, n , > q> o ...35, (09) (x, m , n /> q. ?,) ...45, (6.) (y, n, I > r , p,) ...25, ft) (y> n > / r , R) ... 6, 76j SURFACES OF THE THIRD ORDER. 451 (x) a.a.a,, (f) a 2 ^c 5 , (1) aAc 6 , (p) O^bgCg, (y) MA, (g) 6 2 c 4 a 5) (rn) & 4 c 8 a 6 , - / \. 7. / f*i \ ri f* n \M/ ^2^S 9 , \% ) GI 4 C$ y ( ll ) Cg w^ Og" , (H) c 4 a 8 6 6 , (r) C 2 a 8 6 9 , \ *T / Cl/jCvoCtg y 1 XI Ct J_tZ gtty y a> M, (P/) a Ac 7 , (77) Ma &a, (v) b,b K b 7 , \J / 1 / , to) 6^07, \q// ^2^6^75 I f\ f* f* f* \S/ 1 2 ^S ) (z) c lC6 c 7 , ( n/ ) c 5 aA, ft) c 2 aA, ( X y Cv j CrgCvg j /T \ A /. V 1 /^ iW% ^T) ^ Cl-uC \i// 3 9 8> (y) ^i^&9, (m,) b<c 6 a a , / n i h f* rt \ i / / ^3^9^8 ^ \^/ ^1 ^8 ^9 , (nj) C 4 a 6 6 8 , (r 7 ) C 3 a 9 6 8 . - The preceding method was the one that first occurred to me, and which appears to conduct most simply to the actual analytical expressions for the forty-five planes ; but it is worth noticing that the relations between the lines and planes might have been obtained almost without algebraical developments, if we had supposed that P, instead of representing a proper surface of the second order, had represented a pair of planes. This would have conducted at once to one of the one hundred and twenty forms U, e.g. U = w60 + kfy^. Or changing the notation so as to include k in one of the linear functions, U = ace bdf, and it is indeed obvious a priori, by merely reckoning the number of arbitrary constants, that any function of the third order can be put under this form. If we suppose a = fd) to be the equation of one of the triple tangent planes through the intersection of the planes a and b, the plane a fjJ) meets the surface in the same lines in which it meets the hyperboloid pce df=0, that is, the two lines in the plane are generating lines of different species, and consequently one of them meets the pair of lines cd and ef, and the other of them meets the pair of lines cf and de (where cd represents the line of intersection of the planes c = 0, d = 0, &c.). This suggests a notation for the lines in question, viz. each line may be repre sented by the three lines which it meets, or by the symbols ab.cd.ef and ab.cf.de. Or observing that /u, has three values, and that the same considerations apply mutatis mutandis to the planes through be and ca, the whole system of lines may be repre sented by the notation, ab, ad, af, cb, cd , cf, eb, ed , ef, (ab.cd.ef)!, (ab.cd.ef) 2 , (ab.cd.ef) 3 , (ad.cf.eb) lt (ad . c/". e6 ) 2 , (ad.cf.eb) 3 , (af.cb.ed),, (af. cb .ed\, (af.cb.ed) 3 , (ab.cf.ed\, (ab . cf. ed). 2) (ab . cf. ed ) 3 , (ad.cb. ef\, (ad. cb. e/) 2 , (ad.cb.ef) 3 , (af. cd . eb \ , (af.cd.eb) 2 , (af.cd.eb ) 3 , 572 452 ON THE TRIPLE TANGENT PLANES OF [76 where the last eighteen lines have been divided into two systems of nine each. The five planes through (ab.cd.ef\ may be considered as cutting the surface in ab) (ab.cf.ed\, cd\ (af.cd.eb\, ef ; (ad . cb . ef\, (ad.cf.eb\] (af.cb.ed) 3 , (ad . cf. eb) 3 ; (a/, cb .ed\, (which supposes however that the distinguishing suffixes 1, 2, 3, are added to the different planes according to a certain rule). And similarly for the lines in the planes through the other lines represented by symbols of the like form. The five planes through ab intersect the surface in the lines cb , eb, ad, of, (ab.cd.ef\, (ab.cf.ed) 1} (ab .cd.ef)z, (ab.cf. ed\ , (ab .cd.ef) 3 , (ab . cf. ed) 3 , and similarly for the planes through the other lines represented by symbols of a like form. Observing that , 77, , correspond to b, d, f, and w, 6, B, to a, c, e respectively, ab corresponds to the intersection of w and , i.e. to a^, &c. ; also (ab.cd.ef\, (ab . cd . ef). 2 , (ab.cd.ef) 3 correspond to three lines meeting a l , 6 2 > and c 3 , i.e. to a 5 , Oy, a 9 , &c. ; and the system of the twenty-seven lines as last written down corre sponds to the system, a lt b lt c 1; do , 6 2 , C. 2 , a 5 , a? , ag, b 5 , b 7 , b 9 , 64, The investigations last given are almost complete in themselves as the geometrical theory of the subject: there is however some difficulty in seeing d priori the nature of the correspondence between the planes which determines which are the planes which ought to be distinguished with the same one of the symbolic numbers, 1, 2, 3. 76] SURFACES OF THE THIRD ORDER. 453 There is great difficulty in conceiving the complete figure formed by the twenty- seven lines, indeed this can hardly I think be accomplished until a more perfect notation is discovered. In the mean time it is easy to find theorems which partially exhibit the properties of the system. For instance, any two lines, a^, b 2 , which do not meet are intersected by five other lines, a 2 , b 1} a 5 , a 7 , a g , (no two of which meet). Any four of these last-mentioned lines are intersected by the lines a 1} 6 2 and no other lines, but any three of them, e.g. a s , a?, a g , are intersected by the lines a 1} 6 2 , and by some third line (in the case in question the line c 3 ). Or generally any three lines, no two of which meet, are intersected by three other lines, no two of which meet. Again, the lines which do not meet any one of the lines a s , a,, a 9 , are a,, 3> b 3 , b l} d, c 2 : these lines form a hexagon, the pairs of the opposite sides of which, a 2 , b^, a 3 , c\ b 3 , c 2 , are met by the pairs a 1} 6 2 ; c 3 , a : and a ls 6 2 , respectively, viz. by pairs out of the system of three lines intersecting the system a 5 , a 7 , a 9 . And the lines a 5 , a 7 , a 9 may be considered as representing any three lines no two of which meet. Again, consider three lines in the same triple tangent plane, e.g. a 1} 6 X , c 1? and the hexahedron formed by any six triple tangent planes passing two arid two through these lines, e.g. the planes x, y, z, %, 77, . These planes contain (independently of the lines a lt b 1} Cj) the twelve lines a 2 , a s , a 4 , a s , 6 2 , b s , 6 4 , b s , c 2 , c 3 , c 4 , C 5 . Consider three contiguous faces of the hexahedron, e.g. x, y, z, the lines in these planes, viz. 4 , 6 5 , c 4 , a s , 6 4 , C 5 , form a hexagon the opposite sides of which intersect in a point, or in other words these six lines are generating lines of a hyperboloid. The same property holds for the systems x > *)> % > %> y> % > > 7 7 z - B U ^ for the system ^, 77, f, the six lines are 2 , b 2 , c 2 , and s> ^s, c 3 , which form two triangles, and similarly for the systems , y, z\ x, rj, z; and ao, y, ; so that the twelve lines form four hexagons (the opposite sides of which inter sect) circumscribed round four of the angles of the hexahedron, and four pairs of triangles about the opposite four angles of the hexahedron. The number of such theorems might be multiplied indefinitely, and the number of different combinations of lines or planes to which each theorem applies is also very considerable. Consider the four planes x, , x, x, and represent for a moment the equations of these planes by x + Aw 0, x + Bw = 0, x + Gw = 0, x + Dw = 0, so that A If 1 r A=0, B = j\m -- { n -- , C = j -- -- , , = k V in) V n) P+P P - 13 By the assistance of B ~ = Imn (?- /3) [ln (P ~ a} it is easy to obtain (A-C)(D-B) ^ p -p[l(p-a) + 2mn] [m (p - a) + 2nl] [n(p-a) (A -D)(B-C}~ p+~@ [run (p - a) + 21] [nl (p - a) + 2m] [Im (p-*) + 2n] 454 ON THE TRIPLE TANGENT PLANES OF [76 which remains unaltered for cyclical permutations of I, m, n, i.e. the anharmonic ratio of x, , x, x is the same as that of y, 77, y, y, or z, z, z ; there is of course no correspondence of x to y or to 77, &c., the correspondence is by the general pro perties of anharmonic ratios, a correspondence of the system on, , x, x, to any one of the systems (y, 17, y, y), or (77, y, y, y), or (y, y, y, 77), or (y, y, 77, y), indifferently. The theorems may be stated generally as follows : " Considering two lines in the same triple tangent plane, the remaining triple tangent planes through these two lines respectively are homologous systems." Suppose the surface of the third order intersected by an arbitrary plane. The curve of intersection is of course one of the third order, and the positions upon this curve of six of the points in which it is intersected may be arbitrarily assumed. Let these points be the points in which the plane is intersected by the lines a 1; b 1} a 6 , b 6 , c 6 , a 8 ; or as we may term them, the points a 1} b 1} a 6 , b 6 , c 6 , a^ 1 ) The point c x is of course the point in which the line a^ intersects the curve. The straight lines a 4 6 6 c 8 , b 4 c e a 8 , C 4 a 6 6 8 , and aj) s c 6 , 6 4 c 8 a 6 , C 4 a 8 6 6 , show that c 4 and 6 4 are the points in which a 8 b 6 , and a s c 6 inter sect the curve, and then b 8 and c 8 are determined as the intersections of a 6 c 4 , a 6 6 4 with the curve. The intersection of the lines 6 6 c 8 and b 8 c 6 (which is known to be a point upon the curve by the theorem, every curve of the third order passing through eight of the points of intersection of two curves of the third order passes through the ninth point of intersection) is the point 4 . The systems a 4 , 6 4 , c 4 ; a 6 , b e , c 6 ; a 8 , b s , c 8) determine the conjugate system a s , b 5> c 5 ; a 7 , b 7> c 7 ; a 9 , b 9 , c 9 ; by reason of the straight lines Oia t a 5 , bJ)J) 5 , CiC 4 c s ; a,ia 6 a 7 , bj) 6 b 7 , CiC 6 c 7 ; a^ag, bj) 8 b g , c&Cg, viz. a s is the point where 0^4 intersects the curve, &c. The relations of the systems ( 4 , 6 4 , c 4 ; a s , b s , c 5 ), (a 6 , b 6 , c fi ; ?. b 7 , c 7 ), (a s , b s , c 8 ; a g , b 9 , c 9 ) to the system a a , b 1} c 1 ; a. 2 , b. 2 , c 2 ; a s , b 3 , c 3 are precisely identical. It is only necessary to show how the points a 2 , b 2 , c 2 ; a 3) b 3 , c 3 of the latter system are determined by means of one of the former systems, suppose the system a e , b 6 . c 6 ; a?, b 7 , c 7 ; and to discover a compendious statement of the relation between the two systems. The points a 1} b lt Cj; a 2 , b. 2 , c 2 ; a 3 , b 3 , c 3 ; a 6 , b 6 , c 6 ; a?, b 7 , c 7 , are a system of fifteen points lying on the fifteen straight lines a-Jb^, a 2 6 2 c 2 , a 3 b 3 c 3 , a^as, bJ}J) 3 , C!C 2 c 3 , a-^a^j, bj) 6 b 7 , C^CT, a 3 b 7 c 6 , b 3 c 7 a 6 , C 3 a 7 b 6 , a 2 b 6 c 7 , b^a^, C 2 a 6 b 7 , viz. the nine points a 1} b 1} c^ a 2 , b 2 , c 2 ; a 3 , b 3 , c 3 are the points of intersection of the three lines a j&j d, a 2 b 2 c. 2 , a B b 3 c 3 with the three lines a^a^, 6i& 2 & 3 , 0^X3, and the remaining six points form a hexagon a 6 b 7 c 6 a 7 b e c 7 , of which the diagonals a 6 a 7 , b 6 b 7 , c 6 c 7 pass through the points i> ^i, c 1} respectively, the alternate sides a 6 b 7 , c 6 a 7 , and b 6 c 7 pass through the points c 2 , 6 2 , a 2 respectively, and the remaining alternate sides b 7 c 6 , a 7 b 6 , and c 7 a s pass through the three points a s , b 3 , c 3 respectively. The fifteen points of such a system do not necessarily lie upon a curve of the third order, as will presently be seen : in the actual case however where all the points lie upon a given curve of the third order, and the points a 1} b lt d; a e , b 6 , c 6 ; a 7 , b 7 , c 7 are known, a 2 , b 2 , c 2 ; a 3) b 3 , c 3 are the intersections of the curve with b 6 c 7 , c 6 a 7 , a s b 7 , b 7 c 6 , C 7 a 6 , a 7 b 6 respectively, and the fact 1 In general, the point in which any line upon the surface intersects the plane in question may be repre sented by the symbol of the line, and the line in which any triple tangent plane intersects the plane in question may be represented by the symbol of the triple tangent plane : thus, a x , b i , c are points in the line a-J)^ , or in the line w, &c. 76] SURFACES OF THE THIRD ORDER. 455 of the existence of the lines aj)^, a 2 b< 2 , a 3 b 3 c 3 , &&&*, bJ>J) 3 , G^CZ is an immediate consequence of the theorem quoted above with respect to curves of the third order a theorem from which the entire system of relations between the twenty-seven points on the curve might have been deduced cb priori. But returning to the system of fifteen points, suppose the lines oA^i, a 2 & 2 c 2 , a.Jb z c 3 , and OiC^s, bjb 2 b 3 , and also the point a 6 to be given arbitrarily. The point 07 lies on the line a^, suppose its position upon this line to be arbitrarily assumed (in which case, since the ten points a 1} a 2 , a 3> &!, 6 2 , 6 S , d, c 2 , c s , are sufficient to determine a curve of the third order, there is no curve of the third order through these points and the point a 7 ). If the points b, c 6 , 6 7 , c 7 can be so determined that the sides of the quadrilateral b 6 b 7 c 6 c 7 , viz. b 6 b 7 , b 7 c 6 , C 6 c 7 , c 7 6 6 pass through the points b 1} a 3 , c 1( a 2 respectively, while the angles b 6 , b 7 , c 6 , c 7 lie upon the lines a 7 c 3 , a e c 2 , a 7 b 2 and a 6 b 3 respectively, the required condi tions will be satisfied by the fifteen points in question ; and the solution of this problem is known. I have not ascertained whether in the case of an arbitrary position as above of the point a 7 , it is possible to determine a complete system of twenty- seven points lying three and three upon forty-five lines in the same manner as the twenty-seven points upon the curve of the third order; but it appears probable that this is the case, and to determine whether it be so or not, presents itself as an interesting problem for investigation. Suppose that the intersecting plane coincides with one of the triple tangent planes. Here we have a system of twenty-four points, lying eight and eight in three lines; the twenty-four points lie also three and three in thirty-two lines, which last-mentioned lines therefore pass four and four through the twenty-four points. If we represent by a, b, c, d, a , b , c , d and a, b, c, d, a , b , c , d , the eight points, and eight points 456 ON THE TRIPLE TANGENT PLANES OF SURFACES OF THE THIRD ORDER. [76 which lie upon two of the three lines (the order being determinate), the systems of four lines which intersect in the eight points of the third line are (aV, & b , c c , d d ); (a b, 6 a, cd , dc ) ; (a c, c a, bd , db ); (a d, d a, be , cb ): (aa , 6b , cc , (ab , la, , c d, d c), (ac , ca , d b, I d), (ad , do!, b c, c b), the principle of symmetry made use of in this notation (which however represents the actual symmetry of the system very imperfectly) being obviously entirely different from that of the case of an arbitrary intersecting plane. The transition case where the intersecting plane passes through one of the lines upon the surface (and is thus a double tangent plane) would be worth examining. It should be remarked that the preceding theory is very materially modified when the surface of the third order has one or more conical points; and in the case of a double line (for which the surface becomes a ruled surface) the theory entirely ceases to be applicable. I may mention in con clusion that the whole subject of. this memoir was developed in a correspondence with Mr Salmon, and in particular, that I am indebted to him for the determination of the number of lines upon the surface and for the investigations connected with the representation of the twenty-seven lines by means of the letters a, c, e, b, d, f, as developed above. 77] 457 77. ON THE ORDER OF CERTAIN SYSTEMS OF ALGEBRAICAL EQUATIONS. [From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 132 137.0)] SUPPOSE the variables as, y ... so connected that any one of the ratios x\y\z,... or, more generally, any determinate function of these ratios, depends on an equation of the /i th order. The variables x, y, z ... are said to form a system of the /t th order. In the case of two variables as, y, supposing that these are connected by an equation U=0 (U being a homogeneous function of the order p) the variables form a system of the ^ th order; and, conversely, whenever the variables form a system of the /* th order, they are connected by an equation of the above form. In the case of a greater number of variables, the question is one of much greater difficulty. Thus with three variables x, y, z; if //, be resolvable into the factors /* , p", then, supposing the variables to be connected by the equations 17=0, V=Q, U and V being homogeneous functions of the orders /, / , respectively, they will it is true form a system of the /i th order, but the converse proposition does not hold: for instance, if /* is a prime number, the only mode of forming a system of the ^ order would on the above principle be to assume ^ = ^ / = 1, that is to suppose the variables connected by an equation of the /* th order and a linear equation; but this is far from being the most general method of obtaining such a system. In fact, systems not belonging to the class in question may be obtained by the introduction of subsidiary 1 This memoir was intended to appear at the same time with Mr Salmon s "Note on a Eesult of Elimi nation," (Journal, vol. in. p. 169) with which it is very much connected. iUSTIVERSITYj 458 ON THE ORDER OF CERTAIN SYSTEMS OF ALGEBRAICAL EQUATIONS. [77 variables to be eliminated : the simplest example is the following : suppose a, b, a , V, a", V to be linear functions (without constant terms) of as, y, z, and write b?) =0, b r) =0, b" v =0; equations from which, by the elimination of rj, two relations may be obtained between the variables x, y, z. Suppose, however, from these three equations x, y, z are first eliminated: the ratio : 77 will evidently be determined by a cubic equation ; and assuming : 77 to be equal to one of the roots of this, any two of the three equations may be considered as implying the third ; and will likewise determine linearly the ratios x : y : z. Hence any deter minate function of these ratios depends on a cubic equation only, or the system is one of the third order. But the order of the system may be obtained by means of the equations resulting from the elimination of 77; and since this will explain the following more general example (in which the corresponding process is the only one which readily offers itself), it will be convenient to deduce the preceding result in this manner. Thus, performing the elimination, we have Z = (a 6"-a"6 ) = 0, L = (a"b - ab") = 0, L" = (aV - a b) = 0. Here the equations L = Q, L = 0, L" = 0, are each of them of the second order, and any two of them may be considered as implying the third. For we have identically, aL + a L + a"L" = 0,(i] so that Z = 0, Z =0, gives a"L"=0, or L" = 0. Nevertheless the system is imperfectly represented by means of two equations only. For instance, Z = 0, L = do, of them selves, represent a system which is really of the fourth order. In fact, these equations are satisfied by a" = 0, Z" = 0, (which is to be considered as forming a system of the first order), but these values do not satisfy the remaining equation L" = 0. In other words, the equations L = 0, L = contain an extraneous system of the first order, and which is seen to be extraneous by means of the last equation L" : the system required is the system of the third order which is common to the three equations L = Z = 0, L" = 0. Suppose, more generally, that x, y, z are connected by p + 1 equations, involving p variables f , 77, . . . , 1 Also l)L + b L + b"L" = 0: but since by the elimination of L", taking into account the actual values of L. and L , we obtain an identical equation, these two relations may be considered as equivalent to a single one. 77] ON THE ORDER OF CERTAIN SYSTEMS OF ALGEBRAICAL EQUATIONS. 459 or what comes to the same by the equations (equivalent to two independent relations) a, a , a", ... b, V, V, ... = 0; (where the number of horizontal rows is p). Consider x, y, z, as connected by the two equations a, ... 6, ... = 0, b, ... -0: these form a system of the order p 2 , but they involve the extraneous system a, b, =0. Suppose $(p) is the order of the system in question, then the order of this last system is $(p-l) and hence <j>(p) = p 2 - (f> (p - 1) : observing that 0(2) = 3, this gives directly <f>(p) = %p(p + 1). Hence the order of the system is Suppose x, y, z, connected by equations of the form U = 0, V=0, W=Q; V, V, W being linear in x, y, z, and homogeneous functions of the orders m, n, p respectively in , 77. By eliminating x, y, z, the ratio : 77 will be determined by an equation of the order m + n + p ; and since when this is known the ratios x : y : z are linearly determinable, we have m + n + p for the order //, of the system. Thus, if m = n = p = 2, selecting the particular system C7? 2 = 0, dif = 0, cf + 2^77 + erf = 0, it is possible in this case to obtain two resulting equations of the orders two and three respectively, and which consequently constitute the system of the sixth order without containing any extraneous system. In fact, from the identical equation . (a| 2 + 26^7 + drf) = (ae - 4>bd + 3c 2 ) p, 582 460 ON THE ORDER OF CERTAIN SYSTEMS OF ALGEBRAICAL EQUATIONS. and (ce - d 2 ) (af + 2bn + crj 2 ) + (cd - be) (6f + 2cf / + drf) + (bd - c 2 ) (cf + 2d|77 + erf) = (ace - ad 2 - b*e -c s + 2bcd) f , we deduce [77 ( ae 4<bd + 3c 2 = , \ ace - ad 2 -b 2 e-c 3 + 2bcd = ; which form the system in question, and may for shortness be represented by 7=0, J=0. The three equations in , 77 may be considered as expressing that f drj 3 = 0, t- erf = 0, have a pair of equal roots in common ; in other words, that it is possible to satisfy identically (A + Brj) (af + 3&f77 + 3cr; 2 + dr) s ) + (A % + B y) (6f + 3cp77 + U%rf + erj s ) = 0. Equating to zero the separate terms of this equation, and eliminating A, B, A , B , we obtain a , 3c, d b , 3c, 3d, e a, 36, 3c, d , . b, 3c, 3d, e , . It is not at first sight obvious what connection these equations have with the two, / = 0, J = 0, but by actual expansion they reduce themselves to the following five, =0. 3[2(ce- d 2 )!- 3 [(be- cd)I-3dJ] = 0, [- (ae + 2bd - 3c 2 ) / + 9cJ] = , 3 [(ad - be) I- 3bJ] - , 3[2(oc- 6 2 ) 7 - 3oJ] = ; which are satisfied by 7 = 0, / = 0. By the theorem above given, the equations are to be considered as forming a system of the tenth order; the system must therefore be considered as composed of the system 7 = 0, J=Q, and of a system of the fourth order. The system of the fourth order may be written in the form 2(ac-6 2 ) : ad -be : 2bd-3c* : be - cd : 2(ce-d 2 ) : 3J 3c : d : e : 7: 77] ON THE ORDER OF CERTAIN SYSTEMS OF ALGEBRAICAL EQUATIONS. 461 but to justify this, it must be shown first that these equations reduce themselves to two independent equations ; and next that system is really one of the fourth order. We may remark in the first place, that if u, = a? + 46 3 77 + 6cfY is a perfect square, the coefficients will be proportional to those of ^ ~j 2 ~ ( j^j } C) Thus the conditions requisite in order that u may be a perfect square, are given by the system 2(ac-6 2 ) : (ad -be) : ae + 26c-3c 2 : be-cd : 2(ce-d>) = a : b : 3c : d : e, or these equations are equivalent to two independent equations only (this may be easily verified d posteriori) ; and by writing 3*7 in the form e (ac - 6 2 ) - 2d (ad -bc) + c (ae + 2bd - 3c 2 ) - 26 (be -cd) + a (ce - d 2 ), the remaining equations of the complete system (3) are immediately deduced ; thus the latter system contains only two independent equations. (The preceding reasoning shows that the system (3) expresses the conditions in order that the equations af + 36f 77 + 3c^ 2 + drf = 0, 6f + 3cp77 + 3^ 2 + erf = 0, may have a pair of unequal roots in common : we have already seen that the equa tions / = 0, J = represent the conditions in order that these two equations may have a pair of equal roots in common.) Finally, to verify d posteriori the fact of the system (3) being one only of the fourth order, me may, as Mr Salmon has done in the memoir above referred to, represent the system by the two equations a (ce -d 2 )-e (ac - 6 2 ) = 0, e (ad - be) - 26 (ce - d 2 ) = 0, that is, by ad 2 -e6 2 = 0, These equations contain the extraneous system (a = 0, 6 = 0) and the extraneous systems (6 = 0, d = 0) and (d = 0, e = 0), each of which last, as Mr Salmon has remarked from geometrical considerations, counts double, or the system is one of the 4 th order only. 1 More generally whatever be the order of u, if u contain a square factor, this square factor may easily cPucPu /d%\ 2 be shown to occur in , - - I , . d? drf \dZd7, J 462 [78, 78. NOTE ON THE MOTION OF ROTATION OF A SOLID OF REVOLUTION. [From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 268270.] USING the notation employed in my former papers on the subject of rotation (Cambridge Math. Journal, vol. in. pp. 224232, [6]; and Cambridge and Dublin Math. Journal, vol. I. pp. 167264, [37]), suppose B = A, then r is constant, equal to n suppose ; and writing (A-C)n " /, v\ . -T = V > rC =( l -n) A > also putting 6 = vt + 7, (where 7 is an arbitrary constant) the values of p, q, r are easily seen to be given by the equations (p = M sin#, j q = M cos 0, ^r = n, (where M is arbitrary). And consequently h = A {M 2 + n (n - v) }, k~ = A*{M* + (n-vY\. Also, since a 2 + b 2 + c 2 = & 2 , we may write a = k sin i cos j, b= k cos i cos j, c = k sin j ; k having the value above given, and the angles * , j being arbitrary. 78] NOTE ON THE MOTION OF ROTATION OF A SOLID OF REVOLUTION. 463 From the equations (12) and (15) in the second of the papers quoted, we deduce 2u = k {k + A (n - v) sin j + MA cos j cos (6 + i)}, ^> = k {n sin j + M cos j cos (6 + i)}, V = - vkM A cos j sin (6 + i), (values which verify as they should do the equation (19)). Hence, from the equation (27), 2dv 7 ^ 1 7/1 i writing -=- =dt = - dd, we have 2 tan- 1 = B + - \ d6 1 f v J h + kn sinj + kMcosj cos (6 + i) k~ ^ v J v k + A (n v) sinj + MA cos j cos (0 + i) This is easily integrated ; but the only case which appears likely to give a simple result is when the quantity under the integral sign is constant, or A (h + kn sin j) = k {k + A (n v) sin j}, or that is, A (n v) + k sinj = : whence A(n v) . AM . (n v) C n -l -- = A_ - . or L HJI. / ._ . J M Observing that \TT j is the inclination of the axis of z to the normal to the invariable plane, this equation shows that the supposition above is not any restriction upon the generality of the motion, but amounts only to supposing that the axis of z (which is a line fixed in space) is taken upon the surface of a certain right cone having for its axis the perpendicular to the invariable plane. Resuming the solution of the problem, we have n k k vA which may also be written under the form o T^t 2 i Ai ? . ftc tan l - r = o 1 + -T , k A ( where BI = B + -7 ) . And hence V vAJ H = k tan ^ ( $! + -j ) = k tan i/r, \ A/ where 464 NOTE ON THE MOTION OF ROTATION OF A SOLID OF REVOLUTION. Substituting these values, a = MA sin i, b = M. A cos i, c = A (n v), 2v = A 2 M 2 {1 + cos (0 + i)} = 2M 2 A 2 cos 2 %(0 + i) ; and substituting in the equations (14) the values of X, /*, v reduce themselves to 1 M. A cos T: {k tan ^r sin (9 i) A (n v) cos TT {A; tan ty cos ^ (^ i) + A (n v} sin % (0 i)}, v = tan kt [78 where, recapitulating, 6 = vt + <y, 2i/r = I may notice, in connexion with the problem of rotation, a memoir, " Specimen Inaugurale de motu gyratorio corporis rigidi &c.," by A. S. Rueb (Utrecht, 1834), which contains some very interesting developments of the ordinary solution of the problem, by means of the theory of elliptic functions. 79] 465 79. ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTI S PEOBLEM, AND ON ANOTHER ALGEBRAICAL SYSTEM. [From the Cambridge and Dublin Mathematical Journal, vol. iv. (1849), pp. 270 275.] CONSIDER the equations by 2 + cz 2 + 2fyz = 6 2 a (be - f 2 ) , cz 2 + ax 2 + 2gzx = 6 2 b (ca - g 2 ) , ax 2 + by 2 + 2hxy = 6 2 c (ab -h 2 ); or, as they may be more conveniently written, by 2 + cz 2 + 2fyz = 8 2 a&, cz 2 + ax 2 + 2gzx = 2 b3$, ax 2 + by 2 + 2hxy = 6 2 c(&. The second and third equations give (g 2 <& -h^x 2 - b 2 3$y 2 + c 2 (&z 2 + 2cg<&zx - 2bh3$xy = , hence {(g 2 - h 2 3$) x - Wd&y + cg<z} 2 - 23 OD (- bgy + chz) 2 = 0, and consequently = : dividing this by g\/<& + h\/3B, and writing down the system of equations to which the equation thus obtained belongs, x - V23 y + cV< z=0, x + (AV& -/V) y - x+ b^ y 59 466 ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTl S PEOBLEM, [79 Hence also z - ( W& + MB -g^}x = 0, - (- W& + &V33 +/V)y = ; these equations may be written - ft V + V(&23)} [{ffi + V(Wj y - {ft + ^(833)} *] = 0, ] = 0, ~ )} y] = ; where, as usual, $=gh-af, = hf-bg, 1=fg-ch, K = abc- of 2 - bg* - ch* + 2fgh ; (in fact the coefficient of y in the first equation is - eft) v^: + (a - 2 ) vis - (ffift - ajF) v}, = AV^ + W23 -/ve, as it should be, and similarly for the coefficients of the remaining terms). We have therefore y = {ft or, what comes to the same thing, Now [abc - -fgh - {as readily appears by \\Titing the first of these equations under the form a {< + VOaOD)} {ft + V(^23)} = { JF + V(230T)} {a^ -/jp and comparing the rational term and the coefficients of 79] AND ON ANOTHER ALGEBRAICAL SYSTEM. 