CORNELL UNIVERSITY LIBRARY MATHEMATICS Cornell University Library QA 553.S27 1865 A treatise on the analytic geometry of t 3 1924 001 103 872 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001103872 A TREATISE ANALYTIC GEOMETRY THEEE DIMENSIONS. BY GEOEGE §ALMON, D.D., F.E.S., FELLOW AND SENIOR TUTOR OF TRINITY COLLEGE, DUBLIN. SECOND EDITION. $sbUn: HODGES, SMITH, AND CO., GEAPTON STEEET, BOOKSELLERS TO THE UNIVERSITY. 1865. CAMBRIDGE : Printed by William Metcalfe, Green Street. PREFACE. In the preface to the former edition I expressed the difficulty which I found in trying to avoid two opposite evils, neither of which I have altogether escaped : the evil, namely, of making the book in- conveniently large, or else of omitting some im- portant subjects, or treating them imperfectly. This second edition remains in substance the same as the former. The most important alteration is in Chap, ix., where, following the steps of a corre- sponding chapter in the last edition of my Treatise on Conic Sections, I have expanded and thrown into a more systematic form the theorems which I had given respecting Invariants and Covariants of a system of two equations of the second degree. I have also incorporated the results of a few Memoirs published since my last edition; as, for instance, Schwarz's Memoir On Developables, p. 280; Clebsch's Papers On the Osculating Planes of Curves, p. 291 ; On the Normals to a Surface of the Second Degree, p. 399 ; and On the General Theory of Surfaces, p. 444 ; Cayley's Memoirs On Scrolls, p. 372; and the paper IV PREFACE. in which De Jonquieres, following Chasles, has studied the Properties of Systems of Surfaces, p. 507. I have already expressed my obligations to the friends who originally aided me in the prepara- tion of this work, especially to Dr. Hart and the Messrs. Roberts. I have now to add my thanks to the correspondents who have had the kindness to send me lists of the errors, whether of the press or my own, which they noticed in the former edition. In particular I have to name Mr. Todhunter and Mr. Traill. The list sent by the latter did not reach me until the sheets of about half this volume had been printed off, and although many of his errata had been already noticed by myself, yet several had escaped correction. A list of these will be found at the end of the Table of Contents. Since the above was written, I have been tanta- lized by receiving the Second Volume of Dr. Fiedler's German translation of this work ; exactly when it is too late to make any use of his notes and additions. But it is perhaps better that I have not been tempted into greater fulness of treatment, unless I had fol- lowed Dr. Fiedler's example in dividing the Book into Two Volumes. Trinity College, Dublin, September, 1865. CONTENTS. Tlie following selected course is recommended to Junior Headers : The Theory of Surf aces of the Second Order, pp. 1-100. Confocal Surfaces, Arts. 149-161. The Curvature of Quadrics, pp. 140-144. The General Theory of Surfaces, Chap. XI. The Theory of Curves, Arts. 310-319, 352-354, 358-360, 367-369. And the Chapter on Families of Surfaces, Arts. 409-433. CHAPTER I. THE POINT. Method of co-ordinates .... Properties of projections .... Co-ordinates of point cutting in a given ratio the distance between Co-ordinates of centre of a tetrahedron Distance between two points (rectangular co-ordinates) Direction-cosines of a lines .... Area of a figure in terms of areas of its projections . Angle between two lines in terms of their direction-cosines Perpendicular distance of a point from a line Direction-cosines of the perpendicular to the plane of two lines Transformation of Co-ordinates Distance between two points (oblique co-ordinates) Degree of an equation unaltered by transformation two points TAGE 1 3 5 G 9 9 11 11 CHAPTER II. interpretation oj? equations. Meaning of a single equation ; of a system of two or three equations . 12 Every plane section of a surface of the n tn degree is a curve of the re" 1 degree . 14 Every right line meets a surface of the n ta degree in n points . . 14 Order of a curve in Space defined . . . . .14 Three surfaces of degrees m, n, p, intersect in mnp points . . 15 Cylindrical surfaces defined . . . . . .15 VI CONTENTS. CHAPTER III. THE PLANE. PAGW Every equation of the first degree represents a plane . . . IS Equation of a plane in terms of its direction-cosines and perpendicular from origin 16 Angle between two planes . . . . . . '17 Condition that two planes may be mutually perpendicular . . 17 Equation of plane in terms of intercepts made on axes . . .17 Equation of plane through three points . . . . 18 Interpretation of terms in this equation . . . . .19 Value of determinant whose constituents are the direction-cosines of three right lines ....... 19 Length of perpendicular from a given point on a given plane . . 20 Co-ordinates of intersection of three planes .... 21 Condition that four planes may meet in a point . . . .21 Volume of tetrahedron in terms of co-ordinates of its vertices . . 21 Volume of tetrahedron, the equations of whose faces are given . . 22 Equations of surfaces passing through intersection of given surfaces . 22 The equation of any plane can be expressed in terms of those of four given planes 23 QtTADRIPLANAR CO-ORDINATES .... Anharmonic ratio of four planes ..... The Right Line ...... Equations of a right line include four constants Condition that two lines may intersect .... Direction-cosines of a line whose equations are given Equations of perpendicular from a given point on a given plane Direction-cosines of the bisectors of the angle between two lines . Angle between two lines ..... Conditions that a line may lie in a given plane . . , Number of conditions that a line may lie in a given surface . Inference as to the existence of right lines on surfaces of the second and third Equation of plane drawn through a given line perpendicular to a given plane Equation of plane parallel to two given lines Equations and length of shortest distance between two given lines Properties op Tetrahedra .... Relation connecting the mutual distances of four points in a plane Volume of a tetrahedron in terms of its edges . Relation connecting mutual distances of four points on a sphere Radius of sphere circumscribing a tetrahedron Shortest distance between two opposite sides . , Angle of inclination of two opposite sides 23 23 24 24 24 25 26 26 27 28 28 28 28 29 30 31 31 32 32 33 33 33 CHAPTER IV. PROPERTIES OP QTJADRIOS IN GENERAL. Number of conditions necessary to determine a quadric . . .34 Result of transformation to parallel axes .... 35 Equation of tangent plane at any point . . . . .36 Equation of polar plane ' , . . . .37 Cones defined — tangent cone , , . , , .37 CONTENTS. Vll Locus of harmonic means of radii through a point Every homogeneous equation in x, y, z represents a cone whose vertex is the origin Discriminant of a quadric ..... Co-ordinates of centre ...... Conditions that a quadric may have an infinity of centres Equation of diametral plane ..... Conjugate diameters ...... A quadric has three principal diametral planes .... Formation of equation representing the three principal planes Rectangles under segments of intersecting chords proportional to the rectangles under the segments of a pair of parallel chords Equations of tangent plane and cone, &c., derived by Joachimsthal's method Condition that a plane may touch the surface .... Co-ordinates of the pole of a given plane .... Condition that a line may touch the surface .... This condition derived from the former condition PAGE 37 38 40 41 41 41 42 44 44 45 46 47 47 48 48 CHAPTER V. CLASSIFICATION OF QUADRICS. Functions of coefficients which are unaltered by rectangular transformation . 50 Discriminating cubic . . . . . . .50 Cauchy's proof that its roots are real ..... 51 Ellipsoids . . . . • . ■ .52 Hyperboloids of one and two sheets ..... 53 Asymptotic cones . . . . . . .53 Paraboloids ....... 56 Actual reduction of equation of a paraboloid . . . .57 CHAPTER VI. CENTRAL SURFACES, Equation referred to axes ..... Length of normal ..... Sum of squares of reciprocals of three conjugate diameters is constant Locus of intersection of three tangent planes which cut at right angles Sum of squares of three conjugate diameters is constant . Parallelopiped constant whose edges are conjugate diameters Also gum of squares of projections of conjugate diameters, on any line and on any plane ...... Locus of intersection of tangent planes at extremities of conjugate diameters Quadratic which determines lengths of axes of a central section Edge of concentric cone is an axis in section by its tangent plane " when the quadric is given by the general equation Circular Sections ..... Form of equations of concyclic surfaces Two circular sections of opposite systems lie on the same sphere Umbilics defined ...... Circular sections of paraboloids .... Rectilinear Generators .... Two lines of opposite systems must intersect 58 59 59 60 61 61 63 63 64 64 65 65 66 67 68 68 69 70 Vlll CONTENTS. No two lines of the same system intersect .... Ruled surfaces defined ...... Distinction between developable and skew surfaces A right line whose motion is regulated by three conditions generates a surface Surface generated by a line meeting three director lines . . . Right lines on hyperbolic paraboloid . Four generators of one system cut any generator of the other in a, constant anharmonic ratio ....... Surface generated by lines joining corresponding points on two homographically divided lines ...... Method of constructing hyperbolic paraboloid .... Conditions for Surfaces of Revolution .... Examples of Loci ....... Locus of intersection of three rectangular tangent lines to a quadric Method of finding equation of cone, given its vertex, and a, curve through which it passes ........ Reciprocal cones . .... PAGE 71 72 72 73 74 75 77 77 78 78 81 82 83 CHAPTER VII. METHODS OF ABRIDGED NOTATION. Reciprocal Surfaces .... Degree of the reciprocal of a surface, how measured Reciprocal of a curve in space .... Osculating plane of a curve defined Plane section of one surface answers to tangent cone of reciprocal Reciprocal of quadric, when ellipsoid, hyperboloid, or paraboloid Reciprocal of ruled surface is ruled surface of same degree Reciprocal cones defined .... Sections by any plane of reciprocal cones are reciprocal Eocal lines of a cone denned Reciprocal of sphere ..... Properties of surfaces of revolution obtained by reciprocation Equation of reciprocal of a quadric Tangential Equations .... Tangential equation of a quadric Equation of system of quadrics having common curve All quadrics through seven points pass through an eighth Locus of centres of quadrics touching eight planes " passing through eight points Eour cones pass through the intersection of two quadrics Properties of systems of quadrics having double contact Properties of three quadrics having one plane curve common Similar quadrics ..... Geometrical solution of problem of circular sections Twelve umbilics he three by three on eight lines Quadrics touching along a plane curve Properties of their sections .... Two quadrics enveloped by the same third intersect in plane curves Form of equation referred to self conjugate tetrahedron Simplest form of writing equations of a system of three quadrics . 85 85 86 8G 86 87 87 87 87 90 91 92 93 93 94 95 96 97 97 97 98 98 99 99 99 99 100 CONTENTS. ix CHAPTER VIII. FOCI AND CONFOCAL SURFACES. _ .... PAGE Focus and directrix of a quadric defined . . . . .101 Two kinds of foci ....... 101 Focal conies ....... 103 Analysis of species of focal conies for each kind of quadrics . . 104 Foci of sections normal to a focal conic ..... 106 Focal lines of a cone ...... 107 Focal conies of paraboloids . , ... 108 Focus and directrix property of quadrics .... 109 Tangent cone, whose vertex is a focus, is a right cone . . .110 Reciprocal of a quadric when a surface of revolution . . . m Property of umbilicar foci obtained by reciprocation . . .111 Focal properties of quadrics obtained by reciprocation . . . 112 Focal conies the limits of confocal surfaces .... 114 Three confocals through a point all real and of different species . . 115 Co-ordinates of intersection of three confocals .... 116 Co-ordinates of umbilics ...... H6 Two confocals cut at right angles . . . . .117 Axes of central section in terms of axes of confocals through extremity of conjugate diameter ...... 118 pD constant along the intersection of two confocals .... 120 Locus of pole of fixed plane with regard to a system of confocals . . 120 Axes of tangent cone are the three normals through its vertex . . 121 Transformation of equation of tangent cone to the three normals as axes of co-ordinates ....... 123 Cones circumscribing confocal surfaces are confocal . . . 125 The focal lines of these cones are the generators of the hyperboloid through the vertex ....... 126 Reciprocals of confocals are concyclic ..... 126 Tangent planes through any line to the two confocals which it touches are mutually perpendicular . . . . .127 Two confocals seen from any point appear to cut at right angles . . 127 Normals to tangent planes through a given line generate a hyperbolic paraboloid 127 Chasles's method of obtaining equation of tangent cone . . . 129 Intercept on a bifocal chord between tangent plane and parallel plane through centre ........ 130 Given three conjugate diameters of a quadric to find the axes . . 131 Locus of vertices of right cones enveloping a quadric . . . 132 Locus of intersection of three mutually perpendicular tangent lines . 133 Corresponding points on confocals .... 134 Elliptic co-ordinates ...... 135 Ivory's theorem as to the distance of two corresponding points . . 137 Jacobi's analogue to the plane theorem that the sum of focal distances is constant 137 Locus of points of contact of parallel planes touching a series of confocals . 139 Curvature of Quadrics ...... 139 Radii of curvature of a normal and of an oblique section . . . 140 Line of curvature defined . . ... 1 12 Construction for principal centres of curvature .... 142 Surface of centres : its equation, how found . . . . 1 13 Its sections by principal planes . . . . .143 Equation of its reciprocal . . . . . Ill X CONTENTS. CHAPTER IX. INVARIANTS AND COVARIANTS OP SYSTEMS OF QUADRIOS. MSB Fundamental invariants of a system of two quadrics . . . 145 Condition that a tetrahedron can be inscribed in one quadric, self -conjugate with regard to another ..... Hesse's theorem as to vertices of two self-conjugate tetrahedra Eaure's property of spheres circumscribing self-conjugate tetrahedra Condition that two quadrics should touch .... Equation of surface parallel to a quadric .... Point of contact of two surfaces, a double point on their curve of intersection Stationary contact denned ...... Equation of surface of centres formed .... Condition that a tetrahedron can be inscribed in one quadric having two opposite edges on another ...... Equation of quadric touching four planes .... Invariants of a cone and a quadric ..... Two kinds of equilateral hyperboloids .... Tangential equation of imaginary circle at infinity Two planes at right angles conjugate with regard to imaginary circle . Tangential equation of curve in space .... How to form reciprocal of tangential equation of a conic Equation of cone touching a quadric along a given plane section . Contravariants of a system of two quadrics Two principal covariant quadrics of a system of two quadrics Equation of developable circumscribing two quadrics Its sections by the principal planes ..... Equation of developable generated by intersection of two quadrics Intersection of this developable with either quadric A system of confocals is touched by a common developable . Locus of points whence three rectangular lines or planes can be drawn to a quadric ....... Two quadrics having a common cm-ve can be described to touch a line, and three to touch a plane ..... Properties of confocals deduced from those of a system having a common developable ....... Method of finding equations of focal conies of quadrics given by general equation ...... Quadriplanar equation of system of confocals .... Equation of sphere inscribed in a tetrahedron Equation of c^jSimscribing sphere ..... Condition that quadriplanar equation should represent a sphere , Condition that section should be a parabola or equilateral hyperbola Co-ordinates of foci of a plane section .... Locus of foci of central sections of a given quadric Jacobian of a system of four quadrics .... Seduction of two quadrics to their canonical form Invariants of a system of three quadrics .... Tact-invariants ....... Discriminants of discriminants ..... Property of lines joining corresponding vertices of two conjugate tetrahedra Analogues to Pascal's theorems ..... CONTENTS. XI CHAPTER X. CONES AND SPHERO-CONICS. PAOR Spherical co-ordinatea . . . . . . .184 Cyclic arcs of sphero-conics analogous to asymptotes . . . 188 Sum of focal distances constant ...... 190 Focus and directrix property of sphero-conica .... 191 Difference of squares of reciprocals of axes of central section of a quadric, pro- portional to product of sines of angles it makes with cyclic planes . . 192 Equation in spherical co-ordinates of imaginary circle at infinity . . 194 Equation of sphere inscribed in a tetrahedron . . . .195 Equation of a right cone ...... 196 Investigation of Feuerbach's theorem ..... 197 CHAPTER XI. GENERAL THEORY OF SURFACES. Number of terms in general equation ..... 201 Section of surface by tangent plane has point of contact for a double point . 202 A surface in general has triple tangent planes .... 203 Inflexional tangents defined ...... 204 The indicatrix ; elliptic, hyperbolic, and parabolic points . . . 204 Conjugate tangents ....... 206 Tangent plane at a parabolic point is a double tangent plane . . 207 Double points on a surface ...... 207 Application of Joachimsthal's method .... 209 Number of double tangent lines which can be drawn through a point on a surface 211 Formation of equation of tangent cone to a surface .... 212 Number of inflexional or double tangents which can be drawn through any point ....... 212,213 Degree of reciprocal surface ...... 214 Discriminant of a surface .... . 215 Polar quadric of a parabolic point is a cone . . . .216 Hessian of a surface ...... 217 Number of stationary tangent planes which pass through a point . . 217 Every right line on a surface touches the Hessian . . 218 Curvature of Surfaces ...... 218 Radius of curvature of normal section .... -19 Euler's formula .....-• 221 Meunier's theorem ...... "--3 Two spheres have stationary contact with a surface . . . 223 Values of principal radii at any point .... 224 Locus of points where radii are equal and opposite .... 225 Equation determining directions of principal sections Conditions for an umbilic . Lines of spherical curvature . • • • Number of umbilics on a surface of the ifl 1 order .... 229 Stationary contact implies contact at two points . . . 229 Determination of normals which meet a consecutive normal . . . 230 Bertrand's theory of curvature . 231 Lines of curvature ...•••• -°" 2-26 ■»7 CONTENTS. PAGE Their differential equation ...-■■ 234 Lines of curvature of ellipsoid . . . • .235 Dupin's theorem ....-•• ^" If two surfaces cut at right angles, their intersection, if a line of curvature on one, i3 so on the other ...■•• 236 Locus of centres along a line of curvature is a cuspidal edge on surface of normals 238 Properties of surface of centres . 238 In what cases it can have a double line . . ■ 239 Geodesic lines denned ....•• 240 If a line of curvature be plane, it makes constant angle with tangent plane . 243 Lancret's theorem of variation of angle between tangent plane and osculating plane of line of curvature ..... 244 A geodesic line of curvature must be plane .... 245 CHAPTER XII. CORVES AND DEVELOPABLES. Sec. I. Projective Properties ..... 246 Direction-cosines of tangent to a curve . . . 248 Theory of developables explained . . . 249 Envelope of a plane whose equation contains one parameter . . . 250 Tangent planes to developables touch along a line . . 253 Characteristics ..... • 254 Cuspidal edge of a developable ... . 254 Stationary points and planes ...... 255 Cayley's equations connecting singularities of a curve in space . . 255 Developable generated by tangents is of same degree as reciprocal developable . 258 Nodal curve on developable ..... 260 Seo. II. Classification of Curves ..... 261 A twisted cubic can be described through six points . . . 263 Projection of a twisted cubic has a double point .... 265 Properties of twisted cubics ..... 265 Their different species ...... 269 Singularities of curve of intersection of two surfaces . . . 270 Number of apparent double points of intersection . . . .271 Case of surfaces which touch ... . 272 Equations connecting singularities of curves which together make up intersection of two surfaces ....... 273 Two distinct families of quartics ... . 274 Four- quartics of second family through eight points . . 278 Classification of quintics ... . . 279 Planar developables ....... 280 Eiemann's theorem for plane curves, and its application to eurves in space . 281 Common curve on three surfaces equivalent to how many points of intersection . 283 Singularities of a double curve connected with those of its complementary . 284 Sec. III. Non-Projeotive Properties of Curves . . . 285 Direction-cosines of normal plane ..... 286 Equation of osculating plane ...... 286 The helix ....... 287 Equation of osculating plane of intersection of two surfaces . . . 289 CONTENTS. Condition that four points may lie in a plane Radius of absolute and of spherical curvature Expressions for angle of contact Radius of curvature of intersection of two surfaces . Expression for angle of torsion Osculating right cone . . . . , Rectifying developable .... Rectifying surface is surface of centres of original developable Angle between two successive radii of curvature Cuspidal edge of polar developable is locus of centres of spherical curvature Every curve has an infinity of evolutes These are geodesies on the polar developable "Characteristics of polar developable Radius of sphere through four consecutive points . . Co-ordinates of its centre . History of theory of non-plane curves Sec. IV. Curves traced on Surfaces Differential equation of a geodesic Line joining extremities of indefinitely near and equal geodesies cuts them right angles ....... Radius of geodesic curvature .... pD constant for a geodesic on a quadric Value of the constant the same for all geodesies through an umbilic Mr. M. Roberts's deductions from this theorem . Liouville's transformation of equation pZ> = constant Chasles's proofs of this theorem .... " " and extensions of it . Elliptic co-ordinates ..... Area of surface of ellipsoid .... Second integral of equation of geodesic Length of a geodesic ..... Geodesic polar co-ordinates .... Dr. Hart's proof of Mr. Roberts's expressions Umbilical geodesies do not return on themselves Lines of level . Lines of greatest slope ..... Gauss's theory of curvature of surfaces Measure of curvature unaltered by deformation Total curvature of geodesic triangle on any surface . at PAGE 291 293 294 295 296 297 298 299 299 301 301 302 303 303 304 304 305 , 306 306 307 309 311 312 313 315 316 317 318 319 319 320 323 327 328 328 329 331 337 CHAPTER XHI. FAMILIES OF SURFACES. Equations involving a single arbitrary function Cylindrical surfaces .... Conical surfaces .... Conoidal surfaces . . . • • Surfaces of revolution . Order of differential equation of a family involving n functions Surfaces generated by lines parallel to a fixed plane Or by lines which meet a fixed axis . 339 342 343 345 346 349 352 354 XIV CONTENTS. PAQK Differential equations of ruled surfaces ..... 356 Theory of envelopes ...... 357 Determination of arbitrary functions ..... 359 Partial differential equation of developables .... 363 Their Pro-Hessian ....... 364 Nature of its intersection with the developable . . ■ 364 Tubular surfaces ....... 365 Differential equation of characteristics . . . . 366 Differential equation of geodesic on a cone .... 369 Ruled Surfaces . . ... 372 Normals along a generator generate paraboloid . . . 373 Lines of striction ....... 374 Nature of contact along any generator ..... 375 Double curves generally exist on ruled surfaces . . . 377 Surfaces generated by a line resting on three fixed directors . . . 378 Surfaces generated by a line which meets a curve twice and another curve once 381 " by a line which meets a curve three times . . . 381 Degree of condition that three surfaces should have a line in common . . 383 CHAPTER XIV. SURFACES DERIVED FROM QUADRICS. Equation of wave surface .... Its sections by principal planes .... Apsidal surfaces ..... Polar reciprocal of apsidal, apsidal of reciprocal Degree of reciprocal of wave surface Geometrical investigation of planes which touch along circles Equation in elliptic co-ordinates Expression for angle between tangent plane and radius vector Construction for tangent plane at any point Lines of curvature of wave surface Surface of centres Its sections by principal planes . Its cuspidal curves Its nodal curves Theory of derived pedal surfaces Properties of inverse surfaces Lines of curvature of surface of elasticity First negative pedal of a quadric Problem of finding negative pedals identical with that of finding parallel surfaces 387 388 389 390 391 392 393 397 397 398 399 402 403 404 405 406 408 408 410 CHAPTER XT. SURFACES OF THIRD DEGREE, Cubics having double lines .... 413 Cubics having double points . 4,g Cubics whose tangent cones from points on surface break up into two of second „ , degree 416 Sylvesters canonical form for equation of cubic ; . 417 CONTENTS. XV Corresponding points on the Hessian .... 418 Relation of the five planes to the Hessian ..... 420 Polar cubics of a plane ...... 420 These all touch the Hessian ...... 421 Eight lines on cubics ...... 422 Number of triple tangent planes ..... 423 Schafli's scheme for the twenty-seven lines . . . 425 Involution of six lines in space ...... 425 Condition that five lines should be met by a common transversal . . 426 Analysis of Bpecies of cubics ...... 427 Section by tangent plane, how met by polar plane with regard to Hessian . 428 Invariants and covariants of cubics ..... 429 Method of obtaining contravariants in five letters . . . 431 Five fundamental invariants ...... 434 Equation of surface which determines twenty-seven lines . . 436 CHAPTER XVI. GENERAL THEORY OP SURFACES. Jacobian of four surfaces ... , 437 Degree of tact-invariant of three surfaces .... 438 Degree of condition that two surfaces may touch .... 439 Order of developable enveloping a surface along a given curve . . 439 Of developable generated by a line meeting two given curves . . 440 Contact of Lines with Surfaces ..... 440 Locus of points of contact of doubly inflexional tangents . , , 441 And of triple tangents ...... 443 Number of points at which two tangents are doubly inflexional . . 444 Number of points at which lines can be drawn to meet in five consecutive points 445 Contact of planes with surfaces ..... 446 Locus of points of contact of double tangent planes . . , 449 Theory of Reciprocal Surfaces .... 450 Number of triple tangent planes to a surface .... 454 Effect of multiple lines on degree of reciprocal . . . 458 Application to developables of theory of reciprocals . . , 459 Singularities of developable generated by a line resting twice on a given curve 460 Application to ruled surfaces ...... 461 Hessian of ruled surface where meets the surface . . . 462 APPENDIX. On quaternions ....... 468 On triple orthogonal systems ..... 476 On Clebsch's calculation of surface S . . . . 480 On the order of systems of equations .... 489 On the properties of systems of surfaces ..... 507 ERRATA. FAGS LINE 37, 7, 8, Interchange the words " former" and " latter." 61, 7, for cos a, cos a, read p cos a, p' cos a'. " 23, for Art. 35, ?-eatf Art. 31. 78, 3 from bottom, for "positive, 1 " read "real." 131, 14, for " axis major," read "semi-axis major." " 7 from bottom, for " tangent planes," read "planes parallel to the principal planes." 142, 2, the quantity multiplying - should have the index — 1. " 3, for " square," read " inverse square." 147, last line, for x, y, z, read a, f3, y. 148, 20, for " last article," read " last example." 160, 16, for a', read (3. 189, 5, for " chord," read " point." 210, 13, after " polar surfaces," insert " of the origin." 289, last line but one,yb?* /a, read in, 293, 7 from bottom, for " surface," read " curve." 317, 27, for h and 0, read h and 0. 333, 22, 23, 24, for a', §', y'. read 2a', 2/3', 2y'. ANALYTIC GEOMETRY OF THREE DIMENSIONS. CHAPTER I. THE POINT. 1. We have seen already tow the position of a point C in a plane is determined, by referring it to two co-ordinate axes OX, Y drawn in the plane. To determine the position of any point P in space, we have only to add to our apparatus a third axis OZ not in the plane (see figure next page). Then if we knew the distance, measured parallel to the line OZ, of the point P from the plane XOY, and also knew the x and y co-ordinates of the point C, where PC parallel to OZ meets the plane, it is obvious that the position of P would be completely determined. Thus, if we were given the three equations x = a,y = b, z = c, the first two equations would determine the point C, and then drawing through that point a parallel to OZ, and taking on it a length PC=c, we should have the point P. We have seen already how a change in the sign of a or b affects the position of the point C. In like manner the sign of c will determine on which side of the plane XOY the line PC is to be measured. If we conceive the plane XO Y to be horizontal, it is customary to consider lines measured upwards as positive, and lines measured downwards as negative. In this case then the z of every point above that plane is counted as positive, and of every point below it as negative. It is obvious that every point on the plane has its z = 0. 8 2 THE POINT. The angles between the axes may be any whatever j but the axes are said to be rectangular when the lines OX, OY are at right angles to each other, and the line OZ perpendicular to the plane XOY. space, 2. We have stated the method of representing a point in in the manner which seemed most simple for readers already acquainted with Plane Analytic Geometry. We pro- ceed now to state the same more symmetrically. Our appa- ratus evidently consists of three co-ordinate axes OX, OY, OZ meeting in a point 0, which, as in Plane Geometry, is called the origin. The three axes are called the axes of x, y, Jg, respec- tively. These three axes determine also three co- ordinate planes, namely, the planes XOY, YOZ, ZOX, which we shall call the planes xy, ye, zx respectively. Now since it is plain that PA = GE = a, PB = CD = b, we may say that the position of any point P is known if we are given its three co-ordinates ; viz. PA drawn parallel to the axis of * to meet the plane yz, PB parallel to the axis of y to meet the plane zx, and PC drawn parallel to the axis of z to meet the plane xy. Again, since OD = a, OE=b, OF=c, the point given by the equations x = a, y = b, z — c may be found by the follow- ing symmetrical construction: measure on the axis of x, the length OB = a, and through D draw the plane PBGD parallel to the plane yz : measure on the axis of y, OE= b, and through E draw the plane PA GE parallel to zx : measure on the axis of z, 0F= c, and through F draw the plane PABF parallel to xy : the intersection of the three planes so drawn is the point P, whose construction is required. THE POINT. ^ 3. The points A, B, C, are called the projections of the point P on the three co-ordinate planes ; and when the axes are rectangular they are its orthogonal projections. In what fol- lows we shall be almost exclusively concerned with orthogonal projections, and therefore when we speak simply of projections, are to be understood to mean orthogonal projections, unless the contrary is stated. There are some properties of orthogonal projections which we shall often have occasion to employ, and which we therefore collect here, though we have given the proof of some of them already. (See Conies, p. 319.) The length of the orthogonal projection of a finite right line on any plane is equal to the line multiplied by the cosine of the angle* which it makes with the plane. Let PC, P C be drawn perpendicular to the plane XO Y; and CC is the orthogonal pro- jection of the line PP on that p' plane. Complete the rectangle by drawing PQ parallel to CC', and PQ will also be equal to CC. But PQ = PP cosPPQ. 4. The projection on any plane of any area in another plane is equal to the original area multiplied by the cosine of the angle between the planes. (See Conies, p. 319.) * The angle a line makes with a plane is measured by the angle which the line makes with its orthogonal projection on that plane. The angle between two planes is measured by the angle between the perpendiculars drawn in each plane to their line of intersection at any point of it. It may also be measured by the angle between the perpendiculars let fall on the planes from any point. The angle between two lines which do not intersect, is measured by the angle between parallels to both drawn through any point. When we speak of the angle between two lines, it is desirable to express without ambiguity whether we mean the acute or the obtuse angle which they make with each other. When therefore we speak of the angle between two lines (for instance PP 1 , CC in the figure), we shall understand that these lines are measured in the direction from P to P' and from C to C", and that PQ parallel to CC is measured in B2 4 THE POINT. For if ordinates of both figures be drawn perpendicular to the intersection of the two planes, then, by the last article, every ordinate of the projection is equal to the corresponding ordinate of the original figure multiplied by the cosine of the angle between the planes. But it was proved ( Conies, p. 350,) that when two figures are such that the ordinates corresponding to equal abscissas have to each other a constant ratio, then the areas of the figures have to each other the same ratio. 5. The projection of a point on any line, is the point where the line is met by a plane drawn through the point perpen- dicular to the line. Thus, in figure, p. 2, if the axes be rect- angular, D, E, Fare the projections of the point P on the three axes. The projection of a finite right line upon another right line is equal to the first line multiplied by the cosine of the angle between the lines. Let PP be the given line, and DD' its projection on OX. Through P draw PQ parallel to OX to meet the plane P' CD' ; and since it is perpendicular to this plane, the angle PQP' is right, and PQ = PP cos PPQ. But PQ and DD' are equal, since they are the intercepts made by two parallel planes on two parallel right lines. 6. If there be any three points P, P', P", the projection of PP" on any line will be equal to the sum of the projections on that line of PP' and PP". Let the projections of the three points be D, D\ D'\ then if D' lie between D and D", DD" is evidently the sum of DD' z p Q n jD 7 P > JO c V > the same direction. The angle then between the lines is acute. But if we spoke of the angle between PP' and C'C, we should draw the parallel PQ' m the opposite direction, and should wish to express the obtuse angle made by the lines with each other. When we speak of the angle made by any line OP with the axes, we shall always mean the angle between OP and the positive directions of the axes, viz. OX, OT, OZ. THE POINT. 5 and D'D". If D" lie between D and D', DD" is the difference of DD' and D'D"; but since the direction from D' to D" is the opposite of that from D to D', DD" is still the algebraic sum of DD' and D'D". It may be otherwise seen tbat the projection of P'P" is in the latter case to be taken with a negative sign, from tbe consideration that in this case the length of the projection is found by multiplying P'P" by the cosine of an obtuse angle (see note, p. 3). In general, if there be any number of points P, P', P", P'", &c, the projection of PP'" on any line is equal to the sum of the projections of PP', P'P", P"P", &c. 7. We • shall have constant occasion to make use of the following particular case of the preceding. If the co-ordinates of any point P be projected on any line, the sum of the three projections is equal to the projection of the radius vector on that line. For consider the points 0, D, C, P (see figure, p. 2) and the projection of OP must be equal to the sum of the pro- jections of OD (=«), DC (=#), and CP{=z). 8. Having established those principles concerning projec- tions which we shall constantly have occasion to employ, we return now to the more immediate subject of this chapter. TJie co-ordinates of the point dividing in the ratio in : n the distance between two points x'y'z', x"y"z", are mx" + nx' my" + ny' mz" + nz x = , y = — , 2= • m + n m-\-n m + n The proof is precisely the same as that given at Conies, p. 5, for the corresponding theorem in Plane Analytic Geometry. The lines PJ/, QJS T in the figure there given, now represent the ordinates drawn from the two points to any one of the. co-ordinate planes. If we consider the ratio m : n as indeterminate, we have the co-ordinates of any point in the line joining the two given points. 6 THE POINT. 9. Any side of a triangle is cut in the ratio m : n, and the line joining this point to the opposite vertex is cut in the ratio m + n : I, to find the co-ordinates of the point of section. Ans. _ Ix + mx" + nx" _ ly' + my" + ny"' _ lz' + rnz" + nz'" l + m + n ' * I -{-m + n ' l+m + n This is proved as in Plane Analytic Geometry (see Conies, p. 6). If we consider I, in, n as indeterminate, we have the co-ordinates of any point in the plane determined by the three points. Ex. The lines joining middle points of opposite edges of a tetrahedron meet in a point. The x's of two such middle points are \ ix' + x"), ^ (x" f + x""), and the x of the middle point of the line joining them is \ ix' 4- x" + x'" + x""). The other co-ordinates are found in like manner, and their symmetry shows that this is also a point on the line joining the other middle points. Through this same point will pass the line joining each vertex to the centre of gravity of the opposite triangle. For the x of one of these centres of gravity is J (x' + x" + x'"), and if the line join- ing this to the opposite vertex be cut in the ratio of 3 : 1, we get the same value as before. 10. To find the distance betioeen two points P, P', whose rectangular co-ordinates are x'y'z, x"y"z". Evidently (see figure, p. 3) PP'* = P'Q* + PQ\ But P'Q = z'-z", a,ni PQ'= GC' 2 is by Plane Analytic Geometry = [x — x"Y + («/' — y")'\ Hence PP* = (x'-x'T+{y'-y"y + (z'-s"Y. Cor. The distance of any point x'y'z' from the origin is given by the equation OP' = x' 2 + y' 2 +z'\ 11. The position of a point is sometimes expressed by its radius vector and the angles it makes with three rectangular axes. Let these angles be a, ,S, 7. Then since the co-ordinates x, y, z are the projections of the radius vector on the three axes, we have x = p cosct, y = p cos£?, z = p cosy. And, since x' 1 + y 2 + z 2 - p\ the three cosines (which are THE POINT. 7 sometimes called the direction-cosines of the radius vector) are connected by the relation cos 2 a + cos 2 /3 + cos 2 7 = 1.* The position of a point is also sometimes expressed by the following polar co-ordinates — the radius vector, the angle 7 which the radius vector makes with a fixed axis OZ, .and the angle COD{=) which 00 the projection of the radius vector on a plane perpendicular to OZ (see figure, p. 4) makes with a fixed line OX in that plane. Since then OC = p sin 7, the formulae for transforming from rectangular to these polar co-ordinates are x = p sin7 cos0, y = p SU17 sin<£, z = p COS7. 12. Tlie square of the area of any plane figure is equal to the sum of the squares of its projections on three rectangular planes. Let the area be A, and let a perpendicular to its plane make angles a, /3, 7 with the three axes; then (Art. 4) the projections of this area on the planes yz, zx, xy respectively, are A cosa, A cos$, A C0S7. And the sum of the squares of these three = A*, since cos 2 a + cos 2 /3 + cos 2 7 = 1. 13. To express the cosine of the angle 6 between two lines OP, OP' in terms of the direction-cosines of these lines. We have proved (Art. 10), pp* = ( x - x y+ty-yy+(z-zy. * I have followed the usual practice in denoting the position of a line by these angles, but in one point of view there would be an advantage in using instead the complementary angles, namely, the angles which the line makes with the co-ordinate planes. This appears from the corresponding formulae for oblique axes which I have not thought it worth while to give in the text, as we shall not have occasion to use them afterwards. Let a, f$, y be the angles which a line makes with the planes yz, zx, xy, and let A, B, be the angles which the axis of a- makes with the plane of yz, of y with the plane of a; and of z with the plane of xy, then the formulas which correspond to those in the text, are x sin A = p sin a, ysmB = psmf3, z sin C = p siny. These formula? are proved by the principle of Ait. 7. If we project on a line perpen- dicular to the plane of yz, since the projections of y and of z on this line vanish, the projection of x must be equal to that of the radius vector, and the angles made by x and p with this line are the complements of A and a. 8 THE POINT. But also PP* = p 2 4 p' 2 - 2pp cos 6. And since p 2 = x 2 + / + s 2 , p' 2 = x"' 4 y* 4 z'\ we have pp cos = xx 4 3/2/ + zs', or cos # = cos a cos a' 4 cos /3 cos /3' 4 cos 7 cos 7'. Coe. The condition that two lines should be at right angles to each other is cos a cos a' 4 cos/3 cos/3' + cos 7 cos 7' = 0. ] 4. The following formula is also sometimes useful : sin 2 = (cos/3 COS7' — C0S7 cos/3') 2 + (0037 cosa' — cosa COS7') 2 4 (cosa cos/3' — cos ,8 cosa') 2 . This may be derived from the following elementary theorem for the sum of the squares of three determinants [Lessons on Higher Algebra, Art. 21), but which can also be verified at once by actual expansion, {bo' - cb'f 4 {ca! - ac'f 4 (aV - ba'f = (a 2 4 b* 4 c") (a 2 + b' 2 4 c' 2 ) - {aa' 4 W 4 cc')\ For when a, b, c ; a', b\ c are the direction-cosines of two lines, the right-hand side becomes 1 — cos 2 #. Ex. To find the perpendicular distance from a point x'y'z' to a line through the origin whose direction-angles are a, /3, y. • Let P be the point x'y'z', OQ the given line, PQ the perpendicular, then it is plain that PQ = OP sin POQ; and using the value just obtained for sin POQ, and remembering that x' = OP cos a', &c, we have PQ? = in' cosy - 2' cos/3) 2 + (z' cosa - x' cosy) 2 + (x' cos/3 - 7/ cosa) 2 . 15. To find the direction-cosines of a line perpendicular to two given lines, and therefore perpendicular to their plane. Let a'/3Y, a."ft"ry" be the direction-angles of the given lines, and a/37 of the required line, then we have to find a/3y from the three equations cosa cosa' 4 cos/3 cos/3' 4 cos 7 cos 7' = 0, cosa cosa" 4 cos /3 cos/3"4cos7 cos7" = 0, cos 2 a 4 cos 2 /3 4 cos 2 7 = 1. TRANSFORMATION OF CO-ORDINATES. 9 From the first two equations we can easily derive, by elimi- nating in turn cosa, cos/3, cosy, X cosa = cos/8' cosy" — cos/3" cosy', X cos/3 = cosy' cosa" — cosy" cosa', \ cosy =cosa' cos/8" —cosa" cos8', where X is indeterminate ; and substituting in the third equa- tion, we get (see Art. 14), if 8 be the angle between the two given lines, \*=sin' 2 0. This result may be also obtained as follows : take any two points P, Q, or x ' y'z ', x"y"z", one on each of the two given lines. Now double the area of the projection on the plane of xy of the triangle POQ, is (see Conies, p. 31) xy' ' — y'x", or p'p" (cosa' cos/8" — cosa" cos8'). But double the area of the triangle is p'p" sin#, and therefore the projection on the plane of xy is p'p" sin 8 cosy. Hence, as before, sin# cosy = cosa' cos 8" - cosa" cos 8', and in like manner sin# cosa =cos/8' cosy" — cos8" cosy' ; sin0 cos8 = cosy' cosa" — cosy" cosa'. TRANSFORMATION OF CO-ORDINATES. 16. To transform to parallel axes through a new origin, whose co-ordinates referred to the old axes are x\ y', z'. The formulas of transformation are (as in Plane Geometry) x = X+x, y=Y+y\ s = Z+z. For let a line drawn through the point P parallel to one of the axes (for instance z) meet the old plane of xy in a point O, and the new in a point C ; then PC = PC' + C'C. But PC is the old z, PC is the new z ; and since parallel planes make equal intercepts on parallel right lines, C'C must be equal to the line drawn through the new origin 0' parallel to the axis of z, to meet the old plane of xy. 10 TRANSFORMATION OF CO-ORDINATES. 17. To pass from a rectangular system of axes to another system of axes having the same origin. Let the angles made by the new axes of x, y, z with the old axes be a, /3, y ; a', /3', 7' ; a", /3", 7" respectively. Then if we project the new co-ordinates on one of the old axes, the sum of the three projections will (Art. 7) be equal to the projection of the radius vector, which is the corresponding old co-ordinate. Thus we get the three equations a; = Xcosa + I^cosa' + Z cosa." ] y = X cos/3 + Y cos/3' + Z cos jS" \ (A). z = X cos y + Y C0S7' 4 Z cosy" > We have, of course, (Art. 11) cos 2 a + cos 2 /3 + cos 2 y = 1, cos 2 a' + cos 2 /3' + cos 2 7' = 1, cos 2 a" + cos 2 /3" + cos 2 7" = 1 (B). If the new axes be also rectangular, we have also (Art. 13) cosa 'cos a' + cos/3 cos/3' +cosy cosy' =0] cosa' cosa" + cos/3' cos/3"+ cosy' cosy" = Or ... [G). cosa" cosa + cos/3" cos/3 + cosy" cosy = ® By the help of these relations we can verify that when we pass from one system of rectangular axes to another, we have, as is geometrically evident, x 1 + y 2 + z* = X 2 + Y 2 + Z". When the new axes are rectangular, since a, a', a" are the angles made by the old axis of x with the new axes, &c. we must have cos 2 a + cos 2 a' + cos 2 a" = 1, cos 2 /3 + cos 2 /3' + cos 2 /3" = 1, cos 2 y + cos'V + cos 2 y"= 1 (D), cosa cos/3+cosa' cos/3'-f cosa" cos/3" = 0") cos/3 cosy + cos/8' cosy' + cos/3" cosy" = 0f ■••(-^)j cosy cosa +cosy' cosa' +cosy" cosa" =0J and the new co-ordinates expressed in terms of the old are X=xcosa +y cos/3 +scosy ) F=secosa' +y cos/3' +zcosy' f {&)• Z= x cosa" + y cos/3" + z cosy"-' TRANSFORMATION OF CO-ORDINATES. 11 It is not difficult to derive analytically equations D, E, F, from equations A, B, 0, but we shall not spend time on what is geometrically evident. 18. When we transform rectangular axes to a system not rectangular, let \, /^ y be the angles between the new axes of y and z, of z and x, of x and y respectively, then (Art. 13) co3X = cosa' cosa" + cos/3' cos/3" + C0S7' C0S7", cos/i = cosa" cosa + cos/3" cos/8 +cos7"cosy, cosi'=cosa cos a' + cos /3 cosyS' +COS7 C0S7'. Hence, squaring and adding equations (A) (Art. 17), x* 4 f + z* = X* + T' + Z 2 + 2YZ cos A, + 2ZX cos/* 4 2XY cos v. Thus we obtain the radius vector from the origin to any point expressed in terms of the oblique co-ordinates of that point. It is proved in like manner that the square of the distance between two points, the axes being oblique, is (x 1 - x"f 4 (y' -y"f 4 (z' - zj + 2 (y --y") {z 1 - z") cosX 4 2 («' - 2") (x' - x") cos/* 4 2 [x - x") {y - y") cos v.* 19. The degree of any equation between the co-ordinates is not altered by transformation of co-ordinates. This is proved, as at Conies, p. 9, from the consideration that the expressions given (Arts. 16, 17) for x, y, z, only involve the new co-ordinates in the first degree. * As we shall never require in practice the formula? for transforming from one set of oblique axes to another, we only give them in a note. Let A, B, C have the same meaning as at note, p. 7. and let a, /8, y; a', /J*, y'; a", |8", y" be the angles made by the new axes with the old co-ordinate planes; then by projecting on lines perpendicular to the old co-ordinate planes, as in the note referred to, we find x sin A = X sin a + T sin a' + Z sin a", y sinB = Xsin/3 + l'sin|3' + Zsin/3", z sinC =Xsiny + rsiny' + Zsiny". ( 12 ) CHAPTER II. INTERPRETATION OF EQUATIONS. 20. It appears from the construction of Art. 1 that if we were given merely the two equations x = a, y = b, and if the z were left indeterminate, the two given equations would de- termine the point C, and we should know that the point P lay somewhere on the line PC. These two equations then are considered as representing that right line, it being the locus of all points whose x = a, and whose y — b. We learn then that any two equations of the form x = a, y = b represent a right line parallel to the axis of z. In particular, the equa- tions x = 0, y = represent the axis of z itself. Similarly for the other axes. Again, if we were given the single equation x = a, we could determine nothing but the point D. Proceeding, as at the end of Art. 2, we should learn that the point P lay some where in the plane PBCD, but its position in that plane would be indeterminate. This plane then being the locus of all points whose x = a is represented analytically by that equation. We learn then that any equation of the form x = a represents a plane parallel to the plane yz. In particular, the equation x = denotes the plane yz itself. Similarly, for the other two co-ordinate planes. 21. In general, any single equation between the co-ordinates represents a surface of some hind ; any two simultaneous equations between them represent a line of some kind, either straight or curved; and any three equations denote one or more points. I. If we are given a single equation, we may take for x and y any arbitrary values; and then the given equation solved for z will determine one or more corresponding values of z. In other words, if we take arbitrarily any point G in the plane of xy, we can always find on the line PC one or INTERPRETATION OP EQUATIONS. 13 more points whose co-ordinates will satisfy the given equation. The assemblage then of points so found on the lines PC will form a surface which will be the geometrical representation of the given equation (see Conies, p. 13). II. When we are given two equations, we can, by elimi- nating z and y alternately between them, thrown them into the form y = (a;), z = ifr (x). If then we take for x any ar- bitrary value, the given equations will determine corresponding values for y and z. In other words, we can no longer take the point C anywhere on the plane of xy, but this point is limited to a certain locus represented by the equation y = [x, y) = will be satisfied not only for any point of this curve in the plane of xy, but also for any other point having the same x and y though a different z : that is to say, for any point of the surface generated by a right line moving along this curve, but remaining parallel to the axis of z.* The curve in the plane of xy can only be represented by two equations, namely, z = 0, cj>(x,y)=0. If an equation contain only one of the variables x, we know by the theory of equations, that it may be resolved into n factors of the form x — a = 0, and therefore (Art. 20) that it represents n planes parallel to one of the co-ordinate planes. * A surface generated by a right line moving parallel to itself is called a cylindrical surface. ( 16 ) CHAPTER III. THE PLANE. 25. In the discussion of equations we commence of course with equations of the first degree, and the first step is to prove that every equation of the first degree represents a plane, and conversely, that the equation of a plane is always of the first degree. We commence with the latter proposition, which may be established in two or three different ways. In the first place we have seen (Art. 20) that the plane of xy is represented by an equation of the first degree, viz. z = 0; and transformation to any other axes cannot alter the degree of this equation (Art. 19). We might arrive at the same result by forming the equation of the plane determined by three given points, which we can do by eliminating I, m, n from the three equations given Art. 9, when we should arrive at an equation of the first degree. The following method however of expressing the equation of a plane leads to one of the forms most useful in practice. 26. To find the equation of a plane, the perpendicular on which from the origin =p, and makes angles a, /3, 4 C'*)} ' l(f {AB'-A'BY+(BC'-B'CT+(GA'- C'A)* (A 2 + B* + C*) {A" 2 4 £" 4 C" 2 ) Hence the condition that the planes should cut at right angles hAA' + BB' + CG' = 0. They will be parallel if we have the conditions AB' = A'B, BC' = B'G, CA'=G'A; in other words, if the coefficients A, B, G be proportional to A', B\ C", in which case it is manifest from the last article that the direction of the perpendicular on both will be the same. 29. To express the equation of a plane in terms of the in- tercepts a, 6, c, which it makes on the axes. c sin 18 THE PLANE. The intercept made on the axis of x by the plane Ax + By + Gz + D = is found by making y and z both = 0, when we have Aa+D=0. And similarly, Bb -f D = 0, Cc + D = 0. Substituting in the general equation the values just found for A, B, G, it becomes x y z _ a b c If in the general equation any term be wanting, for instance, if .4 = 0, the point where the plane meets the axis of x is at infinity, or the plane is parallel to the axis of x. If we have both A = 0, Z? = 0, then the axes of x and y meet at infinity the given plane which is therefore parallel to the plane of xy (see also Art. 20). If we have A = 0, B = 0, C=0, all three axes meet the plane at infinity, and we see, as at Gonics, p. 66, that an equation O.x+O.y + 0.z + D = must be taken to re- present a plane at infinity. 30. To find the equation of the plane determined by three points. Let the equation be Ax + By + Cz+D = 0; and since this is to be satisfied by the co-ordinates of each of the given points, A, B, G, D must satisfy the equations Ax' + By' +Cz' + D = 0, Ax" + By" + Gz" + D = 0, Ax'" + By'" + Cz'" + D = 0. Eliminating A : B, G, D between the four equations, the result is the determinant x, i x y-> z , i y", *", i i r,y'",z"\ 1=0. Expanding this by the common rule, the equation is x{yJs'-z") +y"{z'"-d) +y'"(z'-z")} + y {«' (x" - x") + z" (x'" - x') + z'" {x' - x")} + z{x'(y"-y'") +X "(y'"-y') +x'"(y'-y")} = x' (yV " - y'"z") + x" iy'"z' - y'z'") + x"' {y'z" - y"z'\ THE PLANE. 19 If we consider x, y, z as the co-ordinates of any fourth point, we have the condition that four points should lie in one plane. 31. The coefficients of a;, y, z in the preceding equation are evidently double the areas of the projections on the co- ordinate planes of the triangle formed by the three points. If now we take the equation (Art. 26) x cosa + y cos/S + s cosy=j3, and multiply it by twice A, (A being the area of the triangle formed by the three points) the equation will become identical with that of the last article, since A cos a, A cos/3, A cos 7, are the projections of the triangle on the co-ordinate planes (Art. 4). The absolute term then must be the same in both cases. Hence the quantity x' (y"z'" - y'"z") + x" (y" V - yV") + x" &*" ~ f*') represents double the area of the triangle formed by the three points multiplied by the perpendicular on its plane from the origin : or, in other words, six times the volume of the triangular pyramid, lohose base is that triangle, and whose vertex is the origin.* * If in the preceding values we substitute for x', y', z' ; p' cos a', p' cos/3', p' cosy', n > p ; where A? is the sum of the squares of the three numerators. Ex. 4. To find the equation of the plane through the two intersecting lines x — x' _y — y' _z — z' x — x' _y — y' __z — z' cosa — cos/? — cosy ' cosa' — cos/3' ~ cosy' ' The required plane passes through x'y'z' and its perpendicular is perpendicular to two lines whose direction-cosines are given ; therefore, (Art. 15) the required equation is {x — x') (cos/3 cosy' — cosy cos/3') + {y — y') (cosy cos a' — cosy' cosa) + (z — z') (cosa cos/3' — cosa' cos/3) = 0. Ex. 5. To find the equation of the plane passing through the two parallel lines x — x' y — y 1 z — z' x — x 1 ' _ y — y" z — z'' cosa ~ cos/3 — cosy ' cosa — cos/3 — cosy The required plane contains the line joining the given points, whose direction- cosines are proportional to x' — x", y' — y", z' — z 1 ' ; the direction-cosines of the perpendicular to the plane are therefore proportional to {y' — y") cosy - (z' — z") cos ft (z' — z") cosa — (x' — x") cosy, (x' — x") cos/3 — (!/' — y") cosa. These may therefore be taken as the coefficients of x, y, z, in the required equation, while the absolute term determined by substituting x'y'z' for xyz in the equation ia (yV — y"z') cosa + (z'x" — z'V) cos/3 + (x'y" — x"y') cosy. 42. To find the equations of the perpendicular from x'y'z on the plane Ax + By + Cz + D. The direction-cosines of the perpendicular on the plane (Art. 27) are proportional to A, B, C; hence the equations required are x — x'_y — y' z — z' ~A~ = ~^~ == ~C'- 43. To find the direction-cosines of the bisector of the angle between two given lines. As we are only concerned with directions it is of course sufficient to consider lines through the origin. If we take points x'y'z', x'y'z' one on each line, equidistant from the origin, then the middle point of the line joining these points is evidently a point on the bisector, whose equation therefore is xyz x + x" ~ y' + y" ~ z' + z" ' THE EIGHT LINE. 27 and whose direction-cosines are therefore proportional to x'-\rx", y'+y", z' + z"; but since a?', y\ z', x", y", z" are evidently proportional to the direction-cosines of the given lines, the direction-cosines of the bisector are cosa' + cosa", cos/3' + cos/8", cos7' + cos7", each divided by the square root of the sum of the squares of these three quantities. The bisector of the supplemental angle between the lines is got by substituting for the point x"y"z" a point equi-distant from the origin measured in the opposite direction, whose co-ordinates are — x", —y", —z"; and therefore the direction- cosines of this bisector are respectively proportional to cosa' — cosa", cos/3' — cos/3", C0S7' — COS7". N.B. The equation of the plane bisecting the angle between two given planes is found precisely as at Conies, p. 30, and is (x cosa +y cos/3 + zcosy — p) =±{x cosa' + y cos/3' + z COS7' — j>). 44. To find the angle made with each other by two lines x — a y — b s ~ c . x — a_y — b z — e I m no m n Evidently (Arts. 13, 41), . IF + mm' + nn' cos a = ■ COR. The lines are at right angles to each other if W + mm' + nn = 0. Ex. To find the angle between the lines =■ = tton = -tt^t ; ttsv = y, s = 0. Am. 30°. 45. To find the angle between the plane Ax + By -f- Cz + D, and the line x—a_y—b _ z—c I m. n The angle between the line and the plane is the complement of the angle between the line and the perpendicular on the plane, and we have therefore Al + Bin + On sin0 = -/(P + m' + M*) J (A* +& + (?) 28 THE EIGHT LINE. Coe. When Al + Bm + Gn = 0, the line is parallel to the plane, for it is then perpendicular to a perpendicular on the plane. 46. To find the conditions that a line x — mz + a, y = nz + b should be altogether in a plane Ax + By+ Cz + D. Substitute for x and y in the equation of the plane, and solve for z, when we have Aa + Bb + D ~ Am + Bn+ C and if both numerator and denominator vanish, the value of z is indeterminate and the line is altogether in the plane. We have just seen that the vanishing of the denominator expresses the condition that the line should be parallel to the plane ; while the vanishing of the numerator expresses that one of the points of the line is in the plane, viz. the point ab where the line meets the plane of xy. In like manner in order to find the conditions that a right line should lie altogether in any surface, we should substitute for x and y in the equation of the surface, and then equate to zero the coefficient of every power of z in the resulting equation. It is plain that the number of conditions thus resulting is one more than the degree of the surface.* 47. To find the equation of the plane drawn through a given line perpendicular to a given plane. Let the line be given by the equations Ax + By+Cz + D = 0, A'x + B'y + C'z + D '= 0, and let the plane be A"x + B"y+ C"z + D" = Q. Then any plane through the line will be of the form X (Ax + By + Gz + D) + /* [A'x + B'y + C'z +■ D') = 0, * Since the equations of a right line contain four constants, a right line can be determined which shall satisfy any four conditions. Hence any surface of the second degree must contain an infinity of right lines, since we have only three conditions to satisfy and have four constants at our disposal. Every surface of the third degree must contain a finite number of right lines since the number of conditions to be satisfied is equal to the number of disposable constants. A surface of higher degree will not necessarily contain any right line lying altogether in the surface. THE RIGHT LINE. 29 and in order that it should be perpendicular to the plane we must have (\A + iJ,A')A"+(\B + fiB')B" + (XC+iJ,C')C" = 0. This equation determines X : fi ) and the equation of the required plane is {A' A" + B'B" + C 0") {Ax + By + Cz + D) = (AA" + BB" + CC") (A'x + B'y + C'z + D'). When the equations of the given plane and line are given in the form _ x — x v — y' z — z' x cosa + y cos/3 + z COS7 = » : , = - — £7 = T : J 1 sr 1 coga COS/3 COS7 ' we can otherwise easily determine the equation of the required plane. For it is to contain the given line whose direction-angles are a', /3', 7' ; and it is also to contain a perpendicular to the given plane whose direction-angles are a, /3, 7. Hence (Art. 15) the direction-cosines of a perpendicular to the required plane are proportional to cosy3'cos7— cosyScos7', cos7'cosa— C0S7 cosa',cosa' cos /3— cosa cos/3', and since the required plane is also to pass through xy'z\ its equation is (x— x')(cos/8 C0S7'— cos/3' cos7) + (y— ?/')(cos7 cosa' — C0S7' cosa) + [z — z) (cosa cos/6' — cosa' cos/8) = 0. 48. Given two lines to find the equation of a plane drawn through either parallel to the other. First, let the given lines be the intersections of the planes L, M; N, P whose equations are given in the most general form. Then proceeding exactly as in Art. 36, we obtain the identical relation L{A'B"C'")-M(A"B'"C)+N(A" I BC')-P{AB'C")=(A'B"C'"B), the right-hand side of the equation being the determinant, whose vanishing expresses that the four planes meet in a point. It is evident then that the equations L (A'B"C") - M(A"B'"C) = 0, N(A'"BC) - P{AB'G") = represent parallel planes since they only differ by a constant 30 THE EIGHT LINE. quantity ; but these planes pass each through one of the given lines. Secondly, let the lines be given by equations of the form x — x y — y z — z' x — x" y — y" _ z — z" cosa cos/3 cosy ' cosa' cos/3' cosy' Then since a perpendicular to the sought plane is perpendicular to the direction of each of the given lines, its direction-cosines (Art. 15) are the same as those given in the last example, and the equations of the sought parallel planes are (x — x) (cos/3 cosy'— cos/3' cosy) + (y—y')(cosy cosa'— co3y' cosa) + {z — z') (cosa cos/3' — cosa' cos/3) = 0, [x — £e")(cos/3 cosy'— cos/3' cosy) + {y—y") (cosy cosa'— cosy' cosa) + (z — z") (cosa cos/3' - cosa' cos/S) = 0. The perpendicular distance between two parallel planes is equal to the difference between the perpendiculars let fall on them from the origin, and is therefore equal to the difference between their absolute terms, divided by the square root of the sum of the squares of the common coefficients of x, y, z. Thus the per- pendicular distance between the planes last found is (x'—x") (cos/3 cosy'— cos/3' cos cosa = i/ ' cos/S^— p— , cosy=-^^. 32 NOTE ON THE PEOPEET1ES OF TETEAHEDRA. Substituting these values and reducing, we find for the required relation a" p - e 2 ) (rf» -f) + V [e 2 -f) (e 2 - d 2 ) + c 2 {f - d 2 ) [f - e 2 ) + a 2 d 2 (d z -b 2 - c 2 ) + b 2 e 2 {b 2 -a 2 - c 2 ) + c 2 f (c 2 - a 2 - b 2 ) + a'bV = 0. 51 . To express the volume of a tetrahedron in terms of its six edges. Let the sides of the triangle formed by any face ABC be a, 6, c ; the perpendicular on that face from the remaining vertex be p, and the distances of the foot of that perpendicular from A, B, G be d', e, f. Then a, b, c, d\ e, f are connected by the relation given in the last article. But if d, e, f be the remaining edges d 2 = d' 2 +p 2 , e 2 = e' 2 +p'\ f 2 =f' 2 +p 2 ; whence d 2 — e' 2 = d' 2 — e' 2 , &c. and putting in these values, we get - F=p 2 (2a 2 b 2 + 2b 2 c 2 + 2cV - a 4 - 5 4 - c 4 ), where F is the quantity on the left-hand side of the equation in the last article. Now the quantity multiplying p 2 is 16 times the square of the area of the triangle ABC, and since p mul- tiplied by this area is three times the volume of the pyramid, wehavei^=-144F 2 . 52. To find the relation between the six arcs joining four points on the surface of a sphere. We proceed precisely as in Art. 50, only substituting for the formulas there used the corresponding formulas for spherical triangles, and if a, /3, y, 8, e, <£ represent the cosines of the six arcs in question, we get « 2 +/3 2 +7 2 +S 2 +e 2 + 2 -a 2 S 2 - / SV- 7 2 ^+2a / 88 £ + 2/3 7 £<£+27aS4> - 2a/3y - 2ae0 - 2/3 S<£ - 278s = 1. This relation may be otherwise proved as follows: Let the direction-cosines of the radii to the four points be cosa, cos/3, cos 7 , cosa', cos/S', cosy', cosa", cos/3", cosy", cosa'", cos/3"', cosy'". NOTE ON THE PROPERTIES OP TETRAHEDRA. 33 Now from this matrix we can form (by the method of Lessons on Higher Algebra, Art. 20) a determinant which shall vanish identically, and which (substituting cos a a + cos'' ! / 8 + cos'' ! 7 = l, cosa cosa '+ cos/3 cos/3' + cos 7 COS7' = cos a5, &c.) is 1, cosaS, cosac, cosa^ cos 5a, 1, cos 5c, cosbd cosca, coscS, 1, cosc^ cosc?a, cos db, cosdc, 1 which expanded has the value written above. = 0, 53. To find the radius of the sphere circumscribing a tetra- hedron. Since any side a of the tetrahedron is the chord of the arc 2 whose cosine is a, we have a = 1 — —3 , with similar expressions for /S, 7, &c. ; and making these substitutions, the formula of the last example becomes F_ 2a 2 d*bV + WeVf + 2 C yW - a'd* - 5V - c'f _ 4r 6+ 16r 8 ~ ' whence if we have ad+be + cf=28 i S{S-ad){S-be)(S-cf) 36 V The reader may exercise himself in proving that the shortest distance between two opposite sides of the tetrahedron is equal to six times the volume divided by the product of those sides multiplied by the sine of their angle of inclination to each other, which may be expressed in terms of the sides by the help of the relation 2ad cos d = b 2 + e 2 -c i -f 2 . ( 34 ) CHAPTER IV. •PROPERTIES COMMON TO ALL SURFACES OF THE SECOND DEGREE. 54. We shall write the general equation of the second degree ax 2 + by 2 + cz 2 + 2lyz + 2mzx + 2nxy + 2px + 2qy + 2rz + d=0. This equation contains ten terms, and since its signification is not altered if by division we make one of the coefficients unity, it appears that nine conditions are sufficient to determine a surface of the second degree, or as we shall call it for short- ness, a quadric^ surface. Thus if we are given nine points on the surface, by substituting successively the co-ordinates of each in the general equation, we obtain nine equations which are I c sufficient to determine the nine unknown quantities - , - , &c. a 1 a 1 And in like manner the number of conditions necessary to de- termine a surface of the n* degree is one less than the number of terms in the general equation. The equation of a quadric may also (see Art. 37) be ex- pressed as a homogeneous function of the equations of four given planes x, y, 2, a>, ax 2 +by 2 +cs 2 + d ■+2qym+2rzm=0. For the nine independent constants in the equation last written may be so determined that the surface shall pass through nine given points, and therefore may coincide with any given quadric. In like manner (see Conies, p. 67) any ordinary x, y, z equa- tions may be made homogeneous by the introduction of the * The reader will compare the corresponding discussion of the equation of the second degree (Conies, Chap, x.) and observe the identity of the methods now pursued and of many of the results obtained. t In the Treatise on Solid Geometry by Messrs. Frost and Wolstenholme, surfaces of the second degree are called conicoids. PROPERTIES OF THE SECOND DEGREE. 35 linear unit (which we shall call w) ; and we shall frequently employ equations written in this form for the sake of greater symmetry in the results. We shall however for simplicity commence with x, y, z co-ordinates. 55. The co-ordinates are transformed to any parallel axes drawn through a point x'y'z', by writing x + x, y + y', z + z for x, y, z respectively (Art. 16). The result of this substitu- tion will be that the coefficients of the highest powers of the variables (a, 5, c, £, m, n) will remain unaltered, that the new absolute term will be U' (where U' is the result of substituting x', y\ z' for x, y, z in the given equation), that the new coeffi- 7 TJI cient of x will be 2 [ax' + ny -f mz ~\-p) or -7-7- , and in like manner that the new coefficients of y and z will be — r-7 dU' . . . dy and -J7 . We shall find it convenient to use the abbreviations dU dU dU u » u " u ' {m -te>7&>&- 56. We can transform the general equation to polar co- ordinates by writing x = Xp, y = /j,p, z = vp (where, if the axes be rectangular, X, /j, v are equal to cos a, cos/3, cos 7 respec- tively, and if they are oblique (see note, p. 7) X, /a, v are still quantities depending only on the angles the line makes with the axes) when the equation becomes p 3 (aX* + b[i* + cv* + 2lfj.v + 2mvX + 2rik(i) + 2p (j>\ +qfi + rv) + d = 0. This being a quadratic gives two values for the length of the radius vector corresponding to any given direction; in ac- cordance with what was proved (Art. 22), viz. that every right line meets a quadric in two points. 57. Let us consider first the case where the origin is on the surface (and therefore d=0), in which case one of the roots of the above quadratic is p = ; and let us seek the condition that the radius vector should touch the surface at the origin. In this case obviously the second root of the quadratic will also vanish, and the required condition is therefore pX + qp + rv = 0. D2 36 PEOPEETIES COMMON TO ALL SURFACES If we multiply by p and replace \p, pp, vp by x, y, 2, this becomes j>x + qy + rz = 0, and evidently expresses that the radius vector lies in a certain fixed plane. And since \, p, v are subject to no restriction but that already written, every radius vector through the origin drawn in tbis plane touches the surface. Hence we learn that at a given point on a quadric an in- finity of tangent lines can be drawn, that these lie all in one plane which is called the tangent plane at that point ; and that if the equation of the surface be written in the form m 2 + w, = 0, then u x = is the equation of the tangent plane at the origin. 58. We can find by transformation of co-ordinates the equa- tion of the tangent plane at any point x'y'z on the surface. For when we transform to this point as origin, the absolute term vanishes, and the equation of the tangent plane is (Art. 55) xu;+yu;+ e u;=o, or, transforming back to the old axes, {x-x^U' + ^-^U' + iz-^U^O. This may be written in a more symmetrical form by the intro- duction of the linear unit oj, when, since £7 is now a homogeneous function, and since x'y'z is to satisfy the equation of the surface, we have x'U;+y'U; + z'U; + o>'i; = 2U' = 0. Adding this to the equation last found, we have the equation of the tangent plane in the form xVJ + yUJ + zUJ + uU^-O; or, writing at full length, x (ax + ny + mz +p)+y (nx' + by 4 lz' + q) + z (mx' + ly + cz' + r) +px' + qy' + rz' + d= 0. This equation, it will be observed, is symmetrical between xyz and x'y'z') and may likewise be written x'U^y'Q + z'U^to'U^O. OF the se P P P d or, returning to x, y, z co-ordinates, px + ay + rz + d = ; but this is the polar plane of the origin, as may be seen by making a;', y\ z all =0 in the equation written in full (Art. 58). 38 PROPERTIES COMMON TO ALL SURFACES From this definition of the polar plane, it is evident that if a section of a surface be made by a plane passing through any point, the polar of that point with regard to the section will be the intersection of the plane of section with the polar plane of the given point. For the locus of harmonic means of all radii passing through the point, must include the locus of har- monic means of the radii which lie in the plane of section. 61. If the polar plane of any point A pass through B, then the polar plane of B will pass through A. For since the equation of the polar plane is symmetrical with respect to xyz, oe'y'z', we get the same result whether we substitute the co-ordinates of the second point in the equation of the polar plane of the first, or vice versa. The intersection of the polar planes of A and of B will be a line which we shall call the polar line, with respect to the surface, of the line AB. It is easy to see that the polar line of the line AB is the locus of the poles of all planes which can be drawn through the line AB. 62. If in the original equation we had not only d = 0, but also jy, q, r each = 0, then the equation of the tangent plane at the origin, found (Art. 58), becomes illusory since every term vanishes ; and no single plane can be called the tangent plane at the origin. In fact the coefficient of p (Art. 56) vanishes whatever be the direction of p, and therefore every line drawn through the origin meets the surface in two consecutive points, and the origin is said to be a double point on the surface. In the present case, the equation denotes a cone whose vertex is the origin, as in fact does every homogeneous equation in x, y, z. For if such an equation be satisfied by any co- ordinates x\ y\ z', it will also be satisfied by the co-ordinates Jos', hy\ Jcz (where k is any constant), that is to say, by the co-ordinates of every point on the line joining x'y'z 1 to the origin. This line then lies wholly in the surface, which must therefore consist of a series of right lines drawn through the origin. The equation of the tangent plane at any point of the OF THE SECOND DEGREE. 39 cone now under consideration may be written in either of the forms xu;+ y u; + zu; = % x'u^ y 'u 2 +z'u 3 =o. The former form (wanting an absolute term) shews that the tangent plane at every point on the cone passes through the origin; the latter form shews that the tangent plane at any point x'y'z touches the surface at every point of the line joining x'y'z' to the vertex ; for the equation will represent the same plane if we substitute Tex, lcy\ kz' for x', y', z'. When the point x'y'z' is not on the surface, the equation we have been last discussing represents the polar of that point, and it appears in like manner that the polar plane of every point passes through the vertex of the cone, and also that all points which lie on the same line passing through the vertex of a cone have the same polar plane. To find the polar plane of any point with regard to a cone we need only take any section through that point, and take the polar line of the point with regard to that section ; then the plane joining this polar line to the vertex will be the polar plane required. For it was proved (Art. 60) that the polar plane must contain the polar line, and it is now proved that the polar plane must contain the vertex. 63. We can easily find the condition that the general equa- tion of the second degree should represent a cone. For if it does it will be possible by transformation of co-ordinates to make the new j), q, r, d vanish. The co-ordinates of the new vertex must therefore (Art. 55) satisfy the conditions u; =o, u;=o, c; = o, w=o, which last combined with the others is equivalent to Z7' 4 =0. And if we eliminate x', y', z from the four equations ax' + ny + mz' + p = 0, nx + by' + lz' + q = 0, mx + ly' + cz + r = 0, px + qy + rz + d = 0, 40 PROPERTIES COMMON TO ALL SURFACES we obtain the required condition in the form of the determinant a, n, m, P n, h h 1 ra, h c ) r Pi , 2L, 2M, 2N, 2P, 2Q, 2B, to denote the differential coefficients of A taken with respect to a, b, c, &c. Thus A = bed + 2rql - br 2 - cq> - dl\ B = cda + 2pmr — cp' — dm 2 — ar', C = dab + 2nqp — dti' — aq 2 — bp', D = abc + 2lmn — aP — hii 2 — en 1 , L = aqr + dmn — adl + lp 2 — nrp —pmq, M=bpr + dln — bdrn + mq 2 — qnr — lpq } N= cpq + dim — cdn + nr 2 — mqr — rip, P=bmr+cnq —bep 4 Pp — nrl—lmq, Q = alr + enp - caq + m 2 q — mnr— Ipm, B — alq +• bmp — abr + n 2 r — mqn— nip. 64. Let us return now to the quadratic of Art. 56, in which d is not supposed to vanish, and let us examine the condition that the radius vector should be bisected at the origin. It is obviously necessary and sufficient that the coefficient of p in that quadratic should vanish, since we should then get for p values equal with opposite signs. The condition required tllen Is p\ + q/j, + rv = J which multiplied by p shews that the radius vector must lie in the plane px + qy + rz = 0. Hence (Art. 60) every right line drawn through the origin in a plane parallel to its polar plane is bisected at the origin. OP THE SECOND DEGREE. 41 65. If however we had p = 0, q = 0, r = 0, then every line drawn through the origin would be bisected and the origin would be called the centre of the surface. Every quadric has in general one and but one centre. For if we seek by trans- formation of co-ordinates to make the new p, q,r = 0, we obtain three equations, viz. Z7/ = 0, or ax' + ny' + mz' +p = 0, Z7 2 ' = 0, or nx' + by' + h' + q=0, Z7 3 ' = 0, or mx' ■+ ly' + cz' + r = 0, which are sufficient to determine the three unknowns x', y\ z'. P Q Tt The resulting values are x' = -^ , y' = -^ , z' = -=- , where P, Q, B, D have the same meaning as in the last article. If however Z>=0 the co-ordinates of the centre become infinite and the surface has no finite centre. If we write the original equation w 2 +- w, + w = 0, it is evident that D is the discriminant of w a * 66. To find the locus of the middle points of chords parallel to a given line — = — = -. J X ft v If we transform the equation to any point on the locus as origin, the new jp, q, r must fulfil the condition (Art. 64) p\ + q/i + rv = 0, and therefore (Art. 55) the equation of the locus is This denotes a plane through the intersection of the planes U^ U 3 , U s , that is to say, through the centre of the surface. * It is possible that the numerators of these fractions might vanish at the same time with the denominator, in which case the co-ordinates of the centre would become indeterminate, and the surface would have an infinity of centres. Thus if the three planes U{, Ui, U£ all pass through the same line, any point on this line will be a centre. The conditions that this should be the case may be written ", », m, p ii, b, I, 1 in, ', <; r :0, the notation indicating that all the four determinants must = 0, which are got by erasing any of the vertical lines. We shall reserve the fuller discussion of these cases for the next chapter. 42 PROPERTIES COMMON TO ALL SURFACES It is called the diametral plane conjugate to the given direction of the chords. lix'y'z' be any point on the radius vector drawn through the origin parallel to the given direction, the equation of the diametral plane may be written x'lT l+ y'U^z'U^0. If now we take the equation of the polar plane of kx, ky\ kz\ kx' U x + ky U 2 + kz' # s + Z7 4 = 0, divide it by k, and then make k infinite, we see that the diametral plane is the polar of the point at infinity on a line drawn in the given direction, as we might also have inferred from geometrical considerations (see Conies, p. 281). In like manner, the centre is the pole of the plane at infinity, for if the origin be the centre its polar plane (Art. 60) is d = 0, which (Art. 29) represents a plane situated at an infinite distance. In the case where the given surface is a cone, it is evident that the plane which bisects chords parallel to any line drawn through the vertex is the same as the polar plane of any point in that line. In fact it was proved that all points on the line have the same polar plane, therefore the polar of the point at infinity on that line is the same as the polar plane of any other point in it. 67. The plane which bisects chords parallel to the axis of x is found, by making /j, = 0, v = in the equation of Art. 66, to be U^ = 0, or ax + ny + mz +p = 0,* and this will be parallel to the axis of y, if n = 0. But this is also the condition that the plane conjugate to the axis of y should be parallel to the axis of x. Hence if the plane con- jugate to a given direction be parallel to a second given line, the plane conjugate to the latter will be parallel to the former. * It follows that the plane x = will bisect chords parallel to the axis of x, if n = 0, m = 0, p = ; or, in other words, if the original equation do not contain any odd power of x. But it is otherwise evident that this must be the case in order that for any assigned values of y and z we may obtain equal and opposite values of x. OF THE SECOND DEGREE. 43 When w = the axes of x and y are evidently parallel to a pair of conjugate diameters of the section by the plane of xy ; and it is otherwise evident that the plane conjugate to each of two conjugate diameters of a section passes through the other. For the locus of middle points of all chords of the surface parallel to a given line must include the locus of the middle points of all such chords which are contained in a given plane. Three diametral planes are said to be conjugate when each is conjugate to the intersection of the other two, and three diameters are said to be conjugate when each is conjugate to the plane of the other two. Thus we should obtain a system of three conjugate diameters by taking two conjugate diameters of any central section together with the diameter conjugate to the plane of that section. If we had in the equation 1 = 0, m = 0, n = 0, it appears from the commencement of this article that the co-ordinate planes are parallel to three conjugate diametral planes. When the surface is a cone it is evident from what was said (Arts 62, 66) that a system of three conjugate diameters meets any plane section in points such that each is the pole with respect to the section of the line joining the other two. 68. A diametral plane is said to be principal if it be per- pendicular to the chords to which it is conjugate. The axes being rectangular, and \, fj,, v the direction- cosines of a chord, we have seen (Art. 66) that the corresponding diametral plane is X(ax + ny + mz+p) + fi (nx + by + te+ q)+ v (mx+ly+cz + r)=0, and this will be perpendicular to the chord, if (Art. 42) the coefficients of x, y, z be respectively proportional to X, fi, v. This gives us the three equations Xa + fin + vm = JcX, Xn + fib + vl = kfi, Xm + fil + vc = Jcv. From these equations which are linear in X, /i, v, we can eliminate \, fi, v, when we obtain the determinant a — k, n, m «, b — Jc, I m, I, c — h =0, 44 PEOPEETIES COMMON TO ALL SUEFACES which expanded gives a cubic for the determination of k, viz. ¥ - ¥ {a ■+ b + c) + h {ab + be 1 ca - f - »i 2 - ri') - (abc + 2lmn - a? - bn? - cri') = 0. And the three values hence found for Jc being successively substituted in the preceding equations enable us to determine the corresponding values of X, fi, v. Hence a quadric has in general three principal diametral planes, the three diameters perpendicular to which are called the axes of the surface. We shall discuss this equation more fully in the next chapter. Ex. To find the principal planes of 7a; 2 + 6/ + 5z 2 — ixy - iyz = 6. The cubic for h is B - 1W + 99£ - 162 = 0, whose roots are 3, 6, 9. Now our three equations are 7\-2fi = k\, - 2\ + 6p. - 2v = V, - 2/t + 5k = hv. If in these we substitute k = 3, we find 2X = /j. = v. Multiplying by p, and sub- stituting x for \p, &c, we get for the equations of one of the axes 2x = y — z. And the plane drawn through the origin, (which is the centre) perpendicular to this line, is x + 2y + 1z = 0. In like manner the other two principal planes are 2x — 2y + z = 0, 2x + y - 2z = 0.* 69. The sections of a quadric by parallel planes are similar to each other. Since any plane may be taken for the plane of xy, it is sufficient to consider the section made by it, which is found by putting s = in the equation of the surface. But the section by any parallel plane is found by transforming the equation to parallel axes through any new origin, and then making z = 0. If we retain the planes yz and zx, and transfer the plane xy parallel to itself, the section by this plane is got at once by writing z = c in the equation of the surface, since it is evident that it is the same thing whether we write z + c for z : and then make z = 0, or whether we write at once z = c. * It is prayed (Lessons on Higher Algebra, p. 112) that if U denote the terms of highest degree in the equation, and S denote (be - V) x 2 + (ca - m 2 ) y> + [ab - re 2 ) z 2 + 2 (e/- al) yz + 2 (fd - bm) zx + 2(de- ere) xy, then the equation of the three principal planes, the centre being origin, is denoted by the determinant = 0. X, y, z u v u 2 , u, s u s t , >% OF THE SECOND DEGREE. 45 And since the coefficients of x'\ xy, and y* are unaltered by this transformation, the curves are similar. It is easy to prove algebraically, that the locus, of centres of parallel sections is the diameter conjugate to their plane, as is geometrically evident. 70. If p', p" be the roots of the quadratic of Art. 56, their product p'p" is = d divided by the coefficient of p 2 . But if we transform to parallel axes, and consider a radius vector drawn parallel to the first direction, the coefficient of p 2 remains unchanged, and the product is proportional to the new d. Hence if through two given points A, B, any parallel chords be drawn meeting the surface in points B, R' ; S, S', then the products BA.AB', SB.JBS' are to each other in a constant ratio, namely, U' : U" where U', U" are the results of sub- stituting the co-ordinates of A and of B in the given equation. 71. We shall conclude this chapter by shewing how the theorems already deduced from the discussion of lines passing through the origin might have been derived by a more general process, such as that employed [Conies, Art. 91). For sym- metry we use homogeneous equations with four variables. To find the. points where a given quadric is met by the line joining two given points xy'z". Let us take as our unknown quantity the ratio ? : m, in which the joining line is cut at the point where it meets the quadric, then (Art. 8) the co-ordinates of that point are proportional to mx + lx'\ my' + ly", mz + Zz", W + Ico" ; and if we substitute these values in the equation of the surface, we get for the determination of I : to, a quadratic m*W + lmP+FU" = 0. The coefficients of P and m a are easily seen to be the results of substituting in the equation of the surface the co-ordinates of each of the points, while the coefficient of Im may be seen (by Taylor's theorem, or otherwise) to be capable of being written in either of the forms «tu;' + y'u;' + z>u;>+ should touch the surface given by the general equation. If x, y, 2, m be the co-ordinates of the point of contact, and k an indeterminate multiplier, we have (Art. 58) Jca=ax + ny + rnz+pa, kft = nx + by + Iz + qa>, ky=mx+ ly + cz +j-u, k8 =px + qy + rz+ dm, from which equations, together with ax + fty + yz + Sw = 0, we have to eliminate x, y, s, w. Solving for x, y, «, m from these equations, we find Ax = Jc {Aa + N/3 + My + PS), Ay = k(Na+B/3 + Ly + Q8), Az=k[Ma + Ll3-)- Cy+B8), Am = k (Pa + QP + By + DB)* where A, B, C, &c. have the same meaning as in Art. 63. Substituting these values in ax + fiy + yz + dco = 0, we get Aa' -+ B& + Of ■+ JD8" + 2LPy + 2Mya + 2iVa/3 + 2Pa8 + 2 Q@$ + 2ByS = 0, which is the required relation. This condition may also be written a, «» ™, _p, a «i &, z > 2) P m, ', c, r, 7 Pi 2j r, d, s «, & y, 8 = 0. 76. The condition that the surface should be touched by any line ax + f3y + yz + 8eo = 0, a'a; + /3'# + y'z + B'm = 0, is found by eliminating two of the variables between the equa- tions of the line and that of the quadric, and forming the con- dition that the resulting quadratic should have equal roots. The result contains the coefficients of the quadric in the second * These equations also contain the solution of the problem to find the co-ordinates of the pole of a given plane. 48 PROPERTIES OP THE SECOND DEUREE. degree, and is also a quadratic function of the determinants (a/3' -/3a'), (a 7 '- 7 a'), &c. Writing these (a/3'), (a 7 '), &c. the result is found to be S [ah - ri>) (yS'f + 22 [mn - a!) (/3S') ( 7 S') + 2Sw{(aS')( 7 /S')-(a 7 ')(/3S')}, where the sum includes all terms of like form obtained by symmetrical interchange of letters. This condition may also be written a, «, m, p, a, a' n, h h 2> A £' m, h c, »•) 7, 7' Pi 2, r , rf, 8, a- = 0. a, /3, 7 , 8 a', /3', 7 ', S' If in the condition of the last article we write a + Xa' for a, &c, and then form the condition that the equation in \ should have equal roots, the result will be the condition of this article multiplied by the discriminant. For the two planes which can be drawn through a given line to touch a quadric, will coincide either if the line touches the quadric or if the surface has a double point. ( ft ) CHAPTER V. CLASSIFICATION OF QTJADRICS. 77. Our object in this chapter is the reduction of the general equation of the second degree to the simplest form of which it is susceptible, and the classification of the different surfaces which it is capable of representing. Let us commence by supposing the quantity which we called D (Art. 63) not to be = 0. By transforming the equation to parallel axes through the centre, the coefficients j>, q ) r are made to vanish, and the equation becomes ax* + by* + cz'* + 2lyz + 2 mzx + 2nxy + d' — 0, where d' is the result of substituting the co-ordinates of the centre in the equation of the surface. Remembering that 2 w = x' u; + y ' u; + z' u; + »• u;, and that the co-ordinates of the centre make £/",', C^', UJ vanish, it is easy to calculate that ,, _ pP+gQ + rB + dD _ A where A, P, Q, B have the same meaning as in Art. 63. 78. Having by transformation to parallel axes made the coefficients of x, y, z to vanish, we can next make the co- efficients of yz, zx, and xy vanish by changing the direction of the axes, retaining the new origin ; and so reduce the equation to the form \ a'x*+by + c'z 2 + d' = 0. It is easy to shew from Art. 17 that we have constants enough at our disposal to effect this reduction, but the method we shall follow is the same as that adopted, Conies, p. 147, namely, to prove that there are certain functions of the co- efficients which remain unaltered when we transform from one E 50 CLASSIFICATION OF QUADEICS. rectangular system to another, and by the help of these re- lations to obtain the actual values of the new a, b, c. Let us suppose that by using the most general transfor- mation which is of the form x = \x 4 fiy 4 vz, y = \'x 4 p'y 4 v'z, z = \"x 4 p'y 4 v"i, that ax 2 + by 2 + cz 2 + 2lyz + 2mzx 4 2nxy becomes a'x' 4 b'y' + c'z' 4 2l'yz 4 2m zx 4 2n'xy, which we write for shortness U= U. And if both systems of co-ordinates be rectangular, we must have x*+/+z 2 =xyf+z*, which we write for shortness 8=8. Then if k be any constant, we must have U+ kS = U+ kS. And if the first side be re- solvable into factors, so must also the second. The discrimi- nants of Z74 kS and of U+ kS must therefore vanish for the same values of k. But the first discriminant is k 3 - ¥ (a 4 b 4 c) 4 k (ab + be 4 ca — F - m 2 - ra 2 ) — (abc 4 2lmn — aF — bm 2 - en 2 ). Equating then the coefficients of the different powers of k to the corresponding coefficients in the second, we learn that if the equation be transformed from one set of rectangular axes to another, we must have a + b + c = a' + b' + c, bc + ca + ab-F- m 2 -n 2 = b'c' 4 ca 4 a'V - F - m' 2 - n' 2 , abc 4 2lmn -aF- bm' - en 2 = a' b'c' 4 2l'm'n - a'F - b'm' 2 - en' 2 .* 79. The above three equations at once enable us to trans- form the equation so that the new £, m, n shall vanish, since they determine the coefficients of the cubic equation whose roots are the new o, 5, c. This cubic is then ■\a' 3 -(a + b + c)a' 2 +(bc + ca + ab-F-m 2 -n 2 )a' - (abc 4 2lmn — aF - bm 2 — en 2 ) = 0, * There is no difficulty in forming the corresponding equations for oblique co- ordinates. We should then substitute for S (see Art. 18), x 2 + y 2 + z 2 + 2;yz cos A. + 2zx cos/n + 2xy cosv, and proceeding exactly as in the text, we should form a cubic in h, the coefficients of which would bear to each other ratios unaltered by transformation. t This is the same cubic as that found, Art. 68, as the reader will easily see ought to be the case. CLASSIFICATION OF QUADEICS. 51 which may also be written (a' - a) (a' - b) (a' - c) - f(a'-a)-w a (a'-5)- n\a'-c) - 22mn=»0. We give here Cauchy's proof that the roots of this equation are all real. The proof of a more general theorem, in which this is included, will be found in Lessons on Higher Algebra, Lesson XV. Let the cubic be written in the form {a' - a) {(a' - b) (a - c) - F} - m 2 (a -b)- ri 2 (a -c)- 2lmn = 0. Let a, /3 be the values of a' which make (a'—b)(a'-c)-P=0, and it is easy to see that the greater of these roots a is greater than either b or c, and that the less root /S is less than either.