467 Hence, observing the values of yz, zx, xy, we find = ~ {abc - = ~ {abc -fgh - Hence, forming the value of any one of the functions by 2 + cz 2 + Zfyz, cz 2 + ax 2 + 2gxy, ax* + by 2 + 2hxy, we obtain s = O 2 ; or we have It may be remarked that the equations b y 2 + c z 2 + 2f yz = L, c z 2 + a x 2 + 2g yz = M, in which the coefficients are supposed to be such that the functions M (a"x 2 + b"y 2 + 2h"xy) - N (c f z 2 + a x 2 + Zg yz), N(b y 2 + c z 2 +2f yz) - L (a"x 2 + b"y 2 + 2h"yz), L (c f z 2 + a x 2 + ^g yz) - M (b f + c z 2 + 2f yz), are each of them decomposable into linear factors, may always be reduced to a system of equations similar to those which have just been solved. Suppose abc and write */X, \fY, JZ instead of x, y, z. The equations to be solved become IY + cZ + 2rV(F ) = (6 + c) r, cZ +aX+ 2r<J(ZX ) = (c + a) r, (a + b) r, 592 468 ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTl s PROBLEM, [79 where r 2 = - , - , and the solution is a + b + c x =-{ a + b + c-r + VO 2 + a 3 ) - V(V 2 + 6 2 ) - VO 2 + c 2 )}, Ztt y =L{ a + j ) + c-r+ VO 2 + a 2 ) + VO 2 + b 2 ) - VO 2 + c 2 )}, z =L{ a + l> + c-r- V(r 2 + a 2 ) - VO 2 + & 2 ) + V(r 2 + c 2 )}, SEC V(yz) = i {r-a + V(> 2 + a 2 )}, V(zx) = i {r-&+V0 2 + & 2 )}, V(xy) = i {r-c+V(r 2 + c 2 )}, a system of formulae which contain the solution of the problem "In a given triangle to inscribe three circles such that each circle touches the remaining two circles and also two sides of the triangle." In fact, if r denote the radius of the inscribed circle, and a, b, c the distances of the angles of the triangle from the points where the sides are touched by the inscribed circle (quantities which it is well known satisfy the con dition r 2 = -- 7- --- ), also if x, y, z denote the radii of the required circles, there is no CL ~T~ ~T C difficulty whatever in obtaining for the determination of x, y, z, the above system of equations. The problem in question was first proposed and solved by an Italian geometer named Malfatti, and has been called after him Malfatti s problem. His solu tion, dated 1803, and published in the 10th volume of the Transactions of the Italian Academy of Sciences, appears to have consisted in showing that the values first found for the radii of the three circles satisfy the equations given above, without any indi cation of the process of obtaining the expressions for these radii. Further information as to the history of the problem may be found in the memoir "Das Malfattische Problem neu gelost von C. Adams," Winterthur, 1846. In connexion with the preceding investigations may be considered the problem of determining I and m from the equations B( I + 0) 2 - 2H( I + 6) m + (A + l)m 2 = 0, A(m + 0) 2 -2H(m + 6)l + (B + l)l 2 =0; which express that the function 6* U + (Ix + my) 2 , ( U = Aa? + ZHxy + By 2 ), has for one of its factors a factor of U + x 2 , and for the other of its factors a factor of U+y^. There is no difficulty in solving these equations; and if we write 79] AND ON ANOTHER ALGEBRAICAL SYSTEM. 469 the result is easily shown to be - AB. But the problem may be considered as the problem for two variables, analogous to that of determining the conic having a double contact with a given conic, and touching three conies each of them having a double contact with the given conic ; and in this point of view I was led to the following solution. If we assume Bl Hm = u, HI + Am = v, or, what is the same thing, Kl = Au + Hv, Km = Hu + Bv, then putting V = Au" + 2Huv + Bv 2 , the two equations become after some reduction Hence, writing KQ" 1 + ^ V = s 2 , we have s 2 (2 OTl 2 CT2 2 - K (A + 2H + B + 1) + 2 (K - H) s- (OTJCT., + K - H}* : rf, v=Ke + OT 2 5, V + K W + Ks 2 = ; and substituting these values of u, v in the last equation, A (K0 + OTl s) 2 + 2H (KB + nr lS ) (KB + 2 s) + B(K6 + *7 2 s) 2 + K-6* + Ks* = 0, or reducing, K*& (A+2H + B + l) + 2K6s {(A +H}^+(H + B) 2 } + s 2 (A^ + 2H^^. 2 + B^ + K) = ; whence s 2 [{(A + H) 1 + (H + B) ^ 2 } 2 - (A + 2H + B + s 2 {-K (m, - CT 2 ) 2 - (A^,- + ZH-v^ + B.?) -K(A + 2H + B+ 1)}, 470 ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTl s PROBLEM, &C. [79 and therefore KB (A + 2H+B + l) + s{(A+H) ^ + (H + B) v? 2 } = s (^^ 2 + K- H}, giving s = ^ 1 ^ 2 -(A + H)-a: 1 -(H + B)^ 2 + K-H But Kl = Au + Hv = (A + H) K6 + (A^ + H*) s, Km = (H+B}K0 + (Htx^ + Bvr 2 ] s, and substituting the above value of s, we obtain, after some simple reductions, I : m . 6 = ( < BT 1 5r 2 + K H ) (A + H sr 2 ) : ( - S7 1 nro -f- K H ) (B + H S7 2 ) a result which presents itself in a very different form from the one previously obtained if, however, the terms of this proportion be multiplied by the factor K^ 2 + AB- KH they become (as they ought to do) identical with those of the former proportion, and the identical equations to which this process gives rise are not without interest. 80] 471 80. SUE QUELQUES TRANSMUTATIONS DES LIGNES COUEBES. [From the Journal de Mathematiques Pures et Appliquees (Liouville), torn. XIV. (1849) pp. 4046.] JE suppose qu une courbe quelconque soit represented par une Equation homogene entre trois variables oc, y, z, et je me propose d examiner (en ne faisant attention qu aux droites et aux coniques) ce que signifient les transmutations I. = V^ 77 = Vy, W^ II. = x*, 77 = 7/ 2 , = z 2 , III. = -, rj = -, =-, x y z ces quantite s dtant prises dans chaque cas pour de nouvelles coordonne es. Les thdoremes auxquels on se trouve conduit comprennent les the oremes qu obtient M. Wil liam Roberts par sa methode ge nerale de transmutation en e crivant n = , n = 2. ( Voir sa Note sur ce sujet, t. XIIL [1848] de ce Journal, page 209). 1 I. Soit = v x, T] = \y, % = N z. Je supposerai ici et partout dans ce me moire que PQR soit le triangle forme par les droites x = 0, y = 0, z = 0. 1 M. William Eoberts a bien voulu me parler I et6 pass6 a Dublin, avant que cette Note eut paru, des thcoremes qui devaient s y trouver : cela me suggera, et je lui communiquai, peu de jours apres, la deuxieme et la troisieme de mes m^thodes de transmutation. On verra dans la suite que ce sont la premiere et la deuxieme de mes mcthodes qui correspondent aux cas n ^, n = 2, respectivement, mais que la troisieme est aussi tres e troitement liee avec le cas n=. 472 SUR QUELQUES TRANSMUTATIONS DES LIGNES COURSES. [80 Cela pose, en conside rant les nouvelles coordonnees , 77, , on voit tout de suite qu un systeme d dquations telles que | : 77 : =a : (3 : y correspond a un point ; qu une equation lineaire quelconque correspond a une conique qui touche les trois cotes du triangle PQR (ou, si Ton veut, inscrite a ce triangle) ; et qu une equation quelconque du second ordre af + IT? + C? + 2f&i + 2gtf+ 2/Myf = correspond a une courbe du quatrieme ordre qui touche les trois cote s du triangle PQR, chaque cote deux fois. Et reciproquement, de telles coniques et de telles courbes du quatrieme ordre peuvent toujours se reprdsenter par une Equation lineaire ou par une equation du second ordre entre les coordonnees , 77, . On de duit de la cette propriety gdnerale : " Tout thdoreme descriptif qui se rapporte a des points, a des droites, et a des coniques, conduit a un the oreme qui se rapporte d une maniere analogue a des points, a des coniques qui touchent chacune trois droites fixes, et a des courbes du quatrieme ordre qui touchent chacune ces trois droites fixes, deux fois chaque droite." En supposant que les coefficients g, h se rdduisent a zeVo, 1 equation est celle d une conique tangente aux deux droites PQ, PR, et la courbe du quatrieme ordre se rdduit a deux coniques egales et superposes 1 une a 1 autre. De la: " Tout theoreme descriptif qui se rapporte a des points, a des droites, et a une conique, conduit a un theoreme qui se rapporte d une maniere analogue a des points, a des coniques qui touchent chacune trois droites fixes, et a une conique qui touche deux de ces droites fixes." En particulier, en supposant que les deux droites fixes que la conique touche soient les deux droites tangentes a cette conique qui passent par un des foyers, et que la troisieme droite fixe soit a 1 infini : "Tout the oreme descriptif qui se rapporte a des points, a des droites, et a une conique, conduit a un the oreme qui se rapporte d une maniere analogue a des points, a des paraboles qui ont pour foyer commun un point fixe, et a une conique qui a aussi ce point fixe pour un de ses foyers." II. Soit a present ! = ?, 77 = f y %=z~. Un systeme d e quations telles que : 77 : =a : ft : 7 80] SUE QUELQUES TRANSMUTATIONS DES LIGNES COUHBES. 473 correspond a quatre points qui formeront ce que Ton peut nommer un systeme symetrique par rapport aux points P, Q, R, ou tout simplement un systeme syme trique. On voit sans peine que les quatre points d un meme systeme symetrique sont tels, que le quadrilatere forme par ces quatre points a pour points de concours des diagonales et des cote s opposes les points P, Q, R. Pour ne pas interrompre la suite des raisonne- ments, il convient d entrer dans quelques details relatifs a ce sujet. Nous nommerons courbe symetrique par rapport aux points P, Q, R, ou simplement courbe symetrique, toute courbe lieu d un systeme symetrique. Cela pose , il y a une infinite de coniques syme triques ; savoir, toute conique par rapport a laquelle les points P, Q, R sont des points conjugue s (cela veut dire par rapport a laquelle Fun quelconque de ces points a pour polaire la droite menee par les deux autres) est une conique syme trique. Remarquons qu une courbe symetrique du quatrieme ordre peut se reduire a un systeme de deux coniques telles, que les points de chaque systeme syme trique de la courbe soient partage s deux a deux sur les deux coniques. En effet, considdrons une conique telle, que, par rapport a cette conique, le point P ait pour polaire la droite QR. II est facile de construire une autre conique telle, que 1 ensemble des deux coniques soit une courbe syme trique. Pour cela, menons une transversale quelconque PMN qui rencontre la conique donnee aux points M, N : soient M , N les points de rencontre des droites QM, NR et des droites QN, MR; le lieu des deux points M , N (lesquels seront en ligne droite avec le point P) sera la conique dont il s agit. Nous dirons que les deux coniques sont des coniques supplementaires. En revenant a notre but actuel, une equation lineaire quelconque A% + Brj + (7=0 correspond a une conique symetrique. Une e quation du second ordre quelconque = correspond a une courbe symetrique du quatrieme ordre. Et reciproquement, toute conique syme trique, ou toute courbe syme trique du quatrieme ordre, peut se repre senter par une e quation line aire, ou du second ordre, avec les coordonnees , 77, De la cette propriete ge nerale : "Tout theoreme descriptif qui se rapporte a des points, a des droites, et a des coniques, conduit a un theoreme qui se rapporte d une maniere analogue a des systemes syme triques (par rapport a trois points fixes), a des coniques symetriques, et a des courbes symetriques du quatrieme ordre." Dans ce thdoreme on peut, si Ton veut, considerer 1 un des points d un systeme syme trique comme repre sentant le systeme, et substituer aux mots systemes symetriques (par rapport a trois points fixes), le mot points. En supposant que dans la courbe du quatrieme ordre on ait a la fois ac g- = 0, ab h" = 0, il est facile de voir que la courbe du quatrieme ordre se reduit a deux coniques supplementaires. c - GO 474 SUR QUELQUES TRANSMUTATIONS DES LIGNES COURBES. [80 En effet, ces conditions dtant remplies, en re tablissant les valeurs de , 77, on obtient une equation qui se divise en deux e quations, telles que 003 _|_ ty2 + ^2 _ y z _ 0^ 0^2 + ^2 + ^2 _|_ g^ _ Q j qui appartiennent, comme on le voit sans peine, a une paire de coniques supple men- taires. De la, en remarquant qu il est permis de faire abstraction de 1 une de ces coniques: "Tout thdoreme qui se rapporte a des points, a des droites, et a une conique, conduit a un theoreme qui se rapporte dune maniere analogue a des points, a des coniques symetriques (par rapport a trois points fixes), et a une conique telle, que, par rapport a cette conique, 1 un des trois points fixes a pour polaire la droite menee par les deux autres points fixes." Supposons en particulier que les deux points fixes dont nous venons de parler soient les points ou la droite a 1 infini est rencoritree par un cercle qui a pour centre le troisieme point fixe; ce troisieme point fixe sera le centre tant des coniques syme triques par rapport a ces trois points fixes, que de la conique par rapport a laquelle le troisieme point fixe a pour polaire la droite menee par les deux autres points fixes. De plus, les asymptotes de 1 une quelconque des coniques syme"triques formeront avec les droites imaginaires asymptotes du cercle un faisceau harmonique, et de la les asymptotes de la conique dont il s agit seront a angle droit, ou, autrement dit, cette conique sera une hyperbole equilatere. De la enfin: "Tout theoreme descriptif qui se rapporte a des points, a des droites, et a une conique, conduit a un theoreme qui se rapporte d une maniere analogue a des points, a des hyperboles e"quilateres et concentriques, et a une conique concentrique avec ies hyperboles." III. Soit enfin Cette supposition conduit a 1 une des methodes de transmutation donnees par M.^ Steiner parmi les observations generates qui forment la conclusion de son ouvrage intitule* : Systematische Entwickelung &c., [Berlin, 1832], methode qu obtient M. Steiner au moyen de la theorie de 1 hyperboloide gauche. Je la reproduis ici tant pour en faire voir la theorie analytique qu a cause de son analogic avec les methodes I. et II. II est evident d abord qu un systeme d equations telles que 7 correspond a un point ; de meme, une equation lineaire quelconque, telle que Af+Bg + CC-0, correspond a une conique qui passe par les trois points P, Q, R, et une equation quelconque du second ordre, telle que a? + b^- + c? + 2/7?+ 2g& + 2hfr = 0, 80] SUR QUELQUES TRANSMUTATIONS DES LIGNES COURBES. 475 correspond a une courbe du quatrieme ordre qui a les trois points P, Q, R pour points doubles. Reciproquement, de telles coniques et de telles courbes du quatrieme ordre peuvent toujours se repre senter par des equations line"aires et par des Equations du second ordre. De la : "Tout the oreme descriptif qui se rapporte a des points, a des droites, et a des coniques, conduit a un the oreme qui se rapporte d une maniere analogue & des points, a des coniques qui passent par trois points fixes, et a des courbes du quatrieme ordre qui ont ces trois points fixes pour points doubles." En supposant que Ton ait a la fois a = 0, 6 = 0, la courbe du quatrieme ordre se reduit a une conique qui passe par les deux points Q, R (ou, si Ton veut, au systeme forme de cette conique et des droites PQ, PR). De la : "Tout theoreme descriptif qui se rapporte a des points, a des droites, et a une conique, conduit a un the oreme qui se rapporte d une maniere analogue a des points, a des coniques qui passent par trois points fixes, et a une conique qui passe par deux de ces points fixes." On ne peut pas particulariser ce theoreme .de maniere a obtenir des theoremes inte ressants. Mais, en prenant les polaires reciproques des deux theoremes qui viennent d etre obtenus, on a ces proprietes nouvelles : "Tout the oreme descriptif qui se rapporte a des droites, a des points, et a des coniques, conduit a un theoreme qui se rapporte d une maniere analogue a des droites, a des coniques qui touchent chacune trois droites fixes, et a des courbes de la quatrieme classe qui touchent chacune ces memes droites fixes, deux fois chaque droite ;" et: "Tout the oreme qui se rapporte a des droites, a des points, et a une conique, conduit a un theoreme qui se rapporte d une maniere analogue a des droites, a des coniques qui touchent chacune trois droites fixes, et a une conique qui touche deux de ces droites fixes." De la, comme dans la me thode I. : "Tout theoreme qui se rapporte a des droites, a des points, et a une conique, conduit a un theoreme qui se rapporte d une maniere analogue a des droites, a des paraboles qui ont pour foyer commun un point fixe, et a une conique qui a ce meme point fixe pour un de ses foyers ; " theoreme que Ton doit comparer avec le troisieme theoreme que donne la me thode I. Pour completer cette theorie, nous aurions du ajouter les deux theoremes polaires reciproques du premier et du deuxieme the oreme que donne la methode I. ; mais cela se fait sans la moindre peine. Quant au premier et au deuxieme the oreme que donne la me thode II., les polaires re ciproques de ces deux theoremes ne conduisent a rien de nouveau. GO 2 476 81. ADDITION AU MEMOIRE SUE, QUELQUES TRANSMUTATIONS DES LIGNES COURBES. [From the Journal de Mathe matiques Pures et Appliquees (Liouville), torn. XV. (1850), pp. 351356: continued from t. xiv. p. 46, 80.] JE me propose de re sumer ici la thdorie des courbes du quatrieme ordre, auxquelles donne lieu la premiere de mes me thodes de transmutation applique e a une conique quelconque. L equation d une telle eourbe est de la forme ax + by + cz + Zf Jyz + 2g *Jzx + 2h ^ xy = 0, et nous avons deja vu que cette eourbe a pour tangentes doubles les trois droites a = 0, 2/ = 0, z=0 (droites que nous avons represented par QR, RP, PQ). Pour trouver les autres proprie te s de la eourbe, mettons 1 dquation sous la forme I T* fJIJ r puis, en ^crivant, pour plus de simplicite, J j, ^Tf ~L au ^ eu ^ e x > V> 2 cette gfi tij gti equation devient et de la, en mettant 81] ADDITION AU MEMOIRE SUR QUELQUES TRANSMUTATIONS &C. 477 1 dquation de la courbe prend cette forme tres simple w = 0, en se souvenant toujours que les quantite s as, y, z, w satisfont a liquation lindaire qui vient d etre donne e. Je repre senterai dans la suite cette Equation lineaire par OLX + /3y + yz + Sw = 0. II est e vident que la droite w = 0, de meme que les droites QR, EP, PQ, est tangente double de la courbe. De plus, ces quatre droites sont le systeme complet des tangentes doubles, car la courbe a, comme nous aliens le voir, trois points doubles: en effet, la forme rationnelle de 1 dquation est O 2 + f + z * + w 2 - 2yz - 2zx - 2xy - 2xw - 2yw - 2zw) 2 - 64<xyzw = 0, et, au moyen de I identite + y) 2 (z + w) 2 - IQxyzw = (x- y} 2 (z + w) 2 + (x + y) 2 (z - w) 2 - (x - y} 2 (z - w) cette equation rationnelle se transforme en [0 -y? ~(z - wJJ -*(x + y-z-w)[(x + y)(z-w} 2 - (z + w) (x-y) 2 ] = 0, laquelle fait voir que le point (x = y, z = w) est un point double; de la aussi les points (x = z, w = y), (x = w, y = z} sont des points doubles. Remarquons en passant qu en supposant que les coefficients a, 0, y, 8 restent indetermines, les droites x = y, x = z, x = w seront des droites quelconques par les points (a? = 0, y = 0), = 0, z = 0), (x = 0, w=0) respectivement, et ces droites une fois connues, les droites y = z, z = w, w= y seront de termine es, la premiere au moyen des points (y = 0, z = 0), (y=x, z = x} la deuxieme au moyen des points 0=0, w = 0), (z=x, w = x\ et la troisieme au moyen des points (w = 0, y = 0), (w = x, y = x) et les trois droites ainsi de termine es se couperont ne cessairement dans un meme point. Cela revient au theoreme suivant : " Les trois points doubles d une courbe du quatrieme ordre avec trois points doubles sont les centres dun quadrangle dont les cote s passent par les angles du quadrilatere forme par les tangentes doubles de la courbe." Cette proprie te des courbes du quatrieme ordre dont il s agit (je veux dire celle d avoir trois points doubles) aurait du faire partie du the oreme general donne aupara- vant pour cette premiere methode de transmutation. En supposant que la conique a transmuter passe par le point P, on aura a + 8 = 0, et^il suit de la que le point double (x = w, y = w), identique dans ce cas avec le point P, se change en point de rebroussement, et en meme temps que les droites PQ, PR ne sont plus des tangentes doubles proprement dites, mais se rdduisent a des OF THE UNIVERSITT 478 ADDITION AU MEMOIRE SUR QUELQUES [81 tangentes simples qui passent par le point de rebrousseraent. Ajoutorjs que la tangente au point de rebroussement est la droite x=w. En supposant que la conique a transmuter passe a la fois par les deux points P et Q, nous aurons a + S = 0, /3 + 8 = 0. Ici les deux points doubles (x w, y z\ (y = w, x z), identiques dans ce cas avec les points P, Q, deviennent des points de rebroussement, les droites QR, HP, PQ ne sont plus des tangentes doubles proprement dites, mais les droites RP, PQ sont les tangentes simples qui passent par les deux points de rebroussement respectivement, et la droite PQ est la droite menee par les deux points de rebroussement. Ajoutons que les tangentes aux deux points de rebroussement respectivement sont les droites x = w et y w. On sait qu uii cercle quelconque peut s envisager comme conique qui passe par deux points fixes, savoir [les points circulates a 1 infini, c est-a-dire] les points oil 1 infini, conside re comme droite, est rencontre par les deux droites imaginaires aux- quelles se reduit un cercle evanouissant quelconque. En nommant ces droites les axes imaginaires de leur point d intersection, prenons pour les droites PQ, PR les axes imaginaires d un point quelconque P, et pour la droite QR 1 infini. Cela etant, un cercle quelconque sera transmute" dans une courbe du quatrieme ordre ayant deux points de rebroussement aux points ou 1 infini est rencontre par les axes imagi naires du point P, ou, ce qui est la meme chose, d un point quelconque, et ayant de plus un point double. Et le point P, comme point d intersection de deux axes imaginaires tangents de la courbe, est un foyer de la courbe (voyez le Memoire de M. Plucker: Ueber solche Punkte die lei Curven hohern Ordnung als der zweiten den Brennpunkten der Kegelschnitte entsprechen, Journal de M. Crelle, t. X. [1833] pp. 84 91). Cela suffit pour faire voir que la courbe est un lin^on de Pascal ayant le point P pour le foyer qui n est pas le point double. En effet, prenant pour vrai le theoreme . "Les ovales de Descartes ont deux points de rebroussement aux points ou I infini est rencontre" par les axes imaginaires d un point quelconque 1 ," comme cela revient a huit conditions, et qu un ovale de Descartes peut etre determine de maniere a satisfaire a six conditions (ce qui fait en tout quatorze conditions, nombre des conditions qui determinent une courbe du quatrieme ordre), toute courbe du quatrieme ordre avec deux points de rebroussement, tels que nous venous de les mentionner, sera un ovale de Descartes, et si, de plus, la courbe du quatrieme ordre a un point double, elle se reduira en limacon (cas particulier, comme on sait, des ovales de Descartes). Done, en resume, tout cercle est transmute dans un limacon ayant un point fixe pour le foyer qui n est pas le point double, the oreme qui se rapporte a la methode de M. Roberts pour le cas n = ^. L on doit cependant remarquer que cette methode est due a M. Chasles. En effet, on trouve dans la Note citee de VApergu historique, non-seulement la proprie te des droites de se transmuter en des paraboles, mais aussi 1 M. Chasles a remarque en passant (Note XXI. de VAperqu hintorique [Bruxelles, 1839]) que les ovales de Descartes out deux points conjugues imaginaires a I infini, theoreme moins complet que celui que je viens d enoncer. Pour la demonstration du theoreme complet, voyez la Note a la suite de ce memoire. 81] TRANSMUTATIONS DES LIGNES COURSES. 479 celle des cercles de se transmuter en des ovales de Descartes (seulement M. Chasles parait ne pas avoir remarque que ces ovales etaient necessairement des limacons), et c est la lecture de cette Note qui m a appris cette thdorie de la transmutation des cercles. En supposant que la conique a transmuter passe par les trois points P, Q, R, nous aurons a + S = 0, /? + 8 = 0, 7 + S = ; les points doubles, identiques (dans ce cas) avec les points P, Q, R, deviennent des points de rebroussement, et les droites QR, RP, PQ, au lieu d etre des tangentes doubles, sont tout simplement les droites qui passent chacune par deux points de rebroussement. Ajoutons que les tangentes de la courbe, aux trois points de rebrousse ment respectivement, sont les droites ac w=0, y w = 0, z w 0. II y a encore un cas particulier a considerer, savoir celui ou la conique a transmuter est telle que, par rapport a cette conique, les points Q et, R sont situe s chacun dans la polaire de 1 autre ; on a alors f= 0, cas qui dchappe a 1 analyse ci-devant employee. On voit sans peine que les deux points doubles (y w, x z), (z = w, y = z} deviennent ici identiques, ce qui donne lieu a un point d osculation. La droite QR et la droite w ne sont plus des tangentes doubles proprement dites, mais ces droites deviennent 1 une et 1 autre identiques avec la tangente au point d osculation. Note sur les ovales de Descartes. De 1 equation de ces ovales, *J(x a) 2 + y 2 =m *a? + y 2 + n, on tire d abord (1 - m 2 ) (x 2 + y") - 2ax + a? - n 2 = puis (1 - w 2 ) 2 (x 2 + y 2 ) 2 - 4a (1 - m 2 ) x (x 2 + y 2 ) + 2 [a 2 (1 - m 2 ) - n 2 (1 + m 2 )] (x 2 + y 2 ) + ka?a? - 4a (a 2 - n 2 ) x + (a 2 - n 2 ) 2 = 0. Pour trouver la nature de la courbe a 1 infini, mettons x + yi = g, x yi = rj, i = V 1, et introduisons la quantite de maniere a rendre 1 equation homogene. Cela donne (1 - m 2 ) 2 fV - 2o (1 - 7/i 2 ) + 2 [a 2 (1 - m 2 ) - n 2 (1 + m 2 )] &? + a 2 (f + <r)) 2 ? - 2a (a 2 - n 2 ) (! + 77) + (a 2 - n 2 ) 2 ? = 0: ce qui fait voir, sans la moindre peine, qu il y a des points de rebroussement aux points ( = 0, =0), (?? = 0, =0), savoir, aux points oti 1 infini, consider^ comme droite, est rencontre par les droites x + yi = 0, x yi = 0, qui sont les axes imaginaires du point x = 0, y = 0. 480 ADDITION AU MEMOIRE SUE QUELQUES TRANSMUTATIONS &C. [81 Nous pouvons remarquer, en passant, que 1 dquation (1 - ra 2 ) (a? + y 2 ) - 2ox + (a 2 - n 2 ) = 2raw a 2 + y 2 , conduit, avec beaucoup de facility, a une autre proprie te , donne e par M. Chasles dans la Note deja cite e. En effet, en mettant , _ n 2 - a 2 ,_m 2 (?i 2 - 1) , _n ~a(m 2 -!) ~ a(m 2 -!) ~a cette Equation se transforme en (1 - m 2 ) (a? + y 2 ) - Za x + (a 2 - ri 2 ) = 2mW >J x 2 + y 2 , et par la on voit que liquation primitive peut se transformer en (a? - a ) 2 + y 2 = m *x 2 + y* + ri, c est-a-dire, il y a toujours un troisieme foyer de la courbe. II ne reste qu a demontrer que la transform^e (selon la me thode de M. Chasles) d un cercle est toujours un Iima9on. Soit, pour cela, r 2 - 2ar cos + 8 = liquation du cercle ; en mettant Vrar au lieu de r, et %d au lieu de 6, cette equation devient mr 2a Vmr cos ^0 + S = 0, ce qui donne (mr + S) 2 = 2a?mr (1 + cos 0), ou, en mettant r cos x et en rdduisant a une forme rationnelle, [m 2 (x 2 + f) + B 2 - 2a 2 mxJ - 4m 2 (8 - a 2 ) (a? + y 2 ) = 0, ce qui appartient evidemment a un ovale de Descartes. En mettant t/ = 0, 1 equation devient [mV + 2m (3 - 2a 2 ) x + S 4 ] (mx - 8) 2 = 0, c est-a-dire le point mx 8=0, y = Q est point double, ou la courbe est le limacon de Pascal. 82] 481 82. ON THE TEIADIC ARRANGEMENTS OF SEVEN AND FIFTEEN THINGS. [From the Philosophical Magazine, vol. xxxvu. (1850), pp. 50 53.] THERE is no difficulty in forming with seven letters, a, b, c, d, e, f, g, a system of seven triads containing every possible duad ; or, in other words, such that no two triads of the system contain the same duad. One such system, for instance, is abc, ade, afg, bdf, beg, cdg, cef\ and this is obviously one of six different systems obtained by permuting the letters a, b, c. We have therefore six different systems containing the triad abc ; and there being the same number of systems containing the triads abd, abe, abf and abg respectively, there are in all thirty-five different systems, each of them containing every possible duad. It is deserving of notice, that it is impossible to arrange the thirty-five triads formed with the seven letters into five systems, each of them possessing the property in question. In fact, if this could be done, the system just given might be taken for one of the systems of seven triads. With this system we might (of the systems of seven triads which contain the triad abd} combine either the system abd, acg, aef, bee, bfg, dcf, deg, or the system abd, acf, aeg, beg, bef, dee, dfg; (but any one of the other abd systems would be found to contain a triad in common with the given abc system, and therefore cannot be made use of: for instance, the system abd, acg, aef, bcf, beg, dee, dfg contains the triad beg in common with the given abc system): and whichever of the two proper abd systems we select to combine with the given abc system, it will be found that there is no abe system which does not contain some triad in common, either with the abc system or with the abd system. C. 61 482 ON THE TRIADIC ARRANGEMENTS OF SEVEN AND FIFTEEN THINGS. [82 The order of the letters in a triad has been thus far disregarded. There are some properties which depend upon considering the triads obtained by cyclical per mutations of the three letters as identical, but distinct from the triads obtained by a permutation of two letters, or inversion. Thus abc, bca, cab are to be considered as identical inter se, but distinct from the triads acb, cba, bac, which are also identical inter se. I write down the system (equivalent, as far as the mere combination of the letters is concerned, to the system at the commencement of this paper) ade, afg, bdf, bge, cdg, cef, cba, derived, it is to be observed, from a pair of triads, ade, afg, by a cyclical permutation of the e, f, g, and by successively changing the a into b and into c, the remaining triad of the system being the letters a, b, c taken in an inverse order. Let it be proposed to derive the system in the same manner from any other two triads of the system ; for instance, from the triads acb, ade. The process of derivation gives acb, ade, gcd, geb, fee, fbd, fga, ( J ) which is, in fact, the original system. But attempt to derive the system from the two triads ebg, efc, the process of derivation gives ebg, efc, dbf, dcg, abc, agf, ade, which is not the original system, inasmuch as the triads dbf, dcg, abc, agf are in versions of the triads bdf, cdg, cba, afg of the original system. The point to be attended to, however, is, that both triads of the pair dbf, dcg, or of the pair abc, agf, are inversions of the triads of the corresponding pair in the original system ; the pair is either reproduced (as the pair efc, dbf), or there is an inversion of both triads. Where there is no such inversion of the triads of a pair, the system may be said to be properly reproduced; and where there is inversion of the triads of one or more pairs, to be improperly reproduced. There is no difficulty in seeing that the system is properly reproduced from a pair of triads containing in common any one of the letters a, b, c or d, and improperly reproduced from pairs of triads containing in common any one of the letters d, e or f. It is owing to the reproduction, proper or improper, of the system from any pair of duads that it is possible to form a system of "octaves" analogous to the quaternions of Sir William Hamilton; the impossibility of a corresponding system of fifteen imaginary quantities arises from the circumstance of there being always, in whatever manner the system of triads is formed, an inversion of a single triad of some one or more pairs of triads containing a letter in common. When the system is considered as successively derived from different pairs, the system is not according to the previous definition reproduced either properly or improperly. A system of triads having the necessary properties with respect to the mere combination of the letters (viz. that afiy and aSe being any two 1 The order of the letters /, g is selected so as to reproduce the original system so far as the mere combination of the letters is concerned. 