* Then if we substitute in the given cubic a = a, it reduces to -{(a-b) m* + 2lmn + (a - c) «*}, and since the quantity within the brackets is a perfect square in virtue of the relation (a - b) (a - c) = F, the result of sub- stitution is essentially negative. But if we substitute a' = /3, the result is ^ _ p\ m « _ 2 j mn + (<,-£) n% which is also a perfect square, and positive. Since then, if we substitute a = oo , a = a, a' = /3, a = — ao , the results are alternately positive and negative, the equation has three real roots lying within the limits just assigned. The three roots are the coefficients of a? a , y 2 , z 1 in the transformed equation, but it is of course arbitrary which shall be the coefficient of a; 2 or of y, since we may call whichever axis we please the axis of x. 80. Quadrics are classified according to the signs of the roots of the preceding cubic. I. First, let all the roots be positive, and the equation can be transformed to j^ + jy + c ' a » + $ = 0> -(- * We may see this either by actually solving the equation, or by substituting suc- cessively o' = oo , a' = b, a' = c, a' = — oo , when Tve get results +, — — , +, shewing that one root is greater than i, and the other less than c. f I suppose in what follows that d' != -g, Art. 77J is negative. If it were positive we should only have to change all the signs in the equation. If it were = the surface would represent a cone (Art. 63). E2 52 CLASSIFICATION OF QUADRICS. The surface makes real intercepts on each of the three axes, and if the intercepts be a, b, c, it is easy to see that the equation of the surface may be written in the form a? y s z' , a 2 + V + c 2 As it is arbitrary which axis we take for the axis of x, we suppose the axes so taken that a the intercept on the axis of x may be the longest, and c the intercept on the axis of z may be the shortest. The equation transformed to polar co-ordinates is 1 cos" a cos 2 /3 cos 2 7 which (remembering that cos 2 a ■+ cos 2 /3 + cos 2 7 = 1) may be written in either of the forms COS'/tf + - - - 2 C08^7 ] 1 (I i ? a' + \v- ~ a- 1 i n I — ? - 1 vc 2 " 'a*. ""-(?-p) co,,ft from which it is easy to see that a is the maximum and c the minimum value of the radius vector. The surface is con- sequently limited in every direction, and is called an ellipsoid. Every section of it is therefore necessarily also an ellipse. Thus the section by any plane z = /c is — + ^ = 1 5 , and we shall obviously cease to have any real section when k is greater than c. The surface therefore lies altogether within the planes z = ±c. Similarly for the other axes. If two of the coefficients be equal (for instance, a = 6), then all sections by planes parallel to the plane of xy are circles, and the surface is one of revolution, generated by the revolution of an ellipse round its axis major or axis minor, according as it is the two less or the two greater coefficients which are equal. These surfaces are also sometimes called the prolate and the oblate spheroid. If all three coefficients be equal, the surface is a sphere. CLASSIFICATION OF QUADRICS. 53 81. II. Secondly, let one root of the cubic be negative. We may then write the equation in the form x l f z' — + - =1 a b c ' where a is supposed greater than b, and where the axis of z evidently does not meet the surface in real points. Using the polar equation 1 _ cos 2 a cos 2 /3 cos 2 7 it is evident that the radius vector meets the surface or not according as the right-hand side of the equation is positive or negative ; and that putting it = 0, (which corresponds to p = oo ) we obtain a system of radii which separate the diameters which meet the surface from those that do not. We obtain thus the equation of the asymptotic cone x 2 y* z' h =0 2 ~ 7,2 2 — v • a o c Sections of the surface parallel to the plane of xy are ellipses ; those parallel to either of the other two principal planes are hyperbolas. The equation of the elliptic section by the plane x* y 2 H z = k being — A + jt z = 1 + — 2 , we see that we get a real section whatever be the value of k, and therefore that the surface is continuous. It is called the Hyperboloid of one sheet. If a = b, it is a surface of revolution. 82. III. Thirdly, let two of the roots be negative, and the equation may be written a 2 V c 2 The sections parallel to two principal planes are hyperbolas, while that parallel to the plane of yz is an ellipse f z* k* , 7 2 ~ 2 2 b c a It is evident that this will not be real as long as h is within the limits + a, but that any plane x = 1c will meet the surface in a real section provided that k is outside these limits. No 54 CLASSIFICATION OF QUADRICS. portion of the surface will then lie between the planes x = ±a, but the surface will consist of two separate portions outside these boundary planes. This surface is called the Hyperboloid of two sheets. It is of revolution if b = c. By considering the surfaces of revolution, the reader can easily form an idea of the distinction between the two kinds of hyperboloids. Thus if a common hyperbola revolve round its transverse axis the surface generated will evidently consist of two separate portions ; but if it revolve round the conjugate axis it will consist but of one portion, and will be a case of the hyperboloid of one sheet. IV. If the three roots of the cubic be negative, the surface x l f z 2 a 2 + ¥ + 6 l can evidently be satisfied by no real values of the co-ordinates. V. When the absolute term vanishes, we have the cone as a limiting case of the above. Forms I. and IV. then become x 2 y* z* which can be satisfied by no real values of the co-ordinates, while forms II. and III. give the equation of the cone in the form x * tf 2 * The forms already enumerated exhaust all the varieties of central surfaces. Ex. 1. lx 2 + 6f + 5z 2 - 4yz - ixy = 6. The discriminating cubic is a"' — 18a' 2 + 99a' — 162 = 0, and the transformed equation x 2 + 2y 2 + 3z 2 = 2, an ellipsoid. Ex. 2. 11a: 2 + Kfy 2 + 6z 2 - 12^ - iyz + izx = 12. Discriminating cubic a' 3 - 27a' 2 + 180a' — 324 = 0. Transformed equation x 2 + 2y* + 6z 2 = 4, an ellipsoid. Ex. 3. lx 2 - 13y 2 + Gz 2 + 2ixy + Vlyz - ttzx - ± 84. Discriminating cubic a' 3 — 343a' - 2058 = 0. Transformed equation x 2 + 2y 2 — 3z 2 = ± 12, a hyperboloid of one or of two sheets, according to the sign of the last term. Ex. 4. lx 2 + Sy 2 + iz 2 + 6xy + iyz + 8zx = 8. Discriminating cubic is a' 3 — 9a' 2 — 3a' + 20 = 0. By Des Cartes's rule of signs this equation has two positive and one negative root, and therefore represents a hyperboloid of one sheet. CLASSIFICATION OF QUADRIC8. 55 83. Let us proceed now to the case where we have D = 0. In this case we have seen (Art. 65) that it is generally im- possible by any change of origin to make the terms of the first degree in the equation to vanish. But it i3 in general quite indifferent whether we commence, as in Art. 65, by transforming to a new origin, and so remove the coefficients of x, y, z, or whether we first, as in this chapter, transform to new axes retaining the same origin, and so reduce the terms of highest degree to the form a'af + by + c'z*. When D = 0, the first transformation being impossible we must commence with the latter. And since the absolute term of the cubic of Art. 79 is D, one of its roots, that is to say, one of the three quantities a', U, c' must in this case = 0. The terms of the second degree are therefore reducible to the form a'x* ± b'y'. This is otherwise evident from the consideration that D = is the condition that the terms of highest degree should be resolvable into two real or imaginary factors, in which case they may obviously be also expressed as the difference or sum of two squares. In this way the equation is reduced to the form a V ± b'y* + 2p'x + 2q'y + 2r'z + d = 0. We can then, by transforming to a new origin, make the co- efficients of x and y to vanish, but not that of z, and the equation takes the form a > x * ± b y + 2r ' z + d - = _ I. Let r' = 0. The equation then does not contain z, and therefore (Art. 24) represents a cylinder which is elliptic or hyperbolic, according as a and b have the same or different signs. Since the terms of the first degree are absent from the equation the origin is a centre, but so is also equally every other point on the axis of z, which is called the axis of the cylinder. The possibility of the surface having a line of centres is indicated by both numerator and denominator vanishing in the co-ordinates of the centre, Art. 65 (see note p. 41). If it happened that not only r but also d! = 0, the surface would reduce to two intersecting planes. II. If r be not = 0, we can by a change of origin make the absolute term vanish, and reduce the equation to the form a a? ± &'/ + 2r'z = 0. 56 CLASSIFICATION OF QUADEICS. Let us first suppose the sign of V to be positive. In this case while the sections by planes parallel to the planes of xs or yz are parabolas, those parallel to the plane of xy are ellipses, and the surface is called the Elliptic Paraboloid. It evidently extends only in one direction, since the section by any plane z = Jc is da? + b'y" = — ikr , and will not be real unless the right-hand side of the equation is positive. When therefore r is positive, the surface lies altogether on the negative side of the plane of xy, and when r is negative, on the positive side. III. If the sign of V be negative, the sections by planes parallel to that of xy are hyperbolas, and the surface is called a Hyperbolic Paraboloid. This surface extends indefinitely in both directions. The section by the plane of xy is a pair of right lines. IV. If V = 0, that is, if tioo roots of the discriminating cubic vanish, the equation takes the form a'x 2 + 2q'y + 2r'z + d = 0, but by changing the axes y and z in their own plane, and taking for new co-ordinate planes the plane ay + r'z and a plane perpendicular to it through the axis of x, the equation is brought to the form dx 2 + qy -t-cZ = 0, which (Art. 24) represents a cylinder whose base is a parabola. V. If we have also q' = 0, r'=0, the equation a'a? + d=0 being resolvable into factors would evidently denote a pair of parallel planes. 84. The actual work of reducing the equation of a paraboloid to the form dx 2 + b'y 2 + 2r'z = is shortened by observing that the discriminant is an invariant; that is to say, a function of the coefficients which is not altered by transformation of co- ordinates [Higher Algebra, p. 51). Now the discriminant of dx 2 + b'y 2 + 2r'z is simply-a'&V' 2 , which is therefore equal to the discriminant of the given equation. And as d and V are known, being the two roots of the discriminating cubic which do not vanish, r is also known. The calculation of the dis- criminant is facilitated by observing that it is in this case a CLASSIFICATION OF QUADEICS. 57 perfect square {Higher Algebra, p. 124). Thus let us take the example 5a; 2 - if + z* -+ 6zx + ixy + 2a? + % + 6z = 8. Then the discriminating cubic is X 3 — 5X 2 — 14A. = whose roots are 0, 7, and —2. We have therefore a' = 7, b' = - 2. The discriminant in this case is (j> + 2q — 3r) 2 , or putting in the actual values p = l, q = 2,r = 3 is 16. Hence we have 14r' a = 16, r = -77 — r , and the reduced equation is 7x 2 — 2w 2 = —. — ; . V(14)' ^ * V(14) If we had not availed ourselves of the discriminant, we should have proceeded as in Art. 68 to find the principal planes answering to the roots 0, 7, — 2 of the discriminating cubic, and should have found x + 2y - 3z = 0, Ax + y + 2s = 0, x - 2y - z = 0. Since the new co-ordinates are the perpendiculars on these planes, we are to take 4a;+2/ + 2s=A'V(21), x-2y-s = Y*J{&), x+ 2y-3z = Z*J(U), from which we can express x, y, s in terms of the new co- ordinates, and the transformed equation becomes which finally transformed to parallel axes through a new origin gives the same reduced equation as before. If in the preceding example the coefficients p, q, r had been so taken as to fulfil the relation j> + 2q — 3r = 0, the discriminant would then vanish, but the reduction could be effected with even greater facility as the terms in a-, y, z could then be ex- pressed in the form (ix + y + 2a) + X (x - 2y - z). Thus the equation 5x* -y 2 + z* + Gzx+ ixy + 2x+ 2y+2z=% may be written in the form (4a; + y + 2z) s - (a; - 2y - z) 2 + 2 (4a; +■ y + 2a) - 2 (x - 2y - z) = 24, which transformed as before becomes 21a; 2 - 6/ + 2a; V(21) - 2y V(6) = 24, and the remainder of the reduction presents no difficulty. ( 58 ) CHAPTER VI. PROPERTIES OE QUADRICS DEDUCED FROM SPECIAL FORMS OF THEIR EQUATIONS. CENTRAL SURFACES. 85. We proceed now to give some properties of central a; 2 y s " quadrics derived from the equation -^ + |j + -j = 1. This will include properties of the hyperboloids as well as of the ellipsoids if we suppose the signs of b' and of c* to be indeterminate. The equation of the polar plane of the point x'y'z (or of the tangent plane, if that point be on the surface) is (Art. 59) d* + b* + c 2 ~ The length of the perpendicular from the origin on the tangent plane is therefore (Art. 32) given by the equation 1/2 '2 '2 x y z 2 4 ~ 7 4 ~ 4 ' p a b c And the angles a, /3, between x'y'z, and any point on it xyz is + — . But if xyz be taken as the point where the normal meets the plane of xy, we have 3 = 0, and the last of the three preceding equations gives R = -c 2 . Hence the length of the intercept on the normal be- c 2 tween the point of contact and the plane of xy is — . 88. The sum of the squares of the reciprocals of any three rectangular diameters is constant. This follows immediately from adding the equations 1 cos 2 a cos 2 /3 cos 2 7 pa c 1 _ cosV cos a /3' cos 2 y' jF~ IT + ~¥~ + ~c r ~' JL - cosV ' cos 2 ^" cos 2 7" „"« — „* **" A* /." > 60 CONJUGATE DIAMETERS. whence since cos 2 a + cos* a' + cos 2 a" = 1, &c, we have 1 J_ J_ __1 1 1 p p p a C 89. In like manner the sum of the squares of three perpen- diculars on tangent planes, mutually at right angles, is constant, as appears from adding the equations f = cl* cos 2 a + 5 2 cos 2 /3 +c 2 cos 2 7, y 2 =a 2 cos 2 a' +S 2 cos 2 /3' +c 2 cos 2 7', p" 2 = a 2 cos 2 a" + Z> 2 cos 2 /3" + c 2 cosY'. Hence the locus of the intersection of three tangent planes which cut at right angles is a sphere ; since the square of its distance from the centre of the surface is equal to the sum of the squares of the three perpendiculars and therefore to a 2 +6 2 + c 2 . CONJUGATE DIAMETEES. 90. The equation of the diametral plane conjugate to the diameter drawn to the point x'y'z on the surface is xx' vij zz' -^+f- + ^ = °> (Art. 68). It is therefore parallel to the tangent plane at that point, Since any diameter in the diametral plane is conjugate to that drawn to the point x'y'z', it is manifest that when two diameters are conjugate to each other, their direction-cosines are connected by the relation cos a cos a' cos j3 cos/3' cos 7 cos 7' ? + W + ? = Since the equation of condition here given is not altered if we write &a 2 , M\ kc 2 for a\ S 2 , c 2 , it is evident that two lines £C 2 2/ 2 £ 2 which are conjugate diameters for any surface -^ + ^ + — 2 = 1, are also conjugate diameters for any similar surface a' + F + c 2 ~ k - And by making h = we see in particular that any surface and its asymptotic cone have common systems of conjugate diameters. CONJUGATE DIAMETERS. 61 Following the analogy of methods employed in the case of conies we may denote the co-ordinates of any point on the ellipsoid, by a cosX, b cos/i, c cosy, where X, /a, v are the direction-angles of some line ; that is to say, are such that cos 2 A, + cos 2 /*. + cosV =1. In this method the two lines answer- ing to two conjugate diameters are at right angles to each other; for writing cosa = acosX, cos a' = a cos X', &c, the re- lation above written becomes cosX cosX' + cos/a cos/a' + cos v cos v = 0. 91. The sum of the squares of a system of three conjugate semi-diameters is constant. For the square of the length of any semi-diameter x' 2 + y n + z* is, when expressed in terms of X, /i, v, a 2 cos 2 X + V cos'V -|- c* cosV, which when added to a 2 cos 2 X' + J 2 cos 2 // +c 2 cosV, a 2 cos 2 X" + S 2 cos>" + c 2 cosV is equal to a 2 + & 2 +c 2 ; since X, /a, v, &c. are the direction- angles of three lines mutually at right angles. 92. The jparallelopiped whose edges are three conjugate semi- diameters has a constant volume. For if x'y'z', x"y"z'\ &c. be the extremities of the diameters the volume is (Art. 35) or abc x\ y\ z\ x" y'\ z " x" ■>/" z" i V 1 z 1 cosX, COS/4, COSV cosX', COS/i', COS V CO iX", COS/t", CO sv' but the value of the last determinant is unity (see note p. 20) : hence the volume of the parallelopiped is abc. If the axes of any central plane section be a' } &', and j? the perpendicular on the parallel tangent plane, then a'b'jp = abc. 62 CONJUGATE DIAMETERS. For if c' be the semi-diameter to the point of contact, and 6 the angle it makes with p, the volume of the parallelopiped under the conjugate diameters a', '&', c is a'b'c cos#, but c' cos#=j?. 93. The theorems just given may also with ease be deduced from the corresponding theorems for conies. For consider any three conjugate diameters a, b', c\ and let the plane of db' meet the plane of xy in a diameter A, and let C be the diameter conjugate to A in the section a'b', then we have A 2 + G 2 = a' 2 + V* ; therefore a' 2 + b' 2 + c' 2 = A* + C 2 + c'\ Again, since A is in the plane xy, then if B is the diameter con- jugate to A in the section by that plane, the plane conjugate to A will be the plane containing B and containing the axis c, and C, c' are therefore conjugate diameters of the same section as B, c. Hence we have A 2 +C 2 +c' 2 = A i + B 2 + c"; and since, finally, A' + B 2 = d* + b 2 , the theorem is proved. Precisely similar reasoning proves the theorem about the parallelopipeds. We might further prove these theorems by obtaining, as in the note, p. 50, the relations which exist when the quantity 2 2 2 2 2 2 -» + tf> + -is in oblique co-ordinates is transformed to —„ + %s + —, in rectangular co-ordinates. These relations are found to be a? + b*+c* = a' 2 + b' 2 + c' 2 , jy + C V + a 2 b 2 = b' 2 c 12 sin 2 \ + e'V 2 sin> + a' 2 S' 2 sin 2 y, a'b'c* = d'*b n c 2 (1— cos 2 \— cos 2 /*— cosV+2 cosX cos/* cose). The first and last equations give the properties already ob- tained. The second expresses that the sum of the squares of the parallelograms formed by three conjugate diameters, taken two by two, is constant, or that the sum of squares of reciprocals of perpendiculars on tangent planes through three conjugate vertices is constant. 94. The sum of the squares of the projections of three con- jugate diameters on any line is constant. Let the line make angles a, /3, 7 with the axes, then the projection on it of the semi-diameter terminating in the point x'y'z is x cosa + y' cos/3 + z' cos 7, or, by Art. 90, is a cos A, cosa + b cos/i cos/3 + c cosv C0S7. CONJUGATE UtAMETERS. 63 Similarly, the others are a cosV cosa + &cos/i' cos/3 + c cos v' cosy, a cosX" cosa H-6 cos/i" cos/3 + c cosv" cosy; and squaring and adding, we get the sum of the squares a 2 cos 8 a + b 2 cos 2 {3 + c 2 cos 2 y. 95. The sum of the squares of the projections of three con- jugate diameters on any plane is constant. If d, d', d" be the three diameters, 0, 6', 8" the angles made by them with the perpendicular on the plane, the sum of the squares of the three projections is d 2 sm'6 + d' 2 sin 2 0' + d" 2 sin 2 0", which is constant, since d* cos 2 0+ d 12 cos 2 0' + d" 2 cos 2 6" is con- stant by the last article ; and d 2 + d' 2 4 d" 2 by Art. 91. 96. To find the locus of the intersection of three tangent planes at the extremities of three conjugate diameters. The equations of the three tangent planes are - cosA +7 cos/t -f - cosv =1, a be ' - cos A' + f cos u! + - cosv' = 1, a b n c ' - cosA" + > cosit"+- cosv" = l, a be Squaring and adding, we get for the equation of the locus, x 2 y 2 z 2 „ a b c 97. To find the lengths of the axes of the section made by any plane passing through the centre. We can readily form the quadratic, whose roots are the reciprocals of the squares of the axes, since we are given the sum and the product of these quantities. Let a, /3, y be the angles which a perpendicular to the given plane makes with the axes, R the intercept by the surface on this perpendicular ; then we have (Art. 88) «'» + b' 2 + B 2 ~ a 1 + b 2 + c 2 ' 64 CONJUGATE DIAMETERS. 1 1 COS 2 a COS 2 /3 whence 1,1 /l 1 1 cos 2 a cos 2 /3 _ cos 2 7\ ~2 ■" 775 1 ~~2 ' Z2 ' ~~2 2 X2 "~~ a I ) a o \a o c a- o c J i-i /a , „„x 1 P 8 cos 2 a cos 2 /? cos 2 7 wh l le(Art.92)^ = ^ ? = w - +7 ^ + -^. The quadratic required is therefore 1 1 /sin 2 a sin 2 /3 sin 2 7\ cos 2 a cos 2 /3 cos 2 7 ~~4 "a 5 ' 72 ~I 2 J "T~ 722 ~T 2 2~~ ~T" 27 2 == r r V a b c J be ca ab This quadratic may also be written in the form a 2 cos 2 a F cos 2 8 c' cos 2 7 . 2 2 ' 72 2^2 2 « — r — r c — r This equation may be otherwise obtained from the principles explained in the next article- 98. Through a given radius OB of a central quadric we can in general draw one section of which OB shall be an axis. Describe a sphere with OB as radius, and let a cone be drawn having the centre as vertex and passing through the intersection of the surface and the sphere, and let a tangent plane to the cone be drawn through the radius OB, then OB will be an axis of the section by that plane. For in it OB is equal to the next consecutive radius (both being radii of the same sphere) and is therefore a maximum or minimum ; while the tangent line at B to the section is perpendicular to OB, since it is also in the tangent plane to the sphere. OB is therefore an axis of the section. The equation of the cone can at once be formed by sub- tracting one from the other, the equations x l if s 2 , x 2 if s 2 a* V c 2 ' r 2 r 2 r 2 ' when we get '(?-?W(p-3+'(?-a-* If then any plane x cosoc + y cos/3 + s C0S7 have an axis in length = r, it must touch this cone, and the condition that it should touch it, is (Art. 86) o? cos 2 a b'* cos 2 /3 c 2 cos 2 7 which is the equation found in the last article. circular Actions. 65 In like manner we can find the axes of any section of a quadric given by an equation of the form ax* + by" + cz* + 2lyz + 2mzx + 2nxy = 1. The cone of intersection of this quadric with any sphere \(x* + tf + z*) = l is (a - X) x 2 + (b — X) y 2 + (c - X) s 8 + llyz + 2mzx + 2nxy = 0, and we see as before, that if X be the reciprocal of the square of an axis of the section by the plane x cosa + y cos/3 + z cos 7, this plane must touch the cone whose equation has just been given. The condition that the plane should touch this cone (Art. 75) may be written a — X, w, m, cosa 92, b — X, I, cos/8 m, Z, c — X, cosy cosa, cosjS, cos 7 =0, which expanded is X 2 - X {(b + c) cos 2 a +(c + a) cos 2 /3 + (a + b) cos 2 7 - 2l cosjS COS7 — 2m COS7 cosa — 2n cosa cos/3) + (be - P) cos 2 a + (ca — m 2 ) cos 2 /3 + (ab - n*) cos 2 7 + 2 (mw — aZ) coS/S COS7 + 2 (nl — bin) cosy cosa + 2 (Zm — en) cosa cos^S = 0. CIRCULAR SECTIONS. 99. We proceed to investigate whether it is possible to draw a plane which shall cut a given ellipsoid in a circle. As it has been already proved (Art. 69) that all parallel sections are similar curves, it is sufficient to consider sections made by planes through the centre. Imagine that any central section is a circle with radius r, and conceive a concentric sphere described with the same radius. Then we have just seen that 66 CIRCULAR SECTIONS. represents a cone having the centre for its vertex and passing through the intersection of the quadric and the sphere. But if the surfaces have a plane section common, this equation must necessarily represent two planes, which cannot take place unless the coefficient of either x\ y*, or z* vanish. The plane section must therefore pass through one or other of the three axes. Suppose for example we take r = b, the coefficient of y vanishes, and there remains which represents two planes of circular section passing through the axis ofy. The two planes are easily constructed by drawing in the plane of xz a semi-diameter equal to b. Then the plane con- taining the axis of y, and either of the semi-diameters which can be so drawn, is a plane of circular section. In like manner two planes can be drawn through each of the other axes, but in the case of the ellipsoid these planes will be imaginary ; since we evidently cannot draw in the plane of xy a semi-diameter = c, the least semi-diameter in that section being = b ; nor, again, in the plane of yz a semi-diameter = a, the greatest in that section being = b. In the case of the hyperboloid of one sheet c' is negative, and the sections through a are those which are real. In the hyperboloid of two sheets where both & a and c 2 are negative, if we take r' = - c 2 (& 2 being less than c 2 ), we get the two real sections, 'i i\ . ,/i r .? + *j + n?-Fr - These two real planes through the centre do not meet the surface, but parallel planes do meet it in circles. In all cases it will be observed that we have only two real central planes of circular section, the series of planes parallel to each of which afford two different systems of circular sections. 100. Any two surfaces whose coefficients of x\ y 2 , a 2 , differ only by a constant, have the same planes of circular section. Thus Ax* +Bf + Cz* = 1 , and {A + IT) a? + (B +E) f+(C+H) z* = 1 CIRCULAR SECTIONS. 67 have the same planes of circular section, as easily appears from the formula in the last article. The same thing appears by throwing the two equations into the form -a = A cos" a + B cos 2 /3 + cos 2 y, r — = A cos* a + B cos*/3 + C 008*7 + fy from which it appears that the difference of the squares of the reciprocals of the corresponding radii vectores of the two sur- faces is constant. If then in any section the radius vector of the one surface be constant, so must also the radius vector of the other. The same consideration shews that any plane cuts both in sections having the same axes, since the maximum or minimum value of the radiu3 vector will in each correspond to the same values of a, /3, 7. Circular sections of a cone are the same as those of a hyper- boloid to which it is asymptotic. 101. Any two circular sections of opposite systems lie on the same sphere. The two planes of section are parallel each to one of the planes represented by Now since the equation of two planes agrees with the equation of two parallel planes as far as terms of the second degree are concerned, the equation of the two planes must be of the form x* ?-?)+'$-?)+'$-?>'>-• where w, represents some plane. If then we subtract this from the equation of the surface, which every point on the section must also satisfy, we get i(a,* + 2, 2 + s 8 )- Ml =l, which represents a sphere. F2 68 CIRCULAR SECTIONS. 102. All parallel sections are as we have seen similar. If now we draw a series of planes parallel to circular sections the extreme one will be the parallel tangent plane which must meet the surface in an infinitely small circle. Its point of contact is called an umbilic. Some properties of these points will be mentioned afterwards. The co-ordinates of the real umbilics are easily found. We are to draw in the section, whose axes are a and c, a semi-diameter = b, and to find the co-ordinates of the extremity of its conjugate. Now the for- mula for conies b' 2 = a 2 — eV applied to this case gives us j. , a c , x* a 2 a 2 -b 2 j. ., . z> F-c 2 a*-c" Smnlarly c* = a*-c' whence There are accordingly in the case of the ellipsoid four real umbilics in the plane of xz, and four imaginary in each of the other principal planes. 103. It is convenient to add in this place how in like manner we are able to determine the circular sections of the paraboloid given by the equation a?_ tf_2s « a * b*~ c ' Consider a circular section through the origin, and describe a sphere through it having, at the origin, the same tangent plane [z] as the paraboloid ; then (Art. 57) the equation of the sphere must be of the form x * + y l + s ' 2 ~ % rz - And the cone of intersection of this sphere with the paraboloid is 4-SW( 1T ?W=°- This will represent two planes if one of the terms vanishes. It will represent two real planes, in the case of the elliptic GT paraboloid, if we take — = 1, for the equation then becomes 6V = [a' — b'') y'. But in the case of the hyperbolic paraboloid there is no real circular section, since the same substitution EECTILINEAE GENEEATOES. 69 would make the equation of the two planes take the imaginary form 5V + (a 2 + & 2 ) f = 0. Indeed, it can be proved in general that no section of the hyperbolic paraboloid can be a closed curve, for if we take its intersection with any plane z = Ix + my + n, the projection on ,i , j. . a? if 2{lx+my + n) , . , . the plane ot xy is — - ^ = — ^ ^ '■ which is necessarily a hyperbola. EECTILINEAE GENEEATOES. 104. We have seen that when the central section is an ellipse all parallel sections are similar ellipses, and the section by a tangent plane is an infinitely small similar ellipse. In like manner when the central section is a hyperbola, the section by any parallel plane is a similar hyperbola, and that by the tangent plane reduces itself to a pair of right lines parallel to the asymptotes of the central hyperbola. Thus if the equation referred to any conjugate diameters be * 4. £. _ s l - i a' 2 + V 2 c'* ~ ' and we consider the section made by any plane parallel to the plane of xz (y = /3), its equation is a' a c" 2 V* - And it is evident that the value /3 = V reduces the section to a pair of right lines. Such right lines can only exist on the hyperboloid of one sheet, since if we had the equation *L i. - - a" ¥'~ + c' 2 ' the right-hand side of the equation could not vanish for any value of z. It is also geometrically evident that a right line cannot exist either on an ellipsoid, which is a closed surface; or on a hyperboloid of two sheets, no part of which, as we saw, lies in the space included between several systems of two parallel planes, while any right line will of course in general intersect them all. 70 RECTILINEAR GENERATORS. 10/). Throwing the equation of the hyperboloid of one sheet into the form *? * 2 1 f a" c' tr it is evident that the intersection of the two planes lies on the surface ; and by giving different values to X we get a system of right lines lying in the surface ; while, again, we get another system by considering the intersection of the planes What has been just said may be stated more generally as. follows : If a, /3, 7, S represent four planes, then the equation ay = /38 represents a hyperboloid of one sheet, which may be generated as the locus of the system of right lines a = X/3, X7 = 8, or a — XS, X7 = /3. In the case of the equation 2 2 2 X 1/ z 1- = 1 2 * 7.2 2 ) a c ' the lines may be also expressed by the equations - = - cos0 + sin#, %: = - sin# + cos#. a c be 106. Any two lines belonging to opposite systems lie in the same plane. Consider the two lines a - X/3, X7 — S, a -X'S, X7-/3. Then it is evident that the plane a — X/3 + XX'7 — X'S contains both, since it can be written in either of the forms (a - X/3) -1- X' (X7 - 8), a - X'S + X (X'7 - /3). It is evident in like manner that no two lines belonging to the same system lie in the same plane. Since no plane of RECTILINEAR GENERATORS. 71 the form (a - X/3) + h (Xy - S) can ever be identical with (a - \'/3) + h' (X'y - 8) if \ and \' are different. In the same way we see that both the lines SC Z 11 Z - = - cos0-sin0, t = - sin0 + cos#, a c 1 o c ' x z . y z . , - = - cosffi+sinm, *~ = - sm) +| sin£ {0 + 4>)=- cos£ (0 - 0) -sin J (0-). Now this plane is parallel to the second line of the first system CC Z tJ z - = - cos — sin, ^ = - sind + costi, a c T ' b c T r ' but it does not pass through it, for the equation of a parallel plane through this line will be found to be ^cos£(0 + )+|sin£(0 + <£)=?cos£(0-<£)+sin£(0-^), which differs in the absolute term from the equation of the plane through the first line. 107. We have seen that any tangent plane to the hyper- boloid meets the surface in two right lines intersecting in the point of contact, and of course touches the surface in no other point. If through one of these right lines we draw any other plane, we have just seen that it will meet the surface in a new right line, and this new plane will touch the surface in the point where these two lines intersect. Conversely, the tangent plane to the surface at any point on a given right line in the surface will contain the right line, but the tangent plane will in general be different for every point of the right line. Thus } take the surface x = y^ where the line xy lies on the surface, and and ^ represent planes (though the demonstration would equally hold if they were functions of any higher degree). Then using the equation of the tangent plane (x - x') U; + (y - y') U; + (z- z<) U s ' = 0, 72 RECTILINEAR GENERATORS. and seeking the tangent at the point x = 0, y = 0, « = s',we find axf)' = yifr', where ' and i|/ are what and ^ become on sub- stituting these co-ordinates. And this plane will vary as z' varies. All this is different in the case of the cone. Here every tangent plane meets the surface in two coincident right lines. The tangent plane then at every point of this right line is the same, and the plane touches the surface along the whole length of the line. And generally, if the equation of a surface be of the form X + ifijr = 0, it is seen precisely, as above, that the tangent plane at every point of the line xy is a; = 0. 108. It was proved (Art. 104) that the two lines in which the tangent plane cuts a hyperboloid are parallel to the asymp- totes of the parallel central section ; but these asymptotes are evidently edges of the asymptotic cone to the surface. Hence every right line which can lie on a hyperboloid is parallel to some one of the edges of the asymptotic cone. It follows also that three of these lines (unless two of them are parallel,) cannot all be parallel to the same plane ; since, if they were, a parallel plane would cut the asymptotic cone in three edges, which is impossible, the cone being only of the second degree. 109. We have seen that any line of the first system meets all the lines of the second system. Conversely, the surface may be conceived as generated by the motion of a right line which always meets a certain number of fixed right lines.* Let us remark in the first place, that when we are seeking the surface generated by the motion of a right line, it is necessary that the motion of the right line should be regulated by three conditions. In fact, since the equations of a ,right line include four constants, four conditions would absolutely determine the position of a right line. When we are given * A surface generated by the motion of a right line is called a ruled surface. If every generating line is intersected by the next consecutive one, the surface is called a developable. If not, it is called a shew surface. The hyperboloid of one sheet belongs to the latter class ; the cone to the former. RECTILINEAR GENERATORS. 73 one condition less, the position of the line is not determined, but it is so far limited that the line will always lie on a certain surface-locus, whose equation can be found as follows: Write down the general equations of a right line x=mz+p, y=nz + q- then the conditions of the problem establish three relations between the constants m, n, p, q. And combining these three relations with the two equations of the right line, we have five equations from which we can eliminate the four quantities m i n ) Pi it ana " th e resulting equation in x, y, z will be the equation of the locus required. Or, again, we may write the equations of the line in the form x — x' y — y z — z' cos a cos/3 cos 7 ' then the three conditions give three relations between the con- stants x\ y\ z\ a, /3, 7, and if between these we eliminate a, £, 7, the resulting equation in x\ y', z is the equa- tion of the required locus, since x'y'z may be any point on the line. We see then that it is a determinate problem to find the surface generated by a right line which moves so as always to meet three fixed right lines.* For expressing, by Art. 40, the condition that the moveable right line shall meet each of the fixed lines, we obtain the three necessary relations between m i n iVi ?• Geometrically also we can see that the motion of the line is completely regulated by the given conditions. For a line would be completely determined if it were constrained to pass through a given point and to meet two fixed lines, since we need only draw planes through the given point and each of the fixed lines, when the intersection of these planes would determine the line required. If then the point through which the fine is to pass, itself moves along a third fixed line, we have a determinate series of right lines, the assemblage of which forms a surface-locus. 110. Let us then solve the problem suggested by the last article, viz. to find the surface generated by a right line which * Or three fixed curves of any kind. 74 RECTILINEAR GENERATORS. always meets three fixed right lines, no two of which are in the same plane. In order that the work may be shortened as much as possible, let us first examine what choice of axes we must make in order to give the equations of the fixed right lines the simplest form. And it occurs at once that we ought to take the axes, one parallel to each of the three given right lines.* The only question then is where the origin can most symmetrically be placed. Suppose now that through each of the three right lines we draw planes parallel to the other two, we get thus three pairs of parallel planes forming a parallelopiped, of which -the given lines will be edges. And if through the centre of this parallelopiped we draw lines parallel to these edges, we shall have the most symmetrical axes. Let then the equations of the three pairs of planes be x = + a, y = ±b, z = ±c, then the equations of the three fixed right lines will be y = b, z = — c; a = c, x = — a; x = a : y = — b. The equations of any line meeting the first two fixed lines are z + c = X(y — h); z — c = (j,(x + a), which will intersect the third if c + fia + \b = ; or replacing for \ and fi their values, c [x + a) {y-b) + a (z — c) (y — b) + b (z 4 c) (x + a), which reduced is ayz + bzx + cxy + abc = 0. On applying the criterion of p. 54 this is found to repre- sent a hyperboloid of one sheet, as is otherwise evident, since it represents a central quadric and is known to be a ruled surface. The problem might otherwise be solved thus : Assume for the equations of the moveable line x — x y—y' z — z' cos a cos/3 cos 7 ' * We could not do this indeed if the three given right linea happened to be all parallel to the same plane. This case will be considered in the next Article. It will not occur when the locus is a hyperboloid of one sheet, see Art. 108. RECTILINEAR GENERATORS. 75 the three conditions obtained by expressing that this intersects each of the fixed lines are y' — i _z' + c z' — c _x' + a x —a _y' + b cos/8 C0S7 ' cosy cosa ' cosa cos/3 ' We can eliminate a, /3, 7 by multiplying the equations together, and get for the equation of the locus, (a-a)(y-J)(«-c) = (as + a)(y + &)(* + c), or reducing ayz + bzx + cxy + aba = 0, the same equation as before. The following is another general solution of the same pro- blem : Let the first two lines be the intersections of the planes a, /3 ; 7, 8 ; then the equations of the third can be expressed in the form a = Ay + BB, /3 = Cy + DS. The moveable line, since it meets the first two lines, can be expressed by two equations of the form a = A,/3, y = /j,S. Substituting these values in the equations of the third line we find the condition that it and the moveable line should intersect, viz. A/i + B=\{C/t + D). And eliminating \ and fi between this and the equations of the moveable line, we get for the equation of the locus, /3{Ay + BS)=a{Cy + m). 111. From the general theory explained in Art. 105, it is plain that the hyperbolic paraboloid may also have right lines q? 2/ 2 Z lying altogether in the surface. For the equation — — j- 2 = - (Art. 83) is included in the general form ay = ^S; and the surface contains the two systems of right lines * + f = x, xp + fW. a \a bj c The first equation shews that every line on the surface must be parallel to one or other of the two fixed planes — ± | = ; and in this respect is the fundamental difference between right lines on the paraboloid and on the hyperboloid (see Art. 108). 76 KECTILINEAR GENERATORS. It is proved, as in Art. 106, that any line of one system meets every line of the other system, while no two lines of the same system can intersect. We give now the investigation of the converse problem, viz. to find the surface generated by a right line which always meets three fixed lines which are all parallel to the same plane. Let the plane to which all are parallel be taken for the plane of xy, any line which meets all three for the axis of z, and let the axes of x and y be taken parallel to two of the fixed lines. Then their equations are x = 0, z = a; 2/ = 0, z = b; x = my, z = c. The equations of any line meeting the first two fixed lines are x = X(z- a), y — /j,[z — b), which will intersect the third if \[c — a)= nip, [c — b), and the equation of the locus is [a — c) x (z — b) = m (b — c) y (z — a), which represents a hyperbolic paraboloid since the terms of highest degree break up into two real factors. In like manner we might investigate the surface generated by a right line which meets two fixed lines and is always parallel to a fixed plane. Let it meet the lines £c = 0, z = a; y = 0, z = — a, and be parallel to the plane x cosa + ?/ cos/3 + » cos7=2?. Then the equations of the line are x = \{z — a), y = /j,(z + a), which will be parallel to the given plane if cosy + A cosa + fi cos/3 = 0. The equation of the required locus is therefore cosy (s 2 — a") + x cosa [z + a) +y cos/3 (z — a) = 0, which is a hyperbolic paraboloid since the terms of the second degree break up into two real factors. RECTILINEAR GENERATORS. 77 A hyperbolic paraboloid is the limit of the hyperboloid of one sheet, when the generator in one of its positions may lie altogether at infinity. We have seen (Art. 104) that a plane is a tangent to a surface of the second degree when it /meets it in two real or imaginary lines; and (Art. 83) that a paraboloid is met by the plane at infinity in two real or imaginary lines. Hence a paraboloid is always touched by the plane at infinity. 112. Four right lines belonging to one system cut all lines belonging to the other system in a constant anharmonic ratio. For through the four lines and through any line which meets them all we can draw four planes; and therefore any other line which meets the four lines will be divided in a constant anharmonic ratio (Art. 38). Conversely, if two non-intersecting lines are divided homo- graphically in a series of points, that is to say, so that the anharmonic ratio of any four points on one line is equal to that of the corresponding points on the other; then the lines joining corresponding points will be generators of a hyper- boloid of one sheet. Let the two given lines be a, /S ; y, 8. Let any fixed line which meets them both be a = \'/3, y = /i'§; then, in order that any other line a = \/3, y = /*8 should divide them homo- graphically, we must have (Conies, Art. 57) — , = — , , and if we eliminate X between the equations a = X/3, X'y = //\S, the result is X'jSy = /i'a8. 113. In the case of the hyperbolic paraboloid any three right lines of one system cut all the right lines of the other in a constant ratio. For since the generators are all parallel to the same plane, we can draw, through any three generators, parallels to that plane, and all right lines which meet three parallel planes are cut by them in a constant ratio. Conversely, if two finite non-intersecting lines be divided, each into the same number of equal parts, the lines joining corresponding points will be generators of a hyperbolic para- 78 SURFACES 0? REVOLUTION. boloid. By doing this with, threads, the form of this Surface can be readily exhibited to the eye. To prove this directly, let the line which joins two corre- sponding extremities of the given lines be the axis of s; let the axes of x and y be taken parallel to the given lines, and let the plane of xy be half-way between them. Let the lengths of the given lines be a and b, then the co-ordinates of two corresponding points are z = c, x = pa, y = 0, z = -c, x = 0, y = (ii>, and the equations of the line joining these points are xy — h r = /"■, 2c* — pas = pac, whence, eliminating p, the equation of the locus is which represents a hyperbolic paraboloid. SUEPACES OP REVOLUTION. 114. Let it be required to find the conditions that the general equation should represent a surface of revolution. In this case the equation can be reduced (see p. 52), if the surface X* li* £ 2 be central, to the form -y+^±-s = + l, and if the surface ' a' a' ~ c* ~ ' x* v* 2 2 be non-central to the form —, + —„ = — . In either case then a a e when the highest terms are transformed so as to become the sum of squares of three rectangular co-ordinates, the coefficients of two of those squares are equal. It would appear then that the required condition could be at once obtained by forming the condition that the discriminating cubic should have equal roots. Since however the roots of the discriminating cubic are always p ositive ^ its discriminant can be expressed as the sum of squares (see Higher Algebra, p. 134), and will not vanish (the coefficients of the given equation being supposed to be real) SURFACES OF "EVOLUTION. 79 unless two conditions are fulfilled which can be obtained more easily by the following process. We want to find whether it is possible so to transform the equation as to have ax* + by* + cz* + 2lyz + 2mzx + 2nxy = A (X* + Y*) + CZ% but we have (p. 50) x* + y* + z* = X*+Y* + Z*. It is manifest then that by taking X = A, we should have the following quantity a perfect square : (ax* + by* + cz* + 2lyz + 2mzx + 2nxy) - \ (x* + y* + s 2 ), and it is required to find the conditions that this should be possible. Now it is easy to see that when Ax* + By* + Cz* + 2Lyz + 2Mzx + 2Nxy is a perfect square, the six following conditions are fulfilled :* AB=N\ BG=L\ CA = M\ AL = MN, BM=NL, CN=LM; the three former of which are included in the three latter. In. the present case then these latter three equations are (a — X)l — mn, (b — X)m = «?, (c — \)n=Im. Solving for \ from each of these equations we see that the- reduction is impossible unless the coefficients of the given equa- tion be connected by the two relations mn . nl Im I m n If these relations be fulfilled and if we substitute any of these common values for \ in the function (a - \) x* + (b - X) y* + (c - X) z* + 2hjz + 2mzx + Inxy, it becomes, as it ought, a perfect square, viz. lmn(^ + ^- + -)"=(G-A)Z*, \l in nj and since the plane Z= represents a plane perpendicular to the * That is to say, the reciprocal equation vanishes identically. 80 SURFACES OF REVOLUTION. axis of revolution of the surface, it follows that -7 -f — + - = ; I m n represents a plane perpendicular to that axis. In the special case where the common values vanish which have been just found for X, the highest terms in the given equation form a perfect square, and the equation represents either a parabolic cylinder or two parallel planes (see IV. and V., p. 56). These are limiting cases of surfaces of re- volution, the axis of revolution in the latter case being any line perpendicular to both planes. The parabolic cylinder is the limit of the surface generated by the revolution of an ellipse round its minor axis, when that axis passes to infinity. 115. If one of the quantities I, m, n vanish, the surface cannot be of revolution unless a second also vanish. Suppose that we have I and m both = 0, the preceding conditions become m , I a — n -j = 0— n — = c, I m from which, eliminating the indeterminate — , we get (a — c)(b — c) = ri\ This condition might also have been obtained at once by expressing that (a - X) a? +(b-X) y l + (c - X) z* + 2nxy should be a perfect square, and it is plain that we must have X = c ; (a — c) (b — c) = ri. 116. The preceding theory might also be obtained from the consideration that in a surface of revolution the problem of finding the principal planes becomes indefinite. For since every section perpendicular to the axis of revolution is a circle, any system of parallel chords of one of these circles is bisected by the plane passing through the axis of revolution, and through the diameter of the circle perpendicular to the chords, a plane which is perpendicular to the chords. It follows that every plane through the axis of revolution is a principal plane. Now LOCI. 81 the chords which are perpendicular to these diametral planes are given (Art. 68) by the equations (a— Jc)x+ny+mz=0, nx + (b-k) y+h=0,-mx+ly-\- (c-k) 2=0, which when k is one of the roots of the discriminating cubic represent three planes meeting in one of the right lines required. The problem then will not become indeterminate unless these equations all represent the same plane, for which we have the conditions ■k n m ' a — k n n b —k I ' m I c — & ' which expanded are the same as the conditions found already. LOCI. 117. We shall conclude this chapter by a few examples of the application of Algebraic Geometry to the investigation of Loci. Ex. 1. To find the locus of a point whose shortest distances from two given non- intersecting right lines are equal. If the equations of the lines are written in their general form, the solution of this is obtained immediately by the formula of Art. 14. We may get the result in a simple form by taking for the axis of z the shortest distance between the two lines, and choos- ing for the other axes the lines bisecting the angle between the projections on their plane of the given lines ; then their equations are of the form z = c, y = mx ; z = — c, y = — mx, and the conditions of the problem give v ' 1 + m? 1 + m 2 or cz (1 + to 2 ) + mxy = 0. The locus is therefore a hyperbolic paraboloid. If the shortest distances had been to each other in a given ratio, the locus would have been {(1 + X) z + (1 - X) c] {(1 - X) z + (1 + X) o} + j-^ 2 {(1 + X) y + (1 - X) mx} ((1 - X) y + (1 + X) mx] = 0, which represents a hyperboloid of one sheet. Ex. 2. To find the locus of the middle points of all lines parallel to a fixed plane and terminated by two non-intersecting lines. Take the plane x = parallel to the fixed plane, and the plane 2 = 0, as in the last example, parallel to the two lines and equidistant from them ; then the equations of the lines are z = r, y = mx + n; r = — c, y = m'x + n'. a 82 loci. The locus is then evidently the right line which is the intersection of the planes z = 0, 2y = (m + m') x + (n + n'). Ex. 3. To find the surface of revolution generated by a right line turning round a fixed axis which it does not intersect. Let the fixed line be the axis of z, and let any position of the other be x = mz + n, y = m'z + n'. Then since any point of the revolving line describes a circle in a plane parallel to that of xy, it follows that the value of x 2 + y 2 is the same for every point in such a plane section, and it is plain that the constant value expressed in terms of z is (mz + n) 2 + (m'z + n') 2 . Hence the equation of the required surface is x 2 + y 2 = (mz + n) 2 + (m'z + n') 2 , which represents a hyperboloid of revolution of one sheet. Ex. 4. Two lines passing through the origin move each in a fixed plane, remaining perpendicular to each other, to find the surface (necessarily a cone) generated by a right fine, also passing through the origin perpendicular to the other two. Let the direction-angles of the perpendiculars to the fixed planes be a,b,c; a', V, c', and let those of the variable line be a, /3, y ; then the direction-cosines of the intersec- tions with the fixed planes, of a plane perpendicular to the variable line, will be pro- portional to (Art 15) cos |3 cose —cosy cos b, cosy cos a —cos a cose, cos a cos 5 — cos f! cos a. cos /3 cos c' — cos y cos V, cos y cos a' — cos a cos c', cos a cos b' — cos /3 cos a', and the condition that these should be perpendicular to each other is (cos/3 cose — cosy cos b) (cos/3 cose' — cosy cos V) + (cosy cos a — cos a cose) (cosy cos a' — cos a cose') + (cosa cosJ — cos/3 cosa) (cosa cos5' — cos/3 cosa') = 0, which represents a cone of the second degree. Ex. 5. Two planes mutually perpendicular pass each through a fixed line : to find the surface generated by then- line of intersection. Take the axes as in Ex. 1. Then the equations of the planes are X (z — c) + y — mx ; \' (z + c) +y + mx = 0, which will be at right angles if W + 1 — m 2 = ; and putting in for \, X' their values from the pair of equations, we get y 2 - «¥ + (1 - m 2 ) (z 2 - c 2 ) = 0, which represents a hyperboloid of one sheet. If the lines intersect, in which case c = 0, the locus reduces to a cone. Ex. 6. To find the locus of a point, whence three tangent lines, mutually at right x 2 y2 2-2 angles, can be drawn to the quadric - ■ + - + — = 1. ° ' a? b 2 c 2 If the equation were transformed so that these lines should become the axes of co- ordinates, the equation of the tangent cone would take the form Ayz + Bzx + Cxy = 0, since these three lines are edges of the cone. But the untransformed equation of the tangent cone is, see Art. 74, And we have seen (Art. 78) that if this equation be transformed to any rectangular system of axes, the sum of the coefficients of x 2 , y 2 , and z 2 will be constant. We have loci. 83 only then to express the condition that this sum should vanish, when we obtain the equation of the required locus, viz. a? \m + cV + * 2 U 2 + cV + c 2 U 2 + &) ~ a 2 + ¥ + c 2 ' Ex. 7. To find the equation of the cone whose vertex is x'y'z' and which stands on the conic in the plane of xy, ^- + ^- = 1. a 2 o 2 The equations of the line joining any point a/3 of the base to the vertex are a (z' - z) = z'x — x'z, /3 (z' — z) = z'y — y'z. Substituting these values in the equation of the base, we get for the required cone (z'x - x'z) "- (z'y - y'z) 2 _ d- + 4« - {Z ~ *>"' The following method may be used in general to find the equation of the cone whose vertex is x'y'z' us, and base the intersection of any two surfaces U, V. Substitute in each equation for x, x + Xx' ; for y,y + \j/', &c, and let the results be U+ XSU+ ^ &U + &e. = 0, V+ \SV + ¥ <5 2 7 + &c. = 0, then the result of eliminating X between these equations will be the equation of the required cone. For the points where the line joining x'y'z' w' to xyziu meets the surface U are got from the first of these two equations ; those where the same line meets the surface V are got from the second; and when the eliminant of the two equations vanishes they have a common root, or the point xyzm lies on a line passing through x'y'z' u> and meeting the intersection of the surfaces. Ex. 8. To find the equation of the cone whose vertex is the centre of an ellipsoid and base the section made by the polar of any point x'y'z'. Ant. % + '{-„ a- o 2 s 2 fxx' mi' zz'\ 2 c 2 \a? fi 2 l-J Ex. 9. To find the locus of points on the quadric — , + jr 2 + - = 1, the normals at which intersect the normal at the point x'y'z'. Ans. The points required are the intersection of the surface with the cone a 2 (y'z - z'y) (x — x') + i 2 (z'x — x'z) (y - y') + c- (x'y — y'x) (z - z') = 0. Ex. 10. To find the locus of the poles of the tangent planes of one quadric with respect to another. We have only to express the condition that the polar of x'y'z'w', with regard to the second quadric, should touch the first, and have therefore only to substitute C",. U„, U s , U it for a, /3, y, <5 in the condition given Art 75. The locus is therefore a quadric. Ex. 11. To find the cone generated by perpendiculars erected at the vertex of a given cone to its several tangent planes. Let the cone be Lx- + My- + Xz 2 = 0, and any tangent plane is Lx'x + My'y + X'z'; = the perpendicular to which through the origin is 7-7 = -jj- , = yy • *f tllen tlle com " mon value of these fractions be called p, we have x' = -£-, y' = jjy -' = ^r, substitu- G2 84 loci. jj.2 f.1. g2 ting which values in Lx' 2 +My' 2 + Nz' 2 = 0, p 2 disappears, and we have — + j- + _ - o. The form of the equation shews that the relation between the cones is reciprocal, and that the edges of the first are perpendicular to the tangent plane to the second. It can easily be seen that this is a particular case of the last example. If the equation of the cone be given in the form ax 2 + by 2 + cz 2 + 2fyz + Igzx + llixy = 0, the equation of the reciprocal cone will be the same as that of the reciprocal curve in plane geometry, viz. (be -f 1 ) x- + (ca - g 2 ) y 2 + (ab - h 2 ) z 2 + 2 (gh -af)yz + 2 (kf- bg) zx + 2(fg- ch) xy = 0. Ex. 12. A line moves about so that three fixed points on it move on fixed planes : to find the locus of any other point on it. Let the co-ordinates of the locus point P be a, (3, y ; and let the three fixed planes be taken for co-ordinate planes meeting the line in points A, B, C. Then it is easy AB AC to see that the co-ordinates of A are 0, -=^= j3, -=^ y, where the ratios AB : PB, A C: PC are known. Expressing then, by Art. 10, that the distance PA is constant, the locus is at once found to be an ellipsoid. Ex. 13. A and are two fixed points, the latter being on the surface of a sphere. Let the line jo inin g any other point D on the sphere to A meet the sphere again in D'. Then if on OD a portion OP be taken = AD 1 , find the locus of P. [Sir W. E. Hamilton]. We have AD 2 = A0 2 + OD 2 - 1AO.OD cos AOD. But AD varies inversely as the radius vector of the locus, and OD is given, by the equation of the sphere, in terms of the angles it makes with fixed axes. Thus the locus is easily seen to be a quadric of which is the centre. Ex. 14. A plane passes through a fixed line, and the lines in which it meets two fixed planes are joined by planes each to a fixed point ; find the surface generated by the line of intersection of the latter two planes. Ex. 15. The four faces of a tetrahedron pass each through a fixed point. Find the locus of the vertex if the three edges which do not pass through it move each in a fixed plane. The locus is in general a surface of the third degree having the intersection of the three planes for a double point. It reduces to a cone of the second degree when the four fixed points he in one plane. Ex. 16. Find the locus of the vertex of a tetrahedron, if the three edges which pass through that vertex each pass through a fixed point, if the opposite face also pass through a fixed point and the three other vertices move in fixed planes. Ex. 17. A plane passes through a fixed point, and the points where it meets three fixed lines are joined by planes, each to one of three other fixed lines ; find the locus of the intersection of the joining planes. Ex. 18. The sides of a polygon in space pass through fixed points, and all the vertices but one move in fixed planes ; find the curve locus of the remaining vertex. Ex. 19. All the sides of a polygon but one pass through fixed points, the extremities of the free side move on fixed lines, and all the other vertices on fixed planes, find the surface generated by the free side. ( 8* ) CHAPTER VII. methods of abridged notation. The principle op duality and kecipeocal polaks. 118. We shall in this chapter give examples of the appli- cation to quadrics of methods of abridged notation. It is convenient however first to shew that all the equations we employ admit of a two-fold interpretation, and that every theorem we obtain is accompanied by another reciprocal theorem. In fact, the reader can see without difficulty that the whole theory of Eeciprocal Polars explained (Conies, Chap. XV.) is applicable to space of three dimensions. Being given a fixed quadric (S), and any surface S, we can generate a new surfaces by taking the pole with regard to 2 of every tangent plane to 8. If we have thus a point on s corresponding to a tangent plane of 8, reciprocally the tangent plane to * at that point will correspond to the point of contact of the tangent plane to 8. For the tangent plane to s contains all the points on s consecutive to the assumed point; and to it must correspond the point through which pass all the tangent planes of 8 con- secutive to the assumed tangent plane ; that is to say, the point of contact of that plane. Thus to every point connected with one surface corresponds a plane connected with the other, and vice versd ; and to a line (joining two points) corresponds a line (the intersection of two planes). For example the degree of S, being measured by the number of points in which an arbitrary line meets it, is equal to the number of tangent planes which can be drawn to 8 through an arbitrary right line. Thus the reciprocal of a quadric is a quadric, since two tangent planes can be drawn to a quadric through any arbitrary right line (Art. 76). 119. In order to show what corresponds to a curve in space we shall anticipate a little of the theory of curves of double 86 METHODS OF ABRIDGED NOTATION. curvature to be explained hereafter. A cmwe in space may be considered as a series of points in space 1, 2, 3, &c, arranged according to a certain law. If each, point be joined to its next consecutive, we shall have a series of lines 12, 23, 34, &c, each line being a tangent to the given curve. The assemblage of these lines forms a surface, and a developable surface (see note, p. 75), since any line 12 intersects the consecutive line 23. Again, if we consider the planes 123, 234, 345, &c. containing every three consecutive points, we shall have a series of planes which are called the osculating planes of the given curve, and which are tangent planes to the developable generated by its tangents. Now when we reciprocate, it is plain that to the series of points, lines, and planes will correspond a series of planes, lines, and points ; and thus, that the reciprocal of a series of points forming a curve in space will be a series of planes touching a developable. If the curve in space lies all in one plane, the reciprocal planes will all pass through one point, and will be tangent planes to a cone. Thus the series of points common to two surfaces forms a curve. Reciprocally the series of tangent planes common to two surfaces touches a developable which envelopes both sur- faces. To the series of tangent planes (enveloping a cone) which can be drawn to the one surface through any point, cor- responds the series of points on the other which lie in the corresponding plane : that is to say, to a plane section of one surface corresponds a tangent cone of the reciprocal. It easily follows hence, that to a point and its polar plane with respect to a quadric, correspond a plane and its pole with respect to the reciprocal quadric. 120. The reciprocals are usually taken with regard to a sphere whose centre is called the origin of reciprocation. To the origin will evidently correspond the plane at infinity; and to the section of one surface by the plane at infinity will corre- spond the tangent cone which can be drawn to the other through the origin. Thus then, when the origin is without a quadric, that is to say, is such that real tangent planes can be drawn from it to the surface, the reciprocal surface will have real METHODS OF ABJ^DGED NOTATION. 87 points at infinity, that is to say, will be a hyperboloid ; when the origin is inside, the reciprocal is an ellipsoid; when the origin is on the surface, the reciprocal will be touched by the plane at infinity, or what is the same thing (as we shall presently see) the reciprocal is a paraboloid. The reciprocal of a ruled surface (that is to say, of a surface generated by the motion of a right line) is a ruled surface. For to a right line corresponds a right line, and to the surface generated by the motion of one right line will correspond the surface generated by the motion of the reciprocal line.* Hence to a hyperboloid of one sheet always corresponds a hyperboloid of one sheet unless the origin be on the surface when the reci- procal is a hyperbolic paraboloid. 121. When reciprocals are taken with regard to a sphere, any plane is evidently perpendicular to the line joining the corresponding point to the origin. Thus to any cone corre- sponds a plane curve, and the cone whose base is that curve and vertex the origin has an edge perpendicular to every tangent plane of the first cone, and vice versti. In general two cones (which may or may not have a common vertex) are said to be reciprocal when every edge of one is perpendicular to a tangent plane of the other (see Ex. 11, p. 83). For example it appears from the last article, that the tangent cone from the origin to any surface, is in this sense reciprocal to the asymptotic cone of the reciprocal surface. The sections by any plane of two reciprocal cones, having a common vertex, are polar reciprocals iciih regard to the foot of the perpendicular on that plane from the common vertex. For, let the plane meet an edge of one cone in a point P, and the perpendicular tangent plane to the other in the line QR ; let M * Mr. Cayley has remarked that the degree of any ruled surface is equal to the degree of its reciprocal. The degree of the reciprocal is equal to the number of tangent planes which can he drawn through an arbitrary right line. Now it will be formally proved hereafter, but is sufficiently evident in itself, that the tangent plane at any point on a ruled surface contains the generating line which passes through that point. The degree of the reciprocal is therefore equal to the number of generating lines which meet an arbitrary right line. But this is exactly the number of points in which the arbitrary line meets the surface, since every point in a generating line is a point on the surface. 88 METHODS OF ABRIDGED NOTATION. be the foot of the perpendicular on the plane from the vertex 0; then it is easy to see that the line PM is perpendicular to QR ; and if it meet it in 8, then since the triangle P08 is right- angled, the rectangle PM.M8 is equal to the constant OM'\ The curve therefore which is the locus of the point P is the same as that got by letting fall from M perpendiculars on the tangents QR, and taking on each perpendicular a portion in- versely as its length. The following illustrates the application of the principle here established : Through the vertex of any cone of the second degree can be drawn two lines, called focal lines, such that the section of the cone by a plane perpendicular to either line is a conic, having for a focus the point where the plane meets the focal line. For form a reciprocal cone by drawing through the vertex lines perpendicular to the -tangent planes of the given cone ; then this cone has two planes of circular section (Art. 100) ; and, by the present article, the section of the given cone by a plane parallel to either is a conic having for a focus the foot of the perpendicular on that plane from the vertex. What has been just proved may be stated, the focal lines of a cone are perpendicular to the planes of circular section of the reciprocal cone. 122. The reciprocal of a sphere with regard to any point is a surface of revolution round the transverse axis. This may be proved as at Conies, p. 269. It is easily proved that if we have any two points A and B, the distances of these two points from the origin are in the same ratio as the perpendicular from each on the plane corresponding to the other {Conies, Art. 101). Now the distance of the centre of a fixed sphere from the origin, and the perpendicular from that centre on any tangent plane to the sphere are both constant. Hence, any point on the reciprocal surface is such that its distance from the origin is in a constant ratio to the perpendicular let fall from it on a fixed plane; namely, the plane corresponding to the centre of the sphere. And this locus is manifestly a surface of re- volution of which the origin is a focus ; and the plane in question a directrix. METHODS OF ABRIDGED NOTATION. 89 By reciprocating properties of the sphere we thus get pro- perties of surfaces of revolution round the transverse axis. The left-hand column contains properties of the sphere, the right- hand those of the surfaces of revolution. The line joining focus to any point on the surface is perpendi- cular to the plane through focus and the intersection with the direc- trix plane of the tangent plane at the point. The cone whose vertex is the focus and base any plane section is a right cone, whose axis is the line joining the focus to the pole of the plane of section. A particular case of Ex. 2 is " Every plane section of a paraboloid of revolution is projected into a circle on the tangent plane at the vertex." Ex. 1. Any tangent plane to a sphere is perpendicular to the line joining its point of contact to the centre. Ex. 2. Every tangent cone to a sphere is a right cone, the tangent planes all making equal angles with the plane of contact. Ex. 3. Any plane is perpendi- cular to the line joining centre to its pole. Ex. 4. Any plane through the centre is perpendicular to the con- jugate diameter. Ex. 5. The cone whose base is any section of a sphere has circular sections parallel to the plane of section. Ex. 6. Every cylinder envelop- ing a sphere is right. Ex. 7. Any two conjugate* right lines are mutually perpendicular. Ex. 8. Any quadric enveloping a sphere is a surface of revolution; and its asymptotic cone therefore is a right cone. The line joining any point to the focus is perpendicular to the plane joining the focus to the intersec- tion with the directrix plane of the polar plane of the point. Any plane through the focus is perpendicular to the line joining the focus to its pole. Any tangent cone has for its focal lines the lines joining the ver- tex of the cone to the two foci. Every section passing through the focus has this focus for a focus. Any two conjugate lines are such that the planes joining them to the focus are at right angles. If a quadric envelope a surface of revolution, the cone enveloping the former, whose vertex is a focus of the latter, is a cone of revolution. * The polar planes with respect to a quadric of all the points of a line, pass through a right line, which we call the conjugate line. 90 METHODS OF ABRIDGED NOTATION. 123. The equation of the reciprocal of a central surface with regard to any point is found as at Conies, Art. 319. For the length of the perpendicular from any point on the tangent plane is (see Art. 85) p=— =V(« 2 cos 2 a+5 2 cos 2 /3+c 2 cos 2 y) — [x coscx.+y' cos/3+z' cosy), and the reciprocal is therefore (xx + yy' + zz + P) 2 = oV + b 2 y 2 + c 2 z 2 . Thus the reciprocal with regard to the centre is a quadric whose axes are the reciprocals of the axes of the given one. We have given (Ex. 10, p. 83) the method in general of finding the equation of the reciprocal of one quadric with regard to another. Thus the reciprocal with regard to the sphere x 2 + y 2 + s 2 = 7« 2 , is found by substituting x, y, z, — h 2 for a, /3, y, 8 in the tangential equation, Art. 75 ; or more symme- trically, the tangential equation itself may be considered as the equation of the reciprocal with regard to x 2 + y 2 + z 2 + w* = Q; a, /3, y, S being the co-ordinates. The reciprocal of the reciprocal of a quadric is evidently the quadric itself. If we actually form the equation of the re- ciprocal of the reciprocal Ao? + .B/3 2 + &c, the new coefficient of x 2 is BCD + 2i QR - BE' -CQ 2 - BL 2 , which when we sub- stitute for Bj C, &c. their values, will be found to be aA 2 . And A' will in like manner be a factor in every term, so that the reciprocal of the reciprocal is the given equation multiplied by the square of the discriminant (see Lessons on Higher Algebra, Art. 26). 124. The principle of duality may be established indepen- dently of the method of reciprocal polars, by shewing (as at Conies, Art. 208) that all the equations we employ admit of a twofold interpretation ; and that when interpreted as equations in tangential co-ordinates they yield theorems reciprocal to those which they give according to the mode of interpretation hitherto adopted. We may call a, /3, Yj S the tangential co-ordinates METHODS OF ABEft)GED NOTATION. 91 of the plane ax + (3y + yz + Sw. Now the condition that this plane may pass through a given point, being ax' + fiy' + yz' + Bw = 0, conversely, any equation of the first degree in a, jS, 7, 8, Aa + B/3 + Cy + BB = is the condition that this plane may pass through a point whose co-ordinates are proportional to A, B, C, D; and the equation just written may be regarded as the tangential equation of that point. If the tangential co-ordinates of two planes are a, /3, 7, 8; a', /3', 7', 8' it follows, from Art. 36, that a + ka, j3 + &/3', &c. are the co-ordinates of a plane passing through the line of intersection of the two given planes. And again, it follows from Art. 8, that if L = 0, M = be the tangential equations of two points, L + lcM= denotes a point on the line joining the two given ones; and similarly (Art. 9), that L + M£+ k'N de- notes a point in the plane determined by the three points L, M, N. Again, any equation in a, /3, 7, 8 may be considered as the tangential equation of a surface touched by every plane ax + /Sy + 7s + Bw whose co-ordinates satisfy the given equa- tion. If the equation be of the n th order, the surface will be of the n h class, or such that n tangent planes (fulfilling the given relation) can be drawn through any line. For if we substitute in the given equation a + ka", ff + &/S", &c. for a, /8, &c, we get an equation of the n" degree in h ) determining n planes, satisfying the given relation, which can be drawn through the intersection of the planes a'/SyS', a"/3"y"S". 125. The general tangential equation of the second degree ^a 2 + B$ 3 + C7 2 + DS* + 2i/37 + 2 Mya -I- 2Aa/3 + 2Pa8 + 2 Q0S + 2ByB = can be discussed by precisely the same methods as are used (Arts. 71-75). If we substitute a + ka", &c. for a, &c, we get a quadratic in k, which may be written S' + '2l-P+k*S" = 0. If the plane a'P'y'B touch the surface in question, 8' = 0, and one of the roots of the quadratic is k = 0. The second root will be also & = 0, provided that P=0. In other words, the co- 92 METHODS OF ABRIDGED NOTATION. ordinates of any tangent plane consecutive to a'fi'y'B' must satisfy the condition dS' ~ dS' dS' ~ dS' a M^W + td? + B dv =0 - But this equation being of the first degree represents a point, viz. the point of contact of dfi'y'S' through which every con- secutive tangent plane must pass. We may regard the relation just obtained as one connecting the co-ordinates of a tangent plane with those of any plane passing through its point of contact, and from the symmetry of this relation we infer (as in Art. 59) that if a', /3', y', 8' be the co-ordinates of any plane, those of the tangent plane at every point of the surface which lies in that plane, must fulfil the condition dS' a dS' dS' . dS' n a M +/3 dJ3' + ry W + S dV = - But this equation represents a point through which all the tangent planes in question must pass ; in other words, it re- presents the pole of the given plane. We can, by following the process pursued in Art. 75, deduce from the general tangential equation of the second degree the corresponding equation to be satisfied by its points. If the tangential equation of any point on the surface be x'a + y'/3 + z'y + w'S = 0, and a/3yS the co-ordinates of the corresponding tangent plane ; we infer from the equations already obtained, that if X be an indeterminate multiplier, we must have Xx = Aa + N/3 + My + PS ; Xy' = No. + B@ + Ly + Q8, Xz'=Ma+L/3+ Cy + BB; Xw' = Pa+ Q/3 + By + DS. Solving these equations for aftyS, we get the co-ordinates of the polar plane of any assumed point; and expressing that these co-ordinates satisfy the given tangential equation, we get the relation to be satisfied by the x, y, z, w of any point on the sur- face, a relation only differing by the substitution of capital for small letters from that found in Art. 75. It seems unnecessary to give further examples how all the preceding discussions may be adapted to the corresponding METHODS OP ABRIDGED NOTATION. 93 equations in tangential co-ordinates. In what follows, we have only to suppose the abbreviations to denote equations in tan- gential co-ordinates, when we get direct proofs of the reciprocals of the theorems actually obtained. 126. If U and V represent any two quadrics, then U+ W represents a quadric passing through every point common to U and V, and if X be indeterminate it represents a series of quadrics having a common curve of intersection. Since nine points determine a quadric (Art. 54), U+W is the most general equation of the quadric passing through eight given points (see Higher Plane Curves, p. 21). For if U and V be two quadrics, each passing through the eight points, U-\- X V represents a quadric also passing through the eight points, and the constant X can be so determined that the surface shall pass through any ninth point, and can in this way be made to coin- cide with any given quadric through the eight points. It follows then that all quadrics which pass through eight points have besides a whole series of common points, forming a com- mon curve of intersection ; and reciprocally, that all quadrics which touch eight given planes have a whole series of common tangent planes determining a fixed developable which envelopes the whole series of surfaces touching the eight fixed planes. It is evident also that the problem to describe a quadric through nine points may become indeterminate. For if the ninth point lie any where on the curve which, as we have just seen, is determined by the eight fixed points, then every quadric passing through the eight fixed points will pass through the ninth point, and it is necessary that we should be given a ninth point, not bn this curve, in order to be able to determine the surface. Thus if U and V be two quadrics through the eight points, we determine the surface by substituting the co-ordinates of the ninth point in U+XV=0; but if these co-ordinates make U= 0, V= 0, this substitution does not enable us to de- termine X. 127. Given seven points [or tangent planes] common to a series of quadrics, then an eighth point [or tangent plane] common to the whole system is determined. 94 METHODS OF ABKIDOtED NOTATION. For let U, V, W be three quadrics, each of which passes through the seven points, then U+W+ ft.W may represent any quadric which passes through them ; for the constants X, fi may be so determined that the surface shall pass through any two other points, and may in this way be made to coin- cide with any given quadric through the seven points. But U+'kV+ fiW represents a surface passing through all points common to U, V, W, and since these intersect in eight points, it follows that there is a point, in addition to the seven given, which is common to the whole system of surfaces. We see thus that though it was proved in the last article that eight points in general determine a curve of double curva- ture common to a system of quadrics, it is possible that they may not. For we have just seen that there is a particular case in which to be given eight points is only equivalent to being given seven. When we say therefore that a quadric is deter- mined by nine points, and that the intersection of two quadrics is determined by eight points, it is assumed that the nine or eight points are perfectly unrestricted in position.* 128. If a system of quadrics have If a system of quadrics be in- a common curve of intersection, that scribed in the same developable, is to say, if they have eight points that is to say, if they have eight in common, the polar plane of any common tangent planes, the locus fixed point passes through a fixed of the pole of a fixed plane is a right line. right line. For if P and Q be the polar planes of a fixed point with regard to U and V respectively, then P+\Q is the polar of the same point with respect to U+W. In particular, the locus of the centres of all quadrics in- scribed in the same developable, or touching the same eight planes, is a right line. * The reader who has studied Higher Plane Curves, Arts. 22 — 27, will have no difficulty in developing the corresponding theory for surfaces of any degree. Thus if we are given one less than the number of points necessary to determine a surface of the n a degree, we are given a series of points forming a curve through which the surface must pass ; and if we are given two less than the number of points necessary to deter- mine the surface, then we are given a certain number of other points [namely as many as will make the entire number up to n 3 ] through which the surface must also pass. METHODS OP ABEIDGED NOTATION. 95 129. If a system of quadrics pass through a common curve of intersection [or be inscribed in a common developable], the polars of a fixed line generate a hyperboloid of one sheet. Let the polars of two points in the line be P+XQ, P' + \Q\ then it is evident that their intersection lies on the hyper- boloid PQ' = P'Q. 180. If a system pass through a common curve, the locus of the pole of a fixed plane is a curve in space of the third degree. For eliminating X between P+ X Q, P + X Q', P" + X Q" we get the system of determinants P, P', P" Q, Q', Q" which represents a curve of the third degree. For the inter- section of the surfaces represented by PQ' = P' Q, PQ" = P" Q, is a curve of the fourth degree, but this includes the right line PQ, which is not part of the intersection of PQ" = P" Q, P' Q" = P" Q'. There is therefore -only a curve of the third degree common to all three. Reciprocally, if a system be inscribed in the same develop- able, the polar of a fixed point envelopes the developable which is the reciprocal of a curve of the third degree. 131. Given seven points on a Given seven tangent planes to quadric, the polar plane of a fixed a quadric, the pole of a fixed plane point passes through a fixed point. moves in a fixed plane. For evidently the polar of a fixed point with regard to U+ X V+ ft W will be of the form P+ X Q + pR, and will there- fore pass through a fixed point.* 132. .Since the discriminant contains the coefficients in the fourth degree, it follows that we have a biquadratic equation to solve to determine X, in order that U+ X V may represent a cone, and therefore that through the intersection of two quadrics * Dr. Hesse has derived from this theorem - a construction for the quadric passing through nine given points. Crelle, Vol. xxrv. p. 36. Cambridge and Dublin Mathe- matical Journal, Vol. rv. p. 44. See also some further developments of the same problem by Mr. Townsend, ib. Vol. IV. p. 241. 96 METHODS OF ABRIDGED NOTATION. four cones may be described. The vertices of these cones are determined by the intersection of the four planes, U^X'V t , U i + X'V 2 , U S + XV S , U i + X'V 4 , where X' is one of the roots of the biquadratic just referred to ; and they are given as the four points common to the series of determinants, v» v* u« u t = 0. V V V V ' 1) ' Si 'si ' >■ There are four points whose polars are the same with respect to all quadrics passing through a common curve of intersection, namely the vertices of the four cones just referred to. For to express the conditions that xv;+ y v;+zv;+wv; = <), should represent the same plane, we find the very same set of determinants. In like manner there are four planes whose poles are the same with respect to a set of quadrics inscribed in the same developable. 133. If the surface V break up into two planes, the form TJ-V X V= 0, becomes U+ XLM= 0, a case deserving of separate examination.* In general, the intersection of two quadrics is a curve of double curvature of the fourth degree, but the inter- section with Z7of any of the surfaces U+XLM, evidently reduces to the two conies in which Uis cut by the planes L and If. Any point on the line LM has the same polar plane with regard to all surfaces of the system U+ XLM.'f For if P be the polar of any * The case where U also breaks up into two planes has been discussed, p. 70. f There are two other points whose polar planes are the same with regard to all the quadrics, and which therefore (Art. 132) will be vertices of cones containing both the craves of section. It is only necessary that P, the polar plane of one of these points with regard to U should be the same plane as L'M+ LM the polar with regard to LM. Since then the polar plane of the point with regard to U passes through LM, the point itself must lie on the polar line of LM with regard to V, that is to say, on the intersection of the tangent planes where LM meets U. Let this polar line meet fin AA', and LM in BB', then the points required will be FF', the foci of the involution determined by AA', BB'. For since FF' forms a harmonic system either with AA' or with BB', the polar plane of F either with regard to XI or LM passes through F', and vice versa. METHODS OP ABEfbaED NOTATION. 97 point with regard to U, its polar with regard to U+ XLM will be P+ X (L'M+ LM 1 ) which reduces to P, when L = 0, M = 0. Thus, in particular, at the two points where the line LM meets U, all the surfaces have the same tangent plane. The form, then, U+ XLM, may be regarded as denoting a system of quadrics having double contact with each other. Conversely, if two quadrics have double contact, their line of intersection breaks up into two plane curves. For if we draw any plane through the two points of contact and through any point of their intersec- tion, this plane will meet the quadrics in sections having three points common, and having common also the two tangents at the points of contact; these sections must therefore be ideutical. In like manner all surfaces of the system are enveloped by two cones of the second degree. For take the point where the intersection of the two given common tangent planes is cut by any other common tangent plane ; then the cones having this point for vertex, and enveloping each surface, have common three tangent planes and two lines of contact, and are therefore identical. The reciprocals of a pair of quadrics having double contact will manifestly be a pair of quadrics having double con- tact, and the two planes of intersection of the one pair will corre- spond to the vertices of common tangent cones to the other pah - . 134. If there he a plane curve common to three quadrics, each pair must have also another common plane curve, and the three planes of these last common curves, pass through the same line. Let the quadrics be U, U+ LM, £7+ LN, then the last two have evidently for their mutual intersection two plane sections made by L, M- N. 135. Similar quadrics belong to the class now under dis~ cussion. Two quadrics are similar and similarly placed when the terms of the second degree are the same in both (see Conies, p. 208). Their equations then are of the form £7= 0, U+cL = 0. We see then that two such quadrics intersect in general in one plane curve, the other plane of intersec- tion being at infinity. If there be three quadrics, similar and 98 METHODS OP ABRIDGED NOTATION. similarly placed, their three finite planes of intersection pass through the same right line. Spheres are all similar quadrics, and therefore are to be considered as having a common section at infinity, which section will of course be an imaginary circle. A plane section of a quadric will be a circle if it passes through the two points in which its plane meets this imaginary circle at infinity. We may see thus immediately of how many solutions the problem of finding the circular sections of a quadric is susceptible. For the section of the quadric by the plane at infinity meets the section of a sphere by the same plane in four points, which can be joined by six right lines, the planes passing through any one of which meet the quadric in a circle. The six right lines may be divided into three pairs, each pair inter- secting in one of the three points whose polars are the same with respect to the section of the quadric and of the sphere. And it is easy to see that tbese three points determine the directions of the axes of the quadric. An umbilic (Art. 102) is the point of contact of a tangent - plane which can be drawn through one of these six right lines. There are in all therefore twelve umbilics, though only four are real. If a tangent plane be drawn to a quadric through any line, the generators in that tangent plane evidently pass, one through each of the points where the line meets the surface. Thus, then, the umbilics must lie each on some one of the eight generators, which can be drawn through the four points at infinity common to the quadric and any sphere. Or, as Sir W. Hamilton has remarked, the twelve umbilics lie three by three on eight imaginary right lines. A surface of revolution is one which has double contact with a sphere at infinity. For an equation of the form a: 2 +y' i ■+ as 2 = b can be written in the form {of +f + s 2 - r 2 ) + {{a - 1) z* - {b - r*)} = 0, and the latter part represents two planes. It is easy to see then why in this case there is but one direction of real circular sections, determined by the line joining the points of contact of the sections at infinity of a sphere and of the quadric. METHODS OF ABRIDGED NOTATION. 99 136. If the two planes L, M coincide, the form U+XLM becomes Z7+XL 2 which denotes a system of surfaces touching U at every point of the section of U by the plane L. Two quadrics cannot touch in three points without their touching all along a plane curve. For the plane of the three points meets the quadrics in sections having common those three points and the tangents at them. The sections are therefore identical. The equation of the tangent cone to a quadric given p. 46, is a particular case of the form U— 27. Also two concentric and similar quadrics (Z7, U—c 1 ) are to be regarded as enveloping each other, the plane of contact being at infinity. Any plane obviously cuts the surfaces U and U— II in two conies having double contact with each other, and if the section of one reduce to a point-circle, that point must plainly be the focus of the other. Hence when one quadric envelopes another the tangent plane at the umbilic of one cuts the other in a conic of which tlie umbilic is the focus ; and if one surface be a sphere every tangent plane to the sphere meets the other surface in a section of which the point of contact is the focus. Or these things may be seen by taking the origin at the umbilic and the tangent plane for the plane of a;y, when on making s = 0, the quantity U—L* reduces to x*+y* — F, and denotes a conic of which the origin is the focus, and I the directrix. Two quadrics enveloped by the same third intersect each other in plane curves. Obviously U— L 2 , U— M'\ have the planes L- if, L + M for their planes of intersection. 137. The equation aL' + bM* + cN* + dP\ where L, M, N, P represent planes, denotes a quadric such that any one of these four planes is polar of the intersection of the other three. For aH + blP + cN*' denotes a cone having the point LMN for its vertex ; and the equation of the quadric shews that this cone touches the quadric, P being the plane of contact. The four planes form what I shall call a self-conjugate tetrahedron with regard to the surface. It has been proved (Art. 132) that given two quadrics there are always four planes whose poles with regard to both are the same. If these be taken H2 100 METHODS OF ABRIDGED NOTATION. for the planes L, if, JV, P, the equations of both can be transformed to the forms aU + hW + cN* -t- dP* = 0, a'L* + VM 2 + c'N* + dP 2 = 0. It may also be seen, a priori, that this is a form to which it must be possible to bring the system of equations of two quadrics. For P, M, N, P involve implicitly three constants each ; and the equations written above involve explicitly three independent constants each. The system therefore includes eighteen constants, and is therefore sufficiently general to ex- press the equations of any two quadrics. In like manner the equations of three quadrics may be written in the form aU +bM' i + cN* + dP 2 +eQ> = 0, a'L 2 +VSP +c'N 2 + is the product of the equations of two planes. We must discuss separately however the two cases, where these planes are real and where they are imaginary. In the one case the equation is of the form U=LM, in the other U=L*-{-3P. In the first case the directrix (the line LM) is parallel to that axis of the surface through which real planes of circular section can be drawn; for example, to the mean axis if the surface be an ellipsoid. In the second case the line LM is parallel to one of the other axes. * The properties treated of in this chapter were first studied in detail by M. Chasles and by Professor Mao Cullagh, who about the same time independently arrived at the principal of them. M. Chasles' results will be found in the notes to his Aper<;u tlistorique, published in 1837. 102 FOCI AND CONFOCAL SURFACES. We can shew directly that the line LM is parallel to an axis of the surface. For if the co-ordinate planes x and y be any two planes mutually at right angles passing through LM; then since L and M are both of the form \x + fiy, the quantities LM and L* + M* will be both of the form ax* + 2hxy + by*. And, as in plane geometry, it is proved that by turning round the co-ordinate planes x and y, this quantity can be made to take the form Ax'±By*. The equations then, U=LM, U= L* -f M 2 , written in full, are of the form (x - aY +(y- /3) 2 + (* - yf = Ax* ± By*, and since the terms yz, zx, xy do not enter into the equation, the axes of co-ordinates are parallel to the axes of the surface. 139. We shall next examine whether a given central quadric necessarily has a focus, and whether it has more than one. For greater generality instead of taking the directrix for the axis of z, we take any parallel line ; and the equation of the last article becomes (x - a) 2 + (y - ft* + (* - jf = A (x - a'f + B (y - /3') 2 ;* and we are to enquire whether any values can be assigned to «, /3, 7, a, /3', A, B, which will make this identical with a given equation ~* ,.* £ — V — ^ — =1. L^ M + N Now first, in order that the origin may be the centre, we have 7 = 0, a = Aa', /3 = B/3' ; by the help of which equations, elimi- nating a', /3', the form written above becomes (l-A)x* + (l-B)tf + z*=^o? + ±^l3% , . . N L-N N „ M-N whence 1-^-j, ^~jr 5 1 ~ B = W B = Sr'' \-A „ 1-B -a 2 ' A +___^ = iV ; a* 0' ° r L-N + M-N~ * When A and B have opposite signs the planes of contact of the focus with the quadric are real, while they are imaginary when A and B have the same sign. FOCI AND CONFOCAL SURFACES. 