82] ON THE TRIADIC ARRANGEMENTS OF SEVEN AND FIFTEEN THINGS. 483 triads having a letter in common, there shall be triads such as /38, ye, and ijfie, 7778) may easily be found ; the system to be presently given of the triads of fifteen things would answer the purpose. And so would many other systems. Dropping the consideration of the order of the letters which form a triad, I pass to the case of a system of fifteen letters, a, b, c, d, e, f, g, h, i, j, k, I, m, n, o. It is possible in this case, not only to form systems of thirty-five triads containing every possible duad, but this can be done in such manner that the system of thirty- five triads can be arranged in seven systems of five triads, each of these systems containing the fifteen letters 1 . My solution is obtained by a process of derivation from the arrangements ab .jof. dg . eh and db.cd.ef.gh as follows; viz. the triads are iab jac kaf lad mag nae oah icf jfb kbc Ice inch ncd ocg idg jde kdh Igb mbd ngf ofd ieh jhg kge Ihf mfe nhb obe and a system formed with i, j, k, I, m, n o, which are then arranged in the form klo ino jmo Urn jln ijk kmn iab jac lad nae kaf mag oah ncd mdb kbc ocg mch Ice icf mef keg ieh jfb obe ofd jde jgh Ihf nfg khd idg nhb Ibg an arrangement, which, it may be remarked, contains eight different systems (such as have been considered in the former part of this paper) of seven letters such as i, j, k, I, m, n, o ; and seven of other seven letters, such as i, j, k, a, b, c, /( 2 ). The theory of the arrangement seems to be worth further investigation. Assuming that the four hundred and fifty-five triads of fifteen things can be arranged in thirteen systems of thirty-five triads, each system of thirty-five triads containing every possible duad, it seems natural to inquire whether the thirteen systems can be obtained from any one of them by cyclical permutations of thirteen letters. This is, I think, impossible. For let the cyclical permutation be of the letters a, b, c, d, e, f, g, h, i, j, k, I, m. Consider separately the triads which contain the letter n and the letter o ; neither of these systems of triads contains the letter, whatever it is, which forms a triad with n and o. Hence, omitting the letters n, o, we have two different sets, each of them of six duads, and composed of the same twelve letters. And each of these systems of duads ought, by the cyclical permutation 1 The problem was proposed by Mr Kirkman, and has, to my knowledge, excited some attention in the form "To make a school of fifteen young ladies walk together in threes every day for a week so that each two may walk together." It will be seen from the text that I am uncertain as to the existence of a solution to the further problem suggested by Mr Sylvester, " to make the school walk every week in the quarter so that each three may walk together." 2 [I have somewhat altered this sentence so as to express more clearly what appeared to be the meaning of it.] 612 484 ON THE TRIADIC ARRANGEMENTS OF SEVEN AND FIFTEEN THINGS. [82 in question, to produce the whole system of the seventy-eight duads of the thirteen letters. Hence arranging the duads of the thirteen letters in the form ab . be . cd . de . ef .fg . gh . hi . ij .jk . kl . Im . ma ac .bd.ce . df .eg .fh . gi . hj . ik ,jl . km . la . mb ad .be .cf.dg.eh .fi . gj . hk . il .jm .ka.lb. me ae .bf.cg.dh. ei .fj . gk . hi . im .ja .kb . Ic . md of . bg . ch . di . ej .fk . gl . hm . ia . jb . kc . Id . me ag . bh . ci . dj . ek .fl . gm . ha . ib .jc .kd.le . mf and consequently the duads of each set ought to be situated one duad in each line. Suppose the sets of duads are composed of the letters a, b, c, d, e, f, g, h, i, j, k, I, it does not appear that there is any set of six duads composed of these letters, and situated one duad in each line, other than the single set al, bk, cj, di, eh, fg; and the same being the case for any twelve letters out of the thirteen, the derivation of the thirteen systems of thirty-five triads by means of the cyclical permutations of thirteen letters is impossible. And there does not seem to be any obvious rule for the derivation of the thirteen systems from any one of them, or any prima facie reason for believing that the thirteen systems do really exist, it having been already shown that such systems do not exist in the case of seven things. 83] 485 83. ON CUKVES OF DOUBLE CUHVATUKE AND DEVELOPABLE SUEFACES. [From the Cambridge and Dublin Mathematical Journal, vol. V. (1850), pp. 18 22.] This is translated with some slight alterations from the Memoir in Liouville, t. x. (1845) pp. 245 250, 30. 486 [84 84. ON THE DEVELOPABLE SURFACES WHICH ARISE FROM TWO SURFACES OF THE SECOND ORDER. [From the Cambridge and Dublin Mathematical Journal, vol. V. (1850), pp. 46 57.] ANY two surfaces considered in relation to each other give rise to a curve of intersection, or, as I shall term it, an Intersect and a circumscribed Developable 1 or Envelope. The Intersect is of course the edge of regression of a certain Developable which may be termed the Intersect-Developable, the Envelope has an edge of regression which may be termed the Envelope-Curve. The order of the Intersect is the product of the orders of the two surfaces, the class of the Envelope is the product of the classes of the two surfaces. When neither the Intersect breaks up into curves of lower order, nor the Envelope into Developables of lower class, the two surfaces are said to form a proper system. In the case of two surfaces of the second order (and class) the Intersect is of the fourth order and the Envelope of the fourth class. Every proper system of two surfaces of the second order belongs to one of the following three classes: A. There is no contact between the surfaces; B. There is an ordinary contact; C. There is a singular contact. Or the three classes may be distinguished by reference to the conjugates (conjugate points or planes) of the system. A. The four conjugates are all distinct; B. Two conjugates coincide; C. Three conjugates coincide. To explain this it is necessary to remark that in the general case of two surfaces of the second order not in contact (that is for systems of the class A) there is a certain tetrahedron such that with respect to either of the surfaces (or more generally with respect to any surface of the second order passing through the Intersect of the system 1 The term Developable is used as a substantive, as the reciprocal to Curve, which means curve of double curvature. The same remark applies to the use of the term in the compound Intersect-Developable. For the signification of the term singular contact, employed lower down, see Mr Salmon s memoir On the Classification of Curves of Double Curvature, [same volume] p. 32. 84] ON THE DEVELOPABLE SURFACES &C. 487 or inscribed in the Envelope) the angles and faces of the tetrahedron are reciprocals of each other, each angle of its opposite face, and vice versa. The angles of the tetrahedron are termed the conjugate points of the system, and the faces of the tetrahedron are termed the conjugate planes of the system, and the term conjugates may be used to denote indifferently either the conjugate planes or the conjugate points. A conjugate plane and the conjugate point reciprocal to it are said to correspond to each other. Each conjugate point is evidently the point of intersection of the three conjugate planes to which it does not correspond, and in like manner each conjugate plane is the plane through the three conjugate points to which it does not correspond. In the case of a system belonging to the class B, two conjugate points coincide together in the point of contact forming what may be termed a double conjugate point, and in like manner two conjugate planes coincide in the plane of contact (that is the tangent plane through the point of contact) forming what may be termed a double conjugate plane. The remaining conjugate points and planes may be distinguished as single conjugate points and single conjugate planes. It is clear that the double conju gate plane passes through the three conjugate points, and that the double conjugate point is the point of intersection of the three conjugate planes: moreover each single conjugate plane passes through the single conjugate point to which it does not corre spond and the double conjugate point; and each single conjugate point lies on the line of intersection of the single conjugate plane to which it does not correspond and the double conjugate plane. In the case of a system belonging to the class ((7), three conjugate points coincide together in the point of contact forming what may be termed a triple conjugate point, and three conjugate planes coincide together in the plane of contact forming a triple conjugate plane. The remaining conjugate point and conjugate plane may be distin guished as the single conjugate point and single conjugate plane. The triple conjugate plane passes through the two conjugate points and the triple conjugate point lies on the line of intersection of the two conjugate planes; the single conjugate plane passes through the triple conjugate point and the single conjugate point lies on the triple conj ugate plane. Suppose now that it is required to find the Intersect-Developable of two surfaces of the second order. If the equations of the surfaces be T = 0, T = (T, T being homogeneous functions of the second order of the coordinates , 77, , &>), and x, y, z, w represent the coordinates of a point upon the required developable surface : if more over U, U are the same functions of x, y, z, w that T, T are of 77, and X, Y, Z, W; X , Y , Z , W denote the differential coefficients of U, U with respect to x, y, z, w; then it is easy to see that the equation of the Intersect-Developable is obtained by eliminating , 77, w between the equations T = 0, T = 0, Wa> = 0, 488 ON THE DEVELOPABLE SURFACES WHICH ARISE FROM [84 If, for shortness, we suppose F = YZ - T Z, L =XW -X W, =ZX -Z X, M = YW -Y W, S = XY -X Y, N = ZW -Z W, (values which give rise to the identical equation then, X, /JL, v, p denoting any indeterminate quantities, the two linear equations in , ?;, , 6) are identically satisfied by assuming 77 = _ N~\ . + ~L V + Gp , f = M\-Ifi . +Sp, a)=-F\-Gfj,-Hv . and, substituting these values in the equations T = 0, T = 0, we have two equations : A X 2 + Bp? + Cv> + ZFpv + 2Gv\ + 2H\[* + 2L\p + 2M pp + 2Nvp = 0, C v 2 + 2F fiv + 2G v\ + 2H \p + 2L \p + ZM pp + ZN vp = 0, which are of course such as to permit the four quantities A., JJL, v, p to be simul taneously eliminated. The coefficients of these equations are obviously of the fourth order in x, y, z, w. Suppose for a moment that these coefficients (instead of being such as to permit this simultaneous elimination of X, /j,, v, p) denoted any arbitrary quantities, and suppose that the indeterminates X, //., v, p were besides connected by two linear equations, aX + 6/z + cv + dp = 0, a X + ftp + c v + d p = 0; then, putting bc -b c=f, ad -a d = l , caf c a = g, bd b d = in, ab a b = h, cd c d n , (values which give rise to the identical equation If + mg + nh = 0), and effecting the elimination of X, //,, v, p between the four equations, we should obtain a final equation G = 0, in which Q is a homogeneous function of the second order in each of the systems of coefficients A, B, &c., and A , B , &c., and a homogeneous function of the fourth order (indeterminate to a certain extent in its form on account of the identical TWO SURFACES OF THE SECOND ORDER. 489 equation lf+mg + nh = Q) in the coefficients / g, h, I, m, n( l ). But re-establishing the actual values of the coefficients A, B, &c., A , B , &c. (by which means the function n becomes a function of the sixteenth order in x, y, z, w) the quantities /, g, h, I, m, n ought, it is clear, to disappear of themselves; and the way this happens is that the function Q resolves itself into the product of two factors M and ^, the latter of which is independent of /, g, h, I, m, n. The factor M is a function of the fourth order in these quantities, and it is also of the eighth order in the variables x, y, z, w: the factor "^ is consequently of the eighth order in x, y, z, w. And the result of the elimination being represented by the equation ^ = 0, the Intersect-Developable in the general case, or (what is the same thing) for systems of the class (A), is of the eighth order. In the case of a system of the class (B) the equation obtained as above contains as a factor the square, and in the case of a system of the class (C) the cube, of the linear function which equated to zero is the equation of the plane of contact. The Intersect-Developable of a system of the class (B) is therefore a Developable of the sixth order, and that of a system of the class (C) a Developable of the fifth order. The elimination is in every case most simply effected by supposing two of the quantities A,, /u,, v, p to vanish (e.g. v and p) : the equations between which the elimination has to be effected then are and the result may be presented under the equivalent forms (AR + A B - 2HHJ -4>(AB-H*) (A B - H">) = 0, and (AB f - A BJ + 4 (AH - A H) (BH - B H} = 0, the latter of which is the most convenient. These two forms still contain an extraneous factor of the eighth order in x, y, z, w, of which they can only be divested by sub stituting the actual values of A, B, H, A , B , H . A. Two surfaces forming a system belonging to this class may be represented by equations of the form a x 1 + b f + c z 1 + d w 2 = 0, aV + b y 2 + c z 2 + d w 2 = 0, 1 I believe the result of the elimination is n = 4(PI?-Q 2 ) = 0, where, if we write uA + u A = A, &c., the quantities P, Q, R are given by the equation (identical with respect to a, M ) a theorem connected with that given in the second part of my memoir On Linear Transformations (Journal, vol. i. p. 109) [see 14, p. 100]. I am not in possession of any verification a posteriori of what is subsequently stated as to the resolution into factors of the function Q and the forms of these factors. c. 62 ON THE DEVELOPABLE SURFACES WHICH ARISE FROM [84 where x = 0, y = 0, z = 0, and w = 0, are the equations of the four conjugate planes. There is no particular difficulty in performing the operations indicated by the general process given above ; and if we write, in order to abbreviate, be - b c =f, ad -ad = I, ca c a = g, bd b d = m, ab a b h, cd f c d = n , (values which satisfy the identical equation lf+mg + nh = 0), the result after all reductions is l-z* + m-^oc* + tfhWy 4 + l*fWw* + mty-yW + n 2fmg 2 l 2 x 4 z-w 2 + 2 (mg - nh) (nh - If} (If- mg) X 2 y 2 z 2 w 2 = 0, which is therefore the equation of the Intersect-Developable for this case. The discussion of the geometrical properties of the surface will be very much facilitated by presenting the equation under the following form, which is evidently one of a system of six different forms, \m (go? + nw 2 ) (hy 2 gz 2 + lw~) I (fy~ + nw 2 ) ( hx? 4 fz 2 + mw 2 }} 2 4fglmx 2 y 2 (hy 2 gz~ + lw 2 ) ( ha? +fz 2 + mw") = 0. B. Two surfaces forming a system belonging to this class may be represented by equations such as a x- + b y 2 + c z 2 + 2n zw = 0, aV + b f + c z 2 + Zrizw = 0, iu which x = 0, y=0 are the equations of the single conjugate planes, z=Q that of the double conjugate plane or plane of contact, w = Q that of an indeterminate plane through the two single conjugate points. If we write be b c =f, an an = p, ca c a = g, bn b n q , ab a b = h, en c n = r, (values which satisfy the identical equation pf+ qg + rli = 0), the result after all reduc tions is r*h?z K + 2pr 2 A (rh qg) z 4 ^ 2 Zqr 2 h (pf rh) z*y* +p 2 (rh - qg) 2 zW + q 2 (pf-rh) 2 z-y* + %pq (4r 2 A 2 -fgpq) z l &f + 4p*q (rh qg} za?w + 4<pq s zy*w 4<p 2 q 2 (qg pf) zx^yhu - + 8p s q 3 a; 2 y 2 w 2 + 4>p 2 qrh 2 a?y- + 4cpq 2 rh 2 x 2 y* = 0, 84] TWO SURFACES OF THE SECOND ORDER. 491 which is therefore the equation of the Intersect-Developable for systems of the case in question. The equation may also be presented under the form {q (pa? + rz 2 ) (hy- gz 2 + %pzw) p (qy 2 + rz 2 ) ( hx 2 +fz 2 + 2qzw)} 2 + 4p 2 q s x 2 y 2 (hy 2 - gz 2 + 2pzw) (- hx 2 +fz 2 + 2qzw) = 0, which it is to be remarked contains the extraneous factor z 2 . The following is also a form of the same equation, [r (qg pf) z 3 -fp 2 zx 2 + gq 2 zy 2 + 2pq (px 2 + qy 2 ) w} 2 4>pq (px 2 + qy 2 + rz 2 ) {r (hy 2 gz 2 ) (fz 2 hx 2 ) + 2pq (gx- fy 2 ) zio\ = 0. C. Two surfaces forming a system belonging to this class may be represented by equations of the form a x 2 + b y 2 + 2f yz + 2n zw = 0, a x 2 + b y 2 + 2f yz + 2rizw = 0, in which bri b n = Q, af a f=0. In these equations x = is the equation of a properly chosen plane passing through the two conjugate points, y=0 is the equation of the single conjugate plane, z that of the triple conjugate plane, arid w = is the equation of a properly chosen plane passing through the single conjugate point. Or without loss of generality, we may write a (x 2 - 2yz) + j3 (if - 2zw) = 0, a! (x 2 - 2yz) + & (y 2 - 2zw) = 0, where x, y, z and w have the same signification as before 1 . The result after all reduc tions is = 0, which may also be presented under the forms z (if - 2zw) 2 - 4<y (y 2 - 2zw) (x 2 - 2yz) + 8w (x 2 - Zyz} 2 = 0, and z (Sy 2 + 2zw) 2 4>x 2 (y 3 %x 2 tv + Qyzw) = 0. [In these three equations and in the last two equations of p. 495 as originally printed, there was by mistake, an interchange of the letters x and y.] 1 Of course in working out the equation of the Intersect-Developable, it is simpler to employ the equations x 2 -2yz = 0, y^-2zw-0. These equations belong to two cones which pass through the Intersect and have their vertices in the triple conjugate point and single conjugate point respectively. I have not alluded to these cones in the text, as the theory of them does not come within the plan of the present memoir, the immediate object of which is to exhibit the equations of certain developable surfaces but these cones are convenient in the present case as furnishing the easiest means of denning the planes x = 0, iv = 0. If we represent for a moment the single conjugate point by S and the triple conjugate point by T (and the cones through these points by the same letters), then the point T is a point upon the cone S, and the triple conjugate plane which touches the cone S along the line TS touches the cone T along some generating line TM. Let the other tangent plane through the line TS to the cone T be TAT, where M may represent the point where the generating line in question meets the cone S ; and we may consider M as the point of intersection of the line TM with the tangent plane through the line SM to the cone S : then the plane TMM is the plane represented by the equation x = 0, and the plane SMM is that represented by the equation it>=0. We may add that y = is the equation of the plane TSM , and z = that of the plane TSM. 622 492 ON THE DEVELOPABLE SURFACES WHICH ARISE FROM [84 Proceeding next to the problem of finding the envelope of two surfaces of the second order, this is most readily effected by the following method communicated to me by Mr Salmon. Retaining the preceding notation, the equation U + kU = belongs to a surface of the second order passing through the Intersect of the two surfaces U = 0, U 0. The polar reciprocal of this surface U + k U = is therefore a surface inscribed in the envelope of the reciprocals of the two surfaces U 0, U = 0, and consequently this envelope is the envelope (in the ordinary sense of the word) of the reciprocal of the surface U + k U = 0, k being considered as a variable parameter. It is easily seen that the reciprocal of the surface U+kU = Q is given by an equation of the form A + 3B& + 3C& 2 + D& 3 = 0, in which A, B, C, D are homogeneous functions of the second order in the coordi nates x, y, z, w. Differentiating with respect to k, and performing the elimination, we have for the equation of the envelope in question, (AD - BC) 2 - 4 (AC - B 2 ) (BD - C 2 ) - ; or the envelope is, in general or (what is the same thing) for a system of the class (A), a developable of the eighth order. For a system of the class (B) the equation contains as a factor, the square of the linear function which equated to zero is the equation of the plane of contact ; or the envelope is in this case a Developable of the sixth order. And in the case of a system of the class (C) the equation contains as a factor the cube of this linear function ; or the envelope is a developable of the fifth order only. A. We may take for the two surfaces the reciprocals (with respect to x- + y* 4- z* + w 2 = 0) of the equations made use of in determining the Intersect-Developable. The equations of these reciprocals are bed x* -\-cda y* +dabz* + abcw 2 = , Vc d a? + c d a y 2 + d a b z* + a Vc iu" = ; and it is clear from the form of them (as compared with the equations of the surfaces of which they are the reciprocals) that x = 0, y = 0, z=Q, w = 0, are still the equations of the conjugate planes. We have, introducing the numerical factor 3 to avoid fractions, 3 {(b + kb ) (c + kc ) (d + M) x* + (c + kc ) (d + M) (a + ka ) y- + (d + M) (a + ka ) (b + kb ) z* + (a + leaf) (b + kb ) (c + kef) w 2 } = A + 3B& + 3C& 2 + DA; 3 , which determine the values of A, B, C, D. We have in fact A = 3 (bcdx* + cdaf + dabz 2 + abcw 2 ) B= (b cd + bc d + bcd )a? + C= (bcd + b cd + b c d)a;* + D = 3 (Vc d a? + c d a f + d a Vz* + a Vc w*) 84] TWO SURFACES OF THE SECOND ORDER. 493 and these values give (with the same signification as before of/, g, h, I, m, n) 2(AC-B 2 )=^La 2 +Bb 2 + Cc 2 +2Fbc +2Gca +2Hab +2Lad + 2Mbd + 2Ncd + Dd 2 , 2 (BD - C 2 ) = Aa 2 + Bb 2 + Cc 2 + 2Fb c + 2Gc a + 2Ha b + 2La d + 2Mb d + 2Nc d + Dd 2 , AD - BC = Aaa + Bbb + Ccc + F(bc + b c) + G (ca r + c a) + H (ab + a b) + L (ad 1 + a d) + M(M + b d) -f N(cd + c d) + Ddd , where A = n*y* + m 2 x* +f 2 w* + 2fmz 2 w 2 - 2fny 2 w 2 + 2nmy 2 z 2 , B = I 2 z* + n 2 a? + g 2 w* + 2gnx 2 w 2 2glz 2 w 2 + 2ln z 2 tf, C = m 2 x* -f I 2 y 4 + h 2 w* + 2hly 2 iv 2 2hmx 2 w 2 + 2mlx 2 y 2 , D =f 2 x* + g 2 y 4 + h 2 z* - 2ghy 2 z 2 - 2hfz 2 x 2 - 2fg y 2 z 2 , F = I 2 y 2 z 2 , G = m 2 z 2 x 2 , H />^2/y>2/j2 /fc *J(s U . L =f 2 x 2 w 2 , and then 4 (AC - B 2 ) (BD - C 2 ) - (AD - BC) 2 - (BC - F*)f* + (CA - G 2 ) g 2 + (AB - H 2 ) h 2 + (AD - L 2 ) I 2 + (BD - M 2 ) m 2 + (CD - N 2 ) n 2 + 2 (GH - AF)gh + 2 (HF - BG ) hf + 2 (FG - CH)fg -2(MN-DF)mn-2(NL - DG )nl -2(LM-DH)lm + 2(AM-LH)lh +2(BN-NF)mf + 2(CL -NG}ng -2(AN-LG)lg -2(BL - MH)mh- 2(CM - NF) nf + 2 (NH - MG} If +2 (LF -NH)mg + 2 (MG - LF ) nh. Substituting the values of A, B, &c., in this expression, the result after all reductions is f 2 m 2 n 2 x* +g 2 ri 2 l-y 8 + hWnfz 8 +f 2 g 2 h 2 w 8 -4- 2gl 2 n (mg - nh) y 6 z 2 + 2hm 2 l (nh - If) z*x 2 + 2fn 2 m (If- mg) xy- - 2hFm (mg - nh) y 2 z* - 2fm 2 n (nh - If) #& - 2gnH (If- mg)a?y + 2f 2 mn (mg - nh) a*w 2 + 2g 2 nl (nh - If) y e w 2 + 2h 2 lm (If- mg) zw 2 - 2f 2 gh (mg - nh) x 2 w 6 - 2fg 2 h (nh - If) y 2 w e - 2fgh 2 (If- mg) z 2 w +/ 2 (^/ 2 - Qghmn) u^x 4 + g 2 (m 2 g 2 - Ghfnl) uSy 4 + h 2 (n 2 h* - Qlmfg) utz* + I 2 (I 2 / 2 - Qghmn) y*z* + m 2 (m 2 g 2 - Ghfnl) z^x 4 + n 2 (n 2 h 2 - Qlmfg) o*y* + 2gh (ghmn - 3f 2 l 2 ) w*y 2 z 2 + 2hf(hfnl - 3g 2 m 2 ) wtzW + 2fg (fglm - 3A -H 2hm (ghmn - Bf 2 l 2 ) z*x 2 w 2 + 2fn (hfnl - 3g 2 m 2 ) tfy 2 2 + 2gl (fglm - 3h 2 n 2 - fyn (ghmn - 3/ 2 / 2 ) yWw 2 - 2hl (hfnl - 3g 2 m 2 ) z*ifw 2 - 2fm (fglm - 3h 2 n 2 ) - 2mn (ghmn - 3f 2 l 2 ) x*y 2 z 2 - 2nl (hfnl - Sg 2 m 2 ) ^x 2 - 2lm (fglm - 3h 2 n 2 ) - 2 (mg - nh) (nh - If) (If- mg) o?y 2 z 2 w 2 = 0, 494 ON THE DEVELOPABLE SURFACES WHICH ARISE FROM [84 which is therefore the equation of the envelope for this case. The equation may also be presented under the form + (mnx* + nhf + ImzJ (f V + gY + ^ - %% 2 ^ - ^hfzW - 2fgx 2 y 2 ) = ; and there are probably other forms proper to exhibit the different geometrical properties of the surface, but with which I am not yet acquainted. B. Here taking for the two surfaces the reciprocals of the equations made use of in determining the Intersect-Developable, the equations of these reciprocals are ri 2 b x 2 + n *a y* - a b c w- + Zria b zw = 0, which are similar to the equations of the surfaces of which they are reciprocal, only z and w are interchanged,, so that here x = 0, y = are the single conjugate planes, z = is an indeterminate plane passing through the single conjugate points, and w = is the equation of the double conjugate plane or plane of contact. The values of A, B, C, D are A = 3 (n 2 bx* + n 2 ay 2 abcw* + 2nabzw), B = (2nn b + w 2 6 ) x 2 + ...... C = (2nrib f + n 2 b) x 2 + ...... D = 3 (n a b a? + ri 2 a rf - a b c w* + Zn afb zw). Hence, using /, g, h, p, q, r in the same sense as before, we have for 2 (AC B 2 ), 2 (BD C 2 ), (AD BC) expressions of the same form as in the last case (p, q, r being written for I, m, n), but in which A = f 2 w ij r k(fz-w n - + 8qry 2 w- 4<fqzw 3 , B = g-iv 4 + ^p-z^w" + 8prx"w"* + gpzw*, C = hW, D = 2gV + 2/>-y + 4>h?zw- + 4>pqa?y- 8qhx-zw + 8phy-zw + -Ighy-w- + 2fhx-w -, F = - py w 2 , G = q*x-w 2 , H= 0, L = Ipqy-ziu, M = N = - 84] TWO SURFACES OF THE SECOND ORDER. 495 The substitution of these values gives after all reductions the result ftfhW + 4 (pf- qg)fgh 2 zw* + 4 (r 2 h 2 - Qpqgf) h 2 z 2 w* + 2 (q 2 g 2 + 2prfh)fha;W + 2 (p 2 f 2 + 2qrgh)ghy z w* - 16 (pf- qg) z*w 3 - 4 (q 2 g 2 - 4p 2 / 2 - Qpqfg) qhtfzw 3 - 4 (p 2 f 2 - 4q 2 g 2 - 6pqfg)phy 2 zw 3 + I6p 2 q 2 h"z*w* - 8 (pf+ 4<qg) q 2 phx 2 z 2 iv 2 - 8 (qg + 4<pf) pq 2 hy 2 z 2 w 2 + (q 2 g 2 + 8prfli) qWw 2 + (p 2 f 2 + 8qrgh)p 2 y*w 2 + 2 (10r 2 A 2 - pqfg) pqx 2 y 2 w 2 lQp 2 q s hx 2 z s w + 16p s q 2 hy 2 z 3 w + 4 (4>pf+ 5qg}pq z x*zw 4 (4<qg + 5pf)p 3 qy*zw 4 (pf qg) p 2 q 2 x 2 y 2 zw + 4>p*q 2 ifz 2 + 8p s q 3 x 2 y 2 z 2 + 4<pq 4 rx 6 + 4<p*qry 6 + \Zp 2 q 3 rx*y 2 -f I2p 3 q 2 rx~y* = ; which is therefore the equation of the envelope for this case. This equation may be presented under the form + 4<pq (qx 2 -f py 2 ) 2 (qrx 2 + rpy 2 + pqz 2 ) = 0, and there are probably other forms which I am not yet acquainted with. C. The reciprocals of the two surfaces made use of in determining the Intersect- Developable, although in reality a system of the same nature with the surfaces of which they are reciprocals, are represented by equations of a somewhat different form. There is no real loss of generality in replacing the two surfaces by the reciprocals of the cones x 2 2yz, y 2 = 2zw ; or we may take the two conies (x 2 - 2yz = 0, w = 0) and (y 2 - 2ziu = 0, x = 0), for the surfaces of which the envelope has to be found, these conies being, it is evident, the sections by the planes w and x = respectively of the cones the Intersect-Developable of which was before determined. The process of determining the envelope is however essentially different : supposing the plane %x + r)y+z + ww = to be the equation of a tangent plane to the two conies (that is, of a plane passing through a tangent of each of the conies) then the condition of touching the nrst conic gives | 2 297^=0, and that of touching the second conic gives rf- 2&> = 0. We have therefore to find the envelope (in the ordinary sense of the word) of the plane %x + TJI/ + %z + ww = 0, in which the coefficients , 77, , w are variable quantities subject to the conditions The result which is obtained without difficulty by the method of indeterminate multipliers, [or more easily by writing : rj : % : w = Z& 3 : 26 2 : 6 4 : 2] is - 4*,v 2 y 3 = 0, which may also be written under the form 8* (y 2 - %zw? - x- |4/ + 9 (3x 2 - 8i/z) w } = 0. [Another form, containing the factor w, is 4 (y 2 + 2zw) 3 (2y 3 + 27x 2 w 36yzw~) 2 = ().] 