103 Thus it appears that the surface being given, the v constants A and B are determined, but that the focus may lie anywhere on the conic a' ff 5 _, L-N + M-N~ ' which accordingly is called a, focal conic of the surface. Since we have purposely said nothing as to either the signs or the relative magnitudes of the quantities L, M : N, it follows that there is a focal conic in each of the three principal planes, and also that ' this conic is confocal with the corresponding principal section of the surface ; the conies L + M l ' L-N + M-N ' being plainly confocal. Any point a/3 on a focal conic being taken for focus, the corresponding directrix is a perpendicular to the plane of the conic drawn through the point , a „, /3 La _, if/3 These values may be interpreted geometrically by saying that the foot of the directrix is the pole, with respect to the principal section of the surface, of the tangent to the focal conic at the point a/3. For this tangent is 88 . fly _ 1 or — + ^ - 1 L-N + M-N~ ' £ + M ~ ' which is manifestly the polar of a'/3' with regard to j + ^ = 1. Hence, from the theory of plane confocal conies, the line joining any focus to the foot of the corresponding directrix is normal to the focal conic. The feet of the directrices must evidently lie on that conic which is the locus of the poles of the tangents of the focal conic with regard to the corresponding principal section of the quadric. The equation of this conic is , L-N ,M-N_ for if we eliminate a, /3 from the equation of the focal conic and the equations connecting a^, a'/3', we obtain this relation 104 FOCI AND CONFOCAL SURFACES. to be satisfied by the latter pair of co-ordinates. The directrices themselves form a cylinder of which the conic just written is the base. 140. Let us now examine in detail the different classes of central surfaces, in order to investigate the nature of their focal conies and to find to which of the two different kinds of foci the points on each belong. Now it is plain that the equation L-N + M-N will represent an ellipse when N is algebraically the least of the three quantities L, M, N; a hyperbola when N is the middle, and will become imaginary when N is the greatest. Of the three focal conies therefore of a central quadric, one is always an ellipse, one a hyperbola, and one imaginary. In the case of the ellipsoid, for example, the equations of the focal ellipse and focal hyperbola are respectively x y The corresponding equations for the hyperboloid of one sheet are found by changing the sign of c 2 , and those for the hyper- boloid of two sheets by changing the sign both of b'' and c 2 . Further, we have seen that foci belong to the class whose planes of contact are imaginary or are real, according as A L-N and B have the same or opposite signs, and that A = — = — , M-N . B= — =-; — . Now if N be the least of the three, both nume- M rators are positive, and the denominators are also positive in the case of the ellipsoid and hyperboloid of one sheet, but in the case of the hyperboloid of two sheets one of the denomi- nators is negative. Hence, the points on the focal ellipse are foci of the class whose planes of contact are imaginary in the cases of the ellipsoid and of the hyperboloid of one sheet, but of the opposite class in the case of the hyperboloid of two sheets. Next, let N be the middle of the three quantities ; then the two numerators have opposite signs, and the denominators have the same sign in the case of the ellipsoid, but opposite in the FOCI AND CONFOCAL SURFACES. 105 case of either hyperboloid. Hence the points of the focal hyperbola belong to the class whose planes of contact are real in the case of the ellipsoid, and to the opposite class in the case of either hyperboloid. It will be observed then that all the foci of the hyperboloid of one sheet belong to the class whose planes of contact are imaginary; but that the focal conies of the other two surfaces contain foci of opposite kinds, the ellipse of the ellipsoid and the hyperbola of the hyperboloid being those whose planes of contact are imaginary. This is equi- valent to what appeared (Art. 138) that foci of the other kind can only lie in planes perpendicular to that axis of a quadric through which real planes of circular section can be drawn. 141. Focal conies with real planes of contact intersect the surface in real points, while those of the other kind do not. In fact, if the equation of a surface can be thrown into the form Z7=_Z7 + IP, and if the co-ordinates of any point on the surface make U= 0, they must also make L = 0, M= ; that is to say, the focus must lie on the directrix. But in this case the surface could only be a cone. For taking the origin at the focus, the equation x* + 1/* + z* = L 2 -\- J/ 2 , where L and 31 each pass through the origin, would contain no terms except those of the highest degree in the variables, and would there- fore represent a cone (p. 38). The focal conic on the other hand, which consists of foci of the first kind, passes through the umbilics. For if the equa- tion of the surface can be thrown into the form U=LJl, and the co-ordinates of a point on the surface make TJ= 0, they must also make either L or J/= 0. But since the surface passes through the intersection of U, L ; if the point U lies on L, the plane L intersects the surface in an infinitely small circle ; that is to say, is a tangent at an umbilic. From this property Professor Mac Cullagh called focal conies of this latter kind umbilical- focal conies. 142. The section of the quadric by a plane passing through a focus and the corresponding directrix is a conic having the same point and line for focus and directrix. For taking the origin at the focus, the equation is either x 2 +y*-\- z* = LM, or 106 FOCI AND CONFOCAL SURFACES. x 2 + y 2 + z 2 = L 1 + M 2 . And if we make z = 0, the equation of the section is x 2 + y 2 = Im or = I 2 + m\ where Z, m are the sections of L, M by the plane z. But if this plane pass through LM, these sections coincide, and the equation reduces to a; 2 + y 2 — F, which represents a conic having the origin for the focus and Z for the directrix. Since the plane joining the focus and directrix is normal to the focal conic (Art. 139) ; we may state the theorem just proved, as follows : Every plane section normal to a focal conic has for a focus the point where it meets the focal conic. x 2 if z 2 143. If the given quadric were a cone y+ ~r + ttt = 0, the reduction of the equation to the form U= L 2 ± M 2 proceeds exactly as before, and it is proved that the co-ordinates of the a 2 B" focus must fulfil the condition -= Tr + -=r= — r T = 0? which re- L - N M- N ' presents either two right lines or an infinitely small ellipse according as L — N and M— N have opposite or the same signs. In other words, in this case the focal hyperbola becomes two right lines, while the focal ellipse contracts to the vertex of the x 2 v 2 z* cone. For the cone -= + %■„ — » = 0, the equation of the focal a be 1 1 Hnes 13^-^ = 0. The focal lines of the cone, asymptotic to any hyperboloid, are plainly the asymptotes to the focal hyperbola of the surface. The foci on the focal lines are all of the class whose planes of contact are imaginary; but the vertex itself, besides being in two ways a focus of this kind, may also be a focus of the other kind, for the equation of the cone can be written in any of the three forms d'-F , b 2 + c 2 , b 2 ~d 2 , a 2 + c 2 , or =-^-x 2 +^^z\ or = ~^- f + — — z \ The directrix, which corresponds to the vertex considered as a focus, passes through it. FOCI AND CONFOCAL SURFACES. 107 The line joining any point on a focal line to the foot of the corresponding directrix is perpendicular to that focal line. This follows as a particular case of what has been already proved for the focal conies in general, but may also be proved directly. The co-ordinates of the foot of the directrix have been proved to be a! = j — t= , y8' = -^ — =r r , the equations of the line joining this point to a/3 are £_ a. _ ( 1 _J__\ M-N X L-N y ~' xp \M-N L-NJ> and the condition that this should be perpendicular to the focal line /3aj = ay is -= «. + =^ — t- t = 0, which we have already seen is satisfied. In like manner, as a particular case of Art. 142, the section of a cone by a plane perpendicular to either of its focal lines is a conic of which the point in the focal line is a focus. The focal lines of this article are therefore identical with those de- fined (Art. 121). 144. The focal lines of a cone are perpendicular to the cir- cular sections of the reciprocal cone (see Art. 121). For the circular sections of the cone Lx* + My 2 + Nz* = 0, are (see Art. 99) parallel to the planes [L-N)a? + (M-N)tf = 0, and the corresponding focal lines of the reciprocal cone Z + M + J- = ° are aS WC haVe jUSt S ™*T=N + M ! =N = °» and the lines represented by the latter equation are evidently perpendicular to the planes represented by the former. 145. The investigation of the foci of the other species of quadrics proceeds in like manner. Thus for the paraboloids, a? V included in the equation y + T> = 2a. This equation can be written in either of the forms (*-«r+/+ (*-7) 2 =^(*-^«) 2 +(*-7+^r, 108 FOCI AND CONFOCAL SURFACES, where L-M ^^~ M-1 where __. = 2 7 -i. It thus appears that a paraboloid has two focal parabolas, which may easily be seen to be each confocal with the corre- sponding principal section. The focus belongs to one or other of the two kinds already discussed, according to the sign of the fraction — T — . In the case of the elliptic paraboloid therefore, where both L and M are positive, if L be the greater, then the foci in the plane xz are of the class whose planes of contact are imaginary, while those in the plane yz are of the opposite class. But since if we change the sign either of L or of M, the quantity — j — remains positive, we see that all the foci of the hyperbolic paraboloid belong to the former class, a property we have already seen to be true of the hyperboloid of one sheet. It remains true that the line joining any focus to the foot of the corresponding directrix is normal to the focal curve, and that the foot of the directrix is the pole with regard to the principal section of the tangent to the focal conic. The feet of the directrices lie on a parabola and the directrices them- selves generate a parabolic cylinder. To complete the discussion it remains to notice the foci of the different kinds of cylinders, but it is found without the slightest difficulty that when the base of the cylinder is an ellipse or hyperbola there are two focal lines; namely, lines drawn through the foci of the base parallel to the generators of the cylinder, while, if the base of the cylinder is a parabola, there is one focal line passing in like manner through the focus of the base. 146. The geometrical interpretation of the equation U= LM has been already given. We learn from it this property of foci FOCI AND CONFOCAL SURFACES. 109 whose planes of contact are real, that the square of the distance of any point on a quadric from such a focus is in a constant ratio to the prqdvx,t of the perpendiculars let fall from the point on the quadric, on two planes drawn through the corresponding directrix, parallel to the planes of circular section. The corre- sponding property of foci of the other kind, which is less obvious, was discovered by Professor Mac Cullagh. It is, that the distance of any point on the quadric from such a focus is in a constant ratio to its distance from the corresponding directrix, the latter distance being measured parallel to either of the planes of circular section. Suppose, in fact, we try to express the distance of the point x'y'z from a directrix parallel to the axis of z and passing through the point whose x and y are a', /3', the distance being measured parallel to a directive plane z = mx. Then a parallel plane through xyz, viz. z — z' = m [x - x') , meets the directrix in a point whose x and y of course are a', /3', while its z is given by the equation z— z =m («' — x'). The square of the distance required is therefore [x' - a'f + [y' - /3') 2 + nf (x - a') 2 = [y' - /S') 2 + (1 + nf) [x' - a') 2 . In the equation then, of Art. 139, (x - a)' +(y- 13)* + z\= A[x- aj + B{y- ff)% where A and B are both positive and A is supposed greater than jB, the right-hand side denotes B times the square of the distance of the point on the quadric from the directrix, the distance being measured parallel to the plane z = mx where A — B rr? = — ~ — . By putting in the values of A and B t given in Art. 139, it may be seen that this is a plane of circular section, but it is evident geometrically that this must be the case. For consider the section of the quadric by any plane parallel to the directive plane, and since evidently the distances of every point in such a section are measured from the same point on the directrix, the distance therefore of every point in the section from this fixed point is in a constant ratio to its distance from the focus. But when the distances of a variable point from two fixed points have to each other a constant 110 FOCI AND CONFOCAL SURFACES. ratio, the locus is a sphere. The section therefore is the inter- section of a plane and a sphere ; that is, a circle. An exception occurs when the distance from the focus is to be equal to the distance from the directrix. Since the locus of a point equidistant from two fixed points is a plane, it appears as before, that in this case the sections parallel to the directive plane are right lines. By referring to the previous articles it will be seen (see Art. 145) that the ratio we are considering is one of equality (B—\) only in the case of the hyperbolic paraboloid, a surface which the directive plane could not meet in circular sections, seeing that it has not got any. Professor Mac Cullagh calls the ratio of the focal distance to that from the directrix, the modulus of the surface, and the foci having imaginary planes of contact, he calls modular foci.* 147. It was observed (Art. 133) that all quadrics of the form U— LM are enveloped by two cones, and when U repre- sents a sphere, these cones must be of revolution as every cone enveloping a sphere must be. Further, when U reduces to a point-sphere, these cones coincide in a single one, having that point for its vertex ; and we may therefore infer that the cone enveloping a quadric and having any focus for its vertex is one of revolution. This theorem being of importance, we give a direct alge- braical proof of it. First, it will be observed that any equa- tion of the form a? + y' + z* = [ax + by 4 czf represents a right cone. For if the axes be transformed, remaining rectangular, but so that the plane denoted by ax + by + cz may become one of the co-ordinate planes, the equation of the cone will become * In the year 1836 Professor Mac Cullagh published this modular method of generation of quadrics. In 1842 I published the supplementary property possessed by the non-modular foci. Not long after M. Amyot independently noticed the same property, but owing to his not being acquainted with Professor Mac Cullagh's method of generation, M. Amyot failed to obtain the complete theory of the foci. Professor Mac Cullagh has published a detailed account of the focal properties of quadrics, which will be found in the Proceedings of the Royal Irish Academy, Vol. II., p. 446. Mr. Townsend also has published ->• valuable paper (Cambridge and Dublin Mathe- matical Journal, Vol. III., pp. 1, 97, 148) in which the properties of foci, considered as the limits of spheres having double contact with a quadric, are very fully in- FOCI AND CONF(?CAL SURFACES. Ill X +Y 2 + Z i = \X' 2 , which denotes a cone of revolution, since the coefficients of F a and Z* are equal. But now if we form, by the rule of Art. 74, the equation of the cone whose vertex is the origin and circumscribing tf+tf + tf-L'-M*, where L = ax + by + cz + d, M= a'x + b'y + c'z -+ d\ it is found to be (d* + d n ) (a; 8 + f + z* - U - M l ) + {dL + d'Mf = 0, or (d* + d") {x 2 + f + z*) - {d'L - dMf = 0, which we have seen represents a right cone. Cor. Since, in reciprocation, the cone whose vertex is the origin corresponds to the asymptotic cone of the reciprocal surface, it follows from this article, that the reciprocal of a quadric with regard to any focus is a surface of revolution. A few additional properties of foci easily deduced from the principles laid down are left as an exercise to the reader. Ex. 1. The polar of any directrix is the tangent to the focal conic at the corres- ponding focus. Ex. 2. The polar plane of any point on u directrix is perpendicular to the line joining that point to the corresponding focus. Ex. 3. If a line be drawn through a fixed point cutting any directrix of a quadric, and meeting the quadric in the points A, B ; then if .F be the corresponding focus, tan £AFO . tan §BFO is constant. This is proved as the corresponding theorem for plane conies. Conies, p. 197. Ex. 4. This remains true if the point move on any other quadric having the same focus, directrix, and planes of circular section. Ex. 5. If two such quadrics be cut by any line passing through the common direc- trix, the angles subtended at the focus by the intercepts are equal. Ex. 6. If a line through a directrix touch one of the quadrics, the chord intercepted on the other subtends a constant angle at the focus. 148. The product of the perpendiculars from the two foci of a surface of revolution round the transverse axis, on any tangent plane, is evidently constant. Now if we reciprocate this property with regard to any point by the method used in Art. 122, we learn that the square of the distance from the origin of any point on the reciprocal surface is in a constant ratio to the product of the distances of the point from two fixed planes. 11*2 FOCI AND CONFOCAL SURFACES. It appears from Art. 122, Ex. 5, that the two planes are planes of circular section of the asymptotic cone to the new surface, and therefore of the new surface itself. The intersection of the two planes is the reciprocal of the line joining the two foci; that is to say, of the axis of the surface of revolution. The property just proved,* belongs as we know (Art. 146) to every point on the umbilicar focal conic ; hence the reciprocal of any quadric with regard to an umbilicar focus, is a surface of revolution round the transverse axis ; but with regard to a modular focus is a surface of revolution round the conjugate axis. By reciprocating properties of surfaces of revolution, we obtain properties of any quadric with regard to focus and corresponding directrix. It is to be noted that the axis of the figure of revolution of either kind is the reciprocal of the directrix corresponding to the given focus : and is parallel to the tangent to the focal conic at the given focus (see Art. 139). The left-hand column contains properties of surfaces of re- volution, the right-hand of quadrics in general. Ex. 1. The tangent cone whose The cone whose vertex is a focus vertex is any point on the axis is and hase any section whose plane a right cone whose tangent planes passes through the corresponding make a constant angle with the directrix, is a right cone, whose axis plane of contact, which plane is is the line joining the focus to the perpendicular to the axis. pole of the plane of section, and this right line is perpendicular to the plane through focus and directrix. Ex. 2. Any tangent plane is at The line joining a focus to any right angles with the plane through point on the surface is at right the point of contact and the axis. angles to the line joining the focus to the point where the corresponding tangent plane meets the directrix. Ex. 3. The polar plane of any The line joining a focus to any point is at right angles to the plane point is at right angles to the containing that point and the axis. line joining the focus to the point where the polar plane meets the directrix. * It was in this way I was first led to this property, and to observe the distinction between the two kinds of foci. CONFOCAL SURFACES. 113 Ex. 4. Any two conjugate lines Any two conjugate lines pierce are such that the planes joining a plane through a directrix parallel them to the focus are at right to circular sections, in two points angles. (Ex. 7, Art. 122.) which subtend a right angle at the corresponding focus. Ex. 5. If a cone circumscribe a The cone whose base is any plane surface of revolution, one principal section of a quadric and vertex any plane is the plane of vertex and focus has for one axis the line join- axis, ing focus to the point where the plane meets the directrix. Ex. 6. The cone whose vertex The cone is a right cone whose is a focus and base any plane sec- vertex is a focus and base the sec- tion is a right cone. (Ex. 2, tion made by any tangent cone on Art. 122.) a plane through the corresponding directrix parallel to those of the circular sections. CONFOCAL SURFACES. 149. In the preceding section an account has been given of the relations which each focus- of a quadric considered separately bears to the surface. We shall in this section give an account of the properties of the conies which are the as- semblage of foci, and of the properties of confocal surfaces. And we commence by pointing out a method by which we should be led to the consideration of the focal conies of a quadric independently of the method followed in the last section. Two concentric and coaxal conies are said to be confocal when the difference of the squares of the axes is the same for both. Thus given an ellipse -^ + p = 1, any conic is confocal with it whose equation is of the form a"±\" b*±\* ' If we give the positive sign to X\ the confocal conic will be an ellipse; it will also be an ellipse when \' 2 is negative as long as it is less than 6 2 . When \ 2 is between V* and a 2 the confocal curve is a hyperbola, and when X 2 is greater than a 2 the curve is imaginary. If X 2 = V the equation reducing itself to y* = 0, the axis of x itself is the limit which separates con- i 114 CONFOCAL SURFACES. focal ellipses from hyperbolas. But the two foci belong to this limit in a special sense. In fact through a given point x'y can in general be drawn two conies confocal to a given one, since we have a quadratic to determine A 2 , viz. tf-K ' b 2 -X 2 or X 4 - X 2 (a 2 + b 2 - x' 2 - y">) + a 2 b 2 - b 2 x n - a\f = 0. When y = this quadratic becomes (X 2 - J s ) (X 2 - a 2 +o;' 2 ) = 0, and one, of its roots is X 2 = b 2 : but if we have also a;' 2 = a 2 — b'\ the second root is also X 2 = 6 2 , and therefore the two foci are in a special sense points corresponding to the value X 2 = 5 2 . If x 2 y 2 . y 2 in the equation — z — -, + ./ „ = 1, we make X 2 = Z> 2 , y _ = 0, a — X # — X o — X x 2 we get the equation of the two foci — — ^= 1. 150. Now in like manner two quadrics are said to be confocal if the diiferences • of the squares of the axes be the 2 2 2 X 1J Z same for both. Thus given the ellipsoid -5 + fs + -i = 1, any surface is confocal whose equation is of the form x 2 f z 2 « 2 + X 2 ' b 2 ±\ 2 ' c 2 ±X 2 If we give X 2 the positive sign, or if we take it negative and less than c 2 , the surface is an ellipsoid. A sphere of infinite radius is the limit of all ellipsoids of the system, being what the equation represents when X 2 = 00 . When X 2 is between c* and b 2 the surface is a hyperboloid of one sheet. When it is between b 2 and a 2 it is a hyperboloid of two sheets. When X 2 = or a* (J 2 - X 2 ) ( c 2 - V) + y* (c 2 - X 2 ) (a 2 - X") + z' 2 (a 2 - X 2 ) (i» - A 2 ) = (a 2 -X 2 )(& 2 -X 2 )(c 2 -X 2 ). If s' = 0, one of the roots of this cubic is X 2 = c 2 , the other two being given by the equation af (ft« _ x 2 ) + y-» ( « - X 2 ) = (a 2 - X 2 ) (J 2 - X 2 ), and a root of this equation will also be X 2 = c 3 , if x y a 2 -^ + Z, 2 _ c *- L The points on the focal ellipse therefore belong in a special sense to the value X 2 = c 2 . In like manner the plane y = separates hyperboloids of one sheet from those of two, and to this limit belongs in a special sense the hyperbola in that x* s 2 plane —^ — ^ + -5 — y t = 1. The focal conic in the third principal plane is imaginary. 151. The three quadrics which can be drawn through a given point confocal to a given one are respectively an ellipsoid, a hyperboloid of one sheet, and one of two. For if we substitute in the cubic of the last article successively X 2 = « 2 , X 2 = 6 2 , X 2 = c 2 , X 2 = -oo, we get results successively -\ h — , which prove that the equa- tion has always three real roots, one of which is less than c 2 , the second between c 2 and & 2 , and the third between 6 s and a 2 ; and it was shown in the last article that the surfaces corres- ponding to these values of X 2 are respectively an ellipsoid, a hyperboloid of one sheet, and one of two. 152. Another convenient way of solving the problem to describe through a given point quadrics confocal to a given one, is to take for the unknown quantity the primary axis of the sought confocal surface. Then since we are given 12 116 CONFOCAL SURFACES. a! 2 - b' 2 and a 2 - c' 2 which we shall call ft 2 and ft 2 , we have the equation x' 2 y' 2 z' 2 a' 2 + a' 2 -ft' + a' 2 -k 2 ' or a' 6 - a'* {ft' + It + x' 2 + y' 2 + z' 2 ) + a' 2 [ftW + x' 2 (A 2 + F) + y' 2 U l + «%■} - *'W = 0. From this equation we can at once express the co-ordinates of the intersection of three confocal surfaces in terms of their axes. Thus if a"', a"'\ a'" 2 be the roots of the above equation, the last term of it gives us at once x' J ft 2 k 2 = a W" 2 , or (a 2 -6 a )(a 2 -c 2 )" And by parity of reasoning, since we might have taken ¥ or c 1 for our unknown, we have , 2 _ b' 2 b" 2 b'" 2 ,,_ c'V'V" 2 , y -(b 2 -a')(b'-c')> Z ~{c 2 -a 2 )(c 2 -b 2 )' N.B. In the above we suppose b' 2 , b" 2 , &c. to involve their signs implicitly. Thus c" 2 belonging to a hyperboloid of one sheet is essentially negative, as are also b'" 2 and c'" 2 . 153. The preceding cubic also enables us to express the radius vector to the point of intersection in terms of the axes. For the second term of it gives us aF + y' 2 + z' 2 + [a 2 - b 2 ) + {a 2 - c 2 ) = a' 2 + a" 2 + a""\ or x' 2 + y' 2 + z' 2 = a' 2 + b" 2 +c' m . This expression might also have been worked out directly from the values given for x' 2 , y' 2 , z n in the last article, by a process which may be employed in reducing other symmetrical functions of these co-ordinates. For on substituting the preceding values and reducing to a common denominator, x' 2 +y' 2 +z' 2 becomes a' V'V" 2 (b 2 -' 2 + c' 2 ~" ' a" 2 + 6" 2 c" 2 And if we subtract one of these equations from the other, and remember that a"" -a n = b" a - b n =c" 2 — c' 2 , the remainder is /_»« -2\ ) x ' , J/ , _f; ) _ [a -a ) | a , v , 2 + b% „, + c , v , 2 | - u, which was to be proved. At the point therefore where three confocals intersect, each tangent plane cuts the other two perpendicularly, and the tangent plane to any one contains the normals to the other two. 155. If a plane be drawn through the centre parallel to any 118 CONFOCAL SURFACES. tangent plane to a quadric, the axes of the section made by that plane are parallel to the normals to the two confocals through the point of contact. It has been proved that the parallels to the normals are at right angles to each other, and it only remains to be proved that they are conjugate diameters in their section. But (Art. 90) the condition that two lines should be conjugate diameters is cos a cos a' cos j3 cos/3' cos 7 cos 7' _ The direction-cosines then of the normals being p"x p"y' p"e' p'"x' p'"y' p'"z lr> lr ' IF' Ir* ' ' lr i ' ' T 77 * ' ' we have to prove that P F \a"a"*a'"* b' 2 b" 2 b' m + c'V'V" 2 } But the truth of this equation appears at once on subtracting one from the other the equations which have been proved in the last article, x n y"' a' 2 x" z y n z' 2 a'V' 2 + V%" % c'V' 2 ~ ' a'V" 2 + V*b"" + e'V" 2 ~ 156. To find the lengths of the axes of the central section of a quadric by a plane parallel to the tangent plane at the point x'y'z'. From the equation of the surface the length of a central radius vector whose direction-angles are a, /3, 7 is given by the equation 1 _ cos 2 a cos 2 /3 cos 2 7 7 2 ~ ~~1F~ + ~W~ + ~d^ ■ Put for a, /3, 7 the values given in the last article, and we find for the length of one of these axes, p 2 F |aV + ^" 4 + Now we have the equations, a;' 2 , f_ , a' 2 «'V 2 + b' 2 b" 2 + c'V* ~ °» «2 y! «^_ 1 CONFOCAL SURFACES. 119 Subtracting we have x'* y" z-< 1 And substituting this value in the expression already found for p' we get p 2 = a' 2 — a" 2 . In like manner the square of the other axis is a' 2 — a'" 2 . Hence, if two confocal quadrics intersect, and a radius of one be drawn parallel to the normal to the other at any point of their curve of intersection, this radius is of constant length. 157. Since the product of the axes of a central section by the perpendicular on a parallel tangent plane is equal to abc (Art. 54), we get immediately expressions for the lengths P'i P"i P"- We have q"W a „ 2 _ «" 2 5" 2 c" 2 P ~ (a 2 - a" 2 ) (a' 2 - a'" 2 ) ' p ~ (a" 2 - a' 2 ) (a" 2 - a'" 2 ) » p"* = . '-a") (a"'" -a"")' These values might have been also obtained by substituting in the equation 1 x y z ti f4 "^ 7*4 ' '4 5 p a o c 7 the values already found for a;' 2 , y'*, z'' z and reducing the re- sulting value for j?' 2 by the method of Art. 153. The reader will observe the symmetry which exists between these values for ^Z 2 , p"'\ p'"'\ and the values already found for x'\ y'\ z"\ If the three tangent planes had been taken as co-ordinate planes, p', p", p" would be the co-ordinates of the centre of the surface. The analogy then between the values £orp'p"p'" and those for x'y'z' may be stated as follows: With the point x'y'z' as centre three confocals may be described having the three tangent planes for principal planes and inter- secting in the centre of the original system of surfaces. The axes of the new system of confocals are a', a", a" ; b', i", V" ; c, c", c". The three tangent planes to the new system are the three principal planes of the original system. If a central section be parallel to one of these principal 120 CONFOCAL SURFACES. planes (the plane of yz for instance) in the surface to which it is a tangent, it appears from Art. 156 that the squares of the axes are a' - 1\ a? - c\ It follows then that the direction and magnitude of the axes of the section are the same, no matter where the point x'y'z' be situated. The squares of the axes are equal, with signs changed, to the squares of the axes of the corresponding focal conic* , 158. If D be the diameter of a quadric parallel to the tangent line at any point of its intersection with a confocal, and p the perpendicular on the tangent plane at that point, then pD is constant for every point on that curve of intersec- tion. For the tangent line at any point of the curve of inter- section of two surfaces is the intersection of their tangent planes at that point, which in this case (Art. 154) is normal to the third confocal through the point. Hence (Art. 156) D* = a 12 — a'"'\ a' a &'V 2 and therefore (Art. 157) p*D* = -% m which is constant if i ,, i a —a a, a be given. 159. To find the locus of the pole of a given plane with regard to a system of confocal surfaces. Let the given plane be Ax + By + Cz = 1, and its pole ^- then we must identify the given equation with whence ^r^=A gr^T*. j^^O. Eliminating X s between these equations we find, for the equa- tions of the locus, '~B~" ~C *-a> = l-P=*c*. The locus is therefore a right line perpendicular to the given plane. * In the last edition, I said " equal to the axes of the corresponding foeal conies." I owe to Mr. Todhunter the correction of a mistake which was the less excusable as Chasles had given the theorem correctly. The section parallel to the plane of the focal ellipse is a central section of an hyperboloid of two sheets, and is imaginary. The section parallel to the plane of the imaginary focal conic is a central section of an ellipsoid and is real. CONFOCAL SURFACES. 121 The theorem just proved, implicitly contains the solution of the problem, "to describe a surface confocal to a given one to touch a given plane." For since the pole of a tangent plane to a surface is its point of contact, it is evident that but one surface can be described to touch the given plane, its point of contact being the point where the locus line just determined meets the plane. The theorem of this article may also be stated — "The locus of the pole of a tangent plane to any quadric, with regard to any confocal, is the normal to the first surface." 160. To find an expression for the distance between the point of contact of any tangent plane, and its pole with regard to any confocal surface. Let x'y'z' be the point of contact of a tangent plane to the surface whose axes are a, b, c; £, 17, f the pole of the same plane with regard to the surface whose axes are a', b', c'. Then, as in the last article, we have x _ £ y' _ V z' _ £ , , a" -a" , , b n -V , , c' 2 -c* , whence f-aj = — x, r\-y=— r 2 — y, £-s=— -j-s, Qi O C squaring and adding ^-(•■-^{y + ^ + f}' '2 a whence D=- where p is the perpendicular from the centre on the plane. . 161. The axes of any tangent cone to a quadric are the normals to the three confocals which can be drawn through the vertex of the cone. Consider the tangent plane to one of these three surfaces which pass through the vertex x'y'z'; then the pole of that plane with regard to the original surface lies (Art. 61) on the polar plane of x'y'z', and (Art. 159) on the normal to the ex- terior surface. It is therefore the point where that normal meet3 the polar plane of x'y'z', that is to say, the plane of contact of the cone. 122 CONFOCAL SURFACES. It follows, then (Art. 60), that the three normals meet this plane of contact in three points, such that each is the pole of the line joining the other two with respect to the section of the surface by that plane. But since this is also a section of the cone, it follows (Art. 67) that the three normals are a system of conjugate diameters of the cone, and since they are mutually at right angles they are its axes. 162. If at any point on a quadric a line be drawn touching the surface and through that line two tangent planes to any confocal, these two planes will make equal angles with the tangent plane at the given point on the first quadric. For, by the last article, that tangent plane is a principal plane of the cone touching the confocal surface and having the given point for its vertex, and the two tangent planes will be tangent planes of that cone. But two tangent planes to any cone drawn through a line in a principal plane make equal angles with that plane. The focal cones (that is to say, the cones whose vertices are any points and which stand on the focal conies) are limiting cases of cones enveloping confocal surfaces, and it is still true that the two tangent planes to a focal cone drawn through any tangent line on a surface make equal angles with the tangent plane in which that tangent line lies. If the surface be a cone its focal conic reduces to two right lines, and the theorem just stated in this case becomes, that any tangent plane to a cone makes equal angles with the planes containing its edge of contact and each of the focal lines. This theorem, however, will be proved independently in Chap. x. 163. It follows, from Art. 161, that if the three normals be made the axes of co-ordinates, the equation of the cone must take the form Ax* + Bif + (V = 0. To verify this by actual transformation will give us an independent proof of the theorem of Art. 161, and a knowledge of the actual values of -4, i?, C will be useful to us afterwards. The equation of the tangent cone given, Art. 74, is ,,''<' *i \ r ^ *_ _ , v* 4. y. x t - 1 u x jl x yjL . z ± ^ + ^ + 7- 1 JU + F + ?- 1 J = l^ + # + Z- 1 CONFOCAL SURFACES. 123 If the axes be transformed to parallel axes passing through the vertex of the cone, this equation becomes, as is easily seen, U* + 6 2 + c a 1 )w + V + " 2 ; cc x* y' 3 z'» S and hence „ „ ,, „ + * a + - oW ' Fb'"V 2 ' cVV ,a ~(a' 8 -a ! )(a" ! -ay a' 4 + &' 4 + c' 4 ~/ s Lastly, since -* + ^ + -* = -* , a;' 2 y" z' 2 _ 8 and «V + W 4 c-r 0--0-' . x'* jT _*P 5 1 W6 VC oV + i'V + c'V ~ (a' 2 - a 2 ) 2 / 2 (a' 2 - a 8 ) - * It may be observed that this quantity a' 2 -a" 2 ~a" 2 -«'" 2 ' V But it was proved (Art. 156) that the central section of the hyperboloid of one sheet which passes through x'y'z' is a? , s 2 ' "2 '2 "T" "2 ~'"2 ) and the section of the hyperboloid by the tangent plane itself is similar to this, or is also = 0. Hence the focal lines of the system of cones are the generating lines of the hyperboloid which passes through the point — a theorem due to Chasles, Liouville, XI. 121, and also noticed by Jacobi {Crelle, Vol. XII. p. 137). This may also be proved thus : Take any edge of one of the system of cones, and through it draw a tangent plane to that cone and also planes containing the generating lines of the hyperboloid ; these latter planes are tangent planes to the hyper- boloid, and therefore (Art. 162) make equal angles with the tangent plane to the cone. The two generators are therefore such that the planes drawn through them and through any edge of the cone make equal angles with the tangent plane to the cone ; but this is a property of the focal lines (Art. 162). Coe. 1. The reciprocals of a system of confocals, with regard to any point, have the same planes of circular section. For the reciprocals of the tangent cones from that point have the same planes of circular section (Art. 144), and these reci- procals are the asymptotic cones of the reciprocal surfaces. Coe. 2. If a system of confocals be projected orthogonally on any plane, the projections are confocal conies. The pro- jections are the sections by that plane of cylinders perpendicular to it, and enveloping the quadrics. And these cylinders may be considered as a system of enveloping cones whose vertex is the point at infinity on the common direction of their generators. CONFOCAL SbRFACES. 127 168. Two confocal surfaces can be drawn to touch a given line. Take on the line any point x'y'z ; let the axes of the three surfaces passing through it be a', a", a"\ and the angles the line makes with the three normals a, /8, 7. Then it appears, from Art. 165, that a is determined by the quadratic cos 2 a cos 2 /3 cos 2 7 /2 2 ~ 'm 2 ' '"2 2 a —a a —a a —a If a and a' be the roots of this quadratic, the two cones x" 1? z l - n X* y" z' V 2 o 2 ^." 2 « 2 V 2 n 2 ? rt ' 2 n' 2 " 2 n' 2 rt '" 2 n ' 2 have the given line as a common edge, and it is proved, pre- cisely as at Art. 154, that the tangent planes to the cones through this line are at right angles to each other. And since the tangent planes to a tangent cone to a surface, by definition touch that surface, it follows that the tangent planes drawn through any right line to the two confocals which it touches, are at right angles to each other. The property that the tangent cones from any point to two intersecting confocals cut each other at right angles, is sometimes expressed as follows: two confocals seen from any point appear to intersect everywhere at right angles. 169. If through a given line tangent planes be drawn to a system of confocals, the corresponding normals generate a hyper- bolic paraboloid. The normals are evidently parallel to one plane; namely, the plane perpendicular to the given line ; and if we consider any one of the confocals, then, by Art. 159, the normal to any plane through the line contains the pole of that plane with regard to the assumed confocal, which pole is a point on the polar line of the given line with regard to that confocal. Hence, every normal meets the polar line of the given line with regard to any confocal. The surface generated by the normals is therefore a hyperbolic paraboloid (Art. 111). It is evident that the surface generated by the polar lines, just referred to, is the same paraboloid, of which they form the other system of generators. 128 CONPOCAL SURFACES. The points in which this paraboloid meets the given line are the two points where this line touches confocals. A special case occurs when the given line is itself a normal to a surface TJ of the system. The normal corresponding to any plane drawn through that line is found by letting fall a perpendicular on that plane from the pole of the same plane with regard to U (Art. 159), but it is evident that both pole and perpendicular must lie in the tangent plane to Z7 to which the given line is normal. Hence in this case all the normals lie in the same plane. From the principle that the anharmonic ratio of four planes passing through a line is the same as that of their four poles with regard to any quadric, it is found at once that any four of the normals divide homographically all the polar lines correspond- ing to the given line with respect to the system of surfaces. In the special case, now under consideration, the normals will therefore envelope a conic, which conic will be a parabola, since the normal in one of its positions may lie at infinity ; namely, when the surface is an infinite sphere (Art. 150). The point where the given line meets the surface to which it is normal lies on the directrix of this parabola. 170. If a, /3, 7 be the direction-angles, referred to the three normals through the vertex, of the perpendicular to a tangent plane of the cone of Arts. 163, &c, since this perpendicular lies on the reciprocal cone, a, /3, 7 must satisfy the relation (a" - a 2 ) cos 2 a + (a" 2 - a 2 ) cos 2 /3 + (a"" 2 - a 2 ) cos 2 7 = 0, or a' 2 cos 2 a + a" 2 cos 2 /3 + a'" 2 cos 2 7 = a 2 . This relation enables us at once to determine the axis of the surface which touches any plane, for if we take any point on the plane, we know a\ a", a" for that point, as also the angles which the three normals through the point make with the plane, and therefore a 2 is known. 171. If the relation of the last article were proved inde- pendently, we should, by reversing the steps of the demon- stration, obtain a proof without transformation of co-ordinates CONFOCAL SURFACES. 129 of the equation of the tangent cone (Art. 165). The following proof is due to M. Chasles : The quantity a' 2 cos 2 a + a" 2 cos 2 /3 + a'" 2 cos 2 ? is the sum of the squares of the projections on a perpen- dicular to the given plane of the lines a', a", a". We have seen (Art. 157) that these are the axes of a surface having x'y'z' for its centre and passing through the original centre. And it was proved in the same article that three other con- jugate diameters of the same surface are the radius vector from the centre to x'y'z\ together with two lines parallel to two axes of the surface and whose squares are a 2 — S 2 , a 2 — c 2 . It was also proved (Art. 94) that the sum of the squares of the projections on any line of three conjugate diameters of a quadric is equal to that of any other three conjugate diameters. It follows then that the quantity a' 2 cos 2 a + a" 2 cos 2 /3 + a'" 2 cos 2 ? is equal to the sum of the squares of the projections on the perpendicular from the centre on the given plane, of the radius vector, and of two lines whose magnitude and direction are known. The projections of the last two lines are constant, while the projection of the radius vector is the perpendicular itself which is constant if x'y'z belong to the given plane. It is proved then that the quantity a' 2 cos 2 a + a m cos 2 /3 + a'" 2 cos 2 ? is constant while the point x'y'z' moves in a given plane ; and it is evident that the constant value is the a 2 of the surface which touches the given plane, since for it we have cosa=l, cos/3 = 0, cos7 = 0. 172. The locus of the intersection of three planes mutually at right angles, each of which touches one of three confocals is a sphere. This is proved as in Art. 89. Add together _p* =o? cos 2 a +b* cos 2 /3 +c 2 cos 2 ?, f =a' 2 cos 2 a' +J' 2 cos 2 /3' +c' 2 cos 2 ?', p'"* = a" 2 cos 2 a" + 5" 2 cos 2 /3" + c" 2 cos 2 ?", when we get p 2 = a 2 + ¥ + c 2 + {a' 2 - a 2 ) + (a" 2 - a 2 ), K 130 CONFOCAL SURFACES. where p is the distance from the centre of the intersection of the planes. Again, by subtracting one from the other, the two equations f=a 2 cos' 2 a+& 2 cos 2 /3+ c 2 008*7, f=d* cos 2 a + J' 2 cos 2 /3+c' 2 cos 2 y, we learn that the difference of the squares of the perpendiculars on two parallel tangent planes to two confocals is constant and equal a 2 - a 1 . It may be remarked that the reciprocal of the theorem of Art. 89 is that if from any point there be drawn three radii vectores to a quadric, mutually at right angles, the plane joining their extremities envelopes a surface of revolution. If be on the quadric, the plane passes through a fixed point. 173. Two cones having a common vertex envelope two con- focals ; to find the length of the intercept made on one of their common edges by a plane through the centre parallel to the tangent plane to a confocal through the vertex. The intercepts made on the four common edges are of course all equal, since the edges are equally inclined to the plane of section which is parallel to a common principal plane of both cones. Let there be any two confocal cones x y s n x' y' «' a 2 + /3 2 + 7 2 ~ "' a' 2 + /3' 2 + 7' then for their intersection, we have x' y crV (/3 2 - t 2 ) /3 a /3' 2 ( 7 2 - « 2 ) tV (a" - $) ' and if the common value of these be X 2 , we have a? + f + z* = X 2 (a 2 - /3 2 ) (/3 2 - 7 2 ) (a 2 - 7 2 ). Putting in the values of a 2 , /3 2 , 7 2 from the equations of the tangent cones (Art. 168), and determining X 2 by the equation '27'2 '2 (see Art. 157) x' 2 = 7-3 „„. , „ ™ , we get for the square v ' [a -a ){a —a ) 1 L of the required intercept a'W 2 __ («' 2 -a 2 ) (a' 2 -a' 2 )' CONFOCAL SURFACES. 131 If then the confocals be all of different kinds this value shews that the intercept is equal to the perpendicular from the centre on the tangent plane at their intersection. In the particular case where the two cones considered are the cones standing on the focal ellipse, and on the focal hyper- bola, we have a 8 = a 2 — c 2 , a' 2 = a 2 — b 2 , and the intercept reduces to a. Hence, if through any point on an ellipsoid be drawn a chord meeting both focal conies, the intercept on this chord by a plane through the centre parallel to the tangent plane at the point will be equal to the axis-major of the surface. This theorem, due to Prof. MacCullagh, is analogous to the theorem for plane curves, that a line through the centre parallel to a tangent to an ellipse cuts off on the focal radii portions equal to the axis-major. 174. M. Chasles has used the principles just established to solve the problem to determine the magnitude and direction of the axes of a central quadric being given a system of three conjugate diameters. Consider first the plane of any two of the conjugate dia- meters, and we can by plane geometry determine in magnitude and direction the axes of the section by that plane. The tangent plane at P, the extremity of the remaining diameter, will be parallel to the same plane. Now the centre of the given quadric is the point of intersection of three con- focals determined as in Art. 157, having the point P for then- centre. If now we could construct the focal conies of this new system of confocals, then the two focal cones, whose common vertex is the centre of the original quadric, determine by their mutual intersection four right lines. The six planes containing these four right lines intersect two by two in the directions of the required axes, while (Art. 173) the three tangent planes through the point P cut off on these four lines parts equal in length to the axes. The focal conies required are immediately constructed. We know the planes in which they lie and the direction of their axes. The lengths of their axes are to be a 2 -«" 2 , a' 2 -a" 2 ; a 8 - a' 2 , a' - a"\ But now the lengths of the axes of the given K2 132 CONFOCAL SURFACES. section are d' — a"\ a" — a"* (Art. 156), and these latter axes being known, the axes of the focal conies are immediately found. 175. If through any point P on a quadric a chord be drawn, as in Art. 173, touching two confocals, we can find an expression for the length of that chord. Draw a parallel semi-diameter through the centre, the length of which we shall call R. And if through P there be drawn a plane conjugate to this diameter, and a tangent plane, they will intercept (counting from the centre) portions on the diameter whose product = R 2 . ' But the portion intercepted by the conjugate plane is half the chord required, and the portion intercepted by the tangent plane is the intercept found (Art. 173). Hence 2E> V{(a' 2 -a' ! )(a" i -a 12 ) } a'b'c' When the chord is that which meets the two focal conies ; a 2 = a' 2 - b'\ a' 2 = a' 2 - c' 2 , and C= A- ■ 1 a 176. To find the locus of the vertices of right cones which can envelope a given surface. In order that the equation -^ 5 + —^- — „ + —^ — -r, = a —a a —a a — a may represent a right cone, two of the coefficients must be equal ; that is to say, a" = a', or a" = a", or in other words, for the point x'y'z' the equation of Art. 150 must have two equal roots, but from what was proved as to the limits within which the roots lie, it is evident that we cannot have equal roots except when \ is equal to one of the principal axes, or when x'y'z' is on one of the focal conies. This agrees with what was proved (Art. 147). It appears, hence, as has been already remarked, that the reciprocal of a surface, with regard to a point on a focal conic, is a surface of revolution ; and that the reciprocal, with regard to an umbilic, is a paraboloid of revolution. For an umbilic is a point on a focal conic (Art. 141), and since it is on the surface the reciprocal with regard to it is a paraboloid. Another particular case of this theorem is that two right cylinders can be circumscribed to a central quadric, the edges CONFOCAL SURFACES. 133 of the cylinders being parallel to the asymptotes of the focal hyperbola. For a cone whose vertex is at infinity is a cylinder. As a particular case of the theorem of this article, the cone standing on the focal ellipse will be a right cone only when its vertex is on the focal hyperbola, and vice versd. This theorem of course may be stated without any reference to the quadrics of which the two conies are focal conies; that the locus of the vertices of right cones which stand on a given conic is a conic of opposite species in a perpendicular plane. If the equation of one conic be — 2 + yj = l, that of the other will x* _ t a b It was proved (Ex. 8, p. 89) that if a quadric circumscribe a surface of revolution, the cone enveloping the former whose vertex is a focus of the latter is of revolution. From this article then we see that the focal conies of a quadric are the locus of the foci of all possible surfaces of revolution which can circumscribe that quadric. 177. The following examples will serve further to illustrate the principles which have been laid down : Ex. 1, To find the locus of the intersection of generators to a hyperholoid which cut at right angles. The section parallel to the tangent plane which contains the generators must be an equilateral hyperbola, so that (Art. 156) (a" 2 - a' 2 ) + (a" 2 - a'" 2 ) = 0. But (Art. 153) the square of the radius vector to the point is a "2 + J»2 + c »2 _ ((a, a")=0 obtain an equation (p+p, p — p') = 0, from which we can form the equation of the curve on the principal plane which corresponds to the given locus. 180. If the intersection of a sphere and an ellipsoid be pro- jected on either plane of circular section by lines parallel to the least (or greatest) axis, the projection will be a circle. 136 CONFOCAL SURFACES. This theorem is only a particular case of the following: that "if any two quadrics have common planes of circular section, any quadric through their intersection will have the same ;" a theorem which is evident, since if by making z = in U and in V, the result in each case represents a circle, making z = in U+JcV, must also represent a circle. It will be useful, however, to investigate this particular theorem directly. If we take as axes the axis of y which is a line in the plane of circular section and a perpendicular to it in that plane, the y will remain unaltered, and the new x 2 = the old a? + z 2 . But since by the equation of the plane « 2 2 7i 2 7i 2 2 /i 2 ot circular section z =—. .-n »ar, the new x =—, .^ r,x . But for the intersection of J + l + ? =1 ' a ? + f + ** = r% 2 2 7.2 2 we have — — x i -i ^— y=r — c , which, on substituting for a? 2 , V*- 2 It will be observed that to obtain the projection on the planes of circular sections we left y unaltered, and substituted 72 2 2 for a; 2 , -5 5 . 5-j a; 2 . But to obtain the points corresponding Qj — GO .. to any point, as in the last article, we substitute for x'\ —^ 2 x'\ I* a — c and for y*, -73 2 «A Now the squares of the former co-ordinates 5 2 -c 2 have to those of the latter the constant ratio — 5-5 — . Hence ¥ we may immediately infer from the last article that the pro- jection of the intersection of two confocal quadrics on a plane of circular section of one of them is a conic whose foci are the similar projections of the umbilics ; and, again, that given any curve (a, a") on the ellipsoid we can obtain the algebraic equation of the projection of that curve on the plane of circular section. CONFOCAL SURFACES. 137 181. The distance between two points, one on each of two confocal ellipsoids is equal to the distance between the two corre- sponding points. We have (x-Xy+(y-Y)*+(z-ZT = x 2 + tfl-z* + X 2 +Y 2 + Z 2 -2[xX+yY+zZ). Now (Art. 153) x*-iy 2 + z 2 = a 2 -fb' 2 + c" 2 , X 2 + Y 2 + Z i = A 2 + B' 2 + C"\ But for the corresponding points X 12 + Y' 2 + Z' 2 = A 2 + V + c" 2 , x' 2 + y' 2 + z' 2 = a 2 + B n + C" 2 . The sum of the squares therefore of the central radii to the two points is the same as that for the two corresponding points. But the quantities xX, y Y, zZ are evidently respectively equal Ax aX to x'X', y' Y', z'Z', since X' = — , x' = — j- , &c. The theorem of this article, due to Sir J. Ivory, is of use in the theory of attractions. 182. In order to ohtain a property of quadrics analogous to the property of conies that the sum of the focal distances is constant, Jacobi states the latter property as follows : Take the two points C and C" on the ellipse at the extremity of the axis-major, then the same relation p + p = 2a which connects the distances from C and 0" of any point on the line joining these points, connects also the distances from the foci of any point on the ellipse. Now, in like manner, if we take on the principal section of an ellipsoid the three points which corre- spond in the sense explained (Art. 178) to any three points on the focal ellipse, the same relation which connects the dis- tances from the former points of any point in their plane will also connect the distances from the latter points of any point on the surface. In fact, by Art. 181, the distances of the points on the confocal conic from a point on the surface will be equal to the distances of the point on the principal plane 138 CONFOCAL SURFACES. which corresponds to the point on the surface, from the three points in the principal section.* 183. Conversely, let it be required to find the locus of a point whose distances from three fixed points are connected by the same relation as that which connects the distances from the vertices of a triangle, whose sides are a, 5, c, to any point in its plane. Let p, p\ p" be the three distances, then (Art. 50) the relation which connects them is av-/o (p'-n+v^-p*) (p^-n +c* ( P " 2 v) w-p") -a%b 2 +c*-d 2 )p*-b\c*+a i -V)p'*-c%a*+V-c i )p m +a i bV=0. But p 3 — p'\ &c. being only functions of the co-ordinates of the first degree, the locus is manifestly only of the second degree. That any of the points from which the distances are measured is a focus, is proved by shewing that this equation is of the form U+ LM, where U is the infinitely small sphere whose centre is this point. In other words, it is required to prove that the result of making p" = in the preceding equation is the product of two equations of the first degree. But that result is a 2 {p h < - c 2 ) (jT - V) + [by - c'V" 2 ) {p"> - p" 2 +b*- c 9 ). Let now the planes represented by p" — p 2 — c' 2 , p'" 2 — p' 2 — ft* be L and M, then the result of making p' 2 = Q in the equation is d 2 LM + {PL - c*M) [L - M), or 5«i" - ibcLM cos.4 + c l M\ where A is the angle opposite a in the triangle abc. But this breaks up into two imaginary factors, shewing that the point we are discussing is a focus of the modular kind. * Mr. Townsend has shewed from geometrical considerations (Cambridge and Dublin Mathematical Journal, Tol. III., p. 154) that this property only belongs to points on the modular focal conies, and in fact the points in the plane y which correspond to any point x'y'z* on an ellipsoid are imaginary, as easily appears from the formula of Art. 180. Mr. Townsend easily derives Jacobi's mode of generation from MacCtdlagh's modular property. For if through any point on the surface we draw a plane parallel to a circular section, it will cut the directrices corresponding to the three fixed foci in a triangle of invariable magnitude and figure, and the distances of the point on the surface from the three foci will be in a constant ratio to its distances from the vertices of this triangle. And a similar triangle can be formed with its sides increased or diminished in a fixed ratio, the distances from the vertices of which to the point x'y'z' shall be equal to its distances from the foci. CURVATURE OF QUADRICS. 139 184. If several parallel tangent planes touch a series of confocals, the locus of their points of contact is an equilateral hyperbola. Let a, /3, 7 be the direction-angles of the perpendicular on the tangent planes. Then the direction-cosines of the radius . , ,, d l cos a b* cos/3 c 2 cos 7 vector to any point of contact are , , : rp rp rp as easily appears by substituting in the formula (Art. 85) cosa= j ^- T , r cosa' for x and solving for cosoc'. Forming then by Art. 15, the direction-cosines of the perpendicular to the plane of the radius vector and the perpendicular on the tangent plane, we find them to be (J* — c") cos/3 C0S7 (c 2 — a' 2 ) cosy cosa (**- ^O cosa cos/3 rp sin<£ rp sin$ rp s'm

2 ) cosa cos/3. Now these projections being constant for a system of confocal surfaces, we learn that for such a system, both the plane of the triangle and its magnitude is constant. If then CM be the perpendicular on the series of parallel tangent planes and PM the perpendicular on that line from any point of contact P, we have proved that the plane and the magnitude of the triangle GPM are constant, and therefore the locus of P is an equilateral hyperbola of which CM is an asymptote. CURVATURE OF QUADRICS. 185. The general theory of the curvature of surfaces will be explained in Chap. XL, but it will be convenient to state here some theorems on the curvature of quadrics which are immediately connected with the subject of this chapter. 140 CUEVATUEE OF QUADEICS. If a normal section be made at any point on a quadric, its radius of curvature at that point is equal to — , where 8 is the semi-diameter parallel to the trace of the section on the tangent plane, and p is the perpendicular from the centre on the tangent plane. We repeat the following proof by the method of infini- tesimals from Conies, p. 354, which see. Let P, Q be any two points on a quadric ; let a plane through Q parallel to the tangent plane at P meet the central radius CP in R, and the normal at P in 8, then the radius of a circle through the points P, Q having its centre on PS PQ'' is . But if the point Q approach indefinitely near to P, QP is in the limit equal to QB ; and if we denote CP and the central radius parallel to QB by a' and 8, and if P' be the other extremity of the diameter CP, then (Art. 70) 8 2 : a' 2 :: QB' : PB.BP' (=2a'.PB) ; therefore QB 2 = '-. — and the radius of curvature = —, . -^r^. . a a PS But if from the centre we let fall a perpendicular CM on the tangent plane, the right-angled triangle CMP is similar to PBS, and PB : PS :: a : p.' And the radius of curvature is 8' a 8 2 therefore — T . — = — ; which was to be proved. a p p If the circle through PQ have its centre not on PS, but on any line PS', making an angle 6 with PS, the only change PQ' is that the radius of the circle is p, , S' being still on the plane drawn through Q parallel to the tangent plane at P. But PS evidently = PS' cos 6. The radius of curvature is PQ* therefore —p^ cos 6, or the value for the radius of curvature of an oblique section is the radius of curvature of the normal section through PQ, multiplied by cos#. 186. These theorems may also easily be proved analytically. It is proved [Conies, p. 213) that if ax' + 2nxy + by 2 + 2mx = CURVATURE OP QUADRICS. 141 be the equation of any conic, the radius of curvature at the origin is = ■=- . If then the equation of any quadric, the plane of xy being a tangent plane, be ax 1 + 2nxy + by* + 2mxz + 2lyz + cz* 4 2rz = 0, then the radii of curvature by the sections y = 0, x = are • T T respectively - > t • But if the equation be transformed to parallel axes through the centre, the terms of highest degree remain unaltered, and the equation becomes ax 2 + 2nxy + by* + 2mxz -f 2lyz + cz'' = H. TJ Tf The squares of the intercepts on the axes of a; and y are — ,-=-. This proves that the radii of curvature are proportional to the squares of the parallel semi-diameters of a central section. And since, by the theory of conies, the radius of curvature of that section which contains the perpendicular on the tangent plane S* is — , the same is the form of the radius of every other section. The same may be proved by using the equation of the quadric transformed to any normal and the normals to two confocals as axes (see Ex. 2, Art. 166), viz. x 2 y* z* 2p'xy 2p"xz 2x _ ^ + Z^Z a + d'-a m ~ p{a?-a'*) ~ jT^^O + J The radii of curvature of the sections by the planes z = 0, y = 2 -'Si -2 _»2 are respectively , . The numerators are the squares of the semi-axes of the section by a plane parallel to the tangent plane (Art. 156). The equation of the section made by a plane making an angle 6 with the plane of y is found by first turning the axes of co-ordinates round through an angle 0, by substituting y cos 6 - s sin 6, y sin 6 + z cos 6 for y and s, and then making the new z = 0. The coefficient of y l will then become cos 2 sin'g 17a ~r a in ) a — a 142 CUEVATUEE OF QUADEICS. and the radius of curvature is 1 / cos'g sin 2 6> p W-a' % + tf-a" But this coefficient of y' is evidently the square of that semi- diameter of the central section, which makes an angle 6 with the axis y. 187. It follows from the theorem enunciated in Art. 185, that at any point on a central quadric the radius of curvature of a normal section has a maximum and minimum value, the directions of the section for these values being parallel to the axis-major and axis-minor of the central section by a plane parallel to the tangent plane. These maximum and minimum values are called the prin- cipal radii of curvature for that point, and the sections to which they belong are called the principal sections. It appears (from Art. 155) that the principal sections contain each the normal to one of the confocals through the point. The inter- section of a quadric with a confocal is a curve such that at every point of it the tangent to the curve is one of the prin- cipal directions of curvature. Such a curve is called a line of curvature on the surface. In the case of the hyperboloid of one sheet the central section is a hyperbola, and the sections whose traces on the tangent plane are parallel to the asymptotes of that hyperbola will have their radii of curvature infinite ; that is to say, they will be right lines, as we know already. In passing through one of those sections the radius of curvature changes sign ; that is to 3ay, the direction of the convexity of sections on one side of one of those lines is opposite to that of those on the other. 188. The two principal centres of curvature are the two poles of the tangent plane with regard to the two confocal surfaces which pass through the point of contact. For these poles lie on the normal to that plane (Art. 159), and at distances from 2 a 2 _ "2 it = and (Art. 160), but these have been just P V proved to be the lengths of the principal radii of curvature. CURVATURE 0# QUADRICS. 143 We can also hence find, by Art. 160, the co-ordinates of the centres of the two principal circles of curvature, viz. bv yy c'v «"v vy c m z' x = ^r> *=-#-» z =-r> *=?-> *--?-» '=-7- 189. If at each point of a quadric we take the two principal centres of curvature, the locus of all these centres is a surface of two sheets, which is called the surface of centres. We shall show how to find its equation in the next chapter, but we can see h priori the nature of its sections by the principal planes. In fact, one of the principal radii of cur- vature at any point on a principal section is the radius of curvature of the section itself, and the locus of the centres corresponding is evidently the evolute of that section. The other radius of curvature corresponding to any point in the c 2 section by the plane of xy is - , as appears from the for- mula of Art. 185, since c is an axis in every section drawn through the axis of z. From the formulae of Art. 188 the co-ordinates of the corresponding centre are 5 — x\ — „ — y' • that is to say, they are the poles with regard to the focal conic of the tangent at the point x'y' to the principal section. The locus of the centres will be the reciprocal of the principal section, taken with regard to the focal conic, viz. 2 2 72 2 " ^ by 1 ^ — — 1 .«\a ~ /7_2 '2V2 A * {a?-c*f (V- The section then by a principal plane of the surface (which is of the twelfth degree) consists of the evolute of a conic, which is of the sixth degree, and of a conic (it will be found) three times over, this conic being a double line on the surface. The section by the plane at infinity is also of a similar nature. 190. The reciprocal of the surface of centres is a surface of the fourth degree. It will appear from the general theory of the curvature of surfaces, to be explained in Chap. XI., that the tangent plane to either of the confocal surfaces through x'y's' is also a tangent 144 CUEVATUEE OF QUADEICS. plane to the surface of centres. The reciprocals of the intercepts which the tangent plane makes on the axes are given by the equation The relation *.=£. «=y. k=- x"> y n z' 2 !i '2 ' 7 2 7 r'2 ' 2 t n 5 6 « gives between f , 77, £ the relation (r+^+n=(« 2 -«' 2 )(|+|I+^) 5 and the relation x" y" z" 1- '1 — 1 = 1 a' 2 + V 2 + c' 2 gives (a»f + &V + c 2 ? 2 - 1) = (a 2 - a' 2 ) (f + 1; 2 + H- Eliminating a 2 — a' 2 , we have (f + v 2 + ft 1 = f 5 + £ + S) («t + & v + c '^ - !)•* But it is evident (as at Higher Plane Curves, p. 14) that £, 17, £ may be understood to be co-ordinates of the reciprocal surface ; since, if £, 17, f be the co-ordinates of the pole of the tangent plane with regard to the sphere a?+y* + z 3 = 1, the equation x Z+yv-t z £— 1 being identical with that of the tangent plane, £ , 77, £ will be also the reciprocals of the intercepts made by the tangent plane on the axis. * This equation was first given, as far as I am aware, by Dr. Booth, Tangen- tial Co-ordinates, Dublin, 1840. ( 1& ) CHAPTER IX. INVARIANTS AND COVARIANTS OF SYSTEMS OF QUADRICS. 191. It was proved (Art. 132) that there are four values of X for which X U+ V represents a cone. The biquadratic which determines X is obtained by equating to nothing the dis- criminant of X U+ V. We shall write it x 4 A + x 3 e + x' 2 + xe' 4 a' = 0. The values of X, for which X U+ V represents a cone, are evidently independent of the system of co-ordinates in which U and V are expressed. The coefficients A, O, &c. are there- fore invariants whose mutual ratios are unaltered by transforma- tion of co-ordinates. The following exercises in calculating these invariants include some of the cases of most frequent occurrence. Ex. 1. Let both quadrics be referred to their common self-conjugate tetrahedron (Art. 137). We may take U = ax 2 + by 2 + cz 2 + dm 2 , V=x 2 +f + z 2 + w 2 , (see Art. 137, and Conies, Ex. 1, p. 323), then A = abed, = abc + die + dca + dab, & = bc + ca + ab + ad + bd + cd, B' = a + b + c + d, A' = 1. Ex. 2. Let V, as before, be x 2 + y 2 + z 2 + w 2 , and let U represent the general equation. The general value of is a' A + b'B + c'C + 2 , e = - r 2 (a + b) + Idbry, 6' = aa? + i/3 2 + 2ry -(a + b) p 2 , = will be fulfilled, if the edges of a self-conjugate tetrahedron, with respect to either, all touch the other. Ex. 1. The vertices of two self-conjugate tetrahedra, with respect to a quadric, form a system of eight points, such that every quadric through seven will pass through the eighth (Hesse, C'relle, Vol. XX., p. 297). SYSTEMS OF QUADEICS. • 147 Let any quadrie be described through the four vertices of one* tetrahedron, and through three vertices of the second, whose faces we take for x, y, z, w. Then because the quadrie circumscribes the first tetrahedron 6' = 0, or a + b + c + d = (Art. 191, Ex. 2) ; and because it passes through three vertices of xyzw, we have 2 between them, we evidently have the equation of the locus of the centres of curvature of all principal sections. The problem may also be stated thus : If U and U' be any two algebraical equations of the same degree and k a variable parameter, it is generally possible to determine h so that the equation U+kU' = may have two equal roots. But it is not possible to determine k, so that the same equation may have three equal roots, unless a certain invariant relation subsist between the coefficients of Z7and U'. Now the present problem is a particular case of the general problem of finding such an SYSTEMS OF«QUADKICS. 151 invariant relation. It is in fact to find the condition that it may be possible to determine h so that the following biquadratic in X may have three equal roots : a 2 4X 5 a + \ ={Lp+Pl)*+2LP(lp-mq-nr), 0' = 2 (lp -mq- nr) {Lp + PI). And the required condition is 4Ae* = e 3 + 8 A 2 e'. Similarly the condition that it may be possible to find a tetra- hedron having two pairs of opposite edges on the surface of one, and whose four faces touch the other, is 4A'e' = 0' 3 +8A"U This may be derived from the equation examined in the next article. * This problem and its reciprocal appear to answer to the plane problem of finding the condition that a triangle can be inscribed in one conic and circumscribed about another. SYSTEMS OF QUADEICS. 153 1 99. To find the general form of the equation of a quadric which touches the four faces x, y, z, w of the tetrahedron of reference. Since a tangent plane meets a quadric in a conic which breaks up into two right lines, the condition that the plane w should touch ** + y + a * + w * + 2Zy« + 2mzx + 2nxy + 2pxw + 2qyw + 2rzw, is 1 + limn = P + m* -+ n 2 . And in order that the other three planes may touch we must have l+2lqr=P+q*+r' i , l+2mpr=m! i +^+r i , l+2npq=n i +p i +q\ We are about to shew that these conditions cannot be satisfied for the equation of a proper quadric unless / =p, in = q, n = r. We might, for example, satisfy the conditions otherwise by taking one of the coefficients 1= 1, whence it easily follows that we must have m = n, q = r. The equation of the quadric then would be x* + y* + z* + «« s + 2yz + 2mx {y + z) 4 2pxw + 2qw {y + z) = 0, or (y + z + mx + qw) % = (m* — l)x* + 2 (mq —j>) xw+(q'—l) w*. But in virtue of one of the given conditions, the right-hand side of this equation is a perfect square. The quadric therefore breaks up into two planes, a system which may be said to touch any plane, as meeting 4t in two right lines. The conditions in question are most easily discussed by writing Z = cosZ, where L is a real or imaginary angle, &c. The first condition then becomes cos L = cos (21 ± N) whence L±M±N=Q, or = an even multiple of 7r, which will come to the same thing as far as concerns the uses we shall make of the equation. Now if we take all the signs +, the equations L + M+N=0, L + Q + E = 0, M+P+R = 0, N+P+Q = 0, give L — P, M- (), N— B respectively = 0, or = an even multiple of w, whence cosi = cosP, or l=j>, m = q, n= r. If we take one of the terms negative, as, for instance, if we write the last condition N-P+ Q = 0, we easily find L = 0, or 1= 1, a case already considered. The only case distinct from this arises from systems of equations L + M+N=0, L+Q + B = 0, M+P-B = 0, N-P-Q = o } 154 INVAKIANTS AND COYAEIANTS OF whence we have cosJV=cos(Z+lf), cosQ = coa(L+M+P) 1 cosB = cos (M+P), and if we write 2 cosZ = X -f — , &c, the equation becomes A x 2 ^y 2 + z 2 + w 2 + yz(\ + A+zxU + -J+xy(\fi + —j an equation which breaks up into the factors f fix + — y + z 4- fipw j (- x + \y + z-\ w\ . Thus then we have proved that the most general equation of a proper quadric touching the four planes x, y, z, w is x 2 + y 2 + z 2 + w 2 + 2l (yz + xw) + 2m (zx + yw) + 2m (xy + zw) = 0, where 1 + 2 linn = I' + m 2 + n 2 . 200. If V represent a cone we have A' = 0, and we proceed to examine the meaning in this case of ©, 4>, ©'. For simplicity we may take the origin as the vertex of V, or j>\ q\ r\ d' all = 0. "We have then &' = d(a'b'c +2l'm'n -a'P -b'm' 2 -c'ri 2 ), or 6' vanishes either if the cone break up into two planes, or if the vertex of the cone be on the surface of U. Let the cone whose vertex is the origin and which circumscribes Z7, viz. d {ax'' + by 2, + cz 2 + 2lyz + 2mzx + 2nxy) — (px + qy + rz) 2 be written &x 2 + by + zz 2 + i[yz -f 2mza? + 2na;^ = 0, then <& may be written a (5' c ' _ p) + b [c'a' - m.' 2 ) + c (a'V - n' 2 ) + 21 {m'ri - a I') + 2m (n't! - b'ri) + 2n (I'm! - c'ri). Hence by the theory of the invariants of plane conies (Conies, Art. 375) = expresses the condition that it shall be possible to draw three tangent lines to U from the vertex of the cone V, which shall form a system self-conjugate with regard to V. In like manner dQ = a (be - I s ) + J',(ca - m s ), &c, SYSTEMS OF QuADRICS. 155 or 9 vanishes whenever three tangent planes to U can be drawn from the vertex of the cone V which shall form a system self- conjugate with regard to V. The discriminant of the cubic in X will vanish when the cone V touches U. When V represents two planes, both A' and G' vanish. Let the two planes be x and y, then V reduces to n'xy, and $ reduces to n 1 ' 2 (r 2 — «?),, will vanish therefore in this case when the intersection of the two planes touches U. We have 6 = n'N, and its vanishing expresses the condition that the two planes should be conjugate with respect to U; or, in other words, that the pole of either, with regard to U, should lie on the other. For (see Note, p. 47) the coordinates of the pole of the plane x are proportional to A, iV", 31, P, and the pole will therefore lie in the plane y when N=0. The condition e" = 4A is satisfied if either of the two planes touches U. 201. The plane at infinity cuts any sphere in an imaginary circle the cone standing on which, and whose vertex is the origin, is x" + y* + z* = 0. Further, since this cone is also an infinitely small sphere, any diameter is perpendicular to the conjugate plane. If now we form the invariants of x* + y* + z 2 , and the quadric given by the general equation, we get 6 = 0, or A + B + (7=0, as the condition that the origin shall be a point whence three rectangular tangent planes can be drawn to the surface, and = 0, or ad — p~ + bd — (f + cd— r s = 0, as the condition that the origin shall be a point whence three rectangular tangent lines can be drawn to the surface. In particular if the origin be the centre and therefore p, q, r all = 0, and (the surface not being a cone) d not = 0, the cubic is the same as that worked out (Art. 78). The condition = reduces to a + b + c = 0, as the condition that it shall be possible to draw systems of three rectangular asymptotic lines to the surface; and the condition 6 = 0, becomes ^(aJ + &c + ca-Z s -?JJ 3 -n s )=0, as the condition that it shall be possible to draw systems of three rectangular asymptotic planes to the surface. These 156 INVARIANTS AND COVAEIANTS OF two kinds of hyperboloids answer to equilateral hyperbolas in the theory of plane curves (see Ex. 3, Art. 192). Ex. Eveiy equilateral hyperbola which passes through three fixed points passes through a, fourth; what corresponds in the theory of quadrics? It will be seen that the truth of the plane theorem depends on the fact that the condition that the general equation of a. conic shall represent an equilateral hyperbola is linear in the coefficients. Thus then every rectangular hyperboloid (fulfilling the relation a + b + c = 0) which passes through seven points passes through a fixed curve, and which passes through six fixed points passes through two other fixed points. For the conditions that the surface shall pass through seven points together with the given relation enable us to determine all the coefficients of the quadric except one. Its equation therefore containing but one indeter min ate is of the form U + hV which passes through a, fixed curve. And when six points are given the equation can be brought to the form D + kV+ IW which passes through eight fixed points. 202. Since any tangent plane to the cone x' + y' + z* is xx + yy + zz — 0, where x'' + y n + z n = 0, and since any parallel plane passes through the same line at infinity, we see that a? ■+ ft 2 + 7* = is the condition that the plane ax + fty -f- yz + 8 shall pass through one of the tangent lines to the imaginary circle at infinity through which all spheres pass. And therefore a 8 + ft 2 + 7* = may be said to be the tangential equation of this circle. The invariants formed with a 2 + ft 2 + y 2 and the tangential equation of the surface are 6 = A* (a 4- h + c), # = A {be -T + ca- m' + ah- w 2 ), the geometrical meaning of which has been stated in the last article. The condition that two planes should be at right angles viz. aa' + ft ft' + 77' = (Art. 28), being the same as the con- dition that two planes should be conjugate with regard to a? + ft' 4 T 2 , we see that two planes at right angles are con- jugate with regard to the imaginary circle at infinity. 203. In general, the tangential equation of a curve in space expresses the condition that any plane should pass through one of the tangents of the curve. For instance, the condition (Art. 76) that the intersection of the planes ax + fty + yz + Bw, ax + ft'y + 7'z + h'w should touch a quadric, may be considered as the tangential equation of the conic in which the quadric is met by the plane ax + ft'y + ye + S'w. SYSTEMS OF QUADRICS. 157 The reciprocal of a plane curve is a cone (Art. 119), and since an ordinary equation of the second degree, whose discriminant vanishes represents a cone, so a tangential equation of the second degree, whose discriminant vanishes represents a plane conic. From such a tangential equation Aa? + BJ3* + &c. we can derive the ordinary equations of the curve, by first forming the reci- procal of the given tangential equation according to the ordinary rules, (BCD-\-&c.)x*+&c, when we shall obtain a perfect square, viz. the square of the equation of the plane of the curve. And the conic is determined, by combining with this the equation x*(BC- V) + f ( CA - IP) + z 2 (AB - N*) + 2yz (2IN- AL) + 2zx (NL - BM) + 2xy (MI- CN) = 0, which represents the cone joining the conic to the origin. 204. To find the equation of the cone which touches a quadrio U along the section made in it by any plane ax + fiy + , ©', A' all =0. And if we denote by cr the quantity Aa? + Bf3* + &c. (Art. 75), the equation to determine k has three roots = 0, the fourth root being given by the equation kA + a = 0. The equation of the required cone is therefore o-U= &(ax + /3y + yz + 8wy. When the given plane touches U, we have cr = 0, Art. 75, and the cone reduces to the tangent plane itself, as evidently ought to be the case. Under the problem of this article is included that of finding the equation of the asymptotic cone to a quadric given by the general equation. 205. The condition o- = 0, that ax + fiy + yz + Sw should touch U, is a contra variant (see Conies, Art. 380) of the third order in the coefficients. If we substitute for each coefficient a, a+\a, &c, we shall get the condition that ax + fiy + yz + Sw shall touch the surface U+W, a condition which will be of the form a + Xr + XV + XV = 0. The functions cr, cr', t, t' each contain a, /?, &c. in the second degree, and the coefficients of U and V in the third degree. In terms of these functions 158 INVARIANTS AND COVARIANTS OF can be expressed the condition that the plane ax + fty + 72 f 8w should have any permanent relation to the surfaces U, V; as for instance that it should cut them in sections u, v, connected by such permanent relations as can be expressed by relations between the coefficients of the discriminant of u + Xv. Thus if we form the discriminant with respect to X of ct+Xt+XV + XV, we get the condition that ax + fty i- 7s •+ Sv> should meet the surfaces in sections which touch ; or, in other v/ords, the con- dition that this plane should pass through a tangent line of the curve of intersection of U and V. This condition is of the eighth order in a, ft, 7, S, and of the sixth order in the coeffi- cients of each of the surfaces. Thus again t = expresses the condition that the plane should cut the surfaces in two sections such that a triangle self-conjugate with respect to one can be inscribed in the other, &c. The equation o- = may also be regarded as the tangential equation of the surface U; and, in like manner, t = 0, t' = are tangential equations of quadrics having fixed relations to U and V. Thus, from what we have just seen, t = is the envelope of a plane cutting the surfaces in two sections having to each other the relation just stated. And the discriminant of g + Xt + XV + X s a is the tangential equation of the curve of intersection of U and V. Or, again, tr = may be regarded as the equation of the surface reciprocal to £7" with regard to x* + y* + s 2 + w 1, (Art. 123). And, in like manner, a- + \r + XV + XV is the equation of the surface reciprocal to U+ X V. Since, if X varies, U+ X V de- notes a series of quadrics passing through a common curve, the reciprocal system touches a common developable, which is the reciprocal of the curve UV. And the discriminant of a + Xt + XV + XV may be regarded at pleasure as the tan- gential equation of the curve UV, or as the equation of the reciprocal developable. This equation is, as was remarked above, of the eighth degree in the new variables, and of the sixth in the coefficients of each surface. 206. "We can reciprocate the process employed in the last article. If 5 2 , J = A'a* + B'P + 0'>f + D'S*, where A = bed, B=cda, &c, and the reciprocal of a- + \ UV- TU- T V)\ The curve UV is manifestly a double linef on the locus re- presented by this equation, as we otherwise know it to be, and the locus meets U again in the line of the eighth order deter- mined by the intersection of U with T'*-iATV. This is the same line as that found in Art. 207. 210. We can show geometrically (as was stated Art. 207) that a generator of the quadric U at each of the eight points of intersection of the three surfaces U, V, S', (or U, V, T) is also a generator of the developable, and that therefore these eight lines form the locus of the eighth order, U, T' 2 — 4ATK For the surface S' being the locus of the poles with regard to U of the tangent planes to V : the tangent plane to V at one of the eight points in question is also a tangent plane to U, and therefore passes through one of the generators to U at the same point. This generator is therefore the line of intersection of the tangent planes to U and V, and therefore is a generator of the developable in question. * See Cambridge and Dublin Mathematical Journal, Vol. III., p. 171. where, though only the Geometrical proof is given, I had arrived at the result by actual formation of the equation of the developable. See Ibid, Vol. II., p. OS. The equations were also worked out by Mr. Cayley, Ibid, Tol. v., pp. 50, 51. ■(■ It is proved, as at Higher Plane Curves, p. 39, (fee also p. 7'2 of this volume) that when the equation of a surface is U-