496 [85 85. NOTE ON A FAMILY OF CURVES OF THE FOURTH ORDER. [From the Cambridge and Dublin Mathematical Journal, vol. v. (1850), pp. 148 152.] THE following theorem, in a slightly different and somewhat less general form, is demonstrated in Mr Hearn s "Researches on Curves of the Second Order, &c.", Lond. 1846: " The locus of the pole of a line, u + v + w = 0, with respect to the conies passing through the angles of the triangle (u = 0, v = 0, w 0), and touching a fixed line OLU + @v + <yw = 0, is the curve of the fourth order, V {OLU (v + w u)} + V {fiv (w + u v)} + \/ [ryw (u + v w)} = ; " the difference in fact being, that with Mr Hearn the indeterminate line u + v + w = is replaced by the line oo , so that the poles in question become the centres of the conies. Previous to discussing the curve of the fourth order, it will be convenient to notice a property of curves of the fourth order with three double points. Such curves contain eleven arbitrary constants : or if we consider the double points as given, then five arbitrary constants. From each double point may be drawn two tangents to the curve ; any five of the points of contact of these tangents determine the curve, and consequently determine the sixth point of contact. The nature of this relation will be subsequently explained ; at present it may be remarked that it is such that, if three of the points of contact (each one of such points of contact corresponding to a different double point) lie in a straight line, the remaining three points of contact also lie in a straight line. A curve of the fourth order having three given double points and besides such that the points of contact of the tangents from the double points lie three and three in two straight lines, contains therefore four arbitrary constants. Now it is easily seen that the curve in question has three double points, viz. the points given by the equations (u = 0, v w = 0), (v = 0, w u = 0), (w = 0, u v = 0), points which may be geometrically defined as the projections from the angles of the triangle (u = 0, v = 0, w = 0) upon the opposite sides, of the point (u = v = iv] which is the harmonic with respect to the triangle of the given line u + v + w = Q. Moreover, 85] NOTE ON A FAMILY OF CURVES OF THE FOURTH ORDER. 497 the lines u = 0, v = 0, w=0 (being lines which, as we have seen, pass through the double points) touch the curve in three points lying in a line, viz. the given line cm + ftv + yw = 0. Hence the curve in question is a curve with three double points, such that the points of contact of the tangents from the double points lie three and three in two straight lines. Considering the double points as given, the functions u, v, w contain two arbitrary ratios, and the ratios of the quantities a, /3, j being arbitrary, the equation of the curve contains four arbitrary constants, or it represents the general curve of the class to which it has been stated to belong. As to the investigation of the above-mentioned theorem with respect to curves of the fourth order with three double points, the general form of the equation of such a curve is x 2 y 2 z 2 yz zx xy where the double points are the angles of the triangle (a?=0, y = Q, z = 0). It may be remarked in passing, that the six tangents at the double point touch the conic ax 2 + by 2 + cz 2 - 2fyz - Zgxz - 2hxy = 0. To determine the tangents through (y = Q, z=Q), we have only to write the equation to the curve under the form \SS y z) y 2 z 2 yz~ the points of contact are given by the system if z* yz the latter equation (which evidently belongs to a pair of lines) determining the tangents. The former equation is that of a conic passing through the angles of the triangle x = 0, y = 0, z=Q: since the tangents pass through the point (y = 0, z = Q) they evidently each intersect the conic in one other point only. The equation of the tangents shows that these lines are the tangents through the point y=Q, z = to the conic whose equation is aA 2 x 2 + bBy + cCV + ZfBCyz + ZgCAzx + ZhABxy = 0. To complete the construction of the points of contact it may be remarked that the equations which determine the coordinates of these points may be presented under the form Ax= [A } x , 63 498 NOTE ON A FAMILY OF CURVES OF THE FOURTH ORDER. [85 whence also aAx + hBy + gCz = hAx + IBy +fCz = G Kx, -K a, -K a x : or writing for shortness f, ij, instead of the linear functions forming the first sides of these equations respectively, we have _JL_ -V 1. V(-a^) G H from which it follows at once that the equation to the line forming the two points of contact is 1 + 1-0- G + H~ v % and this may again be considered as the line joining the points (f = 0, p+ + // =0 ) and (77-0, =0). Now |=0, 77=0, =0, are the polars of the double points (or angles of the triangle x = 0, y = Q, z = 0) with respect to the last-mentioned conic. The equation of the line joining the point (t/ = 0, z=0) with the point (17 = 0, =0), is easily seen to be GBy - GHz = ; from which it follows, that the lines forming each double point with the intersection of the polars of the other two double points meet in a point, the coordinates of which are given by 1_ J_ J ~ AF BG CH and the polar of this point is the line p+^ + Jf =() - The construction of this line is thus determined ; and we have already seen how this leads to the construction of the lines joining the points of contact of the tangents from the double points. In fact, in the figure, if x/3y be the triangle whose sides are = 0, i?=0, =0, and A, B, C the points of intersection of the sides of this triangle with the line 85] NOTE ON A FAMILY OF CURVES OF THE FOURTH ORDER. 499 the lines in question are Act, B/3, Cy. Moreover, the points of contact upon Aa. are harmonics with respect to A, a, and in like manner the points of contact on J?/3, Cy are respectively harmonics with respect to B, fB and to C, 7. This proves that if three of the points of contact are in the same straight line, the remaining three are also in the same straight line ; in fact, we may consider three of the points of contact as given by f : rj : S=*J(-aK) : G : - H , | : rj : = -F : J(- bK) : H, ::= F : -G :J and the condition that these may be in the same line is V(- a) V(- &) V(- c) + GH V(- a) + HF V(- b) + FG V(- c) = 0, which remains unaltered when the signs of all the roots are changed. The equation just obtained may be considered as the condition which must exist between the coefficients of a b c 2/ 2cr 2h -2 + -0+-2+ + + =0, x 2 y 2 z 2 yz zx xy in order that this curve may be transformable into the form *J{a.u (v + w u}} + V{/3v (w + u v)} + \/{<yw (u + v w)} 0. 632 500 [86 86. ON THE DEVELOPABLE DERIVED FROM AN EQUATION OF THE FIFTH ORDER. [From the Cambridge and Dublin Mathematical Journal, vol. v. (1850), pp. 152 159.] MOBIUS, in his " Barycentrische Calcul," [Leipzig, 1827], has considered, or rather suggested for consideration, the family of curves of double curvature given by equations such as x ; y : z \ w = A : B : C : D, where A, B, C, D are rational and integral functions of an indeterminate quantity t. Observing that the plane Ax+ By + Cz + Dw = may be considered as the polar of the point determined by the system of equations last preceding, the reciprocal of the curve of Mobius is the developable, which is the envelope of a plane the coefficients in the equation of which are rational and integral functions of an indeterminate quantity t, or what is equivalent, homogeneous functions of two variables , 77. Such an equation may be represented by U=a^ n + n^ n ~ 1 r) + ...=0, (where a, b, &c. are linear functions of the coordinates) ; and we are thus led to the developables noticed, I believe for the first time, in my " Note sur les Hyper-deter minants," Crelle, t. xxxiv. p. 148, [54]. I there remarked, that not only the equation of the developable Avas to be obtained by eliminating , 77 from the first derived equations of U= ; but that the second derived equations conducted in like manner to the edge of regression, and the third derived equations to the cusps or stationary points of the edge of regression. It followed that the order of the surface was 2 (n - 1), that of the edge of regression 3 (n 2), and the number of stationary points 4 (n - 3). These values lead at once, as Mr Salmon pointed out to me, to the table, m = 3 (n - 2), n = n, r =2(7i-l), a =0, /3 = 4 (n - 3), g -i(-l)<fi-SX h = i (9n 2 - 53w + 80), x = 2 (n - 2) (n - 3), y =2(n-l)(n-3), 86] ON THE DEVELOPABLE DERIVED FROM AN EQUATION &C. 501 where the letters in the first column have the same signification as in my memoir in Liouville, [30], translated in the last number of the Journal. The order of the nodal line is of course 2 (n 2) (n 3) ; Mr Salmon has ascertained that there are upon this line 6 (n 3) (n 4) stationary points and (n 3) (n 4) (n 5) real double points, (the stationary points lying on the edge of regression, and with the stationary points of the edge of regression forming the system of intersections of the nodal line and edge of regression, and the real double points being triple points upon the surface). Also, that the number of apparent double points of the nodal line is (n - 3) (2w 3 - 18w 2 + 57n - 65). The case of U a function of the second order gives rise to the cone ac b 2 = 0. When U is a function of the third order, we have the developable 4 (ac - b-) (bd - c 2 ) - (ad - be) 2 = 0, which is the general developable of the fourth order having for its edge of regression the curve of the third order, oc-6 2 = 0, 6d-c 2 = 0, ad-bc=0, which is likewise the most general curve of this order: there are of course in this case no stationary points on the edge of regression. In the case where U is of the fourth order we have the developable of the sixth order, (ae - 4<bd + 3c 2 ) 3 - 27 (ace + 2bcd - ad* - fce - c 3 )- = ; having for its edge of regression the curve of the sixth order, ae - 4<bd + 3c 2 = 0, ace + 21>cd - ad 2 - b-e - c 3 = 0, with four stationary points determined by the equations a _ b _ c _ d b c d e The form exhibiting the nodal line of the surface has been given in the Journal by Mr Salmon. I do not notice it here, but pass on to the principal subject of the present paper, which is to exhibit the edge of regression and the stationary points of this edge of regression for the developable obtained from the equation of the fifth order, U = a% r > + o&ffy + 10c V + 1 Odf-y + oegn 4 + frf = ; viz. that represented by the equation rj = = a 4 / 4 + 160a s ce/ a + . . . - 4000&W, [I do not reproduce here this expression for the discriminant of the binary quintic] a result for which I am indebted to Mr Salmon. 502 ON THE DEVELOPABLE DERIVED FROM [86 To effect the reduction of this expression, consider in the first place the equations which determine the stationary points of the edge of regression. Writing instead of : rj the single letter t, these equations are at 2 + 2bt + c=0, bt- + 2ct + d = 0, ct 2 + 2dt + e=0, write for shortness A = 2(bf- 4ce + 3d 2 ), B = af-3be + 2cd, C = 2ae and let a, 3/3, 87, S represent the terms of j a, b, c, d b, c, d, e c, d, e, f viz. a = bdf be" + 2cde c 2 / d 3 , 3^Q = adf ae- bcf -f bde + c-e cd 2 , 3 7 = ac f - ade - b 2 / + bd 2 + bee - c 2 d , B = ace ad- - b~e + 2bcd - c 3 ; it is obvious at first sight that the result of the elimination of t from the four quadratic equations is the system (equivalent of course to three equations), a=0, /3=0, 7 = 0, 8 = 0. The system in question may however be represented under the more simple form (which shows at once that the number of stationary points is, as it ought to be, eight), A=Q, B=Q, (7=0; this appears from the identical equations, (2c + 3d) (bt- + 2ct + d} - (2bt + 4c) (ct 2 2bt + c) - c (bt- + 2ct + d) 36) (ct 2 + 2dt + e) + a(dt 2 +2et+f) = 86] AN EQUATION OF THE FIFTH ORDER. 503 (2bt + 3c) (at 2 + 2bt + c) - (2at + 46) (bt 2 + 2c + d) + a (ct 2 + 2dt + e} = \C, (formulae the first and third of which are readily deduced from an equation given in the Note on Hyperdeterminants above quoted). The connexion between the quantities A, B, C and a, /?, 7, 8, is given by Aa-2Bb + Cc = -65, Ab - 2Bc + Cd = - 67, Ac - 2Bd + Ce =- 6/3, The theory of the stationary points being thus obtained, the next question is that of finding the equations of the edge of regression. We have for this to eliminate t from the three cubic equations, at 3 + Sbt 2 + Sct+d = Q, bt 3 + Set 2 + Sdt + e =0, ct 3 + Sdt 2 + Set +/=0: treating the quantities f\ t 2 , t l , t as if they were independent, we ( at once obtain or as this system may be more conveniently written, 0t + = 0, 7* + 13 = 0, St + 7 = 0. But the most simple forms are obtained from the identical equations, ft (at* + 3bt 2 + Set + d), - (Set +/) (bt 3 + Set 2 + Sdt + e), + (2dt + e) (ct s + Sdt 2 + Set +/) = t 3 (Bt + A ) : (bt + c) (at 3 + Sbt 2 + Set + d), - (at + Sb) (bt 3 + Set 2 + Sdt + e), + a (ct 3 + Sdt 2 +Set+f)= Ct + B: equations which, combined with those which precede, give the complete system 0, &+7 = 0, Bt + A=0, Ct + B=(): 504 ON THE DEVELOPABLE DERIVED FROM [86 or the equations of the edge of regression are given by the system (equivalent of course to two equations), a, /3, /?, y, 7, A, 8, B, B C = 0. The simplest mode of verifying a posteriori that the edge of regression is only of the ninth order, appears to be to consider this curve as the common intersection of the three surfaces of the seventh order : A 3 a - 3A 2 Bb A 3 b - 3A 2 Bc - B 3 d = 0, - B z e = 0, - B* = 0, (which are at once obtained by combining the equation Bt + A = with the cubic equations in t). It is obvious from a preceding equation that if the equations first given are multiplied by fA, 3eA+fB, 2dA eB, and added, an identical result is obtained. This shows that the curve of the forty-ninth order, the intersection of the first two surfaces, is made up of the curve in question, the curve of the fourth order A = 0, B = (which reckons for thirty-six, as being a triple line on each surface), and the curve which is common to the two surfaces of the seventh order and the surface 2dA eB = 0. The equations of this last curve may be written, e (af- 3be + Zed) - 4d (bf- 4ce + 3d 2 ) = 0, e s a - 6e 2 db + 12ed 2 c - 8d 4 = 0, e 3 b - Qe 2 dc + led 3 = ; or, observing that these equations are / (ae - 46cO - 3 (be- - 6ced + 4rf e 2 (ae - 4bd) - 2d (be* - Qced + 4d e (be- - Qced + 4d 3 ) = ; the last-mentioned curve is the intersection of ae 4bd = 0, be 2 ticed + 4rf 3 = 0, where the second surface contains the double line e = 0, d = 0, which is also a single line upon the first surface. Omitting this extraneous line, the intersection is of the fourth order; and we may remark that, in passing, it is determined (exclusively of the double line) as the intersection of the three surfaces ae -4bd = 0, be 2 - Qced + 4d 3 = 0, a~d Qabc + 46 3 = 0, 86] AN EQUATION OF THE FIFTH ORDER. 505 being in fact of the species iv. 4, of Mr Salmon s paper, "On the Classification of Curves of Double Curvature" [Journal, vol. v. (1850), pp. 2346]. But returning to the question in hand ; since 49-9 + 4 + 36, we see that the curve common to the three surfaces of the seventh order, or the edge of regression, is, as it ought to be, of the ninth order. It only remains to express the equation of the developable surface in terms of the functions A, B, C, a, /?, 7 , 8, which determine the stationary points and edge of regression; I have satisfied myself that the required formula is D = (AC - &y -1152 {A (/3S-y 2 )-B(<x&-/3y) + C ( 7 -/3 2 )} = 0, where the quantities a, /3, 7, 8 may be replaced by their values in A, B, C; and it will be noticed that when this is done, the terms of Q are each of them at least of the third order in the last-mentioned functions. I propose to term the family of developables treated of in this paper, planar developables. In general, the coefficients of the generating plane of a developable being algebraical functions of a variable parameter t, the equation rationalized with respect to the parameter belongs to a system of n different planes; the developable which is the envelope of such a system may be termed a multiplanar developable, and in the particular case of n being equal to unity, we have a planar developable. It would be very desirable to have some means of ascertaining from the equation of a developable what the degree of its planarity is. P.S. At the time of writing the preceding paper I was under the impression that the only surface of the fourth order through the edge of regression was that given by the equation AC-B 2 = Q- but Mr Salmon has since made known to me an entirely new form of the function n> the component functions of which, equated to zero, give six different surfaces of the fourth order, each of them passing through the edge of regression. The form in question is 3 Q = LL + 64JO/ - 64NN , where L = a 2 / 2 + 2256V - 32ace 2 - 32b 2 df + 4,80bd* + 480c 3 e - 34a6e/+ 76acdf- 12bc 2 f - I2ad-e - 820bcde-320c-d-, L = 3a 2 /--456V + 64ace 2 + 64Z> 2 d/ - 22abef - Uacdf - 3Qbc-f - 36ad*e + ZObcde, M = IQbcd 2 - I8ad 3 - lobtfe + ZZacde + 6b 2 cf - 9ac 2 / + Zabdf + a 2 ef - 9a6e 2 , M = I0c 2 de - I8c 3 f - Iobd 2 e + 326cc?/+ Gade- - 9b 2 ef + 2acef + a&/ 2 - 9a#/, N = 106 2 d 2 - I5b"ce - I2acd- + 18ac 2 e + abde - 2a 2 e 2 + 6b 3 f - 9abcf + 3a 2 df, N =IW*-I5bd# -I2c*df+18bdy+ beef - 26 2 / 2 where it should be noticed that 64 506 ON THE DEVELOPABLE DERIVED FROM AN EQUATION &C. [86 The expressions of L, L , M, M , N, N as linear functions of A, B, ( (also due to Mr Salmon) are L = (1 1 ae + 28bd - 39c-) A + ( of - 756e + 74cc/) B + ( 1 lb/+ 28ce - 39cf) C f , Jf = 3 (6c -ad)A+ 3 (ae - c-) + (erf - <//) C, M = cd-aA + 3b = 3 (6 2 - ac) ^ + 3 (ad -bc}B+ (bd - ae) C, = (ce - ft/) ^ + 3 (c/ - rfe)5 + 3 (e 2 - df) C. I propose resuming the subject of these forms, and the general theory, in a sub sequent paper. [This was never written.] 87] 507 87. NOTES ON ELLIPTIC FUNCTIONS (FROM JACOBI). [From the Cambridge and Dublin Mathematical Journal, vol. v. (1850), pp. 201 204.] THESK Notes are mere translations from Jacobi s " Note sur une nouvelle application de 1 aualyse des fonctions elliptiques a 1 algebie," [Crclle, t. vn. (1831) pp. 41 43], and from the addition to the notice by him of the third supplement to Legendre s " Theorie des fonctions elliptiques" [Crelle, t. vni. (1832) pp. 413417]. 642 508 [88 ON THE TRANSFORMATION OF AN ELLIPTIC INTEGRAL. [From the Cambridge and Dublin Mathematical Journal, vol. v. (1850), pp. 204 20G.] THE following is a demonstration of a formula proved incidentally by Mr Boole (Journal, vol. II. [1847] p. 7), in a paper " On the Attraction of a Solid of Revolution on an External Point." Let U = then, assuming a. + iy _ v - _ . . I -tay (so that x= 1 gives y 1), we obtain msc (1 - wy) + (m - ina) y 1 - ivy Assume therefore ia + (n imoi) (m ina) = 0, whence - = m-n)+ 2mn (A 2 = 1 + m 4 + w 4 - 2m 2 - 2w 2 - 2m s n 3 ), ON THE TRANSFORMATION OF AN ELLIPTIC INTEGRAL. 509 we find and also whence u- J\ 1 + a2 1, V (1 (n ima) 2 } J _ 1 V[(l y 2 ) {1 (m - ma) 2 that is / ( 1 I 2 = 2 \/|l-(n ~ ma But since 1 - m 2 + n 2 + A n ima. = 1 + m 2 - n 2 + A m ma. = 1 - (n - ima) 2 = - H 2 (A + 1 - m 2 + n 2 ), A 1 + fir = ^ ( A + 1 m 2 n-) ; and therefore 1 + a 2 1 A + 1 m 2 n 2 1 (n ima) 2 m 2 A + 1 m 2 + n 2 I (1 - m 2 - n 2 + A) (1 - m 2 + n 2 - A) _ 2(1 + m 2 -n 2 + A) m 2 (1 m 2 + n 2 + A) (1 m 2 + n 2 A) 4>m 2 consequently Cr = ! V |2(l + m 2 -n 2 + A)}f dy m n j n /[ ,, L /1+m 2 n 2 + P / /r,-, ^a, /T+^-^+AV^TI- Write l+m -n + A A/ ^r /V , _ 2m (1 + m) 2 n 2 then (fa where X and k are connected by the relation that exists for the transformation of the second order, viz. x = ITfc 510 ON THE TRANSFORMATION OF AN ELLIPTIC INTEGRAL. [88 as may be immediately verified ; hence, assuming \z _ y ~ which gives dy - A dz r 1 J we find that is dx . r dz rt 2 } } I ( I " Vi 1 -^ 1 - rr=*P , ) ] J . V(l - *) [|(1 Writing here X = COS 0, 2- = COS then r ^ = r + m 2 ?i 2 2?n cos 0) or if then finally f rffl _ _ /" d(f> J V{ 2 + (* + r cos d) a } ~ J V( 2 + r* + z* - 2ar cos </>) the formula in question. 89] 511 89. ON THE ATTRACTION OF ELLIPSOIDS (JACOBI S METHOD). [From the Cambridge and Dublin Mathematical Journal, vol. v. (1850), pp. 217 226.1 IN a letter published in 1846 in Liouville s Journal (t. XL p. 341) Jacobi says, "II y a quatorze ans, je me suis pose le probleme de chercher 1 attraction d un ellip- soi de homogene exerce e sur un point exterieur quelconque par une methode analogue a celle employe e par Maclaurin par rapport aux points situe s dans les axes principaux. J y suis parvenu par trois substitutions conse cutives. La premiere est une transforma tion de coordonne es; par la seconde le radical v 7 (l w 2 sin 2 /? cos 2 i/r % 2 sin 2 /3sm -i/r) uui entre dans la double integrate transformed est rendu rationnel an moyen de la double substitution m sin /3 cos i/r = sin 77 cos 9, m sin /3 sin ty = sin 77 sin 6 ; la troisieme est encore une transformation de coordonne es. La recherche du sens geome trique de ces trois substitutions m a conduit a approfondir la the orie des surfaces confocales par rapport auxquelles je decouvris quantitd de beaux theoremes dont je communiquai quelques-uns des principaux a M. Steiner. Considerons 1 ellipso ide confocal inene par le point attire P et le point p de 1 ellipsoide propose , conjugue a P. Soient Q et q deux autres points conjugues quelconques situes respectivement sur 1 ellipsoide exte rieur et interieur. Menons de P un premier cone tangent a 1 ellipso ide interieur, de p un second cone tangent a 1 ellipsoide exterieur. Ce dernier, tout imagi- naire qu il est, a ses trois axes reels (ainsi que ses deux droites focales). La premiere substitution ramene les axes de 1 ellipsoide a ceux du premier cdne (c est la substitution employee par Poisson, mais que j avais anteVieurement traitee et meme e tendue a un nombre quelconque de variables dans le me moire De binis Functionibus homogeneis &c. [Crelle, t. xn. (1834) pp. 1 69]). Par la seconde substitution les angles que la droite Pq forme avec les axes du premier cone sont ramenes aux angles que la droite pQ forme avec les axes du second. Par la derniere substitution, on retourne de ces axes aux axes de rellipsoide. La seconde substitution repond a un thdoremo de ge ometrie remarquable, savoir que Les cosinus des angles que la droite Pq forme avec deux des axes du premier cone sont en raison constante avec les cosinus des angles que la droite 512 ON THE ATTRACTION OF ELLIPSOIDS (jACOBl s METHOD). [89 pQ forme avec deux des axes du second cone ; ces deux axes sont les tangents situes respectivement dans les sections de plus grande et de moindre courbure de chaque ellipsoi de, le troisieme axe etant la normale a 1 ellipsoide. Tout cela semble difficile a etablir par la synthese." The object of this paper is to develope the above method of finding the attraction of an ellipsoid. Consider an exterior ellipsoid, the squared semiaxes of which are f+u, g + u, h + u ; and an interior ellipsoid, the squared semiaxes of which are f+u, g + u, h + u. Let u, p, q be the elliptic coordinates of a point P on the exterior ellipsoid, the elliptic coordinates of the corresponding point P on the interior ellipsoid will be u, p, q, and if a, b, c and a, b, c represent the ordinary coordinates of these points (the principal axes being the axes of coordinates), we have (f-g}(f-h) (g + u) (g + q)(g + r) (g-h)(g-f) (h + u) (h +q)(h + r) (h-g)(h-f) I form the systems of equations (u -q)(u- r) b , = _ (q -r)(q- u) (r u) (r q) a = f+u a = /+g a = f+r (f-g)(f-h) _ (h-f}(h-g} r7 ,_( u +f)(u+g)(u+h) (q -r)(q- u) -, = (r+f)(r+g)(r+h} (r -u)(r- q) 7 = 7 = h + u 7 = c,c h + r a= ^ f* = ^+- u 7 = ATi a = 7 = a" = f+r 7 = h + r 89] ON THE ATTRACTION OF ELLIPSOIDS (jACOBl s METHOD). 513 And then writing X = aX, + oi Y, + a"Z,, Z = X=aX,+ a F, + a %, Z = 7X1+ if X, Y, Z are the cosines of the inclinations of a line PQ to the principal axes of the ellipsoids, X lt Y lt Z^ will be the cosines of the inclinations of this line to the principal axes of the cone having P for its vertex, and circumscribed about the interior ellipsoid. In like manner, X, Y, Z being the cosines of the inclinations of a line PQ to the principal axes of the ellipsoids, X 1} F 1} Z t will be the cosines of the inclinations of this line to the principal axes of the cone having P for its vertex and circumscribed about the exterior ellipsoid. Assuming that the points Q, Q are situated upon the exterior and interior ellipsoids respectively, suppose that X lt Y,, Z 1 and X lt Y lt Z, are connected by the equivalent systems of equations, // Y 2 Y 2 ^ 2 \ -\r if \ / / ^*1 -" 1 ^i \ X l = V(w - w) A / I ~= H 1 , V \u-u u-q u-r) X* Y, 2 u u q u r then it will presently be shown that the points Q, Q are corresponding points, which will prove the geometrical theorem of Jacobi. Before proceeding further it will be convenient to notice the formulae u-u f+u g + u Xa Yb Zc _ u-u f + u g + u h + u af \u u u q u r / Xa + F6 Zc Y U-M/ Z 2 F 2 _^_> v + * S + A + w/ ! 2 \f+ u g+u h + u; C. <H \u u u q u 514 ON THE ATTRACTION OF ELLIPSOIDS (jACOBl s METHOD). [89 and the corresponding ones a 2 b 2 c 2 u u 1- f+u g + *i h + u u h+u ai 2 \u-u u-q u-r Xa Yb Zc\*u-u& H -- : -- 1~ i . u-u( X? F, 2 The coordinates of the point Q are obviously a + pX, b + pY, c + pZ (where p=PQ); substituting these values in the equation of the interior ellipsoid, we obtain Xa Yb _Z, ___ + + reducing the coefficients of this equation by the formulse first given, and omitting a factor r , we obtain vu _ u + : t p H that is, (it -it) 2 -w MI-? w-r/ \u-u u-q u-r z l0i FA z l or p = u u u q u r, 1 ~r - ~r w q u r which is easily transformed into u u P (f+u)b l Y l -(f+u)b 1 Y l , (f+u)c l Z 1 -(f+u)c 1 Z 1 and this form remaining unaltered when t* and u are interchanged, it follows that if PQ = ~p, then p = ~p, which is a known theorem. The value of p or p may however be expressed in a yet simpler form ; for, considering the expression a J a 1 X l b^\ c^l _ a^ f a 1 X\ a { ., u u 89] ON THE ATTRACTION OF ELLIPSOIDS (jACOBl s METHOD). 515 we see that X X 1 / a a and similarly Y Y If b Z which are in fact the equations which express that Q and Q are corresponding points. It is proper to remark that supposing, as we are at liberty to do, that P, P are situate in corresponding octants of the two ellipsoids, then if the curve of contact of the circumscribed cone having P for its vertex divide the surface of the interior ellipsoid into two parts M, N, of which the former lies contiguous to P: also if the curve of intersection of the tangent plane at P divide the surface of the exterior ellipsoid into two parts M, N, of which M lies contiguous to the point P ; then the different points of M, M correspond to each other, as do also the different points of N, N. We may now pass to the integral calculus problem. The Attraction parallel to the axis of x is A _ f xdxdydz J (rr? 4- 1/2 4- *2 J the limits of the integration being given by (y + 6) (* + c) . f+u g+u h+u or putting x = rX, y = rY, z = rZ, where X, Y, Z have the same signification, as before, we have doc dy dz = r 2 dr dS, and then A = fXdrdS = fpXdS, where p has the same signification as before: it will be convenient to leave the formula in this form, rather than to take at once the difference of the two values of p, but of course the integration is as in the ordinary methods to be performed so as to extend to the whole volume of the ellipsoid. The expression dS denotes the differen- 652 516 ON THE ATTRACTION OF ELLIPSOIDS (jACOBl s METHOD). [89 tial of a spherical surface radius unity, and if 0, <f> are the parameters by which the position of p is determined, we have In the present case x, Y, Z c dX dY dZ d0 d0 de dX dY dZ <ty df d* x,, Y,, Z, dX, dY, dZ, dY, dY, dY, dX, dY, dZ, dZ, dZ, dZ, 7 <, in terms of X,, Y, ?,*)} we deduce 1 ! Y,, Z, {observing that X, must be But dS = dS,= X, whence dS = -q)(u- r)\ X, which shows that the corresponding elements of the spheres whose centres are P, P, projected upon the tangent planes at P and P respectively, are in a constant ratio. It may be noticed also that if /*, Jl are the masses of the ellipsoids, the ratio in question f(u- g )(u-r)| _ , {(u -q)(u- We have thus ~ X, that is /\(u-q)(u-r)} i \/\( u - q )(u-r)}j q)(u-r)\J X, The value which it will be convenient to use for ~p is that derived from the equation ^" F 2 . Z*_\.^_fXa . Yb Z c\ u-u + u g =0, 89] ON THE ATTRACTION OF ELLIPSOIDS (jACOBl s METHOD). 517 with only the transformation of expressing the radical in terms of X l} viz. Xa | Yb [ Zc | 1 f+u g+u h+u substituting these values and observing that Fj and Z^ are rational functions of X, 7, and Z, but that X 1 is a radical, and that in order to extend the integration to the whole ellipsoid, the values corresponding to the opposite signs of X l will require to be added, the quantity to be integrated (omitting for the moment the exterior constant factor) is _ f+u g+u h+u the integration to be extended over the spherical area 8. Consider the quantity within { }, this is The coefficients of Y and Z vanish, in fact that of Y is W _JL + _ b * a /ft ~ u \ JiL W /( r ~ u\ Ji& f+u g + u a, (f + q) V (q - u) g + q + a, (f + r) V (r - u) g +r , /(g V \q - __ .. \(f+u)(g + u) (S+q)(g + q)V \q -u) ^ (f+r)(g + r)V (r - _ | i ((/+ M) (9 + ) (/+ q) (g + q) (f+ r) (g + r) = and similarly for the coefficient of Z. The coefficient of X is in like manner shown to be tt c? _\_au (f-g}(f-h) ad _ a ~ ~ or the quantity in question is simply a . 518 ON THE ATTRACTION OF ELLIPSOIDS (jACOBl s METHOD). [89 Multiplying this by X,,=aX + a Y+a"Z, the terms containing XY, XZ vanish^ after the - rr __ si %(i integration, and we need only consider the term Z 2 , or what is the same ^ , Z 2 . Hence ^ _ / ^ ") \ /( i __ I . ==r - = f + u g + u h + u The value of the corresponding function A (that is, the attraction of the exterior ellipsoid upon P) is X*dS f+u I f+u g + u h+u the limits being the same, whence A _A _ / f 0* ~ ~V\(u- ~ g) (M - r ) f+ u ggi _ V(M + 9} Vft* q)(u-r}\ f+uaa, J(u + g)J(u + h) or we have A _u ^ ^ ~ ~ ( +7) v< +/) v( + 9) formulae which constitute in fact Ivory s theorem. Let K, K denote the attractions in the directions of the normals at P, P, we have = t XM J and it is important to remark that this is true not only for the entire ellipsoids ; but if Jtfl, jBt denote the attractions of the cones standing on the portions M, N of the surface of the interior ellipsoid, and J$l, J| the attractions of the portions of the exterior ellipsoid bounded by the tangent plane at P, and the portions M, N of the surface of the exterior ellipsoid, then where obviously K = M this theorem is so far as I am aware new. 90] 519 90. NOTE SUE QUELQUES FORMULES RELATIVES AUX CONIQUES. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn, xxxix. (1850), pp. 1-3.] SOIT, comme a 1 ordinaire : gl = EG - F* , 23 = CA -G 2 , = AB - H* , $ = GH-AF, [ .................. (1) (& = HF-BG, %=FG -CH, K = ABC-AF*-BG 2 -CH*+2FGH, et de signons, pour abrdger, les fonctions Aa? + Ef + Cz* + ZFyz +2Gxz + ZHxy, Aax + Bfiy + C^z + F (/3z + jy ) + G (jx + A (ftz - 72 /) a + B (yx - out) 3 + C(ay- fa)* + + 2G (ay - fa par &c. Cela pose.soient ^ +a, 33 + b, < + c, J + f, (ay + fa), - oa) (ay - fa) yy) + 2H (fa - yy) ( yx - *) ; ...- ) Aax+...\ A . . . ; &c. ce que deviennent a, 23, D, J, , ffi, K, en ecrivant A + a 2 , B + /3 2 , C + y*, F+/3y, G + 7 a, H + a/3 an lieu de A, B, C, F, G, H. Alors on aura d abord 520 NOTE SUR QUELQUES FORMULES RELATIVES AUX CONIQUES. [90 et les quantit^s a, b, c, f, g, h seront donne es par 1 dquation ax>+ ...=A(/3z-yyY~+ .................................... (3), ou, si Ton veut, par celle-ci : 1 + ... =A (^z-yy}(fa-yy l } + ........................ (4), (savoir, en conside rant ces Equations comme identiques par rapport a x, y, z et x l , y l , z l respectivement). On obtient sans difficult^ les Equations identiques: (AOLOL, + ...) (Axx 1 +...)- (Aax + . . .) (Aa.x, + ...)= & (fa - yy,} (&z - 7^) + . . . , (5), ,*! +...) = & [A (fa - 77/0 (fa - 7l y) + ...] (6). Comme on sait, la condition sous laquelle la droite Ix + my + nz = touche la conique U= Ax 2 + ... = peut etre pre sente e sous la forme U 2 +...=0. Done la con dition pour que cette droite touche la conique U+(ax + fty + yz) 2 = Q, est &P+...+A(*ym-l3ny i +...=0 ........................... (7). En re duisant au moyen de liquation (gla 2 + . . .) (&P + ...) = K[A (ym - /3n) 2 + ...] (laquelle n est qu un cas particulier de 1 equation (6)), cette condition devient: (al+...?=0 ..................... (8). Pour trouver la condition sous laquelle les coniques U + (ax + @y + yzf = 0, U + (OL,X + &y + ^zf = se touchent, on n a qu a remarquer que 1 equation de la tangente commune est, ou (a -a 1 )x + (0- &) y + (7 - 7^ z = 0, ou (a +a 1 ) a ; + (l3+ &) y + (7 + 7^ z = 0. En ne considerant que la premiere de ces droites, on a pour la condition cherchee : ...=0 .............................. (9); ou, en re duisant au moyen du meme cas particulier de 1 equation (6), on obtient cette condition sous la forme (K + &a 2 +.. .)(K+a* + ...)-(# + &<**+. ..) 2 = ............ (10). On sait que les coordonnees X, Y, Z du pole de la droite Ix + my + nz = 0, par rapport a la conique U=0, peuvent etre trouve es par 1 equation 90] NOTE SUE QUELQUES FORMULES RELATIVES AUX CONIQUES. 521 (conside re e comme identique par rapport a X, p, v}. De la les coordonne es X, Y, Z du p61e de cette meme droite par rapport a la conique U + (ax + @y + 72)* = se trouveront par 1 equation (11) (considered comme identique par rapport a \, ^ v ). Cette equation peut aussi tre presentee sous la forme K(\X+ nY+vZ} = (K + &a? + ...)(<a& + ...)-(&a\+...)OaaJ + ...) ... (12), ce qui peut etre ddmontre facilement an rnoyen d un cas particulier de 1 e quation (6). Si les deux droites lx + my+nz=0, l x+ m y + n z = 0, touchent la conique U=0, lequation de la droite qui passe par les points de contact sera A (mri - m n) X + ... = 0. Done: si^ ces deux droites touchent la conique U+(aa; + 0y + yzy=0, 1 e quation de la droite qui passe par les deux points de contact sera A (mn - m n} x + . . . + (ax + 0y + 7 /> [a ( mn - m n) + /3 (nl - ril) + 7 (lm - I m}] = ... (13). Les formules obtenues seront utiles pour la solution du probleme du memoire suivant, [91]. Je les ai rapproche es ici pour ne pas interrompre cette solution. C. ,,, 06 522 [91 91. SUE LE PROBLEMS DES CONTACTS. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn, xxxix. (1850), pp. 413.] J E me propose ici la solution analy tique du problem e suivant : " Etant donndes trois coniques inscrites a une meme conique : trouver une autre coriique, aussi inscrite a cette conique, qui touche les trois coniques inscrites; et tirer de la les constructions ge ome triques ordinaires." Je commence par re capituler quelques-unes des propriete s d un systeme de trois coniques inscrites a la meme conique. Un systeme de six droites qui passent trois a trois par quatre points, s appelle quadrangle. Les points de rencontre des cotes opposes sont les centres du quadrangle; les cotes du triangle forme par ces trois centres sont les axes du quadrangle. De meme, un systeme de six points situes trois a trois sur quatre droites, s appelle quadrilatere. Les droites qui passent par les angles opposes sont les axes; et les angles du triangle forme" par les trois axes sont les centres du quadrilatere. Deux coniques quelconques se coupent en quatre points qui forment un quadrangle inscrit aux deux coniques. Elles ont quatre tangentes communes qui forment un quadrilatere circonscrit aux deux coniques. Le quadrangle inscrit et le quadrilatere circonscrit ont les meme centres et les memes axes. Si deux coniques sont circoriscrites ou inscrites 1 une a 1 autre, la droite qui passe par les deux points de contact s appelle chorde de contact, et le point de rencontre des deux tangentes communes s appelle centre de contact. Cela pose : les coniques circonscrites a deux coniques donnees, peuvent etre divisees en trois classes : une conique circonscrite appartient a une quelconque de ces trois classes, selon que les points de rencontre des chordes de contact de la conique 91] SUE LE PROBLEME DES CONTACTS. 523 circonscrite et de chacune des deux coniques donnees coincide avec un quelconque des trois centres du quadrangle inscrit, ou du quadrilatere circonscrit ; ou, si Ton veut, selon que la droite qui passe par les centres de contact de la conique circonscrite et des deux coniques donne es, coincide avec un quelconque des trois axes du quadrangle inscrit, ou du quadrilatere circonscrit. En conside rant les deux coniques donne es, et une conique circonscrite, nous dirons que les deux cote s du quadrangle inscrit qui se coupent dans le point d intersection des deux chordes de contact, sont les axes de symptose des deux coniques donnees, et que les deux angles du quadrilatere circonscrit, situs s sur la droite qui passe par les deux centres de contact, sont les centres d homologie des deux coniques donnees. Soient maintenant inscrites trois coniques a la meme conique. En combinant deux a deux ces trois coniques, les six axes de symptose se couperont trois a trois en quatre points que nous nommerons centres de symptose, et les six centres d homologie seront situe s trois a trois sur quatre droites que nous appelerons axes d homologie. Soient U+7* = 0, U+V* = 0, 7+F 3 2 = les equations des trois coniques inscrites, ou U = Aa? + Bf + Cz* + 2Fyz + V 3 =s a 3 x + /% + 7^. Si lx -t- my + nz = est 1 equation d une tangente commune aux coniques U+V^ Q, 7+ F 2 2 = 0, les formules de la "Note sur quelques formules &c." [90], en adoptant la notation de cette note, donneront les Equations t^ui serviront a determiner les valeurs de I, m, n ; on obtient par la 1 expression V(^ + ^ 2 2 +...)(^a 1 U...)-V(^ + ^a 1 2 +...)(^^+...X qui fait voir que liquation lx + my + nz = Q est satisfaite en ecrivant i + OD 71) - 66 2 524 SUB LE PROBLEMS DES CONTACTS. [91 et ces equations, qui peuvent aussi etre presentees sous la forme plus simple Ax + Hy + Gz : Hx + By + Fz : Gx + Fy+Cz (2), correspondent a un centre d homologie des deux coniques U + V? = 0, U + V? = 0. En mettant, pour abr^ger, ^(K + $(%* + ) Pi> *J(K + 1&ai2 + ...) = p 2 , \J(K + Ja 3 2 + . ..) =p s (3), on obtient facilement, pour un des axes d homologie des trois coniques, I dquation Pi, P*> PS = (4), Ax + Hy+Gz. a 1} a 2 , a 3 Hx + By + Fz, ft, ft, ft Gx + Fy + Cz, 7i, 72, 73 et celles des trois autres axes d homologie en peuvent etre tirees en changeant les signes de p l , p*, p s - Remarquons que 1 equation 1,1,1 =0 (5) Gz, a,, a 2 , a 3 Hx + By + Fz, ft, ft, ft Gx + Fy+Cz, 7 1; 73, 73 est celle de la polaire d un des centres de symptose des trois coniques par rapport a la conique circonscrite UQ, savoir de celle qui est donne e par le systeme V 1 = F 2 =F 3 , et que les Equations des trois autres polaires correspondantes se trouveront en changeant les signes de ct } , ft, 7^ ou de a 2 , ft, 73, ou de <z 3 , ft, 73. Cherchons le pole de 1 axe d homologie dont nous venons de trouver 1 equation, par rapport a une quatrieme conique inscrite U + F 2 = (V=ax + fBy + r yz). En exprimant cette equation par Ix + my + nz = 0, on obtiendra les coordonnees de ce pole au moyen de liquation ou, pour abreger, on a mis p = \/ (K + ^a 2 + ...). (Memoire cit^ ; equation (12).) Mais ici on a Ix + my + nz Pi, Pa, Ax+Hy + Gz, Hx+By + Fz, Gx + Fy + Cz, 91] SUE, LE PROBLEME DES CONTACTS, ce qui donne immediatement PI -, PI, P X, i, a 2 , a 3 a, a 1} r, A, A, A ^ 7i, 72, 7s 525 /n I P* \ 7, 7i. 72, 7s et de la on obtient \X + fj,Y + vZ = Pi, Pz, p X, a,, a 2 , /*, A, &, & *> 7i, 72, 7 A A, A, A 7. 7i 72, 7s (7); savoir, en considerant cette equation comme identique par rapport a X, ^, v , on obtient les coordonne es X, Y, Z du point dont il s agit. En prenant particulierement ce pole par rapport a la conique U+ V? = 0, cette equation se re duit a Pi vZ}=Pi Pi, p 2 fM, /9 1; /3 2 , fl Z/ ryj iy ou, si Ton veut, JT, F, Z seront determines par les expressions A, &, A 7i, 72, 73 (8), (9). Le facteur a e te supprime. De la on obtient aussi 1 e quation de la droite rnenee par ce point, savoir par le pole de 1 axe d homologie par rapport a la conique V+Vi=0, et par le centre de syraptose V^=V 2 =V 3 . En effet, cette e quation est (10). y ]? y Ys = 1 , 1 , 1 On pourrait dgalement chercher le point d intersection de la polaire du centre de symptose 7 1 =F 2 =F 3 par rapport a U+V* = Q, et de 1 axe d homologie; ce point serait evidemment le pole de la droite exprime e par la derniere equation, par rapport a U+ Fr = 0. Ces resultats seront utiles pour 1 interprdtation de la formule relative a la conique qui touche les trois coniques donndes et que nous irons chercher maintenant. 526 SUR LE PROBLEME DES CONTACTS. [91 Repre sentons par U+ F 2 =0 1 dquation de cette conique ; liquation (10) du me moire cite donnera les expressions (11), ...= - K -\- pp 3 , auxquelles nous ajouterons 1 equation qui donne la valeur de p, savoir gta 2 +... = -# + p 2 (12). II n y a qu a substituer dans cette derniere Equation les valeurs de a, /3, 7 que donnerit les trois autres. Par la on obtient, pour determiner p, une Equation du second degrd et de la forme K 2 L - 2KpM +p-N - (13) ; c est-a-dire, en faisant M 2 NL = O 2 , A ^ , on aura p = A.K et de la (14), ou A est une quantite connue, dont la valeur sera doimee dans la suite. Pour le moment il suffit de remarquer qu en changeant a la fois les signes de p 1} p. 2 , p a , fi, cette quantitd A ne change que de signe. Au lieu de chercher les valeurs de or, (3, 7, il vaut mieux eliminer ces quantites entre ces dernieres equations et 1 equation V = ax + fty + <yz 0. Cela donne, pour trouver V, 1 dquation F, z, vJk i qui peut aussi tre dcrite comme suit : V, A A x + Hy + Gz t Hx + By + Fz, Fy + Cz, = ... (15), - 1, - 1, Ap 3 - 1 73 = 0. (16). En mettant F = 0, on aura 1 equation de la chorde de contact de la conique cherche e et de la conique circonscrite U0. En remarquant que 1 equation de la 91] SUB LE PROBLEMS DBS CONTACTS. 527 conique cherche e peut etre mise sous la forme V = V U, 1 equation de cdtte conique se presentera sous la forme tres simple : \/-U, Ap l - 1, Ap 2 - 1, Ap a - Ax + Hy + Gz, a u Hx + By + Fz, ft, Gx + Fy+Cz, 7l , ou enfin, si Ton veut, sous la forme plus usitee : a 2 , &, 73 = 0. (17), a lt a., a 2 U + s A^j - 1, A^ 2 -l, Ap s -l ft, &, ft, ^ + % + *, a,, 2 , a 3 7i, 72, 7s Hx + By + Fz, ft, A, ft 8 G^ + ^y+C^, 7l , 72, 7s la premiere de ces deux formes est peut-etre la plus elegante. Les proprie tes geometriques sont absolument independantes de la valeur de la quantite A : mais pour completer la solution, je vais donner 1 expression de cette quantite. Pour cela, remarquons qu en mettant pour abre ger: 72, 7s (19), (pp 3 - K\ on aura d abord 1 equation identique ^Tn (ax + (3y + jz) = pp,- K, pp,-K, pp a -K Hy+Gz, !, a,, a 3 Hx + By + Fz, ft, ft, & Gx + Fy + Cz, 7l , 7,, 7:! ou, ce qui est la meme chose, KTl (ax + fty + yz) = Cela donne Gz) + p (Hx } + v (Gx + Fy + Cz\ = (4X- -f ...), 528 SUR LE PROBLEME DES CONTACTS. [91 c est-a-dire et, en vertu de cette expression, 1 dquation qui sert a determiner p se reduit a Ktt* + AK+...-Kn. *p 2 = ........................... (20). En la comparant avec liquation K 2 L-2KpM +p 2 N=0, on obtient + ...]. 1 M= [A& + 1.+ N = - KI1 2 + [A &P! + l a et de la, par urie transformation deja employ e e, M*-NL = & = IP ( - [A (1 IPI + I 2 p 2 + I 3 p s ) 2 + ...] mais 1 interpretation de ce rdsultat parait etre difficile. En revenant sur liquation trouvee pour la conique qui touche les trois coniques donne es, remarquons que les signes de a 1} /3 l} % , ou de a,, /3 2 , <y 2 , ou de a s , /3 3 , j 3 , peuvent etre changes conjointement. Cela revient en effet a ^crire F,, ou F 2 , ou - F 3 au lieu de + F 1} ou de + F 2 , ou de + F 3 , ce qui ne change pas les coniques inscrites. Mais il est facile de voir qu en changeant a la fois les signes de F l5 F 2 , F 3 , on ne change pas 1 dquation de la conique dont il s agit ; cette Equation ne change non plus, en changeant a la fois les signes de p 1} p 2 , p 3 , fl ; de maniere que liquation trouv^e correspond re ellement a 32 coniques diffdrentes. En distinguant ces 32 coniques par des symboles de la forme (F a , + F s , +FS, p lt p 2 , p 3 , n>: quatre symboles tels que ( F!, F 2> F 3 , ^i, p a , p 3 , O), (- Fj, - F a , - F,, P!, p,, p 3 , fl), ( F!, F a , F g , -p 1} -p a , -p 3 , -H), (- F n -F 2 , - F,, -p!, - p a , - p 9 , - fl), ne se rapporteront qu a une seule conique. Nous appelerons paires de coniques deux coniques quelconques exprime es par des symboles de la forme (F 1} F 2 , V 3 ,p 1>p2) p 3 , 0); done les 32 coniques forment 16 paires, groupies quatre a quatre de deux manieres differentes : savoir, pour former un groupe, quatre paires telles que (+F,, F 2 , V 3 , Pl , p 2 , p s , n), 91] S(JR LE PROBLEME DES CONTACTS. 529 ou quatre paires telles que ,, V 2 , V s , + Pl , p 2 , peuvent etre combinees. Ces deux especes de groupes peuvent etre distingue es par les noms groupes par rapport a un axe d homologie, et groupes par rapport a un centre de symptose. En effet: considerons une paire de coniques, par exemple celle qui est represented par les symboles ( V 1 , V 2 , V 3 , p 1 , p 2 , p s , II). Les Equations des deux coniques de la paire sont les memes aux valeurs de A pres ; il est done Evident que les chordes de contact de ces deux coniques avec la conique circonscrite U=0 doivent se rencontrer au point d intersection des droites 1,1,1 Gz, a 1} a 2 , &3 Hx + By+Fz, ft, &, & Gx + Fy+Cz, 7l , 73, 73 = 0, Pi, P-2, PB Ax + Hy+Gz, !, a 2 , a 3 Hx + By + Fz, fr, fa, & Gx+Fy + Cz, 7l) 70, 73 = 0. La premiere de ces e quations se rapporte a la polaire d un des centres de symptose des trois coniques inscrites par rapport a la conique circonscrite ; la seconde se rapporte a un des axes d homologie. On a done le theoreme suivant : "Les points de rencontre des 16 chordes de contact des paires de coniques sont les points de rencontre des polaires des quatre centres de symptose par rapport a la conique circonscrite, avec les quatre axes d homologie." Cela suffit pour expliquer la maniere dont les deux especes de groupes ont e te distingue es. Cherchons liquation de la droite menee par les points de contact d une des coniques inscrites (par exemple celle que donne 1 equation U+Vi 2 = 0) avec deux coniques de la meme paire. En representant par U + (ax ) 2 = et U+(a x + @ y + y z)- = les e quations de ces deux coniques : les equations des tangentes communes seront =0 et et 1 equation de la droite qui passe par les points de contact de ces deux droites avec la conique U + V? = est : (* + fry + 7i (23). Pour require cette equation, en exprimant par II, 1 1} m^ &c. les valeurs plus haut, mettons \ = h (Ap 1 1) + 1 2 (A.p 2 1) + l 3 (Ap 3 1) &c., et soient X , //, v ce que devien- nent les valeurs de \, /JL, v en ^crivant A au lieu de A. On aura C. 67 530 SUR LE PROBLEME DES CONTACTS. [91 et des expressions pareilles des valeurs de A, , /jf, v. Dela on tire n 2 (#/ - /3 7 ) = O - /" ") & + <X - v f \} |^ + (V - * <&> &c., fiv - fjfv = ( m ,n 3 - m 3 n. 2 ) {(Ap. 2 - 1) (A> 3 - 1) - (A> 2 - 1) (Ap, - 1)} + &c. = n (A 7 - A) [a, (p, -p 3 ) + 2 (p 3 -pi) H- 3 (^ -jp s )], &c. A -A c est-a-dire, en supprimant le facteur commun - ^ : &c. Aussi on aura, en supprimant le meme facteur : 7i (13 - /3) - & (7 - 7) = ( *i Pi + ^2 + Z 3 ^ 3 ) (7 + (ihpi -r n 2 p 2 + n 3 p 3 ) (&A - ! .&). De la on obtient imm^diatement A (/3y - 7) a? + ... = K [V, (p, -p 3 ) + F 2 (p 3 -p,) + V s (p, -p 2 )], et, par une reduction un peu plus difficile, = ( ! P! + lp* + hps) \? (PK + 33 A + ^70 - y (ffi x + JF A + OD7i)] -I- ) [y ( ai + 1^/3, + 7i) - (^i + 23 A et enfin 7^) [i W ~ /3 7) + A (7 a/ ~ 7 /a ) Done, en reunissant ces expressions des trois parties de 1 dquation dont il s agit, cette Equation se reduit a V, [(K + &<*> + . ...)p 2 ] = ..................... (24). 91] En mettant on aura SUR LE PROBLEME DES CONTACTS. 531 au lieu de K + 1&a.?+... et en supprimant le facteur commun p 1} = 0, et en remettant p^ (gta 1 2 + F 3 [# - j ...) = K, on obtient pour 1 equation dont il s agit : 1 , F 2 , 1 , F 3 1 Pi 2 - = ... (25). Cette Equation est celle de la droite mende par les points de contact de deux coniques de la meme paire avec la conique U + V : 2 = 0. Elle est la meme qui a e te deja obtenue pour la droite re sultante d une certaine construction geometrique ; on est done arrive au thdoreme connu suivant : " La droite mende par le pole d un axe d homologie par rapport a une des trois coniques inscrites, et par un centre de symptose, rencontre cette meme conique en deux points qui sont les points de contact de cette conique avec deux coniques de la meme paire ; " ou, ce qui revient au meme : "Le point de rencontre de la polaire d un centre de symptose, par rapport a une des trois coniques inscrites, et d un axe d homologie, est le point de rencontre des tangentes communes de cette meme conique et de deux coniques de la meme paire." 672 532 [92 92. NOTE SUE UN SYSTEMS DE CERTAINES FORMULES. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn, xxxix. (1850), pp. 14, 15.] LES formules dont il s agit se rapportent a la theorie de la composition des formes quadratiques. Je les presente ici pour faire voir la relation qui existe entre elles et quelques formules de mon memoire sur les hyperdeterminants (t. xxx. p. 1), [16]. En adoptant la notation de ce me moire, et en mettant 2A = 2A = (122 nil [212 221 savoir, en mettant pour abre ger, 2B = 2B = 2B" = (222} t (222) t 222J 2(7 = 2(7 = 2C" = [222J (121) [222] 222{ J a=lll, e = 112, 6=211, /=212, c = 121, # = 122, d=22l, h = 222; 2B =ah + bg-de-cf, C =bh-df, ^ A = af-be, 2B f = a h + cf-bg- de, C = ch - dg, \ A " =/0 ~ eh, 2B" = ah + de-bg- cf, C" = bc - ad, A = aa on aura identiquement : AA = A" a? + 2B"ae + C"e 2 .. AB = A"ac + B" (ag + ce) + C"eg , AC = A"c* + 2B"cg + C"g* j (IX (2); (3), (4); t 92] XOTE SUE, UN SYSTEME DE CERTAIXES FORMULES. 533 BA = A"ab + B" (af + be) + C"ef , ^ BB + = A"ad + B" (ah + de) + C"eh , ( * \ BB - = A"be+B"(bg + cf) + C"fg, \ EC = A"cd + B" (ch + dg) + C"gh ; J CA = A"b* + 2B"bf + G"f- , } CB =A"bd+B"(bh+df) + C"fh, I .............................. (6); CG = A"d? + 2B"dh + C"h 2 } = 2 - AC = B 2 - A G = B" 2 - A"C" ....................................... (7) = i {tt 2 A 2 + % 2 + c 2 / 2 + d*e 2 - Zahbg - Zahcf- Zahde - Zbgcf- Zbgde - Zcfde + 4<adfg + 4bech}. En regardant la seconde et la troisieme des equations (5) comme equivalentes avec la seule Equation 2BB = A"(ad+ be) +B"(ah + de + bg + cf) + C" (eh+fg), on trouvera que les systemes (4, 5, 6) r^pondent a la transformation "z* = (Ax* + 2Ba! 1 x s + Cx.?) (A y* + Wy.y, + C y.*) ......... (8), ^ = aacjiCi + by&a + cx^ + z 2 = ex^ +fy 1 as a qui appartient a une the orie dont celle des transformations line aires n est qu im cas particulier. Je profite de cette occasion pour donner une addition a la " Note sur les hyper- determinants " (t. xxxiv.), [54]. J y ai dit ( in.) que je ne pouvais pas expliquer la raison de ce que la courbe du sixieme ordre donne e par les Equations ae 4bd + 3c 2 = et ace + 2bcd ad 2 eb 2 c 3 = 0, ait une osculatrice deVeloppable qui n est que du sixieme ordre, mais que cette reduction s operait en partie au moyen des quatre points de rebroussement de la courbe. En effet, cette courbe, considered comme 1 intersection de deux surfaces, 1 une du second et 1 autre du troisieme ordre, a six droites que dans le memoire [30] cit^ dans cette note j ai nomme droites par deux points : cela suffit pour completer la reduction dont il s agit. M. Salmon, a qui je dois cette remarque, m a fait voir aussi que Fexpression que j ai donne e pour le nombre des points de rebrousse ment de la courbe, dans le cas d une Equation du m i0me ordre, combinee avec les forrnules du me moire mentionne , suffit pour former le tableau complet des singularite s de la famille de surfaces deVeloppables dont il s agissait, savoir: m > n > r, a, /3, ffi h, 3(m-2), m, 2(m-l), 0, 4(m-3), ^(m-l)(m-2), (9?i 2 - 53n + 80), Ici, dans la ligne supdrieure, m, n, r, a, j3, g, A, as, y ont les memes significations que dans le me moire dont je viens de parler; et dans la ligne infe rieure, m est le degre de I e quation primitive. 534 [93 93. NOTE SUR QUELQUES FOEMULES QUI SE RAPPORTENT A LA MULTIPLICATION DES FONCTIONS ELLIPTIQUES. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn, xxxix. (1850), pp. 1622.] LES fonctions {X, p, x, y] =P 6t0 - (j P M as + T P 0>1 y) + (^ P 2<0 ^ + J_ p x>1 ^ + J_ P ^ ou P 00 = l et les autres coefficients sont donnas par liquation a differences 2m + 2) (X - 21 + 2m + 1) P z _ 1>m - Mm [V ~ (2Z + 2m - 4) (X + /*)] P^, ^ = 0, jouent, comme je crois, un role important dans la thdorie des fonctions elliptiques 1 1 La fonction {X, p, x, y} satisfait a liquation , /I A \ ( o^ 2 { n d?U <,d?U\ +(l-4x-4y) x-,-2xy --+f]=.Q \ dx- dx dy dy 2 J qui peut etre tiree de liquation n (n - 1) x*u + (n - 1) (ax - 2x*) f + (1 - ax 2 + x*) - 2n (a 2 - 4) = Q/3c dx" dct (voyez le m^moire cite plus bas), mais qu on obtient plus facilement au moyen de 1 equation a differences a laquelle satisfont les coefficients Pi <m . 93] NOTE SUB QUELQUES FOBMULES QUI SE RAPPOBTENT &C. 535 En effet, en faisant x = \/k sin am M, a = & + T et en reprdsentant par z le ddnominateur de la fonction *Jk sin am nu (ou n est un entier positif quelconque), on aura ... +Z S + ... , cette serie etant continued jusqu au terme z^ ou z^ (n . 1} , selon que n est pair ou impair, et la fonction z dtant donne e par liquation z = - - 2ns, 2ns, , ou cependant les termes qui contiennent des puissances negatives de a doivent etre negliges. Ces formules reviennent a celles que j ai pre sente es dans la "Note sur les fonctions elliptiques" (t. XXXVIL), [67]. En revenant aux fonctions (X, yu,, a?, y}, j ai trouve les deux formules (ou selon la notation de Vandermonde la factorielle p (p -1) ... (p-q + l) est exprimee par [p]). De la, et en calculant la valeur de P 2i2 a 1 aide de 1 equation a differences, on obtient : -^ 0,1 == f^i P 2 = X (X 3), ,2 = p(p 3), 3! o = X(X-4)(X-5), 536 NOTE SUR QUELQUES FORMULES QUI SE RAPPORTENT [93 P M X(X-5)(X-6)(X-7), P 3il = X L (\ - 4) (X - 5) + 114X - 330 P., - X (X - 8) fr - 8) + 152X, + 336 - la premiere partie de cette table se trouve dans la note citde. Nous remarquerons en passant que pour y = 0, on a (X, p, x, 0} = (i + V(i #)) A - On salt que la thdorie de la multiplication des fonctions circulaires depend de la fonction (x + \/(a? 1)) A ou, en faisant %x~ 2 = x, de la fonction ( + V(i X )) A - Cela fait espdrer que Ton parviendra par les fonctions {X, yu., , i/} a la theorie complete de la multiplication des fonctions elliptiques. J ai calcule les valeurs qui servent a trouver les ddnominateurs z de sin am me, ou n est un quelconque des entiers 1, 2, 3, 4, 5, 6, 7. Les voici: z = z =l, z = z l = 64a 3 x w -16a 2 ( + 4a ( (27 5 2 2 = 16a 2 a? 20 - 4a oaf* 6, + z = I, - z, = - 2560 4 + 64a 3 ( - 16a 2 ( + 4a (1520 1 444^ + 105*), 93] A LA MULTIPLICATION DES FONCTIONS ELLIPTIQUES. 537 + 2 2 = 256a 4 # 24 - 64a 3 ( 12# 26 + 24^ 22 ) + 16a 2 ( 54 28 + 210 34 + 2520 20 ) - 4a (112^ + 576# 26 + 1584^ 22 + 1520 18 ) + (105^ 2 + 444 28 + 2400 24 + 7704 20 + 5814 16 ), -z,= - tf 36 , n = 7, z = z + z 1 + z., + z 3 . z<>= 1, ^ = 1024a 5 x u - 256a 4 ( + 64a 3 ( - 16a 2 ( 5425^ 20 + 5040a; 16 + 4a ( 35525 22 + 41300# 18 + 14934 14 (1662570 s1 + 2607500 s0 + 220395^ 16 + 147560 12 + 1304^ + 1960*), z 2 = 4096a 6 x- s - 1024a 5 ( 21^ + 28a- 26 ) + 256a 4 ( 189^ 2 + 4700 s8 + 3500 s4 ) - 64a 3 ( 952^+ 3192^+ 4550a; 26 + 2576 22 ) + 16a 2 (2940 36 + 11200 i r 32 +2l750^ 28 + 25452^ + 123970 s0 ) 4a (5733^ + 22064^ + 44324-r 30 + 82488a; 26 + 96761a; 22 + 409640") + (7007^ + 593880 s6 + 35231^ 2 + 41132a; 28 + 2781730 s4 + 3029180 SO + 949620 16 ), ^3 = 64a 3 ic 42 -16a 2 ( 7x"+ 420 40 ) + 4a (140 46 + 2360 4S + 8190 s8 ) ( 7^ + 3080 44 + 4053# 40 + 9S420 36 ). Pour rassembler tous mes r^sultats, je veux citer ceux que j ai donne dans le Cambridge and Dublin Mathematical Journal [vol. II. (1847), 45, and vol. ill. (1848), 57]. En ^crivant la lettre p au lieu de n (symbole qui represente le carre du nombre n de ce me moire), et en changeant les signes des termes alternatifs, on aura pour solution particuliere de liquation ~ - 2p (a 2 - 4) = c. 68 538 NOTE SUR QUELQUES FORMULES QUI SE RAPPORTENT [93 (savoir pour la solution qui pour p = if se r^duit au d^nominateur de sin am u) la valeur A & " &C " -* 1.2.3.4 * 1.2.3.4.5.6 les coefficients etant determines au moyen de 1 expression C r+2 = - (2r + 1) (2r + 2) (p - 2r) (p-2r- 1) O r + (2r + 2)(p-2r- 2) aC r+1 Cela donne les valeurs particulieres suivantes : &c. [viz. with the change referred to, these are the values of C n , C 3 , ... C 8 given ante p. 299]. On remarquera que dans ces formules le premier terme de C 8 ne contient pas, comme on pourrait 1 attendre, le facteur (p 25). Cela vient de ce que le coefficient C 8 est composd des coefficients des termes correspondants de z et z l} tandis que les coefficients G 7 , &c., sont tout simplement des coefficients de z . La suite des coefficients C office plusieurs discontinues de cette sorte. Par exemple on obtient generalement C r = (-Y+ 1 ( 2-*p (p-l*)...(p-( r - I) 2 ) CSof-* %*-*p (p-! 2 )...(p-(r- 2) 2 ) C r *0>-* + 2 2 - 9 p (p - I 2 ) . . . (p - (r - 3) 2 ) C r *ar~ mais le terme suivant ne contient pas le facteur p (p I 2 ) ... (p (r 4) 2 ). Quant a la loi des coefficients Cy 1 , C r 2 , C r 3 , on a C r a = (r - 3) [n (2r - 7) + (r - 1) (8r - 7)}, C r 3 = (r - 4) (r - 5) {n 2 (4r 2 - 24r + 51) + n (32r :i - 220r 2 + 412r - 255) + 2 (r - 1) (r - 2) (32r 2 - 88r + 51)j. Egalement, en ordonnant la s^rie suivant les puissances descendantes de x, la quantite z ^tant la solution particuliere qui pour p n 2 (n impair) se reduit au denominateur de sin am nu, on aura = (- 1)**- . ^P ,- - A + ft - &c. , 93] A LA MULTIPLICATION DES FONCTIONS ELLIPTIQUES. 