' + vv^j/ + r'x' = 0, where a, e are the tangent planes at that point to U and V, and (© V- A' U) + X 3 A' V\ = {AV+XT' + X 2 T+X 3 A'U)\ But it can easily be verified that this result is divisible, as it ought to be, by A + XB + X^ + X 3 ®' + X i A\ and the quotient is (© UV- TV- A V) + X (* UV- TU- T V) + X 2 (©' UV- TV- A' U') = 0. Thus then we see that ®UV= T'U+AV is the condition that the intersection of the two polar planes should touch Uj while $ UV= TU+ T V is the condition that it should be cut harmonically by the surfaces Z7, V; and again the equation of the developable is i{®UV-T'U- AV'){&UV-TV-A'W)^{^UV-TU-TV)\ 212. The equation ax'+bf+cz 2 +X(x'+f+ 2 2 ) = 1, denotes (Art. 100) a system of concentric quadrics having common planes of circular section. And the form of the equation shows that the system in question has common the imaginary curve in which the point sphere x? + i/ 2 + z 2 meets any quadric of the system. Again, since the tangential equation of the system of confocal quadrics of f z> a+X b+X c+X is aof + ijff + cy' + X («" + £* + + tf + ^y (fx* + tftf + AV - 2ghtfz* - 2A/»V - 2fgxY) + 2/» [g -h)a»+ 2tf (A -/) tf + 2A 2 (/- g) z° + 2 /lA - 3/) ay - 2.9 [gh - 3/ 2 ) xy - 2f[fg - 3A 2 ) asV + 2A (] + 2yz [al — mn) + 2zx (bm — In) + 2xy (en — hn) + 2x {[b + c)p — nq- mr\ + 2y {(c + a) q — np — Ir) + 2z {{a + b)r-lq- np]\ + \*{D(x*+tf+z*)+A+B+C-2Px-2Qy-<2Rz} + \*=0. This is the equation of a series of confocal surfaces, and its discriminant with respect to X will represent the developable considered in the last article. If we write the coefficients of X and X 2 respectively T and T', then T=0 denotes the locus of points whence three rectangular lines can be drawn to touch the given quadric, and T = the locus of points whence three rectangular tangent planes can be drawn to the same quadric. 166 INVARIANTS AND CO VARIANTS OF If the paraboloid — + j- + 2z be treated in the same way, we obtain, as the equation of a system of confocal surfaces, (bx 2 +ay 2 +2abz)+\{x 2 +y 2 +2(a+b)z-ab}+X i {2z-(a+b)}-\ s =Q, and the developable which they all touch is, if we write a—b=m 1 4 (x 2 + yj (x 2 + y 2 + z 2 ) + Wmz (x 2 + y 2 + z 2 ) (x 2 - f) + is (x 2 4- f) (ax 2 + by 2 ) + 16mV + 32mV (a; 2 + y 2 ) + 2im(bx 2 + ay 2 )z 2 +(ax 2 +byy+8m(bx 2 +af) (x*-y 2 )+12, &c, then the coefficient of X in ax + (3y + P 221. The equation of any other sphere can only differ from the preceding by terms of the first degree, which must be of (x it z uo \ - + —, + — +—,), the second p p p p J factor denoting the plane at infinity. If then we add to the equation of the last article the product of these two factors, identify with the general equation of the second degree and eliminate the indeterminate constants, we obtain the conditions that the general equation of the second degree in qnadriplanar co-ordinates aa; 2 + hy' + &c. may represent a sphere, viz. hp' i + cp'" i -'2\p'p" _ cp'" + ap'-2mp"p _ a/ + hp* - 2np/ {bey ~ {ca)' ~ {abf _ a.p'+Ap""-2ppp'" _ bp' 2 +d^'" 2 -2q y y" c p"*+&p"" i -frp"p"" {adf ~ ' {bdy {cd)* 222. It was shown (Art. 205) that by forming the con- dition that ax + fiy + yz + 8w should touch U+ X V, we get an equation in \ whose coefficients are the invariants r piano A, A', 9, 6' of the sections of U and V by the give plane. It was also shown {Conies, Art. 382) that if we forL the invariants of any conic and the pair of circular points at infinity, = is the condition that the curve should be a parabola, 6' = the condition that it should be an equilateral hyperbola, and e' 2 = 4e the condition that the curve should pass through either circular point at infinity. Applying then these principles to any quadric in rectangular co-ordinates and the tangential equation of the imaginary circle o? + (3 i -\-'f 1 172 INVARIANTS AND COVARIANTS OF we get for the condition, 9 = 0, that any section should be a parabola, {be - F) a 2 + (ca - m 2 ) 2 + [ab - n") y* + 2 [mn -at) 0y + 2 (nl- bm) 7a + 2 (Im -en) a/3 = ; for the condition 9' = that it should represent an equilateral hyperbola (b + c) a 2 -I- (c + a) 0* + (a + b) y' -2l0y- 2mya - 2na/3 = 0, while 6' 2 = 49 (a 2 + /3 2 + 7 2 ) is the condition that the plane should pass through any of the four points at infinity common to the quadric and any sphere. 223. We know from the theory of conies that if a = be the tangential equation of a conic, and a = the tangential equation of the two circular points at infinity in its plane, cr + Xcr' = is the tangential equation of any confocal conic. Now the tangential equation of the pair of points where the imaginary circle a 2 +0'+y 2 is met by the plane a'x+0'y+y'z + B'w is evidently (a' 2 + /3' 2 + y' 2 ) (a 2 + /3 2 + y 2 ) - (owe' + 00 + yy'f = 0. Thus then the tangential equation of all conies confocal to the section by a'x + 0y + y'z + B'w of ax 2 + by 2 + cz 2 + dw 2 , is o 2 {{cd0 2 + dby' 2 + bcB' 2 ) + X (0* + y' 2 )} -f 2 {{cdo! 2 + day' 2 + acB' 2 ) + X (a' 2 + y' 2 )} + y 2 {(bda 12 + da0 2 + abB") + X (a' 2 4 2 )} + B 2 (bcu 12 + ca0* + aby' 2 ) -2{ad + X) 0y'0y -2{bd + X) y'a'ya. -2(cd+ X) a'0a0 - 2bca'B'aS - 2ca0B'0B - 2aby'B'yS = 0. If we form the reciprocal of this according to the ordinary rules, we get the square of a!x + 0y + y'z + B'w multiplied by 2 2 + X29' + X 2 (a 2 + 0* + y 2 ) 9 where 2 is the condition that ax + 0y + y'z + B'w should touch the given quadric, and 9', 9 have the same signification as in the last article. By equating the second factor to nothing we obtain the values of X which give the tangential equations of the foci of the plane section in question. SYSTEMS OF QUADRICS. 173 Ex. 1. To find the foci of the section of ix> + y 2 - 4z 2 + 1 by x + y + z. The equation for X is found to be 3X 2 + 2X = 16, whence X = 2 or = — §. The equation of the last article, for the values a' = p = y' = 1, and the given values of a, b, c, d, is a 2 (-3 + 2X) + 2X/?->(5 + 2X)y 2 -16,5 2 -2(4 + X)/3y-2(l + X)ya-l-2(4-X)a/3 = 0. Substituting X = 2 it becomes (a + 2/3 — 37)" — 16<5 2 , whence the co-ordinates of the foci are +\, ± f , +" f . The other value of X gives the imaginary foci. Ex. 2. To find the locus of the foci of all central sections of the quadric ax 2 + by 2 + cz 2 + 1. Making & = 0, the equation for X is found to be o+X 4 + X c+X By the help of this relation the tangential equation of the foci is reduced to the form f aa' pp yy' \ - tea'- + cap! 2 + aby- U + X + 4 + X + c + xJ (a + X) (4 + X) (c + X) "~ " Thus then the co-ordinates of the foci are a' ft" y' ,_ bca 12 + cap 2 + aby' 2 *~'a~+\' S ~b~+\' C_ c + X' W " ~ (a + X) (4 + X) (c + X)" Solving for a, p, y' from the first three equations and substituting in the equation for X, we get (ax 2 + if + cz 2 ) + X (a* + y 2 + c=) = ; solving for X and substituting in the value for 1c 2 . we get the equation of the locus, viz. (i^y^z 2 )[bc^ 2 {{a^)ij 2 +(a,-c)z 2 } 2 + meet in mnp points, the Jacobian which is of the fourth degree meets the intersection of the two quadrics T, W in sixteen points. 225. To reduce a pair of quadrics Z7, V to the canonical form x l + y l + s* + 10 s , ax' + by* + cz* + dw'\ In the first place the constants a, b, c, d are given by the biquadratic AX* - ©X s + *X 2 - &'\ + A' = 0. Then solving the equations x 2 + y* + z*+ w' 2 =Z7, a{bc + cd+db)x i + &c. = T, a(b + c+d)x*±&c. = T', ax* + &c = F, we find a; 2 , y*, «*, w a , in terms of the known functions £7, F, T, T'. Strictly speaking we ought to commence by dividing U and F by the fourth root of A, in order to reduce them to a form in which the discriminant of U shall be 1. But it will come to the same thing if leaving U and V unchanged we divide by A, T and T' as calculated from the coefficients of the given equation. Ex. 1. To reduce to the canonical form 5x"- - Uf - lie 5 - 6u> 2 + 24i/z + 22ias - 20zy + St/m + izw = 0, 25a; 2 - Wtf - 15z 2 - 5w 2 + 38yz + i6zx - 3(% - Wxw + 10yw + 18zm = 0. The reciprocals of these equations are 550a 2 + 1036/3 2 + 850y 2 - SUP + 212O0y + 500ya - 520«/3 - 180ct5 + 2088/3 ( 5+ 1980yS= 0, 3950a 2 + 800|8 2 + 2750y 2 - 9720a 2 + 11200/3y + 4900ya - 4160a/3+ 25920/3S+ 16200yd = 0. And the biquadratic is 8100 {X* - 10X 3 + 35X 2 - 50X + 24} = ; whence a. b, c, d are 1, 2, 3, 4. We then calculate T and T' by the formula T=x* {^(oi-re 2 ) +C"(oc-m 2 ) +£'(ad-p T ) + 2L\al-mn) + 2Q' (aq-np) +2E'(ar-mp)} + 2y« [A' (al - mn) + If (o7 - qr) + Q (ql-br) + R' (W - cq) + J/' (tot - en) + N' (In -bm) + L'(P- Ic) + P (2pl -nr- mq)} + &c, 176 INVARIANTS AND COVAEIANTS OF and dividing T and 7" so calculated by A (= 8100), we write X 2 + F 2 + Z 2 + W = 5a: 2 - XXf - llz 2 - 6w 2 + 24yz + llzx - 20xy + 8yw + izw, X 2 + 2F 2 + 3Z 2 + 4W 2 = 25m 2 - 10y 2 - 15z 2 - 5w 2 + 38yz + 4Gza: - dQxy - Wxm + lOyw + 18zm, 9X 2 + 16F 2 + 21Z 2 + 24 W* = 161a; 2 - 100/ - 135z 2 - 55io 2 + 30%z + 342za: - 250a;!/ - 70xw + YO^w + 126zro, 26X 2 + 38 F 2 + 42Z 2 + 44 W 2 = 280X 2 - 300y 2 - 360z 2 - 170» 2 + 772 + 252zw. Then from 24C- V+ V - T, we get 6X 2 = - 6 {2a: + 3y-2ts- 2wf. And, in like manner, F 2 = - (x + 2y - 3z + 2w) 2 , Z 2 = (3x-y + z -to) 2 , J*' 2 = (a; + y + z + w)\ Ex. 2. It having heen shown that a; 2 , j/ 2 , 2 2 , vfl can be expressed in terms of U, V, T, V, it follows that the square of the Jacobian of these four surfaces can also be expressed as a function of them. We find thus 2 = AT" 4 - 6T 3 T' + ®W 2 - 9'77" 3 - AT 1 + V {(6 2 - 2A4>) T 3 + (9* - 39' A) W + (99' - 4AA') 7T' 2 - A'07" 3 } + Z7((9' 2 - 2A'3>) T' 3 + (9'* - 36A') T'*T+ (99' - 4AA') T'T* - AQ'T 3 } + AF ! {($ 2 - 290' + 2AA') T 2 - (9' - 30A') 77' + *AT'[ + A'P" 2 {(4> 2 - 260' + 2 A A') 7" 2 - (9* - 3A0') TV + A*7 2 J + T {(9' 2 - 2A'4>) F 3 A 2 - (0' 2 - 299' 2 + 50'A'A - 9A') 7 2 £7A + (0 2 * - 2A) TJWA' + (9' 2 * - 2$ 2 A' - 90' A' + 4AA' 2 ) APT 2 - A 2 A'0'F 3 } + A 3 A' 2 V* + A 2 A' 3 U* - PT 3 A 2 {0' 3 - 39'* A' + 30 A' 2 } - U 3 FA' 2 {9 3 - 39*A + 30' A 2 } + AA'C 2 F 2 I* 3 - 34>AA' + 39 2 A' + 30' 2 A - 399'$}. 226. Given three quadrics the locus of a point whose polar planes with respect to all three meet in a line is a curve of the sixth order, which may be called the Jacobian curve of the system. For such a point must evidently satisfy all the equa- tions got by equating to nothing the determinants of the system v» v« u a u< V V V V w» w„ w t , w 4 and it will be shown hereafter that such a system represents a curve of the sixth order. SYSTEMS OF QUADRICS. 177 227. If we form the discriminant of X U+ fi V+ v W, the coefficients of the several powers of X, /j,, v will evidently be invariants of the system U, V, W. There are two invariants however of this system, (which we shall call 7, J) which deserve special attention as being also invariants of any three quadrics of the system X U+ p V+ v W. The invariant which we call 7 vanishes whenever any four of the points of intersection of U, V, W lie in a plane, or, in. other words, whenever it is possible to find values of X, /x, v, which will make \U+pV+vW represent two planes. If, as at Art. 137, we write U= ax* + fo/ 2 + cz 2 + du* + ew 2 , V= a!a? + b'f + cV J + d'u* + eV, TF= aV + by + c"z* + d"u? + e"o\ (where x+>/ + z+ u + v = 0), we shall show that in this case I is the product of the ten determinants (ab'c"), &c. For (ab'c") a; 2 + (db'c") w. 2 + [eb'd') v 1 is evidently a surface of the system X Z7+ fx V+ v W which will reduce to two planes, if one of the determinants (ab'c") vanishes. Thus we see that I is of the tenth order in the coefficients of each of the surfaces. That I is of the tenth degree may be otherwise seen as follows : Let Z7, U\ V, W be four quadrics passing each through the same sis points ; then since through these points twenty planes [ten pairs of planes] can be drawn, it follows that the problem to determine X, yu., v, so that U+XW + fiV+ vW may repre- sent two planes, admits of ten solutions. But X might also be determined by forming the invariant 7 of the system U, V, W, and then substituting for each coefficient a of U, a -I- Xa'. And since there are ten values of X, the result of substitution must contain X in the tenth degree; and therefore 7 must contain the coefficients of U in the same degree. 228. The invariant which we call J vanishes whenever any two of the eight points of intersection of the surfaces U } V, W coincide.* Thus, if at any point common to the three surfaces, * This invariant is called by Mr. Cayley the tact-invariant of a svstem of three quadrics, as that considered Art. 193 is the tact-invariant of a system of two.. N 178 INVARIANTS AND CO VARIANTS OF their three tangent planes pass through a common line, the consecutive point on this line will also be common to all the surfaces. Such a point will also be the vertex of a cone of the system XU+fiV+vW. For take the point as origin and if the tangent planes be x, y, ax + by, the equations of the surfaces are x + w 2 , y + v 2 , ax + by-+ w 2 , where « 2 , v t , w 2 de- note terms of the second degree. And it is evident that a U+ b V— W is a cone having the origin for its vertex. J will be of the sixteenth degree in the coefficients of each of the surfaces. For if in J we substitute for each coefficient a of U, a + Xd where a is the corresponding coefficient of another surface U', it is evident that the degree of the result in A, is the same as the number of surfaces of the system U+ X U' which can be drawn to touch the curve of intersection of V, W; that is to say, sixteen (Cor., Art. 224). 229. If ax 2 + by* + cz 2 + du 2 + ev 2 represent a cone, the co- ordinates of the vertex satisfy the four equations got by diffe- rentiating with respect to x, y, z, u; that is to say, (remem- bering that x-\-y + z + u + vis supposed to =0) ax = ev, by = ev, &c. The co-ordinates of the vertex may then be written - , j , - , -j , - , substituting which values in the condition connecting x, y, z, u, v, we obtain the discriminant of the surface, viz. 11111 - + T + - + j +-=0. a b c a e Thus then if we write the equations of U, V, W in the form here used, the discriminant of XU+ fiV+ vW is 1 ! P ^ i Ti + Tl Ti TTi + & c - = j Xa + fia + va Xb -+■ fib + vb and when X U+ fi V+ v W represents a cone, if we substitute the co-ordinates of the vertex in the equations of one of the surfaces, we get (Xa + fid + vd'f (Xb + fib' + vb") 2 a + 7TTX7J/X^ » + &c - = °> &c - (Xa + fid + vd'f (Xb + fib' + vb"]' SYSTEMS OF QUADEICS. 179 But these equations are the differentials of the discriminant with respect to X, fi, v. Hence we derive the theorem that if we form the discriminant of X U+ y. V+ v W, and then the discriminant of this again with respect to \, /it, v ; J will be a factor in the result. It may be shewn easily that I must also be a factor in this result, and the result is in fact 2V.* 230.f The lines joining ike vertices of any tetrahedron to the corresponding vertices of its polar tetrahedron with regard to a quadric belong to the same system of generators of a hyperboloid of one, sheet, and the intersections of corresponding faces of the two tetrahedra possess the same property. The result of substituting the co-ordinates of any point 1, in the polar of another point 2, is the same as that of sub- stituting the co-ordinates of 2 in the polar of 1. Let this result be called [1, 2]. Let the polar of 1 be called P t . Then it is easy to see that the line joining the point 1, to the intersection ofP 9 ,P 3 ,P 4) is P. P. P t [1, 2] [1, 3] [1, 4] * For this denotes a right line passing through the intersection of P 2) P 9 , P 4 , and whose equation is satisfied by the co-ordinates of 1. The notation will be more compact if we call the four polar planes x, y, z, w, and denote the quantities [1, 1], [1, 2], * An analogous theorem, due to Mr. Cayley, is that if U and V be homo- geneous functions of two variables of the n 01 degree; and if we form the discri- minant of U + X V and then the discriminant of this with respect to X, the result 'will be AB 2 C 3 where A is the result of ehmination between U and V; B (of the degree 2 (» — 2) (ra — 3) in both sets of coefficients) vanishes whenever X. can be so determined that U+ W shall have two pairs of equal factors ; and C (of the degree 3 (» — 2)) vanishes whenever X can be determined so that U+ W shall have three equal factors. In like manner, if U and V be homogeneous functions of three varia- bles, the disciiminant with regard to X of the discriminant of U+ W is still AErC*, where *i (of the degree 3» (« — 1) in each set of coefficients) is the condition that U and V should touch, B vanishes whenever it is possible to determine X so that V + X V may have two double points ; and C, so that it may have a cusp. Lastly, when U, V, W are three conies, the discriminant with respect to X, /i, v of the dis- criminant of XC+ fiV + uW is ABr, where A = is the condition that the curves should intersect and B = is the condition that XU + fiV+uW should ever be a perfect square. t The following theorems, which in the last edition were given in Chap, vii., have been inadvertently omitted from their proper place. N2 180 INVARIANTS AND COVAEIANTS OF [1, 3], [1, 4] by a, w, m,p, that is to say, by the same letters by which we have expressed the coefficients of x'\ xy, xz, xw in the general equation of a quadric. Then the equations of the four lines we are considering are y z w Z W X — ^ — ^ — — = — = — . n in p 1 q n 1 w x y x y z r m I ' — =: — z=Z — . par Now the condition that any line ax + /3y + T) (nivz + rxy) + [pi — nr) (mwy + qzx) = 0. 231. The second part of the theorem is only the polar reciprocal of the first, but, as an exercise, we give a separate proof of it. * This theorem is due to M. Cbasles. The proof here given is by Mr. Ferrers, Quarterly Journal of Mathematics, (Vol. I., p. 241). SYSTEMS OF JJlJADKICS. 181 Let A, B, G, &c. have the same meaning as at p. 40, with reference to the a, b, c, &c. of the last article. Then the equa- tion of the plane containing the three points 1, 2, 3, is easily seen to be Px + Qy + Bz + Dw = 0. And the equations of the four lines are x = 0, Ny + Mz-\- Pw = 0, y = 0, Nx + Lz + Qw = 0, a = 0, Mx + Ly + Bw = 0, w = 0, Px + Qy + Bz = 0. Now the conditions that any line ax + $y + yz + Bw = 0, a'x + fi'y + j'z + S'w = 0, should intersect each of these are found to be N(yB' - 7 '8) + M(b? - 873) + P (#/ - /3' 7 ) = 0, N{Sy' - S'y) + Q ( 7 «' - y'a) + L (aS' - Sat) = 0, M (J3E? - /3'S) + L {lot - S'a) + B (a/3' - a'/S) = 0, ■P (fi'y -Pi) + Q H ~ 7«') + -B 08a' - /3'a) = 0, and, as before, the theorem is proved by the fact that these conditions when added vanish identically. The equation of the hyperboloid is found to be x*MNP+ y'LNQ + z*LMB + w'PQB + xyN(PL +QM)+ yzL{QM+BN) + zxM(PL + BN) + xwP {MQ + BN) +ywQ (LP + NB) + zwB (LP+ QM). As a particular case of these theorems the lines joining each vertex of a circumscribing tetrahedron to the point of contact of the opposite face are generators of the same hyperboloid. 232. Pascal's theorem for conies may be stated as follows : " The sides of any triangle intersect a conic in six points lying in pairs on three lines which intersect each the opposite side of the triangle in three points lying in one right line." M. Chasles has stated the following as the analogous theorem for space of three dimensions : " The sides of a tetrahedron intersect a quadric in twelve points, through which can be drawn four 182 INVARIANTS AND CO VARIANTS OF QUADRICS. planes, each containing three points lying on edges passing through the same angle of the tetrahedron; then the lines of intersection of each such plane with the opposite face of the tetrahedron, are generators of the same system of a certain hyperboloid." Let the faces of the tetrahedron be x, y, z, w, and the quadric nj Xy t*+y* + z* + w*-(l+^yz-(m + ^)zx-(n + ^ -(p+-J xw - (2 +-) v w - ( r + "J zw i then the four planes may be written x = ny +111Z + pw, , y = nx+ Iz + qw, z = mx+ ly +rw, w=jix + qy + rz, whose intersections with the planes a?, y, 2, w, respectively are a system of lines proved in the last article to be generators of the same hyperboloid. ( 183. ) CHAPTER X. CONES AND SPHERO-CONICS. 233. If a cone of any degree be cut by any spbere, whose centre is the vertex of the cone, the curve of section will evidently be such that the angle between two edges of the cone is measured by the arc joining the two corresponding points on the sphere. When the cone is of the second degree, the curve of section is called a sphero-conic. By stating many of the properties of cones of the second degree as properties of sphero-conics, the analogy between them and corresponding properties of conies becomes more striking.* Strictly speaking, the intersection of a sphere with a cone of the n th degree is a curve of the 2n th degree : but when the cone is concentric with the sphere, the curve of intersection may be divided, in an infinity of ways, into two symmetrical and equal portions, either of which may be regarded as analo- gous to a plane curve of the n ta degree. For if we consider the points of the curve of intersection which lie in any hemi- sphere, the points diametrically opposite evidently trace out a perfectly symmetrical curve in the opposite hemisphere. Thus then a sphero-conic may be regarded as analogous either to an ellipse or to a hyperbola. A cone of the second degree evidently intersects a concentric sphere in two similar closed curves diametrically opposite to each other. One of the principal planes of the cone meets neither curve, and if we look at either of the hemispheres into which this plane divides * See M. Chasles's Memoir on Sphero-conics (published in the Sixth Volume of the Transactions of the Royal Academy of Brussels, and translated by Professor Graves, Dublin, 1837), from which the enunciations of many of the theorems in this chapter are taken. See also M. Chasles's later papers Comptes Rendus, March and June, 1860. 184. CONES AND SPHEEO-CONICS. the sphere, we see a closed curve analogous to an ellipse. But if we look at one of the hemispheres into which the sphere is divided by a principal plane meeting both the opposite curves, we see a curve consisting of two opposite branches like a hyperbola. The curve of intersection of any quadric with a concentric sphere is evidently a sphero-conic. 234. The properties of spherical curves have been studied by means of systems of spherical co-ordinates formed on the model of Cartesian co-ordinates. Choose for axes of co-ordi- nates any two great circles OX, OY intersecting at right angles, and on them let fall perpendiculars PM, PN from any point on the sphere P. These perpendiculars are not, as in plane co-ordinates, equal to the opposite sides of the quad- rilateral OMPN; and therefore it would seem that there is a certain latitude admissible in our selection of spherical co- ordinates, according as we choose for co-ordinates the per- pendiculars PM, PN, or the intercepts OM, ON which they make on the axes. M. Gudermann of Cleves has chosen for co-ordinates the tangents of the intercepts OM, ON (see Crelle's Journal, Vol. VI., p. 240), and the reader will find an elaborate discussion of this system of co-ordinates in the appendix to Dr. Graves's translation of Chasles's Memoir on Sphero-conics. It is easy to see however that if we draw a tangent plane to the sphere at the point 0, and if the lines joining the centre to the points M, N, P, meet that plane in points m, n, p ; then Om, On will be the Cartesian co-ordinates of the point p. But Om, On are the tangents of the arcs OM, ON. Hence the equation of a spherical curve in Gudermann's system of co-ordinates is in reality nothing but the ordinary equation of the plane curve in which the cone joining the spherical curve to the centre of the sphere is met by the tangent plane at the point 0. So again, if we choose for co-ordinates the sines of the per- pendiculars PM, PN, it is easy to 3ee in like manner that the equation of a spherical curve in such co-ordinates is only the CONES AND SPftERO-OONICS. 185 equation of the orthogonal projection of that curve on a plane parallel to the tangent plane at the point 0. It seems, however, to us that the properties of spherical curves are obtained more simply and directly from the equa- tions of the cones which join them to the centre, than from the equations of any of the plane curves into which they can be projected. 235. Let the co-ordinates of any point P on the sphere be substituted in the equation of any plane passing through the centre (which we take for origin of co-ordinates), and meeting the sphere in a great circle AB, the result will be the length of the perpendicular from P on that plane ; which is the sine of the spherical arc let fall perpendicular from P on the great circle AB. By the help of this principle the equations of cones are interpreted so as to yield properties of spherical curves in a manner precisely corresponding to that used in interpreting the equations of plane curves. Thus, let a, /3 be the equations of any two planes through the centre, which may also be regarded as the equations of the great circles in which they meet the sphere, then (as at Conies, p. 55) a — ft/3 denotes a great circle such that the sine of the perpendicular arc from any point of it on a is in a constant ratio to the sine of the perpendicular on /3; that is to say, a great circle dividing the angle between a and /3 into parts whose sines are in the same ratio. Thus, again, a — ft/3, a — ft'/3 denote arcs forming with a and /3 a pencil whose anharmonic ratio is p . And a — ft/8, a + ft/3 denote arcs forming with a, /3 a harmonic pencil. It may be noted here that if A' be the middle point of an arc AB, then B', the fourth harmonic to A', A and B, is a point distant from A' by 90°. For if we join these points to the centre G, CA' is the internal bisector of the angle A CB, and therefore GB' must be the external bisector. Conversely, if two corresponding points of a harmonic system are distant from each other by 90°, each is equidistant from the other two points of the system. 186 CONES AND SPHEEO-CONICS. It is convenient also to mention here that if xy'z' be the co-ordinates of any point on the sphere, then xx' + yy' + zz' denotes the great circle having xy'z' for its pole. It is in fact the equation of the plane perpendicular to the line joining the centre to the point xy'z . 236. We can now immediately apply to spherical triangles the methods used for plane triangles (Conies, p. 56, &c). Thus if a, /3, 7 denote the three sides, then, as in plane triangles, la = m/3 = ny denote three lines meeting in a point, one of which passes through each of the vertices : while «j/3 + ny — la, ny + la — m/3, la + m/3 — ny are the sides of the triangle formed by connecting the points where each of these joining lines meets the opposite sides of the given triangle ; and la + rnft + ny passes through the inter- sections of corresponding sides of this new triangle and of the given triangle. The equations a = /3 = y evidently represent the three bi- sectors of the angles of the triangle. And if A, B, C be the angles of the triangle, it is easily proved that as in plane triangles a cos^l = /3 cosJB = y cosC denote the three per- pendiculars. It remains true, as at Conies, p. 57, that if the perpendiculars from the vertices of one triangle on the sides of another meet in a point, so will the perpendiculars from the vertices of the second on the sides of the first. The three bisectors of sides are a sin.4 = /3 sinZ?=7 sin (7. The arc a sinA + fi s'mB+y sinC passes through the three points where each side is met by the arc joining the middle points of the other two ; or, again, it passes through the point on each side 90° distant from its middle point, for a s'mA + ft smB meet 7 in two points which are harmonic conjugates with the points in which a, /3 meet them, and since one is the middle point the other must be 90° distant from it (Art. 235). It follows from what has been just said that the point where a miA + fi smB+y sinC meets any side is the pole of the great circle perpendicular to that side, and passing through its middle point, and hence that the intersection of the three such perpendiculars (that is to say, the centre of CONES AND sAERO-CONICS. 187 the circumscribing circle) is the pole of the great circle a sin^ + yg sinB+y sin(7. The equations of the lines joining the vertices of the triangle to the centre of the circumscribing circle are found to be « = ft 7 sm^B+C-A) sm$[0+A-B) sm^{A + B-C)' 287. The condition that two great circles ax + by + cz, dx + b'y + c'z should be perpendicular is manifestly ad + bb' + cc' = 0. The condition that a i_ _ cos 2 %{0 + ff) sm^jd+ff) a" ~ T~ + a" sin# sin#\ Subtracting, we have 1 _1_ _ /I _ V ¥ ~ d* ~ {, x' z + 1/ 2 + z 1 - /3 2 sec 2 /)', and let us form the condition that \8->r S' should break up into factors. This cubic being \ 3 A + X 2 + \& + A' = 0, we have A = — tan 2 /?, A' = — tan 2 p', © =sec 2 /3 sec 2 /)' sin 2 Z) — 2 tan 2 /) —tan 2 /)', ©' = sec 2 /) sec 2 /)' sin 2 J? - 2 tan 2 /)' - tan 2 /), where D is the distance between the centres. Now the corresponding values for two circles in a plane are A = -j- 2 , A' = -r' 2 , = Z> 2 -2r 2 -/ 2 , 0' = D 2 -2r' 2 -V. Hence, if any invariant relation between two circles in a plane is expressed as a function of the radii and of the distance 02 196 CONES AND SPHEEO-CONICS. between their centres, the corresponding relation for circles on a sphere is obtained by substituting for r, r\ D; tan*-, tan/-', and seer sec/ sinZ>. Thus the condition that two circles in a plane should touch is obtained by forming the discriminant of the cubic equation, and is either D = or D = r + r. The corresponding equation therefore for two circles on a sphere is tanr ±tanr' = seer seer' sinD, or sin D = sin [r + r). Again, if two circles in a plane be the one inscribed in, the other circumscribed about the same triangle, the invariant relation is fulfilled © 2 = 4A©', which gives for the distance between their centres the expression IP = R* — 2Er. The distance therefore between the centres of the inscribed and circumscribed circles of a spherical triangle is given by the formula sec 2 i? sec'V sin 2 Z) = tan 2 i? — 2 tani? tanr. So, in like manner, we can get the relation between two circles inscribed in, and circumscribed about the same spherical polygon. 252. The equation of any small circle (or right cone) in trilinear co-ordinates must (Art. 249) be of the form a 2 sin'J. + yS 2 sm'B+y 2 sin 2 G + 2/37 sm -^ sin(7 cosa+27a sin (7 s'mA cos6+ 2a/3 sin.4 sinB cose = (la + mfi + nyy. If now the small circle circumscribe the triangle a/3y, the coefficients of a 2 , /3'\ and y 2 must vanish, and we must therefore have la + raft + wy = a. smA + /3 sinZ? + 7 sin G. Hence, as was proved before, this represents the polar of the centre of the circumscribing circle. Substituting the values, sin.4, sini?, sinC, for I, m, n; the equation of the small circle becomes /3y tan \a + 70c tan \b + a/3 tan \c = 0. The equation of the inscribed circle turns out to be of exactly the same form as in the case of plane triangles, viz. co?,\A V(a) + cos|5 VQ8) + cos£CV(7) = °- C0NE8 AND SPHEEO-CONICS. 197 The tangential equation of a small circle may either be derived by forming the reciprocal of that given at the commencement of this article, or directly from Art. 237, by expressing that the perpendicular from the centre on Xa + /j,f3 + vy is constant. We find thus for the tangential equation of the circle whose centre is a'/3V and radius p sin 2 p (X 2 + ft? + v 3 - 2(jlv cos A - 2vX cos B - 2X//, cos G) = ( a 'A + /3> + 7 Vr; a form also shewing (see Art. 251) that every circle has double contact with the imaginary circle at infinity. 253. As a concluding exercise on the formulas of this chapter, we investigate Dr. Hart's extension of Feuerbach's* theorem for plane triangles, viz. that the four circles which touch the sides are all touched by the same circle. It is easier to work with the tangential equations. The tangential equations of circles which touch the sides of the triangle of reference must want the terms X 2 , /a 2 , v 2 , and there- fore evidently are X 2 + fi 2 + v 2 — 2fMv cosA — 2vX coaB—2X/i cosG = (X + pi ± v)* ; or fiv cos 2 ^4 + vX cos 2 |i?-r X/m cos 2 |(7=0 (1), fivcos^A-vX sm^B-X/i sin 2 £C=0 (2), -fj,v Bm*$A + v\ cos^B-Xfj, sin a £C=0 (3), — fiv &m*%A-vX sin 2 fB + X/i cos 2 |C=0 (4), all which four are touched by the circle (5) X 2 4- /* 2 + v 2 — 2p,v cos J. — 2vX cos 2? — 2Xfi cos G = {X cos(£- G) + p cos( G- A) + v cos(A - B)f. For the centres of similitude of the cirqles (1) and (5) are given by the tangential equations (X+/* + v)±{X cos(B-C) + fi cos(C-^)+vcos(^-5)} = 0, one of them therefore is X sin 2 ^ - G) + p sin 2 £( G - A) + v mf±[A - B). * In tie Conies and elsewhere I hare erroneously ascribed this theorem to Terquem. 198 CONES AND SPHEEO-CONICS. And (Conies, p. 119) the condition that this point should be on the circle (1) is cos £4 sin ^(B- C) + cos \B £m\(Q-A) + cos£<7 %m\(A-B)^% which is satisfied. It is proved, in like manner, that the circle (5) touches the other three circles. The co-ordinates of the point of contact have been proved to be sin^(B-C), sm"i(G-A), aln'^A- B). 254. The co-ordinates of the centre of Dr. Hart's circle have been proved to be cos(B—C), cos(G—A) 1 cos(A — B). This point therefore lies on the line joining the point whose co-ordinates are cosi? cos C, cosO cos A, cos A cosB to the point whose co-ordinates are sin-SsinC, sinCsin^, sin.4 sini?; that is to say, (Art. 236) on the line joining the intersection of per- pendiculars to the intersection of bisectors of sides. Since cos.4 - cos(B - G) = 2 nn$(A + B-C) sin \{C + A - B) ; the centre lies also on the line joining the point cos .4, cos B, cosC to the point sm(S-B)sm[8-C), sin(S-C) sin(S-A), £n{S-A) sin(S-B). The first point is the intersection of lines drawn through each vertex making the same angle with one side that the per- pendicular makes with the other; the second point is the in- tersection of perpendiculars let fall from each vertex on the line joining the middle points of the adjacent sides. The centre of Dr. Hart's circle is thus constructed as the intersection of two known lines. 255. The problem might also have been investigated by the direct equation. We write a sin.4 = a:, &c. so that the equation of the imaginary circle at infinity is U= 0, where JJ= x l + y* + z* + 2yz cos a + 2zx cos b + 2xy cose. Then the equation of the inscribed circle is U= {x cos (s- d)+y cos (s — b) + z cos (s — c)}' 1 , where 2s = a + b + c. For this equation expanded is af sin 2 (s— a) +y' sm\s—b) +z' sm\s—c) — 2yz sin (s-b) sin(s— c) — 2zx sin (s — c) sin (s — a) — 2xy sin (s — a) sin (s — b) = 0. CONES AND SPHEEO-CONICS. 199 TJ is not altered if we change the sign of either a, b, or c. Consequently, we get three other circles also touching x, y, z if we change the signs of either a, b, or c in the equation of the inscribed circle. All four circles will be touched by T j_ (x cos ^5 cos^c y cos^c cos|-a z cos|a cos^J) 2 ( cos|-a cos \b ' cos^c J ' This last equation not being altered by changing the sign of a, b, or c, it is evident that if it touches one it touches all. Now one of its common chords with the inscribed circle is f . . cosiJ cosAcl f , 7 . x < cos (s — a) — - — — > + y \ cos ls—b)- { v ; cos^a ) J { v cos^-c cosset cosAZ> which reduced is x { , . cosia cos ib + z ^ costs - e) — - — — { v cos£c + -=-? r^-7 n + sin(s-J)— sin (s—c) sin(s— c)— sin(s— a) sin(s— a)-sin(s-J) But the condition that the line Ax +By + Cz shall touch »J(ax) + »J{by) + V(c«) is T + » + Ti • -^PP^S tnis condition, the line we are considering will touch the inscribed circle if sin(s — a) {sin(s-S) - sin(s-c)} +sin(s-J){sin(s-e)— sin(s-a)}+sin(s-c){sin(s— a)-sin(s-5)}=0; a condition which is evidently fulfilled. It will be seen that the condition is also fulfilled that the common tangent in ques- tion should touch *J{x) + \%) + V( a ) 5 that is to say, the sphero- conic which touches at the middle points of the sides; a fact remarked by Sir Wm. Hamilton, and which leads at once to a construction for that tangent as the fourth common tangent to two conies which have three known tangents common. The polar of the centre of Dr. Hart's circle has been thus proved to be asm ^ c ^i^l%^ sin 5 C -^^+ 7 sinC C <^ = 5 cos^a cos|6 cos^c or a tan -|a + /3 tan $ + y tan $c = 0, 200 CONES AND SPHEEO-CONICS. which may be also written a cos (S- A) + /3 cos (S - B) + y cos (S- C) = 0, forms which lead to other constructions for the centre of this circle. The radius of the circle touching three others whose centres are known, and whose radii are r, r', r'\ may be formed by substituting r + B, r +B, r" + B for d, e, / in the formulae of Arts. 51, 52, and solving for B. Applying this method to the three escribed circles I have found that the tangent of the radius of Dr. Hart's circle is half the tangent of the radius of the circumscribing circle of the triangle.* * With reference to what was stated (p. 183) as to the intersection of a cone with a sphere, Mr. Cayley has referred me to a paper by Mbbius, in Abhandlungen der K. Scwks. Gesellschaft, "Vol. I., in which he shows that a concentric cone may meet a sphere either in twin curves, that is, in two detached curves opposite to each other ; or may meet in single curves each of which is its own opposite. Thus a great circle is evidently its own opposite. Or, again, if the cone project a plane curve of the third degree consisting of an oval and infinite braneh, the part projecting the oval meets in twin curves ; and the remaining part in a simple curve. In a cone of odd degree the number of single sheets is odd, as easily appears from what was stated about the ovals of plane curves {Higher Plane Curves, p. 200). ( 201 ) CHAPTER XL GENERAL THEORY OF SURFACES. INTEODUCTOET CHAPTEB. 256. Keseeving for a future chapter a more detailed ex- amination of the properties of surfaces in general, we shall in this chapter give an account of such parts of the general theory as can be obtained with least trouble. Let the general equation of a surface be written in the form, A + Bx+Cy + Dz + Ex* + Fy* + Gz a + 'lllyz + 2Kzx + 2Lxy 4 &c. = 0, or, as we shall write it often for shortness, U + U l + M 2 + M 3 + & C * - ) where w 2 means the aggregate of terms of the second degree, &c. Then it is evident that u consists of one term, u x of three, w 2 of six, &c. The total number of terms in the equation is therefore the sum of n+1 terms of the series 1, 3, 6, 10, &c, The number of conditions necessary to determine a surface, of the w 01 degree is one less than this, or = — ^ ' . The equation above written can be thrown into the form of a polar equation by writing p cosa, p cos/8, p COS7, for £c, y, z, when we obviously obtain an equation of the n m degree, which will determine n values of the radius vector answering to any assigned values of the direction-angles a, /3, 7. 202 GENERAL THEORY OF SURFACES. 257. If now the origin be on the surface, we have u = 0, and one of the roots of the equation is always p = 0. But a second root of the equation will be p = if a, /8, y be con- nected by the relation B cosa + G cos/3 + D cosy = 0. Now multiplying this equation by p it becomes Bx+Cy+Dz=0, and we see that it expresses merely that the radius vector must lie in the plane u, = 0. No other condition is necessary in order that the radius should meet the surface in two coincident points. Thus we see that in general through an assumed point on a surface we can draw an infinity of radii vectores which will there meet the surface in two coincident points ; that is to say, an infinity of tangent lines to the surface ; and these lines lie all in one plane, called the tangent plane, determined by the equation u t = 0. 258. The section of any surface made by a tangent plane ■is a curve having the point of contact for a double point.* Every radius vector to the surface, which lies in the tangent plane, is of course also a radius vector to the section made by that plane; and since every such radius vector (Art. 257) meets the section at the origin in two coincident points, the origin is, by definition, a double point (see Higher Plane Curves, p. 27). We have already had an illustration of this in the case of hyperboloids of one sheet, which are met by any tangent plane in a conic having a double point, that is to say, in two right lines.> And the point of contact of the tangent plane to a quadric of any other species is equally to be con- sidered as the intersection of two imaginary right lines. From this article it follows conversely, that any plane meeting a surface in a curve having a double point touches the surface, the double point being the point of contact. If the section have two double points, the plane will be a double * I had supposed that this remark was first made by Mr. Cayley: Gregory's Solid Geometry, p. 132. I am informed however by Professor Cremona that the point had been previously noticed by the Italian geometer, Bedetti, in a memoir read before the Academy of Bologna, 1841. The theorem is a particular case of that of Art. 19-1. GENERAL THEOET OF SURFACES. 203 tangent plane; and if it have three double points, the plane will be a triple tangent plane. Since the equation of a plane contains three constants, it is possible to determine a plane which will satisfy any three conditions, and therefore a finite number of planes can in general be determined which will meet a given surface in a curve having three double points: that is to say, a surface has in general a determinate number of triple tangent planes. It will also have an infinity of double tangent planes, the points of contact lying on a certain curve locus on the surface. The degree of this curve, and the number of triple tangent planes will be subjects of investi- gation hereafter. 259. Through an assumed point on a surface it is generally possible to draw two lines which shall there meet the surface in three coincident points. In order that the radius vector may meet the surface in three coincident points, we must not only, as in Art. 257, have the condition fulfilled B cosa -I- G cos/3 + D cosy = 0, but also E cos 2 a -+ F cos 2 ,8 + G cos 2 7 + 2H cos/3 cosy + 2if cosy cosa + 2L cosa cos/3 = 0. For if these conditions were fulfilled, A being already supposed to vanish, the equation of the n" degree which determines p, becomes divisible by p 3 , and has therefore three roots = 0. The first condition expresses that the radius vector must lie in the tangent plane u t . The second expresses that the radius vector must lie in the surface w ? = 0, or Ex 2 + Fy* + Gz* + 2Hyz + 2Ksx + 2Lxy = 0. This surface is a cone of the second degree (Art. 62) and since every such cone is met by a plane passing through its vertex in two right lines, two right lines can be found to fulfil the required conditions. Every plane (besides the tangent plane) drawn through either of these lines, meets the surface in a section having the point of contact for a point of inflexion. For a point of 204 GENERAL THEORY OF SURFACES. inflexion is a point, the tangent at which meets the curve in three coincident points (Higher Plane Curves, p. 35). On this account we shall call the two lines which meet the surface in three coincident points, the inflexional tangents at the point. The existence of these two lines may be otherwise perceived thus. We have proved that the point of contact is a double point in the section made by the tangent plane. And it has been proved (Higher Plane Curves, p. 28) that at a double point can always be drawn two lines meeting the section (and therefore the surface) in three coincident points. 260. A double point may be one of three different kinds according as the tangents at it are real, coincident, or imaginary. Accordingly the contact of a plane with a surface may be of three kinds according as the tangent plane meets it in a section having a node, a cusp, or a conjugate point; or in other words, according as the inflexional tangents are real, coincident, or imaginary. Dupin, who first noticed* the difference between these three kinds of contact, stated the matter as follows : Suppose that we confine our attention to points so near the origin that all powers of the co-ordinates above the second may be neglected, then the tangent plane (or a very near plane parallel to it) meets any surface u t + u 2 + u 3 + &c. in the same section in which it meets the quadric ii t + u 2 . And according as the sections of this quadric by planes parallel to the tangent plane are ellipses, hyperbolas, or parabolas, so the section made by the tangent plane is to be considered as an infinitely small ellipse, hyperbola, or parabola. This infinitely small section Dupin calls the indicatrix at the point of contact, and he divides the points of the surface, according to the nature of the in- dicatrix into elliptic, hyperbolic, and parabolic points. We shall presently show that there will be in general on every surface a number of parabolic points forming a curve locus, this curve separating the elliptic from the hyperbolic points. * See Dupin's Dereloppements de Gecmetrie, p. 48. GENERAL THEORY OF SURFACES. 205 If the tangent plane be made the plane of xy, and the equa- tion of the surface be z + Ax* + 2Bxy + Cif + 2Bxz + 2Eyz + Fz* + &c. = 0, it is manifest that the origin will be an elliptic, hyperbolic, or parabolic point, according as B' 1 is less, greater than, or equal to AC* « 261. Knowing the equation of the tangent plane when the origin is on the surface, we can, by transformation of co-ordinates, find the equation of the tangent plane at any point. It is proved precisely as at Art. 58 that this equation may be written in either of the forms , ,. dU' , . ,. dU' , ,, dU' n dU' dU' dU' dU' n x n7 + yw +z ^ r+w M = 262. Let it be required now to find the tangent plane at a point, indefinitely near the origin, on the surface z + Ax* + 2Bxy 4- Gy l + 2Bxz + 2JEyz + Fz* + &c. = 0. We have to suppose x, y so small that their squares may be neglected ; while, since the consecutive point is on the tangent plane, we have 3' = 0: or, more accurately, the equation of the surface shows that z is a quantity of the same order as the squares of x and y'. Then, either by the formula of the last article, or else directly by putting x-\-x\ y+y' for x and y, and taking the linear part of the transformed equation, the equation of a consecutive tangent plane is found to be z + 2 (Ax -f By') x + 2 (Bx + Cy) y = 0. * This is sometimes expressed as follows : When the plane of xy is the tangent plane, and the equation of the surface is expressed in the form z =

a v + -^ x-y A 2 U' + &c. = 0, where A represents the operation d d d d dx dy' dz dw' ' Following the analogy of plane curves we shall call the surface represented by the first polar of the point x'y'z'w. We shall call f , d , Z7/ = 0, we see that if all four differentials be made to vanish by the co-ordinates of any point, then every line through the point meets the surface in two coincident points ; and the point is therefore a double point. The condition that a given surface may have a double point is obtained by eliminating the variables between the four equations ff = 0, &c, and is called the dis- criminant of the given surface [Lessons on Higher Algebra, page 43). The discriminant being the result of elimination between four equations, each of the degree n — 1, contains the coefficients of each in the degree [n — lf, and is therefore of the degree 4 (n — l) 3 in the coefficients of the original equation. It is obvious from what has been said, that when a surface has a double point, the first polar of every point passes through the double point. 216 OENERAL THEORY OF SURFACES. The surfaces represented by Z7„ Z7 2 , &c, may happen not merely to have points in common, but to have a whole curve common to all four surfaces. This curve will then be a double curve on the surface U, and every point of it will be a double point. Now we saw (Art. 258) that the surface represented by the general Cartesian equation of the n th degree will, in general, have an infinity of double tangent planes ; the re- ciprocal surface therefore will, in general, have an infinity of double points, which will be ranged on a certain curve. The existence then of these double curves is to be regarded among the " ordinary singularities" of surfaces (see Higher Plane Curves, page 47). When the point x'y'z'w is a double point, U' and AU' vanish identically ; and any line through the double point meets the surface in three consecutive points if it satisfies the equation A' 2 U' = 0, which represents a cone of the second degree. 278. The polar quadric of a parabolic point on a surface is a cone. The polar quadric of the origin with regard to any surface n , 71—1 , n-2 i f /\ UJlO + U L W + U.JJD + &C. = 0, (where, as in Art. 266, we have introduced w so as to make the equation homogeneous) is found by differentiating n — 2 times with respect to w. Dividing out by (n~ 2) [n — 3). ..3, and making ie = l, the polar quadric is n (n - 1) w + 2 [n - 1) u, + 2w a = 0. Now the origin being a parabolic point, we have seen, Art. 260, that the equation is of the form z + Cf + 2Dzx + 2Ezy + Fz* + &c, [or, in other words, w = 0, and u 2 is of the form u l v 1 + w^]. The polar quadric then is e(n-l + 2Dx + 2Ey + Fz) + Cf = 0. But we have seen (page 40) that any equation represents a cone when it is a homogeneous function of three quantities, each of the first degree. The equation just written therefore represents a cone whose vertex is the intersection of the three GENERAL THEORlToF SURFACES. 217 planes, s, n — 1 + Wx + lEy + Fz, and y. The two former planes are tangent planes to this cone, and y the plane of contact. -79. It follows from the last article that if we form the locus of points whose polar quadrics represent cones, this will meet the surface in the parabolic points. This locus is found by writing down the discriminant of A 2 U' = 0. If a, d' 2 V d*U' b, &c, denote the second differential coefficients , „ , -}-& » &c, the discriminant will be (page 40) abcd+ lalqr -f 2bmpr + 2cnpq + Idhnn — adV — bdm* — cdn* — bep* — caq* — abr" + Pp* -f m*q 2 + n'r' — Imnqr — 2nhp — 2lmpq = 0. This denotes a surface of the degree 4 (n — 2), which we shall call the Hessian of the given surface. In the same manner then as the intersection of a plane curve with its Hessian de- termines the points of inflexion, so the intersection of a surface with its Hessian determines a curve of the degree in (n — 2), which is the locus of parabolic points (see Art. 263.) 280. It follows from what has been just proved that through a given point can be drawn An (n — \) (n — 2) stationary tangent planes (see Art. 263). For since the tangent plane passes through a fixed point, its point of contact lies on the polar surface, whose degree is n — 1 ; and the intersection of this sur- face with the surface U, and the surface determined in the last article as the locus of points of contact of stationary tangent planes, determine -in [n — 1) {n — 2) points. Otherwise thus ; the stationary tangent planes to the surface through any point are also stationary tangent planes to the tangent cone through that point, and if the cone be cut by any plane, these planes meet it in the tangents at the points of inflexion of the section. But the number of points of in- flexion on a plane curve is determined by the formula (Higher Plane Curves, page 91) i — « = 3 (v — /*). But in this case, Art. 273, we have v = n (•»- 1) ! , fi = n (n- 1) ; 218 CURVATURE OF SURFACES. therefore v — fi = n (n — 1) (n — 2), K — n(n— 1) (n — 2). Hence, as before, i = in (n — 1) (n — 2). The number of double tangent planes to the cone is de- termined by the formula 2(t-8) = (v-/*)(v+p-9), and 2$ = «(ra-l)(«-2)(n-3); (v + /i- 9) = w 3 -n !! -9. Hence 2t = ji (m— 1) (n — 2) (n 3 — ri* + n— 12). It follows then that through any point can be drawn t double tangent planes to the surface, where t is the number just de- termined. It will be proved hereafter, that the points of contact of double tangent planes lie on the intersection of the surface with one whose degree is (n — 2) (« 3 — « 2 + n — 12). 281. If a right line lie altogether in a surface it will touch the Hessian and therefore the parabolic curve, (Cambridge and Dublin Mathematical Journal, Vol. IV., p. 255). Let the equation of the surface be xcf> + y-^r = 0, and let us seek the result of making x and y = in the equation of the Hessian, so as thus to find the points where the line meets - ., . d*U d'U d"U „ that surface. JNow evidently -^-j- , -^--. , -= — =— , all contain J dz ' dw dzdw x or y as a factor, and therefore vanish on this supposition. And if we make c = 0, d = 0, r = in the equation of the Hessian, it becomes a perfect square (Ip — mof, showing that the right line touches the Hessian at every point where it meets it. If we make x = 0, y = in Ip — ma, it reduces to -r ~r- — ¥ i ■ It is evident that when the taDErent plane dz dw dw dz touches all along any line, straight or curved, this line lies altogether in the Hessian. The reader can verify this without difficulty, with regard to the surface x

(which is done by substituting e cos—y sin$ for z, and z §m.$-\-y cos<£ for y). If we now make the new ?/ = 0, we shall get the equation (still to rectangular axes) of the section by a plane making an angle with the old plane y = 0, but still passing through the old axis of x ; and this equation will plainly be z cos(j> + Ax' + iBxz sin<£ + Gz l sin 2 + Wxz cos(f> + 2Ez' z sin0 + Fz l cos' + &c, * The illustration of the summit of a mountain pass will enable the reader to conceive how a surface may in two directions sink below the tangent plane, and on the other sides rise above it. The shape of a saddle affords another familiar illustra- tion of the same thing. CURVATURE OP SURFACES. 223 and by the same method as before the radius of curvature is found to be , or is =Bcos(f>, where E is the radius of curvature of the corresponding normal section. This is Meunier's theorem, that the radius of curvature of an oblique section is equal to the projection on the plane of this section of the radius of curvature of a normal section passing through the same tangent line. Thus we see that of all sections which can be made through any line drawn in the tangent plane, the normal section is that whose radius of curvature is greatest; that is to say, the normal section is that which is least curved and which approaches most nearly to a straight line. Meunier's theorem has been already proved in the case of a quadric (see p. 140), and we might therefore, if we had chosen, have dispensed with giving a new proof now; for we have seen that the radius of curvature of any section of M i + % + M s + & c - i s the same as that of the corresponding section of the quadric u l + «,,. 288. It was proved (Art. 194) that if two surfaces u 1 +u i +&c. J «i + « 2 4- &c. touch, their curve of intersection has a double point, the two tangents at which are the intersections of the plane u x with the cone u 2 - t> 2 . When the plane touches the cone, the surfaces have what we have called stationary contact. It is also proved, as at Art. 196, that a sphere has stationary contact with a surface when the centre is on the normal and the radius equal to one of the principal radii of curvature. In fact, the condition for stationary contact between z + ax 2 + 2nxy f by* + &c, z + a'x* + 2n'xy + b'y 2 + &c. is (a-a')(b-b') = (n-n'y, which when n and n both vanish implies either a=a' or b = b'. The surface therefore z + Ax* + Cy* + &c. will have stationary contact with the sphere 2rz + x* -{- y* -{■ z* if r = — -. or — -,; but these are the values of the principal radii. 289. The principles laid down in the last article enable us to find an expression for the values of the principal radii at any point; the axes of co-ordinates having any position. 224 CURVATURE OF SURFACES. If we transform the equation to any point x'y'z on the surface as origin, it becomes dU' dU' dU' 1 ( d d d\\ T , , X a^ + ^W +Z '^ r+ ^[ X d^' + lJ d^ + Z d7) U+&C - or if we denote the first differential coefficients by L, M, iV, and the second by a, b, c, &c. 2 (Lx+My+Nz) + ax* + by 2 + cz 2 + 2lyz + 2msx + 2nxy + &c. = 0. The equation then of any sphere having the same tangent plane is 2(Lx + My + Nz) + X(x 2 + y 2 -+z 2 ) = 0, and the sphere will have stationary contact with the quadric if X be determined so as to satisfy the condition that Lx+My+Nz shall touch (a — X) x* + (b — X) y 2 + (c — X) z 2 + 2lyz + 2mzx + 2nxy. This condition is ot, L 1, M c-X, N N =0, which expanded is {(j_X)( c -X)-r} J L 2 +{(o-X)(a-X)- m '' ! }ilf !! +{(a-X)(J-X)-« 2 }iV ra +2{rnn-(a-X)l}MN+2{nl-(b-X)m}NL-t2{lm-(c-X)n}LM=0, or X is given by the quadratic (L 2 + M 2 + N 2 ) X 2 - {(b + c) L 2 + (c + a) M 2 + (a + b) N 2 - 21MN- 2mNL - 2nLM] X + {be - V) L 2 + (ca - m 2 ) M 2 + (ab - ri*) N 2 + 2 (mn - al) MN+ 2 [nl-bm) NL + 2(lm- en) LM= 0. Now if r be the radius of the sphere X (x 2 + y 2 + z 2 ) +2 (Lx + My + Nz), we have r 2 = r-g . We therefore find the principal X If Ti , JJT2 , XTV\ radii by substituting — for X in the preceding quadratic. a — X, n, n, b-X, m, h L, M, CURVATURE OF SURFACES. 225 The absolute term in the equation for X may be simplified by writing for L ) il/", N their values from the equations (n — 1) L = ax ■+ ny + mz +pw, &c, when the absolute term reduces to — -, where S is the («-l) Hessian, written at full length, Art. 279. We might have seen a prion' that for any point on the Hessian, the absolute term must vanish. For since the directions of the principal sections bisect the angles between the inflexional tangents ; when the inflexional tangents coincide, one of the principal sections coin- cides with their common direction, and the radius of curvature of this section is infinite, since three consecutive points are on a right line. Hence one of the values of X (which is the reciprocal of r) must vanish. By equating to nothing the coefficient of \ in the preceding quadratic, we obtain the equation of a surface of the degree 3« — 4, which intersects the given surface in all the points where the principal radii are equal and opposite : that is to say, where the indicatrix is an equilateral hyperbola. The quadratic of this article might also have been found at once by Art. 98, which gives tbe axes of a section of the quadric ax 2 + by* + cz' 2 + '2hjz + 2mzx + 2nxy = 1 made parallel to the plane Lx + My + A; = 0. 290. From the equations of the last article we can find the radius of curvature of any normal section meeting the tangent plane in a line whose direction-angles are given. For the centre of curvature lies on the normal, and if we describe a sphere with this centre, and radius equal to the radius of curvature, it must touch the surface, and its equation is of tbe form 2 (Lx + My + Xz) + \ (x* + y s + s 2 ) = 0. The consecutive point on that section of the surface which we are considering satisfies this equation, and also the equation «! + « a = °> 2 (Lx + My + Nz) 4 ax* + by* + cz' + 2bjz + 2msx 4- 2nxy = 0. Q 226 CURVATURE OF SURFACES. ' Subtracting, we find ax* + by* + cz* + 2 lyz + 2mzx + 2nxy x'+y* + z* And since this equation is homogeneous, we may write for x, y, z the direction-cosines of the line joining the consecutive A . . , . , s ^(L* + M* + N) point to the origin. As in the last article A, = — . Hence */(L' + M* + N*) a cos 2 a+&cos' 2 /3+ccos s 7-+ 2£cos/3cos7+2mcos7cosa+2wcosa cos/3 * The problem to find the maximum and minimum radius of curvature is therefore to make the quantity ax 1 + by* + cz* + llyz + 2mzx + 2nxy a maximum or minimum, subject to the relations Lx + My + Nz = 0, a?+y*+s*=l. And thus we see again that this is exactly the same problem as that of finding the axes of the central section of a quadric by a plane Lx + My + Nz. 291. In like manner the problem to find the directions of the principal sections at any point is the same as to find the directions of the axes of the sectioD by the plane Lx + My + Nz of the quadric ax" + by* +- cz* + 2lyz 4 2mzx ■+ 2nxy = 1. Now given any diameter of a quadric, one section can be drawn through it having that diameter for an axis; the other axis being plainly the intersection of the plane perpen- dicular to the given diameter with the plane conjugate to it. Thus if the central quadric be U= 1, and the given diameter pass through x'y'z', then the diameter perpendicular and con- jugate is the intersection of the planes xx'+yy' + zz' = 0, x'U^+y'U^ s'Z7 3 = 0. If the former diameter lie in a plane Lx' ■+ My' + Nz\ the latter diameter traces out the cone which is represented by the determinant obtained on eliminating x'y'z' from the three preceding equations: viz. [Ms - Ny) TJ X + (Nx - Lz) U 2 + (Ly - Mx) U 3 = 0. CURVATURE OF SURFACES. 227 And this coDe must evidently meet the plane Lx + My + Nz in the axes of the section hy that plane. Thus then the directions of the principal sections are determined as the inter- section of the tangent plane Lx + My + Nz with the cone (Mm — Ny) (ax + ny + mz) + (Nx — Lz) (nx + by + Iz) + (Ly — Mx) (rnx + ly + cz) = 0, or (Mm - Nn) x> + (Nn - LI) f + (LI - Mm) s a + {L(b-c)- nM+ mN) yz + [Ln + M(o-a)- NT] zx + {- Lm + Ml+N(a - b)} xy = 0. 292. The methods used in Art. 289 enable us also easily to find the conditions for an umbilic* If the plane of xy be the tangent plane at an umbilic the equation of the surface is of the form z + A (x* + f) + 2Dxz + 2Eyz + Fz> + &c. = ; and if we subtract from it the equation of any touching sphere, viz. z + \(x i + y* + z' I )=0, it is evidently possible so to choose X (namely, by taking it = A) that all the terms in the remainder shall be divisible by z. We see thus that if u x + u 2 -f &c. represent the surface, and m, + Xu 2 any touching sphere, it is possible, when the origin is an umbilic, so to choose X that w 2 — \v 2 may contain Mj as a factor. We see then by transformation of co-ordinates as in Art. 289, that any point x'y'z will be an umbilic if it is possible so to choose X that (a - X) x" + (b-\)y* + (c- X) z 2 + 2lyz + 2mzx + 2nxy may contain as a factor Lx + My + Nz. If so, the other factor must be a — X b — X c — X * We might find the condition for an umbilic by forming the condition that the quadratic of Art. 289 should have equal roots. But, as at p. 51, this quadratic having its roots always real is one of the class discussed, Higher Algebra, p. 134; whose dis- criminant can be expressed as the sum of squares. If therefore we only consider real tunbilics, the result of equating the discriminant to nothing is equivalent to two conditions, which can be more easily obtained as in the text. Q2 228 CUEVATUEE OF SURFACES. Multiplying out and comparing the coefficients of yz, zx, xy, we get the conditions ( & - x )^-+( c - x )^= 2Z ) (c-\)-^+(a-X)j i = 2m i (a-\) J+(J-\)^=2b. Eliminating ~k between these equations we obtain for an umbilic the two conditions bN*+cM'-2lMN _ cL i + aN*-2mLN _ aM* + I V- 2 nLM Since there are only two conditions to be satisfied, a surface of the n th degree has in general a determinate number of umbilics ; for the two conditions, each of which represents a surface, combined with the equation of the given surface de- termine a certain number of points. It may happen however that the surfaces represented by the two conditions intersect in a curve which lies (either wholly or in part) on the given surface. In such a case there will be on the given surface a line, every point of which will be an umbilic. Such a line is called a line of spherical curvature. 293. There is one case in which the conditions of the last article are not applicable in the form in which we have written them. They appear to be satisfied by making L = 0, bW i + cM 2 -2U[N o = r™ — 575 ? whence we might conclude that the N* + M* ° surface L = must always pass through umbilics on the given surface. Now it is easy to see geometrically that this is not the case, for L (or Z7j) is the polar of the point yzw with respect to the surface, so that if L necessarily passed through umbilics it would follow by transformation of co-ordinates that the first polar of every point passes through umbilics. On referring to the last article, however, it will be seen that the investigation tacitly assumes that none of the quantities L, M, N vanish ; for if any of them did vanish, some of the equations which we have used would contain infinite terms. Supposing CURVATURE OF SURFACES. 229 then L to vanish, we must examine directly the condition that My + Nz may be a factor in (a - X) * 2 + (b - X) y' + (c - X) s s + Hyz + imzx + 2nxy. We must evidently have X = a, and it is then easily seen that bN* + cJ\P - 2L1AV , „ . we must, as betore, have a = ™ — Tr; , while m jst z -t ip addition, since the terms Imzx + 2nxy must be divisible by My + JSfz, we must have Mm = Nn. Combining then with the two conditions here found, L = 0, and the equation of the surface, there are four conditions which, except in special cases, cannot be satisfied by the co-ordinates of any points. If we clear of fractions the conditions given in the last article, it will be found that they each contain either L, M, or JV^ as a factor. And what we have proved in this article is that these factors may be suppressed as irrelevant to the question of umbilics.* We now proceed to draw some other inferences from what was proved (Art. 288) ; namely, that the two principal spheres have stationary contact with the surface. 294. JVIien two surfaces have stationary contact, they touch in two consecutive points. The equations of the two surfaces being s + ax* + Inxy + by 3 + &c. = 0, z + a'x' + in'xy + b'y 2 + &c, * From what has been said we can infer the number of umbilics which a surface of the n 01 degree will in general possess. We have seen that the umbilics are deter- mined as the intersection of the given surface with a curve whose equations are of the form — ,= „ = —, . Now if A, B, C be of the degree ?, and A', B", C" of the A Jr C degree m, then AB' - BA', AC - CA' are each of the degree I + m, and intersect in a curve of the degree (1 + i»)». But the intersection of these two surfaces includes the curve A A' of the degree An which does not he on the surface BC — CB'. The degree therefore of the curve common to the three surfaces is P +Jm + 111-. In the present case / = 3» — 4. nt = 2m - 2, and the degree of the curve would seem to be 19„2 _ 46n + 2S. But we have seen that the system we are discussing includes three curves such as L, a (M- + .V 3 ) - {!>X- + cJP - 2LVX) which do not pass through umbilics. Subtracting therefore from the number just found 3 (» — 1) (3» — 4), we see that the umbihcs are determined as the intersection of the given surface with a curve of the degree (10h 2 - 2an + 16), and therefore that the number of umbilics is in general » (10t> 2 - 25» + 16). 230 CUKVATUKE OF SURFACES. the tangent planes at a consecutive point are (Art. 262) s + 2 (ax + ny') x + 2 (nx' + by') y = 0, z + 2 (ax + m'?/') a; + 2 (nV + b'y') y = 0. That these may be identical, we must have ax + ny = ax + ny ', nx + by = nx + b'y', and eliminating £c' : y' between these equations, we have (a — a) (b — b') = (n — m') 2 , which is the condition for stationary contact. The sphere, therefore, whose radius is equal to one of the principal radii touches the surface in two consecutive points; or two consecutive normals to the surface are also normals to the sphere, and consequently intersect in its centre. Now we know that in plane curves the centre of the circle of curvature may be regarded as the intersection of two consecutive normals to the curve. In surfaces the normal at any point will not meet the normal at a consecutive point taken arbitrarily. But we see here that if the consecutive point be taken in the direction of either of the principal sections, the two consecutive normals will intersect, and their common length will be the corresponding principal radius. On account of the importance of this theorem we give a direct investigation of it. 295. To find in what cases the normal at any point on a surface is intersected by a consecutive normal. Take the tangent plane for the plane of xy, and let the equation of the surface be z + Ax 2 + 2Bxy + Gtf + 2Dxz + ZEyz + Fz' 2 + &c. = 0. Then we have seen (Art. 262) that the equation of a consecutive tangent plane is z + 2 (Ax + By') x+2 (Bx' + Cy') y-0, and a perpendicular to this through the point x'y' will be a*- 3 ** _ y-y' _ 9 , Ax'+By' Bx' + Cy' This will meet the axis of z (which was the original normal) if x' y' Ax' + By' ~ Bx' + Cy" CURVATURE OF SURFACES. 231 The direction therefore of a consecutive point whose normal meets the given normal is determined by the equation Bx* + {0 - A) x'y' - By" = 0. But this is the same equation (Art. 285) which determines the directions of maximum and minimum curvature. At any point on a surface therefore there are two directions, at right angles to each other, such that the normal at a consecutive point taken on either, intersects the original normal. And these directions are those of the two principal sections at the point. Taking for greater simplicity the directions of the principal sections as axes of co-ordinates ; that is to say, making .5 = in the preceding equations, the equation of a consecutive normal becomes , = ' , = 2z, whence it is easy to see that the normals corresponding to the points y' = 0, x' = intersect the axis of z at distances respectively z = —r , z = —= . The inter- 2ji 2 cepts therefore on a normal by the two consecutive ones which intersect it are equal to the principal radii.* 296. We may also arrive at the same conclusions by seek- ing the locus of points on a surface, the normals at which meet a fixed normal which we take for axis of z. Making x = 0, y = in the equation of any other normal we see that the point where it meets the surface must satisfy the condition jY = 4f. The curve where this surface meets the given surface has the extremity of the given normal for a double * M. Bertrami, in his theory of the curvature of surfaces, calculates the angle made by the consecutive normal with the plane containing the original normal and the consecutive point x'y'. Supposing still the directions of the principal sec- tions to be axes of co-ordinates, the direction-cosines of the consecutive normal are proportional to 2.4a;', 2Ci/, while those of » tangent line perpendicular to the radius vector are proportional to — y', x', 0. Hence the cosine of the angle between these two lines, or the sine of the angle which the consecutive normal makes with the normal section, is proportional to (C — A) x'y'; or, if a be the angle which the direction of the consecutive point makes with one of the principal tangents, is proportional to (C — A) sin la. When a = or = 90°, this angle vanishes and the consecutive normal is in the plane of the original normal. 232 CURVATURE OF SURFACES. point, the two tangents to which are the two principal tangents to the surface at that point. (See Ex. 9, p. 83). The special case where the fixed normal is one at an umbilic deserves notice. The equation of the surface heing of the form z + A (x 2 + f) + &c. = 0, the lowest terms in the equa- tion xU i =yU„ when we make 2 = 0, will be of the third degree, and the umbilic is a triple point on the curve locus. Thus while every normal immediately consecutive to the normal at the umbilic meets the latter normal, there are three directions along any of which the next following normal will also meet the normal at the umbilic* 297. A line of curvature^ on a surface is a line traced on it such that the normals at any two consecutive points of it intersect. Thus starting with any point if on a surface, we may go on to either of the two consecutive points iV, N' whose normals were proved to intersect the normal at M. The normal at N, again, is intersected by the consecutive normals at two points P, P', the element NP being a continuation of the element MN while the element NP' is approximately per- pendicular to it. In like manner we might pass from the point P to another consecutive point Q and so have a line of curva- ture MNPQ. But we might evidently have pursued the same process had we started in the direction MN'. H«nce, at any point M on a surface can be drawn two lines of curvature ; these cut at right angles and are touched by the two " prin- cipal tangents" at M. A line of curvature will ordinarily not be a plane curve, and even in the special case where it is plane * Sir W. R. Hamilton has pointed out (Elements of Quaternions, Art. 411) how this is verified in the case of a quadric. He has proved that the two imaginary generators (see p. 98) through any umbilic are lines of curvature, the third line of curvature through the umbilic being the principal section in which it lies. In fact for a point on *t principal section, the cone (Ex. 9, p. 83) breaks up into two planes. The normal therefore at such a point only meets the normals at the points of the prin- cipal section, and at the points of another plane section. For the umbilic the latter plane is a tangent plane and the section reduces to the two imaginary generators. The normals along either lie in the same imaginary plane. At every point on either generator, distinct from the umbilic, the two directions of curvature coincide with the line, which is perpendicular to itself. (Conies, p. 336). t The whole theory of lines of curvature, umbilics, &c. is due to Monge. See his "Application de 1' Analyse a la Geometrie," p. 124, Liouville's Edition. CURVATURE OF SURFACES. 233 it need not coincide with a principal normal section at M, though it must touch such a section. For the principal section must be normal to the surface, and the line of curvature may be oblique. A very good illustration of lines of curvature is afforded by the case of the surfaces generated by the revolution of any plane curve round an axis in its plane. At any point P of such a surface one line of curvature is the plane section passing through P and through the axis, or, in other words, is the generating curve which passes through P. For all the normals to this curve are also normals to the surface, and being in one plane, they intersect. The corresponding principal radius at P is evidently the radius of curvature of the plane section at the same point. The other line of curvature at P is the circle which is the section made by a plane drawn through P perpendicular to the axis of the surface ; for the normals at all the points of this section evidently intersect the axis of tho surface at the same point, and therefore intersect each other. The intercept on the normal between P and the axis is plainly the second principal radius of the surface. The generating curve which passes through P is a prin- cipal section of the surface, since it contains the normal and touches a line of curvature ; but the section perpendicular to the axis is not a principal section because it does not contain the normal at P. The second principal section at that point would be the plane section drawn through the normal at P and through tho tangent to the circle described by P. The example chosen serves also to illustrate Meunier's theorem; for tho radius of the circle described by P (which, as we have seen, is an oblique section of the surface) is the projection on that plane of the intercept on the normal between P and the axis, and we have just proved that this intercept is the radius, of curvature of the corresponding normal section. 298. It was proved (Art. 291) that the direction-cosines of the tangent line to a principal section fulfil the relation (J/COS7— iVcos/3) (a cosa + H cos/3 + »? cos 7) + (iVcosa — L COS7) [n cosa + b cos/3 + I C0S7) + [L cos/8 - M cosa) {m cosa + I cos/3 + c COS7) = 0. 234 CUEVATUEE OF SUEFACES.' Now the tangent line to a principal section is also the tangent to the line of curvature; while, if ds be the element of the arc of any curve, the projections of that element upon the three axes being dx, dy, dz, it is evident that the cosines of the angles which ds makes with the axes are -7- , ~, - r . ds ds 7 ds The differential equation of the line of curvature is therefore got by writing dx, dy, dz for cosa, cos/3, cosy in the preceding formula. This equation may also be found directly as follows (see Gregory's Solid Geometry, p. 256) : Let «, /3, 7 be the co- ordinates of a point common to two consecutive normals. Then, if xyz be the point where the first normal meets the surface, by the equations of the normal we have — == — = — ~- = — r^- : or if we call the common value of L M N these fractions 6, we have a = x+L8, /3 = y + M8, y = z+W. But if the second normal meet the surface in a point x + dx, y + dy, z + dz, then expressing that a/87 satisfies the equations of the second normal, we get the same results as if we differen- tiate the preceding equations, considering a/3y as constant, or dx + Ld6 + 0dL = 0,dy + Md6 + 6dM= 0, dz + NdO + 6dN= 0, from which equations eliminating 0, dQ, we have the same determinant as in Art. 291, viz. dx, dy, dz L, M, N dL, dM, dN = 0. Of course dL =adx+ ndy + mdz, dM= ndx + bdy + Idz, dN= mdx+ Idy + cdz. Ex. To find the differential equation of the lines of curvature of the ellipsoid 7vr dz c x° y' z' , Here we have L-- M- 1 *=;, az.-% dM~% CUEVATUEE OF SUEFACES. 235 Substituting these values in the preceding equation it becomes, when expanded, (5" - c 2 ) xdydz + (c 2 - a 2 ) ydzdx + (a 2 - tf) zdxdy = 0. Knowing as we do that the lines of curvature are the inter- sections of the ellipsoid with a system of concentric quadrics (Art. 187), it would be easy to assume for the integral of this equation Ax 1 + By* + Cz' z = 0, and to determine the constants by actual substitution. If we assume nothing as to the form of the integral we can eliminate z and dz by the help of the equation of the surface, and so get a differential equation in two variables which is the equation of the projection of the lines of curvature on the plane of xy. Thus, in the present case, multiplying by -5 and reducing by the equation of the ellipsoid and its differential, we have {{b*-6>)xay+{c*-a*)ydx}\^ + ?^J = (a 2 -& 2 ) jl- J - |] 2 -c*) . a 2 (a 2 -J 2 ) orwntmg ^^r^f-^ -^r 1 ^£)H**-W-B) d -l-xy = 0, the integral of which (see Boole's Differential Equations, Ex. 3, p. 135) is aa, r ' x y 1 B~BC = AO+1 1 or the lines of curvature are projected on the principal plane into a series of conies whose axes a', b' are connected by the relation a *(a*-c 2 ) b K {b*-c?) _ a*(a 2 -F) + F(b*-a' J )~ It is not difficult to see that this coincides with the account given of the lines of curvature in Art. 187. 299. The theorem that confocal quadrics intersect in lines of curvature is a particular case of a theorem due to Dupin,* * Developpements de Geome'trie, cinqui&ne Meinoire. The demonstration here given is by Professor W. Thomson : see Gregory's Solid Geometry, p. 2(33. Cambridge Mathematical Journal, Vol. IV., p. 62. 236 CURVATURE OF SURFACES. which we shall state as follows: If three surfaces intersect at right angles, and if each pair also intersect at right angles at their next consecutive common point, then the directions of the intersections are the directions of the lines of curvature on each. Take the point common to all three surfaces as origin, and the three rectangular tangent planes as co-ordinate planes ; then the equations of the surfaces are of the form x + ay* + 2byz + cz l + 2dzx + &c. = 0, y + a'z 2 + 2b' zx + ex 1 + 2d'zy + &c. = 0, s + a"x 2 + 2b" xy + c'y' + &c. = 0. At a consecutive point common to the first and second surfaces, we must have x = 0, y = 0, z = z' where z is very small. The consecutive tangent planes are (1 + 2dz) x + 2bz'y + 2cz'z = 0, 2b' zx + (1 + 2d'z) y + 2a Ys = 0. Forming the condition that these should be at right angles and only attending to the terms where z' is of the first degree, we have b + V = 0. In like manner, in order that the other pairs of surfaces may cut at right angles at a consecutive point, we must have b' + b" = 0, b" + b = 0, and the three equations cannot be ful- filled unless we have b, b', b" each separately = ; in whiph case the form of the equations shows (Art. 295) that the axes are the directions of the lines of curvature on each. Hence follows the theorem in the form given by Dupin ; namely, that if there be three systems of surfaces, such that every surface of one system is cut at right angles by all the surfaces of the other two systems, then ilie intersection of two surfaces belonging to different systems is a line of curvature on each. For, at each point of it, it is, by hypothesis, possible to draw a third surface cutting both at right angles. 300. If two surfaces cut at right angles,* and if their inter- section is a line of curvature on one, it is also a line of curvature on the other. * This is also true if they cut at any constant angle. CURVATURE OP SURFACES. 237 Proceeding as in the last article, and taking the origin at any point of their intersection, we must, in order that they may cut at right angles, have b + V = 0, whence if b = 0, b' = 0. Otherwise thus : the direction-cosines of the tangent planes of the two surfaces heing proportional to L, M, N; L', M', N' ; the direction-cosines of their line of intersection are propor- tional to MN'-M'N, NL'-N'L, LM'-L'M; and in order that this intersection should be the direction of a line of curva- ture on the first surface, we must have the condition fulfilled (Art. 298) MN'-M'N, NL'-N'L, LM'-ML' L, M, N I dL, dM, dN = 0, which expanded is (LL' + MM' + NN') (LdL + MdM+ NdN) - (L* + ill- + N*) [L'dL + M'dM+ N'dN) = 0. If the two surfaces are at right angles, we have LL' + MM' + NN' = 0, and the condition just written reduces to L'dL + M'dM+ N'dN= 0, from which two equations we infer LdL' + MdM' ■+ NdN' = ; but this is the condition that the line of intersection should be a line of curvature on the second surface. 301. A line of curvature is, by definition, such that the normals to the surface at two consecutive points of it intersect each other. If then we consider the surface gene- rated by all the normals along a line of curvature, this will be a developable surface (Note, p. 72) since two consecutive gene- rating lines intersect. The developable generated by the nor- mals along a line of curvature manifestly cuts the given surface at right angles. The locus of points where two consecutive generators of a developable intersect is a curve whose properties will be more 238 CURVATURE OF SURFACES. fully explained in the next chapter, and which is called the cuspidal edge of that developable. Each generator is a tan- gent to this curve, for it joins two consecutive points of the curve; namely, the points where the generator in question is met by the preceding and by the succeeding generator (see Art. 119). Consider now the normal at any point M of a surface ; through that point can be drawn two lines of curvature MNPQ, &c, MN'PQ\ &c. : let the normals at the points M, N, P, Q, &c. intersect in (7, D, E, &c, and those at M, N', P', Q' in C, D', E' ; then it is evident that the curve CDE, &c. is the cuspidal edge of the developable generated by the normals along the first line of curvature, while G'D'E' is the cuspidal edge of the developable generated by the normals along the second. The normal at M, as has just been ex- plained, touches these curves at the points G, C which are the two centres of curvature corresponding to the point M. What has been proved may be stated as follows: The cuspidal edge of the developable generated by the normals along a line of curvature, is the locus of one of the systems of centres of curvature corresponding to all the points of that line. 302. The assemblage of the centres of curvature C, 0" answering to all the points of a surface is a surface of two sheets called the -surface of centres (see Art. 189). The curve CDE lies on one sheet while G'D'E' lies on the other sheet. Every normal to the given surface touches both sheets of the surface of centres: for it has been proved that the normal a M touches the two curves CDE, G'D'E', and every tangent line to a curve traced on a surface is also a tangent to the surface. Now if from a point, not on a surface, be drawn two con- secutive tangent lines to a surface, the plane of those lines is manifestly a tangent plane to the surface ; for it is a tangent plane to the cone which is drawn from the point touching the surface. But if two consecutive tangent lines intersect on the surface, it cannot be inferred that their plane touches the CUKVATUEE OF SURFACES. 239 surface. For if we cut the surface by any plane whatever, any two consecutive tangents to the curve of section (which, of course, are also tangent lines to the surface) intersect on the curve, and yet the plane of these lines is supposed not to touch the surface. Consider now the two consecutive normals at the points M, N, these are both tangents to both sheets of the surface of centres. And since the point C in which they intersect is on the first sheet but not necessarily on the second, the plane of the two normals is the tangent plane to the second sheet of the surface of centres. The plane of the normals at the points 31, N' is the tangent plane to the other sheet of the surface of centres. But because the two lines of curvature through 31 are at right angles to each other, it follows that these two planes are at right angles to each other. Hence, the tangent planes to the surface of centres at the two points C, 0", where any normal meets it, cut each other at right angles. 303. It is manifest that for every umbilic on the given surface, the two sheets of the surface of centres have a point common ; or, in other words, the surface of centres has a double point ; and if the original surface have a line of spherical curvature, the surface of centres will have a double line. The two sheets will cut at right angles every where along this double line. This however is not the only case where tbe surface of centres has a double line. A double point on that surface arises not only when the two centres which belong to the same normal coincide, but also when two different normals intersect, and the point of intersection is a centre of curvature for each. It was shewn, p. 229, that a surface of the n tti degree possesses ordinarily a definite number of umbilics, and, therefore, in general not a line of spherical curvature. Hence a double line of the first kind is not among the ordinary singularities of the surface of centres. But that surface will in general have a double line of the second kind. Through any point several normals can be drawn to a surface : every point on the surface of centres is a centre of curvature for one of these normals, 240 CURVATURE OF SURFACES. a locus of points on the surface will be each a centre of cur- vature for two normals, and there will even be a definite number of points each a centre of curvature for three normals.* 304. It is convenient to define here a geodesic line on a surface, and to establish the fundamental property of such a line; namely, that its osculating plane (see Art. 119) at any point is normal to the surface. A geodesic line is the form assumed by a strained thread lying on a surface and joining any two points on the surface. It is plain that the geodesic is ordinarily the shortest line on the surface by which the two points can be joined, since, by pulling at the ends of the thread, we must shorten it as much as the interposition of the surface will permit. Now the resultant of the tensions along two consecutive elements of the curve, formed by the thread, lies in the plane of those elements, and since it must be de- stroyed by the resistance of the surface, it is normal to the surface ; hence, the plane of tivo consecutive elements of the geo- desic contains the normal to the surface.^ The same thing may also be proved geometrically. In the first place, if two points A, C in different planes be connected * The possibility of double lines of the second kind was overlooked by Monge and by succeeding geometers ; and oddly enough first came to be recognized in con- sequence of M. Kummer's having had a model made of the surface of centres of an ellipsoid (see Monatsberichte of the Berlin Academy, 1862). Instead of finding the sheets, as he expected, to meet only in the points corresponding to the umbilies, he found that they intersected in a curve, and that they did not cut at right angles along this line. Of course when the existence of the double line was known to be a fact its mathematical theory was evident. M. Clebsch had, on purely mathematical grounds, independently arrived at the same conclusion in an elaborate paper on the normals to an ellipsoid, of equal date with Kummer's paper, though of later pub- lication. A discussion of the surface of centres of an ellipsoid, founded on Clebsch's paper, will be given in Chapter xiv. f I have followed Monge in giving this proof, the mechanical principles which it involves being so elementary that it seems pedantic to object to the introduction of them. For the benefit of those who would prefer a purely geometrical proof I add one or two in the text. For readers familiar with the theory of maxima and minima it is scarcely necessary to add that a geodesic need not be the absolutely shortest line by which two points on the surface may be joined. Thus, if we consider two points on a sphere joined by a great circle, the remaining portion of that great circle, ex- ceeding 180° is a geodesic though not the shortest line connecting the points. The geodesic however will always be the shortest line if the two points considered be taken sufficiently near. CURVATUKE 0? SURFACES. 241 by joining each to a point B in the intersection of the two planes, the sum of AB and BC will be less than the sum of any other joining lines AB', B'C, it AB and BC make equal angles with TT, the intersection of the planes. For if one plane be made to revolve about TT' until it coincide with the other, AB and BC become one right line, since the angle TBA is supposed to be equal to T'BC; and the right line AC is the shortest by which the points A and C can be joined. It follows then that if AB and BC be consecutive elements of a curve traced on a surface, that curve will be the shortest line connecting A and C when AB and BC make equal angles with BT, the intersection of the tangent planes at A and C. We see then that AB (or its production) and BC are con- secutive edges of a right cone having BT for its axis. Now the plane containing two consecutive edges is a tangent plane to the cone ; and since every tangent plane to a right cone is perpendicular to the plane containing the axis and the line of contact, it follows that the plane ABC (the osculating plane to the geodesic) is perpendicular to the plane AB, BT which is the tangent plane at A. The theorem of this article is thus established. M. Bertrand has remarked (Liouville, t. Sill., p. 73, cited by Cayley, Quarterly Journal, Vol. I., p. 186) that this funda- mental property of geodesies follows at once from Meunier's theorem (see Art. 287). For it is evident, that for an inde- finitely small arc the chord of which is given, the excess in length over the chord is so much the less as the radius of curvature is greater. The shortest arc therefore joining two indefinitely near points A, B, on a surface is that which has the greatest radii of curvature, and we have seen that this is the normal section. 305. Returning now to the surface of centres, I say that the curve CDE (Art. 302), which is the locus of points of inter- section of consecutive normals along a line of curvature, is a geodesic on the sheet of the surface of centres on which it lies. For we saw (Art. 302) that the plane of two consecutive 242 CURVATURE OF SURFACES. normals to the surface (that is to say, the plane of two con- secutive tangents to this curve) is the tangent plane to the second sheet of the surface of centres and is perpendicular to the tangent plane at G to that sheet of the surface of centres on which C lies. Since then the osculating plane of the curve CDE is always normal to the surface of centres, the curve is a geodesic on that surface. 306. We have given the equations connected with lines of curvature on the supposition that the equation of the surface has been given, as it ordinarily is, in the form <£ (x, 3/, z) = 0. As it is convenient, however, that the reader should be able to find here the formulae which have been commonly employed, we shall conclude this chapter by giving the principal equations in the form given by Monge and by most subsequent writers, viz. when the equation of the surface is in the form z = cj>(x, y). We use the ordinary notations dz =pdx + qdy, dp = rdx + sdy, dq — sdx + tdy. We might derive the results in this form from those found already ; for since we have JJ— cf> [x, y) — z = 0, we have dU_ dU_ d _£__i dx ~ P ' dy -!? ' de ~ h with corresponding expressions for their second differential coefficients. We shall, however, repeat the investigations for this form as they are usually given. The equation of a tangent plane is z-z=p{x-x') + q{y-y), and the equations of the normal are [x — x') +p [z — z') = 0, y — y' + q { z — «') = 0. If then a/37 be any point on the normal and xyz the point where it meets the surface, we have (a - x) +p (7 - s) = 0, (#-#) + q{j-z) = 0. And if a/37 also satisfy the equations of a second normal, the differentials of these equations must vanish, or dx +pdz = (7 — z) dp, dy + qdz =(y — z)dq; CURVATURE OF SURFACES. 243 whence, eliminating (7 — »), we have the equation of condition (dx +pdz) dq = (dy + qdz) dp. Putting in for dz, dp, dq their values already given, and arranging, we have & {(1 +f) s - M t} + d £{(l +q y-[l+f)t} - {(l+p*)s-pqr}=0. This equation determines the projections on the plane of xy of the two directions in which consecutive normals can be drawn so as to intersect the given normal. 307. From the equations of the preceding article we can also find the lengths of the principal radii. The equations dx +pdz = (7 — z) dp, dy + qdz = (y — s) dq, when transformed as above become {1 +p 2 — (7 — z) r] dx + {pq - (7 - z) s\ dy = 0, {l + q*-(y-z)t}dy+{pq-(y-z)s}dx = 0, whence eliminating dx : dy, we have ^- z f{rt-s")-^-z){{l + q i )r-2pqs+{l + p i )t} + {l+p i +q i )^0. Now 7 — z is the projection of the radius of curvature on the axis of z ; and the cosine of the angle the normal makes with that radius being —rr- 5 ^ , we have i?=(7- 2 )V(l+/ + 2 2 )- Eliminating then 7-2 by the help of the last equation, B is given by the equation R* (rt -J)-B {(1 + q 2 )r - 2pqs + (1 +/) *} V(l + j>" + ? 8 ) + {l+p* + q7 = 0. 308. From the preceding theorems can be deduced Joachimsthal's theorem (see Crelle, Vol. xxx., p. 347) that if a line of curvature be a plane curve, its plane makes a constant angle with the tangent plane to the surface at any of the points where it meets it. Let the plane be z = 0, then the equation of Art. 306 (dx + pdz) dq = [dy + qdz) dp R2 244 CURVATURE OF SURFACES. becomes dxdq = dydp. But we have also pdx-\-qdy = Q, con- sequently pdp + qdq = ; p* + q'' = constant. But p'" + q* is the square of the tangent of the angle which the tangent plane makes with the plane xy; since cos 7= -7- = s-, . r ■*' V(l+f+2') Otherwise thus (see IAouville, Vol. XI., p. 87) : Let MM', M'M" be two consecutive and equal elements of a line of curvature, then the two consecutive normals are two perpen- diculars to these lines passing through their middle points i, /', and C the point of meeting of the normals is equidistant from the lines MM', M'M". But if from G we let fall a perpen- dicular CO on the plane MM'M", will be also equidistant from the same elements; and therefore the angle GIO = GTO. It is proved then that the inclination of the normal to the plane of the line of curvature remains unchanged as we pass from point to point of that line. More generally let the line of curvature not be plane. Then, as before, the tangent planes through MM' and through M'M" make equal angles with the plane MM'M". And evidently the angle which the second tangent plane makes with a second osculating plane M'M"M'" differs from the angle which it makes with the first by the angle between the two osculating planes. Thus we have Lancret's theorem, that along a line of curiature the variation in the angle between the tangent plane to the surface and the osculating plane to the curve is equal to the angle between the two osculating planes. For example, if a line of curvature be a geodesic it must be plane. For then the angle between the tangent plane and osculating plane does not vary, being always right : therefore the osculating plane itself does not vary. From the same prin- ciples we obtain a simple proof of the theorem of Art. 300. 309. Finally, to obtain the radius of curvature of any normal section. Since the centre of curvature a/37 ^ es on the normal, we have (a-a;)+jj(y-«)=0, (£~y) + q (7 -*) = 0. Further, we have (a-z) 2 +(/3- j J* dy + ™ • The radius of curvature therefore of a normal section whose projection on the plane of xy is parallel to y = tnx is ±v(i + / +g 8 ) (i+/)+ , 2 r:^ +gV2 . v r a ' r + 2sm + tm The conditions for an umbilic are got by expressing that this value is independent of m, and are r s t ( 246 ) CHAPTER XII. CURVES AND DEVELOPABLES. SECTION I. PROJECTIVE PROPERTIES. 310. It was proved (p. 13) that two equations represent a curve in space. Thus the equations U= 0, V= represent the curve of intersection of the surfaces Z7, V. The degree of a curve in space is measured by the number of points in which it is met by any plane. Thus, if Z7, V be of the m m and n th degrees respectively, the surfaces which they represent are met by any plane in curves of the same degrees, which intersect in mn points. The curve UV is therefore of the km" degree. By eliminating the variables alternately between the two given equations, we obtain three equations which are the equations of the projections of the curve on the three co-ordinate planes. Any one of the equations taken separately represents the cylinder whose edges are parallel to one of the axes, and which passes through the curve (Art. 24). The theory of elimination shows that the equation («/, z) = obtained by eliminating x between the given equations is of the mn a degree. And it is also geometrically evident that any cone or cylinder* standing on a curve of the r n degree is of the r tb degree. For if we draw any plane through the vertex of the cone [or parallel to the generators of the cylinder] this plane meets the cone in r lines ; namely, the lines joining the vertex to the r points where the plane meets the curve. * A cylinder is plainly the limiting case of a cone, whose vertex is at infinity. PROJECTIVE PROPERTIES OF CURVES. 247 311. Now, conversely, if we are given any curve in space and desire to represent it by equations, we need only take the three plane curves which are the projections of the curve on the three co-ordinate planes ; then any two of the equations $ (#) z ) = 0, ty (z, x) = 0, x i x ~ y) = will represent the given curve. But ordinarily these will not form the simplest system of equations by which the curve can be represented. For if r be the degree of the curve, these cylinders being each of the r th degree, any two intersect in a curve of r 2 degree ; that is to say, not merely in the curve we are considering but in an extraneous curve of the degree r s — r. And if we wish not merely to obtain a system of equations satisfied by the points of the given curve, but also to exclude all extraneous points, we must preserve the system of three projections; for the projection on the third plane of the extraneous curve in which the first two cylinders intersect will be different from the projection of the given curve. It may be possible by combining the equations of the three projections to arrive at two equations U= 0, V— 0, which shall be satisfied for the points of the given curve, and for no other. But it is not generally true that every curve in space is the complete intersection of two surfaces. To take the simplest example, consider two quadrics having a right line common, as, for example, two cones having a common edge. The intersection of these surfaces, which is in general of the fourth degree, must consist of the common right line, and of a curve of the third degree. Now since the only factors of 3 are 1 and 3, a curve of the third degree cannot be the complete intersection of two surfaces unless it be a plane curve ; but the curve we are considering cannot be a plane curve,* for if so any arbitrary line in its plane would meet it in three points, but such a line could not meet either quadric in more than two, and therefore could not pass through three points of their curve of intersection. * Curves in space which are not plane curves have commonly been called "curves of double curvature." In what follows, I use the word "curve" to de- note a curve in space, which ordinarily is not a plane curve, and I add the adjec- tive •' twisted" when I want to state expressly that the curve is not a plane curve. 248 PROJECTIVE PROPERTIES OF CORVES. 312. If a curve be either the complete or partial inter- section of two surfaces U, V, the tangent to the curve at any point is evidently the intersection of the tangent planes to the two surfaces, and is represented by the equations xu;+ y u; + zu;+wu;=o, xv;+ y v:+zv; + wv;=o. The direction-cosines of the tangent are plainly proportional to MN'-M'N, NL'-N'L, LM'-L'M, where L, if, &c. are the first differential coefficients. An exceptional case arises when the two surfaces touch, in which case the point of contact is a double point on their curve of intersection. All this has been explained before (see Art. 104). As a particular case of the above, the projection of the tangent line to any curve is the tangent to its projec- tion ; and when the curve is given as the intersection of the two cylinders y = [z), x = y{r(z) 1 the equations of the tan- gent are T'lis may be otherwise expressed as follows: Consider any element of the curve ds ; it is projected on the axes of co- ordinates into dx, dy, ds. The direction-cosines of this element are therefore ~ , ■—■ , ~ , and the equations of the tangent are x — x - y~j~ -. z-z dx dy dz ds ds Is Since the sum of the squares of the three cosines are equal to unity, we have ds* = dx 2 + dy 2 + dz 1 . We shall postpone to another section the theory of normals, radii of curvature, and in short everything which involves the consideration of angles, and in this section we shall only consider what may be called the projective properties of curves. 313. The theory of curves is in a great measure* identical with that of developables, on which account it is necessary to PROJECTIVE PROPERTIES OF CURVES. 249 enter more fully into the latter theory. In fact it was proved (Art. 119) that the reciprocal of a series of points forming a curve is a series of planes enveloping a developable. We there showed that the points of a curve regarded as a system of points 1, 2, 3, &c. give rise to a system of lines ; namely, the lines 12, 23, 31, &c. joining each point to its next consecutive, these lines being the tangents to the curve : and that they also give rise to a system of planes, viz. the planes 123, 234, &c. containing every three consecutive points of the system, these planes being the osculating planes of the curve. The as- semblage of the lines of the system forms a surface whose equation can be found when the equation of the curve is given. For the two equations of the tangent line to the curve involve the three co-ordinates x', y', z, which being connected by two relations are reducible to a single parameter; and by the elimination of this parameter from the two equations, we obtain the equation of the surface. Or, in other words, we must eliminate x'y'z' between the two equations of the tangent and the two equations of the curve. We have said (Art. 119) that the surface generated by the tangents is a developable, since every two consecutive positions of the generating line intersect each other. The name given to this kind of surface is derived from the property that it can be unfolded into a plane without crumpling or tearing. Thus imagine any series of lines Aa, Bb, Gc, Dd, &c. (which for the moment we take at a finite distance from each other) and such that each inter- sects the consecutive in the points a, b, c, &c. ; and suppose a surface to be made up of the faces AaB, BbC, CcD, &c, then it is evident that such a surface can be developed into •a plane by turning the face AaB round aB as a hinge until it formed a continuation of BbC; by turning the two, which we had thus made into one face, round cC until they formed a continuation of the next face, and so on. In the limit when the lines Aa, Bb, &c. are indefinitely near, the assemblage of plane elements forms a developable which, as just explained, can be unfolded into one plane. The reader will find no difficulty in conceiving this from the examples of developables with whieh he is most familiar, 250 PROJECTIVE PROPERTIES OF CURVES. viz. a cone or a cylinder. There is no difficulty in folding a sheet of paper into the form of either surface and in un- folding it again into a plane. But it will easily be seen to be impossible to fold a sheet of paper into the form of a sphere (which is not a developable surface) ; or, conversely, if we cut a sphere in two it is impossible to make the portions of the surface lie smooth in one plane. 