539 ou les coefficients D sont (tonne s par 1 expression + (2r + 3) (p - 2r - 3) aD r+1 - 2p (of - 4) ce qui donne les valeurs particulieres suivantes : A = (P- i) , D t = 2(p-l)G> + &c. [viz. with the change referred to, these are the values of D 1} D 2 , ... D 6 given ante pp. 364, 365]. Les memes remarques sont applicables aux coefficients D ; seulement la discon- tinuite a lieu ici des le coefficient _D 4 . II parait que c est cause de cette discontinuity que le signe ndgatif se presente aux premiers termes des coefficients _D 4 , &c. En effet, on a gdneralement : ~ty (p- ( 2r ~ 3) 2 ) r(r-l)(p + 4r - 2) a. r ~ 2 + &c. ; ici la discontinuity se prdsente deja dans le terme suivant, qui ne contient pas le facteur (p l)(p 9) ... (p (2r 5) 2 ). Et c est precisement le terme suivant qui devient negatif dans les expressions de D 4 , D 5 et D 6 . Mais tout cela est moins important que la thdorie des series partielles z g) sur lesquelles les recherches ulte rieures seront k fonder. PROBLEME. Conner la solution de 1 equation a differences + m(p + 2l- 2m + 2) (p + 21 - 2m + 1) P i<m -i - 161m (\fj, - (21 + 2m - 4) (\ + /*)) P^,,^, = 0, dans laquelle P 0>0 = 1. (Voyez la "Note sur quelques formules &c.") 682 540 [94 94. NOTE SUR L ADDITION DES FONCTIONS ELLIPTIQUES. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn. XLI. (1851), pp. 5765.] SoiT, pour observer autant que possible la syme trie : ?/ Su = V& sin am -77- v# n U L>u = cos am 7 , V# n U (JTU= A am -7 , VK K K + iK et soit pour abreger : Su = x, Sv = y, &c. Cu= "" x ~ Gu= Cu Gu = l - Cela pose, les me thodes d Abel donnent les expressions suivantes de sin am d une somme quelconque d arcs : savoir a / \9 @ 3 Qyn-i. (H) ^2(5) /fcnirai S (u + v + . . .) = ( iyi-i L ^ c ^> y 2J [i, e\...e-- n -\ 0, s , ... ^-30] 94] NOTE SUE, L ADDITION DES FONCTIONS ELLIPTIQTJES. 541 pour un nombre impair 2n I d arcs, et _ [1, 6\ ... 6*\ 0, , ... S (U + v + --)--[0 f 3}> ^ 0mr-i t @ } 0,^ _ _ _ 271 - 2 pour un riombre pair 2w d arcs. Dans ces expressions les symboles dans lesquelles entrent les lettres 6, , sont censes repre senter les determinants, dont on obtient les termes en changeant successivement ces lettres en x, X\ y, Y; &c. J ai trouve qu on a aussi , . _ [e,, && ... ffi&,, ee> a , &&, ... ^- 3 @j -; ~ [ 1, 2 , ... G-\ 6 , 6 s ,... 6-* ] pour un nombre impair 2nl d arcs, et r( x_[0Q,. & lt ...ffi% lt % ~[0, 6 s , ... e-\ , pour un nombre pair 2?i d arcs. Les valeurs correspondantes de G(u + v+...) se trouvent en dchangeant les symboles / et @ /7 . Particulierement pour la somme de trois arcs on a : \6 d 3 I Pour r^duire ces expressions a une forme qui soit encore applicable au cas ou deux quelconques des quantites u, v, w sont egales, il n y a qu a multiplier les termes des fractions a droite par - cela donne, apres une reduction un peu difficile : - H [ 0, 6\ @ ] = (xYZ + yZX + zXY) - xyz (a - x> - y 2 - z" + aft/W ), ft [@ /( ffi,, e& lf ] = (1 - kxhfz^ X, Y,Z t - (yzX + zxY+xyZ) XJ tl Z tl , ft [ 1, 2 , 0&]= 1 - fz 2 - zW - a?y *+ax*y i z *-xyz(xYZ+yZX + zXY\ 542 NOTE SUE, L ADDITION DES FONCTIONS ELLIPTIQUES. [94 de maniere qu en e crivant M = 1 _ gpyi - yW - zW + ax-y-z 1 - xyz (xYZ + yZX + zXY), on a xYZ + yZX + zXY- xyz (a - a? - y* - z 2 M. 1 - M x,?,P,, - k (y zX + Y + *yv> x ?, z < M Les memes formules peuvent etre trouvdes plus simplement en e crivant u + \v + #, v + ^v + ^8 au lieu de u, v. La somme u + v + w se change par la en u + v + w + (v + 8), et les fonctions S (u + v + w), C (u + v + w), G (u + v + w) deviennent S (u + v + w), C (u + v + w), G (u + v + w). De plus x, X /y X tt , X et y, F /} F //( F se changent 1 -i X,, x s\-r) ell - x X t X . 1 -i Y,, . .. Y / Y . ., . - et , 77 - , i\/k - , , et 1 ( a; a, y VA; y y y 8 (u + v + w) = C (ll + V + W)= y = k a? , 1 , xX _^_ /y>3 /y ^ J| . t// j l/ j ^^ JL i. 2/ 2 , 1 , -yY 7/ 3 , 7/ , - F 0" ^^ * > ^ > Z 1 , 2 , ^^ a^Z,,, Z /7 , kxX, /v>3 V^ "V w/ j vU , *~ -/\- v/ 2 F F y * M* a kyY, ^ y > _ 7 2,, ^,, & 1 , ^ 2 , ^ -or, z,, \* X *_ /y3 rtrt _ Jf . t// j * * * /F,, F, , \yY lt y*> y, -Y ^ ~" ^ l *^l zZ, 1 , ^ 2 , zZ Ces formules conduisent aux formes re duites que Ton obtient en multipliant par le facteur beaucoup plus simple xY+yX En passant, il y a a noter les equations identiques x, X s , X = x 3 , x, -X y, f, Y y\ y, -Y z, z 3 , Z 1, z\ zZ , &c. 94] NOTE SUE, L ADDITION DBS FONCTIONS ELLIPTIQUES. 543 auxquelles conduit la methode qui vient d etre explique"e. Aussi en multipliant les valeurs de C(u + v + w), G(u+v + w) on obtient liquation _ VX YZ + AyzX + Mzx Y + dans laquelle A = II M = n N =n n = - a + 2 + 2 ) + (2a 2 - 4) ^y 2 ^ 2 ( 2 + + z*) - Pour le cas de quatre arcs, je n ai trouve que le sin am de la somme. En effet on a 1, tf 2 , a?*, I, t\ t\ tT x, x 3 , X, y?X y, f, Y, y*Y z, z 3 , Z, z 2 Z t, t 3 , T, t*T ou les termes de la fraction sont a multiplier par n = - yX} (xZ + zX) (xT+ tX) (yZ + zY} (zT+ tZ} (tY+ yT) (x> - f} ( 2 -* 2 ) (a? - e 2 ) (f - z 2 ) (z* - ?) (? - f) Mais il est plus simple de se servir de la forme S(u + v + w +p) = - 1 x*, x z , 1 , - xX zZ tT x 3 , x , x 2 X, X y\ y, -fY> -Y z, z 3 , Z, z -Z t , t 3 , T, VT Ton obtient de la meme maniere que la forme analogue pour trois arcs. Ici le (yX+xY)(zT+tZ) 2222 que facteur est et Ton obtient, toute reduction faite, 544 NOTE SUB L ADDITION DES FONCTIONS ELLIPTIQUES. [94 = (1 - afyW) (xYZT + yZTX + zTXY + tXYZ} - {(a - x- - y 2 - z 2 - 1- + y*zH 2 + zW + t 2 x 2 y 2 + x 2 y 2 z 2 - ax-y~z~t 2 ) x (Xyzt + Yztx + Ztxy + Txyz)}, = 1 - x 2 y 2 - x~z 2 - xH 2 - y 2 z 2 - zH 2 - t 2 y 2 - (x 2 y 2 + x 2 z 2 + x 2 F + y 2 z 2 + z 2 t 2 + t 2 y 2 ) X 2 y 2 z 2 t 2 + x A y*z*V + a. (x 2 y 2 z 2 + y 2 z 2 t 2 + zH 2 x 2 + t 2 x 2 y 2 ) + a. (x 2 + y 2 + z- + V) X 2 y 2 z 2 t 2 + (2 - 2a 2 ) x 2 y 2 2 2 t 2 - (x 2 Y 2 + y 2 X 2 ) ztZT - (x 2 Z 2 + z 2 X 2 ) yt YT - (x z T- + t 2 X 2 ) yz YZ -(y 2 Z 2 +z 2 Y II y a a remarquer qu en employant la premiere valeur de S(u + v + w+p) et le facteur correspondant, on aurait trouve le meme num^rateur, et aussi le meme denomi- nateur, ce qui donne lieu a des Equations identiques, semblables a celles qui ont lieu pour le cas de trois arcs. En mettant x 2 = a, T X = A, &c. on voit K Revenons a 1 expression x, X s , X z, z 3 , Z qui donne le numerateur de S ( qu il s agit d effectuer la division de 1, a, A (B+G)(C + A) 1, b, B 1, c, C par le produit (b c) (c a) (a b), les fonctions A, B, C denotant des racines carrees de fonctions rationnelles d une forme particuliere. Or, en supposant toujours que les carres de A, B, C soient des fonctions rationnelles, et d ailleurs d une forme quelconque, cela peut se faire dans tous les cas particuliers au moyen de liquation identique 1, a, A 1, b, B I, c, C (A 2 + B 2 + C 1, a, A 2 1, b, B 2 1, c, C 2 BC+CA+ AB) - 1, a, A 4 1, b, B* 1, c, <7 4 94] NOTE SUE I/ ADDITION DES FONCTIONS ELLIPTIQUES. De meme le ddnominateur de 8 (u + v + w) depend de liquation analogue 1, a, aA (B + C) (C + A) (A + B) 1, b, bB I, c, cC 545 1, a, aA 2 1, b, bB 2 l, c, cC 2 (A 2 AB)- 1, a, aA* I, b, bB* 1, c, cC* et le numerateur et le denominateur de S(u + v + w + p) dependent de I e quation 1, a, a 2 , a A I, b, b 2 , bB 1, c, c 2 , cC = M 1, d, d 2 , dD 1, a, a 2 , aA 2 N 1, b, b 2 , bB 2 1, c, c 2 , cC 2 1, d, d 2 , dD 2 1, a, a 2 , a A* 1, b, b 2 , bB 4 1, c, c 2 , cC* I, d, d 2 , dD* 1, a, a 2 , aA 6 1, b, b 2 , bB s 1, c, c 2 , cC 6 1, d, d 2 , dD s dans laquelle M = 2ab 2 c 2 + ... +a s b 2 + ... + a 3 bc + ... + 3a 2 bcd + ... , F = (a + b + c + d)(a 2 + b 2 + c 2 + d 2 ) + (abc + bed + cda + dab) , P = a + b + c + d et de 1 equation I, a, A, aA I, b, B, bB I, c, C, cC 1, d, D, dD = (a 2 b 2 + ...+ abc 2 + ... + Zabcd) 1, a, A 2 , a A 2 1, b, B 2 , bB 2 I, c, C 2 , cC 2 1, d, D 2 , dD 2 1, a, A\ a A* 1, 6, B*, bB* I, c, C*, cC* 1, d, D*, dD* mais je n ai pas encore trouve la loi gene rale de ces Equations. a OF THE UNIVERSITT 546 NOTE SUE, L ADDITION DES FONCTIONS ELLIPTIQUES. Pour faciliter 1 usage des symboles 8, C, G, je veux exprimer par cette notation les proprietes les plus simples des forictions elliptiques. Cela me donnera aussi I opportunite d arranger d une maniere particuliere les formules qui se rapportent a la somme ou a la difference de deux arcs. On a d abord Chi = 1-1. S 2 u , k G 2 u = 1 k . S 2 u , S u = Ou . Git , C u = -}.Su.Gu, K G u = -k.Su.Cu, 8 (0) =0, (0) =1, G (0) =1, 8 (0) = 1, C (0) = 0, G (0) - 0, 8 (- u) =-8 (it), G (- u) = C (u), G (- u) = G(v), 8 (i^) = , 8 Qv + &) = oo , Y to iu)= 0, 7(^)=-, Cfti; +*):= oo v)= k , G(&)= 0, . , v \/k . Cu 1, G (u + v 8 (u + mv C (u + mv + n8) = (- !) . Cu G (u + mv + ntf) = (- l) n . Gu 94] NOTE SUR L ADDITION DES FONCTIONS ELLIPTIQUES. 547 1 ik Dans ces equations les symboles S, C, G, v, X, k, k , i et 8, G, C, X, v, j- , -r- , i peuvent etre echange s les uns d avec les autres. Les formules fondamentales qui se rapportent a deux arcs sont Su .Cv.Gv + Sv. Cu . Gu $< + *>- I-&H.&V Cu.Cv-.Su.Gu.Sv.Gv Gu.Gv-k.Su.Cu.8v.Gv l-Vu.Vv -> auxquelles on ajoutera : . (1 + S 2 u . S 2 v) (Cu . Gu . Cv . Gv) - Su .Sv(a- 2Shi - 2S 2 v + a S 2 u . 8 2 v) Mais pour trouver toutes les formes differentes de ces equations, mettons pour abreger (en supposant comme auparavant Su = x, &c.) : A=a;7 , A = yX, B = X I Y I , B =- R = I -kx* - kf + xhf, S = X Y , S =-jxy , T=xX tl Y t , T =-yY /t X t , U = xX / Y //) U = yT t X H> K=l- x 2 f- Alors on aura , A + A _ P _ U+U _T-T K ~ A- A ~ B -B ~ C -C" ,_B + B U-U Q S + S K A -A B - B C-C" C+C T+T S- S R 692 548 NOTE SUE L ADDITION DES FONCTIONS ELLIPTIQUES. [94 et les valeurs correspondantes de S (u v), C (u v), G (u v) se trouveront en echangeant les signes de A , B , C , S , T, U . [Reverting to the functions sin am, cos am, A am, or say sn, en, dn, instead of S, C, D, and introducing Dr Glaisher s very convenient notation s 1} C T , d for the sn, en, dn of u, and s. 2 , c 2 , d 2 for those of v, the formulae just obtained may be written / \ _ i SoCjC?! Si 2 S 2 2 SjCjC^a + S. 2 C 2 d 1 , x _ CiC 2 SjdjSA s&dv s^d-i 1 &! 2 s? + k 1 k^s-iS^ sfiidv s 2 Cidi CjCa + s^s^ d^ + k^ _ Sjd^ S^C-L _ c^c^ + k^s^ _ 1 k 2 sf k*s + k^sJ , , , _ j 1 viz. we have thus a fourfold representation of the addition-equation for each of the three functions.] De ces formules il peut etre tire 7 un grand nombre d dquations identiques; par exemple celles-ci : (A 2 -A *) = KP, &c., S 2 -S 2 = QR, &c., (B + B ) (C - C ) = K(S + S ), &c., (B - B ) (G +C ) = K(S- S ), &c., BC-B C = KS, &c., B C-BC = KS , &c., S T U + S TU +8T U + STU +8T U + S TU + S T U = PQR, (A - A ) (S + S ) = (C-C )(U- U ), &c., (A+A )(S -S )= (C + C ) (U + U \ &c., (A - A ) (T - T ) = P (C - C ), &c., (A+A )(T + T ) = P(C + C ), &c., (A-A )(U+ U } = P(B- B \ &c., (A+A XU- U ) = P (B + B ), &c., &c., &c., 94] NOTE SUR L ADDITION DES FONCTIONS ELLIPTIQUES. 549 et ces equations donnent imme diatement et dans les formes les plus simples, les formules qui se rapportent aux sommes et aux produits des fonctions de u + v et u v, par exemple savoir au moyen de la premiere de ces equations identiques: p de maniere que toutes ces formules peuvent etre considerees comme comprises dans les equations fondamentales et dans ce systeme d equations identiques. 550 [95 95. NOTE SUE QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn. XLI. (1851), pp. 6672. Continued from t. xxxvm. p. 104, 70.] VII. EN considerant les soixante droites auxquelles donne lieu le the oreme de Pascal, et en appliquant ce the oreme aux hexagones differents qui peuvent etre formes par six points sur une meme conique, M. Kirkman a trouve" que ces soixante droites se coupent trois a trois non seulement dans les vingt points de M. Steiner (points que M. Kirkman nomrne les points g\ mais aussi dans soixante points h. II a trouve aussi qu il y a quatre-vingt-dix droites J, dont chacune contient deux des points h et un des quarante cinq points p, dans lesquels s entrecoupent, deux a deux, les droites mendes par deux quelconques des six points. Les recherches e"tendues que M. Kirkman a faites dans la geome trie de position, paraitront dans un nume ro prochain du Cambridge and Dublin Mathematical Journal, [t. v. (1850), pp. 185200]. En attendant, M. Kirkman a publid dans le Manchester Courier du 27 i0me Juin 1849, vingt cinq theoremes qui contiennent les resultats de ses recherches. Moi, j ai depuis trouve que les soixante points h sont situds trois a trois sur vingt droites X. Tous ces theoremes peuvent etre demontres assez facilement quand on connait la maniere suivant laquelle les points et les droites doivent etre combine s en construisant les points et les droites h, J, &c, Cela se fait alors d une maniere tres simple, au moyen d une notation que je vais expliquer. Reprdsentons les six points sur la conique par 1, 2, 3, 4, 5, 6. En combinant ces points deux a deux par les droites 12, 13 &c., les systemes tels que 12, 34, 56 95] NOTE SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 551 peuvent etre represented par les combinaisons binaires des six symboles a, b, c, d, e, f, et au moyen de la table qui se trouve III. de ce memoire, savoir la table 12. 34 56 = ac 13.45.62 = ab 14 .56 23 = bd 12. 35 64 = be 13 . 46 . 25 = cd 14 .52 36 = ae 12. 36 45 = df 13 . 42 . 56 = ef 14 .53. 62 = cf 15 . 62 . 34 = de 16 .23. 45 = ce 15 .63.42=5c 16 .24. 53 = ad / 15 . 64 . 23 = af 16 .25. 34 = bf: le symbole ac denote ici 1 ensemble des droites 12, 34, 56; et ainsi de suite. On voit que pour obtenir les six cotes d un quelconque des soixante hexagones, il n y a qu a combiner les droites correspondantes, par paires, telles que ab, ac, qui ont une lettre en commun. Cela pose, les hexagones, ou, si Ton veut, les droites derivees de ces hexagones au moyen du theoreme de Pascal (droites que je nommerai droites de Pascal), peuvent etre represented par les symboles ab . ac, &c., conformement a la table que voici: 216345 =fb .fd 453 = ec . eb 534 = ad . ac 354 = da . df 213456 = ab.ac 214356 = cf.ca 215346 = ed .e b 564 ef.el> 563 = db.df 463 =fa.fd 645 = dc.df 635 = ea . eb 634 = cb .ca 465 = cd . ca 365 = ae . ac 364 = bc .be 546 = ba.be 536 =fc .fd 436 = de . df 654 =fe .fd 653 = bd . be 643 af . ac 3U256 = ea.ef 315246 = c 6. cd 31 6245 = ad. ab 562 = bd . ba 462 = a/.a6 452 ce.cd 625 = cf.cd 624 = ed.ef 524 =fb . fe 265=/c./e 264 = de . dc 254 = bf . ba 526 ae.ab 426 bc.ba 425 = da . dc 652 = db . dc 642 =fa .fe 542 = ec .ef 416235 = ce . cf 516234 = ec .ed 352 = ad. ae 342 = 6/.6c 523 = bf.bd 423 = ad . af 253 =fb .fc 243 = da.de 325 = ec . ea 324 = ce .cb 532 = da . db 432 =/6 .fa 543 = ce . ca 415236 = af . ae = cb . cf 263 = ed . ea 326 =fa .fc 632 = bc .bd 552 NOTE SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [95 Remarquons maintenant que les droites de Pascal qui passent par un point p, tel que 12.45, sont ca ce, ba.be, ac . ab, ec.eb. Cela dtant, le point 12.45 peut etre repre sente par la notation cb . ae, et de cette maniere le systeme complet des points p est repre sente par la table suivante : 14 . 35 = ab . de 23 . 45 = ad . bf 23. 46 = 6c.de 12. 36 = 06. ce 14.56 = a/.ce 23 . 56 = ae . cf 12. 4,5 = ae.be 15.23 = be.cd 24.35 = 6/.ce 24>. 36 = af.de 12. 46 = ad. c/ 12. 56 = bf.de 13.24>=ac .bd 13 .2Q = bd.ec 13. 45= cf.de 13 . 46 = ae . bf 13 . 56 = ad. be 14 . 23 = ac . ef 14. 26 = ad. be 15. 24 = ae . df 15 . 26 = ac . bf 15 . 34 = ab . cf 15.36 = ad .fe 16 . 23 = ab . df 16 . 24 = cf . be 16. 25= ac.de 16.34 = ae .cd 16.35 = bc .ef 16 . 45 = af . bd 24 . 56 = ab . ed 25 . 34 = ad .ce 25.36 = bd.cf 25 . 46 = ab . ef 26 . 35 = ae . bd 26.4<5 = cd.ef 34 . 56 = be . df 35 . 46 = ac . df 36 . 45 = ac . be. Enfin les droites 12, &c. peuvent tre repre sente es par des symboles tels que ac.be.df, &c., et au moyen de la table suivante, qui est pour ainsi dire la reciproque de la table (A) : ac . be .df=12 ab.ed.fc =62 ae. df cb = 36 ac . bd fe = 56 ab .ef .cd = 13 ae . dc bf=52 ac . bf ed = 34 ab . ec . df= 45 ae . db fc = U ad.fb . ce= 16 af. be . ed = 15 ad .fc . eb = 53 af. be .dc = 64 ad ,fe . be = 24 af. bd.ce = 32. II y a a remarquer qu une droite de Pascal ab.ac contient les points be . ad be ae bc.af et que par un point ab.cd passent les droites (les cdte s opposed d un hexa^one) ac.bd.e/, ab.bc.ef, et les droites de Pascal ca . cb, da.db, ac . ad, bc.bd. Cela pose en combmant les proprie^s deja connues d avec celles que j ai enoncdes au commencement J cette section, en particularisant en meme temps les combinaisons qui donnent lieu aux point* et droites g, h, J, & c . et en adoptant une notation convenable pour ces points et droites, on trouvera ce qui suit: (a) Les droites ab .be, bc.ca, ca.ab se rencontrent dans un meme point abc qui est un des vmgt points g, et que j ai de not^ par ce symbole, III. de ce memoire. 95] NOTE SUE, QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 553 (/3) Les points abc, abd, abe, abf sont situs s sur une meme droite ab qui est une des quinze droites de M. Steiner ou de M. Plticker, et que j ai de notee ( III.). Je nommerai droites / ces droites. (7) Les droites ab . ac, ac . ad, ad . ab se rencoritrent dans un meme point a . ef qui est un des soixante points h de M. Kirkman. (S) Les points b . cd, c . db, d . be sont situes sur la meme droite {bed} qui est une de mes vingt droites X. (e) Les points ab . cd, e . ab, f. ab sont situes sur une meme droite (ab) cd qui est un des quatre-vingt-dix points J de M. Kirkman. Quant aux the oremes (a) et (/3), je vais reproduire dans la notation de cette section les demonstrations de M. Pliicker. Voici le principe de la demonstration du theoreme (a): principe qui s applique aussi, comme nous le verrons, aux demonstrations des theoremes (y, 8, et e). Supposons qu il s agit de demontrer ge neralement que trois droites X, X , X" se rencontrent dans un meme point, et supposons que ces droites sont determine es : X au moyen des points A , B , C , X A , B , C , X" A", B", C"- formons d abord la table A A", B B", G C", () < A" A , B"B , C"C , (A A , BB , CC , ou A A", fec. sont les droites qui passent par les points A et A", c. ; et puis la table B B". C C", C C".A A", A A". B B", (])) < B"B . C"G , C"C . A" A , A" A . B"B , (BB .CC , CC .AA , AA .BB , ou B B" . C C", &c. sont les points d intersection des droites B B" et C C", &c. On sait que si les points de 1 une quelconque des colonnes verticales de cette derniere table sont situes sur la meme droite, les droites X, X , X" se couperont dans un meme point; et reciproquement. Precise ment de la meme maniere on ddmontrerait que trois points X, X , X" sont situs s sur une meme droite ; seulement A, B, &c. seraient des droites, A A", &c. des points ; et ainsi de suite. Or les droites du theoreme (a) sont determine es, ab . be au moyen des points ac . be, ac .bf, ac . bd, be . ca ba . ce, ba . cf, ba . cd, ca . ab cb . ae, cb . af, cb . ad ; C. 70 554 NOTE SUR QTJELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [95 done la table (o) se r^duit a ac.be. df, ac . bf. de, ac . bd . ef, ba.ce.df, ba.cf.de, ba.cd.ef, cb .ae. df, cb . of. de, cb .ad. ef, et la table (])) a be.df, bf.de, bd.ef, ce . df, cf. de, cd . ef, ae . df, af. de, ad . ef; et les points de la premiere colonne verticale de cette table sont situes sur la droite ed . ef, ceux de la deuxieme colonne verticale sur la droite fd .fe, et ceux de la troisieme colonne verticale sur la droite de . df: 1 existence de 1 une quelconque de ces droites fait voir la verite" du thdoreme dont il s agit. Pour de montrer le theoreme (/3), consideVons a part un quelconque des points abc, abd, abe, abf; par exemple le point abf. On peut envisager ce point comme determine par les droites ab . af, ab . bf, et ces droites contiennent : ab . af les points bf. ac, bf. ad, bf. ae, ab . bf af. be, af. bd, af. be. Or bf.ac et af. be sont situes sur la droite ab.de. cf; bf.ad et af. bd sur la droite ab.ec.df; et bf. ae et af. be sur la droite ab . cd . ef. De plus, les points bf.ac, bf.ad, bf.ae sont situes sur les droites ab .be, ab .bd, ab.be respectivement, et les points af. be, af.bd, af.be sont situds sur les droites ab . ac, ab.ad, ab.ae respectivement. Done on peut repre senter les points de la droite ab . af par les symboles (ab.de.cf) (ab.bc), (ab . ec . df) (ab . bd), (ab . cd . ef) (ab . be), et les points de la droite ab . bf par les symboles (ab . de . cf) (ab . ac), (ab . ec . df) (ab . ad), (ab . cd . ef) (ab . ae). Maintenant Les droites de meme que les droites se rencontrent sur la droite ab . bd, ab . ae ab . be, ab . ad, ab.de. cf, ab.be, ab . ac ab.bc, ab.ae, ab.ec.df, ab . be, ab .ad ab . bd, ab . ac, ab . cd . ef, c est a dire, il existe un systeme de trois hexagones dont les cote s sont (ab.de. cf, ab.ec.df, ab . cd . ef, ab.bc, ab.bd, ab.be), (ab.de. cf, ab.ec.df, ab . cd . ef, ab.ae, ab.ad, ab.ae), (ab .be , ab .bd , ab .be , ab . ac, ab . ad, ab . ae). 95] NOTE SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. 555 Ces hexagories ont pour angles les memes six points. Or 1 existence de 1 une ou de 1 autre des droites ab . of, ab . bf suffit pour faire voir que ces six points sont situds sur la meme conique : done les cote s opposes du troisieme hexagone se rencontrerit dans trois points situes sur la meme droite. De plus, on voit aisement que les hexagones sont precise ment tels, qu en vertu du thdoreme (a), les trois droites, auxquelles donnent lieu ces hexagones, se rencontrent dans un meme point ; et ce point sera evidemment le point abf. Mais les cote s opposes du troisieme hexagone, savoir les droites ab . be et ab . ac ; ab . bd et ab .ad ; ab . be et ab . ae, se rencontrent dans les points abc, abd, abf: done les quatre points abc, abd, abe, abf sont situes sur la meme droite : theoreme dont il s agissait. Pour demontrer le theoreme (7), on n a qu a conside rer les droites de ce theoreme com me determine es, ab . ac par les points be . ad, be . ae, be . af, ac . ad cd . ab, cd . ae, cd . af, ad . ab bd . ac , bd . ae, bd . af. La table (0) se reduit alors a be . ad . ef, da . de, da . df, cd . ab . ef, ba . be, ba . bf, bd .ac . ef, ca . c e, ca . cf, et la table (])) a (da . df) (da . de), af. de, ae . df, (ba . bf) (ba . be), af. be, ae . bf, (ca. cf) (ca . ce), af . ce, ae . cf. Or les points de la deuxieme colonne verticale sont situds sur la droite ea . ef. et les points de la troisieme colonne verticale sur la droite fa.fe. L existence de 1 une ou de 1 autre de ces droites fait voir que les droites ab . ac, ac . ad, ad . ab se rencontrent dans un meme point a . be. Pour demontrer le theoreme (S), il est evident que les points de la premiere colonne verticale de la table qui vient d etre pre sentee, sont situe s sur la meme droite. Mais ces points sont prdcisement les points d .be, b .cd, c.db; le the oreme est done de montre . [En remarquant que les trois droites sur lesquelles sont situes les neuf points de la table generale ())) se rencontrent dans un meme point, cette demonstration de 1 existence des droites X fait voir que la droite {bed} passe par le point d intersection des droites ae.af, ef.fa, savoir par le point afe; ou bien que chacune des vingt droites X passe, non seulement par trois points h, mais aussi par un seul point g. Ce theoreme est du a M. Salmon, qui independamment de mes recherches a trouve 1 existence des vingt droites X. 8 aout 1849. Inserted here from Crelle, t. XLI. p. 84.] 702 556 NOTE SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION. [95 Enfin, pour demontrer le theoreme (e), nous pouvons considerer les points de ce theoreme comme determines, ab.cd par les droites be . bd, ad.be . ef, ac . bd . ef, f.ab fc.fd, fc .fe , fd.fe , e . ab ec . ed, ec .fe , ed .fe La table (0) se reduit alors a cd . ef, f .ce, f . de, cd . eb, cf. eb, df. eb , cd. bf, ce .bf, de.bf, et la table (D) a fe , ce. cf, de . df, fe .fb , cb . ce, db . de, fe.eb, cb.cf, db.df. Or les droites de la premiere colonne verticale de cette table se rencontrent dans le point bf. be, celles de la deuxieme colonne verticale dans le point c . ad, et celles de la troisieme colonne verticale dans le point d . ac ; le theoreme dont il s agit est done demontre. Dans cette demonstration on aurait aussi pu e changer les lettres a, b. Les theoremes (a) et (7) peuvent etre e nonces par le seul th^oreme suivant : "Etant donnes six points sur la meme conique, et menant par ces points neuf droites, de maniere que chaque droite passe par deux points et que par chaque point il passe trois droites : on formera avec ces neuf droites trois hexagones differents dont chacun a les six points pour angles. Les droites de Pascal, auxquelles donnent lieu ces trois hexagones, se rencontreront dans un meme point." En supposant que le systeme de neuf droites contient toujours un meme hexagone, il est possible de completer de quatre manieres differentes le systeme des neuf droites; savoir, on peut aj outer aux cote s de 1 hexagone 1 les trois diagonales de 1 hexagone 2, 3 ou 4, en menant une quelconque de ces diagonales et deux droites, chacune par deux angles alternes de 1 hexagone. Ces quatre systemes donnent lieu au point g, et aux trois points h, qui se trouvent sur la droite de Pascal, correspondante a 1 hexagone dont il s agit; savoir le premier systeme donne lieu au point g, et les trois derniers systemes au point h. 96] 557 96. MEMOIKE SUE, LES CONIQUES INSCEITES DANS UNE MEME SURFACE DU SECOND ORDRE. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn. XLI. (1851), pp. 7386.] EN conside rant une surface quelconque du second ordre, le probleme se presents : d examiner les proprietes des coniques inscrites dans cette surface et des cones circon- scrits. La plupart de ces proprietes est peut-etre connue 1 ; cependant je crois qu on ne les a pas encore de veloppe es systematiquement. Je me propose de donner ici 1 analyse des proprietes les plus simples d un tel systeme de coniques, et la solution du probleme analogue au probleme des tactions qui se presente ici, ainsi que quelques theoremes relatifs au passage a un systeme de coniques situees dans un meme plan et inscrites dans une meme conique, en me re servant pour une seconde partie de ce memoire les deVeloppements ulteYieurs concernant ce passage, et la solution complete du probleme analogue au probleme de Malfatti, generalise par M. Steiner. Remarquons d abord que les coniques inscrites et les cones circonscrits, ainsi que les plans des coniques inscrites et les sommets des cones circonscrits, sont des figures reciproques par rapport a la surface du second ordre. En considerant deux coniques inscrites quelconques, et les cones circonscrits correspondants, on remarquera que les plans des coniques inscrites se rencontrent dans une droite. Je la nommerai Droite de symptose. Les sommets des cones circonscrits seront situes dans une droite que je nommerai Droite d homologie. Ces deux droites seront eVidemment rdciproques 1 une a 1 autre. II se trouvera sur la droite d homologie deux points dont chacun est le sommet d un cone qui passe par les deux coniques inscrites. Ces deux points peuvent etre nommes Points d homologie. De meme il passera par la droite de symptose deux plans, 1 Voyez le memoire de M. Steiner " Einige geometrische Betrachtungen " Journal t. i. [1826] pp. 161 184, et un memoire de M. Olivier, Quetelet, Corresp. Math. t. v. [1829]. 558 MEMOIRE SUR LES CONIQUES INSCRITES [96 qui sont les plans des coniques dans lesquelles se coupent les deux cones circonscrits. Ces deux plans peuvent etre nomme s Plans de symptose. Les plans de symptose et les points d homologie ne sont pas seulement des figures rdciproques : les deux plans de symptose passent aussi par les deux points d homologie, chacun par le point re ciproque de 1 autre plan ; c est a dire : les plans de symptose sont des plans conjugues par rapport a la surface du second ordre, et les points d homologie sont des points conjugues par rapport a cette meme surface. Remarquons aussi qu en considerant le systeme forme par les plans des coniques inscrites et les plans tangents a la surface mene s par la droite de symptose, on trouvera que les plans de symptose sont les plans doubles (ou si Ton veut les plans auto- conjugues) de 1 involution. De meme, en considerant le systeme forme par les sommets des cones circonscrits et par les points de leur intersection avec la surface de la droite d homologie, on trouvera que les points d homologie sont les points doubles (ou auto-conjugues) de 1 involution. Les deux cones circonscrits qui ont pour sommets les deux points d homologie, peuvent etre nommes Cones d homologie; de meme, les deux coniques inscrites, situees dans les deux plans de symptose, peuvent etre nomme es Coniques de symptose. (En passant, nous remarquerons que ces coniques de symptose correspondent aux Potenzkreise de M. Steiner.) II est Evident que les cones d homologie et les coniques de symptose sont des figures reciproques. En considerant trois coniques inscrites, et les c6nes circonscrits correspondants, on verra que les plans des coniques inscrites se rencontrent dans un point que je nommerai Point de symptose. Les sommets des cones circonscrits seront situe s dans un plan que je nommerai Plan d homologie. Ce point et le plan seront re ciproques 1 un a 1 autre. En combinant deux a deux les coniques inscrites ou les cones circon scrits, cela donne lieu a trois droites de symptose qui passent chacune par le point de symptose, et a trois droites d homologie situees chacune dans le plan d homologie. II existe aussi six plans de symptose qui se coupent trois a trois dans quatre droites, aretes d une pyramide quadrilatere qui a pour axes les trois droites de symptose. Les quatre droites dont il s agit, peuvent etre nominees Axes de symptose. II existe egalement six points d homologie, situes trois a trois dans quatre droites, cote s d un quadrilatere qui a pour axes les trois droites d homologie. Les quatre droites dont il s agit, peuvent etre nominees Axes d homologie. La pyramide et le quadrilatere sont des figures reciproques, et il convient de remarquer (quoique cela soit assez evident) qu il y a ici trois points d homologie, non-situe s dans un des cote s du quadrilatere, mais contenus dans trois points de symptose qui se coupent dans une arete de la pyramide. Par 1 un quelconque des axes d homologie il passe deux plans dont chacun touche les trois coniques inscrites; de meme il se trouve sur 1 un quelconque des axes de symptose deux points, dont chacun est un point d intersection des trois cones circonscrits. Cela constitue la solution du probleme : Trouver la conique inscrite, ou le cone circonscrit, qui touche trois coniques inscrites, ou trois cones circonscrits. II y a huit solutions de ce probleme. Avant d aller plus loin, je vais indiquer quelques-unes des formules analytiques correspondantes a la theorie qui vient d etre expliquee. 