314. The plane AaB containing two consecutive gene- rating lines is evidently, in the limit, a tangent plane to the developable. It is plain that we might consider the surface as generated by the motion of the plane AaB according to some assigned law, the envelope of this plane in all its positions being the developable. Now if we consider the developable generated by the tangent lines of a curve in space, the equa- tions of the tangent at any point x'y'z are plainly functions of those co-ordinates, and the equation of the plane containing any tangent and the next consecutive (in other words, the equation of the osculating plane at any point x'y'z) is also a function of these co-ordinates. But since x'y'z are connected by two relations, namely, the equations of the curve ; we can eliminate any two of them, and so arrive at this result, that a developable is the envelope of a plane whose equation contains a single variable parameter. To make this statement better understood we shall point out an important difference between the cases when a plane curve is considered as the envelope of a moveable line, and when a surface in general is considered as the envelope of a moveable plane. 315. The equation of the tangent to a plane curve is a function of the co-ordinates of the point of contact ; and these two co-ordinates being connected by the equation of the curve, we can either eliminate one of them, or else express both in terms of a third variable so as to obtain the equation of the tangent as a function of a single variable parameter. The converse problem to obtain the envelope of a right line whose equation includes a variable parameter has been discussed, Higher Plane Curves, p. 93. Let the equation of any tan- PROJECTIVE PROPERTIES OF CURVES. 251 gent line be u = 0, where u is of the first degree in x and y, and the constants are functions of a parameter k. Then the line answering to the value of the parameter a -)- h is , du h d*u h* „ , ,, . , „ . , ,. . , u+-y- - + ts «-= + <* c -i aQ d the point ot intersection or these da. 1 da 1.2 ' r ,...,,, ,. „ du h d 2 u two lines is given by the equations u = 0, -7- + — -=-, + &c. = 0. And, in the limit, the point of intersection of a line with the next consecutive (or, in other words, the point of contact of any line with its envelope) is given by the equations u — 0, -j— = 0. If from these two equations we eliminate a we obtain the locus of the points of intersection of each line of the system with the next consecutive ; that is to say, the equation of the envelope of all these lines. It is easy to prove that the result of this elimination represents a curve to which u is a tangent. We get that result, if in u we replace a by its value, in terms of x and y, derived from the equation —=0. Now, if we differen- . du fdu\ du da. , du (du\ du da tiate, we have 7= 7 +77™ r= r +tti ' ax \dx) da ax ay \dyj da dy ' where (t-j, f-^-J are the differentials of u on the supposition that a is constant. And since -7- = it is evident that t- , — da dx ' dy are the same as on the supposition that a is constant. It follows that the eliminant in question denotes a curve touched by u. If it be required to draw a tangent to this curve through any point, we have only to substitute the co-ordinates of that point in the equation u = 0, and determine a so as to satisfy that equation. This problem will have a definite number of solutions, and the number will plainly be the number of tan- gents which can be drawn to the curve from an arbitrary point; that is to say, the class of the curve. For example, the envelope of the line aa 8 + 35a 2 +3ca + <2=0, where a, b, c, d, are linear functions of the co-ordinates, is plainly a curve of the third class. 252 PROJECTIVE PROPERTIES OF CURVES. 316. Now let us proceed in like manner with a surface. The equation of the tangent plane to a surface is a function of the three co-ordinates, which being connected by only one relation (viz. the equation of the surface), the equation of the tangent plane, when most simplified, contains two variable parameters. The converse problem is to find the envelope of a plane whose equation u = contains two variable parameters a, /3. The equation of any other plane answering to the values a + h, /3 + k will be /, du , du\ 1 / 7 „ d\ „ \ Now, in the limit, when h and k are taken indefinitely small, they may preserve any finite ratio to each other k = Xh. We see thus that the intersection of any plane by a consecutive one is not a definite line, but may be any line represented by the equations m = 0, j + X j^=0, where X is indeterminate. But we see also that all planes consecutive to u pass through ,i • • i ,i ,• ^ du „ du the point given by the equations u = 0, -j- = 0, -^ = 0. From these three equations we can eliminate the parameters a, /3, and so find the locus of all those points where a plane of the system is met by the series of consecutive planes. It is proved, as in the last article, that the surface represented by this eliminant is touched by u. If it be required to draw a tangent plane to this surface through any point, we have only to substitute the co-ordinates of that point in the equation u = 0. The equation then containing two indeterminates a and /3 can be satisfied in an infinity of ways; or, as we know, through a given point an infinity of tangent planes can be drawn to the surface, these planes enveloping a cone. Suppose, however, that we either consider /3 as constant, or as any definite function of a, the equation of the tangent plane is reduced to contain a single parameter, and the envelope of those particular tangent planes which satisfy the assumed con- dition is a developable. Thus, again, we may see the analogy between a developable and a curve. When a surface is con- PROJECTIVE PROPERTIES OF CURVES. 253 sidered as the locus of a number of points connected by a given relation, if we add another relation connecting the points we obtain a curve traced on the given surface. Bo when we con- sider a surface as the envelope of a series of planes connected by a single relation, if we add another relation connecting the planes we obtain a developable enveloping the given surface. 317. Let us now see what properties of developables are to be deduced from considering the developable as the envelope of a plane whose equation contains a single variable parameter. In the first place it appears that through any assumed point can be drawn, not as before an infinity of planes of the system, forming a cone ; but a definite number of planes. Thus, if it be required to find the envelope of aa? + Sbd' -t 3ca + d, where a, b, c, d represent planes, it is obvious that only three planes of the system can be drawn through a given point, since on substituting the co-ordinates of any point we get a cubic for a. Again, any plane of the system is cut by a consecutive plane in a definite line; namely, the line m = 0, -^-=0; and, if we eliminate a between these two equations, we obtain the sur- face generated by all those lines, which is the required developable. . It is proved, as at Art. 315, that the plane u touches the developable at every point which satisfies the equations w = 0, ^->=0; or, in other words, touches along the whole of the line da of the system corresponding to u. It was proved (Art. 107) that in general when a surface contains a right line the tangent plane at each point of the right line is different. But in the case of the developable the tangent plane at every point is the same. If x be the plane which touches all along the line xy, the equation of the surface can be thrown into the form x(j> + y*\}r = (seep. 72).* * It seems unnecessary to enter more fully into the subject of envelopes in general, since what is said in the test applies equally if «, instead of representing a plane, denote any surface whose equation includes a variable parameter. Monge calls the curve « = 0, — = 0, in which any surface of the system is intersected by the con- 254 PKOJECTIVE PROPERTIES OF CURVES. 318. Let us now consider three consecutive planes of the system, and it is evident as before that their intersection satisfies the equations u — 0, -=- = 0, -^ = 0. For any value of a, the point is thus determined where any line of the system is met by the next consecutive. The locus of these points is got by eliminating a between these equations. We thus obtain two equations in x, y, z, one of them being the equation of the developable. These two equations represent a curve traced ou the developable. Thus it is evident that starting with the definition of a developable as the envelope of a moveable plane, we are led back to its generation as the locus of tangents to a curve. For the consecutive intersections of the planes form a series of lines, and the consecutive intersection of the lines are a series of points forming a curve to which the lines are tangents. We shall presently show that the curve is a cuspidal edge* on the developable. 319. Four consecutive planes of the system will not meet in a point unless the four conditions be fulfilled u = 0, -r = 0, 1 da. ' d 2 u n d 3 u T . . ... - . -=-„ = 0, -, 8 = 0. It is in general possible to find certain values of a, for which this condition will be satisfied. For if we eliminate x } y 1 a, we get the condition that the four secutive, the characteristic of the envelope. For the nature of this curve depends only on the manner in which the variables x, y, z enter into the function u } and not on the manner in which the constants depend on the parameter. Thus, when u represents a plane, the characteristic is always a right line, and the envelope is the locus of a system of right lines. When u represents a sphere, the characteristic, being the intersection of two consecutive spheres, is a circle ; and the envelope is the locus of a system of circles. And so envelopes in general may be divided into families according to the nature of the characteristic. * Monge has called this the " arete de rebroussement," or "edge of regression" of the developable. There is a similar curve on eveiy envelope, namely, the locus of points in which each " characteristic" is met by the nest consecutive. The part of the characteristic on one side of this curve generates one sheet of the envelope, and that on the other side generates another sheet. The two sheets touch along this curve which is their common limit, and is a cuspidal edge of the envelope. Thus in the case of a cone the parts of the generating lines on opposite sides of the vertex generate opposite sheets of the cone, and the cuspidal edge in this case reduces itself to a single point, namely, the vertex. PROJECTIVE PEOPEETIE3 OF CUKVES. 255 planes, whose equations have been just written, shall meet in a point. This condition is that a function of a is equal to nothing, whence we shall in general get a determinate number of values of a for which the condition is satisfied. There are therefore in general a certain number of points of the system through which four planes of the system pass; or, in other words, a certain number of points in which three consecutive lines of the system intersect. We shall call these, as at Higher Plane Curves, p. 28, the stationary points of the system ; since in this case the point determined as the intersection of two consecutive lines, coincides with that determined as the inter- section of the next consecutive pair. Eeciprocally, there will be in general a certain number of planes of the system which may be called stationary planes. These are the planes which contain four consecutive points of the system ; for, in such a case, the planes 123, 234 evidently coincide. 320. We shall now show how, from Pliicker's equations con- necting the ordinary singularities of plane curves,* Mr. Cayleyt has deduced equations connecting the ordinary singularities of developables. We shall first make an enumeration of these singularities. We speak of the "points of the system," the " lines of the system," and the " planes of the system" as ex- plained (Art.' 119). Let m be the number of points of the system which lie in any plane ; or r in other words, the degree of the curve which generates the developable. Let n be the number of planes of the system which can be drawn through an arbitrary point. We have proved (Art. 317) * These equations are as follows : see Higher Plane Curves, p. 91. Let fi be the degree of a curve, v its class, 8 the number of its double points, x that of its double tangents, k the number of its cusps, t that of its points of inflexion; then i/ = n(fi-l)-2$-3K; n= v (k - 1) - 2t - 3t, i = 3fi Qi - 2) - 68 - 8k; k = 3v (» - 2) - 6t - 8t. Whence also t - k = 3 (» - /j.) ; 2 (t - S) = (u - fi) {v + fi - 9). f See Iiouville's Journal, Yol. X., p. 245 ; Cambridge and Dublin Mathematical Journal, Vol. v., p. 18. 256 PROJECTIVE PROPERTIES OF CURVES. that the number of such planes is definite. We shall call this number the class of the system. Let r be the number of lines of the system which intersect an arbitrary right line. It is plain that if we form the con- dition that w, -5- and any assumed right line may intersect, the result will be an equation in a, which gives a definite number of values of a. Let r be the number of solutions of this equation. We shall call this number the rank of the system, and we shall show that all other singularities of the system can be expressed in terms of the three just enumerated. Let a. be the number of stationary planes, and j3 the number of stationary points (Art. 319). Two non-consecutive lines of the system may intersect. When this happens we call the point of meeting a " point on two lines," and their plane a "plane through two lines." Let x be the number of "points on two lines" which lie in a given plane, and y the number of "planes through two lines" which pass through a given point. In like manner we shall call the line joining any two points of the system a " line through two points," and the intersection of any two planes a " line in two planes." Let g be the number of " lines in two planes" which lie in a given plane, and /* the number of " lines through two points" which pass through a given point. The developable has other singularities which will be deter- mined in a subsequent chapter, but these are the singularities which Plucker's equations (note, p. 255) enable us to determine. 321. Consider now the section of the developable by any plane. It is obvious that the points of this curve are the traces on its plane of the "lines of the system," while the tangent lines of the section are the traces on its plane of the " planes of the system." The degree of the section is therefore r, since it is equal to the number of points in which an arbitrary line drawn in its plane meets the section, and we have such a point whenever the line meets a " line of the system." PROJECTIVE PROPERTIES OF CURVES. 257 The class of the section is plainly n. For tlie number of tangent lines to the section drawn through an arbitrary point is evidently the same as the number of "planes of the system" drawn through the same point. A double point on the section will arise whenever two " lines of the system" meet the plane of section in the same point. The number of such points by definition is x. The tangent lines at such a double point are usually distinct, because the two planes of the system corresponding to the lines of the system intersecting in any of the points x are commonly different. The number of double tangents to the section is in like manner g ; since a double tangent arises whenever two planes of the system meet the plane of section in the same line. The m points of the system which lie in the plane of section are cusps of the section. For they are double points as being the intersection of two lines of the system ; and the tangent planes at these points coincide, since the two consecutive lines, intersecting in one of the points ?», lie in the same plane of the system. This proves, what we have already stated, that the curve whose tangents generate the developable is a cuspidal edge on the developable ; for it is such that every plane meets that surface in a section which has as cusps the points where the same plane meets the curve. Lastly, we get a point of inflexion (or a stationary tangent) wherever two consecutive planes of the system coincide. The number of points of inflexion is therefore a. We are to substitute then in the formulae, note p. 255, /M = r, v = n : S = x, T=y, « = m, i = a. And we have ,i= r (r- 1) - 2x - 3m ; r= n {n - 1) - 2g- 3a, a = Br (>• - 2) - 6x - 8?» ; m = 3» [n -2)-6g- 8a, whence also m - a = 3 (r - n) ; 2 (x - g) = (r - n) (r + n - 9). 322. Another system of equations is found by considering the cone whose vertex is any point and which stands on the given curve. It appears at once by considering the section 258 PROJECTIVE PROPERTIES OF CURVES. of a cone by any plane that the same equations connect the double edges, double tangent planes, &c. of cones, which con- nect the double points, double tangents, &c. of plane curves. The edges of the cone which we are now considering are the lines joining the vertex to all the points of the system ; and the tangent planes to the cone are the planes connecting the vertex with the lines of the system, for evidently the plane containing two consecutive edges of the cone must contain the line joining two consecutive points of the system. The degree of the cone is plainly the same as the degree of the curve and is therefore m. The class of the cone is the same as the number of tangent planes to the cone which pass through an arbitrary line drawn through the vertex. Now since each tangent plane contains a line of the system, it follows that we have as many tangent planes passing through the arbitrary line as there are lines of the system which meet that line. The number sought is therefore r.* A double edge of the cone arises when the same edge of the cone passes through two points of the system, or 8 = h. The tangent planes along that edge are the planes joining the vertex to the lines of the system which correspond to each of these points. A double tangent plane will arise when the same plane through the vertex contains two lines of the system ; or t = y. A stationary or cuspidal edge of the cone will only exist when there is a stationary point in the system ; or k — /3. Lastly, a stationary tangent plane will exist when a plane containing two consecutive lines of the system passes through the vertex ; or t = n. Thus we have fi = m, v = r, S = h, T = y, k = /3, t = n. Hence by the formulae (note p. 255) r= m{m-l)-2h-?j^; in = r (r - 1) -2?/-3w, n = 3m (m - 2) - 6h - 8/3 ; /3 = 3r (r - 2) - &ij - 8m. * It ia easy to see that the class of this cone is the same as the degree of the developable which ia the reciprocal of the points of the given system. Hence, the degree of the developable generated hy the tangents to any curve is the same as the degree of the developable which is the reciprocal of the points of that curve, see note, p. 87. PROJECTIVE PROPERTIES OF CURVES. 259 Whence also (n - /3) = 3 (?• - m) ; 2 (y - h) = (•>• - mi) (r + »i - 9). And combining these equations with those found in the last article, we have also a - /3 = 2 (n - m) ; x -y = n- m ; 2(g-h) = (n- m)(n + m - 7). •Pluckcr's equations enable us, when three of the singularities of a plane curve are given, to determine all the rest. Now three quantities r, m, n are common to the equations of this and of the last article. Hence, when any three of the singu- larities which we have enumerated, of a curve in space, are given, all the rest can he found.* 323. To illustrate this theory, let us take the developable which is the envelope of the plane at + hot*' 1 + %Jc (k - 1) cr 2 + &c. = 0, where t is a variable parameter, a, b, c, &c. represent planes, and 7c is any integer. The class of this system is obviously k, and the equation of the developable being the discriminant of the preceding equation, its degree is 2(&— 1); hence r = 2 (k — 1). Also it is easy to see that this developable can have no stationary planes. For in general if we compare coefficients in the equations of two planes, three conditions must be satisfied in order that the two planes may be identical. If then we attempt to determine t so that any plane may be identical with the consecutive one, we find that we have three conditions to satisfy, and only one constant t at our disposal. Having then k= I; r — 2 (k - 1), a = 0, the equations of the last two articles enable us to determine the remaining singu- larities. The result is m = 3(&-2); /3 = 4(£-3); x = 2 (&- 2) (fc-3) ; y = 2(£-l)(fc-3); g = i 2 {k- 1) (fc-2) ; * = £ (9# -53& + 80). * It is to tie observed that besides those -which we count as the ordinary sin- gularities, a curve may have others. Thus if it have 6 points at which three points of the system lie in a right line, the tangent line at every such point is doubly a line of the system, and meets in a cusp the section of the developable by any plane. Instead then of having, as in Art. 3*21, k = -i, we have k = -i + 8. And in like manner (Art. 322) we have • = n + 6. S2 260 PROJECTIVE PROPERTIES OF CURVES. The greater part of these values can be obtained independently as at Higher Plane Curves, p. 94. But in order to economize space we do not enter into details. 324. The case considered in the last article, which is that when the variable parameter enters only rationally into the equation, enables us to verify easily many properties of de- velopables. Since the system w = 0, -J = is obviously re- ducible to at* + [k - 1 ) U™ + &c. = 0, It"' 1 + (&-!) <^" 2 + &c. = 0, and the system u = 0, -j- = 0, -^ — is reducible to a r i +{k-z) br 3 + &c. = o, &r 2 + {h - 2) cf- s + &c. = 0, cf+(7c-2)dt t -'> + &c. = 0; it follows that a is itself a plane of the system (namely, that corresponding to the value £=<»), ab is the corresponding line, and abc the corresponding point. Now we know from the theory of discriminants (see Higher Algebra, p. 47) that the equation of the developable is of the form a(f> + b*yfr = 0, where y}r is the discriminant of u when in it a is made = 0. Thus we verify what was stated (Art. 317) that a touches the develop- able along the whole length of the line ab. Further, ^r is itself of the form bj> + c 2 -^'. If now we consider the section of the developable by one of the planes of the system (or, in other words, if we make a = in the equation of the develop- able), the section consists of the line ab twice and of a curve of the degree r — 2 ; and this curve (as the form of the equation shows) touches the line ab at the point a5c,*and consequently meets it in r — 4 other points. These are all " points on two lines," being the points where the line ab meets other lines of the system. And it is generally true that if r be the rank of a developable each line of the system meets r — 4 other lines of the system. The locus of these points forms a double curve on the developable, the degree' of which is x, and the other properties of which will be given in a subsequent chapter, where we shall also determine certain other singularities of "■ the developable. CLASSIFICATION OF CURVES. 261 We add here a table of the singularities of some special sections of the developable. The reader, who may care to examine the subject, will find no great difficulty in establishing them. I have given the proof of the greater part of them, Cambridge and Dublin Mathematical Journal, Vol. v., p. 24. Section by a plane of the system //. = r-2, v = n-l, t = «, K = m-3, r=g-n + 2, B = x~2r+8. Cone whose vertex is a point of the system A*=m-1, v=r-2, * = n-8, k = /3, T =y-2r+8, Z=fi-m + 2. Section by plane passing through a line of the system /* = r-l, v = n, t = a + ], K = m-2, r=g-l, § = x -r + ±. Cone whose vertex is on a line of the system fi = m, v=r-l, i = n-2, *=j8 + l, r<=y-r+4, S = h-l m Section by plane through two lines /* = r-2, v = n, * = oc + 2, « = m-4, T =g-2, 8 = a;-2r+9« Cone whose vertex is a point on two lines fj, = m, v = r-2, t = w-4, k = /3+2, T =#-2r + 9, 8 = h- 2 fc Section by a stationary plane j*=r-3, v=n-2, t=oc-l, /c=m-4, r=y-2n + 6, S=a;-3r+13„ Cone whose vertex is a stationary point fi=m-2, v=r-3, *=»-4, «=/3-l, T=y-3r+13, S=A-2?n+6. SECTION II. CLASSIFICATION OF CURVES. 325. The following enumeration rests on the principle that a curve of the degree r meets a surface of the degree p in pr points. This 'is evident when the curve is the complete intersection of two surfaces whose degrees are m and n. For then we have r = mn and the three surfaces intersect in mnp points. It is true also by definition when the surface breaks up into p planes. We shall assume that, in virtue of the law of continuity, the principle is generally true. The use we make of. the principle is this. Suppose that we take on a curve of the degree r, as many points as are sufficient to determine a surface of the degree p ; then if the number of points so assumed be greater than pr, the surface 262 CLASSIFICATION OF CURVES. described through the points must altogether contain the curve ; for otherwise the principle would be violated. We assume in this that the curve is a proper curve of the degree r, for if we took two curves of the degrees m and n (where m+n = r), the two together might be regarded as a complex curve of the degree r, and if either lay altogether on any surface of the degree p, of course we could take on that curve any number of points common to the curve and surface. All this will be sufficiently illustrated by the examples which follow. 326. There is no line of the first degree hut the right line. For through any two points of a line of the first degree and any assumed point we can describe a plane which must alto- gether contain the line, since otherwise we should have a line of the first degree meeting the plane in more points than one. In like manner we can draw a second plane containing the line, which must therefore be the intersection of two planes ; that is to say, a right line. There is no proper line of the second degree but a conic. Through any three points of the line we can draw a plane, which the preceding reasoning shows must altogether contain the line. The line must therefore be a plane curve of the second degree. The exception noted at the end of the last article would occur if the line of the second degree consisted of two light lines not in the same plane ; for then the plane through three points of the system would only contain one of the right lines. In what follows we shall not think it necessary to notice this again, but shall speak only of proper curves of their respective orders. 327. A curve of the third degree must either be a plane cubic or the partial intersection of two quadrics, as explained, Art. 311* * Non-plane curves of the third degree appear to have been first noticed by Mobius in his Barycentrie Calculus, 18:27. Some of their most important properties are given by M. Chasles in Note XXXIII. to his Apercu IHstorique, 1837, and in a paper in Liouville's Journal for 1857, p. 397. More recently the properties of these curves have been treated of by M. Schrbter, Crelle, Vol. lvi., and by Professor Cremona of Milan, Crelle, Vol. Lvni., p. 138. Considerable use has been made of the latter paper in the articles which immediately follow. CLASSIFICATION OP CURVES. 263 For through seven points of the curve and any two other points describe a quadric ; and, as before, it must altogether contain the curve. If the quadric break up into two planes, the curve may be a plane curve lying in one of the planes. As we may evidently have plane curves of any degree we shall not think it necessary to notice these in subsequent cases. If then the quadric do not break up into planes, we can draw a second quadric through the seven points, and the intersection of the two quadries includes the given cubic. The complete intersection being of the fourth degree, it must be the cubic together with a right liue ; it is proved therefore that the only non-plane cubic is that explained, Art. 311. 328. The cone containing a curve of the m 01 degree and whose vertex is a point on the curve, is of the degree m — 1 ; hence the cone containing a cubic and whose vertex is on the curve is of the second degree, in- can thus describe a twisted cubic through six given jwints. For we can describe a cone of the second degree of which the vertex and five edges are given, since evidently we are thus given five points in the section of the cono by any plane, and can thus determine that section. If then we are given six points a, b, c, d, e, f, we can describe a cone having the point, a for vertex, and the lines ab, ac, ad, ac, af for edges ; and in like manner a cone having b for vertex and the lines ba, be, bd, be, bf for edges. The intersection of these cones consists of the common edge ab and of a cubic which is the required curve passing through the six points. The theorem that the lines joining six points of a cubic to any seventh are edges of a quadric cone, leads at once to the following by Pascal's theorem : " The' lines of intersection of the planes 712, 74o; 723, 756 ; 734, 761 lie in one plane." Or in other words, " the points where the planes of three con- secutive angles 567, 671, 712 meet the opposite sides lie in one plane passing through the vertex 7.'"* Conversely if this * M. Cremona adds that when the six points are fixed and the seventh variable, this plane passes through a fixed chord of the cubic 264 CLASSIFICATION OF CURVES. be true for two vertices of a heptagon it is true for all the rest : for then these two vertices are vertices of cones of the second degree containing the other points, which must there- fore lie on the cubic which is the intersection of the cones. 329. A cubic traced on a hyperboloid of one sheet meets all its generators of one system once, and those of the other system twice. Any generator of a quadric meets in two points its curve of intersection with any other quadric, namely, in the two points where the generator meets th'e other quadric. Now when the intersection consists of a right line and a cubic, it is evident that the generators of the same system as the line, since they do not meet the line, must meet the cubic in the two points ; while the generators of the opposite system, since they meet the line in one point, only meet the cubic in one other point. Conversely we can describe a system of hyperboloids through a cubic and any chord which meets it twice. For take seven points on the curve, and an eighth on the chord joining any two of them ; then through these eight points an infinity of quadrics can be described. But since three of these points are on a right line, that line must be common to all the quadrics, as must also the cubic on which the seven points lie. 330. The question to find the envelope of at 3 — 3bi* + Set — d (where a, b, c, d represent planes and t is a variable parameter) is a particular case of that discussed, Art. 323. We have r = 4, m = n = 3, a = /3 = 0, x = y — Q, g = h=l. Thus the system is of the same nature as the reciprocal system, and all theorems respecting it are consequently two-fold. The system being of the third degree must be of the kind we are considering ; and this also appears from the equation of the envelope {ad - be)* = 4 (6 2 - ac) (c 2 - b d) , for it is easy to see that any pair of the surfaces ad — be, b 2 — ac, c 2 — bd, have a right line common, while there is a cubic common to all three, which is a double line on the envelope. It appears from the table just given that every plane con- tains one "line in two planes"; or that the section of the CLASSIFICATION OF CURVES. 265 developable by any plane has one double tangent; while re- ciprocally through any point can be drawn one line to meet the cubic twice ; the cone therefore, whose vertex is that point, and which stands on the curve, has one double edge ; or in other words, the cubic is projected on any plane into a cubic having a double point. The three points of inflexion of a plane cubic are in one right line. Now it was proved (Art. 322) that the points of in- flexion correspond to the three planes of the system which can be drawn through the vertex of the cone. Hence the three points of the system which correspond to the three planes which can be drawn through any point 0, lie in one plane passing through that point.* Further it is known that when a plane cubic has a conjugate point, its three points of inflexion are real ; but that when the cubic has a double point, the tangents at which are real, then two of the points of inflexion are imaginary. Hence if the chord which can be drawn through any point meet the cubic in two real points, then two of the planes of the system which can be drawn through are imaginary. Reciprocally, if through any line two real planes of the system can be drawn, then any plane through that line meets the curve in two imaginary points, and only one real one."(" 331. These theorems can also be easily established alge- braically; for the point of contact of the plane at 3 -3bt*+Sct-d, being given by the equations at=b, bt=c, ct=d, may be denoted by the co-ordinates a=l, b = t, c = f, d=f. Now the three values of t answering to planes passing through any point are given by the cubic a'f - 3b't* + 'Set - d' = 0, whence it is evident from the values just found, that the points of contact lie in the plane ad— 3b'c + 3c'b — d'a = 0. But this plane passes through the given point. Hence the intersection of three planes of the system lies in the plane of the corresponding points. The equation just written is unaltered if we interchange accented and unaccented * Chasles, Llourille, 1S57. Schroter, Crelte, Vol. lvi. t Joacbimsthal, Crelle, Vol. LVI., p. 15. Cremona, Crelle, Vol. LVIII., p, 146. 266 CLASSIFICATION OP CURVES. letters. Hence if a point A be in the plane of points of contact, corresponding to any point B, B will be in the plane in like manner corresponding to A. And again, the planes which thus correspond to all the points of a line AB pass through a fixed right line, namely the intersection of the planes corresponding to A and B. The relation between the lines is plainly reci- procal. To any plane of the system will correspond in this sense the corresponding point of the system ; and to a line in two planes corresponds a chord joining two points. The three points where any plane Aa + Bb + Cc + Dd meets the curve have their th given by the equation Dt + Ctf + Bt + A = 0, and when this is a perfect cube, the plane is a plane of the system. From this it follows at once, as Joachimsthal has remarked, that any plane drawn through the intersection of two real planes of the system meets the curve in but one real point. For in such a case the cubic just written is the sum of two cubes and has but one real factor. 332. We have seen (Art. 130) that a twisted cubic is the locus of the poles of a fixed plane with regard to a system of quadrics having a common curve. More generally such a curve is expressed by the result of the elimination of A between the system of equations \a = a, \b = b', Ac = c. Now since the anharmonic ratio of four planes whose equations are of the form \a = a, X'a = a, &c. depends only on the coefficients A, V, &c. (see Conies, Art. 59), this mode of obtaining the equation of the cubic may be interpreted as follows : Let there be a system of planes through any line aa, a homographic system through any other line bb', and a third through cc', then the locus of the intersection of three corre- sponding planes of the systems is a twisted cubic. The lines aa', bb', cc are evidently lines through two points, or chords of the cubic. Reciprocally, if three right lines be homo- graphically divided, the plane of three corresponding points envelopes the developable generated by a twisted cubic, and the three right lines are " lines in two planes" of the system. The line joining two corresponding points of two homo- graphically divided lines, touches a conic when the lines are CLASSIFICATION OF CURVES. 267 in one plane, and generates a hyperboloid when they are not. Hence given a series of points on a right line and a homo- graphic series either of tangents to a conic or of generators of a hyperboloid, the planes joining each point to the corre- sponding line envelope a developable as above stated. Ex. If the four faces of ;<, tetrahedron pass through fixed lines, and three ver- tices move in fixed lines, the locus of the remaining vertex is a twisted cubic. Any number of positions of the base form ii system of planes which divide homo- graphically the three lines on which the comers of the base move, whence it follows that the three planes which intersect in the vertex are corresponding planes of three homographic systems. 333. From the theorems of the last article it follows con- versely that "the planes joining four fixed points of the system to any variable line through two points form a constant an- harmonic system" and " four fixed planes of the system divide any ' line in two planes' in a constant anliarmonic ratio." It is very easy to prove these theorems independently. Thus we know that the section of the developable by any plane -4* of the system, consists of the corresponding line a of the system twice, together with a conic to which all other planes of the system are tangents. Thus then the anharmonic property of the tangents to a conic shows that four of these planes cut any two lines in two planes, AB, A in the same anharmonic ratio ; and in like manner A is cut in the same ratio as CD. As a particular case of these theorems, since the lines of the system are both lines in two planes and lines through two points ; four fixed planes of the system cut all the lines of the system in the same anharmonic ratio ; and the planes joining four fixed points of the system to all the lines of the system are a constant anharmonic system. Many particular inferences may be drawn from these theorems, as at Conies, p. 282, which see. Thus consider four points a, /8, 7, S; and let us express that the planes joining them to the lines a, b, and aft, cut the line 78 homographically. Let the planes A, B meet 70 in * It is often convenient to denote the planes of the system by capital letters, the corresponding lines by italics, and the corresponding point by Greek letters. 268 CLASSIFICATION OF CURVES. points t, t'. Let the planes joining the line a to /3, and the line b to a meet 78 in k, h'. Then we have If the points t, k' coincide, it follows from the first equation that the points Jc, t' coincide, and from the second that the points t, t', 7, § are a harmonic system. Thus we obtain Prof. Cremona's theorem, that if a series of chords meet the line of intersection of any plane A with the line joining the corresponding point a to any line b of the system, then they will also meet the line of intersection of the plane B with the line joining /9 to a ; and will be cut harmonically where they meet these two lines and where they meet the curve. The reader will have no difficulty in seeing when it will happen that one of these lines passes to infinity, in which case the other line becomes a diameter. 334. We have seen that the sections of the developable by the planes of the system are conies. We may therefore investigate the locus of the centres of these conies, or more generally the locus of the poles with respect to these conies of the intersections of their planes with a fixed plane. Since in every plane we can draw a " line in two planes" we may suppose that the fixed plane passes through the intersection of two planes of the system A, B. Now consider the section by any other plane C; the traces on that plane of A and B are tangents to that section, and the pole of any line through their intersection lies on their chord of contact, that is to say, lies on the line joining the points where the lines of the system a, b meet G. But since all planes of the system cut the lines a, b homographically, the joining lines generate a hyperboloid of one sheet, of which a and b are generators. However then the plane be drawn through the line AB, the locus of poles is this hyperboloid. But further, it is evident that the pole of any plane through the intersection of A, B lies in the plane which is the harmonic conjugate of that plane with respect to those tangent planes. The locus therefore which we seek is a plane conic. It is plain also from the construction that since the poles when any plane CLASSIFICATION OF CUKVES. 269 A + \B is taken for the fixed plane, lie on a conic in the plane A — XB; conversely the locus when the latter is taken for fixed plane is a conic in the former plane.* 335. In conclusion, it is obvious enough that cubics may- be divided into four species according to the different sections of the curve by the plane at infinity. Thus that plane may either meet the curve in three real points ; in one real and two imaginary points; in one real and two coincident points, that is to say, a line of the system may be at infinity; or lastly, in three coincident points, that is to say, a plane of the system may be altogether at infinity. These species have been called the cubical hyperbola, cubical ellipse, cubical hyper- bolic parabola, and cubical parabola. It is plain that when the curve has real points at infinity, it has branches proceeding to infinity, the lines of the system corresponding to the points at infinity being asymptotes to the curve. But when the line of the system is itself at infinity, as in the third and fourth cases, the branches of the curve are of a parabolic form pro- ceeding to infinity without tending to approach to any finite asymptote. Since the quadric cones which contain the curve become cylinders when their vertex passes to infinity, it is plain that three quadric cyclinders can be described containing the curve, the edges of the cylinders being parallel to the asvmptotes. Of course in the case of the cubical ellipse two of these cylinders are imaginary: in the case of the hyper- bolic parabola there are only two cylinders, one of which is parabolic, and in the case of the cubical parabola there is but one cylinder which is parabolic. It follows, from Art. 330, that in the case of the cubical ellipse the plane at infinity contains a real line in two planes, which is imaginary in the case of the cubical hyperbola. That is to say, in the former case, but not in the latter, two planes of the system can be parallel. From the anharmonic property we infer that in the case of the cubical parabola three planes of the system divide in a constant ratio all the lines of the * The theorems of this article are taken from Prof. Cremona's paper. 270 CLASSIFICATION OF CUEVES. system. In this case all the planes of the system cut the developable in parabolas. The system may be regarded as the envelope of xt - 3yf + Szt - d where d is constant. For further details we refer to Prof. Cremona's Memoir. 336. We proceed now to the classification of curves of higher orders. We have proved (Art. 325) that through any curve can be described two surfaces, the lowest values of whose de- grees in each case there is no difficulty in determining. It is evident then on the other hand that if commencing with the simplest values of ft, and v we discuss all the different cases of the intersection of two surfaces whose degrees are p, and v, we shall include all possible curves up to the r tu order, the value of this limit r being in each case easy to find when fi and v are given. With a view to such a discussion we commence by investigating the characteristics of the curve of intersection of two surfaces.* We have obviously m = pv, and if the surfaces are without multiple lines and do not touch, as we shall suppose they do not, their curve of intersection has no multiple points (Art. 194), and therefore (3 — 0. In order to determine completely the character of the system, it is necessary to know one more of its singularities, and we choose to seek for r, the degree of the developable generated by the tangents. Now this developable is got by eliminating x'y'z' between the four equations £7'=o, v=o, u;x+u; y +u;z+u; w =o, f>+f;^+f:>+f>=o. These equations are respectively of the degrees //., v, fi — 1, v — 1 : and since only the last two contain xye, these variables enter into the result in the degree fiv (v — 1) + (IV (ft, — 1) =/£V (fl + V — 2). Otherwise thus : the condition that a line of the system should intersect the arbitrary line ax + (3y + yz + Biv, ax + f3'y + y'z + S'w * The theory explained in the remainder of this section is taken from a paper dated July, 1849, which I published in the Cambridge and Dublin Mathematical Journal, Vol. v,, p. 23. CLASSIFICATION OF CURVES. 271 «, p, 7, a', ft', 7', v» v« u a , V V V 8 S' v. 0, which is evidently of the degree fi + v-2. This denotes a surface which is the locus of the points, the intersection of whose polar planes with respect to U and V meet the arbitrary line. And the points where this locus meets the curve UV are the points for which the tangents to that curve meet the arbitrary line. Having then m = /j,v, /3 = 0, r = (iv(/Ji+ v — 2) 7 we find, by Art. 3-2-2, n = 3fiv{(i + v-3), a = 2fiv(3fi + 3v-10), 2h = /iv{[i-l)[v-l), 2,j = fiv {pv (3/i + 3v - 9) a - 22 [/i + v)+ 71}, 2.i- = fiy {pv (p + v- 2)- - 4 (/t + v) + 8}, 2// = ftv {/J-v (fj, + v-2Y- 10 {fi + v) + 28}. 337. We verify this result by determining independently h the number of "lines through two points" which can pass through a given point, that is to say, the number of lines which can be drawn through a given point so as to pass through two points of the intersection of U and V. For this purpose it is necessary to remind the reader of the method employed, p. 83, in order to find the equation of the cone whose vertex is any point and which passes through the intersection of U and V. Let us suppose that the vertex of the cone is taken on the curve so as to have both U and V= for the co- ordinates of the vertex. Then it appears, from p. 83, that the equation of the cone is the result of eliminating X between 8 U+ A S 2 U+ -^- 8 s U+ &c. = 0, * r +h* v+ U3i 8T+&c. = 0. These equations in X are of the degrees /* — 1, v-1; 8U 7 8*U 7 &c, contain the co-ordinates x'y'z\ xyz in the degrees ft- 1, 1 ; 272 CLASSIFICATION OF CURVES. fM — 2, 2, &c. A specimen term of the result is ($U)»~ 1 V I T 1 . Thus it appears that the result contains the variables xyz in the degree v— l + v(/i — 1) = fiv — 1 ; while it contains xyz' in the degree (/* - 1) (v - 1). Every edge of this cone of the degree fiv— 1, whose vertex is a point on the curve, is of course a "line through two points." If now in this cone we consider the co-ordinates of any point xyz on the cone as known and xyz as sought, this equation of the degree (/a — l)(v — 1) combined with the equations U and V determine the "points" belonging to all the "lines through two points" which can pass through the assumed point. The total number of such points is therefore /av(/x — 1) (y- 1), and the number of lines through two points is of course half this. The number determined in this article, I call the number of apparent double points in the intersection of two surfaces, for to an eye placed at any point two branches of a curve appear to intersect if any line drawn through the eye meet both branches. 338. Let us now consider the case when the curve UV has also actual double points ; that is to say, when the two surfaces touch in one or more points. Now in this case, the number of apparent double points remains precisely the same as in the last article, and the cone, standing on the curve of intersection and whose vertex is any point, has as double edges the lines joining the vertex to the points of contact in addition to the number determined in the last article. It is easy to see that the investigation of the last article does not include the lines joining an arbitrary point to the points of contact. That investigation determines the number of cases when the radius vector from any point has two values the same for both surfaces, but the radius vector to a point of contact has only one value the same for both, since the point of contact is not a double point on either surface. Every point of contact then adds one to the number of double edges on the cone, and therefore diminishes the degree of the de- velopable by two. This might also be deduced from Art. 336 since the surface generated by the tangents to the curve of CLASSIFICATION OF CURVES. 273 intersection must include as a factor the tangent plane at a point of contact, since every tangent line in that plane touches the curve of intersection. If the surfaces have stationary contact at any point (Art. 195) the line joining this point to the vertex of the cone is a cuspidal edge of that cone. If then the surfaces touch in t points of ordinary contact and in |3 of stationary contact, we have m = ftv, /8 = /3, 2h = fiv(fi — 1) (v — 1) +2*, r = ftv (ft + v - 2) - 2t - 3/3, and the reader can calculate without difficulty how the other numbers in Art. 336 are to be modified. We can hence obtain a limit to the number of points at which two surfaces can touch if their intersection do not break up into curves of lower order ; for we have only to subtract the number of apparent double points from the maximum number of double points which a plane curve of the degree ftv can have (Higher Plane Curves, p. 31). 339. We shall now show that when the curve of inter- section of two surfaces breaks up into two simpler curves, the characteristics of these curves are so connected that, when those of the one are known, those of the other can be found. It was proved (Art. 337) that the points belonging to the " lines through two points" which pass through a given point are the intersection of the curve UV with a surface whose degree is (ft— 1) (v— 1). Suppose now that the curve of inter- section breaks up into two whose degrees are in and »»', where m + m' = ftv, then evidently the "two points" on any of these lines must either lie both on the curve m, both on the curve m' } or one on one curve and the other on the other. Let the number of lines through two points of the first curve be h, those for the second curve A', and let H be the number of lines which pass through a point on each curve, or, in other words, the number of apparent intersections of the curves. Considering then the points where each of the curves meets the surface of the degree (ft — 1) (v — 1), we have obviously the equations m(ft-l)(v-l) = 2h+H, m'(ft-l)(v-l) = 2h'-hff, whence 2 (A - A') = (m - rri) (ft - 1) (v - 1 ). T 274 CLASSIFICATION OF CURVES. Thus when m and h are known m' and Ji can be found. To take an example which we have already discussed, let the intersection of two quadrics consist in part of a right line (for which in - 1, A' = 0), then the remaining intersection must be of the third degree in = 3, and the equation above written determines h=l. 340. In like manner it was proved (Art. 336) that the locus of points, the intersection of whose polar planes with regard to U and V meets an arbitrary line, is a surface of the degree fj, + v - 2. The first curve meets this surface in the t points where the curves m and in intersect (since U and V touch at these points) and in the r points for which the tangent to the curve meets the arbitrary line. Thus then m (fi 4 v — 2) = r ■+ t, m [fj,+ v — 2) = r +t, (in — m') (fi + v — 2) = i — >•', an equation which can easily be proved to follow from that in the last article. The intersection of the cones which stand on the curves m, in consists of the t lines to the points of actual meeting of the curves and of the H lines of apparent intersection ; and the equation H+ 1 = mm is easily verified by using the values just found for H and t, remembering also that m'=jiv — m, r = m [m— 1) — 2k. 341. Having now established the principles which we shall have occasion to -employ, we resume our enumeration of the different species of curves of the fourth order. Every quartic curve lies on a quadric. For the quadric determined by nine points on the curve must altogether contain the curve (Art. 325). It is not generally true that a second quadric can be described through the curve ; there are therefore two principal families of quartics, viz. those which are the intersection of two quadrics, and those through which only one quadric can pass.* We commence with the curves of the first family. The character- * The existence of this second family of quartics was first pointed out in the Memoir already referred to. CLASSIFICATION OF CURVES. 275 istics of the intersection of two quadrics which do not touch are (Art. 336) mi = 4, n = 12, r = 8, a=16, /3 = 0, as =16, y = 8, #=38, 7i = 2. Several of these results can be established independently. Thus we have given (Art. 209) the equation of the developable generated by the tangents to the curve, which is of the eighth degree. It is there proved also that the developable has in each of its four principal planes a double line of the fourth order, whence x = 16* Again, it is shown, p. 163, that the equation of the osculating plane is Tu = T'v, (« and v being the tangent planes to U and V at the point), which contains the co-ordinates of the point of contact in the third degree. If then it be required to draw an osculating plane through any assumed point, the points of contact are determined as the intersections of the curve UV with a surface of the third degree, and the problem therefore admits of twelve solutions ; « = 12. Lastly, every generator of a quadric containing the curve is evidently a "line through two points" (Art. 339). Since then we can describe through any assumed point a quadric of the form U+ X V, the two generators of that quadric which pass through the point are two lines through two points ; or h = 2. The lines through two points may be otherwise found by the following construction, the truth of which it is easy to see : Draw a plane through the assumed point 0, and through the intersection of its polar planes with respect to the two quadrics, this plane meets the quartic in four points which lie on two right lines intersecting in 0. A quartic of this species is determined by eight points (Art. 126). 342. Secondly, let the two quadrics touch : then (Art. 338) the cone standing on the curve has a double edge more than in the former case, and the developable is of a degree less by two. Hence ■»i = 4, n = 6, r = 6; g = 6, h = 3; a = 4, /3 = 0; .r = 6, y=4. * It has been mentioned (p. 161) that the developable circumscribing tn-o quadrics has, as double Hues, a conic in each of the principal planes. The number y = 8 is thus accounted for. T2 276 CLASSIFICATION OF CURVES. Thirdly, the quadrlcs may touch at a stationary point, when we have m = 4, n = 4, r = 5 ; g = 2, h = 2 ; a = 1, /3 = 1 ; a? = 2, */ = 2. This system* may be expressed as the envelope of af+Gcf + ldt+e, where t is a variable parameter. The envelope is (ae + 3c 2 ) 3 = 27 (ace - ad' - c 3 ) 2 , which expanded contains a as a factor and so reduces to the fifth degree. The cuspidal edge is the intersection of ae + 3c z , ice - 3d*. Since a cone of the fourth degree cannot have more than three double edges, two quadrics cannot touch in more points than one, unless their curve of intersection break up into simpler curves. If two quadrics touch at two points on the same generator, this right line is common to the surfaces, and the intersection breaks up into a right line and a cubic. If they touch at two points not on the same generator, the intersection breaks up into two plane conies whose planes intersect in the line joining the points. 343. If a quartic curve be not the intersection of two quadrics it must be the partial intersection of a quadric and a cubic. We have already seen that the curve must lie on a quadric, and if through thirteen points on it, and six others which are not in the same plane,f we describe a cubic surface, it must contain the given curve. The intersection of this cubic with the quadric already found must be the given quartic together with a line of the second degree, and the apparent double points of the two curves are connected by the relation h-h' = 2. as appears on substituting in the formula of Art. 339 the values m = 4, tri = 2, /j, = 3, v = 2. When the line of the second degree is a plane curve (whether conic or two right lines), we have * I owe this remark to Mr. Cayley. t This limitation is necessary, otherwise the cubic might consist of the quadric and of a plane. Thus if a curve of the fifth order lie in a quadric it cannot be proved that a, cubic distinct from the quadric can contain the given curve - see Cambridge and Dublin Mathematical Journal, Vol, v. p, 27. CLASSIFICATION OF CURVES. 277 n = ; therefore h = 2, or the quartic is one of the species already examined having two apparent double points. It is easy to see otherwise that if a cubic and quadiic have a plane curve common, through their remaining intersection a second quadric can be drawn; for the equations of the quadric and cubic are of the form zio = u 2 , zv i = u a x, which intersect on i\ = xw. If, however, the cubic and quadric have common two right lines not in the same plane, this is a system having one apparent double point, since through any point can be drawn a transversal meeting both lines. Since then h'=l, h = 3 or these quartics have three apparent double points, and are therefore essentially distinct from those already discussed which cannot have more than two. The numerical character- istics of these curves are precisely the same as those of the first species in Art. 342, the cone standing on either curve having three double edges, and the difference being that one of the double edges in one case proceeds from an actual double point, while in the other they all proceed from apparent double points. This system of quartics is the reciprocal of that given by the envelope of a? + 4bf ■+ 6c? + 4dt + e. Moreover, this latter system has, in addition to its cuspidal curve of the sixth order, a nodal curve of the fourth which is of the kind now treated of. It is proved, as in Art 329, that these quartics are met in three points by all the generators of the quadiic on which they lie, which are of the same system as the lines common to the cubic and quadric, and are met once by the generators of the opposite system. The cone standing on the curve, whose vertex is any point of it, is then a cubic having a double edge, that double edge being one of the generators passing through the vertex of the quadiic which contains the curve. Thus while any cubic may be the projection of the inter- section of two quadiics, quartics of this second family can only be projected into cubics having a double point. The quadric may be considered as the surface generated by all the "lines through three points" of the curve. It is plain from what has been stated, that evert/ quartic, having three 278 CLASSIFICATION OF CURVES. apparent double points, may be considered as the intersection of a quadric with a cone of the third order having one of the generators of the quadric as a double edge. 344. Mr. Cayley has remarked that it is possible to de- scribe through eight points a quartic of this second family. We want to describe through the eight points a cone of the third degree having its vertex at one of them, and having a double edge, which edge shall be a generator of a quadric through the eight points. Now it was proved (Art. 341) that if a system of quadrics be described through eight points all the generators at any one of them lie on a cone of the third degree, which passes through the quartic curve of the first family determined by the eight points. Further, if 8, 8', S" be three cubical cones having a common vertex and passing through seven other points, \S+fiS'+vS" is the general equation of a cone fulfilling the same conditions ; and if it have a double edge, A,#, + /*$/ + vS", passes through that edge. Eliminating then X, /^, v between the three differentials, the locus of double edges is the cone of the sixth order s, (sjs: - s:'sj ■+ s, [8: &: - «') + s 3 (s;s: - s^) = o. The intersection then of this cone of the sixth degree with the other of the third determines right lines, through any of which can be described a quadric and a cubic cone fulfilling the given conditions. It is to be observed, however, that the lines connecting the assumed vertex with the seven other points are simple edges on one of these cones and double edges on the other, and these (equivalent to fourteen intersections) are irrelevant to the solution of the problem. Four quartics there- fore can be described through the points. 345. Mr. Cayley has directed my attention to a special case of this second family of quartics which I had omitted to notice. It is, when the curve has a linear inflexion of the kind noticed, note, p. 259 ; that is to say, when three consecutive points of the curve are on a right line. Such a point obviously cannot exist on a quartic of the first family ; for the line joining the three points must then be a. generator of both quadrics, whose CLASSIFICATION OF CURVES. 279 intersection would therefore break up into a line and a cubic; and would no longer be a quartic. Let us examine then in what case three consecutive planes of the system af+ibf+Gcf+idt+e can pass through the same line. If such a case occurs we may suppose that we have so transformed the equation that the singular point in question may answer to t = oo ; the three planes a, b, c, must therefore pass through the same line ; or c must be of the form Xa + fib. But we may then transform the equa- tion further by writing for t,t+6, and determining 6 so that the quantity multiplying b in the coefficient of f shall vanish. The system then is the envelope of a plane at+lbf+QXatf-t-idt+e. A still more special case is when X vanishes, or when the plane reduces to at 4, + ±b? + idt + e ; it is obvious then that we have two points of linear inflexion ; one answering to t = oo , the other to t = 0. The developable in this latter case is {ae-ibd) 3 = 2T(ad* + eby; which has for its edge of regression the intersection of ae — ibd with ad 2 + eb s ; but this consists of a curve of the fourth degree with the lines ab, de. This system then is one whose reciprocal is of the same nature; for we have m = » = 4, ft. = k = 3, a; = y = 4. And the section of the developable by any plane has six cusps, viz. the four poiuts where the plane meets the cuspidal edge, and the two where it meets the double generators ab. de. In the case previously noticed where c does not vanish but is equal to \a, there is but one point of linear inflexion ; the envelope in question is then the reciprocal of a system for which m = 4, « = 5, r = 6, 7«=3, & = 4, .r = 5, y=4. Another special case to be considered is when a curve has a double tangent ; such a line being doubly a line of the system is a double line on the developable. But this case does not occur in curves of the fourth order.* 346. There is no difficulty in carrying on this enumeration to curves of higher orders.f This is done with regard to curves * For other properties of curves of the fourth order, see papers by M. Chasles, Comptes Hernias, Vols. Liv. and lv. ; and by II. Cremona, Memoirs of the Bologna Academy, 18(51. + To complete the enumeration of curres up to the fourth order, it would be necessary to classify, according to their apparent double points, improper systems made 280 CLASSIFICATION OF CUKVES. of the fifth order in the memoir already cited. It is easy to see that besides plane quintics we have, I., quintics which are the partial intersection of a quadric and a cubic, the remaining intersection being a right line. These quintics have four ap- parent double points, and may besides have two actual nodal or cuspidal points. We may have, II., quintics with five apparent double points, and which may besides have one actual nodal or cuspidal point ; these curves being the partial inter- section of two cubics, the remaining intersection being a quartic of the second class. We may have, III., quintics with six apparent double points being the partial intersection of two cubics, the remaining intersection being an improper quartic with four apparent double points. To these may be added, IV., quintics with six apparent double points which are the partial intersection of a quadric and a quartic surface ; the re- maining intersection being three lines not in the same plane. 347. Instead of proceeding, as we have done, to enumerate the species of curves arranged according to their respective orders, we might have arranged our discussion according to the order of the developables generated, and have enumerated the different species of developables of the fourth, fifth, &c. orders. This is the method followed by Chasles, who has enumerated the species of developables up to the sixth order [Comptes Bendus, Vol. LIV.), and by Schwarz (Crelle, Vol. LXIV., p. 1) who has carried on his enumeration to the seventh order. Schwarz's discussion contains the answer to the following ques- tion started by Mr. Cayley : The equation considered, Art. 323, where the parameter enters rationally, denotes a single plane whose envelope is a class of developables which Mr. Cayley calls planar developables ; on the other hand, if the parameter up of simpler curves of lower orders. Thus, we have, for m = 2, h = 1, two lines not in the same plane ; m — 3, h = 1 , a conic and a line once meeting It ; h = 2, a conic and line not meeting it ; h = 3, three lines, no two of which are in the same plane ; m ~ 4, h = 2, a plane cubic and line once meeting it, or a twisted cubic and line twice meeting it, or two conies having two points common j m = 4, h — 3, a plane cubic and line not meeting it, or a twisted cubic and line once meeting it, or two conies having one point common ; m = 4, h = 4, a twisted cubic and non-intersecting line, or two non-intersecting conies ; h = 5, a conic and two lines meeting neither the conic nor each other ; h = 6, four lines no two of which are in the same plane. CLASSIFICATION OF CURVES. 281 entered by radicals, the equation rationalized would denote a system of planes whose envelope would therefore be called a multiplanar developable : now it is proposed to ascertain con- cerning each developable, what is, in this sense, the degree of its planarity. M. Schwarz has answered this question by shewing that the developables of the first seven orders are all planar. His method depends on the following principle established by Riemann * C relic, Liv., 133 : the co-ordinates of a plane curve can be expressed rationally as functions of a single variable, if it have its maximum number of double points; that is to say, if \ (^ - 1) - 2) - (« + S) = ; if this number be = 1, the co- ordinates can be expressed in terms of one variable, and the square root of an integral function of the third or fourth order in this variable ; and generally, it is on this numberf that the degree of simplicity of the expression of the co-ordinates of a curve in terms of one variable depends. By considering then a section of a developable, Schwarz shews that it is planar if £ (r - 1) (r - 2) - (m + x) = 0, Uplanar if it = 1, &c. 348. The discussion of the possible characteristics of a de- velopable of given order, depends on the principle (p. 261) that the section by a plane of the system is a curve of degree »• — 2 having in — 3 cusps. Thus, if the developable be of the fifth order the section by a plane of the system is a cubic ; and as this can have no more than one cusp, the edge of regression is at most of the fourth degree. And it cannot be of lower degree, since we have already seen that twisted cubics generate developables only of the fourth order. Hence the only de- velopablesj of the fifth order are those, considered p. 276, generated by a curve of the fourth order. * See also Clebsch's paper, Crelle, lxiii. 189. f Clebsch shows that this number is not only the same for a curve and its re- ciprocal, but also for auy curve (such as the evolute) derived so that one point of the derived curve answers to one point of the original. In like manner for curves of double curvature, the number }(r_ 1) (r - -2) - {m + x) = i(r - 1) (r - 2) - (» + y) = J(i»- 1) (in -2)-(h + fl = i (» - 1} (h - 2) - (i + o) is the same for every system derived in the manner just explained. J The properties of these developables are treated of by Professor Cremona, ComjJtes Bendus, Vol. LIT., p. 60-1. 282 CLASSIFICATION OF CUEVES. In the same manner the section of a developable of the sixth order by a plane of the system is a quartic, which may have one, two, or three cusps. We have therefore w = 4, 5, or 6 ; and, in like manner, n is confined within the same limits ; and therefore, p. 261, the section by the plane of the system is at most of the fifth class. Now a curve of the fourth degree with one cusp must have two other double points if it is only of the fifth class : and, if it have two cusps, it must have one other double point. In any case, therefore, this quartic has three double points, and the developable is planar. The case when the quartic has only one cusp (or m = 4) has been already considered. The edge of regression has a nodal point ; and the system is the reciprocal of the system which envelopes at + ibf + Get + idt + Affl = 0, where there is a double plane of the system answering to t — and also to t = co . If, again, the quartic section have three cusps, it is of the third class, and therefore for the developable n = 4. This then is also a case already discussed, Art. 343, the developable being the envelope of at + ibf + 6ct ■+ 4tdt + e = 0. Lastly, when the quartic has two cusps, it must, as we have seen, also have a double point, and therefore be of the fourth class. Hence n = 5, From the preceding formulae the charac- teristics of a system for which in = n = 5, r = 6, are g = k = 4 1 x = y = 5, a = /3 = 2; and, if we take the two stationary planes answering to t = co , t = 0, the system is the envelope of at + 5Xat + lOcf + lOdt + 5/jft +/= 0. M. Schwarz has noticed that the stationary tangent planes may be replaced by a triple tangent plane ; that is to say, the system may be the envelope of at + b\at + Wftaf + lOdt + bet +/= 0. I have not examined with any care the theory of the effects of triple points of the curve of intersection of two surfaces on the number of its apparent double points. But (considering CLASSIFICATION OF CURVES. 233 the case where X. and ji vanish in the equation last written) if we make h and e = in the equations which I have given {Cambridge and Dublin Mathematical Journal, v. 158) for the edge of regression of the developable which results as the envelope of a quintic, the edge of regression is found to be the intersection of 2tf — odf, with af 2 — 12(T<3. And this intersection is the right line ef with a curve of the fifth order, having the point dcf for a triple point. For this being a double point on each surface is a quadruple point on their curve of intersection ; and since the right line passes through the point def, the re- maining curve has a triple point at that point. 349. We shall conclude this section by applying some of the results already obtained in it, to the solution of a problem which occasionally presents itself : " Three surfaces whose degrees are /x, v, p have a certain curve common to all three ; how many of their fivp points of intersection are absorbed by the curve? In other words, in how many points do the surfaces intersect in addition to this common curve?" Now let the first two surfaces intersect in the given curve, whose degree is ?m, and in a complementary curve fiv — m, then the points of intersection not on the first curve must be included in the (fiv — in) p intersections of the latter curve with the third surface. But some of these intersections are on the curve w, since it was proved (Art. 340) that the latter curve intersects the complementary curve in m (//. + v - 2) - r points. Deducting this number from (fj.v — m) p we find that the sur- faces intersect in fivp-m(fi + v + p-2) + r points which are not on the curve m; or that the common curve absorbs" m (u, + v + p — 2) — r points of intersection. In precisely the same way we solve the corresponding question if the common curve be a double curve on the sur- face p. "We have then to subtract from the number [fiv — m) p, •2 {m ffi + v — ^—r} points, and we find that the common curve diminishes the intersections by m (p + 2/t + 2v - 4) - 1r points. These numbers expressed in terms of the apparent double points of the curve m are m{/i + v + p--m-i) + 2h and m (p + 2/* -t- 2v - 2m - 2) + ih. 284 CLASSIFICATION OF CURVES. 350. The last article enables us to answer the question: " If the intersection of two surfaces is in part a curve of order m which is a double curve on one of the surfaces; in how many points does it meet the complementary curve of inter- section?" Thus, in the example last considered, the surfaces fi, p intersect in a double curve m and a complementary curve lip — 2m; and the points of intersection of the three surfaces are got by subtracting from (fip - 2m) v the number of inter- sections of the double curve with the complementary. Hence [lip — 2m) v-i = p,vp - m (p + 2/a + 2v — 4) + 2r, whence i = m (p + 2[t — 4) — 2r. We can verify this formula when the curve m is the complete intersection of two surfaces Z7, V whose degrees are k and I. Then p is of the form AW + BUV+ CV* where A is of the degree p — 2k, &c, and /*. is of the form D U+ EV where D is of the degree /i — k. The intersections of the double curve with the complementary are the points for which one of the tangent planes to one surface at a point on the double curve coincides with the tangent plane to the other surface. They are therefore the intersection of the curve UV with the surface AE'-BDE+CD 1 which is of the degree p + 2fi-2 [k+l). The number of intersections is kl [p + 2/4 — 2 (k + 1)} which coin- cides with the formula already obtained on putting kl=m, kl(k + l-2)=r. 351. From the preceding article we can show how, when two surfaces partially intersect in a curve which is a double curve on one of them, the singularities of this curve and its complementary are connected. The first equation of Art. 340 ceases to be applicable because the surface //. + v - 2 altogether contains the double curve, but the second equation gives us m (/* + v — 2) = 2t + r = r + 2m (fi + 2v — 4) - 4r, whence 4r — r = (2m — m')(fi + v—2) + 2m (y — 2). In like manner we find that the apparent double points of the two curves are connected by the relation 8h-2h' = (2m-m.') -1) (v-1) - 2m(v-l). NON-PROJECTIVE PROPERTIES OF CURVES. 285 Thus when a quadric passes through a double line on a cubic the remaining intersection is of the fourth degree, of the sixth rank, and has three apparent double points. SECTION III. NON-PROJECTIVE PROPERTIES OF CURVES. 352. As we shall more than once in this section have occasion to consider lines indefinitely close to each other, it is convenient to commence by showing how some of the formulae obtained in the first chapter are modified when the lines considered are indefinitely near. We proved (Art. 14) that the angle of inclination of two lines is given by the formula sin 2 = (cos/3 cosy' - cosjS' cosy) 2 + (cosy cosa' — cosy' cosa) 2 + (cosa cos$' — cosa' cos/3) 2 . When the lines are indefinitely near we may substitute for cosa', cosa + 8 cosa, &c, and put sin#= 80, when we have 86' = (cosyS 8 cosy — cosy 8 cos/3) a -f (cosy 8 cosa — cosa 8 cosy) 2 -f (cosa ScosyS- cos^S Scosa) 2 . If the direction-cosines of any line be - , — , - where r ' r ' r f + m * + }i 2 = j- a , the preceding formula gives r<86* = (mSn - n8mf + (n8l - 18nf + (18m - mSZ) 2 . Since we have cos 2 a + cos 2 /3 + cos s y = 1 ; cosa Scosa + cosj3 Scos^ + cosy Scosy = 0, if we square the latter equation and add it to the expression for S0 2 , we get another useful form 86* = {8 cosa) 2 4 (8 cos/3) 2 + (S cosy) 2 . It was proved (Art. 15) that cos/S cosy' — cos/3' cosy, &c. are proportional to the direction-cosines of the perpendicular to the plane of the two lines. It follows then that the direc- tion-cosines of the perpendicular to the plane of the consecutive lines just considered are proportional to mSn — n8m t nSl — ISn, 18m - mhl, the common divisor being r*80. 286 NON-PROJECTIVE PEOPEETIES OF CURVES. Again, it was proved (Art. 43) that the direction-cosines of the line bisecting the external angle made with each other by two lines are proportional to cos a — cos a, cos /3 — cos /3', cos 7 — cos 7', &c. Hence when two lines are indefinitely near, the direction-cosines of a line drawn in their plane, and perpendicular to their common direction are proportional to S cosot, 8 cos/3, S C0S7, the common divisor being 80. 353. We proved (Art. 312) that the direction-cosines of doc t In dz a tangent to a curve are -=- , -j- , -=- , while, if the curve be given as the intersection of two surfaces, these cosines are proportional to MN' - MN, NL - N'L, LM' - L'M, where L, M, &c. denote the first differential coefficients. An infinity of normal lines can evidently be drawn at any point of the curve. Of these, two have been distinguished by special names ; viz. the normal which lies in the osculating plane, which is commonly called the principal normal ; and the normal perpendicular to that plane, which being normal to two consecutive elements of fhe curve, has been called by M. Saint- Venant the Binormal. All the normals lie in the plane perpendicular to the tangent line, viz. [x — x) dx + [y — y) dy+iz — z') dz = in the one notation ; or in the other (MN' - M'N) [x - x') + {NL' - N'L) [y - y') + {LM'-L'M)(z-z')=0. 354. Let us consider now the equation of the osculating plane. Since it contains two consecutive tangents of the curve, its direction-cosines (Art. 352) are proportional to dyd'z — dzd'*y, dzd 2 x — dxd' 2 z 1 dxd 2 y — dyd' 2 x, quantities which for brevity we shall call X, Y, Z. The equa- tion of the osculating plane is therefore X(x-x')+Y(y-y'}+Z(z-z') = 0, NON-PEOJECTIVE PROPERTIES OP CURVES. 287 The same equation might have been obtained (by Art. 30) by forming the equation of the plane joining the three con- secutive points x'y'z' • x + dx, y' + dy\ z + dz' ; x + Ux + d*x, y' + <2dy' + d\j\ z + 2dz' + d*z. In applying this formula we may simplify it by taking one of the co-ordinates at pleasure as the independent variable, and so making d*x, d*y or d*s = 0. 355. In order to be able to illustrate by an example the application of the formula? of this section, it is convenient here to form the equations and state some of the properties of the helix or curve formed by the thread of a screw. The helix may be defined as the form assumed by a right line traced in any plane when that plane is wrapped round the surface of a right cylinder.* From this definition the equations of the helix are easily obtained. The equation of any right line y = mx ex- presses that the ordinate is proportional to the intercept which that ordinate makes on the axis of x. If now the plane of the right line be wrapped round a right cylinder so that the axis of x may coincide with the circular base, the right line will become a helix, and the ordinate of any point of the curve will be proportional to the intercept, measured along the circle, which that ordinate makes on the circular base, counting from any fixed point on it. Thus the co-ordinates of the projection on the plane of the base, of any point of the helix are of the form x = a cos 0, y = a sin 0, where a is the radius of the circular base. But the height z has been just proved to be proportional to the arc 0. Hence the equa- tions of the helix are x = a cos j , y = a sin -=- , whence also x 2 + y' = d\ We plainly get the same values for x and y when the arc in- creases by 27r, or when z increases by 2irh ; hence the interval between the threads of the screw is 2Trh. * Conversely *■ heUx becomes a right line when the cylinder on which it ia traced ia developed into a plane, and is therefore a geodesic on the cylinder (Art. 304). 288 NON-PKOJECTIVE PROPERTIES OF CURVES. Since we have a . z , v -, -, a z 7 x 7 dx = -r sm- dz = ~j_dz, dy = ^ cosj-dz = ^dz, we have ds*= a * dz\ It follows that ■£ is constant, or the angle made by the tangent to the helix with the axis of z (which is the direction of the generators of the cylinder) is constant. It is easy to see that this is the same as the angle made with the generators by the line into which the helix is developed when the cylinder is developed into a plane. The length of the arc of the curve is evidently in a constant ratio to the height ascended. The equations of the tangent are (Art. 312) x — x y—y'_ z — z y x h If then x and y be the co-ordinates of the point where the tangent pierces the plane of the base, we have from the pre- ceding equations (x - x'f +{y-y'Y~ (a" + y'*) J = « 2 J , or the distance between the foot of the tangent and the pro- jection of the point of contact is equal to the arc which measures the distance along the circle of that projection from the initial point. This also can be proved geometrically, for if we imagine the cylinder developed out on the tangent plane, the helix will coincide with the tangent line, and the line joining the foot of the tangent to the projection of the point of contact will be the arc of the circle developed into a right line. Thus then the locus of the points where the tangent meets the base is the involute of the circle. The equation of the normal plane is y'x — x'y = h(z — z'). To find the equation of the osculating plane, we have d*x = —72 ocdz*, d'y = — p ydz'\ d*z = 0, N0N-PE0JECT1VE PRO?ERTIES OF CURVES. 289 whence the equation of the osculating plane is h (y'x — x'y) + a* (z — z') = 0. The form of the equation shows that the osculating plane makes a constant angle with the plane of the base. We leave it as an exercise to the reader to find the tangent, normal plane, and osculating plane of the intersection of two central quadrics. 356. We can give the equation of the osculating plane a form more convenient in practice when the curve is given as the intersection of two surfaces U, V. Since the osculating plane passes through the tangent line, its equation must be of the form \ [Lx 4 My 4 Nz 4 Pw) = ft {L'x 4 M'y + N'z + P'w), where Lx + &c. is the tangent plane to the first surface. This equation is identically satisfied by the co-ordinates of a point common to the two surfaces, and by those of a consecutive point; and on substituting the co-ordinates of a second con- secutive point, we get t t=Ld*x+Md>y+Nd i z+Pd*w 1 \=L'd*x+M'd 2 y+Nd i z+P'd 2 w. But differentiating the equation Ldx 4 Mdy 4 Ndz + Pdw = 0, we get Ld*x + Md*y + Nd*z + Pd'w = - V\ where V = adx* + bdy' + cdz* + ddw* -f lldydz 4 2mdzdx + 2ndxdy + 2pdxdw -f 2qdydw + Srdzdw, where a, 5, &c. are the second differential coefficients. Now dx, &c. satisfy the equations Ldx + Mdy 4 Ndz 4 Pdw = 0, 11 dx 4 M'dy 4 N'dz + Pdw = ; and since we may either, as in ordinary Cartesian equations, take w as constant ; or else x, or y, or z ; or more generally may take any linear function of these co-ordinates as constant ; wo may therefore add to the two preceding equations the arbitrary equation adx 4 fidy 4 ydz 4 Sdio = 0. u 290 NON-PKOJECTIVE PROPERTIES OF CURVES. Now it can easily be verified that if we substitute in any quadric the intersection of three planes Lx + My+Nz+Pw, L'x+M'y + N'z + P'w, ax + By + yz 4 Bw, the residt U' will be proportional to the determinant (see p. 48) a, «, in, Pi L, L\ a », ft, h ft % s Now this determinant may be reduced by subtracting from the fifth column multiplied by (m — 1) the sum of the first four columns, multiplied respectively by x, y, z, w; when the whole of the fifth column vanishes except the last row which becomes — (ax + By + yz + ozo). In like manner we may then subtract from the fifth row multiplied by (m — 1) the sum of the first four rows multiplied respectively by x, y, z, w, when in like manner the whole of the fifth row vanishes except the fifth column which is — {ax + fiy+yz + Bw). Thus the determinant reduces to (ax + fty + yz + Bwf [m - 1)" a, «, m, p, L »j ft, l , q.i M' »2, h c, r, N' Pi 1, r, d, P L\ M\ N', P If we call the determinant last written 8 and the corresponding determinant for the other equation 8', the equation of the osculating plane is S' r ,,.„.„, 8 (Lx 4 My+ Nz + Pw) ■■ [m-iy [L'x + M'y+N'z + P'w).* * This equation is due to M. Hesse, see CreUe's Journal, Vol. in, NON-PROJECTIVE PROPERTIES OF CURVES. 291 This equation has been verified in the case of two quadrics, see note, p. 163. Ex. 1. To find the osculating plane of ax 2 + bf + cz- + dm 2 , a'x 2 + b'y 2 + c'z 2 + d'w 2 . Ans. (flit - ba') (ac' - ca') (ad' - da') x' s x + (ba' - b'a) (be' - b'c) (bd' - b'd) y'*y + (ca' - c'a) (cb' - c'b) (cd' - c'd) z' 3 z + (da' - d'a) (db' - d'b) (dc' - d'e) ro' 3 w = 0. Ex. 2. To find the osculating plane of the line of curvature x 2 y 1 z- . x 2 y 2 z 2 a' + b* + c 5 ~ ' a 7 " 2 + V 2 + 7* ~ l ' . a" 2 xx' b" 2 mj c" 2 zz' , 357. The condition that four points should lie in one plane, or, in other words, that a point on the curve should be the point of contact of a stationary plane, is got by substituting in the equation of the plane through three consecutive points, the co-ordinates of a fourth consecutive point. Thus from the equation of Art. 351 the condition required is the determinant d*x(dyd*z - dzd*y) + d 3 y (dzd*x-dxd*z) +d 3 z[dxd*y - dyd 2 z)=0. If a curve in space be a plane curve, this condition must be fulfilled by the co-ordinates of every point of it. When the curve is given as the intersection of two surfaces C, F, Clebsch works out as follows (see Crelle, lxiii. 1) the condition for a point of osculation. Writing for brevity 8=(m-iyT, 8'={n-iyT the equation given in the last article for the osculating plane is (T'L-TL')x+(TM-TM , )y+{T'y-TX')z+(TP-TF)w=Q, and the equation of a consecutive osculating plane differs from this by terms ( T'dL + LdT- TdL' - L'dT) x + &c. = 0. And in order that the two planes may coincide, introducing an arbitrary differential dt, we must have the four equations T'dL 4 LIT - TdL - LdT= ( TL - TL') dt, &c If now we write T= AL' + BM' 4 CN' + DP, T = A'L + B'M+ CX+ B'P, U2 292 NON-PROJECTIVE PROPERTIES OF CURVES. A-* — R-iiLL & c where .4, 5, &c. are proportional to minors of the determinant 8, and where in fact ■dL n *dar we must have AL + BM+ CN+ DP= 0, AdL + BdM+ CdN+ DdP= 0, A'L' + &c. = 0, J.VZL' + &c. = ; for if in the determinant S, we substitute for the last column, either L, M, N, P; or dL, dM, dN, dP, it is easy to see that the determinant vanishes. Multiply then the four equations last considered by A, B, C, D respectively, and add ; and we have, after dividing by T, dT+ -| (§ dL' + §, dM' + §, dN' + § dP) = Tdt, which we may write dT+^d(T) = Tdt, where by d ( T) we may mean the differential of T considered merely as a function of L', M', N', P' ; a, b, &c. being regarded as constants. Similarly we have dT + \d(T) = Tdt. Let us now write at full length for dT, T x dx + T s dy + &c; and elimi- nate dx, dy, dz, dw, dt between the two equations just obtained, and the three conditions, p. 289, which connect dx, dy, dz, dw, when we obtain the required condition in the form of a de- terminant ^, + OT, T 2 + \{T^ T a + ^(T 3 ), T t +${T A \ T t;+\{t;), t;+\{t;), t; + ^{t;), t;+%[t;), t =0. Now T is a function of x, y, z, w of the degree 3m + 2n — 8 but when regard is paid only to the xyzw which enter into L, M', &c, (T) is of the degree 2 (n — 1) ; if therefore we multiply the first four columns by x, y, z, w respectively and subtract from 3(m + n — 3) times the last column, the first four terms of the last column vanish ; and the equation just L, M, N, p, L', W, N', Py «j 0, y, 8, NON-PROJECTIVE PROPERTIES OF CURVES. 293 written may be reduced by cancelling the fifth row and column of the determinant. The condition that we have just obtained is of the degree &m 4 6« - 20 in the coefficients of U and V, as might be inferred from the value of a, Art. 336. If the surfaces IT and V are quadrics, and therefore the coefficients «, b, &c. really constant, (TJ, (T 8 ) &c. are identical with T^ T 3l &c. and the condition that we have obtained is the result of equating to nothing the Jacobian of the four surfaces T, T\ U, V. 358. We shall next consider the circle determined by three consecutive points of the curve, which, as in plane curves, is called the circle of curvature. It obviously lies in the oscu- lating plane : its centre is the intersection of the traces on that plane, by two consecutive normal planes; and its radius is commonly called the radius of absolute curvature, to dis- tinguish it from the radius of spherical curvature, which is the radius of the sphere determined by four consecutive points on the curve, and which will be investigated presently. If through the centre of a circle a line be drawn perpendicular to its plane, any point on this line is equidistant from all the points of the circle, and may be called a pole of the circle. Now the intersection of two consecutive normal planes, evidently passes through the centre of the circle of curvature, and is perpendicular to its plane. Monge has therefore called the lines of intersection of two consecutive normal planes, the polar lines of the surface. It is evident that all the normal planes en- velope a developable of which these polar lines are the generators, and which accordingly has been called the polar surface. We shall presently state some properties of this surface. The polar line is evidently parallel to the line called the Binomial (Art. 353). 359. In order to obtain the radius of curvature we shall first calculate the angle of contact, that is to say, the angle made with each other by two consecutive tangents to the curve. The direction-cosines of the tangent being -^- , -~- , -^ , 294 NON-PROJECTIVE PROPERTIES OF CURVES. it follows from Art. 352 that dd the angle between two con- secutive tangents is given by either of the formulae or &w=x 2 +r a + z i! , where X= dyd^z - dzd'*y, &c. The truth of the latter formula may be seen geometrically : for the right-hand side of the equation denotes the square of double the triangle formed by three consecutive points (Art. 31) ; but two sides of this triangle are each ds, and the angle between them is dd, hence double the area is ds'dd. If now ds be the element of the arc, the tangents at the extremities of which make with each other the angle dd, then since the angle made with each other by two tangents to a circle is equal to the angle that their points of contact subtend at its centre, we have pdd — ds. And the element of the arc and the two tangents being common to the curve and the circle of curvature, the radius of curvature is given by the formula ds . „ ds 1 '%+w+m . . ds" or p - X *+Y* + Z*' Ex. To find the radius of curvature of the helix. Using the formulae of Art. 355, a 2 + h- we find p — ; or the radius of curvature is constant. * By performing the differentiations indicated, another value for dd 2 is found without difficulty, ds 2 d6 2 = (d 2 xf + {d 2 yf + (d 2 z) 2 - {d 2 s) 2 . This formula may also be proved geometrically. Let AB, BC be two consecutive elements of the curve ; AD a line parallel and equal to BC; then since the projections of BC on the axes are dx + d 2 x, dy + d 2 y, dz + d 2 z, it is plain that the projections on the axes of the diagonal BD are d 2 x, d 2 y, d 2 z, whence BD 2 = {d 2 x) 2 + (dd d d~s>d8 d Js> orp ~dT' p ~a7' p ~dT- If x\ y\ z' be the co-ordinates of a point on the curve, and x, y, z those of the centre of curvature, then the projec- tions of the radius of curvature on the axes are x' — x,y'— y, z' — s] but they are also p cos a, p cos/3, p cosy. Putting in then for cos a, cos/3, cos 7 their values just found, the co-ordinates of the centre of curvature are determined by the equations , dx ,dy -.dz d-r=- d-f- d-r , „ as , . j as , ..as *-*=p-2Tiy-y=p — . *~*=p — • 361. When a curve is given as the intersection of two surfaces which cut at right angles, an expression for the radius of curvature can be easily obtained. Let r and r be the radii of curvature of the normal sections of the two surfaces, the sections being made along the tangent to the curve; and let be the angle which the osculating plane makes with the first normal plane : then, by Meunier's theorem, we have p = r cos0 and also p = r sin$, whence ~a = -^ + 75 • The same equations determine the osculating plane by the T formula tan<£ = - . T r If the angle which the surfaces make with each other be a>, the corresponding formula is sin*&) 112 cosw = ""2 T ~7a p- r r rr We can hence obtain an expression for the radius of cur- vature of a curve given as the intersection of two surfaces. 296 NON-PROJECTIVE PROPERTIES OF CURVES. We may write L* + M* + 2P = IP, L" + M'* + N'* = R' 2 ; and we have LL + MM' + NN' COS CO = ■ RR' . , {MN' - M'Nf + {NL' - N 'Lf + [LM' - L'Mf bib 01 WW2 We must then substitute in the formula of Art. 290, cosa = MN'-M'N , cos/3 = NL'-N'L RR sina> ' RR' sinai The denominator of that formula becomes a, n, in, L, L' n, b, I, M, M' in, I, c, N, N' L, M, N l; w, n' which reduced, as in Art. 356, becomes LM'-L'M cos7 = — vtv—- • ' RR sin to [m-iy 7 S. And we In like manner (m- 1)* R*R' 2 sin* co S {n-iyR 2 R' 3 sm i co S' Whence —, = -. —, r , 6n , 4 . 