96] DANS UNE MEME SURFACE DU SECOND ORDRE. 559 Ecrivons, pour abreger : U = Aa? + Bif + Cz V= ox + (3y +yz + 8w, et representons par 21, 53, (, 3), %, , >, S, S DR, de -4, 5, (7, D, F, G, H, L, M, N. Soit de plus X = Ax + Hy + Gz + Lw, + 2Lasw les coefficients du systeme inverse a + = 2la 2 Cela pose, en prenant U =Q pour Equation de la surface du second ordre, et V 1 = 0, Fg = 0, F 3 = pour les Equations des plans des coniques inscrites, on obtient pour 1 un des plans de symptose des coniques inscrites ( U = 0, V l = 0), (U0, V 2 = 0) 1 dquation tres simple p^V^ =piV 2 = 0. De la on tire pour les coordonnees du point d homologie, qui est le reciproque de ce plan de symptose, les equations X : Y : Z : W=&i-& En formant dgalement les expressions des coordonndes d un point d homologie des deux autres paires de coniques inscrites, on obtient pour equation de 1 un des axes d homologie : X, Y, Z, W =0; Pi, <*!> @1, 7l> g l Pa, OBI @a, J-2> $2 PS, a s, @ 3 , 7s> ^s savoir, en choisissant quatre colonnes verticales quelconques de cette formule, on trouve que les determinants que Ton obtient, sont tous egaux a zdro. Nous ajouterons que la droite qui, par rapport a la conique (U=0, V 1 0), est reciproque de cet axe d homologie, est donn^e par les Equations Vi = 0, F 2 : F 3 = p,p z - SM, - ... : p,p 3 - Sl^o, - ... ; il est clair que cette droite rencontre la surface du second ordre en deux points situes dans les plans des coniques inscrites qui, au moyen de 1 axe d homologie dont 1 equation vient etre donnee, sont determines de maniere a toucher les trois coniques inscrites donnees. Mais sans se servir des equations de cette droite, on peut determiner 1 equation des deux plans mends par 1 axe d homologie dont il s agit, de maniere a toucher la conique inscrite (U0, ^ = 0); et la symetrie du resultat fera voir que ces deux plans touchent aussi deux autres coniques inscrites. La recherche de cette Equation etant un peu difficile, je la donnerai en detail, en supposant cependant counu le theoreme suivant : En ecrivant v = ~\a + py + vz + pw, v = \ x + pfy + v z + p w : les plans menes par la droite (v = 0, v=0) de maniere qu ils touchent la conique inscrite (7=0, V=0), sont par 1 equation f [21 (Xu - X u) 2 +...]- [8l (\v - \ v) + . . .] 3 = 0. 560 MEMOIRE SUR LES COXIQUES IXSCRITES Pour appliquer ce the oreme an probleme dont il s agit, nous n avons qu a substituer Fj au lieu de V, et qu a ecrire a, b , c , d X, Y, Z, W Pi, i, &, 7i. Si v = a , b , c , d X, Y, Z, W Pi, i, &, 7i> Si ou les coefficients a, b, c, d; a , b , c , d sont des quantite s quelconques. Reduisons d abord 1 expression 21^ (Xi/ \ v ) + . . . . Pour cela, mettons dans les valeurs de v, v , les expressions 21 i + . . . , ^^ + . . . , &c. a la place de x, y, ... : les quantites X, Y, Z, W deviennent alors Ka. l} Kft-L, K<y l , K^ (ou comme a 1 ordinaire K est le determinant forme par les quantites A, B, ...), et Ton obtient ainsi, aux signes pres : A + ... =Kp l a , b , c , d a , b , c , d i, A, 71, ^i et de la i (Xv x - Vu) + . . . = ou = a , b , c a , b , c , d X, Y, Z, W Pi, <*i, A, 7i> ^ a, c, a , 6 , c , c? X, Y, Z, W PI formule qui au moyen des propriety s des determinants se re duit a X, Y, Z, W a , 6 , c , d, a , 6 , c , d , Pi, a i, A, 71, S 1} Passons a 1 expression 21 (\v \ v) + ... , que nous mettrons sous la forme i {A [21 (\v -\ vy-+. ..] 2 +...]. En prenant des quantite s quelconques a, b, C, U, on obtient par une analyse semblable 96] DANS UNE MEME SURFACE ou B = a, b , c , tr a , b , c , d - a , b , c , d X, Y, Z, W Pi, a i, 0i, 7i. Si Pi, i, 0i, 7i, I 561 formule qui se reduit a Pi, a , b , t , & Z, F, Z, W a , b , c , & a , 6 , c , c? Pi, i> A, 71, $1 a , 6 , c , d a , b , c , d a , b , c , d X, Y, Z, W Pi De la, en supprimant les facteurs constants de Q et de Q, on obtient a , b , c , & X, Y, Z, W Pi, expression qui sert a definir les fonctions f, 77, f, w. L equation qu il s agissait de trouver devient X, Y, Z, W 2 =0, ou il faut avoir egard que Ton a X Ax + . . . , &c. Savoir, liquation qu on vient d ecrire se decompose necessairement en facteurs lindaires qui, egale s a zero, donnent les Equations des plans des coniques inscrites qui touchent chacune les trois coniques inscrites donne es. Nous avons obtenu ce re sultat en traduisant en analyse une construction geome - trique ; mais on y peut aussi parvenir en considerant le probleme d une maniere purement analytique. En effet : soient comme plus haut, U = 1 equation de la surface du second ordre, F x = 0, F 2 = 0, F 3 = les Equations des plans des trois coniques inscrites donne es, F = 1 dquation du plan de la conique inscrite qui touche chacune de ces trois coniques. La condition pour que cette conique touche la conique inscrite situde dans le plan V l = 0, est Slaotj + ... =ppi- On a done les trois equations . . . = pp 3t Au lieu de tirer de ces equations les quantites a : /9 : 7 : 8, nous ajouterons an systeme la nouvelle Equation out; + . . . = 0, C. 71 5(>2 MEMOIRE SUE LES CONIQUES INSCRITES [96 par laquelle il sera possible d eliminer les quatre quantites a, /3, y, 8. En attribuant a X, ... la meme signification qu auparavant, nous mettrons les quatre equations sous la forme Ecrivons de plus OU + ...)*. + ...=pp 2 , (21 a + ...) a + ... =p, ou les quantites a, b, C, fo sont arbitraires. En eliminant de ces equations les fonctions (2l + ...), puis en mettant a la place de p la quantite a gauche de 1 equation, on obtient, a un facteur constant pres, a, b , t , U X, F, Z, W pi, i, A, 7i, g i cela donne d abord ! + ...=! X, Y, Z, W , puis, en ecrivant ^> 2 = 2la 2 + . . . = ...)+ ...], on a et a , 1) , t , a Z, 7, ^, W et de la enfin, en substituant dans 1 equation 8laa x + . . . = p^ , on obtient comme plus haut 1 equation X, F, Z, W 3 = 0. II est clair que cette analyse peut etre appliquee a la solution d un nombre quelconque d equations de la forme 2laa 1 + ... = pp^. 96] DANS UNE MEME SURFACE DU SECOND ORDRE. 563 En revenant a la thdorie geometrique, considerons un point quelconque que nous prendrons pour point de projection : le cone qui passe par une conique inscrite quel conque aura (comme on sait) un contact double avec le cdne qui a pour sommet le point de projection. Le plan de contact sera le plan mene par le point de projection et par la droite d intersection du plan de la conique inscrite et du plan reciproque au point de projection. En conside rant plusieurs coniques inscrites ayaiit une droite de symptose commune, tous les cones auxquels donnent lieu ces coniques inscrites, auront pour aretes communes les deux droites menees par le point de projection aux points dans lesquels la surface est rencontre e par la droite de symptose commune. Aj on tons que les plans de contact des cones dont il s agit, avec le cone circonscrit, rencontrent le plan des deux aretes communes dans une droite fixe, savoir dans 1 une ou 1 autre des droites doubles (ou auto-conjuguees) de 1 involution formee par les deux aretes communes et par les droites dans lesquelles le plan de ces deux aretes communes rencontre le cone circonscrit. De plus, en conside rant les plans tangents rnenes par 1 une ou par 1 autre des deux aretes communes, ces plans tangents forment un systeme homologue a celui des plans des coniques inscrites. En considerant en particulier 1 une ou 1 autre des coniques de symptose de deux coniques inscrites quelconques : le plan tangent du cone correspondant est le plan double (ou auto-conjugue) de 1 involution formee par les plans tangents des cones qui correspondent aux deux coniques inscrites (c est a dire par les plans tangents qui passent par 1 arete commune dont il s agit), et par les plans tangents de la surface du second ordre mene s par cette meme arete commune. C est la en effet la propriete qui conduit a la construction des coniques de symptose de deux coniques situees dans le meme plan et considere es comme inscrites dans une conique donnee. 712 564 [97 97. NOTE SUR LA SOLUTION DE L EQUATION x 257 -l=0. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn. XLI. (1851), pp. 8183.] SOIT p m la m ifime puissance d une racine quelconque (1 unite excepte e) de liquation x- 57 1 = 0, et representons par a une racine quelconque (1 unite exceptee) de 1 equation a 258 _ i = o. En posant 1 equation (Po + *pi +%...+ a 255 > 255 ) 2 = M (p + tfp, + a*p a ... + ap, 55 ), on sait que la quantite M peut etre exprime e en fonction rationnelle de a. Cette fonction une fois connue, donnera tout de suite la valeur de 1 expression (p n + ap l + a?p 2 ... + a 235 ^) 256 en fonction rationnelle de a, et cela suffit pour resoudre 1 equation dont il s agit. Une solution du probleme a ete donnee depuis longtemps par M. Richelot qui commence par supposer que a. soit une racine primitive de 1 equation a 128 1=0. Cette solution est comprise, comme cas particulier, dans celle que je vais donner. La question est d ailleurs interessante, a cause de son rapport avec la theorie des nombres. En effet, quoiqu en tant que je sache Ton n a pas encore trouve la regie pour former a priori la valeur de M, il est clair que les recherches de MM. Jacobi et Kummer doivent conduire a cette regie. Le re sultat ici bas pourra servir pour la verifier. Voici la valeur que j obtiens pour la fonction M : M = - 2 + 2a - 2a 4 + 2a 5 + 2a 7 + 2a 9 - 2a 10 - 2a 14 - 2a 16 + 2a 21 + 2a 23 + 2a 25 - 2a 26 - Ztf* + 2^ - 2a 30 + 2a : - Za 34 - 2a 36 + 2a 37 - 2a 38 + 2a 45 + 2a 47 - 2a 48 + 2a 49 - 2a 50 + 2a 51 + 2a 53 - 2a w - 2a i0 + 2a 61 - 2a w + 2a 65 - 2a 66 + 2a 67 - 2a 68 + 2a 69 - 2a 72 - 2a 74 + 2a 75 - 2a 76 + 2a 79 - 2a 80 - 2a 82 2a 84 2a 86 + 2a 89 2a 92 2a w 2a 100 + 2a 101 2a 104 2a 10ti - 2a 108 + 2a 109 + 2a lu + 2a 113 2a 114 + 2a 115 + 2a 117 + 2a 119 2a 122 + 2a J - 3 2a 124 + 2a 127 2a 128 + 2a 129 - 2a 130 - 2a 134 + 2a 135 - 2a 138 + 2a 141 + 2a 145 + 2a 147 + 2a 151 + 2a 153 - 2a lr>4 2a 156 a 160 + 2a 161 + 2a 163 2a 164 + 2a 165 2a 166 + 2a 171 + 2a 177 + 2a 181 2a 182 + 2a 183 2a 186 + 2a 187 + 2a 189 2a 190 2a 192 + 2a 195 2a 198 2a 198 + 2a 199 2a 206 - 2a 212 + 2a 213 - 2a 214 + 2a 215 - 2a 218 + 2a 217 - 22 220 + 2a 221 + 2a 223 + 2a" 5 - 2a 226 + 2a 227 - 2a 228 + 2a 229 + 2a 233 2a 234 + 2a 233 2a 236 + 2o^ 2a 238 + 2a 239 2K 240 + 2a 243 - 2a M - 2^. + ^w _ 2<x^ + 2a 249 - 2a 232 - 2a 2M . 97] NOTE SUR LA SOLUTION DE LIQUATION X 257 - 1 = 0. 565 Reprdsentons par M ce que devient M, en supposant a 128 + 1 = 0, nous obtiendrons M = 2a 2 - 2a 4 + 2a 5 + 2a 6 + 2a 9 - 2a 13 - 2a 14 - 2a 16 - 2a 17 - 2a 19 + 2a 21 + 2a 29 - 2a 30 + a 32 - 2a 34 - 2a 35 - 2a 43 + 2a 45 + 2a 47 - 2a 48 - 2a 50 + 2a 51 - 2a 55 + 2a 58 - 2a 59 - 2a 60 + 2a 62 + 2a 65 - 2a 66 + 2a 69 + 2a 70 - 2a 71 - 2a 72 - 2a 74 + 2a 75 - 2a 76 + 2a 78 + 2a 79 - 2a 80 - 2a 82 - 2a 85 - 2a 87 + 2a 88 - 2a 98 - 2a 94 - 2a 95 - 2a 97 + 2a 98 - 2a" - 2a 104 - 2a 105 - 2a 107 + 2a 110 + 2a 112 + 2a 113 - 2a 114 + 2a 116 + 2a 117 + 2a 118 + 2a 120 - 2a 121 - 2a 122 + 2a 123 + 2a 126 + 2a 127 . Soient M 1} M{ ce que devient M en supposant successivement a 128 1 = 0, a 64 + 1 = 0, et soient M 2 , M. 2 ce que deviennent M ou Mj_ en supposant successivement a 64 1 = 0, a 32 +l=0, et ainsi de suite, jusqu a M 7 , M 7 qui seront ce que deviennent M ou M l} &c. en supposant successivement a 2 1 = 0, a+l = 0; nous aurons : M l = - 4 + 4a - 2a 2 - 2a 4 + 2a 5 - 2a 6 + 4a 7 + 2a 9 - 4a 10 + 2a 13 - 2a 14 - 2a 16 + 2a 17 4- 2a 19 + 2a 21 + 4a 23 + 4a 25 - 4a 26 - 4a 28 + 2a 29 - 2a 30 - a 32 + 4a 33 - 2a 34 + 2a 35 - 4a 36 + 4a 37 - 4a 38 + 2a 43 + 2a 45 + 2a 47 - 2a 48 + 4a 49 - 2a 50 + 2a 51 + 4a 53 - 4a r>4 + 2a 55 - 2a 58 + 2a 59 - 2a 60 + 4a 61 - 2a fi2 - 4a w + 2a 65 - 2a 66 + 4a 67 - 4a 68 + 2a 69 - 2a 70 + 2a^ - 2a 72 - 2a 74 + 2a 75 - 2a r(i - 2a 78 + 2a 79 - 2a 80 - 2a 82 - 4a 84 + 2a 85 - 4a 86 + 2a 87 - 2a 88 + 4a 89 - 4a"- + 2a 93 - 2a M + 2a 95 + 2a 97 - 2a 98 + 2a" - 4a 100 + 4a 101 - 2a 104 + 2a 105 - 4a llKi + 2a 107 - 4a 108 + 4a 109 2a 110 + 4a nl 2a 112 + 2a 113 2a 114 + 4a 115 2a 116 + 2a 117 - 2a 118 + 4a 119 - 2a 120 + 2a 121 - 2a 122 + 2a 123 - 4a 124 - 2a 126 + 2a 127 . MI = 2a - 4a 3 + 2a 4 + 2a 7 + 2a 8 + 2a 9 - 2a 10 - 2a n + 2a 12 + 2a 13 - 2a 15 + 2a 17 + 2a 18 + 2a 19 + 4a 20 + 4a 22 + 2a 23 + 2a 24 - 4a 26 - 2a 31 - a :ia + 2a 33 - 4a 38 + 2a 4(> - 23 41 + 4a 42 + 4a - 2a 45 + 2a 46 - 2a 47 + 2a 49 - 2a 51 + 2a 52 + 2a 33 - 2a M - 2a 55 + 2a 56 - 2a 57 + 2a 60 + 4a 61 - 2a 63 . M, = - 8 + Ga - 4a 2 + 4a 3 - Ga 4 + 4a 5 - 4a 6 + Ga 7 - 2a 8 + 2a 9 - Ga 10 + 2a n - 2a 12 + 2a 13 - 2a 14 + 2a 15 - 4a 16 + 2a 17 - 2a 18 + 2a 19 - 4a 20 + 4a 21 - 4a 22 + Ga 23 - 2a 24 + 8a 25 - 4a 2(i - Sa 28 + 4a 29 - 4a 30 + 2a 31 - a 32 + Ga 33 - 4a** + 4a 35 - Sa 3 " + 8a 37 - 4a 38 - 2a 4U + 2a 41 - 4a 42 + 4a 43 - 4a + Ga 43 - 2a 46 + Ga 47 - 4a 48 + 6a 4!1 _ 4 a so + g a 5i _ 2052 + g a 53 _ g a 54 + o a s5 _ ^se + %& - 4a 58 + 4a 59 - Ga BO + 4a 61 - 4a (i2 + 2a 63 . M a = -7 + 2a 4 - 4a 5 + Ga 7 - 2a 10 - 2a u + 2a 12 - 4a 13 - 2a 14 - 4a 15 - 4a 17 + 2i 18 _ 4 a i _ 2a 20 2a 21 + 2a- 2 + Ga 25 4a 27 2 * *. 566 NOTE SUR LA SOLUTION DE LIQUATION X 1 = 0. [97 M, = -9 + 12a -8a 2 + 8a 3 - 14a 4 + 12a 5 - 8a B + 6a 7 -4a H + 4a 9 -10a 10 + 6a u - 6a 12 + 8a 13 - (ja 14 + 8a 15 - 8a 16 + 8a 17 - 6a 18 + 8a 19 - 6a 20 + 10a 21 - lOa 22 + I2& 3 - 4a 24 + lOa 25 - 8a 2(i + 4a 27 -14a 28 + 8a 29 - 8a 30 + 4a 31 . j|// = - 1 + 4a - 2a 2 - 8a* + 2a 5 + 2a 6 - 6a 7 - 8a - Ga 9 - 2 a 10 4- 2a n + 8a 12 + 2a 14 + 4a 15 . J/ 4 = - 17 + 20a - 14a 2 + 16a 3 - 20a 4 + 22a 5 - ISa 6 + 18a 7 - 8a 8 + 14a 9 - ISa 10 + lOa 11 - 20a 12 + 16* 1:! - 1 4a 14 + 12a 15 . Ml = - 9 + 6a + 4a 2 + 6a 3 + 6a 5 - 4a 6 + 6a 7 . JI/5 = - 25 + 34a - 32a 2 + 26a 3 - 40a 4 + 38a 6 - 32a 6 + 30a 7 . Jlf s = -15-4a-4a 3 . Jl/" B = - 65 + 72a - 64a 2 + 56a 3 . / 7 =-129 + 128a. --257. Ces differentes expressions dtant trouvdes, supposons qtie a soit une racine primitive, et repre sentons par F, F lt F 3 , ... F 7 ce que deviennent Jlf, If/, Tlf/ ... ^// en substituant a, a 2 , a 4 , ... a 128 au lieu de a (F=M / , ^ = -257); nous aurons cette equation constitue la solution dont il s agit. Ajoutons encore les formules beaucoup plus simples qui correspondent a liquation x 1 -1= 0. En supposant a 16 - 1 = 0, nous aurons M = - 2 - a 4 + 2a e + 2a 7 - 2a 8 + 2a fl - 2a 10 + 2a 13 - 2a 14 . M = - 2a + 2a 2 - a 4 + 2a 6 + 2a 7 . Mi = - 4 + 2a - 2a 2 - a 4 + 4a 5 - 2a 6 + 2a 7 . 7H/ = - 3 - 2a - 2a 3 . M a = - 5 + 6a - 4a 2 + 2a 3 . JJ// = - 1 + 4 a . I/, = - 9 + 8a. ^a = -17; et de la, en supposant que a. soit une racine primitive: (p* + api+... +OL 15 p 15 ) lB = (- 2 + 2a 2 - a 4 + 2a B + 2a 7 ) 8 . (- 3 - 2a 2 - 2a 6 ) 4 . (- 1 + 4a*) a . 17. Pour a; 5 - 1 = on obtient sans peine Oo + P! + tfpz + a?p z y = (- 2 - a 2 + 2a 3 ) 2 . 5 ; ou a est une racine primitive de liquation a 4 - 1 = 0, savoir &= + {. 98] 567 98. NOTE RELATIVE A LA SIXIEME SECTION DU "MEMOIRE SUR QUELQUES THEOREMES DE LA GEOMETRIE DE POSITION." [From the Journal fur die reine und angewandte Mathematik (Crelle), torn. XLI. (1851), p. 84.] This Note which is thus entitled in Crelle, but which in fact refers to the seventh section of the Memoir, is inserted in its proper place, ante p. 555. 568 [99 99. NOTE SUE QUELQUES FOKMULES QUI SE KAPPORTENT A LA MULTIPLICATION DES FONCTIONS ELLIPTIQUES. [From the Journal fur die reine und angewandte MatUematik (Crelle), torn. XLI. (1851), pp. 8592.] Ex revenant sur liquation {- l\-m^-\-(l- my 2 } Pi <m + I (X - 2Z + 2m + 2) (X - 2 + 2m + 1) P^ <m 2m + 2) (p+2l- 2m + dans laquelle P 0(0 = l, les expressions que j ai donndes pour P^ , P (1 [93, see p. 535] peuvent etre ecrites comme suit : P M = M [X - Z - l]-i + l\ [x - /]^ l8Z - 16 + -- + _ ( _ 1Q ^ _ \ A t / equations qui peuvent etre representees par Pl,o = Ql,o, PI,I = Qi,i + RI,I; ic-r~ i X + /i ce qui conduisent a la forme -PZ,2= Ql, 2 + ^,3; 1 C ; -- HjRi, 2;2/ ^ X + p (X ou les coefficients P ne contiennent que la seule quantite X, et oil Q M est line fonction integrale du Z i6me ordre par rapport a X, et du second ordre par rapport a /*. 99] NOTE SUR QUELQUES FORMULES QUI SE RAPPORTENT &C. 569 Cela donne \p , >^> 2 {"\ &-1.2 + ^-1,2; ! ^^ + Rlr-l.l; 2 "\ 2 2 - 321 (X,* - 2Z (X + /*)} &-i,i 4- ft-Mi = 0, ce qui se reduit a {- l\-^ + (l- 2) } Q z , 2 - (I - 2) (X - I + + Z (X - 2Z + 6) (X - 2 + 5) |Q|_ M + ^z-i.aji ^^ + ^-1,2; + 2 (/A + 2Z - 2) (yu + 2Z - 3) Q u + 2 O - X + 4^ - 5) \pR ttl . x + 2 (X - 22 + 2) (X - 2 + 3) R ltl - 20 ^_ 1>1; x + 32XjR t _ 1 , 1 . x j = 0. En ne faisant attention d abord qu aux termes qui contiennent des puissances nega tives de X + /U-, nous obtenons et La premiere equation, en calculant la constante arbitraire au moyen de -R 2(2>2 = 200, donne 1 expression de E 2i2j2 = 200 se trouve par celle de P 2 , 2 qui peut etre ecrite sous la forme P 2>2 = X (X - 3) /a 0* - 3) + 152X/* + 336 - 40X> + (40\ 2 - 1156) ^ + En substituant la valeur de -R J>2 . 2 et celles de c. 72 570 NOTE SUR QUELQUES FORMULES QUI SE RAPPORTENT [99 dans la seconde Equation, on obtient - 20l\ (X - 21 + 3) [X - 1] 1 1 - 1201 (l-l)\s[\-l + i]-3 = o, c est a dire - 20l\ [\ - I] 1 -* {6 (I - 1)X 2 (X - I + 1) + (X- 21 + 4) (\ - 21 + 3) 2 (X -21 + 2)}. Mettons Ri <2 , 2 = l(l-l)\[\-l + l] l ~ 3 j ; cela donne 20 + (X - 21 + 4) (X - 21 + 3) 2 (X - 21 + 2)], ce qui devient, quelques reductions faites, _ 20 (I - 2) (I - 3) 20 (I - 2) (I - 4) \-l+l \-l+2 et de la on tire ^ = C - 340J + 1 X (/ -\- L Or .R 2 , 2 . , = 2^, = 40X 2 - 1156 ; done ^ 2 - 20\ 2 - 578, et de la C = 62 - 60X ; done enfin, en restituant la valeur de -R- - I 1 X I -\- L Passons a 1 expression de Q l ]2 . Elle donne {- l\-2^ + (l- 2) 2 } Q L< , + l(\-2l + 6)(\-2l + 5) &_ lj2 + 2 (/* + 21 - 2) (/, + 21 - 3) Q fil - 32 {X^ - 2^ (X + ^)} Q,_ ltl - 20 ^_ lil; j = ; ou la derniere ligne se reduit a -21(1- 1) M 2 [X - * + !]-- 62 - 20^ - 120X - I l-l X-J+1 99] A LA MULTIPLICATION DES FONCTIONS ELLIPTIQUES. 571 Done on a {l\ + 2p - (I - 2) 2 } Q lt , - I (X - 21 + 6) (X - 2Z + 5) Q,_ li2 2) = 2 (p, + 21 - 2) (p + 21 - 3) \fj,\ [X - I - I] 1 - 1 + l\ [X - IJ-* (181 - 16 + - ^^ .- + ir(l-84 + - g < < - 2 >< - 8 )) \ A, 6 -(- 1 / - 201 (41-5+p- x) x> [x - rp- 2 Equation qui peut etre represented par {l\ + 2 f i-(l-2y }Q lta -l(\-2l ou les valeurs de H, 33, (2T, 13 sont donndes par les formules \ - 32ZX (X - 20 [X - IJ~* - 20ZX 2 [X - l] l ~ 2 , OT = 2 (2 - 2) (2Z - 3) X [X - i - I] - 1 + 2 (4* - 5) JX [X - ^] z - 2 J18Z - 16 + 2( ^ ( + 64Z 2 X 2 [X - l] l ~* - 32^ (Z - 1) (X - 20 X [X - Z + I] - 3 \181 - 34 + 2 ^ ( x i + - 20Z (4^ - 5 - X) X 2 [X - l] l ~ 2 2) 20 (< - 1) (t - 2) - a (t - 1) V [X - < + 1]- 68 - 20Z I 6 J. = 2 (2Z - 2) (21 - 3) a [X - J] - 2 (l82 - 16 + 2 ^- 1 )^ C 2 (I 2} (I + 64- (i - 1) X 2 [X - I + l] l ~ s \ 181 - 34 + -^-r i Sans m arreter a reduire ces expressions aux formes les plus simples, j ecris en substituant cette valeur, les termes qui contiennent p? se detruisent, et la compa- raison des autres termes donne 2I t - 6X [X - I - I] 1 - 1 + {l\ -(I- 2) 2 } X [X - I - I] - 1 2l + o)\[\- l] l ~- - i3 = 0, 72 2 572 NOTE SUR QUELQUES FORMULES QUI SE RAPPORTENT [99 2J t + [l\ -(I- 2) 2 } {- 3X [X - J - l]i-i + 7;} - I (X - 21 + 6) (X - 2Z + 5) {- 3X [X - ] z ~ 2 + 7^} - OP = 0, {ZX -(I- 2) 2 } J, - Z (X - 21 + 6) (X - 21 + 5) J^ -39 = 0. Les valeurs de 7 Z et J t peuvent etre tirees sans integration de la premiere et de la seconde de ces Equations. La valeur ainsi trouvee de J z satisfera a la troisieme equation (ce qui cependant doit etre verifie a posteriori). II m a paru plus simple de tirer la fonction I t de la premiere Equation, et celle de J t en integrant la troisieme Equation ; alors ce sera la seconde Equation qu il y a a verifier. En effet on obtient 7, = ix x - i-* mi + 4 - L equation qui sert a determiner J t devient, en substituant la valeur de {l\-(l-2Y\ J t -l(\- 2^ + 6) (X- 2J + 5)/i_ 1 = 41(1- 1) (21 - 3) X [X - 1]^ \181 - 16 + 2 (^ ~ [ X i - 1) X 2 [X - J + I] - \181 - 34 + X t + 1 En ecrivant on trouve [d_(J_2))F,-(!-2)(X-J+2)F^ . 8 ( - 3) P- " + )-" + *) _ -!)(<- 8) X i + l X I , ( X t + En faisant cette equation se rdduit a {l\ -(I- 2 )} M l - (I - 2) (X - I + 2 (X fc + l)(X X + 128ZX (9 - 17) + 128 ^ X L -|- J_ et cela donne tout de suite la valeur de B L , et apres quelques reductions celle de A t . 99] A LA MULTIPLICATION DES FONCTIONS ELLIPTIQUES. 573 On obtient ainsi B l = 2(l-2)(l- 3) (I - 4) (21 - 3). En substituant ces valeurs, on a, toute reduction faite: c est a dire IMt -(1-2) M^ = 1296P - 25201 + 192, Mj-M^ = 648Z-624, ou enfin Mj = 324Z 2 - 300Z - 528 , M^ = 324Z 2 - 948Z + 192 : valeurs qui (comme cela doit etre) se changent 1 une dans 1 autre en entrechangeant les quantites I, I \. De la on obtient valeur qu on trouverait aussi par 1 autre proce de indique ci-dessus. Or nous avons \/j, done enfin, en r^unissant les valeurs des diffe rentes parties de P M , on obtient 574 NOTE SUR QUELQUES FORMULES QUI SE RAPPORTENT [99 Equation qui fait suite aux Equations 7 nZ-i_L A f-x /~]Z-2jl7 1 -"V -^/v "/( ir^ r^ m-9. ^ t 1J + l\ [_A_ tj < lot ID Je vais essayer maintenant a chercher d une maniere plus systdmatique les terraes de Pi, m qui ne contiennent pas la quantite /i, ou bien a chercher la solution de I dquation + I (X - 21 + 2m + 2) (\ - 21 + 2m + 1) P z _ 1>m + 2w (I - m + 1) (21 - 2m + 1) P t>m _ 1 + 32lm (l + m-2) PI^^-I = 0. En supposant que p _ rnm -\ r-\ / i i iz-m-i 5 1 ^^.^.P ^i.-ra ^L^- s p^j:^riy (ou la sommation se rapporte a p, nombre qui doit etre etendu depuis p = jusqu a p = m), on obtiendra sans peine + 2m (2i - 2m + 1) [X - 21 + 2m? ou p s ^tend seulement jusqu a m - 1 dans la troisieme et dans la quatrieme ligne. Pour re duire la premiere ligne, j ecris l\ + (l- m) 2 = -l(\-l + m -p)+ [ m "- - ( m ce qui r^duit le terme general a 99] A LA MULTIPLICATION DES FONCTIONS ELLIPTIQUES. 575 expression qui, en ecrivant r + 1 au lieu de p dans le premier terme, et r dans le second terme, pent etre remplace e par La seconde ligne donne tout de suite le terme general /o\ (7 m) Ai_ l<m>r+l Pour reduire la troisieme ligne, je mets [X - 21 + 2m] 2 - [\ - I + m - p] 2 - 2 [I - m -p + I] 1 [X - I + m - p -I] 1 + [I - m-pj, ce qui re duit le terme general a 2^ - 2m + 1) [I - m - p]"- expression qui (en ecrivant r + 1 au lieu de p dans le premier terme, r dans le second terme et r 1 dans le troisieme terme) peut etre remplace e par (y) 2m (2 - 2ire + 1) [I - m - r + I] 2 Pour reduire la quatrieme ligne, je mets X = (X I + m p) + (I m +p), ce qui re duit le terme general a expression qui (en e crivant r + 1 au lieu de p dans le premier et r dans le second terme) peut etre remplace e par 576 NOTE SUE QUELQUES FORMULES QUI SE RAPPORTENT &C. [99 Done en re unissant les expressions (a), (/3), (7), (8) et eri faisant attention que le coefficient du terme qui contient [A, I + m l] r au denominateur, doit se re duire a , on obtient, en arrangeant encore les termes d une maniere convenable : 2m (21 - 2m + 1) Ai <rin _ l<r+l + 32lm (l + m- 2) ^_i, m _i >r+1 1 A ~ ( - a - - 4m (21 - 2m + 1) (I - m - r + 1) A^ m ^ 1<r + 32lm (l + m-2)(l + m-r) ^_ 1>m _ M . + (m 2 - (m + r) I) A^ m<r 2m (21 -2m + l)(l-m-r + l)(l- m - r) ^, m _ 1>r _ 1 = 0, ou r s dtend depuis r = jusqu a r = m, en reduisant a zero les termes pour lesquels le troisieme suffixe est ndgatif ou plus grand que le second suffixe. Par exemple dans le cas de r = m, on obtient 1 e quation tres simple qui (sous la condition Ai i0t0 = l) donne sans peine la valeur ge nerale de ^z, m>m , savoir: A t , m , m =[2l-m-l] m [l-m- l] m : valeur qui peut etre presente e sous d autres formes en considerant a part les deux cas de m pair et de m impair. En supposant m = l, m = 2, on obtient AM = 2 (I ~ 1) (I ~ 2), A^ <2 = 2(21- 3) (I - 2) (I - 3) (I - 4) : valeurs qui servent a verifier des resultats deja trouve s. 100] 577 100. NOTE SUE LA THEOEIE DES HYPERDETERMINANTS. [From the Journal fur die reine und angewandte Mathematik (Crelle), torn. XLII. (1851), pp. 368371.] DANS la thdorie dont il s agit, je suis parvenu a un theoreme qui pourra, a ce qu il me parait, conduire a des deVeloppements interessants. Je ne considere ici que le cas d une fonction homogene a deux variables, et en me servant des nouveaux termes de M. Sylvester, je nomme Covariant d une fonction donne e, toute fonction qui ne change pas de forme en faisant subir aux variables des transformations lineaires quelconques, et Invariant toute fonction des seuls coefficients qui a la propriete mentionnee. Cela pose , soit U une fonction donne e quelconque, du degre n par rapport aux variables, et, comme a 1 ordinaire, contenant des coefficients arbitraires a, b, c, &c. Soit Q un covariant quelconque (y compris le cas particulier ou Q est un invariant) de la fonction V, s le degre de Q par rapport aux variables, r le degre de cette meme fonction Q par rapport aux coefficients. En supposant que la fonction U ait un facteur 6 V (ou = Ix + my est une fonction lindaire des variables), ou autrement dit, en supposant 1 dquation U6 V V, je dis que le covariant Q contiendra ce meme facteur 6 e leve a la puissance TV \ (rn s). En effet, en se rappelant la me thode dont je me suis servi dans la seconde partie de mon memoire sur les Hyperdeterminants (t. xxx. de ce Journal, [16]) (je suppose que le lecteur ait ce me moire sous les yeux), on verra que cette fonction Q, supposee, comme plus haut, du degre r par rapport aux coefficients, sera ne cessairement de la forme _s Q = 12 a 13 23 V ... U,U 2 ... U r , puisque les coefficients n entrent dans Q que par les fonctions C/i, U 2 , &c. Or Q etant du degre s par rapport aux variables, on obtient s = rn 2 (a + /3 + 7 + ...), c est a dire: a + + 7 . . . = (rn - s). Cela pose , puisque U= V V, on aura de meme U 1 = 6 1 "V 1 , U<, = V V, &c. Les expressions 12, &c. qui entrent dans 1 expression de Q contiennent d Xl , d Vl , &c. : symboles c. 73 578 NOTE SUR LA THEORIE DES HYPERDETERMIXANTS. [100 qui doivent etre remplace s par 9 a , 1 + 9e 1 , 9,,, + ^, &c. en supposant (comme il est permis) que les nouveaux symboles d Xl , d Vl , &c. ne se rapportent plus a 6-! V l , &c., mais seulement a Fi, &c. Cela donne 12 = (d xi + Id 6l ) (9, /2 + w9,,) - (9,, + Z9 fl .) (d yi + md ei ) c est a dire : 12 est une fonction lineaire par rapport a 9^, 90 2 , et il en sera de meme pour les expressions analogues 13, 23, &c. : done le nombre des differentiations par rapport aux quantites 6 1} 6, &c., prises ensemble, ne surpasse pas a + J3 +y + ... , ou ^(rns). Or 1 expression a differentier contient le facteur 0-fOj ... #/; done, en remettant, apres les differentiations, au lieu de l} 2 ,...0 r , la fonction Q coritiendra le facteur 6 e leVe a la puissance TV \ (rn s). Tout cela suppose implicitement que Ton ait TV \(rn s)<s. Or le meme raisonnement, modifid tres peu, fait voir aussi que pour rv ~ 2 ( nr ~ s ) > s > ou plus simplement pour r(v-fyi)>\s, la fonction Q doit s evanouir d elle-meme, savoir en etablissant entre les coefficients de U les relations qui expriment 1 existence du facteur 0". De la on tire le the oreme suivant : Etant donn^e une fonction U du degre n, tout covariant du degre r par rapport aux coefficients et du degre s par rapport aux variables, s evanouit en supposant que la fonction U ait un facteur 6 pour lequel r (v ^n) > ^s ; et en particulier : Un invariant quelconque de la fonction U s evanouit en supposant que la fonction U ait un facteur 6", pour lequel v > ^n. En mettant ?i=2m ou 2m + 1, Yinvariant s evanouit en supposant que U ait le facteur 6 m+1 . Les conditions pour que la fonction U ait un tel facteur, se trouvent en e galant a z^ro les coefficients differentiels de U du m icme ordre par rapport aux variables x, y, et en eliminant ces variables. Mais avant d aller plus loin il convient d entrer dans quelques details de la theorie d une telle elimination. Je prends 1 exemple le plus simple, et je suppose que 1 on ait a e liminer cc, y des equations ax + by = 0, bx +cy = 0, ex + dy = 0. On est habitue* a dire que ce systeme equivaut a deux equations entre les seuls coefficients : mais cela n est juste que dans un sens qui manque de precision. Le systeme equivaut plutot a deux relations entre les coefficients, et ces deux relations soiit exprimees par les trois equations, bd c" = 0, be ad =0, ac b" = 0. II n est pas 100] NOTE SUR LA THEORIE DES HYPERDETERMINANTS. 