6 p l (m — If R 6 R i sm 6 co + S" 2SS' cosw [n - If R'R" sin 6 o> (m - 1)" (n - 1)" R°R' B sin 6 362. Let us now consider the angle made with each other by two consecutive osculating planes, which we shall call the angle of torsion, and denote by dij. The direction-cosines of the osculating plane being proportional to X, Y, Z; the second formula of Art. 352 gives (T+ Y 2 +Z*) W= ( YdZ-ZdY) l + (ZdX-XdZy+ (XdY- YdX)\ Now Y= dzd 2 x — dxd 2 z, Z= dxd^y — dyd 2 x, d Y= dzd s x — dxd s z, dZ— dxd 3 y — dyd 3 x. NON-PROJECTIVE PROPERTIES OF CURVES. 297 Therefore (Lessons on Higher Algebra, p. 26) YdZ-ZdY^Mdx, where If is the determinant Xd»x+YdSj + Zd 3 z. Hence ( X* + Y* + Z i )' dr," = J/W, d Mds V X'+Y' + Z'' This formula may be also proved geometrically. For M denotes six times the volume of the pyramid made by four consecutive points, while X~ + Y'+Z" denotes four times the square of the area of the triangle formed by three consecutive points. Now if A be the triangular base of a pyramid, A' an adjacent face making an angle t) with the base, s the side common to the two faces, and p the perpendicular from the vertex on s, so that 2A'=sp: then for the volume of the pyramid we have 3 Y=Ap sin?? and 6 Vs = 2Aj>s sin i? = AAA' sin 17. Now in the case considered, the common side is ds, and in the limit A = A' ■ hence 6 Vds = + ifl- arf-i . f , (x 2 + y 1 - arf) x coslnz ± v — ^ '- !■ + y sm \nz ± i « M- = a. Since t his equation becomes impossible when k 2 + y 2 < a 1 , it is plain that no part of the surface lies within the cylinder on which the helix is traced. 366. We shall now speak of the polar developable generated by the normal planes to the given curve. Fourier has re- marked, that the " angle of torsion" of the one system is equal to the " angle of contact" of the other, as is sufficiently obvious since the planes of this new system are perpendicular to the lines of the original system, and vice versa. The reader will observe however that it does not follow that the -y of , ds one system is equal to the -y- of the other, because the ds is not the same for both. Since the intersection of the normal planes at two con- secutive points Kj K' of the curve is the axis of a circle of which K and K' are points (Art. 358), it follows that if any point D on that line be joined to K and if', the joining lines are equal and make equal angles with that axis. NON-PROJECTIVE PROPERTIES OF CURVES. 301 _ It is plain that three consecutive normal planes intersect in the centre of the osculating sphere; hence the cuspidal edge of the polar developable is the locus of centres of spherical cur- vature. In the case of a plane curve this polar developable reduces to a cylinder standing on the evolute of the curve. 367. Every curve has an infinity of evolutes lying on the polar developable;* that is to say, the given curve may be generated in an infinity of ways by the unrolling of a string wound round a curve traced on that developable. Let MM', MM", &c. denote the successive elements of the curve, K, K', &c. the middle points of these elements, then the planes drawn through the points K perpendicular to the elements are the normal planes. The lines AB, A'B', &c. are the lines in which each normal plane is intersected by the consecutive; these lines being the generators of the polar developable, and hence tangents to the cuspidal edge RS of that surface. Draw now at pleasuref any line KD in the first normal plane, meeting the first generator in D; join DK' which beiDg in the second normal plane will meet the second generator A'B', say in D'. In like manner, let K"D' meet A'B" in D". We get thus a curve DD'D" traced on the polar developable which is an evolute of the given curve. For the lines DK, D'K', &c. the tangents to the curve DD'D", are normals to the curve * See Monge, p. 396. f This figure is taken from Leroy's Geometry of Three dimensions. 302 NON-PEOJECTIVE PROPERTIES OF CURVES. KK'K", and the lengths DK=DK, D'K' = D'K", &c. (see Art. 366). If therefore DK be a part of a thread wound round DD'D", it is plain that as the thread is unwound the point K will move along the given curve. Since the first line DK was arbitrary the curve has an infinity of evolutes. A plane curve has thus an infinity of evolutes lying on the cylinder whose base is the evolute in the plane of the curve. For example, in the special case where this evolute reduces to a point; that is, when the curve is a circle, the circle can be described by moving round a thread of constant length fastened to any point on the axis passing through the centre of the circle. In the general case, all the evolute curves DD'D", &c. are geodesies on the polar developable. For we have seen (p. 241) that a curve is a geodesic when two successive tangents to it make equal angles with the inter- section of the corresponding tangent planes of the surface ; . and it has just been proved (Art. 366) that DK, DK which are two successive tangents to the evolute make equal angles with AB which is the intersection of two consecutive tangent planes of the developable. An evolute may then be found by drawing a thread as tangent from K to the polar develop- able, and winding the continuation of that tangent freely round the developable. 368. The locus of centres of curvature is a curve on the polar developable, but is not one of the system of evolutes. Let the first osculating plane MM'M" meet the first two normal planes in KG, K'C, then G is the first centre of curvature: and in like manner the second centre is G', the point of inter- section of KG', K"G', the lines in which the second oscu- lating plane M'M"M'" is met by the second and third normal planes. Now the radii KG, KG' are distinct, since they are the intersections of the same normal plane by two different osculating planes, KG' will therefore meet the line AB in a point I which is distinct from G. Consequently the two radii of curvature KG, KG' situated in the planes P, P have no common point in AB the intersection of these planes; two NON-PEOJECTIVE PROPERTIES OF CURVES. 303 consecutive radii therefore do not intersect, unless in the case where two consecutive osculating planes coincide. The centres of curvature then not being given by the suc- cessive intersections of consecutive radii; these radii are not tangents to the locus of centres. Any radius therefore KG would not be the continuation of a thread wound round CO' C", and the unwinding of such a thread would not give the curve KK'K", except in the case where the latter is a plane curve.* 369. To find the radius of the sphere through four con- secutive points. Let R be the radius of any sphere, p the radius of a section by a plane making an angle t) with the normal plane at any point; then, by Meunier's theorem, Bcosr] = p- and for a consecutive plane making an angle •9 + 877, we have Sp = - E sin 17 St;. Hence B 2 = p ! +(-£-). We have then only to give in this expression to p and dq the values already found (Arts. 360, 362). -j- is obviously the length of the perpendicular distance from the centre of the sphere to the plane of the circle of curvature. 370. To find the co-ordinates of the centre of the osculating sphere. Let the equation of any normal plane be (a - x) dx + (/? -y) dy + (7 - z) dz = 0, where xyz is the point on the curve, and a/Sy any point on * The characteristics of the polar developable may be investigated by arguments similar to those used Sir/her Plane CWr&j. Art. 116. They are »' = 7» + r, be two consecutive and equal elements of the curve. Produce nt = 7nn, and let fall tq perpendicular to the surface : join nq and qp. Then since nt makes an infinitely * Appendix to Monge, p. J7i>. x2 308 CUEVES TRACED ON SURFACES. small angle -with the surface, its projection nq is equal to it. nq is the second element of the normal section, and is also the second element of the geodesic production of ran. If now 6 be the angle of contact tnp, and ff be tnq the angle of contact of the normal section we have tp = 6ds, tq = 6'ds. Now the angle qtp (= <£) is the angle between the osculating plane of the curve and the plane of normal section, and since tq = tp cos we have & = cosrf) and -^ = which is Meunier's theorem ; B being M p the radius of curvature of the normal section and p that of the given curve. Now; in like manner, pnq being 6" the geodesic angle of contact, we have pq = 8"ds and pq = tp sm • fx'x" ijy" z'z" \ n , ,. , at xy z is p \—p- + ^p- + — - 1) , and the perpendicular from x'y'z' on the tangent plane at x"y"z" is , fx'x" «'«" z'z" \ If now from the points A, B there be drawn lines AT, BT to any point T on the intersection of the tangent planes at A and B, and if AT make an angle t with the intersection of the planes, the angle between the planes being a> • then the perpendicular from A to the intersection of the planes is A T sin i and from A on the other plane is A T sin i sin co. In like manner the perpendicular from B on the tangent plane &tA lsBTs'mi' sin and — ^ + ^?Z_ i . c ^ a f. = L t fL + ±'??L + ± ds ds ds ds ds ds d' ds" b'* ds* c 2 us But the right-hand side of the equation denotes the reciprocal of the square of a central radius whose direction-cosines are dx dy dz ds 1 ds ' ds' The geometric meaning therefore of the integral we have found is pD = constant.* 378. The constant pD has the same value for all geodesies which pass through an umhilic. For at the umbilic the p is etc of course common to all, being = -j- ; and since the central section parallel to the tangent plane at the umbilic is a circle, the diameter parallel to the tangent line to the geodesic is * Dr. Hart proves the same theorem as follows : Consider any plane section of an ellipsoid, let w be the perpendicular from the centre of the section on the tangent line, d the diameter of the section parallel to that tangent, i the angle the plane of the section makes with the tangent plane at any point. Then along the section rod is constant, and it is evident that pD is in a, fixed ratio to wd sini. Hence along the section pD varies as sini and will be a, maximum where the plane meets the surface perpendicularly. But a geodesic osculates a series of nor- mal sections; therefore, for such a line pD is constant, its differential always vanishing. Cambridge and Dublin Mathematical Journal, Vol. IV., p. 84. 312 CURVES TRACED ON SURFACES. constant ; being always equal to the mean axis b. Hence for a geodesic passing through an umbilic, we have pD = ac. Let now any point on a quadric be joined by geodesies to two umbilics, since we have just proved that pD is the same for both geodesies, and since at the point of meeting the p is the same for both, the D for that point must also have the same value for both; that is to say, the diameters are equal which are drawn parallel to the tangents to the geodesies at their point of meeting. But two equal diameters of a conic make equal angles with its axes; and we know that the axes of the central section of a quadric parallel to the tangent plane at any point are parallel to the directions of the lines of cur- vature at that point. Hence, the geodesies joining any point on a quadric to two umbilics make equal angles with the lines of curvature through that point.* It follows that the geodesies joining any point to the two- opposite umbilics, which lie on the same diameter, are con- tinuations of each other ; since the vertically opposite angles are equal which these geodesies make with either line of cur- vature through the point. It follows also (see Art. 373) that the sum or difference is constant of the geodesic distances of all the points on the same line of curvature from two umbilics. The sum is constant when the two umbilics chosen are interior with respect to the line of curvature ; the difference when for one of these um- bilics we substitute that diametrically opposite so that one of the umbilics is interior, the other exterior to the line of curvature. If A, A' be two opposite umbilics, and B another umbilic, since the sum PA + PB is constant and also the difference PA' -PB; it follows that PA + PA' is constant; that is to say, all the geodesies which connect two opposite umbilics are of equal length. In fact, it is evident that two indefinitely near geodesies connecting the same two points on any surface must be equal to each other. * This theorem and its consequences developed in the following articles are due to Mr, Michael Roberts, Lioucille, Vol. XI., p. 1. CURVES TRACED ON SURFACES. 313 379. The constant pD has the same value for all geodesies which touch the same line of curvature. It was proved (Art. 158) that pD has a constant value all along a line of curvature; but at the points where either geodesic touches the line of curvature both p and D have the same value for the geodesic and the line of curvature. Hence then a system of lines of curvature has properties completely analogous to those of a system of confocal conies in a plane ; the umbilics answering to the foci. For example, two geodesic tangents drawn to one from any point on another make equal angles with the tangent at that point. Dr. Graves's theorem for plane conies holds also for lines of curvature, viz. that the excess of the sum of two tangents to a line of cur- vature over the intercepted arc is constant, while the intersection moves along another line of curvature of the same species (see Conies, p. 355). 380. The equation pJO = constant has been written in another convenient form.* Let a', a" be the primary semi-axes of two confocal surfaces through any point on the curve, and let i be the angle which the tangent to the geodesic makes with one of the principal tangents. Then since a 2 — a' 2 , a" — a" 2 (Art. 156) are the semi-axes of the central section parallel to the tangent plane, any other semi-diameter of that section is given by the equation 1 cos s, i sin 2 «s" v,-i • 1 (q a -a ,2 )(a a -Q , A „ t ^ while, again, - a = ^— — J^ (Art. 157). The equation therefore pD = constant is equivalent to (a 2 - a") cos 2 t' + (a 2 - a" 2 ) sin 2 i = constant, or to a' cos 2 i + a"' 1 sin 2 i = constant. 381. The locus of the intersection of two geodesic tangents to a line of curvature, which cut at right angles, is a sphero-conic. * By Iiouvflle, Vol, ix., p. 401. 314 CURVES TRACED ON SURFACES. This is proved as the corresponding theorem for plane conies. If a, a" belong to the point of intersection, we have a" 2 cos 2 i+a"' 2 sm 2 a = constant, a 1 ' 2 sinV+a"* cos%' = constant, hence a' 2 + a'" 2 = constant ; and therefore (Art. 153) the distance of the point of intersection from the centre of the quadric is constant. The locus of inter- section is therefore the intersection of the given quadric with a concentric sphere. The demonstration holds if the geodesies are tangents to different lines of curvature ; and, as a par- ticular case, the locus of the foot of the geodesic perpendicular from an umbilic on the tangent to a line of curvature is a sphero-conic. 382. To find the locus of intersection of geodesic tangents to a line of curvature loMch cut at a given angle (Besge, Liouville, Vol. XIV., p. 247). The tangents from any point whose a\ a" are given, to a given line of curvature are determined by the equation a' 2 cos 2 «'-f a" 2 sin 2 i=/3; and since they make equal angles with either of the principal tangents through that point, i the angle they make with one of these tangents is half the angle they make with each other. We have therefore tan^ yr';) ; Un^'^-OV(^ > 1 V(a - /3) ' a 2 + a" 2 - 2/3 ' ( a * + a" 2 - 2/3) 2 tan 2 = 4/3 (a' 2 + a'" 2 ) - 4a'V 2 - 4/3 2 . This is reduced to ordinary co-ordinates by the equations (Arts. 152, 153) a' 2 + a"' 2 = x 2 + f + z 2 + b 2 + c 2 - d 2 : a' 2 a" 2 = gVzgjjgzg) 7 a 2 ' whence it appears that the locus required is the intersection of the quadric with a surface of the fourth degree.* * Mr. Michael Roberts has proved (Liouville, Vol. xv., p. 291) by the method of Art. 179, that the projection of this curve on the plane of circular sections is the locus of the intersection of tangents at a constant angle to the conic into which the line of curvature is projected. CURVES TRACED ON SURFACES. 315 383. It was proved (Art. 168) that two confocals can be drawn to touch a given line; that if the axes of the three surfaces passing through any point on the line be a, a', a" and the angle the line makes with the three normals at the point be a, /3, 7; then the axis-major of the touched confocal is determined by the quadratic cos 2 at cos 2 /S cos 8 7 a* - a 2 + ^T* + £53? = °' Let us suppose now that the given line is a tangent to the quadric whose axis is a, we have then cosa = 0, since the line is of course at right angles to the normal to the first surface; and we have cos/3 = sin 7, since the tangent plane to the sur- face a contains both the line and the other two normals. The angle 7 is what we have called * in the articles immediately preceding. The axis then of the second confocal touched by the given line is determined by the equation sin 3 i cos 8 i • .» _ 11 + — 5 = 0, or a" 2 cos'i+ a" 2 sin 2 ^= a 2 . If then we write the equation of a geodesic (Art. 381) a' 2 cosV+a" 2 sin 2 * = a 2 , we see from this article that that equa- tion expresses that all the tangent lines along the same geodesic touch the confocal surface whose primary axis is a.* The geodesic itself will touch the line of curvature in which this confocal intersects the original surface; for the tangent to the geodesic at the point where the geodesic meets the confocal is, as we have just proved, also the tangent to the confocal at that point. The geodesic therefore and the intersection of the confocal and the given surface have a common tangent. The osculating planes of the geodesic are plainly tangent planes to the same confocal ; since they are the planes of two consecutive tangent lines to that confocal. The value of pD for a geodesic passing through an umbilic is ac (Art. 378) ; and the corresponding equation is therefore a' 2 cosV + a" 2 surV=a 2 — J*. Now the confocal, * The theorems of thia article are taken from M. Chasles'i Memoir, LicmviUe, Vol. xi., p. 5. 316 CUEYES TRACED ON SURFACES. whose primary axis is >J(d' -V), reduces to the umbilicar focal conic. Hence, as a particular case of the theorems just proved, all tangent lines to a geodesic which passes through an umbilic, intersect the umbilicar focal conic. Conversely, if from any point on that focal conic recti- linear tangents be drawn to a quadric and those tangents produced geodetically on the surface, the lines so produced will pass through the opposite umbilic; the whole lengths from to the umbilic being equal. 384. From the fact (proved p. 127) that tangent planes drawn through any line to the two confocals which touch it are at right angles to each other, we might have inferred directly, precisely as at Art. 305, that tangent lines to a geodesic touch a confocal. For the plane of two consecutive tangents to a geodesic being normal to the surface is tangent to the confocal touched by the first tangent. The second tangent to the geodesic therefore touches the same confocal ; as, in like manner, do all the succeeding tangents. Having thus established the theorem of the last article, we could, by reversing the steps of the proof, obtain an independent de- monstration of the theorem pD = constant. 385. The developable circumscribed to a quadric along a geodesic has its cuspidal edge on another quadric, which is the same for all geodesies touching the same line of curvature. For any point on the cuspidal edge is the intersection of three consecutive tangent planes to the given quadric, and the three points of contact, by hypothesis determine an oscu- lating plane of a geodesic which (Art. 383) touches a fixed confocal. The point on the cuspidal edge is, the pole of this plane with respect to the given quadric; but the pole with respect to one quadric of a tangent plane to another lies on a third fixed quadric. 386. M. Chasles has given the following generalization of Mr. Eoberts's theorem, Art. 378. If a thread fastened at two fixed points on one quadric A be strained by a pencil moving CURVES TRACED ON SURFACES. 317 along a confocal B (so that the thread of course lies in geo- desies where it is in contact with the quadrics and in right lines in the space between them), then the pencil will trace a line of curvature on the quadric B. For the two geodesies on the surface B, which meet in the locus point P, evidently make equal angles with the locus of P; but these geodesies have as tangents the rectilinear parts of the thread which both touch the same confocal ; therefore (Art. 383) the pD is the same for both geodesies, and hence the line bisecting the angle between them is a line of curvature. A particular case of this theorem is that the focal ellipse of a quadric can be described by means of a thread fastened to two fixed points on opposite branches of the focal hyperbola. 387. Elliptic Co-ordinates. The method used (Arts. 381, 382) in which the position of a point on the ellipsoid is defined by the primary axes of the two hyperboloids intersecting in that point, is called the method of Elliptic Co-ordinates (see p. 135 and Higher Plane Curves, p. 276). It being more convenient to work with unaccented letters, I follow M. Liouville* in denoting the quantities which we have hitherto called a, a" by the letters fi, v; and in this notation the equation of the lines of curvature of one system would be of the form fi = constant, and those of the other v = constant. The equation of a geodesic (Art. 380) would be written //cosV+ v 8 sin** = /*'*; and when the geodesic passes through an umbilic, we have /*'" = o?-V = h\ It will be remembered (Art. 149) that /j, lies between the limits h and Jc, and v between the limits k and 0. Throwing the equation of a geodesic into the form ft? + v 2 tan 2 i = /*'" (1 + tanV) ; we see that it is satisfied (whatever be /a') by the values fj? = v 2 , tan' J t'=— 1. Whence it follows that the same pair of imaginary tangents, drawn from an umbilic, touch all the lines of curvature,! a further analogy to the foci of plane conies. * I cannot, however, bring myself to imitate him in calling the axis of the ellipsoid/); and his denoting the quantities a-— b-, a 2 — c 2 (which we call A 2 , i 2 ) by the letters b-, c 2 , seems likely to confuse. f ITr. Roberts, Liouvilk, Vol. XV., p. 289. 318 CURVES TRACED ON SURFACES. 388. To express in elliptic co-ordinates the element of the arc of any curve on the surface. Let us consider first the element of any line of curvature, p = constant. Let that line be met by the two consecutive hyperboloids, whose axes are v and v + dv ; then, since it cuts them perpendicularly, the intercept between them is equal to the difference between the central perpendiculars on the tangent planes to the two hyperboloids. But (Art. 172) (p"+dp") 2 -2}" 2 = (v + dvy-v* or p"dp" = vdv. Now we have proved that dp" = do; the element of the arc we are seeking, and n^V _ y»(A'-v»)(fl'-0 P ~ (a 2 -a" 2 ) (a' 2 -a" 2 ) (a 2 - v*) (^ - v 2 ) ' Hence da = ^—^-^ dv . In like manner the element of the arc of the line of curvature v = constant is given by the formula Now if through the extremities of the element of the arc ds of any curve, we draw lines of curvature of both systems, we form an elementary rectangle of which da, do are the sides and ds the diagonal. Hence _ (a 2 -//)(// -v 2 ) , y- W-f ) aS ~ {p?-h 2 ) (tf - /**) ^ + (A 2 - v 2 ) (ft" - v 2 ) ' 389. In like manner we can express the area of any portion of the surface bounded by four lines of curvature ; two lines /tt , ^ 2 , and two v t , v 2 . For the element of the area is d da * ~ VIO^-A")^- S) (A 2 - v 2 ) [V - v*)} t* ' the integral of which is pi ju, 2 VK - A'''') dp [ "' V(« 2 - v 2 ) t?v ' J ft V{(/*' - A 2 ) (A* - gj ■/», Vt(A 2 - Q g - v 2 )} ' * The area of the surface of the ellipsoid was thus first expressed by Legendre, Traite des Fonctions EUiptiques, Vol. I., p. 352. CUEVES TRACED ON SURFACES. 319 i So, in like manner, we can find the differential equation of the orthogonal trajectory of a curve whose differential equation is Mdfi + JSfdv. For the orthogonal trajectory to Pda + Qda is , . , da- da . plainly -p — ; since do-, da are a system of rectangular co-ordinates. But Md/t, + Ndv can be thrown without difficulty into the form Pda + Qda by the equations of the last article. The equation of the orthogonal trajectory is thus found to be a} - u!' da a 2 - v 2 dv (/*" - A 2 ) [k* - /*■) M (A 2 - v 2 ) {tf - v 2 ) N '' 390. The first integral of a geodesic /t 2 cosV+v 2 sin 2 * = /// 2 can be thrown into a form in which the variables are separated and the second integral can be obtained. That equation gives -MSfc? But tan^- da - - the angle which the radius vector makes with the line joining the umbilics. Now the equation (Art. 390) of a geodesic passing through an umbilic gives the sum of two integrals equal to a constant. This constant can- not be a function of p since it remains the same as we go along the same geodesic : it must therefore be a function of u> only : and if we pass from any point to an indefinitely near one, not on the same geodesic radius vector, we shall have _vVV)^ vV - v 2 ) dv (/*■ - W) v]F - ^) ± W - *) 7(¥^7) ~ $ W dw - We shall determine the form of the function by calculating its value for a point indefinitely near the umbilic, for which fi = v = h. The left-hand side of the equation then becomes >v/(Ht«) x limit of {&? + *£?) • Now if we P ut /jl = Ji+t], v = h-B, the quantity whose limit we want to find dr) d& 18 2hn + v* ~ 2kz - e* ' whlch > as ^ and e tend to vanish, becomes the limit of I (^ - ^ e ) or of ±d log 2 . CURVES TRACED ON SURFACES. 321 Now since the angle external to the vertical angle of the triangle formed by the line joining any point to two umbilics, is bisected by the direction of the line of curvature, that external angle is double the angle i in the formula yu,' 2 cosV+ v' sin 2 i' = K*. In the limit when the vertex of the triangle approaches the umbilic, the external angle of the triangle becomes a>, and we have at the umbilic {h + t))* cos'^co + {h- s) 2 sin 2 £w = K\ and in the limit tanHw = — . 2 e Using this value, the limit of the left-hand side of the equation is 4\/(J^)^ lo s tan >)- We have therefore */((?-/**) dp *J(d'-v*)dv _ 1 //a 5 -, (jf - tf) V(^ 2 - /*") (A" - v') V(&" - v 2 ) ~ h V U" - * And the constant which occurs in the integrated equation of a geodesic through an umbilic is of the form <^V(^) lostan>+a 393. If P, Q be two consecutive points on a curve, and if PP' be drawn perpendicular to the geodesic radius vector OQ, it is evident that PQ 2 = PP' 2 + P Q\ Now since (Art. 372) OP=OP\ we have P'Q = dp, while PP' being the element of an arc of a geodesic circle, for which p is constant (or dp = 0), must be of the form Pd 2 - c 2 ) sin 2 w sin 2 w (Art. 152); therefore p _ # ,,* sma) In this investigation it is not necessary to assume the result of the last article. If we substitute for the right-hand side of the equation in the last article an undetermined function of &>, it is proved in like manner that P = y

\ Hence P= sin/3. But in the sphere y= sin p sin w. The function therefore which multiplies y is 1 sin to ' 394. Consider now the triangle formed by joining any point P to the two umbilics 0, 0'. Then for the arc OP we have the function P= -P— and for the arc O'P, connecting: sin a> ' ° P with the other umbilic, we have the function P' = . smw ' and P : P' :: sinta' : sinw, an equation analogous to that which expresses that the sines of the sides of a spherical triangle are proportional to the sines of the opposite angles ; since P and P in the rectification of arcs on the ellipsoid answer to smp, s'mp' on the sphere. 395. Again, if P be any point on a line of curvature we know (Art. 378) dp±dp ' : =0, where p and p are the distances from the two umbilics. Now if 6 be the angle which the radius vector OP makes with the tangent, the perpendicular CURVES TEACED ON SURFACES. 323 element Pda> is evidently dp tanfl. But the radius vector O'P makes also the angle 6 with the tangent. Hence, we have dot du>' Pda> ± Pdoi' = 0, or > = <>• sin a> sin to whence tan ^co tan ^to' is constant when the sum of sides of the triangle is given; and tan|' in a given ratio when the difference of sides of the triangle is given. Thus then the distance between two umbilics being taken as the base of a triangle, when either the product or the ratio of the tangents of the halves of the base angles is given ; the locus of vertex is a line of curvature.* From this theorem follow many corollaries : for instance, " If a geodesic through an umbilic meet a line of curvature in points P, _P', then (according to the species of the line of curva- ture) either the product or the ratio of tan |P0'0, tan \P'0' is constant." Again, " if the geodesies joining to the umbilics any point P on a line of curvature meet the curve again in P, P", the locus of the intersection of the transverse geodesies O'P, OP" will be a line of curvature of the same species." 396. Mr. Roberts's expression for the element of an arc perpendicular to an umbilical geodesic has a^_ been extended as fol- lows by Dr. Hart : Let OT, OT be two consecutive geodesies touching the line of curvature formed by the intersection of the surface with a confocal B, dco the angle at which they intersect ; then the tangent at any point T of either * This theorem, as well as those on which its proof depends, (Art. 392, 4c). is due to Mr. M. Roberts, to whom this department of Geometry owes so much, (Liouville, Tola. SIIT., p. 1, and IV., p. 275), T2 324 CURVES TRACED ON SURFACES. geodesic touches B in a point P (Art. 383) ; and if TT be taken conjugate to TP, the tangent plane at T passes through TP (Art. 262) and the tangent line to the geodesic at T touches the confocal B in the same point P. We want now to express in the form Pda> the perpendicular distance from T' to TP. Let the tangents at consecutive points, one on each geodesic, intersect in P' and make with each other an angle d'. Let normals to the surface on which the geodesies are drawn at the points T„ T,' meet the tangents PT, PT at the points T 2 , TJ, then since the difference between T X T^ T^TJ is infinitely small of the third order, PT 2 d§ and P'T t d = I) da> ; and there- fore the distance we want to express PTd

\ ■ 1 n cos i (a + 6) . trigonometry cos £ (A + B) = sin \ (J *-j - gives us m the limit dd = cosad\jr. But evidently j sin ad->jr = sin 9d. Hence dO d sm0 tana' Now we know (Art. 383) that the tangent line at any point of a geodesic passing through an umbilic, if produced, goes to meet the plane of the umbilics in a point on the focal hyper- bola ; and the osculating plane of the geodesic at that point will be the plane joining the point to the corresponding tangent of the focal hyperbola. We know also (Art. 3 76) that the cone circumscribing an ellipsoid and whose vertex is any point on the focal hyperbola is a right cone. Let now PP be an element of an umbilical geodesic pro- duced to meet the focal hyperbola in H. Let P'P" be the consecutive element meeting the focal hyperbola in H' '; then if Eli, H'h' be two con- secutive tangents to the focal hyperbola ; PBk, 326 CURVES TEACED ON SURFACES. P'H'K will be two consecutive osculating planes. Imagine now a sphere round H', and consider the spherical triangle formed by radii to the points h, Ji, P. Then if d be the angle hH'h', the angle of contact of the focal hyperbola; 6 the angle between the osculating plane and hH'h' the plane of the umbilics, while hE'P is a the semi-angle of the cone ; then the spherical triangle is that considered in our lemma, , , dQ d in terms of a; and this we may regard as a problem in plane geometry, for a is half the angle included between the tangents from H to the principal section in the plane of the umbilics, while d. But p = ; — ■ . Hence da = — tana — -f- or da = tana — ~r . p p ab But a' 2 = a* + (a 2 - a" 2 ) cot 2 a, b" = F + (a 2 - a" 2 ) cot" a. TT dcf> a"b"da Hence — — = — tana *J{a 2 -a"* + d 2 tan 2 a) V(« 2 -a"* + F tan B a) ' In the case under consideration the axes of the touched ellipse are a, c ; while the squares of the axes of the confocal hyperbola are a 2 - b 2 , W — c 2 . Hence we have the equation dd_ vV-Z. 2 ) V^-c'Va sin 6 V(&* + « 2 tan 2 a) *J(b'' + c 2 tan 2 a) " CURVES TRACED ON SURFACES. 327 Integrating this, and taking one limit of the integral at the umbilic where we have 6 = a>, and a = \tt • we have I tanjfl = f« V(a 8 -5 a )V(y-cVa 8 tan |a> J ^ V(&" + a 8 tan 2 a) V(&* + c 2 tan 2 a) ' If then / be the value of this integral ; we have tan£0 = & tan^co, where h = e I . Now this integral obviously does not change sign between the limits ± %tt, that is to say, in passing from one umbilic to the other. If then to' be the value of 6 for the umbilic opposite to that from which we set out ; at this limit / has a value different from zero, and k a value different from unity ; and we have tan|" = & 2 tan -|to, and so it will pass and repass for ever making a series of angles the tangents of whose halves are in continued pro- portion.* 398. If we consider edges belonging to the same tangent cone, whose vertex is any point H on the focal hyperbola, a (and therefore h) is constant ; and the equation tan£0 = & tan-|o> d9 dot .. T , . gives -r— -j = . JNow since the osculating: plane of the sin a sin to ° r geodesic is normal to the surface, and therefore also normal to the tangent cone, it passes through the axis of that cone. If then we cut the cone by a plane perpendicular to the axis, the section is evidently a circle whose radius is -r~ , and the sine/' element of the arc is A— -. , or '-. — . Now this element, being: sin 6/ ' sin to ° the distance, at their point of contact, of two consecutive sides of the circumscribing cone, is what we have called (Art. 393) Pdco, and we have thus from the investigation of the last article an independent proof of the value found for P (Art. 393). * The theorems of thia article are Dr. Hart's, Cambridge and Dublin Mathe- matical Journal, Vol. it., p. S2 ; but in the mode of proof I have followed Mr. William Roberts, Liouville, 1857, p. "213. 328 CURVES TRACED ON SURFACES. 399. Lines of level. The inequalities of level of a country can be represented on a map by a series of curves marking the points which are on the same level. If a series of such curves be drawn, corresponding to equi-different heights, the places where the curves lie closest together evidently indicate the places where the level of the country changes most rapidly. Generally, the curves of level of any surface are the sections of that surface by a series of horizontal planes, which we may suppose all parallel to the plane of xy. The equations of the horizontal projections of such a series are got by putting z = c in the equation of the surface; and a differential equation common to all these projections is got by putting dz=0 in the differential equation of the surface, when we have JJ^dx + U 2 dy = 0. We can make this a function of x and y only, by eliminating the z which may enter into the differential coefficients, by the help of the equation of the surface. Lines of greatest slope. The line of greatest slope through any point is the line which cuts all the lines of level perpendicularly ; and the differential equation of its projection therefore is TJ x dy - U 2 dx = 0. The line of greatest slope is often denned as that, the tan- gent at every point of which makes the greatest angle with the horizon. Now it is evident that the line in any tangent plane which makes the greatest angle with the horizon is that which is perpendicular to the horizontal trace of that plane. And we get the same equation as before by expressing that the projection of the element of the curve (whose direction- cosines are proportional to dx, dy) is perpendicular to the trace whose equation is U^x-^+U^y-y'j-U/^O* * It is evident that the differential equation of the curve, which is always per- pendicular to the intersection of the tangent plane, [whose direction-cosines are as L, M, JV] by a fixed plane whose direction-cosines are a, b, c, is dx, dy, dz L, M, N , b, c =0. CURVES TRACED ON SURFACES. 329 Ex. To find the line of greatest slope on the quadric Aa? + Bf + Cz* = D. The differential equation is Axdy = Bydx, which integrated, gives (-,)* = (lY. where the constant has been determined by the condition that the line shall pass through the point * = »', ,j = y>. The line of greatest slope is the intersection of the quadric by the cylinder whose equation has just been written, and will be a curve of double curvature except when x',j lies in one of the principal planes when the equation just found reduces to x = or y = 0. 400. We shall conclude this chapter by giving an account of Gauss's theory of the curvature of surfaces* In plane curves we measure the curvature of an arc of given length by the angle between the tangents, or between the normals, at its extremities ; in other words, if we take a circle whose radius is unity, and draw radii parallel to the normals at the ex- tremities of the arc, the ratio of the intercepted arc of the circle to the arc of the curve affords a measure of the cur- vature of the arc. In like manner if we have a portion of a surface bounded by any closed curve, and if we draw radii of a unit sphere parallel to the normals at every point of the bounding curve, the area of the corresponding portion of the sphere is called by Gauss the total curvature of the portion of the surface under consideration. And if at any point of a surface we divide the total curvature of the superficial element adjacent to the point by the area of the element itself, the quotient ir called the measure of curvature for that point. 401. We proceed to express the measure of curvature by a formula. Since the tangent planes at any point on the surface, and at the corresponding point on the unit sphere are by hypothesis parallel ; the areas of any elementary portions on each are proportional to their projections on any of the co-ordinate planes. Let us consider then their projections on the plane of xy, and let us suppose the equation of the surface to be given in the form z = (x, y). If then x, y, z be the co-ordinates of any point on the surface, X, Y, Z those of the corresponding point on the unit sphere, x + dx, x+ &r, X+dX, X+SX } &c, the co-ordinates of two adjacent points on each : then the areas of the two elementary * His paper has been reprinted in the appendix to Lkniville's edition of Monge, 330 CURVES TRACED ON SURFACES. triangles formed by- the points considered, are evidently in the ratio dXB Y- d YBX :dxBy- dy Bx. But dX, dY, SX, BY are connected with dx, dy, &c, by the same linear transformations, viz. JV dX , dX , , v dY dY dX -dx dx + -dy d ^ dY =J^ dx + lTy d y> BX=^ hx + d f Sy, BY^Bx + ^By: dx «y dx dy " ■ •whence by the theory of linear transformations, or by actual multiplication, dXBY-dYBX^dxSy-dyBx)^ f~f g), , 4 , .. + dXdY dXdY. ., . and the quantity -t— -7- — 5 5— 1S the measure ot curvature. 1 dx dy dy dx Now X, Y, Z beiDg the projections on the axes of a unit line parallel to the normal are proportional to the cosines of the angles which the normal makes with the axes. We have therefore P X= —tt- — ; s— sr , Y = ■ dX _ (1 + g 8 ) r -pqs dX _ (l+q*)s-pqt dX (1+/ + 2T ' ^ (1+/ + 2 2 ) 1 ' dY _ (l + p*)s-pqr dY _[l J tp' 1 )t -pqs dx (1+/+2T ' dy (i+f + qi ' dX dY dX dY (rt-s 2 ) whence -^ dx dy dy dx (1 +p' + q s f ' But from the equation of (Art. 307, p. 243) it appears that the value just found for the measure of curvature is -^y* ■> where B and B! are ike two principal radii of curvature at the point. 402. It is easy to verify geometrically the value thus found. For consider the elementary rectangle whose sides are in the directions of the principal tangents. Let the lengths of the CUEVES TEACED ON SURFACES. 331 sides be X, X', and consequently its area XX'. Now the normals at the extremities of X intersect, and if they make with each other an angle 6, we have 8 = -~ where R is the corresponding radius of curvature. But the corresponding normals of the sphere make with each other, by hypothesis, the same angle; and their length is unity. If therefore ji be the length of the element on the sphere corresponding to X, we have -5 = /"■• ■\ ' 'i In like manner we have -^7 = m' ; and ^-, = -s™ : which was K XX Jxti to be proved. 403. Gauss has proved that if a surface supposed to be flexible but not extensible be deformed in any way : (that is to say, if the shape of the surface be changed, yet so that the distance between any two points measured along the surface remains the same) then the measure of curvature at every point remains unaltered. We have had an example of such a change in the case of a developable surface which is such a deformation of a plane (Art. 313). And the measure of cur- vature vanishes for the developable as well as for the plane, one of the principal radii being infinite (Art. 364). To establish the theorem in general, let us suppose that any point on the surface instead of being given by three co-ordinates connected by the equation of the surface is given by two independent co-ordinates. Let dx = adu + a'dv, dy = bdu -+ b'dv, dz — cdu + c'dv, then ds> = dx' + dif + da" = (a" + b* + c 2 ) du 1 + 2 [ad + bb' + cc) dudv + (a" z •+ b"* + c" 2 ) dv\ If we write this equation ds* = Edi? + 2Fdudv + Gdv\ what we want to prove is that the measure of curvature, or that the product of the principal radii, is a function of E, F, G. In fact, let x'y'z' denote the point of the deformed surface corresponding to any point xyz of the given surface. Then 332 CURVES TRACED ON SURFACES. x, y', z are given functions of x, y, z, and can therefore also be expressed in terms of u and v. And the element of any arc of the deformed surface can be expressed in the form ds' 2 = E'du 2 + 2F'dudv + G'dv\ But the condition that the length of the arc shall be un- altered by transformation, manifestly requires E=E', F=F', G = G'. Any function therefore of E } F, G is unaltered by such a deformation as we are considering. Now it will be remembered (see p. 224) that the principal radii are given by a quadratic, in which the coefficient of V is (L 2 + M' + N 2 ) 2 ; and the absolute term is (be - F) V + (ca - m 2 ) M 2 + (ab - n 2 ) N 2 + 2 (mn - al) MN+ 2 (nl - bm) NL + 2 (lm - en) LM* We shall separately express each of these quantities in terms of E, F : G. 404. Now if we substitute in the equation of the surface Ldx + Mdy + Ndz = 0, the values of dx, dy, dz given in the last article, and remember that since u and v are independent variables, the coefficients of du and dv must vanish separately, we have La+Mb + Nc=0, La + Mb' + Nc =0. Consequently we have L = \(bc'-b'c), M=X(ca'-c'a), N=\(ab' -a'b), where A, is indeterminate, and L 2 + M 2 + N 2 = X 2 {{a 2 + b 2 + c 2 ) (a' 2 + b' 2 + c' 2 ) - {aa' + bb' + cc') 2 }, = \ 2 (EG-F 2 ). (See Lessons on Higher Algebra, Art. 26). 405. Let us now examine the result of making in the absolute term, given Art. 403, the same substitution, viz. * We use Roman letters in order that the a, b, o of p. 224 may not be confounded with «, b, c used in a different sense in this article. CURVES TEACED ON SUEFACES. 333 L=\(bc' — b'c), &c. Now an equation which we had occasion to use in the theory of conies (see Conies, Art. 386) enables us to write this result in a more simple form. Let us write down the equation of a conic a.r 2 + by* + cs 2 + 2\yz 4 2mzx 4 2na-y = 0, and substituting for x,y,z; a + Tea', b + W, c + led ; let us write the result V+ 2lcV + k 2 U\ then UU' - F 2 = (be - V) {be' - cb'Y + (ca - m 2 ) {ca' - da) 2 + &c. In fact, either side of this equation, equated to nothing, ex- presses the condition that the line joining the points abc, a'b'c' should touch the conic. The equation however may be verified by actual multiplication. What we want to calculate then is X'{UU'-V 2 ) where U = aa* + W 4 cc' + 2\bc 4 2mca + 2na5, U' = a«' 2 + b&' 2 -I- cc' 2 4 2lZ>'c' + 2mc'a' + 2na'b', V= &ad + hbb' + ccc 4 1 {be + b'c) 4 m {ca 4 ca) + n {ab' + a'b). Now let us differentiate the equation Ldx + Mdy + Ndz = 0, and we get LtPx + MPy + MTs = - {ndx 2 4 bdy 2 + cdz 2 + 2\dydz + 2mdzdx 4 2ndxdy). If now we write d 2 x = adit? 4 a' dud i) 4 a"dv*, d 2 y = pdu 2 + /3'dudv 4 P"dv\ d 2 z = / ' , dE ' dv 1 . da dF dv 1 aa ^ du ' 1 d6? J(E) du is the element of the curve, passing through the point, for which v is constant; and /J(G)dv is the element of the curve for which u is constant. If these two curves intersect at an angle to, then since ds is the diagonal of a parallelogram of which *J(E) du, *J{G) dv are the sides, we have coso)= ,,,-,,-» -, while the area of the parallelogram being \f\EG) dadd sin. Hence the first integral is da> (l - -j- J . This may be written in a more convenient form as follows : Let be the angle which any radius vector makes with the element of a geodesic ab. Now since aa = Pd is the angle between the z 338 CURVES TRACED ON SURFACES. two extreme radii vectores which we consider, and 0', 6" are the corresponding values of 0. If we call A, B, C the internal angles of the triangle formed by the two extreme radii and by the base, we have a> = A, 6'=B, 6" = ^ — C, and the total curvature is A + B+G—ir. Hence the excess over 180° of the sum of the angles of a geodesic triangle is measured by the area of that portion of a unit sphere which corresponds to the directions of the normals along the sides of the given triangle. The portion on the unit sphere corresponding to the area enclosed by a geodesic returning upon itself is half the sphere. For if the radius vector travel round so as to return to the point whence it set out the extreme values of & and 8" are equal, while a has increased by 27r. The measure of cur- vature is therefore 2ir or half the surface of the sphere.* * For some other interesting theorems, relative to the deformation of surfaces, see Mr. Jellett's paper "On the Properties of Inextensible Surfaces," Transactions of the Royal Irish Academy, Vol. XXII. I have not happened to meet with what would appear to be valuable memoirs by MM. Boivr and Bonnet, on the Theory of Surfaces applicable to one another, to one of which was awarded the Prize of the French Academy in 1860. ( 339 ) CHAPTER XIII. FAMILIES OF SURFACES. 409. Let the equations of a curve (x, y, z, c„ c.;. . .cj = Q, f (as, y, *, c J} c„. . .c„) = 0, include m parameters, or undetermined constants: then it is evident that if n equations connecting these parameters be given, the curve is completely determined. If, however, only n — 1 relations between the parameters be given, the equa- tions above written may denote an infinity of curves : and the assemblage of all these curves constitutes a surface whose equation is obtained by eliminating the n parameters from the given b+1 equations ; viz. the n — 1 relations, and the two equations of the curve. Thus, for example, if the two equa- tions above written denote a variable curve, the motion of which is regulated by the conditions .that it shall intersect w — 1 fixed directing curves, the problem is of the kind now under consideration. For by eliminating x, y } z between the two equations of the variable curve and the two equations of any one of the directing curves, we express the condition that these two curves should intersect, and thus have one relation between the n parameters. And having n - 1 such relations we find the equation of the surface generated, in the manner just stated. We had (Art. 109) a particular case of this problem. Those surfaces for which the form of the functions and i/r is the same, are said to be of the same family, though the equations connecting the parameters may be different. Thus if the motion of the same variable curve were regulated by several different sets of directing curves, all the surfaces generated would be said to belong to the same family. In several important cases the equations of all surfaces belonging to the same family can be included in one equation involving z2 340 FAMILIES OP SURFACES. one or more arbitrary functions ; the equation of any individual surface of the family being then got by particularizing the form of the functions. If we eliminate the arbitrary functions by differentiation, we get a partial differential equation, common to all surfaces of the family, which ordinarily is the expression of some geometrical property common to all surfaces of the family, and which leads more directly than the functional equa- tion to the solution of some classes of problems. 410. The simplest case is when the equations of the variable curve include but two constants.* Solving in turn for each of these constants, we can throw the two given equations into the form u = c , v = c 2 ; where u and v are known functions of a*, y, z. In order that this curve may generate a surface we must be given one relation connecting c,, c 2 , which will be of the form c t = $ (c 2 ) ; whence putting for c, and c 2 their values, we see that, whatever be the equation of connection, the equa- tion of the surface generated must be of the form u — (v). We can also in this case readily obtain the partial diffe- rential equation which must be satisfied by all surfaces of the family. For if U= represents any such surface, U can only differ by a constant multiplier from u — (v), and differerrdating with two similar equations for the differentials with respect to y and z. Eliminating then X and cf>' (u), we get the required partial differential equation in the form of a determinant u« u„ Us = 0. In this case u and v are supposed to be known functions of the co-ordinates ; and the equation just written establishes a relation of the first degree between U i: U^ U s . If the equation of the surface were written in the form * If there were but one constant the elimination of it would give the equation of a definite surface, not of a family of surfaces. FAMILIES OP SURFACES. 341 2 -$(«,«/)= ; we- should have Z7 3 = l, U x = -p, U 2 = -q, where p and q have the usual signification, and the partial differential equation of the family is of the form Pp + Qq = B, where P, Q, R are known functions of the co-ordinates. And conversely the integral of such a partial differential equation, which (see Boole's Differential Equations, p. 322) is of the form u= («), geometrically represents a surface which can be gene- rated by the motion of a curve whose equations are of the form w = c 1 , w = c 2 . The partial differential equation affords the readiest test whether a given surface belongs to any assigned family. We have only to give to £f , U 2 , U 3} their values derived from the equation of the given surface, which values must identically satisfy the partial differential equation of the family if the surface belong to that family. 411. If it be required to determine a particular surface of a given family u = (t>), by the condition that the surface shall pass through a given curve, the form of the function in this case can be found by writing down the equations u = c„ v = c 2 , and eliminating x, y, z between these equations aDd those 0- the fixed curve, when we find a relation between c t and c 2 , or between u and v, which is the equation of the required surface. The geometrical interpretation of this process is that we direct the motion of a variable curve u = c iy v = c a by the condition that it shall move so as always to intersect the given fixed curve. All the points of the latter are therefore points on the surface generated. If it be required to find a surface of the family u= [a'x + b'y + c'z). Writing ax -+ by + cz for w, and a'x + b'y + c'z for v in the equation of Art. 410, we see that the partial differential equa- tion of cylindrical surfaces is [be' - b'c) L\ + {ca' - c'a) £7 + (ab' - a'b) U a = 0, or (Ex. 3, p. 26) U x cosa + Z7 2 cos/3 + Z7 8 C0S7 = 0, where a, ft, 7 are the direction-cosines of the generating line. Eemembering that Z7 1? C 2 , U a are proportional to the direction-cosines of the normal to the surface, it is obvious that the geometrical mean- FAMILIES OP SUEFACES. 343 ing of this equation is that the tangent plane to the surface is always parallel to the direction of the generating line. Ex. 1. To find the equation of the cylinder whose edges are parallel to x = h, y = mz, and which passes through the plane curve z = 0, (x, y) = 0. Ans. (»). We now proceed to the case when the equations of the generating curve include more than two parameters. By the help of the equations connecting these parameters, we can, in terms of any one of them, express all the rest ; and thus put the equations of the generating curve into the form > F{x,y, z, c, (c), yjr{c), &c.j = 0, f{x,y, z, c, <£ (c), f (c), &c.}= 0. FAMILIES OF SURFACES. 349 The equation of the surface , generated is obtained by elimi- nating c between these equations ; and, as has been already stated, all surfaces are said to be of the same family for which the form of the functions F and / is the same, whatever be the forms of the functions <£, ^ , & c . But since evidently the elimination cannot be effected until some definite form has been assigned to the functions , ^, &c. it is not generally possible to form a single functional equation including all sur- faces of the same family: and we can only represent them, as above written, by a pair of equations from which there remains a constant to be eliminated. We can however elimi- nate the arbitrary functions by differentiation and obtain a partial differential equation, common to all surfaces of the same family ; the order of that equation being, as we shall presently prove, equal to the number of arbitrary functions 0, yjr, &c. It is to be remarked however that in general the order of the partial differential equation obtained by the elimination of a number of arbitrary functions from an equation is higher than the number of functions eliminated. Thus if an equation in- clude two arbitrary functions , yjr, and if we differentiate with respect .to x and y which we take as independent variables, the differentials combined with the original equation form a system of three equations containing four unknown functions ', ■yjr'. The second differentiation (twice with regard to x, twice with regard to y, and with regard to x and y) gives us three additional equations ; but then from the system of six equations it is not generally possible to eliminate the six quantities , ty, <£', ijr', <£", \}r". We must therefore pro- ceed to a third differentiation before the elimination can be effected. It is easy to see, in like manner, that to eliminate n arbitrary functions we must differentiate 2n — l times. The reason why, in the present case, the order of the differential equation is less, is that the functions eliminated are all functions of the same quantity. 418. In order to show this it is convenient to consider first the special case, where a family of surfaces can be expressed hy a single functional equation. This will happen when it is 350 FAMILIES OF SURFACES. possible by combining the equations of the generating curve to separate one of the constants so as to throw the equations into the form u = c l ; F(x, y, z, c v c 2 ...ej = 0. Then express- ing, by means of the equations of condition, the other constants in terms of c t , the result of elimination is plainly of the form F{x, y, z, it, (u), i/r (u), &c.} =0. Now if, we denote by F lt the differential with respect to x of the equation of the surface, on the supposition that u is con- stant, we have „ _ dF tt v d p TT IP dF Now in these equations, the derived functions $', ty', &c. only enter in the term -=- ; they can therefore be all eliminated together; and we can form the equation, homogeneous in V, v» u a 3. u» v, F» K, K w„ «„, W 3 which contains only the original functions 0, fy, &c. If we write this equation V= 0, we can form from it in like manner the equation F„ V V V 21 V S = 0, which still contains no arbitrary functions but the original M » + -P"» + m ( M 2 + 2 u a) = °- It is convenient in practice to choose for one of the equations representing the generating curve, its projection on the plane of xy\ then since this equation does not contain z, the value of m derived from it will not contain p or q, and the first differential equation will be of the form p + qm = Bj B being also a function not containing p or q. The only terms then containing r, s, or * in the second differential equation are those derived from differentiating p + qm-, and that equation will be of the form r + 2sm + tm* = S, where S may contain x, y, z, p, q, but not r, s, or t. If now 352 FAMILIES OF SURFACES. we had only two functions to eliminate, we should solve for these constants from the original functional equation of the surface, and from p + qm = E ; and then substituting these values in to and in S, the form of the final second differential equation would still remain r 4 2sto' -f tm'" 2 = 8 1 , where to' and 8' might contain as, y, z, p, q. In like manner if we had three functions to eliminate, and if we denote the partial differentials of z of the third order by a, /3, y, 8, the partial differential equation would be of the form a + 3m/3 + Zm' («), and for c g ty [z). Since in forming the partial differential equation we are to regard z as constant, we may as well leave the equations in the form z = c v y = cx + c. These give us p + qm = 0, to = c 2 . According as we eliminate c 3 or c 2 , these equations give us p + qc 2 = 0, px\ qy = qc 3 . There are therefore two equations of the first order, each containing one arbitrary function, viz. P+ 1$ ( a ) = °) px+qy = qf (z). To eliminate completely arbitrary functions, differentiate p> + qm = 0, remembering that since to = c 2 it is to be regarded as constant when we get r + 2sm + tm? = 0, and eliminating m by means of p + qm = 0, the required equa- tion is q'r-^pqs+p^t-O, FAMILIES OF SUEFACES. 353 Next let the generating line be parallel to ax + by + cz ; its equations are ax + by + cz = Cj, y=^c i x+c s - 1 and the functional equation of the family of surfaces is got by writing for c 2 and c a , functions of ax + by + cz. Differentiating, we have a + cp + m (b + cq) = 0, w. = c 2 . The equations got by eliminating one arbitrary function are therefore a + cp + (b + cq) {ax + by + cz) = 0, (a + cp) x + (b + cq) y=(b + cq)y}r [ax -\-by + cz). Differentiating a-f bm + c(p + mq), and remembering that m is to be regarded as constant, we have r + 2sm + brri' = 0, and introducing the value of m already found (b + cqf r - 2 [a + cp) (b + cq) s+(a + cp)*t = 0. 421. This equation may also be arrived at by expressing that the tangent planes at two points on the same generator intersect, as they evidently must, on that generator. Let a, /3, 7 be the running co-ordinates, x, y, z those of the point of contact ; then any generator is the intersection of the tan- gent plane ry-z=p(a-x) + q(/3-y), with a plane through the point of contact parallel to the fixed plane a (a - x) + b (/3 -y) + c (7 - z) = 0, whence (a + cp) (a - x) + [b + cq) [fi-y) = 0. Now if we pass to the line of intersection of this tangent plane with a consecutive plane, a, /9, 7 remain the same, while x i V-i z i Pi $ var 7- Differentiating the equation of the tangent plane, we have (rdx + sdy) (a-x) + (sdx + tdy) (f3-y)= 0, And eliminating a-x, @-y, (b -t cq) (rdx + sdy) = (a + cp) (sdx + tdy). A A 354 FAMILIES OF SURFACES. But since the point of contact moves along the generator which is parallel to the fixed plane, we have adx + bdy + cdz = 0, or (a + cp) dx + (b + cq) dy = 0. Eliminating then dx, dy from the last equation, we have, as before, (b + cqf r-2(a + cp)(b + cq) s + (a + cpf t = 0. 422. Surfaces generated by lines which meet a fixed axis. This class also includes the family of conoids. In the first place let the fixed axis be the axis of z ; then the equations of the generating line are of the form y = c t x, z = c 2 x + c 8 ; and the equation of the family of surfaces is got by writing in the II latter equation for c 2 and c s , arbitrary functions of - . Diffe- rentiating, we have »i = c 1 , p + niq = c 2 , whence 2>x + qy = x(£j, and z-px- qy = ^ (f) • Differentiating again, we have r + 2sm + tm 2 = 0, and putting for m, its value =c, = -, the required differential equation is rx'* + 2sxy + ty* = 0. This equation may also be obtained by expressing that two consecutive tangent planes intersect in a generator. As, in Art. 421, we have for the intersection of two consecutive tan- gent planes (rdx + sdy) (a. — x) + (sdx + tdy) (fi — y) = 0. But any generator lies in the plane ay=fix, or (a— x)y = (/3— y) x. Eliminating therefore x (rdx + sdy) + y (sdx -f- tdy) = 0. dv Q v But -£■ = — = — . Therefore, as before, rx 2 + 2sxy + ty* = 0. dx a x ' ° J More generally let the line pass through a fixed axis a/3, where a. = ax + by+cz + d, /3 = dx + b'y + c'z + d'. Then tbe equations of the generating line are a = e,/3, y = c i x-\- c s , and the equation of the family of surfaces is y = x -5 + i/r ^ . Diffe- rentiating, we have m = c a , a + cp + m (b + cq) — c, [d -\-c'p + m (b' + c'q)}. FAMILIES OP SURFACES. 355 Differentiating again, we have r + Ism + trri 1 = 0, and putting in for in from the last equation, the required partial differential equation is {[a + cp)p-{a: + c'p)a>;'t -2{(a + cp)fi- {a' + c» a} {[I + cq) - [V + c'q) a} s + {{b+cq)$- {V + c'q) a} 2 r = 0. 423. If the equation of a family of surfaces contain n arbitrary functions of the same quantity, and if it be required to determine a surface of the family which shall pass through n fixed curves, we write down the equations of the generating curve u = c„ F(x, y, z, c , c 2 , &c.) = 0, and expressing that the generating curve meets each of the fixed curves, we have a sufficient number of equations to eliminate c„ c 2 , &c. Thus to find a surface of the family x + y§ (a) + yfr (a) = which shall pass through the fixed curves y = a, F(x, z) = 0; y = —a, F t (x, z) = 0. The equations of the generating line being z = c„ x=yc i + c a , we have, by substitution, F(ac. 2 + (m), where m is one of the roots of r + Ism + tm? = 0. The other two first integrals are y — mx = \{r(m), and z —px — mqx = x(m). The three second integrals are got by eliminating m, from any pair of these equations. 425. Envelopes. If the equation of a surface include n parameters connected by n - 1 relations, we can in terms of any one express all the rest, and throw the equation into the form z = F{x,y, c,(c), f (c), &c.}. dT? Eliminating c between this equation and -j- = 0, we find the etc envelope of all the surfaces obtained by giving different values to c. The envelopes so found are said to be of the same family as long as the form of the function F remains the same, no matter how the forms of the functions , ^, &c. vary. dW The curve of intersection of the given surface with -j- is the characteristic (see p. 254) or line of intersection of two con- secutive surfaces of the system. Considering the characteristic as a moveable curve from the two equations of which c is to be eliminated, it is evident that the problem of envelopes is included in that discussed, Art. 417, &c. If the function F contain n arbitrary functions , \jr, &c, then since -y- contains ', yjr', &c, it would seem, according to the theory previously explained, that the partial differential equation of the family 358 FAMILIES OF SURFACES. ought to be of the 2n tu order. But on examining the manner in which these functions enter, it is easy to see that the order reduces to the w th . In fact, differentiating the equation z = F, we get _, dF „ dF rfF 1 but since -j- = 0, we have p = F 1 ,q = F 2 , where, since F t and F t are the differentials on the supposition that c is constant, these quantities only contain the original functions , ty and not the derived ', yjr'. From this pair of equations we can form another, as in Art. 424, and so on, until we come to the n m order, when, as easily appears from what follows, we have equations enough to eliminate all the parameters. 426. We need not consider the case when the given equation contains but one parameter, since the elimination of this between the equation and its differential gives rise to the equation of a definite surface and not of a family of surfaces. Let the equation then contain two parameters a, 6, connected by an equation giving & as a function of a, then between the three equations z — F^p = F t , q — F 2 , we can eliminate a, b, $nd the form of the result is evidently f(x, y, z,j>, q) = 0. For example, let us examine the envelope of a sphere of fixed radius, whose centre moves along any plane curve in the plane of xy. This is a particular case of the general class of tubular surfaces which we shall consider presently. Now the equation of such a sphere being and the conditions of the problem assigning a locus along which the point a/3 is to move, and therefore determining j3 in terms of a, the equation of the envelope is got by eliminating a between (x-af+ti-tWy + z^r*, (a:-a) + {y-$(a)}$'(a)=0. Since the elimination cannot be effected until the form of the function is assigned, the family of surfaces can only be ex- pressed by the combination of two equations just written. FAMILIES OF SURFACES. 359 We might also obtain these equations by expressing that the surface is generated by a fixed circle, which moves so that its plane shall be always perpendicular to the path along which its centre moves. For the equation of the tangent to the locus of a/3 is dB J/-0=fa[x-a) or y- 0(a) = £'(«) (a>- a). And the plane perpendicular to this is 0* - «) + $°-{y -0 («)} = <>, as already obtained. To obtain the partial differential equa- tion, differentiate the equation of the sphere, regarding a, B as constant, when we have x - a + pz = 0, y — B + qz = 0. Solving for x — a, y — B and substituting in the equation of the sphere, the required equation is s 2 (l+/+ 2 2 )=r*. We might have at once obtained this equation as the geo- metrical expression of the fact that the length of the normal is constant and equal to r, as it obviously is. 427. Before proceeding further we wish to show how the arbitrary functions which occur in the equation of a family of envelopes can be determined by the conditions that the surface in question passes through given curves. The tangent line to one of the given curves at any point of course lies in the tangent plane to the required surface ; but since the en- veloping surface has at any point the same tangent plane as the enveloped surface which passes through that point, it follows that each of the given curves at every point of it touches the enveloped surface which passes through that point. If then the equation of the enveloped surface be z = F(x,y, Cl ,c 2 ...cJ, the envelope of this surface can be made to pass through n — 1 given curves ; for by expressing that the surface whose equa- tion has been just written touches each of the given curves, we obtain n — 1 relations between the constants c„ c a , &c, which combined with the two equations of the characteristic 360 FAMILIES OF SURFACES. enable us to eliminate these constants. For example, the family of surfaces discussed in the last article contains but two constants and one arbitrary function, and can therefore be made to pass through one given curve. Let it then be required to find an envelope of the sphere which shall pass through the right line x = inz, y = 0- The points of intersection of this line with the sphere being given by the quadratic (mz - a) 2 + /3 s + a 2 = r s , or (1 + »i 2 ) a 2 - 2?nza + a 2 -f /3 2 - r" = 0, the condition that the line should touch the sphere is (1 W)(a 2 + /3 2 -r 2 )=™V. We see thus that the locus of the centres of spheres touching the given line is an ellipse. The envelope required then is a kind of elliptical anchor ring, whose equation is got by eliminating a, /8 between (x - a) 2 + (y - /3) 2 + a 2 = r 2 , (1 + m 2 ) (a 2 + /3 2 - r 2 ) = mV, (z-a)c?a4 (y-/3)d/3 = 0, ado. + (1 + »i 2 ) 0d/3 = 0, from which last two equations we have (l+m 2 )/3(a:-a) = a(?/-/3). The result is a surface of the eighth degree. 428. Again, let it be required to determine the arbitrary function so that the envelope surface may also envelope a given surface. At any point of contact of the required sur- face with the fixed surface a =/(*, y), the moveable surface z = F (x, y, c„ c 2 , &c.) which passes through that point, has also the same tangent plane as the fixed surface. The values then of^> and q derived from the equations of the fixed surface and of the moveable surface must be the same. Thus we have fi=Fiif 2 = Fzi an( * if between these equations and the two equations z = F, z =f, which are satisfied for the point of contact, we eliminate x, y, z, the result will give a relation between the parameters. The envelope may thus be made to envelope as many fixed surfaces as there are arbitrary FAMILIES OF SUliFACES. 361 functions in the equation. Thus, for example, let it be re- quired to determine a tubular surface of the kind discussed (Art. 427), which shall touch the sphere a: 2 + f + z l = R*. This surface must then touch (x - a) 2 + (y - /3) 2 + a" = r 2 . We have therefore - = X ^ 2 _ ^Z^ . conditions which imply 2 = 0, z z ' z z ' * J ' a? a; — a _ — = -5 or pa? = ay. Eliminating x and y by the help of these equations, between the equation of the fixed and move- able sphere, we get 4 (a 2 + /3 2 ) E> = (R 2 -r* + a 2 + /3 2 ) 2 . This gives a quadratic for a 2 + j3 2 , whose roots are (R + r) 2 ; showing that the centre of the moveable sphere moves on one or other of two circles, the radius being either R + r. The surface required is therefore one or other of two anchor rings, the opening of the rings corresponding to the values just assigned. 429. We add one or two more examples of families of en- velopes whose equations include but one arbitrary function. To find the envelope of a right cone whose axis is parallel to the axis of z, and whose vertex moves along any assigned curve in the plane of xy. Let the equation of the cone in its original position be z* = m 2 (x' + y') ; then if the vertex be moved to the point a, j3, the equation of the cone becomes z' = m* {(x — a) 2 + (y - /3) 2 }, and if we are given a curve along which the vertex moves, /3 is given in terms of a. Differentiating we have jpz = m 2 (x - a), qz = rri' (y — /3) ; and eliminating we have p i +q* = m\ This equation expresses that the tangent plane to the surface makes a constant angle with the plane of xy, as is evident from the mode of generation. It can easily be deduced hence that the area of any portion of the surface is in a constant ratio to its projection on the plane of xy. 430. The families of surfaces, considered (Arts. 426, 429), are both included in the following: "To find the envelope of a surface of any form which moves without rotation, its motion being directed by a curve along which any given point of the surface moves." Let the equation of the surface in its original position be z = F{x,y), then if it be moved without turning 362 FAMILIES OP SURFACES. so that the point originally at the origin shall pass to the position a/87, the equation of the surface will evidently he z — y=F(x — a,y — fi). If we are given a curve along which the point afiy is to move, we can express a, /3 in terms of 7, and the problem is one of the class to be considered in the next article, where the equation of the envelope includes two arbitrary functions. Let it be given however that the directing curve is drawn on a certain known surface, then, of the two equations of the directing curve, one is known and only one arbitrary, so that the equation of the envelope includes but one arbitrary function. Thus if we assume /3 an arbitrary function of a, the equation of the fixed surface gives 7 as a known function of a, /3. It is easy to see bow to find the partial differential equation in this case. Between the three equations z-y=F(x-a,y-/3),p=F 1 {x-a,y-/3),q = F 2 (x-a,y-/3), solve for x — a,y — fi, z — 7, when we find x-a=f(p, q), y-P = Y{p, q), z-7 = Y(p, ?)• If then the equation of the surface along which a/37 is to move be T (a, /3, 7) = 0, the required partial differential equation is T {x-fip, q), y-Y(p, q), *-yfa q)} =0. The three functions /, '/, "/, are evidently connected by the relation oV s f=pdf+qa r 'f. It is easy to see that the partial differential equation just found is the expression of the fact that the tangent plane at any point on the envelope, is parallel to that at the corre- sponding point on the original surface. Ex. To find the partial differential equation of the envelope of a sphere of con- stant radius whose centre moves along any curve traced on a fixed equal sphere x 1 + y z + z z — r 2 . The equation of the moveable sphere is (x — of + (1/ — /3) 2 + (2 — y) 2 = r 2 , whence x - a. +p (a - 7) = 0, y - ft + q (z - y) = 0, and we have _ — — j »* „ _ — qr _ r If we write 1 +p* + f = p * it is easy to see, by actual differentiation, that the relation is fulfilled 'J=-*(9-*ffl- FAMILIES OF SURFACES. 3(53 The partial differential equation U [xp +prf + (yp + qrf + (zp - rf = ph-% 01 (x' + f + si) (l +i ,2 + j2)j + 2 (j>x + qy-z)r = 0. 431. We now proceed to investigate the form of the partial differential equation of the envelope, when the equation of the moveable surface contains three constants connected by two relations. If the equation of the surface be 2 = F{x, y, a, b, c), then we have p = F s , q = F 2 . Differentiating again, as in Art. 419, we have r + sm = F n + mF n , s + tm = F a + mF a ; and eliminating m, the required equation* is (r-F il )(t-FJ = (s-FJ\ The functions F tl , F K , F m contain a, b, c, for which we are to substitute their values in terms of p, q, x, y, z derived from solving the preceding three equations, when we obtain an equa- tion of the form Br + 2Ss + Tt + U[rt - s*) = V, where B, 8, T, Uj V are connected by the relation BT+ UV= 8\ 432. The following examples are among the most important of the cases where the equation includes three parameters. Developable Surfaces. These are the envelope of the plane z = ax+by + c, where for b and c we may write

(p). Any surface therefore is a developable surface if p and q are con- nected by a relation independent of x } y, z. Thus the family (Art. 429) for which p 2 + q* = m 2 , is a family of developable surfaces. We have also z — px — qy = yfr (p), which is the other first integral of the final differential equation. This last is got by differentiating again the equations p = a, q = b, when we have r + sm = 0, s + tm = 0, and eliminating m, rt — s 2 = 0, which is the required equation. * I owe to Professor Boole my knowledge of the fact that when the equation of the moveable surface contains three parameters, the partial differential equation is of the form stated above. See his Memoir, Phil. Trans. 1862, p. 437, 364 FAMILIES OF SUEFACES. By comparing Arts. 289, 307 it appears that the condition rt = s 2 is satisfied at every parabolic point on a surface. The same thing may be shewn directly by transforming the equation rt-s 2 = into a function of the differential coefficients of U, by the help of the relations U l+F U 3 = 0, U i + qU 3 = 0, U u 4 2U lt p + U aP '=-rU ti U l2 +pU 23 + qU l3+M U 33 = -sU 3 ; when the equation rt - s 2 is found to be identical with the equa- tion of the Hessian. We see now then that every point on a developable is a parabolic point, as is otherwise evident, for since (Art. 324) the tangent plane at any point meets the surface in two coincident right lines, the two inflexional tangents at that point coincide. The Hessian of a develop- able must therefore always contain the equation of the surface itself as a factor. The Hessian of any surface being of the degree in — 8, that of a developable consists of the surface itself, and a surface of 3« — 8 degree which we shall call the Pro-Hessian. In order to find in what points the developable is met by the Pro-Hessian, I form the Hessian of xu + y 2 v, and I find that we get the developable itself multiplied by a series of terms in which the part independent of x and y is v j -j-j- -j—% — I , , ) • . This proves that any generator xy meets the Pro-Hessian in the first place where xy meets v ; that is to say, twice in the point on the cuspidal curve (»z), and in r — 4 points on the model curve (%) ; and in the second place, where the generator meets the Hessian of u considered as a binary quantic ; that is to say, in the Hessian of the system formed hy these r — 4 points com- bined with the point on (m) taken three times; in which Hessian the latter point will be included four times. The intersection of any generator with the Pro-Hessian consists of the point on (»w) taken six times, of the r — i points on (a;), and of 2 (r — 5) other points.* * Mr. Caylsy has calculated the equation of the Pro-Hessian (Quarterly Journal, Vol. vi. p. 108) in the case of the developables of the fourth and fifth orders, and of. FAMILIES OF SURFACES. 365 433. Tubular Surfaces. Let it be required to find the differential equation of the envelope of a sphere of constant radius, whose centre moves on any curve. We have, as in Art. 430, (a - a)» + (y -£)*+(*- 7 )' = .#*, x-a.+p(z-y)=Q, y-/8 + 2 (*-7)=0, whence 1 +p* + (z - 7 ) r + m {pq + [z - y) s] = 0, pq + (s - y) s + m {1 + q 2 + (z - 7) t} = 0. And therefore {l+p*+ (z-y)r} {l + q> + (z- y) t] = {pq + (z-y)s}\ Substituting for z — y its value j- — —, jrj (Art. 400) this becomes E i (rt-s*)-B{(l+q i )r-2pqs^l+f)t}J{l+/+q*)+(l+f+q>y=0, which denotes, Art. 306, that at any point on the required envelope one of the two principal radii of curvature is equal to R as is geometrically evident. 434. We shall briefly show what the form of the diffe- rential equation is when the equation of the surface whose envelope is sought contains four constants. We have, as before, in addition to the equation of the surface the three equations p = F„ q = F a (r - F n ) (t - FJ = {s - FJ. Let us, for shortness, write the last equation pr = a i , and let us write a - F n] = A,j3- F at = B,y- F m = C,S-F^ = D; then, diffe- rentiating pr = cr 2 , we have (A + Bm) T-h(G+ Dm) p - 2 {B+ Cm) a- = 0. Substituting for m from the equation a + rrn = 0, and remember- ing that pr = c 2 , we have At 3 - ZBo-t' + 3 Ga\ - Da 3 = 0, that of the sixth order considered, p. 276. The Pro-Hessian of the developable of the fourth order is identical with the developable itself. In the other two cases the cuspidal curve is a cuspidal curve also on the Pro-Hessian, and is, counted six times in the intersection of the two surfaces. I suppose it may be assumed that this is generally true. The nodal curve is but a simple curve on the Pro-Hessian, and therefore is only counted twice in the intersection. 366 FAMILIES 'OF SURFACES. in which equation we are to substitute for the parameters im- plicitly involved in it, their values derived from the preceding equations. The equation is therefore of the form a ■+ 3/3m + Sym 2 + Bm 3 = U, where m and U are functions of x, y, z, p, q : r, s, t. In like manner we can form the differential equation when the equa- tion of the moveable surface includes a greater number of parameters. 435. Having in the preceding articles explained how partial differential equations are formed, we shall next show how from a given partial differential equation can be de- rived another differential equation satisfied by every charac- teristic of the family of surfaces to which the given equation belongs (see Monge, p. 53). In the first place, let the given equation be of the first order; that is to say, of the form f(xi y, z, p, q) = 0. Now if this equation belong to the en- velope of a moveable surface, it will be satisfied not only by the envelope but also by the moveable surface in any of its positions. This follows from the fact that the envelope touches the moveable surface, and therefore that at the point of contact x, y, z, p, q are the same for both. Now if x, y, z be the co-ordinates of any point on the characteristic, since such a point is the intersection of two consecutive positions of the moveable surface, the equation f(x, y, z, p, q) = will be satisfied by these values of x, y, z, whether p and q have the values derived from one position of the moveable surface or from the next consecutive. Consequently, if we differentiate the given equation, regarding p and q as alone variable, then the points of the characteristic must satisfy the equation Pdp + Qdq = 0. Or we might have stated the matter as follows: Let the equation of the moveable surface be z = F(x, y, a), where the constants have all been expressed as functions of a single parameter a. Then (Art. 425) we have p = F l (x, y, a), q = F 2 (cc 5 y, a), which values of p and q may be substituted in the given equation. Now the characteristic is expressed by FAMILIES OF*SURFACES. 867 combining with the given equation its differential with respect to a : and a only enters into the given equation in consequence of its entering into the values for p and q. Hence we have, as before. P^ + Q$ = . da da Now since the tangent line to the characteristic at any point of it, lies in the tangent plane to either of the surfaces which intersect in that point, the equation dz =pdx +- qdy is satisfied, whether p and q have the values derived from one position of the moveable surface or from the next consecutive. We have therefore J- dx + -^ dy = 0. And combining this equation with that previously found, we obtain the differential equation of the characteristic Pdy — Qdx = 0. Thus if the given equation be of the form Pp + Qq = B, the characteristic satisfies the equation Pdy - Qdx = 0, from which equation combined with, the given equation and with dz =pdx -f- qdy, can be deduced Pdz = Bdx, Qdz = Rdy. The reader is aware (see Boole's Differential Equations, p. 322) of the use made of those equations in integrating this class of equations. In fact, if the above system of simultaneous equa- tions integrated give u = c t , v = c 2 , these are the equations of the characteristic, or generating curve, in any of its positions, while in order that v may be constant whenever u is constant, we must have u = (v). Ex. Let the equation be that considered (Art. 426), viz. z 2 (1 +p' + q 2 ) = r 2 , then any characteristic satisfies the equation pdy = qdx, which indicates (Art. 400) that the characteristic is always o, line of greatest slope on the surface, as is geome- trically evident. 436. The equation just found for the characteristic generally includes p and q, but we can eliminate these quantities by com- bining with the equation just found, the given partial diffe- rential equation and the equation dz =pdx + qdy. Thus, in the last example, from the equations z l (1 +p" + q') = r\ qdx =pdy, we derive z* [dx 2 + dy' + dz 2 ) = r a {dx 1 4 dy*). The reader is aware that there are two classes of differential equations of the first order, one derived from the equation of 368 FAMILIES *P SURFACES. a single surface, as, for instance, by the elimination of any constant from an equation £7=0, and its differential Ufa -+ Ufa 4- Ufa = 0. An equation of this class expresses a relation between the direction-cosines of every tangent line drawn at any point on the surface. The other class is obtained by combining the equations of two surfaces, as, for instance, by eliminating three constants between the equations U— 0, V= and their diffe- rentials. An equation of this class expresses a relation satisfied by the direction-cosines of the tangent to any of the curves which the system U, V represents for any value of the con- stants. The equations now under consideration belong to the latter class. Thus the geometrical meaning of the equation chosen for the example is that the tangent to any of the curves denoted by it, makes with the plane of xy an angle whose cosine is - . This property is true of every circle in a vertical plane whose radius is r; and the equation might be obtained by eliminating the constants a, /3, m, between the equations (x - a) 2 + (y - /S) a + z> = r% x - a + m (y - /3) = 0. 437. The differential equation found, as in the last article, is not only true for every characteristic of a family of surfaces, but since each characteristic touches the cuspidal edge of the surface generated, the ratios dx : dy : dz are the same for any characteristic and the corresponding cuspidal edge ; and consequently the equation now found is satisfied by the cuspidal edge of every surface of the family under consideration. Thus in the example chosen, the geometrical property expressed by the differential equation not only is true for a circle in a vertical plane, but remains true if the circle be wrapped on any vertical cylinder; and the cuspidal edge of the given family of surfaces always belongs to the family of curves thus generated. Precisely as a partial differential equation in p } q (express- ing as it does a relation between the direction-cosines of the tangent plane), is true as well for the envelope as for the par- FAMILIES OF SURFACES. 3C9 ticular surfaces enveloped ; so the total differential equations here considered are true both for the cuspidal edge and the series of characteristics which that edge touches. The same thing may be stated otherwise as follows: the system of equations U= 0, -j- = which, when a is regarded as constant, represents the characteristic, represents the cuspidal edge when a is an unknown function of the variables to be eliminated by means d? U of the equation -=-5- = 0. But evidently the equations U= 0, -j- = have the same differentials when a is considered as da. variable, subject to this condition, as if a were constant. Thus, in the example of the last article} if in the equations (x-af + (y - fif + z* = r% (x-a.) + m (y - /S) = 0, we write /3=(/>(a), m = (p'(ct), and combine with these the equation 1 -f ' (a) 2 = (y — /3) " (a), the differentials of the first and second equations are the same when a is variable in virtue of the third equation, as if it were constant ; and therefore the differential equation obtained by eliminating a, /3, m between the first two equations and their differentials, on the supposition that these quantities are constant, holds equally when they vary according to the rules here laid down. And we shall obtain the equations of a curve satisfying this differential equation by giving any form we please to cj> (a) and then eliminating a between the equations {x-aY + [y-4>{a)] i + z i = r\ x- a + ' (a) [y - <£(«)} = 0, * It is convenient to insert here a remark made by Mr. M. Roberts, viz. that if in the equation of any surface we substitute for x, x + \dx, for y, y + Xcly, for z, z + \dz, and then form the discriminant with respect to \, the result will be the differential equation of the cuspidal edge of any developable enveloping the given surface. In fact it is evident -(see Art.. 271) that the discriminant expresses the condition that the tangent to the curve represented by it touches the given surface. Thus the general equation of the cuspidal edge of developables circumscribing a sphere is (x 1 + y- + z 2 - a?) (cbs 1 + dy 2 + dz-) = (xdx + ydy + zdz)", or (ydz — zdy)- + (zdx - xdz)"- + (xdy — ydx) 2 = a- (dx- + dy- + dz 2 ). In the latter form it is evident that the same equation is satisfied by a geodesic BB 370 FAMILIES OF SURFACES. 438. In like manner can be found the differential equation of the characteristic, the given equation being of the second order (see Monge, p. 74). In this case we can have two consecutive surfaces, satisfying the given differential equation, and touching each other all along their line of intersection. For instance, if we had a surface generated by a curve moving so as to meet two fixed directing curves, we might conceive a new surface generated by the same curve meeting two new directing curves, and if these latter directing curves touch the former at the points where the generating curve meets them, it is evident that the two surfaces touch along this line. In the case supposed then the two surfaces have x, y, z, p, q common along their line of intersection and can differ only with regard to r, s, t. Differentiate then the given differential equation considering these quantities alone variable, and let the result be Rdr + Sds + Tdt = 0. But since p and q are con- stant along this line, we have drdx + dsdy = 0, dsdx + dtdy = 0. Eliminating then dr, ds, dt, the required equation for the cha- racteristic is Rdtf - Sdxdy + Tdx' = 0. In the case of all the equations of the second order, which we have already considered, this equation turns out a per- fect square. When it does not so turn out, it breaks up into two factors, which, if rational, belong to two independent characteristics represented by separate equations ; and if not, denote two branches of the same curve intersecting on the point of the surface which we are considering. 439. In fact when the motion of a surface is regulated by a single parameter (see Art. 316), the equation of its envelope, as we have seen, contains only functions of a single quantity, and the differential equation belongs to the simpler species just referred to. But if the motion of the surface be regulated traced on any cone whose vertex is the origin. For if the cone be developed into a plane, the geodesic will become a right line, and if the distance of that line from the origin be a, then the area of the triangle formed by joining any element ds to the origin is. half ads, but this is evidently the property expressed by the preceding equation. FAMILIES OF SURFACES. 371 by two parameters, its contact with its envelope being not a curve, but a point; then the equation of the envelope will in general contain functions of two quantities, and the diffe- rential equation will be of the more general form. As an illustration of the occurrence of the latter class of equations in geometrical investigations, we take the equation of the family of surfaces which has one set of its lines of curvature parallel to a fixed plane, y = mx. Putting dy = mdx in the equation of Art. 306, the differential equation of the family is m'{{l + q^s-pqt} + m[(l-\-q i )r-[l-\-f)t}-{{l+f)s-pqr}=0. As it does not enter into the plan of this treatise to treat of the integration of such equations, we refer to Monge, p. 161 for a very interesting discussion of this equation. Our object being only to show how such differential equations present themselves in geometry, we shall show that the preceding equation arises from the elimination of a, /3 between the follow- ing equation and its differentials with respect to a and /3 : (x - a) 2 + [y - Pf + [z - (a + m/3)}* = f{i3- mo)\ Differentiating with respect to a and /3, we have (x — a) + [z- ) $' = m yfr'yjr, [y-P) + m i z ~ ) 4>' = ~ V^i whence (cc — a) + m (y - @) + (1 + m?) (z - ) ' = 0. But we have also (x-a)+p(s-)=0, (y-0) + 2(*-0=<>, whence [x - a) + m {y - /3) + (p + mq) {z-) = 0. And by comparison with the preceding equation, we have p + m.q = (1 + m?) ' (a + mfi). If then we call a + mfi, y, the problem is reduced to eliminate 7 between the equations x + my-y+{p + mq){z-(y)} = 0, p + mq = (1 + m*) £'(7). Differentiating with regard to x and y, we have (1 +f + mpq) + (r + ms) {z-(f> (7)} = {1 + [p + mq) '} y t , [m (1 4 q*) +pq) + {s + rnt) {z - (7)} = {1 + {p + mq) 0'} 7s , but from the second equation r ■+ ms : s + mt : : 7, : 7 2 . BB2 372 RULED SURFACES. Hence the result is (1 +p 2 + mpq) (s + mt) = {m (1 + q 1 ) +pq} [r + rns), as was to be proved. RULED SURFACES.* 440. On account of the importance of ruled surfaces, we add some further details as to this family of surfaces. The tangent plane at any point on a generator evidently contains that generator, which is one of the inflexional tangents (Art. 259) at that point. Each different point on the gene- rator has a different tangent plane (Art. 107) which may be constructed as follows: We know that through a given point can be drawn a line intersecting two given lines ; namely, the intersection of the planes joining the given point to the given lines. Now consider three consecutive generators, and through any point A on one, draw a line meeting the other two. This line, passing through three consecutive points on the surface, will be the second inflexional tangent at A, and therefore the plane of this line and the generator at A is the tangent plane at A. In this construction it is supposed that two consecutive generators do not intersect, which ordinarily they will not do. There may be on the surface, however, singular generators which are intersected by a consecutive generator, and in this case the plane containing the two consecutive generators is a tangent plane at every point on the generator. In special cases also two consecutive generators may coincide, in which case the generator is a double line on the surface. 441. The (inharmonic ratio of four tangent planes passing through a generator is equal to that of their four points of con- tact. Let three fixed lines A, i?, C be intersected by four transversals in points aa'a'a", bb'b"b"\ cc'c'c". Then the an- harmonic ratio \bb'b"b'") = [cc'c'c'"}, since either measures the ratio of the four planes drawn through A and the four trans- * The theorems in this section are principally taken from M. Chasles's Memoin Quetelet's Correspondance, t. XI., p. 50, and from Mr. Cayley's paper, Cambridge and Dublin Mathematical Journal, Vol. vii., p. 171. See also his Memoir, Philosophical Transactions, 1863, p. 453. RULED SURFACES. 373 versals. In like manner [cc'c'c'"} = {aa'a"a'"} either measuring the ratio of the four planes through B (see Art. 112). Now- let the three fixed lines be three consecutive generators of the ruled surface, then by the last article, the transversals meet any of these generators A in four points, the tangent planes at which are the planes containing A and the transversals. And by this article it has been proved that the anharmonic ratio of the four planes is equal to that of the points where the transversals meet A. 442. We know that a series of planes through any line and a series at right angles to them form a system in involution, the anharmonic ratio of any four being equal to that of their four conjugates. It follows then, from Art. 441, "that the system formed by the points of contact of any plane, and of a plane at right angles to it, form a system in involution ; or, in other words, the system of points where planes through any generator touch the surface, and where they are normal to the surface, form a system in involution. The centre of the system is the point where the plane which touches the surface at infinity, is normal to the surface ; and by the known properties of in- volution, the distances from this point of the points where any other plane touches and is normal, form a constant rect- angle. 