579 vrai que deux de ces Equations embrassent necessairement la troisieme. En effet, la premiere et la seconde equations sont satisfaites en ecrivant c = 0, d = 0, mais ces valeurs sont absolument dtrangeres a la question, et ne satisfont pas a la troisieme equation, de maniere que toutes les trois equations sont necessaires pour exprimer les relations entre les coefficients. C est pourquoi je dis que ces trois Equations sont des resultats distincts de 1 e limination. Et de meme, pour un systeme quelconque d e qua- tions, le nombre des resultats distincts de 1 e limination n est pas gdneralement a beaucoup pres si faible que le nombre des relations entre les coefficients. Qu on veuille consulter sur ce sujet mon mdmoire "On the order of certain systems of algebraical equations," Cambridge and Dublin Mathematical Journal, t. iv. [1849] pp. 132 137 [77], et le mdmoire de M. Salmon " On the Classification of curves of double curvature," t. v. [1850] pp. 23 46. Je reviens a 1 objet de cette note, et je suppose qu en 6galant a zero les coefficients differentiels du m leme ordre de la fonction V, les equations P = 0, Q = 0, R = 0, &c. forment le systeme entier des rdsultats distincts de 1 elimination. Un invariant quelconque / s evanouira en supposant P = 0, Q = 0, R = 0, &c. II doit done exister une e quation telle que \I = aP + fiQ + yR + ..., ou X, a, /3, 7... sont des fonctions rationnelles et integrales des coefficients. Mais de plus, la fonction A, doit etre purement nume rique, ou ce qui est le meme, doit se rdduire a V unite, car autrement / = serait un rdsultat de 1 elimi nation different des resultats P = 0, Q = 0, R = 0, &c., et ces Equations ne seraient plus le systeme entier des resultats distincts.. Done enfin: un invariant quelconque / sera exprime par une e quation telle que a, /3, 7, ... etant des fonctions integrales et rationnelles des coefficients. Les resultats que je viens d obtenir s accordent parfaitement avec ceux dans ma " Note sur les hyperdeterminants," t. xxxiv. [1847] pp. 148 152 [54]. En effet, j y ai fait voir qu en supposant qu une fonction ax* + 4>ba?y + 6cx*y* + 4>dxy 3 + ey 4 ait un facteur (ax + @y) 3 , 1 elimination des variables entre les equations aa? + %bxy + c y" = 0, bx* + 2cxy + dy" = 0, ex" + %dxy + ey 2 = 0, donne lieu aux equations ae 4>bd + 3c 2 = 0, ace + 2bcd ad* b-e c 3 = ; et les fonctions e galees a zero sont en effet les seals invariants de la fonction du quatrieme ordre. J ajoute que la theorie actuelle fait voir aussi que dans le cas dont il s agit, la de rivee (ax^ + 2bxy + cy-} (ex" + Zdxy + ey*) (bx 2 + 2cacy + dy-}"*, ou son deVeloppement (ac - 6 2 ) x 4 + 2 (ad - be) x 3 y + (ae + 2bd - 3c 2 ) xy + 2 (be- cd) xf + (ce - d 2 ) y 4 , se reduit (a un coefficient constant pres) a (ax + /3y) 4 : et au cas ou la fonction donnee du quatrieme ordre est supposee etre un carre, cette fonction et la de rive e qui vient d etre e crite sont ^gales a un facteur constant pres; resultat dont je me suis servi ailleurs. 732 581 NOTES AND REFERENCES. 1. As to the history of Determinants, see Dr Muir s " List of Writings on Determinants," Quart. Math. Jour. vol. xvn. (1882), pp. 110149; and the interesting analyses of the earlier papers in course of publication by him in the R. S. E. Proceedings, vol. xili. (188586) et seq. The (new ?) theorem for the multiplication of two determinants was given^ by Binet in his " Memoire sur un systeme de formules analytiques &c." Jour. Ecole Polyt. t. x. (1815), pp. 29112. An expression for the relation between the distances of five points in space, but not by means of a determinant or in a developed form, is given by Lagrange in the Memoir " Solutions analytiques de quelques problemes sur les pyramides triangulaires," Mem. de Berlin, 1773: the question was afterwards considered by Carnot in his work " Sur la relation qui existe entre les distances respectives de cinq points quelconques pris dans 1 espace, suivi d un essai sur la the orie des trans- versales," 4to Paris, 1806. Carnot projected four of the points on a spherical surface having for its centre the fifth point, and then, from the relation connecting the cosines of the sides and diagonals of the spherical quadrilateral, deduced the relation between the distances of the five points: this is given in a completely developed form, containing of course a large number of terms. Connected with the question we have the theorem given by Staudt in the paper " Ueber die Inhalte der Polygone und Polyeder," Crelle t. xxiv. (1842), pp. 252 256 ; the product of the volumes of two polyhedra is expressible as a rational and integral function of the distances of the vertices of the one from those of the other polyhedron. More general determinant-formulae relating to the "powers" of circles and spheres have been subsequently obtained by Darboux, Clifford and Lachlan : see in particular Lachlan s Memoir, "On Systems of Circles and Spheres," Phil. Trans, vol. CLXXVII. (1886), pp. 481625. 2 and 3. The investigation was suggested to me by a passage in the Me canique Analytique, Ed. 2 (1811), t. I. p. 113 (Ed. 3, p. 106) ; after referring to a formula of Laplace, whereby it appeared that the attraction of an ellipsoid on an exterior point depends only on the quantities IP - A 2 and O 2 - A- which are the squares of 582 NOTES AND REFERENCES. the eccentricities of the two principal sections through the major semiaxis A, Lagrange remarks that, starting from this result and making use of a theorem of his own in the Berlin Memoirs 1792 93, he was able to construct the series by means of the development of the radical 1 -r- Jx~ + y 2 + z 2 2by 2cz + 6 3 + c 2 in powers of b, c, preserving therein only the even powers of b and c, and transforming a term such as Hb 2m c- n into a determinate numerical multiple of %irABC . H(B 2 A 2 J n (C 2 A 2 ) n . It occurred to me that Lagrange s series must needs be a series A c, W> T\n W S~i n A a^ +B -si f +0 reducible to his form as a function of B 2 - A 2 , C 2 - A 2 , in virtue of the equation (d 2 d 2 d 2 \ da? + db 2 + dc 2 ) ^ ( a ^ ^ = satisfied by the function (/> (I wrote this out some time before the Senate House Examination 1842, in an examination paper for my tutor, Mr Hopkins): and I was thus led to consider how the series in question could be transformed so as to identify it with the known expression for the attraction as a single definite integral. I remark that my formulae relate to the case of n variables: as regards ellip soids the number of variables is of course = 3 : in the earlier solutions of the problem of the attraction of ellipsoids there is no ready method of making the extension from 3 to n. The case of n variables had however been considered in a most able manner by Green in his Memoir "On the determination of the exterior and interior attractions of Ellipsoids of variable densities," Camb. Phil. Trans, vol. v. 1835, pp. 395430 (and Mathematical Papers, 8vo London, 1871, pp. 187222); and in the Memoir by Lejeune-Dirichlet, " Sur une nouvelle me thode pour la determination des integrates multiples," Liouv. t. iv. (1839), pp. 164168, although the case actually treated is that of three variables, the method can be at once extended to the case of any number of variables: it is to be noticed also that the methods of Green and Lejeune-Dirichlet are each applicable to the case of an integral involving an integer or fractional negative power of the distance. This is far more general than my formulas, for in them the negative exponent for the squared distance is = J?i, and, by differen tiation in regard to the coordinates a, b,... of the attracted point, we can only change this into %n+p, where p is a positive integer. But in 28, the radical contained in the multiple integral is ___ - ? wne re s is integer or fractional, and by a like process of expansion and summation I obtain a result depending on a single integral ^(l-uYu^^^du , , . on J o {(+h* u \ , IF" nd m , retaining throughout the general function < (c^ oc 1} ...) and making the analogous transformation of the multiple integral itself, I express the integral V = jdx l ...dx n in terms of an integral ^T n +> (I - T 2 ) k +fWdT, where W= t d^ ... dx n ^( ai - x,T, ...). NOTES AND REFERENCES. 583 I recall the fundamental idea of Lejeune-Dirichlet s investigation; starting with an r r r CCr integral 1 1 1 Udxdydz over a given volume he replaces this by \\ \pUdxdydz where p J J J J J J is a discontinuous function, = 1 for points inside, and = for points outside, the given volume; such a function is expressible as a definite integral (depending on the form of the bounding surface) in regard to a new variable : the limits for x, y, z, may now be taken to be oo , GO for each of the variables x, y, z, and it is in many cases possible to effect these integrations and thus to express the original multiple integral as a single definite integral in regard to 0. I have not ascertained how far the wholly different method in Lejeune- Dirichlet s Memoir " Sur un moyen general de verifier 1 expression du potentiel relatif a une masse quelconque homogene ou heterogene," Crelle, t. xxxii. (1846), pp. 80 84, admits of extension in regard to the number of variables, or the exponent of the radical. 4. As noticed p. 22, the investigation was suggested to me by Mr Greathead s paper, "Analytical Solutions of some problems in Plane Astronomy," Camb. Math. Jour. vol. I. (1839), pp. 182 187, giving the expression of the true anomaly in multiple sines of the mean anomaly. I am not aware that this remarkable expression has been elsewhere at all noticed except in a paper by Donkin, " On an application of the Calculus of Operations in the transformation of trigonometrical series," Quart. Math. Journal, vol. in. (1860), pp. 1 15 ; see p. 9, et seq. 5. In a terminology which I have since made use of: The Postulandum or Capacity (G) of a curve of the order r is = and the Postulation (V) of the condition that the curve shall pass through k given points is in general = k. If however the k points are the mn intersections of two given curves of the orders m and n respectively, and if r is not less than m or n, and not greater than m + n 3, then the postulation for the passage through the mn points, instead of being = mn, is = mn ^ (m + n r 1) (m + n r 2). Writing 7 = m + n r, and 8 -| (7 1) (7 2), the theorem may be stated in the form, a curve of the order r passing through mn 8 of the mn points of intersection will pass through the remaining B points. The method of proof is criticised by Bacharach in his paper, "Ueber den Cayley schen Schnittpunktsatz," Math. Ann. t. 26 (1886), pp. 275 299, and he makes what he considers a correction, but which is at any rate an important addition to the theorem, viz. if the S points lie in a curve of the order 7 8, then the curve of the order r through the mn 8 points does not of necessity nor in general pass through the 8 points. See my paper "On the Intersection of Curves," Math. Ann. t. xxx. (1887), pp. 85 90. 6. The formulae in Rodrigues paper for the transformation of rectangular coordi nates afterwards presented themselves to me in connexion with Quaternions, see 20 ; and again in connexion with the theory of skew determinants, see 52. 584 NOTES AND REFERENCES. 8. A correction to the theorem (18), p. 42, is made in my paper "Notes on Lagrange s theorem," Camb. and Dubl. Math. Jour. vol. vi. (1851), pp. 37 45. 10. This paper is connected with 5, but it is a particular investigation to which I attach little value. The like remark applies to 40. 12. The second part of this paper, pp. 75 80, relates to the functions obtained from n columns of symbolical numbers in such manner as a determinant is obtained from 2 columns, and which are consequently sums of determinants : they are the functions which have since been called Commutants ; the term is due to Sylvester. 13. In modern language: Boole (in his paper "Exposition of a general theory of linear transformations," Camb. Math. Jour. vol. ill. (1843), pp. 1 20 and 106 119) had previously shown that a discriminant was an invariant ; and Hesse in the paper "Ueber die Wendepunkte der Curven dritter Ordnung," Crelle, t. xxvili. (1844), pp. 68 96, had established certain covariantive properties of the ternary cubic function. I first proposed in this paper the general problem of invariants (that is, functions of the coefficients, invariantive for a linear transformation of the facients), treating it by what may be called the " tantipartite " theory : the idea is best seen from the example p. 89, viz. for the tripartite function U = aacjyiZj. we have a function of the coefficients which is simultaneously of the forms H a, b, c, d e> / 9, h H a, b, e, f c, d, g, h H a, c, e, g b, d, /, h and as such it is invariantive for linear transformations of the (# 1} x 2 ), (y lt y 2 ), (z 1} z 2 ). Passing from the tantipartite form to a binary form, I obtained for the binary quartic the quadrinvariant (/ = ) ae 4>bd + 3c 2 : as noticed at the end of the paper, the remark that there is also the cubin variant ( / = ) ace ad 2 b 2 e c 3 + 2bcd was due to Boole. The two functions present themselves, but without reference to the invariantive property and not in an explicit form, in Cauchy s Memoir " Sur la deter mination du nombre des racines rdelles dans les equations algebriques," Jour. Ecole Polyt. t. x. (1815), pp. 457548. In p. 92 it is assumed that the invariant called 6u is the discriminant of the function U = ax 1 y l z 1 w 1 . . . + px 2 y 2 z 2 w 2 : but, as mentioned in [ ], the assumption was incorrect. This was shown by Schlafli in his Memoir, "Ueber die Resultante eines Systemes mehrerer algebraischen Gleichungen," Wiener Denks. t. iv. Abth. 2 (1852), PP- 1 74: see pp. 35 et seq. The discriminant is there found by actual calculation to be a function (not of the order 6 as is 6u, but) of the order 24, not breaking up into factors; in the particular case where the coefficients a, ...p are equal, 1, 4, 6, 4, 1 of them to a, b, c, d, e respectively, in such wise that changing only the variables the function becomes = (a, b, c, d, e$x, y)*, then the discriminant in question does break NOTES AND REFERENCES. 585 up into factors, the value in this case being J 6 (I 3 27 J 2 ) of the order 24 as in the general case, but containing the factor I 3 27e7 2 which is the discriminant of the binary quartic. 14. In this paper I developed what (to give it a distinctive name) may be called the " hyperdeterminant " theory, viz. the expressions considered are of the form where after the differentiations the variables (x i} y^, (# 2 , y 2 ), ... are to be or may be put equal to each other : it is to be noticed that although in the examples I chiefly consider constant derivatives, or invariants, the memoir throughout relates as well to covariants as invariants. The theory is to be distinguished from Gordan s process of Ueberschiebung, or derivational theory, viz. this may be considered as dealing exclusively, or nearly so, with the single class of derivatives (V, W} a , = 12 V-jW z : the theorem that all the covariants of a binary function can be obtained successively by operating in this manner on the function itself and a covariant of the next inferior degree was a very important one. 15. Eisenstein s theorem may be stated as follows: the function a 2 d? + 4ac 3 Qabcd + 4<b 3 d 36 2 c 2 (which is the discriminant of the binary cubic (a, b, c, d^x, 7/) 3 ) is automorphic, viz. it is converted into a power of itself when for a, b, c, d we substitute the differential coefficients ~ , -~ , -~ , ~ of the function itself. It is remarkable, da db dc dd see 54, that the function is automorphic in a different manner, viz. the Hessian determinant formed with the second differential coefficients ~ , &c., is also equal to da 2 a power of the function itself. The first part of the paper relates to the function a-h? + b 2 g 2 + . + 4>bceh which had presented itself to me, 13, in the theory of linear transformations, and which is in like manner automorphic for the change a, b, ... into -r- , -^ , &c. The function however occurs in connexion with the arithmetical theory da db of the composition of quadratic forms, Gauss, Disquisitiones Arithmetical (1801), and see 92. The second part gives for the binary quartic covariant an automorphic formula analogous to those previously obtained by Hesse for the ternary cubic, viz. the Hessian of any linear function of the quartic and its Hessian, is itself a linear function of the quartic and its Hessian, the coefficients depending on the invariants 7, J of the quartic form. 16. This is a mere reproduction of 13 and 14, and requires no remark. 19 and 23. These papers contain a mere sketch of the application of the doubly infinite product expression of the elliptic function sn u to the problem of transformation. As noticed in 23, I purposely abstained from any consideration of the infinite limiting values of m and n. c. 74 586 NOTES AND REFERENCES. 20. The discovery of the formula q(ix +jy + kz) q 1 = iaf+jy + kz , as expressing a rotation, was made by Sir W. R. Hamilton some months previous to the date of this paper. As appears by the paper itself, I was led to it by Rodrigues formulae, see 6. For the further development of the theory, see 68. 21. The system of imaginaries i lt i 2> ... i 7 had presented itself to J. T. Graves about Christmas 1843, see his paper " On a Connection &c." Phil. Mag. vol. xxvi. (1845), pp. 315 320. They are called by him Octads, or Octaves. 24 and 25. These papers precede the researches of Eisenstein on the same subject. Crelle, t. xxxv. (1847). It was I think right that the theory of the doubly infinite products should have been investigated as in these papers : but the investigation is in some measure superseded by the beautiful theory of Weierstrass, viz. he takes for the element of the product (not a mere linear function of u, but) a linear function multiplied by an exponential factor, <ru = uTL 1(1 + ] e~w~* (\ w) w = 2m&> + 2mV where the ratio w : &> is imaginary, and the product extends to all positive or negative integer values of m, m (the simultaneous values 0, excluded) : in consequence of the introduction of the exponential factor the form of the bounding curve becomes immaterial, and the only condition is that it shall be ultimately every where at an infinite distance from the origin. The general theory of Weierstrass in regard to the exponential factor is given in the Memoir, " Zur Theorie der eindeutigen analytischen Functionen," Berlin. Abh. 1876 (reprinted, Abhandlungen aus der Functionenlehre, 8 Berlin, 1886) : and the application to Elliptic Functions is made in his lectures, edited by Schwarz, Formeln und Lehrsdtze u. s. w. 4 Gb tt. 1883. See also Halphen, Theorie des Fonctions Elliptiques, Paris, 1887. 26. The geometrical results in regard to corresponding points on a cubic curve are many of them due to Maclaurin. See his "De linearum geometricarum proprieta- tibus generalibus tractatus," published as an Appendix to his Treatise on Algebra, 5 Ed. Lond. 1788. (See also De Jonquieres " Melanges de Ge ometrie pure," 8 Paris, 1856.) But the theorem in the " Addition " was probably new : the curve of the third class touched by the line PP is the curve called by me the Pippian (as represented by the contravariant equation P U = 0), but which has since been called the Cayleyan of the cubic curve. 27. The theory of the conies of involution was so far as I am aware new. 28 and 29. See 2, 3. 30. As noticed in the paper, the investigation is directly founded upon that of Pliicker for the singularities of a plane curve. It is to be observed that the definition of a " line through two points " ligne menee par deux points (non consecutifs en general) du systeme, does not exclude actual double points, for the line through an actual double point is a line through two points, coincident indeed, but not consecutive ; but NOTES AND REFERENCES. 587 it would have been proper to notice the distinction between actual and apparent double points (first made by Dr Salmon, to whonj the term apparent double point, adp, is due): and the like in regard to the lines in two planes. In the translation of this paper, 83, there are two footnotes signed G. S. (Dr Salmon), giving for plane curves the formula t-K = B(v-fi) and 2 (T - 8) = (v - /A) (v + p - 9) : and for curves of double curvature and developable surfaces the analogous formulae a ft = 2 (n m), x y (n m) and 2(g h) = (n m)(n + m 7). Also in the second set of six equations, p. 210, the last three equations are replaced by a = 2m (3m - 7) - 3 (4& + 5/3), # = |m(3m-7)(3m 2 - 5m- 7) + | (6A + 8/3) (6A + 8/3 + 1)- 3m (m- 2) (6/^ + 8/3) + 19/1 + 24/3, viz. these are the equations serving to express oc, a, g in terms of m, h, /3. For the discussion of some singularities not considered in the present paper see Zeuthen, " Sur les singularite s ordinaires des courbes geome triques a double courbure," Comptes Rendus, t. LXVII. (1868), pp. 225233, and " Sur les singularites ordinaires d une courbe gauche et d une surface deVeloppable," Annali di Matem. t. in. (1869-70), pp. 175217. 40. See 10. 41 and 44. For demonstrations of Sir W. Thomson s theorem for the value of the definite integral f r , - r= - , X } i( . - . ,^ +1 see his papers "Demonstration d un J {(x - of + . . . + u 2 } 1 (of + . . . + v 2 ) t+] theoreme d Analyse" and "Extrait d une lettre a M. Liouville," Liouv. t. x. (1845), pp. 137147 and 364367, also the paper "On certain definite integrals suggested by problems in the theory of Electricity," Comb. Math. Jour. t. II. (1845), pp. 109121. 45. I am not aware that the equation Jk sn u = H (u) ^ (u) had been previously demonstrated otherwise than by the circuitous process employed in the Fundamenta Nova. The series z = \ + d .3 ... + C r - + ... , which is the solution of the differen- 1 . 2 1 . tial equation x 2 z + ax -/- + -=-? - 2 (a 2 - 4)~ = 0, and in which C lt C 2 , ... denote the coeffi- cLoc doc* (jjOC cients of the highest powers of n in the expressions given p. 299, is in fact (as remarked by me in a later paper, Liouv. t. vn. (1862)) the Weierstrassian function Al (*). 47. The surface here considered, the Tetrahedroid, is the general homographic transformation of the wave surface. It is a special case of the 16-nodal quartic surface considered by Kummer in his Memoir, "Algebraische Strahlen-systeme," Berl. Abh. 1866, pp. 1 120, and in various papers in the Berliner Monatsberichte. 742 588 NOTES AND REFERENCES. 48. The expressions f 2 a; = S (a 6) 2 (x c) (x d) . . . for the Sturmian functions in terms of the roots, or (to use Sylvester s term), say the endoscopic expressions of these functions, were obtained by him, Phil. Mag. vol. xv. (1839). It was interesting to express these in terms of the sums of powers Si, S 2 , &c., that is in terms of symmetrical functions of the coefficients, but for the actual expression of the Sturmian functions in terms of the coefficients the process is a very circuitous one, and the proper course is to start directly from the exoscopic expressions as linear functions of fa, f x also due to Sylvester (see his Memoir " On a theory of the syzygetic relations of two rational integral functions &c.," Phil. Trans, t. CXLIII. (1853), pp. 407 548), which is what is done in the subsequent paper 65. 49. I attach some value to this paper as a contribution to the theory of the Gamma function. 50. 55, 70, 95, 98. The general theorem of I. was given in a very different and less suggestive notation, by an anonymous writer, " Theoremes appartenant a la ge ometrie de la regie," Gergonne t. ix. (1818-19), pp. 289 291 ; viz. the statement is in effect as follows: considering in a plane or in space any n points 1, 2, 3...w; then joining these in order, take 12 any point in the line 1, 2 ; 23 any point in the line 2, 3, and so on to n l.n. Take then 123 the intersection of the lines 1, 23 and 12, 3; 234 the intersection of the lines 2, 34 and 23, 4; and so on to n %.n I.n. Take then 1234 the intersection of the lines 1, 234; 12, 34; and 123, 4 (viz. these three lines will meet in a point) : 2345 the intersection of the lines 2, 345 ; 23, 45 ; 234, 5 (viz. these three lines will meet in a point)... and so on to n 3.n2.n l.n. And so on for 12345, &c. up to 123... n; in the successive constructions we have four, five, ... and finally n lines which in each case meet in a point. A proof is given by Gergonne, t. XL (1820-21). A large part of this paper relates to the theory of the relations to each other of the 60 Pascalian lines derived from the hexagons which can be formed with the same six points upon a conic : the literature of the question is very extensive, and I hope to refer to it again in the Notes to another volume. 54. See for an addition which should have been printed with this paper, the last paragraph of 92. 63. Boole s theorem of integration which is here demonstrated is a very remarkable one, and it would be very interesting to investigate the general forms, or a larger number of particular forms, for the functions P, Q satisfying the condition mentioned in the theorem. It may be remarked that the demonstration is very closely connected with Lejeune-Dirichlet s method for the determination of certain definite integrals referred to in 2 and 3, we have a triple integral the real part of which is =fP + Q^ n+q or 0, according as P is or is not comprised between the limits and 1. It includes Boole s formula mentioned in 44 and in 64. 65. See 48. 66. The method here employed of establishing the theory of Laplace s coefficients in n -dimensional space by means of rectangular coordinates, has I think some advantage NOTES AND REFERENCES. 589 over the ordinary one, in which angular coordinates are introduced. The method is not given in Heine s Kugelfunctionen, Berlin 1878-81. 67. 93, and 99. These papers relate to the equation of differences, see p. 540, derived from Jacobi s equation n (n - 1) x 2 z + (n - 1) (ax - 2a?) ~ + (1 - ao 2 + a?) ^ - 2n (of - 4) ~ = : cLx ax doi the investigation seems to show that the integration cannot be effected in any tolerably simple form. 68. See 20. 69. This paper relates to the theory of the functions considered by Jacobi, and to which the name "Pfaffian" has since been given. It is shown that the definition of a determinant may be so extended as to include within it the Pfaffian : and it is proved that a symmetrical skew determinant is the square of a Pfaffian. 71. I doubt whether the theorem is true except in the cases n = 3 and n = 4. 76. As mentioned at the conclusion of the Memoir the whole subject was developed in a correspondence with Dr Salmon. Steiner s researches upon Cubic Surfaces are of later date, viz. we have his Memoir, "Ueber die Flachen dritten Grades" (read to the Berlin Academy 31 January 1856), Crelle, t. LIII. (1857), pp. 133 141 and Werke, t. n. pp. 651659. 77. Contains the general definition of the order of a system of equations. 81. Contains the remark that the Cartesian has a cusp at each of the two circular points at infinity. 82. The Problem of the fifteen school girls was proposed by Kirkman, Lady s and Gentleman s Diary, 1850 : it is a particular case of the Prize-question proposed by him in the Diary for 1844. A great deal has been written on the subject. 83. See 30. 92. See 15. 94. This paper contains my fourfold formula for the addition of the elliptic functions sn, en, and dn. 100. I remark that the terms co variant, invariant, here referred to as introduced by Sylvester, were first employed (together with many other valuable new terms) in the first part of his paper "On the Principles of the Calculus of Forms," Camb. and Dubl. Math. Jour. vol. vn. (1852), pp. 5297. I hope to give in the next volume, in connexion with my Introductory Memoir on Quantics, a review of the earlier history of the subject. END OF VOL. i. CAMBRIDGE : PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS. LD-9 80m 11/80 (Bay View) RETURN WIUL THI8 WILL DAY Cl 2( LOAN PERIOD HOME USE ALL BOOKS MAY Renewals and Re Books may be Re JUL20 AUTO WSLM.2 7 RETURN TO the circulation desk of any University of California Library or to the NORTHERN REGIONAL LIBRARY FACILITY Bldg. 400, Richmond Field Station University of California Richmond, CA 94804-4698 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 2-month loans may be renewed by callina (510)642-6753 1-year loans may be recharged by brinqina books to NRLF Renewals and recharges may be made 4 days prior to due date J3yE_ASJ5TAMPED BELOW AUG 2 6 2005 DD20 12M 1-05 FORM NO. DD6, UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720 > b UNIVERSITY OF CALIFORNIA LIBRARY U.C. BERKELEY LIBRARIES II " COOSSbBOBb