443. The normals to any ruled surface along any generator, generate a hyperbolic paraboloid. It is evident that they are all parallel to the same plane, namely, the plane perpendicular to the generator. We may speak of the anharmonic ratio of four lines parallel to the same plane, meaning thereby that of four parallels to them through any point. Now in this sense the anharmonic ratio of four normals is equal to that of the four corresponding tangent planes, which (Art. 441) is equal to that of their points of contact, which again (Art. 442) is equal to that of the points' where the normals meet the generator. But a system of lines parallel to a given plane and meeting a given line generates a hyperbolic paraboloid, if the anharmonic ratio of any four is equal to that of the 374 EULED SURFACES. four points where they meet the line. This proposition follows immediately from its converse, which we can easily establish. The points where four generators of a hyperbolic paraboloid intersect a generator of the opposite kind, are the points of contact of the four tangent planes which contain these gene- rators, and therefore the anharmonic ratio of the four points is equal to that of the four planes. But the latter ratio is measured by the four lines in which these planes are inter- sected by a plane parallel to the four generators, and these intersections are lines parallel to these generators. 444. The central points of the involution (Art. 442) are, it is easy to see, the points where each generator is nearest the next consecutive, that is to say, the point where each generator is intersected by the shortest distance between it and its next consecutive. The locus of the points on the generators of a ruled surface, where each is closest to the next consecutive, is called the line of striction of the surface. It may be remarked, in order to correct a not unnatural mistake (see Lacroix, Vol. III., p. 668), that the shortest distance between two consecutive generators is not an element of the line of striction. In fact if A a, Bb, Oc be three consecutive generators, ah the shortest distance between the two former, then b'c the shortest distance between the second and third will in general meet Bb in a point b' distinct from b, and the element of the line of striction will be ah' and not ab» Ex. 1. To find the line of striction of the hyperbolic paraboloid « 2 W~ 2. Any pair of generators may be expressed by the equations x y . a b X a y b~ i x y a b X a y . b~ l X y Both being parallel to the plane t , their shortest distance is perpendicular to this plane, and therefore lies in the plane a? — b 2 1 which intersects the first generator in the point z = -k — t» r~ ■ EULED SURFACES. 375 When the two generators approach to coincidence, we have for the co-ordinates of the point where either is intersected by their shortest distance l_ a 2 -b 2 1 ' 1. — " n . jo i 1 a 2 + b 2 \ 2 ' a b a 2 + b 2 \ andhence (a 2 + V) ( x - + f) = ( a 2 - b 2 ) (*-f), Or * + J^ = . \a bj ' \a bj a 3 b 3 The line of striction is therefore the parabola in which this plane cuts the surface. The same surface considered as generated by the lines of the other system hag another line of striction lying in the plane *-£ = a? b s Ex. 2. To find the line of striction of the hyperboloid a 2 b 2 c 2 ~ Ans. It is the intersection of the surface with a 2 A 2 b 2 B 2 _ c 2 C 2 x 2 y 2 ~ z 2 * where A^ 1 + l, B^ + l, C =±-±. 445. Given any generator of a ruled surface, we can de- scribe a hyperboloid of one sheet, which shall have this gene- rator in common with the ruled surface, and which shall also have the same tangent plane with that surface at every point of their common generator. For it is evident from the con- struction of Art. 440 that the tangent plane at every point on a generator is fixed, when the two next consecutive gene- rators are given, and consequently that if two ruled surfaces have three consecutive generators in common, they will touch all along the first of these generators. Now any three non- intersecting right lines determine a hyperboloid of one sheet (Art. 76) ; the hyperboloid then determined by any generator and the two next consecutive will touch the given surface as required. In order to see the full bearing of the theorem here enun- ciated, let us suppose that the axis of z lies altogether in any surface of the n tti degree, then every term in its equation must contain either x or y ; and that equation arranged according to the powers of x and y will be of the form u n _ x x + o r J + u n y + v^xy + w M _y + &c. = 0, where «„_,, v n _ t denote functions of z of the (n — l) m degree, &c. 376 RULED SURFACES. Then (see Art. 107) the tangent plane at any point on the axis will be u ,x + v ,y = 0, where u , denotes the result of sub- n— 1 u-lt/ / n—l stituting in u n _ x the co-ordinates of that point. Conversely, it follows that any plane y = mx touches the surface in n — 1 points, which are determined by the equation u n _ x + mv n ^ = 0. If however u ,, v , have a common factor u„, so that the ji-17 n—l el terms of the first degree in x and y may be written u e [u^f^x + v n _ v _{y) = 0, then the equation of the tangent plane will be u\ y x + v' 7r _ p _,y = 0, and evidently in this case any plane y = mx will touch the surface only in n— p— 1 points. It is easy to see that the points on the axis for which u p = are double points on the surface. Now what is asserted in the theorem of this article is, that when the axis of z is not an isolated right line on a surface, but one of a system of right lines by which the surface is generated, then the form of the equation will be « ,_ 2 {ux + vy) + &c. = 0, so that the tangent plane at any point on the axis will be the same as that of the hyperboloid ux + vy, viz. ux + v'y — 0. And any plane y = mx will touch the surface in but one point. The factor u n _ % indicates that there are on each generator n — 2 points which are double points on the surface. 446. We can verify the theorem just stated, for an im- portant class of ruled surfaces, viz., those of which any gene- rator can be expressed by two equations of the form at + bf 1 - 1 + ci m - 2 + &c. = 0, a'f + b'f- 1 + cT* + &c. = 0, where a, a\ b, b', &c. are linear functions of the co-ordinates, and t a variable parameter. Then the equation of the surface obtained by eliminating t between the equations of the gene- rator (see Higher Algebra, p. 34), may be written in the form of a determinant, the first row and first column of which are identical, viz., {ab'), (ac), (ad 1 ), &c. Now the line ad is a generator, namely, that answering to t = oo ; and we have just proved that either a or a' will appear in every term, both of the first row and of the first column. Since then every term in the expanded determinant contains a factor from the RULED SURFACES. 377 first row and a factor from the first column, the expanded determinant will be a function of, at least, the second degree in a and a\ except that part of it which is multiplied by (ab'), the term common to the first row and first column. But that part of the equation which is only of the first degree in a and a determines the tangent at any point of ad '; the ruled surface is therefore touched along that generator by the hy- perboloid ab' — bd = 0. If a and b (or d and b') represent the same plane, then the generator act! intersects the next consecutive, and the plane a touches along its whole length. If we had b = lca, b' = 7cd, the terms of the first degree in a and a' would vanish, and ad would be a double line on the surface. 447. Returning to the theory of ruled surfaces in general, it is evident that any plane through a generator meets the surface, in that generator and in a curve of the (n — l) tb degree meeting the generator in n— 1 points. Each of these points being a double point in the curve of section is (Art. 258) in a certain sense a point of contact of the plane with the surface. But we have seen (Art. 445) that only one of them is properly a point of contact of the plane ; the other n — 2 are fixed points on the generator, not varying as the plane through it is changed. They are the points where this generator meets other non-consecutive generators, and are points of a double curve on the surface. Thus then a skew ruled surface in general has a double curve which is met by every generator in n—2 points. It may of course happen that two or more of these n _ 2 points may coincide, and that the multiple curve on the surface may be of higher order than the second. In the case considered in the last article it can be proved (see Appendix on the Order of Systems of Equations) that the multiple curve is of the order £ (m + n — 1) (m + n — 2), and that there are on it £ (m + n — 2) (m. + n — 3) (m + n - 4) triple points. A ruled surface having a double line will in general not have any cuspidal line unless the surface be a developable, and the section by any plane will therefore be a curve having double points but not cusps. 378 RULED SURFACES. 448. Consider now the cone whose vertex is any point, and which envelopes the surface. Since every plane through a generator- touches the surface in some point, the tangent planes to the cone are the planes joining the series of gene- rators to the vertex of the cone. The cone will, in general, not have any stationary tangent planes : for such a plane would arise when two consecutive generators lie in the same plane passing through the vertex of the cone. But it is only in special cases that a generator will be intersected by one con- secutive ; the number of planes through two consecutive gene- rators is therefore finite; and hence one will, in general, not pass through an assumed point. The class of the cone, being equal to the number of tangent planes which can be drawn through any line through the vertex, is equal to the number of generators which can meet that line, that is to say, to the degree of the surface (see note, p. 87). We have proved now that the class of the cone is equal to the degree of a section of the surface; and that the former has no stationary tangent planes as the latter has no stationary, or cuspidal, points. The equations then which connect any three of the singularities of a curve prove that the number of double tangent planes to the cone must be equal to the number of double points of a section of the surface ; or, in other words, that the number of planes containing two generators which can be drawn through an assumed point, is equal to the number of points of intersection of two generators which lie in an assumed plane.* 449. We shall illustrate the preceding theory by an enu- meration of some of the singularities of the ruled surface gene- rated by a line meeting three fixed directing curves, the degrees of which are «i„ m 2 , m 3 .f The degree of the surface generated is equal to the number of generators which meet an assumed right line; it is there- fore equal to the number of intersections of the curve m with * These theorems are Mr. Cayley's. Cambridge and Dublin Mathematical Journal, Vol. Til,, p. 171. t I published a discussion of this surface, Cambridge and Dublin Mathematical Journal, Vol. VIII., p. 45. RULED SUEFACES. 379 the ruled surface having for directing curves the curves m 2 , m a and the assumed line ; that is to say, it is m l times the degree of the latter surface. The degree of this again is, in like manner, «i 2 times the degree of the ruled surface whose directing curves are two right lines and the curve «i 3 , while by a repe- tition of the same argument, the degree of this last is 2m a . It follows that the degree of the ruled surface when the generators are curves m„ m 2 , wi 3 , is 2«i 1 w. 2 »i 3 . The three directing curves are multiple lines on the surface, whose orders are respectively »2 2 w 3 , i m s m rn^rn^. For through any point on the first curve pass m 2 m 3 generators, the inter- sections namely of the cones having this point for a common vertex, and resting on the curves m 2 , in a . 450. The degree of the ruled surface, as calculated by Art. 449, will admit of reduction if any pair of the directing curves have points in common. Thus if the curves «i 2 , rn a have a point in common, it is evident that the cone whose vertex is this point, and base the curve in 1 will be included in the system, and that the order of the ruled surface proper will be reduced by m x , while the curve m 1 will be a multiple line of degree only mjn s — 1. And generally if the three pairs made out of the three directing curves have common respectively a, ft, 7 points, the order of the ruled surface will be reduced by m,a 4 mji + m 3 y,* while the order of multiplicity of the directing curves will be reduced respectively by a, /3, 7. Thus if the directing lines be two right lines and a twisted cubic, the surface is in general of the sixth order, but if each of the lines intersect the cubic, the order is only of the fourth. If each intersect it twice the surface is a quadric. If one intersect it twice and the other once, the surface is a skew surface of the third degree on which the former line is a double line. Again, let the directing curves be any three plane sections of a hyperboloid of one sheet. According to the general theory the surface ought to be of the sixteenth order, and let us see how a reduction takes places. Each pair of directing curves * My attention was called by Mr. Cayley to this reduction which takes place when the directing curves have points in common. 380 EULED SURFACES. Lave two points common ; namely, the points in which the line of intersection of their planes meets the surface. And the complex surface of the sixteenth order consists of six cones of the second order, together with the original quadric reckoned twice. That it must be reckoned twice, appears from the fact that the four generators which can be drawn through any point on one of the directing curves, are two lines belonging to the cones, and two generators of the given hyperboloid. In general, if we take as directing curves three plane sec- tions of any ruled surface, the equation of the ruled surface generated will have, in addition to the cones and to the original surface, a factor denoting another ruled surface which passes through the given curves. For it will generally be possible to draw lines, meeting all three curves, which are not gene- rators of the original surface. 451. The order of the ruled surface being 2m l m i m s , it follows, from Art. 447, that any generator is intersected by 2w 1 wz a ?w a — 2 other generators. But we have seen that at the points where it meets the directing curves, it meets [ m t m a ~ 1 ) + [ m a m i ~ *) + [ m i m t - !) other generators. Conse- quently it must meet 2m 1 wi a m a - [m 2 m a + wyn, + m^) + 1 gene- rators, in points not on the directing curves. We shall establish this result independently by seeking the number of generators which can meet a given generator. By the last article, the degree of the ruled surface whose directing curves are the curves m„ m 2 , and the given generator, which is a line resting on both, is 2m 1 m 2 — m 1 — m a . Multiplying this number by wt 8 , we get the number of points where this new ruled surface is met by the curve m B . But amongst these will be reckoned (mm — 1) times the point where the given generator meets the curve m. Sub- tracting this number then, there remain 2»i 1 to 2 to 3 - m i m s — rnpi z — mjn^ + 1 points of the curve m s , through which can be drawn a line to meet the curves «i l5 m 3 , and the assumed generator. But this is in other words the thing to be proved. 452. We can examine in the same way the order of the surface generated by a line meeting a curve «j, twice, and RULED SURFACES. 381 another curve w 2 once. It is proved, as in Art. 449, that the order is w 2 times the order of the surface generated by a line meeting m l twice, and meeting any assumed right line. Now if \ be the number of apparent double points of the curve m , that is to say, the number of lines which can be drawn through an assumed point to meet that curve twice, it is evident that the assumed right line will on this ruled surface be a multiple line of the order A„ and the section of the ruled surface by a plane through that line, will be that line \ times, together with the ^m 1 (m t — 1) lines joining any pair of the points where the plane cuts the curve m x . The degree of this ruled surface will then be \ + ^m l (m i - 1), and, as has been said, the degree will be m 2 times this number, if the second director be a curve m % instead of a right line. The result of this article may be verified as follows : Con- sider a complex curve made up of two simple curves «i t , m ki ; then a line which meets this system twice must either meet both the simple curves, or else must meet one of them twice. The number of apparent double points of the system is \ + K + m i m 2 j* ana< th e order of the surface generated by a line meeting a right line, and meeting the complex curve twice, is i ( m i + ™JK + m 2 -l)+h l + \ + mjn^ = {^m, (»», - 1) + \} + {*»» ll K-l)+^}+2m I m | . 453. The order of the surface generated by a line which meets a curve three times may be calculated as follows, when the curve is given as the intersection of two surfaces U, V: Let xy'z'vj be any point on the curve, xyzw any point on a gene- rator through x'y'z'w ; and let us, as at p. 271, form the two equations S U + ^\$ 2 U' + &c. =0, S V + ^XS' 2 V + &c. = 0. Then if the generator meet the curve twice again, these equations must have two common roots. If then we form the conditions that the equations shall have two common roots ; and * Where I use h in these formulas Mr. Cayley uses r, the rank of the system, substituting for h from the formula r — m (m — 1) — 2A. And when the system is a complex one we hare simply B - r, + r 2 . 382 EULED SURFACES. between these and XT = 0, V = 0, eliminate x'y'z'w, we shall have the equation of the developable ; or rather that equation three times over, since each generator corresponds to three diffe- rent points on the curve UV. But since V and V do not contain xyzw, the order of the result of elimination will be the product of pq the order of V, V by the weight of the other two equations; (see Appendix on the Order of Systems of Equations.) If, then, we apply the formulae given in that appendix for finding the weight of the system of conditions, that two equations shall have two common roots ; putting m=p-l, n = q — l, X = 0, X =p , /x = 0, /J,' = q, the result is ^{pq — 2) [2pq — 3 {p + q) + 4}, and the order of the required developable is this number multiplied by \pq. But the inter- section of U : V is a curve (see p. 271), for which m = pq, 2h=pq(p — l)[q — 'i), whence pq [p + q) = m" + in — 2h. Sub- stituting these values, the order of the developable expressed in terms of to and h is i (to - 2) (&h + to - to' 2 ), or (to -2)h — \m. (to - 1) (to - 2), a number which may be verified as in the last article. 454. The ruled surfaces considered in the preceding articles have all a certain number of double generators. Thus if a line meets the curve to, twice, and also the curves to 2 and to 3 , it belongs doubly to the system of lines which meet the curves to,, to 2 , to 3 and is a double generator on the corresponding sur- face. But the number of such lines is evidently equal to the number of intersections of the curve m 3 with the surface gene- rated by the lines which meet to, twice, and also m 2 , that is to say, is to 2 to 3 {|-to, (to, — 1) + A,} ; the total number of double generators is therefore |to,to 2 to 3 (to, -f m 2 + to 3 - 3) + \m 2 m 3 + hjn^ + A 3 to,to 2 . In like manner the lines which meet to, three times, and also to 2 belong triply to the system of lines which meet to, twice, and also to 2 ; and the number of such triple generators is seen by the last article to be to 2 (to, — 2) \ - |-to,to 2 (to, — 1 ) (m, - 2). The surface has also double generators whose number we shall determine presently, being the lines which meet both m 1 and to 2 twice. RULED SURFACES. 383 Lastly, the lines which meet a curve four times, are multiple lines of the fourth order on the surface generated by the lines which meet the curve three times. We can determine the number of such lines when the curve is given as the intersection of two surfaces, but will first establish a principle which admits of many applications. 455. Let the equations of three surfaces U, F, W contain xyzw in the degrees respectively X, V, X" ; and x'y'z'w' in degrees fi, fi', /t", and let the XX'X" points of intersection of these surfaces all coincide with x'y'z'w' ; then it is required to find the order of the further condition which must be fulfilled in order that they may have a line in common. When this is the case, any arbitrary plane ax + (3y + yz + Sw must be certain to have a point in common with the three surfaces (namely, the point where it is met by the common line), and therefore the result of elimination between Z7, V, W, and the arbitrary plane, must vanish. This result is of the degree XX'X" in u(5y8, and fiX'X" + fi'X"X + fi"XX' in x'y'z'w'. The first of these numbers (see appendix on the Order of Systems of Equations) we call the order, and the second the weight of the resultant. Now, since the resultant is obtained by multiplying together the results of substituting in ax + fiy + yz + Sw, the co-ordinates of each of the points of intersection of U, V, W, this resultant must be of the form n [ax + /%' + yz + Sw') xx '*-". The condition ax' + /3y' + yz + Sw = 0, merely indicates that the arbitrary plane passes through x'y'z'w\ in which case it passes through a point common to the three surfaces, whether they have a common line or not. The condition therefore that they should have a common line is n = ; and this must be of the degree fjbX'X" + fi'X'X + fi"XX' — XX'X" ; that is to say, the degree of the condition is got by subtracting the order from the weight of the equations U, V, W. 456. Now let x'y'z'w be any point on the curve of intersec- tion of two surfaces U, V, xyzw any other point ; and, as in Art. 453, let us form the equations 8 U+ ^XS a U+ &c. = 0, 384 RULED SURFACES. 8 V+ |XS 2 V+ &c. = 0. If x'y'z'w be a point through which a line can be drawn to meet the curve in four points, and xyzw any point whatever on that line, these two equations in A will have three roots common. And therefore if we form the three conditions that the equations should have three roots common, these conditions considered as functions of xyzw, denote surfaces having common the line which meets the curve in four points. But if x'y'z'w had not been such a point, it would not have been possible to find any point xyzw distinct from x'y'z'w', for which the three conditions would be fulfilled ; and therefore in general the conditions denote surfaces having no point common but x'y'z'w. The order then of the condition which x'y'z'w' must fulfil, if it be a point through which a line can be drawn to meet the curve in four points, is, by the last article, the difference between the weight and the order of the system of conditions, that the equa- tions should have three common roots. But (see appendix on the Order of Systems of Equations) the weight of this system of conditions is found by making m = p — 1, n = q — 1, X =p, yu = q, X' = q = 0, to be I WZ* - W CP + 2) + W + fyq (p + qT + \hpq (p + q) - \Zpq - 66 [p + q) + 108} ; while the order of the same system is i {PY-¥Y {p + q) + 2fq* + 2pq [p + qf-Zpq ( ?+ ? )-f 13p 2 -36}. The order then of the condition 11 = to be fulfilled by x'y'z'w, being the difference of these numbers, is ^{2/q 3 -6pY < j > +s)+^ps(p+sY+ l8 Ps{p+s)-^i>s- e Hp+s)+^ The intersection of the surface n with the given curve deter- mines the points through which can be drawn lines to meet in four points ; and the number of such lines is therefore \ of the number just found multiplied hj pq. As before, putting pq = m, pq [p 4- q) = m 2 + m — 2h ; the number of lines meeting in four points is found to be ii {- m* + 18m 3 - 71m* + 78?n - iSmli + I32h + 12& 2 ).* * It may happen, as Mr. Cayley has remarked, that the surface n may altogether contain the given curve, in which case an infinity of lines can he drawn to meet RULED SURFACES. 385 From this number can be derived tbe number of lines wbich meet both of two curves twice. For, substitute in tbe formula just written m l + m 2 for m, and h t -+ \ + m^ for h, and we have the number of lines which meet the complex curve four times.* But from this take away the number of lines which meet each four times, and the number given (Art. 454) of those which meet one three times and the other once ; and the re- mainder is the number of lines which meet both curves twice, viz. \h a + {m^ (m l — 1) (m a - 1). 457. Besides the multiple generators, the ruled surfaces we have been considering have also nodal curves, being the locus of points of intersection of two different generators. I do not know any direct method of obtaining the order of these nodal curves ; but Mr. Cayley has succeeded in arriving at a solution of the problem by the following method. Let m be one of the curves used in generating one of the surfaces we have been considering, M the degree of that surface, $ (m) the degree of the aggregate of all the double lines on that surface ; then if we suppose m to be a complex curve made up of two simple curves m, and m 2 ; the surface will consist of two surfaces M 1} M 2 having as a double line the intersection of M and M 3t in addition to the double lines on each surface. Thus then (m) must be such as to satisfy the condition $ (*», + wi s ) = (to,) + (m a ) + -Mi-Mj. Using then the value already found for M l in terms of m l? solving this functional equation, and determining the constants involved in it by the help of particular cases in which the problem can be solved directly, Mr. Cayley arrives at the conclusion, that the order of the nodal curve distinct from the multiple generators, is in the case of the surface generated by a line meeting three curves »»,, m a m^ % ftt,Myw s {4jw,»8 a w2g - (wyw„ + m i m 1 + m^) — 2 (m,+ rn 2 + «i 3 ) + 5}, in four points. Thus the curve of intersection of a ruled surface by a surface of the p"> order is evidently such that every generator of the ruled surface meets the curve in p points. CO 386 RULED SURFACES. in the case of the surface generated by a line meeting m, twice and m 2 once, is «. UK K - 2) K - 3) + im t (*n - 1) {m t - 2) (m, - 3)} + m 2 (m-1) ^h'+^mf-m- Ij+Jm, (m,-l) «-5wyfl0)}, and in the case of the surface generated by a line meeting m 1 three times, is \\*m x (m, - 5) - \\ [m t l - 5m, 3 + 5m* - 49 jm, + 120) + A K" - 6m i 5 + Sl^j* - 270m. 3 + 868^ - 408^,). ( 387 ) CHAPTER XIV. SURFACES DERIVED FROM QUADRICS. THE WAVE SURFACE. 458. Before proceeding to surfaces of the third degree, we think it more simple to treat of surfaces derived from quadrics, the theory of which is more closely connected with that explained in preceding chapters. We begin by defining, and forming the equation of, Fresnel's Wave Surface.* If a perpendicular through the centre be erected to the plane of any central section of a quadric, and on it lengths be taken equal to the axes of the section, the locus of their ex- tremities will be a surface of two sheets which is called the wave surface. Its equation is at once derived from Arts. 97, 98, where the lengths of the axes of any section are ex- pressed in terms of the angles which a perpendicular to its plane makes with the axes of the surface. The same equa- tion then expresses the relation which the length of a radius vector to the wave surface bears to the angles which it makes with the axes. The equation of the Wave Surface is therefore aV by -W){X*-Tc*) QQ'W- VjjF-fj,*) BR'(h*-v*)(k*-v*) (V-/**)(V-,,») + (x 2 -^)^ 2 -/) + {X *-v*)(S-v*) -°' which is satisfied if P=0, $ = 0, i? = 0. Hence any surface THE WAVE SURFACE. 395 v = constant cut8 at right angles any whose equation is of the form (\, fi) = 0. The hyperboloid therefore, v = constant, cuts at right angles one sheet of the wave surface, while it meets the other in a line of curvature on the hyperboloid. 468. The plane of any radius vector of ike wave surface and the corresponding perpendicular on the tangent plane, makes equal angles with the planes through the radius vector and the nodal lines. For the first plane is perpendicular to OR (Art. 461) which is an axis of the section QOR of the generating ellipsoid, and the other two planes are perpendicular to the radii of that section whose lengths are h, the mean axis of the ellipsoid, and these two equal lines make equal angles with the axis. The planes are evidently at right angles to each other, which are drawn through any radius vector, and the perpendiculars on the tangent planes at the points where it meets the two sheets of the surface. Reciprocating the theorem of this article we see that the plane through any line through the centre and through one of the points where planes perpendicular to that line touch the surface, makes equal angles with the planes through the same line and through perpendiculars from the centre on the planes of circular contact (Art. 465). 469. If the co-ordinates of any point on the generating ellipsoid be x'y'z', and the primary axes of confocals through that point a', a" ; then the squares of the axes of the section parallel to the tangent plane are a* — a", a 1 — a" 2 , which we shall call p 2 , p 12 . These then give the two values of the radius vector of the wave surface, whose direction-cosines are P?L t PL , PI . We shall now calculate the length and the a* ' b ' ' C* direction-cosines of the perpendicular on the tangent plane at either of the points where this radius vector meets the surface. It was proved (Art. 462) that the required perpendicular is equal and perpendicular to the perpendicular on the tangent plane at the point where the ellipsoid is met by one of the axes of the section ; and the direction-cosines of this axis are 396 THE WAVE SURFACE. *-rr , ^Mr , ^75- • The co-ordinates of its extremity are then these several cosines multiplied by p, and the direction-cosines of the corresponding perpendicular of the ellipsoid are p aV 2 ' 9 Vb n ' P cV" 1 f x'* y' 2 z'* where _ = p y* |_ + ^ + __ Now if the quantity within the brackets be multiplied by (a? — a'*Y, we see at once that it will become —, + —% . Hence P 2 - p y » ana ^ -_p« + _p* • This then gives the length of the perpendicular on the tangent plane at the point on the wave surface which we are considering. Its direction-cosines are obtained from the con- sideration that it is perpendicular to the two lines whose direction-cosines are respectively it 1 it 1 a 1 11 11 11 px_ py_ p_z_^ px p yy Pn P± a » 2 , & »* » ™ , Jrp ^i, , jrp Vb „ , jrp -^ . Forming by Art. 15 the direction-cosines of a line perpendicular to these two, we find, after a few reductions, ^'(i-C), &(i-£\, ^'(i-C). pp \ a"V pp \ 6 V P/3 \ J In fact it is verified without difficulty that the line whose direction-cosines have been just written is perpendicular to the two preceding. It follows hence also, that the equation of the tangent plane at the same point is - , ( i --S) + ^( i -f-) + -'( i ~?)=^ In like manner the tangent plane at the other point where the same radius vector meets the surface is aW -_(, it is evident from the expression for tan#, that p' is the distance to the radius vector from the point where the perpendicular on the tangent plane meets this plane. Thus we have the con- struction, "Draw a tangent plane to the generating ellipsoid perpendicular to the given radius vector, from its point of contact let fall perpendiculars on the two planes of Art. 468, then the lines joining to the centre the feet of these perpendiculars, are the perpendiculars- required. 398 THE WAVE SURFACE. We obtain by reciprocation a similar construction, to de- termine the points where planes parallel to a given one touch the two sheets of the surface. Ex. 1. To transform the equation of the surface, as at p. 125, so as to make the radius vector to any point on the surface the axis of z, and the axes of the corre- sponding section of the generating ellipsoid the axes of x and y. Am. (x 2 +f + z 2 ) {p 2 z 2 + (p' 2 + p 2 ) x> + (p" 2 + p' 2 ) f + typ'xz + 1pp"yz + 1p'p"xy\ -p 2 z 2 (p 2 + p' 2 ) - x 2 (pY + p' 2 p' 2 +p" V + jo 2 / 2 ) - y* {pV +p' 2 p' 2 + p"V + pV) - ipp'p n xz - ipp' W +i>VV 2 = o. It is easy to see that if we make x and y = in the equation thus transformed, we get for z 2 the values p 2 and p' 2 as we ought. If we transform the equation to parallel axes through the point z = p, the linear part of the equation becomes 2pp (p 2 - p' 2 ) (pz +p'x), from which the results already obtained as to the position of the tangent plane may be independently established. Ex. 2. To transform similarly the equation of the reciprocal of the wave surface \ 2 obtained by writing — for a, &a, in the equation of the wave surface. Ans. {x 2 + y 2 + z 2 ) {p 2 p' 2 x 2 +p 2 p 2 y 2 - 2pp'p' 2 xz - 1pp"p 2 yz + z 2 {p' 2 p' 2 +p" 2 p 2 + p 2 p' 2 )} - X« (p 2 +p" 2 + p' 2 ) x 2 - \" {p 2 +p' 2 + p 2 ) y 2 - X 4 (p' 2 +p" 2 + p 2 + p' 2 ) z* + 2X»"ar^ + Vtfpp'xz + 2\. 4 pp"yz + X s = 0. X 2 We know that the surface is touched by the plane z — — , and if we put in this value for z, we find, as we ought, a curve having for a double point the point y = 0, jt/X 2 x = - — . If in the equation of the curve we make y = 0, we get from which we learn that that chord of the outer sheet of the wave surface which joins any point on the inner sheet to the foot of the perpendicular from the centre on the tangent plane is bisected at the foot of the perpendicular. The inflexional tangents are parallel to \p' V 2 +p 2 Go' 2 - p 2 )} x 2 - 2p'p"p 2 xy + \p' 2 p 2 + p 2 (p' 2 - p 2 )} y 2 , a result of which I do not see any geometrical interpretation.* * I have no space for a discussion what the lines of curvature on the wave surface are not, though a hasty assertion on this subject in Crelle's Journal has led to interesting investigations by M. Bertrand, Comptes Rendus, Nov. 1858 ; Combescure and Brioschi, Tortolini's Annali di Matematica, Vol. n., pp. 135, 278. It is worth while to cite an observation of Brioschi, that if in the plane Ix + my + nz =

, as in Art. 403, then the plane will envelope a surface in which curves of the families « = constant, v = constant, will, THE SURFACE $F CENTRES. 399 472. The Surface of Centres. We have already showed (Art. 197) how to obtain the equation of the surface of centres of a quadric. We consider the problem under a somewhat more general form as it has been discussed by Clebsch (Crelle, Vol. 62., p. 64), some of whose results we give, working with the canonical form ; and we refer to his paper for fuller details and for his method of dealing with the general equation. By the method of Art. 218, we may consider the normal to a sur- face as a particular case of the line joining the point of contact of any tangent plane to the pole of that plane with respect to a certain fixed quadric. The problem then of drawing a normal to a quadric from a given point may be generalized as follows : Let it be required to find a point xyzw on a quadrie ■Z7, (ax* + by* + cz'' + dw\ such that the pole, with respect to another quadric F, (a?+y* + 2 s + w 2 ), of the tangent plane to U at xyzw., shall lie on the line joining xyzw to a given poirjt sc'y'z'vf. The co-ordinates of any point on this latter line may be written in the form x — X#, y' — Xy, z' — Xz, w — Xio, and expressing that the polar plane of this point, with regard to V, shall be identical with the polar plane of xyzw, with respect to U, we get the equations x' = (a + X)x, y' '= (b + X) y, a' = (c + \)3, w''=[d+X)w. And since xyzw is a point on Z7, \ is determined by the equation by' 2 cz"' duP f . -. \i + /. . ->\a + /j , -w\ = 0. (a + X)*^ {b + Xf " [c + xy (d + Xf When X is known, a;, y, a, w are determined from the preceding system of equations, and since the equation in \ is of the sixth degree, the problem admits of six solutions. If we form the at their intersection, be touched by conjugate tangents of the surface, if the condition be fulfilled, l, m, n, q> hi m 2> TC » i hv m m n vn $12 = °- where the suffixes 1, 2, denote differentiation with respect to u and » respectively : while the curves will cut at right angles if (P + m* + »") {% + mi»»2 + V2) = Ifli + mm i + nn i) ( l h + mm i + ""»)• 400 THE SURFACE OF CENTRES. discriminant, with regard to \, of this equation, we get the locus of points x, y\ z\ w for which two values of \ coincide, and rejecting a factor a/yVV 2 (which indicates that two values, coincide for all points on the principal planes) we shall have a surface of the twelfth degree answering to the surface of centres. 473. The problem of finding the surface of centres itself is easily made to depend on an equation of like form ; for (Art. 188) the co-ordinates of a centre of curvature answering to any point x'y'z' on an ellipsoid, are _ a 'v _vy _t± X ~ a 2 ' V ~ W ' Z ~ c 2 ' Solve for x', y\ z' from these equations, and substitute in the equations satisfied by x'y'z', viz. x n y" z'* , x'* y n z n „ a , + v 1- c * - h , « + yw I" cV , - ", and write for a"\ a 2 - A 2 , &c, and we get JV cV i ^ i i (a 2 - A 2 ) 2 T {V - A 2 ) 2 T (c 2 - A 2 ) 2 aV &y cV (a 2 - A 2 ) 3 + (6 2 - A 2 ) 3 + (c 2 - A 2 ) 3 ~ These two equations represent a curve of the fourth degree, which is the locus of the centres of curvature answering to points on the intersection of the given quadric with a given confocal. The surface of centres is got by eliminating A 2 be- tween the equations ; or (since the second equation is the diffe- rential of the first with respect to A 2 ) by forming the discriminant of the first equation. 474. I first showed in 1857 [Quarterly Journal, Vol. II., p. 218) that the problem of finding the surface of centres was reducible to elimination between a cubic and a quadratic, and Clebsch has proved that the same reduction is applicable to the problem considered in its most general form. In fact, let A denote the discriminant of fiU+W; viz. for the canonical THE SURFACE OP CENTRES. 401 form, (Art. 137) (a/*-f X) {bfi + X) (c/» + X) ((X), must evidently be divisible by a ; and if after the division we make a = 0, it can be proved that the remaining factor is yjr (a) (a) 3 multiplied by the discriminant of (X). Thus then, the section of Clebsch's surface by the principal plane w is the conic * The factor X is added to make Q as well as A a biquadratic function in /i : \. D D 402 THE SURFACE OF CENTRES. ax' by' c# 2 + a + -, -^ taken three times, together with the curve of the sixth degree, which is the reduced discriminant of ax'' by* cz 2 {a+\y {b+xy {c + xy Clebsch has remarked that this conic and curve touch each other, and the method we have adopted leads to a simple proof of this. For evidently the discriminant of ax 2 by* cz' (a + \y ' (5 4\) 2 ' (c+xy may be regarded as the envelope of all conies which can be represented by this equation, and therefore touches every parti- cular conic of the system in the four points where it meets the conic represented by the differential of the equation with re- gard to X, viz. ox 2 bif cz' Ja + X) 3 + Jb + Xj + (c + X) 3 The co-ordinates of these points are ax'' = (a + X) 3 (b- c), by' = {b + xy (c — a), cz' =(c + X) 3 (a — b); and the equations of the common tangents at them to the conic and its envelope are In the case under consideration X = — d. If then we use the abbreviations (a-b) (a-c) (a- d)=- A', (b-a) (b- c) (b-d) = -B\ (c-a){c-b)(c-d)=-C 2 , {d - a)(d- b)(d-c) = - D\ the equations of the common tangents to the conic, and the envelope curve, are xa$ «J* sc* The reasoning used in this article can evidently be applied to other similar cases. Thus the surface parallel to a quadric (p. 1 48) is met by a principal plane in a curve of the eighth order and a conic, taken twice, which touches that curve in four points; and again, the four right lines (Art. 207, p. 161) touch the conic in their plane. THE SURFACE OF CENTRES. 403 476. Beside the cuspidal conies in the principal planes, there are other cuspidal conies on the surface, which are found by investigating the locus of points for which the equation of the sixth degree (Art. 472) has three equal roots. Differentiating that equation twice with regard to X, we arrive at a system of equations reducible to the form by 2 cz l dw' «V Vtf cV d'W (a + X) 4 [b + X) 4 (c + X) 4 {d + xy «v by c v (a + Xf {b + X) 2 ' {c+xy ' (d + X) 2 a^x tyy d-z d^w weget _ ± _| ± _ ± _. = 0; whence we learn that the locus which we are ^investigating consists of curves situated in one or other of eight planes ; and that these planes meet the principal planes in the common tangents to the conic and envelope curve considered in the last article.* But if we eliminate X between the three equations a + X = aV^*, b + X = tfy k B l , c+X = cV C\ so as to form a homogeneous equation in x, y, z, we get jA i {b-c)x i +tfB i (c-a)y i + JC i {a-b)z k = 0, * The existence of these eight planes may be also inferred from the consideration, that the equation of the reciprocal of the wave-surface is of the form (Art. 190) V = VW, which has therefore as double points, the eight points of intersection of U, V, W. The surface of centres then has eight imaginary double tangent planes, which touch the surfaces in conies (see Art. 205). dd2 404 THE SURFACE OF CENTRES. which denotes a cone of the second degree touched by the planes x, y, s. Hence the cuspidal curves in the eight planes, are conies which touch the cuspidal conies in the principal planes. 477. There will be a nodal curve on the surface answering to the points for which the equation of Art. 472 has two pairs of equal roots. Now we saw (Art. 474) that the condition for a single pair of equal roots is got by eliminating m between a quadratic and a cubic equation, namely, the 8 and T of the biquadratic A + inkH. If we write these equations a + bm + cm* = 0, A + Bm + Cm* + Dm" = 0, it will be found that the degrees in x, y, s, w of these coefficients are respectively 0, 2, 4 ; 0, 2, 4, 6 ; and the result of elimination is, as we know, of the twelfth degree. Now the condition that the equation of Art. 472 may have two pairs of equal roots, is simply that this cubic and quadratic may have two common values of m. Generally, if the result of eliminating an inde- terminate in between two equations denotes a surface, the system of conditions that the equations shall have two common roots will represent a double curve on that surface. Thus the result of eliminating m between two quadratics a + hm 4- can 2 , a' + b'm+c'm* is (ac — ca')* + (ba — b'a)(bc — b'c) — 0. But if we remember that a (be — b'c) = b (ac — ca) -f c (ba — S'a), this result may be written a (ac — ca')* — b (ac — ca) (ba — b'a) + c (ba — b'a)* = 0, showing that the intersection of ac' — ca\ ba — b'a, (which must separately vanish if the equations have both roots common) is a double curve on the surface. And to come to the case immediately under consideration, if we have to eliminate between a + bm + cm* = 0, A + JBm+ Cm* + Dm* = 0, we may substitute for the second equation that derived by multiplying the first by A, the second by a, and subtracting, viz. (Ba - I A) -f (Ca-cA)m + Dam* = 0, THE SURF-fCE OF CENTRES. 405 and thus, as has been just shewn, the result of elimination may be written aP 2 - hPQ + cQ' = 0, where P= bcA - acB + a*D, Q = {ac- V) A + ab'B- a" G. We thus see that the curve PQ is a double curve on the surface of centres: but since P is of the sixth degree and Q of the fourth, the nodal curve PQ is of the twenty-fourth. Further details will be found in Clebsch's paper already referred to. 478. We have discussed, p. 148, the problem of finding the equation of a surface parallel to a quadric. The locus of the feet of perpendiculars let fall, from any fixed point, on the tangent planes of a surface is a derived surface to which French mathematicians have of late thought it worth while to give a distinctive name, "podaire," which we shall translate as the pedal of the given surface. From the pedal may, in like manner, be derived a new surface, and from this another, &c. forming a series of second, third, &c. pedals. Again, the envelope of planes drawn perpendicular to the radii vectores of a surface, at their extremities, is a surface of which the given surface is the pedal, and which we may call the first negative pedal. The surface derived in like manner from this is the second negative, and so on. Pedal curves and surfaces have been studied in particular by Mr. W. Koberts, Liouville, Vols. x. and XII., by M. Tortolini, and by Mr. Hirst, Tortolini's Annali, Vol. II., p. 95. We shall here give some of their results, but must omit the greater part of them, which relate to problems con- cerning rectification, quadrature, &c., which, on account of want of space, cannot be included in this treatise. If Q be the foot of the perpendicular from on the tangent plane at any point P, it is easy to see that the sphere described on the diameter OP touches the locus of Q; and consequently the normal at any point Q of the pedal passes through the middle point of the corresponding radius vector OP. It immediately follows hence that the perpendicular OS on the tangent plane at Q lies in the plane PO Q, and makes the angle Q OR = PO Q, so that the right-angled triangle QOB is similar to POQ; and if we call the angle Q OR, a, so that the first perpendicular Q 406 SURFACES DERIVED FROM QUADRICS. is connected with the radius vector by the equation p = pcosa, then the second perpendicular OR will be p cos^a, and so on.* It is obvious that if we form the polar reciprocals of a curve or surface A and its pedal B, we shall have a surface a which will be the pedal of b ; hence if we take a surface 8 and its successive pedals $,, S 2 , ...#„, the reciprocals will be a series 8', 8_ l7 8^ ».S'_ n , the derived in the latter case being negative pedals. It is also obvious that the first pedal is the inverse (Higher Plane Curves, p. 239) of tbe polar reciprocal of the given sur- face (that is to say, the surface derived from it by substituting in its equation, for the radius vector, its reciprocal) ; and that the inverse of the series S ti S 2 , ...8 will be the series S',S'_ X ,...S'^. 479. As we shall not have opportunity to return to the general theory of inversion, we give in this place the following statement (taken from Hirst, Tortolini, Vol. II., p. 165) of the principal properties of inverse surfaces. (1) Three pairs of corresponding points on two inverse surfaces lie on the same sphere, (and two pairs of corresponding points on the same circle) which cuts orthogonally the unit sphere whose centre is the origin. (2) By the property of a quadrilateral inscribed in a circle the line ab joining any two points on one curve makes the same angle with the radius vector Oa, that the line joining the corresponding points a'b' makes with the radius vector Ob'. In the limit then, if ab be the tangent at any point a, the corresponding tangent on the inverse curve makes the saro«> angle with the radius vector. (3) In like manner for surfaces, two corresponding tangent planes are equally inclined to the radius vector, the two cor- * Thus the radius vector to the n th pedal is of length p cos" a, and makes with the radius vector to the curve the angle na. Using this definition of the method of derivation Mr. Roberts has considered fractional derived curves and surfaces. Thus for »=£, the curve derived from the ellipse is Cassini's oval. An analogous surface may be derived from the ellipsoid. SURFACES DERIVE^) FROM QUADRICS. 407 responding normals lying in the same plane with the radius vector, and forming with it an isosceles triangle whose base is the intercepted portion of the radius vector. (4) It follows immediately from (2) that the angle which two curves make with each other at any point is equal to that which the inverse curves make at the corresponding point. (5) In like manner it follows from (3) that the angle which two surfaces make with each other at any point is equal to that which the inverse surfaces make at the corresponding point. (6) The inverse of a line or plane is a circle or sphere passing through the origin. (7) Any circle may be considered as the intersection of a plane, and a sphere A through the origin. Its inverse there- fore is another circle, which is a sub-contrary section of the cone whose vertex is the origin, and which stands on the given circle. (8) The centre of the second circle lies on the line joining the origin to a the vertex of the cone circumscribing the sphere A along the given circle. For a is evidently the centre of a sphere B which cuts A orthogonally. The plane therefore which is the inverse of A cuts B' the inverse of B orthogonally, that is to say, in a great circle, whose centre is the same as the centre of B'. But the centres of B and of B' lie in a right line through the origin. (9) To a circle osculating any curve, evidently corresponds a circle osculating the inverse curve. (10) For inverse surfaces, the centres of curvature of two corresponding normal sections lie in a right line with the origin. To the normal section a at any point m corresponds a curve a situated on a sphere A passing through the origin; and the osculating circle c of a! is the inverse of c the osculating circle of a. If now a, be the normal section which touches a at the point m, then by Meunier's theorem, the centre of c' is the projection on its plane of the centre of c t the oscu- lating circle of a r But the normal m'c, evidently touches the 408 SURFACES DERIVED FROM QUADR1CS. sphere A at m\ so that c, is the vertex of the cone circum- scribed to A along c', and theorem (10) therefore follows from theorem (8). (11) To the two normal sections at m whose centres of curvature occupy extreme positions on the normal at m, will evidently correspond two sections enjoying the same pro- perty; therefore to the two principal sections on one surface correspond two principal sections on the other, and to a line of curvature on one, a line of curvature on the other. a? y* z' 2 480. The first pedal of the ellipsoid -5+T5 + — = 1, being the inverse of the reciprocal ellipsoid, has for its equation «v + &y + cv = [x' + f + z y. This surface is Fresnel's " Surface of Elasticity." The inverse of a system of confocals cutting at right angles is evidently a system of surfaces of elasticity cutting at right angles; the lines of curvature therefore of the surface of elasticity are determined as the intersection with it of two surfaces of the same nature derived from concyclic quadrics. The origin is evidently a double point on this surface, and the imaginary circle in which any sphere cuts the plane at infinity is a double line on the surface. 481. Mr. Cayley first obtained the • equation -of the first negative pedal of a quadric, that is to say, of the envelope of planes drawn perpendicular to the central radii at their extremities. It is evident that if we describe a sphere passing through the centre of the given quadric, and touching it at any point x'y'z, then the point xyz on the derived surface which corresponds to x'y'z', is the extremity of the diameter of this sphere, which passes through the centre of the quadric. We thus easily find the expressions «=*(*-£), y=y(»-p), *=*'( 2 -?); where t = x n + y n + z''\ SUEFACES DEEIVElf FROM QUADEICS. 409 Solving these equations for x\ y', z' and substituting their values in the two equations xx' + yy' + zz'^x^+f + z", K + € + ^ = 1, Or C 2 2 '2 X If z we get ■ -i s£— f- — — — = t K-) (•-*) K) Now the second of these equations is the differential, with respect to £, of the first equation ; and the required surface is therefore represented by the discriminant of that equation, which we can easily form, the equation being only of the fourth degree. If we write this biquadratic At + iBf + 6 Gf + iDt 4 E, it will be found that A and B do not contain x, y, s, while G, D, E contain them, each in the second degree. Now the discriminant is of the sixth degree in the coefficients, and is of the form A + B*ty ; consequently it can contain x, y, -z only in the tenth degree. This therefore is the degree of the surface required. It appears, as in other similar cases, that the section by one of the principal planes s, consists of the discriminant of x* y „ t + „ t~ ' 2- -j 2-ja a o which is a curve of the sixth degree, and is the first negative pedal of the corresponding principal section of the ellipsoid, together with the conic, counted twice, obtained by writing t= 2c 2 , in the last equation. This conic, which is a double curve on the surface, touches the curve of the sixth degree in four points. The double points on the principal planes evidently answer to points on the ellipsoid, for which t = x' i + y" 1 + z''' = 2a 2 or 2& 2 or 2c 2 . There is a cuspidal conic at infinity, and besides, a finite cuspidal curve of the sixteenth degree. 410 SUEFACES DERIVED FEOM QUADRICS. The reader will find [Philosophical Transactions, 1858, and Tortolini, Vol. II., p. 168) a discussion by Mr. Cayley of the different forms assumed by the surface and by the cuspidal and nodal curves according to the different relative values of a 2 , b'\ c 2 . 482. Mr. W. Koberts has solved the problem discussed in the last article in another way, by proving that the problem to find the negative pedal of a surface, is identical with that of forming the equation of the parallel surface. The former problem is to find the envelope of the plane xx + yy + zz = x +y + z where x', y', z satisfy the equation of the surface. The second problem, being that of finding the envelope of a sphere whose centre is on the surface and radius = k, is to find the envelope of (x-xy + (y-y'y+(z-z'y = Jc% or 2xx + 2yy' + 2zz' = »" + f + z' - li + x n + y' 2 + z'\ Now in finding this envelope the unaccented letters are treated as constants, and it is evident that both problems are particular cases of the problem to find, under the same conditions, the envelope of ax + by' + cz' = x n + y n + z ri + d. And it i3 evident that if we have the equation of the parallel surface, we have only to write in it for F, a? + y* -f « 8 , and then \x, \y, \z for x,y,z- when we have the equation of the negative pedal. Thus having obtained (p. 148) the equation of the parallel to a quadric, we can find by the substitutions here explained, the equation of the first negative, the origin being anywhere, as easily as when the origin is the centre. Further, if we write for k, h + k\ and then make the same substitution for Jc, we obtain the first negative, the origin being anywhere, of the parallel to the quadric, a problem which it would probably not be easy to solve in any other way. Having found, as above, the equation of the first negative of a quadric, we have only to form its inverse, when we have the equation of the second positive pedal of the reciprocal quadric (Art. 478). SURFACES DEKIVED^ROM QUADEICS. 411 Ex. 1. To find the envelope of planes drawn perpendicularly at the extremities of the radii vectores to the plane ax + by + cz + d. Here the parallel surface consists of a pair of planes, whose equation is (ax + by + cz + a) 2 = lc 2 , that of the envelope is therefore (ax + by + cz + id) 2 = x 2 + y 2 + z 2 . Ex. 2. To find, in like manner, the first negative of the sphere (x - a) 2 + (y - p) 2 + (z - y) 2 = r 2 . The parallel surface consists of the pair of concentric spheres (x - a) 2 + (y - §) 2 + (z- y) 2 = (r ± h) 2 . The envelope is therefore (x - 2a) 2 + (y - 2/3) 2 + (a - 1y) 2 = [2r ± J(x 2 + y 2 + z 2 )} 2 , which denotes a quadric of revolution. ( 412 ) CHAPTER XV. SURFACES OF THE THIRD DEGREE. 483. The general theory of surfaces, explained p. 201, &c., gives the following results, when applied to cubical surfaces. The tangent cone whose vertex is any point, and which en- velopes such a surface is, in general, of the sixth degree, having six cuspidal edges and no ordinary double edge. It is con- sequently of the twelfth class, having twenty-four stationary, and twenty-seven double tangent planes. Since then through any line twelve tangent planes can be drawn to the surface, any line meets the reciprocal in twelve points ; and the reciprocal is, in general, of the twelfth degree. Its equation can be found as at Higher Plane Curves, p. 99. The problem is the same as that of finding the condition that the plane ax + fiy + yz + Bw should touch the surface. Multiply the equation of the surface by B 3 , and then eliminate Bw by the help of the equation of the plane. The result is a homogeneous cubic in x, y, z, containing also a, yS, 7, 8 in the third degree. The discriminant of this equation is of the twelfth degree in its coefficients, and therefore of the thirty-sixth in afiyB : but this consists of the equation of the reciprocal surface multiplied by the irrelevant factor B M . The form of the discriminant of a homo- geneous cubical function in x, y, z is 64# 3 = T* [Higher Plane Curves, p. 190). The same then will be the form of the re- ciprocal of a surface of the third degree, 8 being of the fourth, and T of the sixth degree in «, /9, 7, B ; (that is to say, 8 and T are contravariants of the given equation of the above degrees). It is easy to see that they are also of the same degree in the coefficients of the given equation. SURFACES OF THE f HIED DEGREE. 413 484. Surfaces may have either multiple points or multiple lines. When a surface has a double line of the degree p; then any plane meets the surface in a section having p double points. There is, therefore, the same limit to the degree of the double curve on a surface of the w " degree, that there is to the number of double points on a curve of the n th degree. Since a curve of the third degree can have only one double point ; if a surface of the third degree has a double line, that line must be a right line.* A cubic having a double line is necessarily a ruled surface, for every plane passing through this line meets the surface in the double line, reckoned "twice, and in another line; but these other lines form a system of generators resting on the double line as director. If we make the double line the axis of z, the equation of the surface will be of the form (ax s + 3bx 2 y + 3cxf + dy") + z (a'x* + ib'xy + ctf) + (aV + 2b" xy + c"y*) = 0, which we may write w 3 + zu^ + v 2 = 0. At any point on the double line there will be a pair of tangent planes »'m + » = 0. But as z -varies this denotes a system of planes in involution (Conies, p. 295). Hence the tangent planes at any point on the double line, are two conjugate planes of a system in involution. There are two values of z\ real or imaginary, which will make z'u 2 + v 2 a perfect square ; there are therefore two points on the double line at which the tangent planes coincide; and any plane through either of which meets the surface in a section having this point for a cusp. If the values of these squares be X* and Y'\ it is evident that u 2 and v 2 can each be expressed in the form IX* + mY i . If then we turn round the axes so as to have for co-ordinate planes, the planes X, Y, that is to say, the tangent planes at the cuspidal points ; then every term * If a surface have a double or other multiple line, the reciprocal formed by the method of the last article would vanish identically ; because then every plane meets the surface in a curve having a double point, and therefore the plane ax + fy + yz + Sw is to be considered as touching the surface, independently of any relation between a, /3, y, S. The reciprocal can be found in this case by eliminating x, y, *, w between u = 0, a - u u P = u 2 > 7 = M 3> $ = 'V 414 SURFACES OP THE THIRD DEGREE. iii the equation will be divisible by either a? or y'\ and the equation may be reduced to the form zx* = wy 2 .* In this form it is evident that the surface is generated by lines y = \x,z = X 2 w ; intersecting the two directing lines xy, zw, and the generators join the points of a system on zw to the points of a system in involution on xy : homographic with the first system. Any plane through zw meets the surface in a pair of right lines, and is to be regarded as touching the surface in the two points where these lines meet zw. Thus then as the line xy is a line, every point of which is a double point, so the line zw is a line, every plane through which is a double tangent. The reciprocal of this surface, which is that considered Art. 450, is of like nature with itself. The tangent cone whose vertex is any point, and which envelopes the surface, consists of the plane joining the point to the double line, reckoned twice, and a proper tangent cone of the fourth order. When the point is on the double line the cone reduces to the second order. 485. There is one case, to which my attention was called by Mr. Cayley, in which the reduction to the form zx* = wy" is not possible. If w 2 and v 2 , in the last article, have a common factor, then choosing the plane represented by this for one of the co-ordinate planes, we can easily throw the equation of the surface into the form y 3 + x {zx + wy) = 0. The plane x touches the surface along the whole length of the double line, and meets the surface in three coincident right lines. The other tangent plane at any point coincides with the tangent plane to the hyperboloid zx + wy. This case may be considered as a limiting case of that considered in the last * It is here supposed that the planes X, Y, the double planes of the system in inrolution, are real. We can always, however, reduce to the form w (a? ± y 2 ) + 2zxy, the upper sign corresponding to real, and the lower to imaginary, double planes. In the latter case the double line is altogether "really" in the surface, every plane meeting the surface is a section having the point where it meets the line for a real node. In the former case this is only true for a limited portion of the double line, sections which meet it elsewhere having the point of meeting for a con- jugate point; the two cuspidal points marking these limits on the double line. A right line, every point of which is a cusp, cannot exist on a cubic unless when the surface is a, cone. SURFACES OP THE f HIED DEGREE. 415 article; viz., when the double director xy coincides with the single one wz. The following generation of the surface may be given : Take a series of points on xy, and a homographic series of planes through it; then the generator of the cubic through any point on the line, lies in the corresponding plane, and may be completely determined by taking as director a plane cubic having a double point where its plane meets the double line and such that one of the tangents at the double point lies in the plane which corresponds to the double point considered as a point in the double line.* 486. The argument which proves that a proper cubic curve cannot have more than one double point does not apply to surfaces. In fact the line joining two double points, since it is to be regarded as meeting the surface in four points, must lie altogether in the surface ; but this does not imply that the surface breaks up into others of lower dimensions. The con- sideration of the tangent cone however supplies a limit to the number of double points on the surface. We have seen (Art. 273) that the tangent cone is of the sixth degree, and has six cuspidal edges, and it is known that a curve of the sixth degree having six cusps can have only four other double points. Since then every double point on the surface adds a double edge to the tangent cone, a cubical surface can at most have four double points. When a surface has a double point, the line joining this point to any assumed point is, as has been said, a double edge of the tangent cone from the latter point ; and it is easy to see that the tangent planes along this double edge are the planes drawn through this line to touch the cone generated by the tan- gents at the double point. If then this cone break up into two planes, it follows that such a point entails a cuspidal edge on the tangent cone through any assumed point. A cubic then can have only three such biplanar double points. The reciprocal of a cubic then having one or more double points may be of any degree from the tenth to the third, each ordinary * The reader is referred to an interesting geometrical memoir on cubical ruled surfaces by Cremona, " Atte del Beale Istituto Lombardo," Vol. II., p. 291. 416 SURFACES OF THE THIRD DEGREE. double point reducing the degree by two, and each biplanar by three. If the two planes of contact at a biplanar point coincide, the line joining this uniplanar node to any assumed point will be a triple edge on the tangent cone through that point, and the degree of the reciprocal will be reduced by six. Dr. Schafli [Phil. Trans. 1863, p. 201) has added the following cases to those noticed by me ; (1) biplanar nodes where a plane different from both nodal planes touches the surface along the nodal edge, which lowers the class of the surface by four ; (2) biplanars where one of the two nodal planes touches the surface along the nodal edge, which lowers the class by jive ; (3) where one nodal plane osculates along the nodal edge, which lowers the class by six ; (4) uniplanars where the nodal plane touches the surface along a line, which lowers the class by seven ; (5) where the nodal plane osculates along a line, which lowers the class by eight. Ex. 1. "What is the degree of the reciprocal of xyz — w 3 ? Ans. There are three biplanar points in the plane w, and the reciprocal is a cubic. Ex. 2. "What is the reciprocal of - H 1 h — = ? x y z w Ans. This represents a cubic having the vertices of the pyramid xyzw for double points; and the reciprocal must be of the fourth degree. The equation of the tangent plane at any point x'y'z'w' can be thrown into the Ix my nz pw torm ^2 + Zn + jr 2 + ^5 = 0, whence it follows that the condition that ax + fy + yz + Sw should be a tangent plane is (fe) J + (m/?) 5 + (my) J + (pS)* = 0, an equation which, cleared of radicals, is of the fourth degree. Generally the re- ciprocal of ax" + by" + cz" + dw n is of the form " n n n Aa.*- 1 + B/3"- 1 + Cy" 1 " 1 + DS~ l = 0, (Higher Plane Curves, p. 102.) The tangent cone to this surface, whose vertex is any point on the surface, being of the fourth degree, and having four double edges, must break up into two cones of the second degree. The properties of the reciprocal of this surface, which is of the fourth degree, and is such that every tangent plane cuts it in two conies, have lately been studied by Steiner, Kummer, Weierstrass, Schroter, Cremona, (see Crelle, Vols, lxiii. lxiv.). A cubic having four double points is also the envelope of M ! + b/P + cy z + 2ip 7 + Imya + 2na(3, where a, b, c, 1, m, n represent planes; and a : y, /3 : y are two variable parameters. SURFACES OF THE THIRD DEGREE. 417 It is obvious that the envelope is of the third degree ; and it is of the fourth class ; since if we substitute the co-ordinates of two points we can determine four planes of the system passing through the line joining these points. Generally the envelope of act" + J/3" + &c. is of the degree 3 (« - l) 2 and of the class re 2 . The tangent cone from any point is of the degree 3re (n — 1). It has a cuspidal curve whose order is the same as the order of the condition that D + \V may represent a plane curve having a cusp, U and V denoting plane curves of the re tB order; or, in other words, is equal to the number of curves of the form U+W+fiW which can have a cusp. The surface has a nodal curve whose order is the same as the number of curves of the form U+W+fiW which can have two double points. For these numbers, see Appendix on the order of systems of equations. 487. The equation of a cubic having no multiple point may- be thrown into the form ax 3 + by 3 + cz" + dv 3 + ew 3 = 0, where x, y, z 1 v, w represent planes, and where for simplicity we suppose that the constants implicitly involved in x, y, &c. have been so chosen, that the identical relation connecting the equa- tions of any five planes (Art. 37) may be written in the form x + y + z+v + w = 0. In fact the general equation of the third degree contains twenty terms and therefore nineteen indepen- dent constants, but the form just written contains five terms and therefore four expressed independent constants, while besides the equation of each of the five planes implicitly involves three constants. The form just written therefore contains the same number of constants as the general equation. This form given by Mr. Sylvester in 1851 (Cambridge and Dublin Mathematical Journal, Vol. VI., p. 199) is most convenient for the investi- gation of the properties of cubical surfaces in general.* * It was observed (Higher Plane Curves, Art. 18) that two forms may apparently contain the same number of independent constants, and yet that one may be less gene- ral than the other. Thus when a form is found to contain the same number of constants as the general equation, it is not absolutely demonstrated that the general equation is reducible to this form ; and Clebsch has noticed a remarkable exception in the case of curves of the fourth order. In the present case, though Mr. Sylvester gave his theorem without further demonstration, he states that he was in possession of a proof that the general equation could be reduced to the sum of five cubes, and in but a single way. Such a proof has been published by Mr. Clebsch (Crelle, Vol. lix., p. 193). He erro- neously ascribes the theorem in the text to Sterner, who gave it in the year 1856 (Crelle, Vol. Mil., p. 133). It chanced that surfaces of the third order were studied in this country a few years before German mathematicians turned their attention to this subject ; and consequently, though, as might be expected from his ability, M. Steiner's investigations led him to several important results, these had been almost all well known here some years before. EE 418 SURFACES OF THE THIED DEGREE. 488. If we write the equation of the first polar of any point with regard to a surface of the n th order x'L + y'M + z'N + w'P = 0, then, if it have a double point, that point will satisfy the equations ax + ny + mz +pw = 0, nx' + by' + lz' + qw' = 0, mx + ly + cz' + rw = 0, px + qy 4 rz'+ dw = 0, where a, b, &c. denote second differential coefficients corre- sponding to these letters, as we have used them in the general equation of the second degree. Now if between the above equations we eliminate x'y'z'w', we obtain the locus of all points which are double points on first polars. This is of the degree 4 (n — 2) and is in fact the Hessian (Art. 279). If we eliminate the xyzw which occur in a, b, &c, since the four equations are each of the degree (n — 2), the resulting equation in x'y'z'w will be of the degree 4 [n — 2) 3 , and will represent the locus of points whose first polars have double points. Or, again, .ST is the locus of points whose polar quadrics are cones, while the second surface, which we shall call J, is the locus of the vertices of such cones. In the case of surfaces of the third degree, it is easy to see that the four equations above written are sym- metrical between xyzw and x'y'z'w' ; and therefore that the surfaces H and J are identical. Thus then if the polar quadric of any point A with respect to a cubic be a cone whose vertex is -S, the polar quadric of B is a cone whose vertex is A. The points A and B are said to be corresponding points on the Hessian (see Higher Plane Curves, p. 154, &c.) 489. The tangent plane to the Hessian of a cubic at A is the polar plane of B with respect to the cubic. For if we take any point A' consecutive to A and on the Hessian, then since the first polars of A and A' are consecutive and both cones, it appears (as at Higher Plane Curves, p. 155) that their inter- section passes indefinitely near B, the vertex of either cone; therefore the polar plane of B passes through AA' ; and, in like manner, it passes through every other point consecutive to A. It is therefore the tangent plane at A. And the SURFACES OF THE THIRD DEGREE. 419 polar plane of any point A on the Hessian of a surface of any degree is the tangent plane of the corresponding point B on the surface J. In particular the tangent planes to U along the para- colic curve, are tangent planes to the surface J: that is to say, in the case of a cubic the developable circumscribing a cubic along the parabolic curve, also circumscribes the Hessian. If any line meet the Hessian in two corresponding points A, B, and in two other points C, J), the tangent planes at A, B inter- sect along the line joining the two points corresponding to C, D. 490. We shall also investigate the preceding theorems by means of the canonical form. The polar quadric of any point with regard to ax 3 -+■ by 3 + cz* + dv 3 + ew 3 is got by substituting for w its value — (x + y + z + v), when we can proceed according to the ordinary rules, the equation being then expressed in terms of four variables. We thus find for the polar quadric ax'x* + by'y 2 + cz' z' + dv'v* + ew'v? = 0. If we differentiate this equation with respect to x, remembering that dw — — dx, we get ax'x = ew'w ; and since the vertex of the cone must satisfy the four differentials with respect to x, y, z, v, we find that the co-ordinates x, y', z\ v\ w of any point A on the Hessian are connected with the co-ordinates x, y, z, v, w of B, the vertex of the corresponding cone, by the relations ax'x = by'y = cz'z — dv'v — ew'w. And since we are only concerned with mutual ratios of co- ordinates, we may take 1 for the common value of these quan- tities and write the co-ordinates of B. — ;, ^—71 —, -n » — , • 7 ax by cz 1 dv ' eio Since the co-ordinates of B must satisfy the identical relation x 4. y + z + v H- w = 0, we thus get the equation of the Hessian 1 111 1 „ — + T-+-+ T-+— =0, ax by cz dv ew or bcdeyzvw + cdeazvwx + deab vwxy -I- edbcwxyz + dbcdxyzv = 0. This form of the equation shows that the line vw lies altogether in the Hessian, and that the point xyz is a double point on the Hessian; and since the five planes x, y, z, v, w give rise to ten combinations, whether taken by twos or by threes, we have EE 2 420 SURFACES OF THE THIRD DEGREE. Mr. Sylvester's theorem that the five planes form a pentahedron whose ten vertices are double points on the Hessian and whose ten edges lie on the Hessian. The polar quadric of the point xyz is dv'v 1 + ew'w'\ which resolves itself into two planes inter- secting along vw, any point on which line may be regarded as the point B corresponding to xyz ; thus then there are ten points ichose polar quadrics break up into pairs of planes ; these points are double points on the Hessian, and the intersections of the corresponding pairs of planes are lines on the Hessian. It is by proving these theorems independently* that the reso- lution of the given equation into the sum of five cubes can be completely established. The equation of the tangent plane at any point of the Hessian may be written x y z v w L -K_. J L . I — ax by cz dv ew which, if we substitute for x, — -, , &c., becomes ' ' ax ' ' ax n x + by'*y + cz'*z + dv''*v + ew' z w = 0, but this is the polar plane of the corresponding point with regard to U (Art. 489). 491. If we consider all the points of a fixed plane, their polar planes envelope a surface, which (as at Higher Plane Curves, p. 152) is also the locus of points whose polar quadrics touch the given plane. The parameters in the equation of the variable plane enter in the second degree ; the problem is therefore that considered (Ex. 2, Art. 486) and the envelope is a cubic surface having four double points. The polar planes of the points of the section of the original cubic by the fixed plane are the tangent planes at those points, consequently this polar cubic of the given plane is inscribed in the developable formed by the tangent planes to the cubic along the section by * It will appear from the appendix "cm the order of systems of equations," that a symmetric determinant of p rows and columns, each constituent of which is a function of the n tb order in the variables, represents a surface of the np th degree having $p {p* — 1) n 3 double points ; and thus that the Hessian of «. surface of the n - " degree always has 10 (n — 2) 3 double points. SURFACES OF THE THIRD DEGREE. 421 the given plane {Higher Plane Curves, Art. 161). The polar plane of any point A of the section of the Hessian by the given plane, touches the Hessian (Art. 489) and is therefore a common tangent plane of the Hessian and of the polar cubic now under consideration. But the polar quadric of P, being a cone whose vertex is A, is to be regarded as touching the given plane at A ; hence B is also the point of contact of the polar plane of A with the polar cubic. We thus obtain a theorem of Steiner's that the polar cubic of any plane touches the Hessian along a certain curve. This curve is the locus of the points B corresponding to the points of the section of the Hessian by the given plane. Now if points lie in any plane Ix + my + nz + pv + qw, the corresponding points lie on I m n i) q the surface of the fourth order \- = — | h -r- + — . Now ax by cz av ew the intersection of this surface with the Hessian is of the sixteenth order, and includes the ten right lines xy, ew, &c. The remaining curve of the sixth order is the curve along which the polar cubic of the given plane touches the Hessian. The four double points lie on this curve ; they are the points whose polar quadrics are cones touching the given plane. 492. If on the line joining any two points x'y'z', x"y"z", we take any point x' + \x", &c, it is easy to see that its polar plane is of the form P„ + 2\P 12 + A, 2 P 22 , where P n , P 22 are the polar planes of the two given points, and P 12 is the polar plane of either point with regard to the polar quadric of the other. The envelope of this plane, considering \ variable, is evidently a quadric cone whose vertex is the inter- section of the three planes. This cone is clearly a tangent cone to the polar cubic of any plane through the given line, the vertex of the cone being a point on that cubic. If the two assumed points be corresponding points on the Hessian, P vanishes identically ; for, the equation of the polar plane, with respect to a cone, of its vertex vanishes identically. Hence the ■polar plane of any point of ike line joining two corresponding points on the Hessian passes through the intersection of the tangent 422 SURFACES OP THE THIRD DEGREE. planes to the Hessian at these points.* In any assumed plane we can draw three lines joining corresponding points on the Hessian ; for the curve of the sixth degree considered in the last article meets the assumed plane in three pairs of corre- sponding points. The polar cubic then of the assumed plane will contain three right lines ; as will otherwise appear from the theory of right lines on cubics which we shall now explain. 493. We said, note, p. 28, that a cubical surface necessarily contains right lines, and we now enquire how many in general lie on the surface.")" In the first place it is to be observed that if a right line lie on the surface, every plane through it is a double tangent plane because it meets the surface in a right line and conic ; that is to say, in a section having two double points. The planes then joining any point to the right lines on the surface are double tangent planes to the surface and therefore also double tangent planes to the tangent cone whose vertex is that point. But we have seen (Art. 483) that the number of such double tangent planes is twenty-seven. This result may be otherwise established as follows: let us suppose that a cubic contains one right line, and let us examine in how many ways a plane can be drawn through the right line, such that the conic in which it meets the surface may break up into two right lines. Let the right line be wz ; let the equation of the surface be wU=zV; let us substitute w=/mz, divide out by z, and then form the dis- criminant of the resulting quadric in x, y, z. Now in this quadric it is seen without difficulty that the coefficients of x 1 , xy, and y" only contain /jl in the first degree ; that those of xz and yz contain (i in the second degree, and that of z' in * Steiner saya that there are one hundred lines such that the polar plane of any point of one of them passes through a fixed line, but I believe that his theorem ought to be amended as above. f The theory of right lines on a cubical surface was first studied in the year 1849 in a correspondence between Mr. Cayley and me, the results of which were published, Cambridge and Dublin Mathematical Journal, Yol. IV., pp. 118, 252. Mr. Cayley first observed that a definite number of right lines must lie on the surface ; the determination of that number as above, and the discussions in Art. 496 were supplied by mc. SURFACES OF THE THIRD DEGREE. 423 the third degree. It follows hence that the equation obtained by equating the discriminant to nothing is of the fifth degree in /J> : and therefore that through any right line on a cubical surface can be drawn five planes, each of which meets the surface in another pair of right lines ; and consequently, every right line on a cubic is intersected by ten others. Consider now the section of the surface by one of the planes just referred to. Every line on the surface must meet in some point the section by this plane, and therefore must intersect some one of the three lines in this plane. But each of these lines is inter- sected by eight in addition to the lines in the plane ; there are therefore twenty-four lines on the cubic besides the three in the plane ; that is to say, twenty-seven in all. We shall hereafter show how to form the equation of a surface of the ninth order meeting the given cubic in those lines. 494. Since the equation of a plane contains three inde- pendent constants, a plane may be made to fulfil any three conditions, and therefore a finite number of planes can be determined which shall touch a surface in three points. We can now determine this number in the case of a cubical surface. We have seen that through each of the twenty-seven lines can be drawn five triple tangent planes : for every plane intersecting in three right lines touches at the vertices of the triangle formed by them, these being double points in the section. The number 5 x 27 is to be divided by three, since each of the planes contains three right lines ; there are therefore in all forty-five triple tangent planes. 495. Every plane through a right line on a cubic is obviously a double tangent plane ; and the pairs of points of contact form a system in involution. Let the axis of z lie on the surface, and let the part of the equation which is of the first degree in x and y be (aa a + bz + c) x + (a'z 2 + b'z + c')y; then the two points of contact of the plane y =/ix are determined by the equation (as 2 + bz + c) + fi (a'z 2 + b'z + c) = 0, but this denotes a system in involution (Oonics, p. 295). It 424 SURFACES OF THE THIRD DEGREE. follows hence, from the known properties of involution, that two planes can be drawn through the line to touch the surface in two coincident points : that is to say, which cut it in a line and a conic touching that line. The points of contact are evidently the points where the right line meets the parabolic curve on the surface. It was proved (Art. 281) that the right line touches that curve. The two points then where the line touches the parabolic curve, together with the points of contact of any plane through it, form a harmonic system. Of course the two points where the line touches the parabolic curve may be imaginary. 496. The number of right lines may also be determined thus. The form ace - bdf, (where a, b, &c. represent planes) is one which implicitly involves nineteen independent constants, and therefore is one into which the general equation of a cubic may be thrown.* This surface obviously contains nine lines [ab, cd, &c). Any plane then a=fib which meets the surface in right lines meets it in the same lines in which it meets the hyperboloid pee — df. The two lines are therefore generators of different species of that hyperboloid. One meets the lines cd, ef; and the other the lines cf de. And, since fi has three values, there are three lines which meet ab, cd, ef. The same thing follows from the consideration that the hyper- boloid determined by these lines must meet the surface in three more lines (Art. 339). Now there are clearly six hyperboloids, ab, cd, ef; ab, cf de, &c. which determine eighteen lines in addition to the nine with which we started, that is to say as before, twenty-seven in all. If we denote each of the eighteen lines by the three which it meets, the twenty-seven lines may be enumerated as follows : there are the original nine ab, ad, af cb, cd, cf, eb, ed, ef: to- gether with (ab.cd.ef)^ [ab.cd.ej) z , [ab.cd.ef) 3 , and in like manner three lines of each of the forms ab.cf.de, ad.be. ef, ad.be.cf af.bc.de, af.be.cd. The five planes which can be * It will be found, in one hundred and twenty ways, SURFACES OF THE THIRD DEGREE. 425 drawn through any of the lines ab are the planes a and b, meeting respectively in the pairs of lines ad, of; be, be ; and the three planes which meet in {ab.cd.ef) x , {ab.cf.de)/, (ab.cd.rf) t , {ab.cf.de)./, {ab.cd.ef\, {ab.cf.de) s . The five planes which can be drawn through any of the lines (ab.cd.ef)^ cut in the pairs of lines, ab, {ab.cf.de) 1 ; cd, (af.cd.be)^ ef, {ad. beef)/, and in {ad. be. cf ")„ {af.bc.de)/, {ad.be.cf) 3 , {af.be. de)^ 497. Prof. Schafli has made a new arrangement of the lines {Quarterly Journal of Mathematics, Vol. n., p. 116) which leads to a simpler notation, and gives a clearer conception how they lie. Writing down the two systems of six non- intersecting lines ab, cd, ef, {ad.be. cf) x , {ad.be. cf)^ {ad.be. cf) s , cf,be,ad,{ab.cd.ef\, {ab.cd.ef)^, {ab.cd.ef)/, it is easy to see that each line of one system, does not intersect the line of the other system which is written in the same vertical line, but that it intersects the five other lines of the second system. We may write then these two s) 7 stems «,) a » «3l «4> ff '5> ««> K K \l h *l Kl & 6> which is what Schafli calls a " double-six." It is easy to see from the previous notation that the line which lies in the plane of a t , b^ is the same as that which lies in the plane of « 2 , b t . Hence the fifteen other lines may be represented by the notation c 12 , c M , &c, where c 12 lies in the plane of a v , b t , and there are evidently fifteen combinations in pairs of the six numbers 1, 2, &c. The five planes which can be drawn through c 12 are the two which meet in the. pairs of lines «A> a Al aIld th ° Se Which m f et m V»l C S6 C 4B> C S6 C 45- The ™ are evidently thirty planes which contain a line of each of the systems a, b, c: and fifteen planes which contain three c lines. It will be found that out of the twenty-seven lines can be constructed thirty-six " double-sixes." 498. We can now geometrically construct a system of twenty-seven lines which can belong to a cubical surface. We 426 SUEFACES OF THE THIRD DEGREE. may start by taking arbitrarily any line a t and five others which intersect it, Z> 2 , Z> s , & 4 , 6 6 , 6 6 . These determine a cubical surface, for if we describe such a surface through four of the points where o, is met by the other lines and through three more points on each of these lines, then the cubic determined by these nineteen points contains all the lines, since each line has four points common with the surface. Now if we are given four non-intersecting lines, we can in general draw two transversals which shall intersect them all; for the hyperboloid determined by any three meets the fourth in two points through which the transversals pass* Through any four then of the lines b 3 , b v 5 6 , b s we can draw in addition to the line a t another transversal « B , which must also lie on the surface since it meets It in four points. In this manner we construct the five new lines a , a , a,, ol, a,. If we then take another transversal 27 37 4? 67 6 meeting the four first of these lines, the theory already ex- plained shows that it will be a line 5, which will also meet the fifth. We have thus constructed a " double-six." We can then immediately construct the remaining lines by taking the plane of any pair a,& 2 , which will be met by the lines J,, a 2 in points which lie on the line c^. 499. M. Schafli has made an analysis of the different species of cubics according to the reality of the twenty-seven * If the hyperboloid touches the fourth line, the two transTeraala reduce to a single one, and it is evident that the hyperboloid determined by any three others of the four lines also touches the remaining one. This remark I believe is Mr. Cayley's. If we denote the condition that two lines should intersect by (12), then the condition that four lines should be met by only one transversal is expressed by equating to nothing the determinant - (12), (13), (14) (21), - (23), (24) (31), (32), - (34) (41), (42), (43), - . The vanishing of the determinant formed in the same manner from five lines, is the condition that they are all met by a common transversal. The vanishing of the similar determinant for six lines, expresses that they are connected by a relation which has been called the " involution of six lines ;" and which will be satisfied when the lines can be the directions of six forces in equilibrium. The reader will find several interesting communications on this subject by Messrs. Sylvester and Cayley, and by M. Chasles, in the Comptes Rendus for 1861, Premier Semestre. SURFACES OF THE THIRD DEGREE. 427 lines. He finds thus five species : A. all the lines and planes real; B. fifteen lines and fifteen planes real; G. seven lines and five planes real; that is to say, there is one right line through which five real planes can be drawn, only three of which contain real triangles ; D. three lines and thirteen planes real : namely, there is one real triangle through every side of which pass four other real planes; and, E. three lines and seven planes real. I have also given {Cambridge and Dublin Mathematical Journal, Vol. IV., p. 256) an enumeration of the modifications of the theory when the surface has one or more double points. It may be stated generally that the cubic has always twenty- seven right lines and forty-five triple tangent planes, if we count a line or plane through a double point as two, through two double points as four, and a plane through three such points as eight. Thus, if the surface has one double point, there are six lines passing through that point, and fifteen other lines one in the plane of each pair. There are fifteen treble tangent planes not passing through the double point. Thus 2 x 6 + 15 = 27 ; 2 x 15 + 15 = 45. Again, if the surface have four double points, the lines are the six edges of the pyramid formed by the four points (6x4), together with three others lying in the same plane, each of which meets two opposite edges of the pyramid. The planes are the plane of these three lines 1, six planes each through one of these lines and through an edge (6 x 2), together with the four faces of the pyramid (4x8). The reader will find the other cases discussed in the paper just referred to, and in a later memoir by Schafli in the Philo- sophic Transactions for 1863. 500. It is known that in a plane cubic the polar line, with respect to the Hessian, of any point on the curve, meets on the curve the tangent at that point. Clebsch has given as the corresponding theorem for surfaces, The polar plane, with respect to the Hessian, of any point on the cubic, meets the tangent plane at that point, in the line which joins the three points of inflexion of the section by the tangent plane. It will be re- 428 SUEFACES OF THE THIRD DEGREE. membered that the section by a tangent plane is a cubic having a double point, and therefore having only three points of inflexion lying on a line. If w be this line, xy the double point, the equation of such a curve may be written x*+y*-\- Gxyw = 0. Writing then the equation of the surface, (the tangent plane being z), x s + y 3 + &xyw + zu = 0, where u is a complete function of the second degree u = dz* + 6pxio + Qqyio + 3rzw + &c, where we have only written the terms which we shall actually require ; and working out the equation of the Hessian, we find the terms below the second degree in x, y, z to be d 2 w 4 + d{r — %pq) zuf. The polar plane then of the Hessian with respect to the point xyz is idw + (r — 2pq) z, which passes through the intersection of zw as was to be proved. If the tangent plane pass through one of the right lines on the cubic, the section by it consists of the right line x and a conic, and may be written x s + Qxyw = ; and, as before, the polar plane of the point xyz with respect to the Hessian passes through the line w, a theorem which may be geometrically stated as follows : When the section by the tangent plane is a line and a conic, the polar plane, with respect to the Hessian, of either point in which the line meets the conic, passes through the tangent to the conic at the other point. If the tangent plane passes through two right lines on the cubic, the section reduces to xyw, and the polar plane still passes through w, that is to say, through the third line in which the plane meets the cubic. If the point of contact is a cusp, it is proved in like manner that the line through which the polar plane passes is the line joining the cusp to the single point of inflexion of the section. The conclusions of this article may be applied with a slight modification to surfaces of higher degree than the third: for if we add to the equation of the surface with which we have worked, terms of higher degree in xyz than the third, these will not affect the terms in the equation of the Hessian which are below the second degree in x, y, z. And the theorem is that the polar plane, with respect to the Hessian, of any point on a surface intersects the tangent plane at that point, in the INVARIANTS AND COVAEIANTS OF A CUBIC. 429 line joining the points of inflexion of the section by the tangent plane of the polar cubic of the same point. INVARIANTS AND COVARIANTS OP A CUBIC. 501. We shall in this section give an account of the principal invariants, covariants, &c. that a cubic can have. We only suppose the reader to have learned from the Lessons on Higher Algebra, or elsewhere, some of the most elementary properties of these functions. An invariant of the equation of a surface is a function of the coefficients, whose vanishing expresses some permanent, property of the surface^ as for example that it has a nodal point. A covariant, as for example the Hessian, denotes a surface having to the original surface some relation which is independent of the choice of axes. A contravariant is a relation between a, £, , 5> = abcde2abm 2 y* z - In order to apply the method indicated (Arts. 502, 503) it is necessary to have a contravariant; and for this purpose I have calculated the contravariant a which occurs in the equation of the reciprocal surface, which, as we have already seen, is of the form 64<7 3 = t\ The contravariant o- expresses the condition that any plane ax + $y + &c. should meet the surface in a cubic for which Aronhold's invariant S vanishes. It is of the fourth degree both in a, /3, &c. and in the coefficients of the cubic. In the case of four variables the leading term is a" multiplied by the S of the ternary cubic got by making INVARIANTS AND COViMtlANTS OF A CUBIC. 433 x = m the equation of the surface. The remaining terms are calculated from this by means of the differential equation (Lessons on Higher Algebra, p. 70). The form being found for four variables, that for five is calculated from it as in Art. 501. I suppress the details of the calculation which though tedious presents no difficulty. The result is s/{bcde) + \/(cdea) + >J(deab) + >J(eabc) + \]{dbcd) = 0. Clearing of radicals, the result, expressed in terms of the principal invariants, is (A 2 - UBf = 16384 (Z> + 2A C), 507. The cubic has four fundamental covariant planes of the orders 11, 19, 27, 43 in the coefficients, viz. L = f2ax, L' = f'2bcdex 1 L" = f2d 2 x^ L"'=f^,d 3 x. Every other covariant, including the cubic itself, can in general be expressed in terms of these four, the coefficients being invariants. The condition that these four planes should meet in a point, is the invariant F of the one hundredth degree. ■ There are linear contravariants the simplest of which, of the thirteenth degree, has been already given ; the next being of the twenty-first, f2 (a — b)(a — j3)] the next of the twenty- ninth, tlLcde {a — b) (a — /3), &c. There are covariant quadrics of the sixth, fourteenth, twenty- second, &c. orders ; and contravariants of the tenth, eighteenth, &c, the order increasing by eight. There are covariant cubicsof the ninth order 2,tcde(a+b)zuv, and of the seventeenth, ^2aV, &c. If we call the original cubic U, and this last covariant F, since if we form a covariant or invariant of U+W, the coefficients of the several powers of \ are evidently covariants or invariants of the cubic : it follows that given any covariant or invariant of the cubic we are discussing, we can form from FF2 436 INVARIANTS AND COVAEIANTS OF A CUBIC. it a new one of the degree sixteen higher in the coefficients, by performing on it the operation „ d , a d „ d ™ d , d da db dc dd de Of higher covariants we only think it necessary, here to mention one of the fifth order, and fifteenth in the coefficients txyzvw which gives the five fundamental planes : and one of the ninth order, © the locus of points whose polar planes with respect to the Hessian touch their polar quadrics with respect to Z7. Its equation is expressed by the determinant, p. 47, if a, /S, &c. denote the first differential coefficients with respect to the Hessian, and a, b, &c. the second differentials with respect to the cubic. The equation of a covariant whose intersection with the given cubic determines the twenty-seven lines is © = 4H$, where * has the meaning explained, Art. 504. We shall give M. Clebsch's proof of this at the end of the volume. I had verified the form, which had been suggested to me by geometrical considerations, by examining the following form, to which the equation of the cubic can be reduced, by taking for the planes x and y the tangent planes at the two points where any of the lines meets the parabolic curve, and two determinate planes through these points for the planes to, 2, z*y + w 2 x 4 2xyz + 2xyw + ax'y + by*x + cx*z + dy'w — 0. The part of the Hessian then which does not contain either x or y is «V: the corresponding part of is — 2 {cz b + dw b ) , and of is - 8wV (ca c -f div b ). The surface ©-4H$ has therefore no part which does not contain either x or y, and the line xy lies altogether on the surface, as in like manner do the rest of the twenty-seven lines.* * This section is abridged from a paper which I contributed to the Philosophical Transactions, 1860, p. 229. Shortly after the reading of my memoir, and before its publication, there appeared two papers in Crelle's Journal, Yol. lviii., by Professor Clebsch of Carlsruhe, in which some of my results were anticipated : in particular the expression of all the invariants of a cubic in terms of five fundamental : and the expression given above for the surface passing through the twenty-seven lines. The method however which I pursued was different from that of Professor Clebsch, and the discussion of the covariants, as well as the notice of the invariant F, I believe were new. Clebsch has expressed his last four invariants as functions of the coefficients of the Hessian. Thus the second is the invariant (1234)'' of the Hessian, &c. ( 437*) CHAPTER XVI. GENERAL THEORY OF SURFACES. 508. We shall in this chapter proceed, in continuation of Art. 281, with the general theory of surfaces, and shall first state for surfaces in general a few theorems proved for quadrics (Art. 224, &c). The locus of the points whose polar planes with regard to four surfaces U, F, W, T {whose degrees are m, n, p, q) meet in a point, is a surface of the degree m+n+p+q—i; which we call the Jacobian of the system. For its equation is evidently got by equating to nothing the determinant whose constituents are the four differential coefficients of each of the four surfaces. If a surface of the form \ Z7+ /j, V+ v W touch T, the point of contact is evidently a point on the Jacobian, and must lie somewhere on the curve of the degree q(m+n+p+q— 4) where the Jacobian meets T. In like manner, pq (m+n +p + q — 4) surfaces of the form \U+ pV, can be drawn so as to touch the curve of intersection of T, W; for the point of contact must be some one of the points where the curve TW meets the Jacobian. It follows hence that the tact-invariant of a system of three surfaces U, V, W; that is to say, the condition, that two of the mnp points of intersection may coincide, contains the coefficients of the first in the degree np (2m + n +p — 4) ; and in like manner for the other two surfaces. For if in this condition we sub- stitute for each coefficient a of U, a + \a, where a' is the corresponding coefficient of another surface U' of the same degree as U, it is evident that the degree of the result in \, is the same as the number of surfaces of the form Z7+ \ V which can be drawn to touch the curve of intersection of V, W.* * Moutard, Terqmm's Annates, Vol. six., p. 58. 438 GENERAL THEORY OF SURFACES. I had arrived at the same result otherwise thus: (see Quarterly Journal, Vol. I., p. 3S&) Two of the points of inter- section coincide if the curve of intersection TJV touch the curve TJW. At the point of contact then the tangent planes to the three surfaces have a line in common : and these planes therefore have a point in common with any arbitrary plane ax + fiy + 72: + 810. The point of contact then satisfies the determinant, which has for one row, a, /3, 7, 8 ; and for the other three the four differentials of each of the three surfaces. The condition that this determinant may be satisfied by a point common to the three surfaces is got by eliminating between the determinant and U, V, W. The result will contain a, /3, 7, S in the degree mnp j and the coefficients of U in the degree np (m + n + jy — 3) + mnjy. But this result of elimination con- tains as a factor the condition that the plane ax + /3y + 73 + &w may pass through one of the points of intersection of U", V 7 W. And this latter condition contains a, /3, 7, 8 in the degree mnp r and the coefficients of U in the degree np. Dividing out this factor, the quotient, as already seen, contains the coefficients of Uin the degree np (2m + n +p> — 4). 509. The locus of points whose polar planes with regard to three surfaces have a right line common, is, as may be inferred from the last article, the Jacobian curve denoted by the system of determinants v* u« v e u< K, K, v« v t w>, w 23 w a w t = 0. But this curve (see Appendix) is of the order (m' 2 + n' 2 +p'"' + m'?i -f rip +p'm'), where in is the order of Z7J, &c, that is to say, m' — m — 1. If a surface of the form X U+ fi V touch W, the point of contact is evidently a point on the Jacobian curve, and therefore the number of such surfaces which can be drawn to touch W, is equal to the number of points in which this curve meets W, that is to say, is p times the degree of that curve. Reasoning GENERAL THEORY OP SURFACES. 439 then as in the last article we see that the tact-invariant of two surfaces U, V, that is to say, the condition that they should touch, contains the coefficients of U in the degree n (n' 2 + 2m'n' + 3m' 2 ) or n(7^ + 2mn + S7n i -in-8m+6), and in like manner contains the coefficients of U in the degree m (m 2 + 2m«H- 3ri*-4m-"8n + 6). Moutard, Terquem, Vol. xix., p. 65. We add, in the form of examples, a few theorems to which it does not seem worth while to devote a separate article. Ex. 1. Two surfaces U, V of the degrees m, n intersect ; the number of tangents to their curve of intersection which are also inflexional tangents of the first surface, is mn (3m + In— 8). The inflexional tangents at any point on a surface are generating lines of the polar quadric of that point ; any plane therefore through either tangent touches that polar quadric. If then we form the condition that the tangent plane to V may touch the polar quadric of U, which condition involves the second differentials of U in the third degree, and the first differentials of V in the second degree, we have the equa- tion of a surface of the degree (3m + 2n — 8) which meets the curve of intersection in the points, the tangents at which are inflexional tangents on U. Ex. 2. In the same case to find the degree of the surface generated by the in- flexional tangents to U at the several points of the curve UV. This is got by eliminating x'y'z'w', between the equations U' = 0, V' = 0, AU' = 0, A z U' = 0, which are in x'y'z'w' of the degrees respectively m, n, m—1, m — 2, and in xyzm of the degrees 0, 0, 1, 2. The result is therefore of the degree mn (3m — 4). Ex. 3. To find the degree of the developable which touches a surface along its intersection with its Hessian. The tangent planes at two consecutive points on the parabolic curve, intersect in an inflexional tangent (Art. 2G3) ; and, by the last ex- ample, since n — 4 (m — 2), the degree of the surface generated by these inflexional tangents is 4m (m — 2) (3m — 4). But since at every point of the parabolic curve the two inflexional tangents coincide, and therefore the surfaces generated by each of these tangents coincide, the number just found must be divided by two, and the degree required is 2m (m — 2) (3m — 4). Ex. 4. To find the characteristics, as at p. 261, of the developable which touches a surface along any plane section of a surface whose degree is m. The section of the developable by the given plane is the section of the given surface, together with the tangents at its 3m (m — 2) points of inflexion. Hence we easily find jit = 6m (m — 2), v = m (m — 1), r — m (3m — 5), a — 0, /3 = 2m (5m — 11), &c. Ex. 5. To find the characteristics of the developable which touches a surface of the degree m along its intersection with a surface oftegree n. Ans. v — mn (m — ,1), o = 0, r = mn (3m + n — 6), whence the other singularities are found as at p. 261. 440 CONTACT OF LINES WITH SURFACES. Ex. 6. To find the characteristics of the developable touching two given surfaces, neither of which has multiple lines. A m. v = mn (m - l) 2 (n - 1) J ; a = 0, r = ma (»» - 1) (n - 1) (m + » - 2). Ex. 7. To find the characteristics of the curve of intersection of two developables. The surfaces are of degrees r and r', and since each has a nodal and cuspidal curve of degrees respectively x and m, x' and m', therefore the curve of intersection has rx' + r'r and mn' + r'm actual nodal and cuspidal points. The cone therefore which stands on the curve and whose vertex is any point, has nodal and cuspidal edges in addition to those considered at p. 271 ; and the formulae there given must then be modified. We have as there p. = rr' ; but the degree of the reciprocal of this cone is p = rr' (r + »•' - 2) - r (1x' + 3m') - r' (2x + 3m), or, by the formulas of p. 257, p = rn' + nr'. In like manner v — nr' + a'r + Srr'. Ex. 8. To find the characteristics of the developable generated by a line meeting two given curves. This is the reciprocal of the last example. We have therefore v ~ rr', p = rm' + mr', p = fir' + ft'r + drr'. CONTACT OF LINES WITH SURFACES. 510. We now return to the class of problems proposed in Art. 266, viz. to find the degree of the curve traced on a surface by the points of contact of a line which satisfies three conditions. The cases we shall consider are : (A) to find the curve traced by the points of contact of lines which meet in four con- secutive points ; (B) when a line is an inflexional tangent at one point and an ordinary tangent at another, to find the degree of the curve formed by the former points ; and ( C) that of the curve formed by the latter; (D) to find the curve traced by the points of contact of triple tangent lines. To these may be added: (a) to find the degree of the surface formed by the lines A ; (b) to find the degree of that formed by the lines considered in (B) and ( G) ; (c) to find the degree of that generated by the triple tangents. Now to commence with problem A ; if a line meet a surface iii four consecutive points we must at the point of contact not only have U' = 0, but also AW = 0, A 2 Z7' = 0, A s U' = 0. The tangent line must then be common to the surfaces denoted by the last three equations. But since the six points of intersection of these surfaces are all coincident with x'y'z'w\ the problem is a case of that treated in Art. 455. Since then, by that article, the condition FT = 0, CONTACT OF LINES Avft-H SUEFACES. 441 that the three surfaces should have a common line, is of the degree \'\"fj, + X"\fi' + W'fj." — W'}J' • substituting \ = 1, V = 2, X" = 3; f* = n-l, fi' = n-2, fj," = n-3; we find that 17 is of the degree (lira -24). The points of con- tact then of lines which meet the surface in four consecutive points : or (as we may call them) of double inflexional tangents ; he on the intersection of the surface with a derived surface B of the degree \\n — 24.* 511. The equation of the surface generated by the double inflexional tangents is got by eliminating x'y'z'w between V' = 0, AZ7' = 0, A a *7' = 0, A 3 £7' = 0; which result, by the ordinary rule, is of the degree n (n - 2) (n - 3) + In (w - 1) (n - 3) + 3ra (n - 1) (n - 2) = 6n 3 - 22w 2 + 18n. Now this result expresses the locus of points whose first, second, and third polars intersect on the surface ; and since if a point be anywhere on the surface, its first, second, and third polars intersect in six points on the surface, we infer that the result of elimination must be of the form U 6 M=Q. The degree of M is therefore 2n(«-3)(3n-2). 512. We can in like manner solve problem B. For the point of contact of an inflexional tangent we have U' = 0, A V = 0, A z V = : and if it touch the surface again, we have * I gave this theorem in 1849 (Cambridge and Dublin Journal, Vol. IV., p. 260). I obtained the equation in an inconvenient form (Quarterly Journal, Vol. I., p. 336) : and in one more convenient (Philosophical Transactions, 1860, p. 229) which I shall presently give. But I substitute for my own investigation the very beautiful piece of analysis by which Professor Clebsch performed the elimination indicated in the text, Crelle, Vol. lviii., p. 93. As the calculation is long, and the method, which is applicable to other problems also, deserves to be studied, I have thought, it better to place it by itself in an appendix than to introduce it here. Mr. Cayley has observed that exactly in the same manner as the equation of the Hessian is the transformation of the equation rt — s 2 which is satisfied for every point of a developable, so the equation S = is the transformation of the equation (p. 356) which is satisfied for every point on a ruled surface. 442 CONTACT OF LINES WITH SURFACES. besides W' = 0, where W is the discriminant of the equation of the degree re — 3 in X : /x, which remains when the first three terms vanish of the equation, p. 209. For W then we have X" = (re + 3) (re — 4), ft," = (re — 3) (re - 4) ; and having. S7 as in the last article, X=l, fx = n — 1; X' = 2, pl = n — 2, we have for the degree of n 2 (re - 3) (re - 4) + (re - 2) (re + 3) (re - 4) + 2 (re - 1) (re + 3) (re - 4) - 2 (re + 3) (re - 4). The degree then of the surface which passes through the points B is (re - 4) {Zri 1 + 5re - 24). The equation of the surface generated by the lines (b) which are in one place inflexional and in another ordinary- tangents is found by eliminating x'y'z'io' between the four equations U' = Q, At7' = 0, A 2 E/" = 0, W' = 0, and from what has been just stated as to the degree of the variables in each of these equations the degree of the resultant is re (re - 2) (re - 3) (re - 4) + 2« (re - I) (re - 3) (re - 4) + re (re - 1) (re - 2) (re + 3) (re - 4) = re (re - 4) (re 3 + 3re 2 - 20w + 18). But it appears, as in the last article, that this resultant contains as a factor, U in the power 2 (re + 3) (re - 4). Dividing out this factor the degree of the surface (b) remains re (re - 3) (re - 4) (re 2 + 6re - 4). 513. In order that a tangent at the point x'y'z'w' may elsewhere be an inflexional tangent, we must have A U' = 0, (an equation for which X = l, /i = n — 1), and besides we must have satisfied the system of two conditions that the equation of the degree re — 2 in X : fi, which remains when the first two terms vanish of the equation, p. 209, may have three roots all equal to each other. If then X', fi! • X", /a" be the degrees in which the variables enter into these two conditions, the order of the surface which passes through the points (G) is, by Art. 455, X>" + X>' + [n-2) X'X". But (see Appendix on the order of systems of equations) X'X" = (re - 4) (re 2 + re + 6), X>" + X>' = (re - 2) (re - 4) (re + 6). CONTACT OP LINES WITH SUHFACE8. 443 The order of the surface G is therefore (n - 2) (n - 4) (w 2 + 2re + 12). The locus of the points of contact of triple tangent lines is investigated in like manner, except that for the conditions that the equation just considered should have three roots all equal, we substitute the conditions that the same equation should have two distinct pairs of equal roots. It will be proved in the Appendix that for this system of conditions we have XV = | (re - 4) (n - 5) (w 2 + 3w + 6), X>" + X'V = (re - 2) (re - 4) (re - 5) (n + 3). The order of the surface which determines the points (2?) is, therefore, -|- (re — 2) (re - 4) (re — 5) (re* + 5re + 12). To find the surface generated by the triple tangents we are to eliminate xy'z'w between U' = 0, AZ7' = 0, and the two conditions, the order of the result being nfi'fi" + n (n - 1) (X'/n" -f yJ'fi!) : but since this result contains as a factor U : in order to find the order of the surface (c) we are to subtract reX'X" from the_ number just written. Substituting the values just given for X'X", X'fi" H- X'V ; and for ft ft", f (n - 2} (n - 3) (» - 4) {n - 5), we get for the order of the surface (c), n(n-3)(n- 4) [n - 5) (re 2 + 3re - 2), a number which probably ought to be divided by three. 514. There remains to be considered another class of problems, viz., the determination of the number of tangents which satisfy four conditions. The following is an enumera- tion of these problems. To determine: (a) the number of points at which both the inflexional tangents meet in four con- secutive points; (/3) the number of lines which meet in five consecutive points; (7) the number of lines which are doubly inflexional tangents in one place, and ordinary tangents in another; (S) of lines inflexional in two places; (s) inflexional in one place and ordinary tangents in two others; (£) of lines which touch in four places. 444 CONTACT OF LINES WITH SURFACES. The first of these problems has been solved as follows by Clebsch, Crelle, Vol. lxiii., p. 14. It was proved, Art. 500, that the points of inflexion of the section by the tangent plane at any point on a surface, of the polar cubic of that point, lie on the plane xH x + yH 2 -f zH s + wH^. Let it be required now to find the locus of points xy'z'w on a surface such that the line joining xy'z'w' to one of these points of inflexion may meet any assumed line, as for instance that joining the points x"y"z"w", x"y"'z"'w" : this is, in other words, to find the condition that co-ordinates of the form Xx ■+ fix" + vx", Xy' + /iy" + vy'", &c. may satisfy the equation of the tangent plane A U, of the polar with respect to the Hessian AH', and of the polar cubic A*£T. But if we substitute co-ordinates Xx + fix" + vx" in the equa- tions of any two planes P, Q; we determine X, fi, v to be respectively proportional to P'Q'"-P"Q", P"Q'-PQ"\ P Q" - P" Q'. In the present case P and Q being A V, AH, we have P = and Q proportional to H, and the co-ordinates of the intersection of the planes AC/"', AH', and the plane joining the three points, .are {P'Q'"-P"Q")x'+ P"Q'x"-P' Q'x"', &c. } and since P" is of the degree n — 1 and Q" of the degree 4re — 9 in xy'z'w', these values of the co-ordinates are of the degree bn — 9 in xy'z'w . The result then of substituting these co- ordinates in A 8 U is of the degree 3 (5n — 9) + (n — 3) = IGn — 30. But if we substitute co-ordinates of the form \x' + fix" + vx"' in A 3 U, the coefficient of X 3 vanishes since x'y'z'w' is on U, and the coefficients of X z /n, XV vanish since fix' + vx" is in the tangent plane. And since fi and v have the common factor Q or H, the result is divisible by H'\ and the quotient is of the degree 8ra - 14. This then is the degree of the locus required. Now I say that the points at which two doubly inflexional tangents can be drawn belong to this locus. At any one of these points the doubly inflexional tangents evidently both lie on the polar cubic of that point, and their plane will therefore - intersect that cubic in a third line which, as we saw (Art. 500), lies in the plane AH'. Every point on that line is to be con- sidered as a point of inflexion of the polar cubic ; and therefore the plane through the point x'y'z'w' and any arbitrary line must CONTACT OP LINES WBTH SURFACES. 445 pass through a point of inflexion. The points then whose number we are investigating, and which are evidently double points on the curve US, are counted doubly among the n (lln — 24) (8m — 14) intersections of the curve US with the locus determined in this article. Let us examine now what other points of the curve US can belong to the locus. At any point on this curve the doubly inflexional tangent lies in the polar cubic, the section of which by the tangent plane consists of a line and conic ; and since all the points of in- flexion of such a system lie in the line, the doubly inflexional tangent itself is in this case the only line joining x'y'z'w' to a point of inflexion. And we have seen, Art. 511, that the number of doubly inflexional tangents which can meet an assumed line is 2m (n- 3) (3m — 2). We have then the equation 2a 4 2m [n - 3) (3m - 2) = n (11m - 24) (8m - 14), whence a = n (41k*- 162m + 162), which is the solution of the problem proposed. 515. I investigate as follows the number of points /S through which a line can be drawn to meet the surface in five con- secutive points. For such points, it is evident by the method already pursued that we must have the conditions satisfied U' = 0, AU' = 0, A a Z7' = 0, A 3 Z7' = 0, A*U' = 0. Let us elimi- nate xyzw between the last four equations, and the result is of the form Z7 6 $, where

denotes the surface generated by the doubly inflexional tangents, and from a consideration of this case I have been led to conclude that the general form of. is U\jr + # 4 . The factor Z7 6 in the eliminant of the four sur- faces is accounted for by the fact that for every point on U, the first, second, third, and fourth polars have six points com- mon. If, however, the point be on the curve US, since AC', A a C, A S C have a line common, there will be a seventh point common to them with A 4 C. Thus we see that the curve US ought to form part of the locus <£. But again, the points for which the four polars have eight points common are either the 446 CONTACT OF PLANES WITH SURFACES. points on US for which a line meets the surface in five con- secutive points, or the double points on US, or the points where US meets H and for which the two inflexional tangents coincide. All these points ought to lie on the intersection of US with ijr. Thus I have been led to form the equation n (43m - 96) (11m - 24) = 8» (41m 2 - 162w + 162) + 4n(n-2) (llre-24)+/3, whence /3 = m (m- 4) (103 m — 204); but I own that this result needs confirmation. The other problems stated in the last article have not yet been solved. CONTACT OP PLANES WITH SURFACES. 516. We can discuss the cases of planes which touch a surface, in the same manner as we have done those of touching lines. Every plane which touches a surface meets it in a section having a double point : but since the equation of a plane includes three constants, a determinate number of tan- gent planes can be found which will fulfil two additional conditions. And if but one additional condition be given, an infinite series of tangent planes can be found which will satisfy it, those planes enveloping a developable, and their points of contact tracing out a curve on the surface. It may be re- quired either to determine the number of solutions when two additional conditions are given, or to determine the nature of the curves and developables just mentioned, when one additional condition is given. Of the latter class of problems we shall consider but two, viz., the discussion of the case when the plane meets the surface in a section having a cusp ; or, when it meets in a section having two double points. Other cases have been considered by anticipation in the last section, as for example, the case when a plane meets in a section having a double point, one of the tangents at which meets in four consecutive points. 517. Let the co-ordinates of three points be x'y'z'w', x"y"z"w", xyzw ; then those of any point on the plane through the points" will be \x + fix" + vx, ~ky' + /j,y" + ry, &c. : and if CONTACT OF PLANES WfTH SURFACES. 447 we substitute these values for xyzw in the equation of the surface, we shall have the relation' which must be satisfied for every point where this plane meets the surface. Let the result of substitution be [IT] = 0, then [Z7] may be written x" v + x">a„ w + \;-va tr + K" 2 (M„ + v&)' w + &c. = o, where A„ = ," -^ l + y" |, + s" A + M ," ^ ; _ d d d d dx " dy' ds' dw ' The plane will touch the surface if the discriminant of this equation in X, //., v vanish- If we suppose two of the points fixed and the third to be variable, then this discriminant will represent all the tangent planes to the surface which can be drawn through the line joining the two fixed points. We shall suppose the point x'y'z'w to be on the surface, and the point x"y"z"w" to be taken anywhere on the tangent plane at that point: then we shall have Z7' = 0, A lt U' = 0, and the discriminant will become divisible by the square of A U'. For of the tangent planes, which can be drawn to a surface through any tangent line to that surface, two will coincide with the tangent plane at the point of contact of that line. If the tangent plane at x'y'z'w be a double tan- gent plane, then the discriminant we are considering, instead of being, as in other cases, only divisible by the square of the equation of the tangent plane, will contain its cube as a factor. In order to examine the condition that this may be so, let us for brevity write the equation [U] as follows, the coefficients of X", X"' 1 /^ being supposed to vanish, TX^v -H-X"- 2 C + T{ ) = 0, where is the discriminant when T vanishes as well as V and A„ U'. In order that the discriminant may be divisible 448 CONTACT OF PLANES WITH SURFACES. by T 3 , some one of the factors which multiply 2 12 must either vanish or be divisible by T. 518. First then let A vanish. This only denotes that the point x"y"z"w" lies on the polar quadric of x'y'z'w : or, since it also lies in the tangent plane, that the point x"y"z"w" lies on one of the inflexional tangents at x'y'z'w'. Thus we learn that if the class of a surface be p, then of the p tangent planes which can be drawn through an ordinary tangent line, two coincide with the tangent plane at its point of contact, and there can be drawn p — 2 distinct from that plane : but that if the line be an inflexional tangent, three will coincide with that tangent plane, and there can be drawn only p — 3 distinct from it. If we suppose that x'y'z'w' has not been taken on an inflexional tangent, A will not vanish, and we may set this factor aside as irrelevant to the present discussion. We may examine at the same time the conditions that T should be a factor in B? — A G, and in . The problem which arises in both these cases is the follow- ing: Suppose that we are given a function V, whose degrees in x'y'z'w', in x"y"z"w", and in xyzw are respectively (A, ytt, fi). Suppose that this represents a surface having as a multiple line of the order /*, the line joining the first two points; or, in other words, that it represents a series of planes through that line : to find the condition that one of these planes should be the tangent plane T whose degrees are («— 1, 0, ]). If so, any arbitrary line which meets T will meet V, and therefore if we eliminate between the equations T=0, 7=0, and the equations of an arbitrary line ax + by + cz + dw = 0, ax + b'y + c'z + d'w = 0, the resultant R must vanish. This is of the degree fi in abed, in a'b'c'd', and in x"y"z"w", and of the degree /a (n — 1) + \ in x'y'z'w'. But evidently if the assumed right line met the line joining x'y'z'w, x"y"z"w", B would vanish even though T were not a factor in V. The condition (Jf=0), that the two lines should meet, is of the first degree in all the quantities we are considering: and we see now that B is of the form CONTACT OF PLANES WITH SURFACES,' 449 M^M'. R' remains a function of ac'y'z'w' alone, and is of the degree /*(«-2) + \. 519. To apply this to the case we are considering, since the discriminant of [U] represents a series of planes through x'y'z'w, x"y"z"w", it follows that B'- AG and both represent planes through the same line. The first is of the degrees {2 (n - 2), 2, 2}, while is of the degrees (n - 2) (rc a - 6), n - 2n* -{ n — 6, n s — 2n* + n — 6, as appears by subtracting the sum of the degrees of T 2 , A, and (B'' — AGf from the degrees of the discriminant of [27], which is of the degree n(n— 1)* in all the variables. It follows then from the last article that the condition {H= 0) that T should be a factor in B* - AG is of the degree 4(w — 2), and the condition [K=0) that T should be a factor in is of the degree (n — 2) [n 3 — ri' + n— 12). At all points then of the intersection of U and H the tan- gent plane must be considered double. H is no other than the Hessian ; the tangent plane at every point of the curve UH meets the surface in a section having a cusp, and is to be counted as double (Art. 263). The curve UK is the locus of points of contact of planes which touch the surface in two distinct points. 520. Let us consider next the series of tangent planes which touch along the curve UH. They form a developable whose degree is p = 2n («- 2) (3?i-4), Ex. 3, p. 439. The class of the same developable, or the number of planes of the system which can be drawn through an assigned point, is v = in{n — 1) (w — 2). For the points of contact are evidently the intersections of the curve UK with the first polar of the assigned point. We can also determine the number of stationary planes of the systerm If the equation of U, the plane z being the tangent plane at any point on the curve UJFJ, be z + 2/ 2 -t-w 8 + &c. = 0, it is easy to show that the direction d''u of the tangent to UH is in the line -j-j- = 0. Now the tan- gent planes to U are the same at two consecutive points proceeding along the inflexional tangent y. If then u 5 do C T G 450 . THEORY OF RECIPROCAL SURFACES. not contain any term a; 3 , (that is to say, if the inflexional tan- gent meet the surface in four consecutive points) the direction of the tangent to the curve UH is the same as that of the inflexional tangent : and the tangent planes at two consecutive points on the curve UH will be the same. The number of stationary tangent planes is then equal to the number of inter- sections of the curve UH with the surface 8. But since the curve touches the surface, as will be seen in the Appendix, we iave ' + 2ft' + 1', or 3jo' + ft' + 1', and the latter form is manifestly to be rejected. of the nature of the reciprocal of a cubic, led me to the theory in the form here explained. Some few additional details will be found in a memoir which I contributed to the Transactions of the Royal Irish Academy, Vol. XXIII., p. 461. THEORY OF RECIPROCAL SURFACES. 453 But considering the curve V as made up of the twenty-seven lines, the points t' occur each on three of these lines: we are then led to the formula V (n' - 2) = p + 2/3' + 3t'. The example we are considering does not enable us to determine the coefficient of 7 in the second formula A, because there are no points 7 on the reciprocal of a cubic. Lastly, the two hundred and forty points in which the curve c meets the second polar are made up of the twelve points er', and the fifty-four points /3'. Now the equation 12a +- 545 = 240 only admits of the systems of integer solutions (11, 2), or (2, 4), and the latter is manifestly to be preferred. In this way we are led to assign all the coefficients of the equations (.4) except those of 7. 524. Let us now examine in the same way the reciprocal of a surface of the n tb order, which has no multiple points. We have then ra' = w(«-l) 2 , n — 2= (n — 2)(« 2 +l), a =n(n-l); and for the nodal and cuspidal curves we have (Art. 280) b' = \n (n- 1) {n- 2) (w 3 -«* + n - 12), c' = in(n-l) («-2). The number of cuspidal edges on the tangent cone to the reciprocal, answering to the number of points of inflexion on a plane section of the original, gives us k =3n(n— 2). The points p and cr', answer to the points of intersection of an assumed plane with the curves UK and TJH (Art. 519) : hence p = n (n — 2) [n 3 — ri l + n — 12) , a = 4w [n — 2) . Substitute these values in the formula a' (ri — 2) = k + p -t- 2a-', and it is satisfied identically, thus verifying the first of formulae (.4). We shall next apply to the same case the third of the formulae (A), It was proved (Art. 520) that the number of points /3' is 2n (n— 2) (llw — 24). Now the intersections of the nodal and cuspidal curves on the reciprocal surface answer to the planes which touch at the points of meeting of the curves US, and UK on the original surface. If a plane meet the surface in a section having an ordinary double point and a cusp, since from the mere fact of its touching at the latter point it is a double tangent plane, it belongs in two ways to the system which touches along UK; or, in other words, it is a stationary 454 THEORY OF RECIPROCAL SURFACES. plane of that system. And since evidently the points /3' are to be included in the intersections of the nodal and cuspidal curve, the points U, H, K must either answer to points ft' or points 7'. Assuming, as it is natural to do, that the points /3 count double among the intersections of UHK we have 7 ' = n {4(«-2)}.{(w-2)(n s -n*+«-12)}-4w(«-2)(lln-24) = An {n - 2) (n - 3) (ri 1 + Sn- 16). But if we substitute the values already found for c', ri, =(25 + 3c) (2m 8 -2n -25 -3c- 1). But the existence of nodal and cuspidal curves on the surface causes also a diminution in the number of double and cuspidal edges in the tangent cone. From the diminution in the degree of the reciprocal surface just given must be subtracted twice the diminution of the number of double edges and three times that of the cuspidal edges. Now from formulas A, we have K =(a-rb-c) [n - 2) + 6y8 + 4? + 3£. But since if the surface had no multiple lines the number of cuspidal edges on the tangent cone would be (a + 25 + 3c) [n— 2), the diminution of the number of cuspidal edges is K= (35 + 4c) [n - 2) - 6/3 - 4? - St. Again, from the first system of equations (Art. 525), we have {a -2b- 3c) (w - 2) (n - 3) = 28 - 8k - 18h - 12 [5c], * In order to verify the theory it would be necessary to show that this number B' coincides with what may be deduced from Ex. 5, p. 439. In the first place the developable generated by the cuspidal curve on the reciprocal surface corresponds with that which envelopes the given surface along US, and which, by the example cited, ought to be of the degree 28re (n — 2) 2 , but if we subtract from this the number ft', we get the value already determined. In like manner, if we take the surface enveloping the given surface along UK (Art. 520) and subtract from the degree de- termined, as in the example cited, 4y' + ft' + Gt', we get not R' but $R'. Possibly this- may be because all the tangent planes which envelope the developable in question are double tangent planes; but it must be owned that there are points in all this theory which need further explanation. THEORY OF RECIPROCAL SURFACES. 457 and putting for [be] its value, 28 = (a-25- 3c) (re-2) (n-3)+8k+18h + 125c - 36/3 - 24? - 12t. But if the surface had no multiple lines 28 would = (a + 25 + 3c) (w - 2) (n - 3). The diminution then in the number of double edges is given by the formulas 2H= (45 -l- 6c) [n - 2) (n - 3) - &h -, 18h - 125c + 36/3 + 24y + 1 2», The entire diminution then in the degree of the reciprocal D - 3iT— 2H is, when reduced, « (75 + 12c) - 45 2 - 9e' 2 - 85 - 15o + Sk + lSh- 18/8- 12Y-12t + 9«. 527. The formula} -B, reduced by the formulas a -f 25+3c = n (»- 1), become a(— 4w + 6) = 28 — a a — 4/3 — 9a- •> 5(-4w + 6)=4^-25 2 -9y8-67-3;-2pi (C). c(-4n + 6) = 6A-3c*-6/8-47-2i- 3o-J To each of these formulas we add four times the corre- sponding formula A • and we simplify the results by writing for a' — a -^-28 — 3k, ri the degree of the reciprocal surface, by giving B the same meaning as in Art. 525, and by writing for c 2 — c — 2h — 3/3, 8 the order of the developable generated by the curve c ; when we obtain the formulas in a more convenient shape, vi?. ri — a = k, — a ~\ 2.5 = 2/3-/3-3* 1 (Z>). 3#+c=£ + 5<7-2.d From the first of equations A and D we may also obtain the equatfon (n — 1) a = ri + p + 3cr, the truth of which may be seen from the consideration that a, the curve of simple contact from any one point, intersects the first polar of any other point, either in the ri points of contact of tangent planes passing through the line joining the two points, or else in the p points where a meets 5, or the a points where it meets c, since every first polar passes through the curves 5, c. 458 THEORY OF RECIPROCAL SURFACES. 528. The effect of multiple lines in diminishing the degree of the reciprocal may be otherwise investigated. The points of contact of tangent planes which can be drawn through a given line are the intersections with the surface of the curve of degree (« — l) 2 which is the intersection of the first polars of any two points on the line. Now let us first consider the case when the surface has only an ordinary double curve of degree b. The first polars of the two points pass each through this curve, so that their intersection breaks up into this curve b and a complemental curve d. Now in looking for the points of contact of tangent planes through the given line, in the first place, instead of taking the points where the complex curve b + d meets the surface, we are only to take those in which d meets it, which causes a reduction bn in the degree of the reciprocal. But, further, we are not to take all the points in which d meets the surface : those in which it meets the curve h being to be rejected ; those being in number 25 (n — 2) - r (Art. 339) where r is the rank of the system b. Now these points consist of the r points on the curve b } the tangents at which meet the line through which we are seeking to draw tangent planes to the given surface, and of 25 [n — 2) — 2r points at which the two polar surfaces touch. These last are cuspidal points on the double curve b ; that is to say, points at which the two tangent planes coincide, and they count for three in the intersections of the curve d with the given surface, since the three surfaces touch at these points ; while the r points being ordinary points on the double line only count for two. The total reduction then is nb + 2r + 3 {2b (n - 2) - 2r] ~b(7n- 12) - 4r, which agrees with the preceding theory. If the curve b, instead of being merely a double curve, were a multiple curve on the surface of the order p of mul- tiplicity, I have found for the reduction of the degree of the reciprocal (see Transactions of the Royal Irish Academy r , Vol. xxiii., p. 485) b (p - 1) (3p + 1) n - 2bp {? - I), -p* (p - 1) r, DEVELOPABLE SURFACES. 459 for the reduction in the number of cuspidal edges of the cone of simple contact h{B [p - l) a « - p (p- 1) (2p - 1)] -p [p - 1) [p - 2) r, and for twice the reduction in the number of its double edges 2bp (p - 1) «* - h [p - l) [Up - 8) n + bp [ p - 1) [8p - 2) -p" [p - l) 2 b 2 +p(p-l) (4p - 6) r* DEVELOPABLE SUEFAOES. 529. The theory just explained ought to enable us to account for the fact that the degree of the reciprocal of a developable reduces to nothing. This application of the theory both verifies the theory itself and enables us to determine some singularities of developables not given, p. 256. We use the notation of the section referred to. The tangent cone to a developable consists of n planes; it has therefore no cuspidal edges and \n [n — 1) double edges. The simple line of contact [a) consists of n lines of the system each of which meets the cuspidal edge m once, and the double line x in [r — 4) points. The lines m and x intersect at the a points of contact of the stationary planes of the system; for since there three con- secutive lines of the system are in the same plane, the inter- section of the first and third gives a point on the line x.f We have then the following table. The letters on the left- hand side of the equations refer to the notation of this Chapter and those on the right to that of Chapter xn. : p=w(r-4), which is the product of two equal factors taken in opposite order. If the arcs a and b be each 90°, then indeed the plane of AB will be the same as that of CD, but the direction of the rotations in the two products will be opposite. If then we multiply to- gether two rectangular quaternions A, B, (that is, such that the rotation is through a right angle) we see from Art. 5 that if A.B be of the form I cos 6 + I sin 0. V, then B.A will be of the form I cos# - 1 smO.V. Two quaternions thus related are said to be conjugate quaternions: that is, when one is of the form scalar + vector : and the other, the same scalar — the same vector. It follows as a particular case of the last, that when 6 = 90°, the product of two rectangular quaternions whose planes are at right angles to each other, gives A.B = — B.A. As this is a fundamental theorem we shall presently prove it independently. 8. It is seen without difficulty that the multiplication of quaternions is a distributive operation: viz., that the product of the quaternions ( ^ — -J - is the sum of the 470 ON THE CALCULUS OF QUATERNIONS. several products ^ - , ^ - , &c. : and that the same thing is r A /t A, .(J, true if the order of multiplication be reversed. Hence then if we have two quaternions,* each expressed in the form [a + U+cJ+dK) [a' + VI+cJ+d'K), the product is the sum of the sixteen term3 got by combining each of the first four terms with each of the second four, care however being taken to attend to the order of the multiplication. Let us then examine the meaning of the terms II, IJ, &c, which occur in such a product. Now if we remember that I denotes a rectangular rotation round the axis of x as axis, and that the effect of such a rotation would be to change a line in the direction of the axis of y to that of z, and one in the direction of z into the negative direction of y, we can write down the equations Ij=k, Ik=-j. In like manner, Jk = i, Ji= — k; Ki—j, Kj = — i. Let us now consider the effect of two of these operations performed consecutively. If we first operate on j with 7, and then again with I on the result k, we get Pj = -j, or P = — 1. In like manner J 2 = — 1, 2T* = — 1, and since it is evidently true, no matter what line be taken for the axis of rotation, that the effect of twice turning round a right angle is to reverse the position of the line operated on ; it follows that the square of every rectangular quaternion may be said to be — 1. Again we have seen that Ij=k, Jk = i; hence Jlj—i; but Kj = — i ; hence JI= — K. In like manner, from the equa- tions Ji= — k, Ik = —j, Ki=j, we conclude IJ=K. Hence IJ= K=-JI. In like manner JK=I=-KJ; KI= J= - IK. If now we compare the equations Ij = k, IJ=K, &c, we shall find that the equations which represent the effect of the operations J, J, K on the lines i, J } k, are exactly the same in form as those which denote the effects of the successive performance of these operations. Now since in the practice of this calculus we are concerned with the laws according to * It is also true, though it is not to be taken for granted, that when we take the continued product of three quaternions (qq') q" = q (q'q"). ON THE CALCULUS «F QUATEBNIONS. 471 which the symbols combine with each other rather than with their interpretation, it is found unnecessary to keep up the distinction of notation between I, '/, K; i, j, k. Whatever propositions are true of the symbols in the one sense, are equally true in the other, and, by interpreting some vectors as lines and others as rotations, we can give a variety of significations to the same equation all of which will be equally true. We shall then understand i to denote at pleasure either a unit line measured along the direction of the axis of x, or a rotation through a right angle round that axis. In like manner a rectangular rotation round any unit vector a is re- presented by the letter a as already stated in Art. 5. We shall write the general form of a quaternion a + bi+cj + dk; and we shall combine these symbols according to the laws s; 2 =_/* = 7c* = — 1 ; ij = k = —ji ; jk = i= —hj ; hi =j = — ih. In forming the continued product of a number of factors the order must be carefully attend, to, except that if a scalar or number is one of the factors, its order is indifferent, and it may be brought to the left hand as a multiplier of the whole. Thus, if a, j3, 7 be any three unit vectors, or rectangular qua- ternions, and if we multiply /3y by a/3, the result a/3 2 — a) a = or S(pa) = a 2 . But a 2 is a scalar, and we may therefore divide by it under the sign S, and write the equation in the form s(-\ — l. This equation may also be inferred from what was stated in a previous article, viz., that the scalar part of the above fraction denotes the projection of the line p on the line a, divided by a. In like manner the equation S [ — ] = 1, which expresses that the projection of the fixed line a on the direction p is in length equal to p, obviously represents the sphere described on the vector a as diameter. Again, the equation Sl-j S [ ~ J = 1, in the first place represents a cone, because if it is satisfied for any value of p, it will also be satisfied for the value mp, where m is any scalar. Secondly it passes through the intersection of S- = 1, S- = 1 : a p it is therefore the cone whose base is the circle represented by the two equations just written. Ex. 3. To find the product of two quaternions. We have only to multiply out a + bi + cj + dk, a' + b'i + c'j + d'k. We may form a clearer conception of the result by separating the scalar and vector parts, and writing the two quaternions S + V, S' + V, when the product is SS' + SV + S'V+ VV. Now if it be required to find the scalar part of the product (since SV" and S'V are mere vectors), it is SS' + S (VV), or the scalar of the product is the product of the scalars + the scalar part of the product of the vectors. Thus let a, /3, y be three radii veotores of a sphere ; then it is an identical equation a ay that - = - 3 . Now if a, b, c be the sides of the spherical triangle formed by the extremities of these vectors ; cos a, cos b, cose are the scalars of the three quaternions, and the scalar part of the product of the vectors on the right-hand side of the equation is the product of their tensors sin a, sin b, into the cosine of the angle between them, thus we have the fundamental formula of spherical trigonometry cose = cosfl cos& + sina sinfi cosC. 9. We can, in like manner, form the product of three vectors. It is found, without difficulty, by actual multipli- cation, that if ix+jy+kz, ix' \jy +hz\ ix" +jy" + kz" be the ON THE CALCULUS oV QUATEKNIONS. 473 three vectors, the scalar part of the product is the determinant whose three rows are x, y, z; x, y', z ; x", y", z". Hence if a > 0> 7 oe the three vectors, the condition that they should lie in one plane is 8 (afiy) =0 (Note, p. 19). This is also evident from the consideration that if 8 (aBy) — 0, then a/37 i g a pure vector, but a/3y = a.S(/3y) + aV(/3y) ; there- fore a.V{By) is a pure vector, or a is perpendicular to V(/3y) y and therefore is in the plane of and 7. Q.e.d. Thus we can find the equation of the plane passing through the extremity of three vectors a, 0, 7. By hypothesis, the lines joining the extremity of any variable vector terminating in the plane, with the extremities of the assumed vectors, lie in the plane. We have, therefore, 8[p — a) (p — 0) (p — 7) = 0. In expanding this we may omit such terms as Sp^y, because p* is a scalar, and p*y a mere vector whose scalar is nothing. The expanded product is then 8 (p@y + a/37 + a @p) = 8&fty, and the vector perpendicular to the plane is V[a - 7) (0 - 7) = 7(07 + 7a + a/3). Eeturning to the product of the three vectors, it is also found by actual multiplication, that V (o#y) = aS[8y) - S (7a) + yS (aft, an equation of great use. In connection with this, the following identical equation may be given, 88 (a/87) = aS (8yS) + 8S (yaS) + yS (a0S), as also that, the vector part of the product VaSVyS may be written in either of the forms aS (^78) + 88 (yaS) or - yS (a/38) + 8S (afiy). In fact, Fa/3 denotes a line perpendicular to a and ; the vector now required must therefore lie in the plane, both of a and 0, and of 7 and 8. 10. As an example of the method of applying this calculus to a geometrical problem, we shall investigate the problem to find the equation of the surface generated by a line resting on three directing lines. In the first place we may follow a 474 ON THE CALCULUS OP QUATERNIONS. process proceeding after the analogy of the co-ordinate methods. It is seen immediately by substituting = [la + ma!) for a in the equation of a plane through three points, that the equation of the plane through the extremity of the vector just written, and through a fixed line, e.g., through the extremities of the vectors /3, 7, is of the form I A + mB = 0, where A denotes the plane through ot/67, and B that through a'/ty. If then we join any assumed point on the vector a— a' to the other two lines we get the equation of two planes in the form IA-+ mB—Q, IA' -t mB' = 0, from which, eliminating I, m, we get the locus in the form AB = BA'. Otherwise thus, we are to express the condition that, if we join by planes any assumed point on the locus to the three lines, the joining planes have a line common. The vectors perpendicular to these planes will then be co-planar. Let then the first line be parallel to the line a, and pass through the extremity of a vector a' ; then the vector perpendicular to the plane through this line being perpendicular to a and to a' — p is Va [a! — p), and the required equation is S{V*(a , -p).V/3(P-p).Vy{r/-p)}=0 i which is reduced and expanded by the last article. 11. We give one more example to shew how infinitesimals are introduced into this calculus. The equation of any sphere is p 2 = -c'\ Now let the line joining the extremity of p to an indefinitely near point be dp, then the next consecutive radius vector is p + dp, and we have (p +dpf =-c 2 , or expanding and neglecting the square of dp, pdp + dp . p = 0, or &{P' dp) = 0, which indicates that the radius p is perpendicular to the tangent line dp. ON THE CALCULUS 6*F QUATERNIONS. 475 Very much more must be said if it were intended to give any complete account of this Calculus, as, for example, the method of finding the equations of tangents and normals, lines of curvature, geodesies; &c. Bat enough has been said to dispose the reader to give credit to the assertion that there is no geometrical problem to which it may not be applied ; that it is very rich in transformations; and that its processes though constantly following the analogies of the co-ordinate methods, are by no means slavishly dependent on that system. ( 476 ) APPENDIX II. ON SYSTEMS OF ORTHOGONAL SURFACES.* It might be thought, from Dupin's theorem, that being given a series of surfaces, involving a parameter, it would be always possible to determine two other systems, each containing a parameter, and cutting the surfaces of the given system at right angles, and along their lines of curvature. This, how- ever, i3 not the case. In order that a given family of surfaces, with a parameter, may form one of a triple orthogonal system, an equation, or equations, of condition must be satisfied. M. Serret arrives at the conclusion (see Liouville, Vol. xn., p. 241) that in order that the equation F{x, y, z) = oc, where a is a parameter, may be one of a triple orthogonal sj'stem, the function must satisfy two partial differential equations of the sixth order. We give Serret's investigation of the par- ticular case where the given function is the sum of three functions of x, y, s respectively. Let an equation then be given of the form X+Y+Z=a (1). It is required to determine the condition, to which these functions must be subject, in order that the surfaces (1) may have a pair of conjugate orthogonal systems. Suppose that are these systems, and it is evident by the- conditions of the problem that we have (X', Y\ Z' being the first derived functions of JT, F, Z) * For the following appendix, on a subject which I had omitted in the preceding treatise, I am almost entirely indebted to a manuscript note kindly placed at my disposal by the Bev. W. Roberts, as well as to his papers published in the Comptes Rendus. ON SYSTEMS OF OKTlft>GONAL SUEFACES. 477 „, dp v ,d(3 „ d/3 dx dy dz x>p + rp + *p=o (2). dx dy dz d/3dy d/3 dy d/3 dy_ dx dx dy dy dz dz Proceeding to integrate the first two of these equations by the ordinary methods of partial differential equations, we find that /3 and 7 are functions of u and v, where _ rdx [dy _ [dx [dz U -JX'~JY" V -]X'~JZ'- Consequently the third of equations (2) becomes (d^,d^\(Zy dy\ X^dpdy X^dpdy_ ■ \du + dv) \du + dv) + Y ri du du + Z n dv dv "" { h Now u and v being functions of «, «/, e, we may regard y and z as functions of u, v, and x. Hence x enters (3) as an indeterminate parameter, and the quantities /3 and 7 must satisfy not only (3) but also the derivatives of (3) obtained by differentiating it on the supposition that x alone is variable. Differentiating (3) with respect to x on this hypothesis, and remembering that dy_T dz_Z_ dx~ Z" dx~ X"> we find, X" - Y" d£ dy X"-Z" dp dy Y ri du du + Z' 2 dv dv~ ' X", F", Z" being the second derived functions of X, Y, Z. Differentiating once more, and denoting the third derived functions by X'", Y", Z", we get X'X" - T T" - 2 Y" {X" - Y") d]3 dy Y" du du XX'" - Z'Z" - 1Z" {X" - Z") dp dy_ + Z* 'dv dv~ Hence at once results the equation of condition sought, namely, XX'" ( Y" - Z") + Y' Y'" {Z" - X") + Z'Z" [X" - Y") + 2 (X" - Y") ( Y" - Z") [Z" - X") = 0. 478 ON SYSTEMS OF ORTHOGONAL SURFACES. This relation expresses the condition that a family of sur- faces, of the particular form represented by equation (1), should form one of a triple orthogonal system. It was first given by M. Bouquet, Liouville, Vol. xi., p. 446, but the above proof has been taken from M. Serret's memoir. Even when the equations of condition are satisfied by an assumed equation it does not seem easy to determine the two conjugate systems. Thus M. Bouquet observed that the con- dition just found is satisfied when the given system is of the form x m y n z'' = a, but he gave no clue to the discovery of the conjugate systems. This lacuna has been completely supplied by M. Serret, who has shown much ingenuity and analytical power in deducing the equations of the conjugate systems, when the equation of condition is satisfied. The actual results are, however, of a rather complicated character. We must con- tent ourselves with referring the reader to his memoir, only mentioning the simplest case obtained by him, and which there is no difficulty in verifying a posteriori. He has shown that the three equations, x ' represent a triple system of conjugate orthogonal surfaces. The surfaces (a) are hyperbolic paraboloids. The system (/3) is composed of the closed portions, and the system (7) of the infinite sheets, of the surfaces of the fourth order, («■ _ y y - 2 /3 2 (s 2 + f + 2x>) + /3 4 = o. M. Serret has observed that it follows at once from what has been stated above, that in a hyperbolic paraboloid, of which the principal parabolas are equal, the sum or difference of the distances of every point of the same line of curvature from two fixed generatrices is constant. Mr. W. Boberts, expressing in elliptic co-ordinates the condition that two surfaces should cut orthogonally, has sought for systems orthogonal to 1+ M + N= a, where L, M, N are functions of the three elliptic co-ordinates respectively. He ON SYSTEMS OF ORTlftGONAL SURFACES. 479 has thus added some systems of orthogonal surfaces to those previously known [Comptes Bendus, September 23, 1861). Of these perhaps the most interesting, geometrically, is that whose equation in elliptic co-ordinates is nv = a.\, and for which he has given the following construction. Let a fixed point in the line of one of the axes of a system of confocal ellipsoids be made the vertex of a series of cones circumscribed to them. The locus of the curves of contact will be a determinate surface, and if we suppose the vertex of the cones to move along the axis, we obtain a family of surfaces involving a parameter. Two other systems are obtained by taking points situated on the other axes as vertices of circumscribing cones. The surfaces belonging to these three systems will intersect, two by two, at right angles. It may be readily shown that the lines of curvature of the above mentioned surfaces (which are of the third order) are circles, whose planes are perpendicular to the principal planes of the ellipsoids. Let A, B, be two fixed points, taken re- spectively upon two of the axes of the confocal system. To these points two surfaces intersecting at right angles will corre- spond. And the curve of their intersection will be the locus of points M on the confocal ellipsoids, the tangent planes at which pass through the line AB. Let P be the point where the normal to one of the ellipsoids at M meets the principal plane containing the line AB, and because P is the pole of AB in reference to the focal conic in this plane, P is a given point. Hence the locus of M, or a line of curvature, is a circle in a plane perpendicular to the principal plane con- taining AB. ( 480 ) APPENDIX III. CLEBSCH'S CALCULATION OF THE SURFACE S.» 1. In this appendix we give the calculation referred to p. 441, by which the equation is determined of a surface which meets a given surface at the points of contact of lines which meet it in four consecutive points. It was proved, Art. 510, that in order to obtain this equation it is necessary to eliminate between the equation of an arbitrary plane, and the functions A 27', A a 0", A S C'. We perform this elimination by solving for the co-ordinates of the two points of intersection of the arbitrary plane, the tangent plane AC7 7 , and the polar quadric A 2 V ; substituting these co-ordinates successively in A 3 £7', and multiplying the results together. I write with M. Clebsch, the four co-ordinates of the point of contact x t , x s , x 3 , x t ; the running co-ordinates ?/,, y 2 , y s , y t ; the differential coefficients «,, m 2 , m 9 , m 4 ; the second and third differential coefficients being denoted in like manner by sub- indices, as m ]2 , w 123 . Through each of the lines of intersection of A V\ A 2 TJ\ we can draw a plane, so that by suitably determining t i: t^ t s , t t we can, in an infinity of ways, form an equation identically satisfied A* & + {t lVl + tj/ t + t^y 3 + tj 4 ) A IT = [pjfi + PUT, +A& + AK.) fajfi + IJf, + 2Jf, + M<) ■■■i A )- We shall suppose this transformation effected; but it is not necessary for us to determine the actual values of <„ &c, for it will be found that these quantities will disappear from the result. Let the arbitrary plane be c 1 y l -f c^/ 2 + c 3 y s + c„y 4 , then it is evident that the co-ordinates of the intersections of * See Note, p. 441. CLEBSCH'S CALCULATrof OF THE SURFACE 8. 481 the arbitrary plane, the tangent plane u^+u^ + u^ + u^ and A 2 Z7', are the four determinants of the two systems C l) C » C 3) C 4 U X1 %1 U *l M 4 Pi, Pi, Pa Pi , <\l C 2 ) C 3 ) c * «„ M 2) M 3 ) W 4 ft) ft) ft) ft These co-ordinates have now to be substituted in A 3 IT', which we write in the symbolical form {a 1 y l + aj/ 2 + a^ 3 + aj t ) s ; where a, means -5— , &c, so that after expansion we may substitute for any term apj^yj^ « 1JB y 1 y i y„ &c. It is evident then that the result of substituting the co-ordinates of the first point in A 3 Z7', may be written as the cube of the symbolical determinant Sa^w^, where after cubing we are to substitute third differential coefficients, for the powers of the \c t u t qJ', (where h l is a symbol used in the same manner as a t ). The eliminant required may therefore be written (Sa lCAA )'(2J AM4 ) 3 = 0* The above result may be written in the more symmetrical form l2oA«. ft)' (ZKW&Y + (Si.c.w, jjJ' (Sa^u^J' = 0. For since the quantities a, b, are after expansion replaced by differentials, it is immaterial whether the symbol used originally were a or b; and the left-hand side of this equation when expanded is merely the double of the last expression. We have now to perform the expansion, and to get rid of p and q by means of equation A. We shall commence by thus banishing p and q. * The reason why We use a different symbol for -5— , Ac. in the second deter- minant, is because if we employed the same symbol, the expanded result would evidently contain sixth powers of a, that is to say, sixth differential coefficients. We avoid this by the employment of different symbols, as in Mr. Cayley's " Hyperde- terminant Calculus," (Lessons on Higher Algebra, p. 79) with which the method here used is substantially identical, II 482 clebsch's calculation of the surface 8. 2. Let us write F= (2a lCA p t ) (2Z>,c 2 w 3 ? 4 ), G = ( S&.e.u.jpJ (Sa.c.u.jJ . The eliminant is F* + G 3 = Q, or (F+ Gf - 3FG (F+ G) = 0. We shall separately examine F+ 6r, and FG, in order to get rid of p and q. If the determinants in F were so far ex- panded as to separate the p and q which they contain, we should have F= (mj), + m 2 p 2 + m 3 p s + m iPi ) {n^ + n 2 q 2 + ?i s q s + n t q 4 ), G = [n lPi + n iPi + n s p s + n iPi ) (m.g-, 4 m A + m s q 3 + m^), where, .for example, m i is the determinant Sa^Wg, and n t is SJjCjMj. If then i, j be any two subindices the coefficient of m t rij in F+G is (Piqj+Pjqi)- And we may write F+G = SSot^ (_p,2/ + M.O; where both * and j are to be given every value from 1 to 4. But, by comparing coefficients in equation A, we have whence f + # = 222«i.w,w li; - + SSwyjj (^ + £«.). Now it is plain that if for every term of the form p^j+pjqi we substitute £■«,- + tju n the result is the same as if in F and G we everywhere altered p and %*> M ») M 24> « 25 C » ". M 8,i w s 2 > M 3S) W S4> a «> C *1 M 3 M 4D M «! M 4S) M 44> °«J C 4, M 4 K, K ^ »« C „ C 2> C 3 > C 4 U„ «,, M„, M. For since this determinant must contain a constituent from each clebsch's calculation of the surface S. 483 of the last three rows and columns it is of the first degree in «,„ &c., and the coefficient of any term u u is -{2a.f a u^b ] c i u t +S,a 1 c i u^bf s u} or - (»!,«< + wy?,). In the determinant just written the matrix of the Hessian is bordered vertically with a, c, u; and horizontally with 5, c, u. As we shall have frequently occasion to use determinants of this kind we shall find it convenient to denote them by an abbreviation, and shall write the result that we have just arrived at, \6, c, uj 3. The quantity FG is transformed in like manner. It is evidently the product of Ki>. + m ,p t ■+ m s p 3 + v 4 ) Ksi + m A + m & + ™a)> and (n iPi + n,p % + n 3 p 3 + n 4 p 4 ) faj, + n A + n s q 3 + » A ). Now if the first line be multiplied out, and for every term ( V\i* +i , a2'i) we substitute its value derived from equation -4, it appears, as before, that the terms including t vanish, and it becomes SSm,^, which, as before, is equivalent to ( ' ' ) , where the notation indicates the determinant formed by border- ing the matrix of the Hessian both vertically and horizontally with a, c, m» The second line is transformed in like manner : and we thus find that (F+ Gf - 3FG (F+ G) = transforms into /a, c, u\ r /a, c, wy _ 3 /a, c, u \b, c, u) \ \i>, c, u) . \«, c, uj \bj c, u, It remains to complete the expansion of this symbolical ex- pression; and to throw it into such a form that we may be able to divide out c,a5, + cje t + c^ + c t x v We shall for short- ness write a, Z>, «, instead of a,*, + a 2 £C 2 + a a x B + a^, \x t + &c, cp i + &c. 4. On inspection of the determinant, p. 482, which we have called W °' J ? it appears that, since u n x, + u lt x t + u ls x, + u,^ = (« - 1) w,, &c, 1 12 b i e i U \^ . 484 clebsch's calculation of the surface 8. this determinant may be reduced by multiplying the first four columns by £c l5 x# x s , x t) and subtracting their sum from the last column multiplied by (« -1), and similarly for the rows; when it becomes M ,.> u n1 M 13> M 14> «,> Cu u ni u >rt M 2 3! M 2 4> «•> c «» 1 ll 3l1 M 32) W S3> M 34) «37 C 3> M 4.J M «) M 43) M 44> 0, C 4l o (n-I)» -Z> c >» c * C S ! C 4> o, o, — c 0, 0, 0, 0, -a, — c ich partially expanded is " {^="1)" I'ffl- ac {b)- J t) 4 a& 0' where denotes the matrix of the Hessian bordered with a single line, vertically of a's and horizontally of Fb. In like manner we have 1 5, C, M (»-in 1 k :)- 2ac + « 2 C»-i)T 26c [ Wtf Now as it will be our first object to get rid of the letter a, we may make these expressions a little more compact by writing cb l — be, = d t) &c, when it is easy to see that = c Thus \b, c, u -KV 0> u -0- {n-iy\d)> fa,c,u\ \b, c, u) '0 ■c 1 (n - l) 2 ( W ~ \d, and the equation of the surface, as given at the end of Art. 3, may be altered into d Mc -2ac + & c - tne corresponding differential coefficient. Then, in the first place, it is evident that a 3 = n (n - 1) (n - 2) u = ; d\ = (n - 1) (« - 2) w u &c. Hence a* Q) = ( M - 1)(« - 2) Q') . But the last determinant is reduced as in many similar cases, by subtracting the first four columns multiplied re- spectively by sb 1? sb s , « 3 , a; 4 from the fifth column, and so causing it to vanish except the last row. Thus we have a' £)=-(«- 2) ifc. Again, f ) is (see Lessons on Higher Algebra, p. 124)=— 2 -5 — a m a „- We have therefore Lastly, it is necessary to calculate a ( ) ( ,) • Now if U mn denote the minor obtained from the matrix of the Hessian by erasing the line and column which contains u mn ; it is easy to see that a{~\ Qj=-(»-2)SCr v C^M„ n o/Z„ where the numbers m i n iVi 1 are eaca to rece i ve in turn all the values 1, 2, 3, 4. But, see Lessons on Signer Algebra^ Art. 28, rdU m UU=TJU„ a -B- J vq mp nq mn P? J*. * mn Substituting this, and remembering that 2 U mn u mn = &H, we have •0GH-»)*Q- Making then these substitutions we have {•©-Q}{'0--0 + '0} 3(3 + 4(.-,)w(3-(.-.)wQ. 486 clebsch's calculation op the sukface S. But attending to the meaning of the symbols d x , &c, we see that d or dp x + d^ + d s x a + d i x i vanishes identically. If then we substitute in the equation which we are reducing the values just obtained it becomes divisible by c s , and is then brought to the form 6. To simplify this further we put for d its value when it becomes Now this is exactly the form reduced in the last article, ex- cept that we have b instead of a, and a in place of d. We can then write down <)H<)v- 2 >m:)-<»- 2 M:)- But (Art. 5 J the last term in both these can be reduced to 12 (»— 2fH s c( J. Subtracting then, the factor c 3 divides out again, and we have the final result cleared of irrelevant factors, expressed in the symbolical form fi\ f. fb\" fb\ (a\ 0*0-© = 0. a J 7. It remains to shew how to express this result in the ordinary notation. " In the first place we may transform it by the identity (see Art. 76, and Lessons on Higher Algebra, Art. 28) whereby the equation becomes clebsch's calculation of the surface 8. 487 Now ( J f J I J expresses the eovariant which we have before called 0. For giving to U mn the same meaning as before, the sym- bolical expression expanded, may be written 2 U mn JJ n U r ,u mr u fl!i1 where each of the suffixes is to receive every value from 1 to 4. But the diiferential coefficient of H with respect to x r can easily be seen to be 2 U u ,., so that is 2 IT, -= ,— , which is, in another notation what we have called 0, p. 436. The eovariant 8 is then reduced to the form © — 4i?l>, where W \a, b) =r)r:)=2u mn u M>Ap u n where TJ pg n denotes a second minor formed by erasing two rows and two columns from the matrix of the Hessian, a form scarcely so convenient for calculation as that in which I had written the equation, Philosophical Transactions, 1860, p. 239. For surfaces of the third degree Clebsch has observed that <£ reduces, as was mentioned before, to 2 UH. where JET de- t i mn Tiini inn notes a second differential coefficient of H. 8. To find the points on a surface where a line can be drawn to meet in five consecutive points, we have to form the condition that the intersection of A U\ A'' U', and an arbitrary plane should satisfy A l U', as well as A*W. M. Clebsch has applied to A 4 ?7' the same symbolical method of elimination which has been here applied to A S V. He has succeeded in dividing out the factor c 6 from this result : but in the final form which he has found, and for which I refer to his memoir, there remain c symbols in the second degree, and the result being of the degree 14n — 30 in the variables, all that can be concluded from it is that through the points which I have called /3, (p. 443) an infinity of surfaces can be drawn of the degree 14w — 30. We can say therefore that the number of such points does not exceed n(lln- 24) (14m — 30). 9. The surface S touches the surface H along a certain curve. Since the equation 8 is of the form - 4.3 = 0, it is sufficient to prove that touches H. But since © is got 488 clebsch's calculation of the surface 8. by bordering the matrix of the Hessian with the differentials of the Hessian, © = is equivalent to the symbolical expression f J = 0. But, by an identical equation already made use of, we have fc, H\ _ (H\ Kc, H) ~ \H) where c is arbitrary. Hence touches H along its inter- section with the surface of the degree In— 15, ( 1. It is proved then that 8 touches H, and that through the curve of contact an infinity of surfaces can pass of the degree In — 15. We have made use, p. 450, of the theorem that the curves US and UH touch each other. " '•--'''H:)-(f)' ( 489 ) APPENDIX IV. ON THE ORDER OF SYSTEMS OF EQUATIONS. 1. We have showed, p. 271, how to determine the cha- racteristics of a curve given as the intersection of two surfaces ; but it has been remarked (p. 247) that there are many curves which cannot be so represented. There is no algebraic curve, however, which may not be represented by means of the equations of a system of surfaces ; because (Art. 325) by taking m large enough we can always find a number of surfaces of the m tb degree each of which shall entirely contain the curve. But any two surfaces of the system will not define the curve, for their intersection will in general consist of the curve in question and an extraneous curve besides ; so that the curve is usually not the complete intersection of any two, but only that part of the intersection which is common to all the rest. The object of this appendix is to show how, when a system of equations is given denoting surfaces which pass through a common curve, the characteristics of that curve can be de- termined. In like manner if we are given r points in space, we can always, by taking m large enough, determine a number of surfaces of the m™ degree which shall pass through the given points. But ordinarily the intersection of three such surfaces will consist of the given points and extraneous points besides ; and we cannot define the given points except by a system of more than three equations, the given points being the only ones which satisfy all the equations. Conversely, it is the object of this appendix, when such a system of equations is given, to ascertain the number of points which satisfy all. 490 ON THE OEDEE OP SYSTEMS OF EQUATIONS. 2. The simplest illustration of this is to take four planes a -f Xoc, b + X/3, c + Xy, d + \S ; where a, a, &c. represent planes, and X is an indeterminate coefficient; then if we form the condition that these four planes should meet in a point, this condition is known to be of the fourth degree in X. It follows that four values of X can be found for which these equations will represent planes meeting in a point. And obviously the four points so found must satisfy any of the six equations (such as a/3 = ba), which are got by eliminating X between any pair of the given equations. Yet these all represent surfaces of the second degree, any three of which intersect in eight points. It follows then that the system of equations z, b, c, d = 0, a, 0, y, 8, denotes a system of surfaces having four points in common; but that any three surfaces of the system intersect not only in these four points but in four extraneous points. In general then, suppose we are given »* + 3 equations involving r para- meters, it is evident that by elimination of the variables we get a sufficient number of equations to determine systems of values of the parameters for which the equations will denote surfaces having a point in common. It is evident also that such points must satisfy the equations got by eliminating the parameters between any r + 1 of the given equations. And yet any three of these latter equations will denote surfaces intersecting not only in these points common to all but in certain extraneous points besides. 3. In like manner if we had been given the three planes a + Xa, b + X/3, c + Xy, it is obvious that we may give to X an infinity of values, to every one of which corresponds a point which is the intersection of the three corresponding planes. It is obvious also that the locus of all these points must be a curve common to all the surfaces a/3 — ba, by- c/3, ca. — ay. But it was proved, p. 263, that though any two of these surfaces intersect in a curve of the fourth degree, there is only a cubic common to all three. And in general if we are ON THE OKDEE OP SYSTEMS OP EQUATIONS. 491 given r ■+ 2 equations, involving r parameters, an infinity of systems of values of these parameters can be determined for which the equations will denote surfaces having a point in common. The locus of these points will be a curve, which will be common to all the surfaces got by eliminating the parameters between any r+1 of the equations. Yet any two such surfaces will intersect not only in this curve but in an extraneous curve. Let us suppose then that we have r + 1 equations, involving r parameters in the first degree. The elimination of these gives rise to a system of determinants a, b, c, a, V , c', = 0, where the number of horizontal rows is supposed to be r, and vertical r + 1. We propose to determine the characteristics of the curve which is common to the surfaces represented by all these determinants.* 4. Let us commence with the simplest case a, b, c a, b , c We suppose the functions a, b, &c. to be of any degree, but we suppose the degrees df the corresponding functions in either the same row or the same column to be equi-different ; so that for example ab' and a'b will be of the same degree. Let X, fi, v ; X', ft', v' be the degrees of these functions, and such that X H- /*' = X' + p, &c. ; then the two determinants of the system ab' — a'b, ac — a'c, representing surfaces of the degrees re- spectively X + fi', X' + v, intersect in a curve whose degree is (X + fi) (X' + v). But these surfaces evidently have common the curve of intersection of a, a' which does not lie on the surface represented by the third determinant of the system, be — b'c. The order then of the curve common to all three is (X + /*') (X' + v) - XX' = Xv + X>' + fi'v. * The first paper treating of the class of problems considered in this appendix was by Mr. Cayley (Cambridge and Dublin Mathematical Journal, iv. 134) ; where he shows that when a, b, a', V, &c. are of the first degree in the variables eliminated, the order of the system is %r (r + 1). I extended his method and applied it to the other problems here treated of, Quarterly Journal, I. 246. 492 ON THE OJRDER OF SYSTEMS OF EQUATIONS. Take next the system with three rows and four columns, b, b', b'\ V" Let us write at full length the determinants obtained by the omission of the third and fourth columns, viz. a (be — b'c) + b" (ca — c'a) + c" (ab' — a'b) ; a" (be - b'c) + b'" (ca - c'a) + c" (ab' - a'b). Then these obviously represent surfaces of the degrees X + /j,' + v", X + ij! + v"\ which have common the curve common to the three surfaces be — b'c, ca — ca, ab' — a'b, whose order has been just determined. And since this curve does not lie on the surfaces represented by the other determinants of the system under consideration, the curve common to all four is of the order (X + //.' -f v") [X + // + v") — (Xv + X'/j,' ■+ fi'v). Having thus determined the order of a system with three rows and four columns, we can, in like manner, thence derive the order of a system with four rows and five columns. Pro- ceeding thus step by step I arrive by induction at a general formula which I establish by shewing that if it is true for a system with h rows, it is true for a system with k+ 1. 5. The formula at which I arrive may be written most simply and generally as follows : Let the orders of the several functions be represented by the letters which denote them, a + a, b + a, c + a, «?+ a, &c. a + /3, b + /3, c + /3, d+/3, &c. a + 7, b + y, c -+ 7, d + 7, &c. &c let p denote the sum of the quantities a, /3, 7, &c, and q denote the sum of their products in pairs ; and let P and Q denote the corresponding sums for the quantities a, b, c, &c. ; then I say that the order of the system is Q+pP+p' — q. For if we consider the two determinants obtained by omitting the first and second columns alternately, these will be of the orders^ + P—a, p + P—b, and the surfaces have common the curve represented ON THE ORDER OF SYSTEMS OF EQUATIONS. 493 by the system obtained by omitting the first two columns. I am about to prove, that if the formula just given holds for that lower system, it holds for the given system. Since, however, in the lower system the number of rows is greater than the number of columns, we must, in order to apply the formula, write the rows as columns and the columns as rows; and thus the new P and Q will be the old p and q. The new p will evidently be P-a — b; and the new q which is the sum of the products in pairs of c, d, &c. can easily be seen to be Q - {a + b) P4 a' + ab + b\ Hence by the formula, the order of the lower system is P*-Q-P{a + b)->rab+p{P-a-b) + q. Subtracting this number from the product [p + P— a) {p + P— b) we obtain, as we ought, Q+p/P+p* — q. Since then the formula we have given is true, as can easily be verified, for the case of two rows and three columns, it is true generally. In applying it we generally have either a or a — 0. When all the rows are of the same order we have a, /3, &c. all = 0, and therefore p, q both = 0, and the order of the system is Q. 6. Next let it be required to find the order of the develop- able generated ^by the curve considered in the last article. As before, commencing with the simplest case, I was led by induction to a general formula, which I verify by showing that it is true for a system with h rows, if true for a system with k—l. Let B be the sum of the products in threes of the quantities a, b, &c. (Art. 5) ; and r the corresponding sum for the quantities a, /3, 7, &c. ; then I say that the order of the developable in question is p=(p*-q+pP+Q){P+2p-2)-q(p+P) + B + r. It was proved (p. 274) that the ranks of two systems which together make up the intersection of two surfaces, are connected by the relation p — p'=(m — m')(p, + v-2). Now we have just seen that the intersection of two surfaces whose degrees are p,=p + P-a, v=p + P-b, is made up of 494 ON THE ORDER OF SYSTEMS OF EQUATIONS. the curve whose degree is m —p* — q +pP+ Q, and of the curve of lower order whose degree is m'=P' i — Q+pP+q—p'(p+P)+q\ where we have written p = a + b, q' = ab. Now to find by the formula of this article, assumed true, the rank of the lower system, we are to write for the new B, the old r ; and for the new r, R-p'Q + [p n - q) P- (p'" - Zp'q). We have then, after a little reduction, p'=(P*+pP+q)(2P+p-'Z)-Q(ZP+'2p-2)+R+r-p'{P+p){ZP+p-2) + 2p' ( Q - q) + (P+p) p" + 2q' (P+p - 1) -p'q. To this value of p add (m — m'){/j, + v-2); that is to say, {f - P° -t- 2 (Q - q) +p' {P+p) - q'} (2p + 2P- 2 - p'), and the result is the value already given for p. As then the formula can easily be verified for the case of two rows, it is generally true. 7. Let us next consider a system such as a, b, c, d, e, / b\ d\ e, f tt 7 tf n 7tr ii pit a , b . , c , d , e , / til 7 /// lit Till III Pill a , b , c , d , e ,f where the number of columns exceeds the number of rows by two, and let us examine how many points are common to all the surfaces represented by the determinants of the system. Now any three surfaces (ad'e'f"'), (bd'e'f'"), (cd'e"/ 1 ") have common the curve d, d', d", d'" and if «i, n, p be the degrees of the surfaces, (i and p the degree and rank of the curve, then (see p. 283) the surfaces will intersect in points not on this curve, in number mnp — /J.(m + n + p — 2) + p. Now if we represent the orders of the several functions in the same way as in Art. 5, it is easy to see that the degrees of the three surfaces are P+p —b—c, P+p — c — a, P+p — a—b; ON THE ORDER OF SYSTEMS OF EQUATIONS. 495 so that if we -write a + b + c = p', hc + ca + ah = q, abc = r, we are to substitute for mnp, (P+pf - 2p* (P+p)* + (P+p) (p'> + J) - (p'q' - r') ; and for m + n + p - 2, 3P+ 3p - 2 - 2p '. The order and rank of the curve, fi and p, are found from the formulae of Arts. 5 and 6, by writing for P, Q and R ; p, q and r ; and for p, q, r respectively, P-F, Q ~ p'P+p'* - q, B-p'Q + [p- - q') P- ( F ' s - 2p'q' + r'). We find thus, as in Arts. 5, 6, p = P*-Q +pP+ q -p' (P+_p) + q', p=(P*+pP+q)(2P+p-2)-Q(3P+2p-2)+R+r-p'(P-+p)(3P-±p-2) + ¥ ( Q - 2) + (P+P) f + 2-Z ' {P+p - 1) -p'i - r. Add then to the value of p just found, the values given for rnnp — /j,[m + n +p — 2), and we find the required result, viz. R + pQ+[p !! -q)P+p 3 -2pq + r. If the several rows are of the same degrees, that is if a, /3, &c. all = 0, then the number of points represented by the system is R. The correspondence of this result with that of Art. 5, may be made more manifest by writing the symmetric function a' + a/3 + /3 2 + &c, p z , and a 3 + a 2 /3 + a/3 2 + /3 3 + &c.,p 3 ; then the result of Art. 5 is Q+pP+p 2 , and of this article R-\-p Q+p^P+p s ; and we are led to expect that the order of a system where the columns exceed the rows by three, will be found to be S+pR J rp i Q J rp !i P+p i , a result however which I have not taken the trouble to verify. 8. It may be deduced hence that the surface represented by any symmetrical determinant has a determinate number of double points. Let the sum, sum of products in pairs, and sum of products in threes of the degrees of the leading terms a , a as , a 33 , &c. be P, Q, R, then the number of such double points is \ (PQ — R). We have the identical equation (Lessons on Higher Algebra, (Ex. 1, p. 26) A u A m - [A^Y = CA, where A n means the minor obtained by erasing from the given determinant the 496 ON THE ORDER OF SYSTEMS OP EQUATIONS. line and column containing a 1)5 A is the determinant itself, and G is the second minor obtained by erasing the two lines and columns which contain a, , a 22 . Now it is evident that the surface represented by A n A 22 — (A^f has as double points the intersections of A n , A^ A u ; and the degrees of these being respectively P—a, P-b, P—^(a + b), the number of double points is the product of these three numbers. Let the sum, and sum of product of pairs, of the terms exclusive of a and b, be denoted byy, g", then the product (P-a)(P-b){P-i(a + b)} is \ {PQ +p" Q + {p'° - q") P +1 r -fq"}. These are then double points on the complex system OA; and are therefore either double points on C, double points on A, or points of intersection of C and A. Now if we erase from the matrix the first two rows, all the determinants of the remaining system (of which C is one) have common a number of points, which can be calculated by the formula of the last article, by writing %(c + a), %(c + b), &c, |-(c? + a), \{d+ 5), &c, for the degrees of the rows. The result is J (B +p" Q + {f* - q") P + (p"° - 2p"q" + r"). But the points whose number has been just found are points at which A uJ A 22 , A n touch, and they each count for four among the intersections of these surfaces. Subtracting then four times the number just found from the total number of intersections, we get x {p Y-r" + PQ-B), whence we learn that if the number of double points on the surface represented by the symmetrical determinant G is \{pq— r"), that of those on the surface A is \{PQ — B), and the first theorem being established in the simplest case the other is generally true. 9. There is still another question which may be proposed concerning the curves, Art. 4. Let there be four surfaces whose degrees are X„ X 2 , X 3 , X 4 , and whose coefficients contain any new variable in the degrees /i„ /tt 2 , p 3} fi^ then the elimi- ON THE ORDER OF SYSTEMS OP EQUATIONS. 497 nant of these four equations contains the new variable in the degree W (a*,\ + /*A) + W (/*A + a*A)- Now W, \ 3 \ are the orders of the curve of intersection of the first and second, and third and fourth surfaces re- spectively ; and if we call /&,\ 2 + /* a \, fi s \ + fj, t \ the weights of the same curves, we can assert that the weight of the con- dition that two curves may intersect is the sum of the products of the weight of each curve by the order of the other. Now we have seen what is the order of the curve denoted by a system of determinants, such as Art. 4 ; it remains to enquire what is the weight of the same system. It is easy to see that when a curve breaks up into two simpler curves the weight of the complex curve is equal to the sum of the weights of its components. We may therefore proceed as in Art. 4, and the following is the result : Let the functions contain the new variable in the degrees a + a, V + a', c ■+ a, &c., a + B', b' + B', &c, then the formula for the weight is derived from that for the order by performing on the latter the operation* a -7- + V -jt 4 &c. + a' -=- + B' -77s + &c. This applies to the da do da dp rr results both of Art. 5 and Art. 7. Thus the former becomes S {ah') +pF +p'P+ 22 (aa) + 2 (a/3'), which may also be written [P+p) (P'f /) + 2 (aa') - 2 {aa'). 10. We propose next to investigate the order and weight of the system of conditions that the two equations at + If- 1 + cf - 2 + &c. = 0, a'f + h'f- 1 + c't n - 2 ■+ &c. = 0, may have two common roots. It is evident that in order that this should be the case, two conditions must be fulfilled; and if t be a parameter, and a, S, &c. functions of the co-ordinates, these conditions will represent a curve in space. But in point of fact, we obtain not two, but a system of conditions, no two of which suffice to define the given curve. These conditions * Mr. Cayley has given this simple form of stating my result. KK 498 ON THE OEDEE OF SYSTEMS OP EQUATIONS. are [Lessons on Higher Algebra, Art. 33) the determinants of the system a, h, c, .. where the first line is repeated n — 1 times, and the second m — 1 times ; there are m + n -2 rows, and m + n — 1 columns. The problem is then a particular case of that considered, Art. 4. We suppose the degrees of the functions introduced to be equi-different : that is to say, if the degrees of a, a be X, ytt, we suppose those of b, b' to be X + a, /j, + a ; of c, c to be X + 2a, /u. + 2a, &c. We may write the formula of Art. 5 in the some- what more convenient form Q+pP+ | (p 2 + s 2 ), where s 2 is the sum of the squares of a, /3, &c. To apply it to the present case we may take for the quantities a, b, c, &c. 0, a, 2a, &c. ; and for the quantities a, /?, 7, &c. of Art. 5, X, X — a, X — 2a, &c. P is then the sum of m + n — 2 terms of the series a, 2a, 3a, &c, and is therefore, if we write m + , (&-!)( &- 2) 1.2 a. In the same case Q is the sum of products in pairs of these quantities, and is therefore = (&-3)(fe-2)(ft-l)(3&-4) , 1.2.3.4 * ' Again p is the sum of n — 1 terms of the series X, X — a, X — 2a, &c, and of in — 1 terms of the series fi, fx, — a, /a — 2a, &c. We have then j> = ( n _ l) X + (m - 1) /a - \a. \{n - 1) [n - 2) + (m - 1) (m - 2)}. In like manner s 2 is the sum of the squares of the same quan- tities, and is («-l)X 2 -(- (ra-l) / u, 2 -Xa(«-l)(n f(« -1)(«- 2) (2n- 3) + «' 1.2.3 + 2) - /ia.[m—l) [m- 2) [m - 1) [m - 2) (2m - 3) 1.2.3 ON THE ORDER OF SYSTEMS OP EQUATIONS. 499 Collecting all the terms, the order of the required system is found to be hn{n-l)X%^ m {rn-l) l ju i +[m-\){n-\)'K l jL+\n{n-l){2m-l)\a. 4 ^m(m— 1) (2n — l)fia. + lmn (m - 1) [n — 1) a". If the eliminant of the equations at m + bf'- 1 + &c. = 0, df + &c. = 0, represent a surface, the curve here considered is a double curve on that surface. If all the functions a, &, &c. are of the first degree, the surface generated is a ruled surface ; and writing X — /jl = 1 and a = in the preceding formula, we find that the order of the double curve is $[m + n— 1) [m + n — 2). If the two equations considered are of the same degree, that is to say, if m = «, we may write X + fA=p, X/j. = £, and the same formula gives for the degree of the double curve \n [n — 1) (p + no) {p + (n - 1) a} — (n — 1) q. If the order in which the un-eliminated variables occur in any term be denoted by the accented letters corresponding to those which express their degrees in the variables to be eliminated, then (Art. 9) the formula for the weight of the system is obtained from that for the order, by performing on it the operation V -5T- + u -r- + a! -r- . In other words, the weight is aX d/j, da. n(n-l) XX' + m(m-l) fifi' 4 (m~l)(n-l) (X/t' + X» + \n [n - 1) (2»z - 1) (Xa! + X'a) + \m (m - 1) (2re - 1) (/ml' + //a) + mn (m - 1) [n — 1) aa'. 11. We can in like manner determine the order of the system of conditions that the equations at m + &c, at + &c. may have three common roots. When geometrically inter- preted these conditions represent triple points on the surface represented by the eliminant of the two equations. The con- ditions are represented by a system of determinants, the matrix for which is formed as in the last article, save that the line o, b, c is repeated n - 2 times, and the line a, b', c, m — 2 times ; and the matrix consists of m + n — 2 columns and K K2 500 ON THE ORDER OP SYSTEMS OF EQUATIONS. to + n — 4 rows. The order of the system is calculated from Art. 7, and is found to be n (n — l)(»i— 2)^_ mini— l)(ra-2) , , , ,., .. . .... K 1-2 _3 -*•' + V ^11 V+i(«-l)(»-2)(«-2)XV + |(to-1) (m -2) (m-2) V 2 + £(»i-1)»i(m-1) (»i-2)X, 2 a + -^(«-l)w(w2— l)(»i-2) / u, 2 a+|(»n-2)(«-2){w(n-l) + w(»2— ^JX/ita + {£w [n - 1) (w - 2) to (to - 2) + -J-n [n - 1) (n - 2)} a 2 \ + {-|to (to - 1) (to - 2) n [n - 2) + -J n (to - 1) (to - 2)} a 8 /* + \m (to - 1) (to - 2) n (n - 1) (n - 2) a 3 . In the case where the surface is a ruled surface, putting a = 0, \ = /jl = 1, we get for the number of triple points (to -f n — 2) (to + n — 3) (to + w — 4) UU The order of the developable generated by the double curve (Art. 10) is calculated in like manner by the formula of Art. 6, but the number so found must be reduced by four times the number of triple points just found, which are also triple points on that curve. Thus in the case of the ruled surface the rank of the double curve is 2 (m + n — 2) (m + n — 3). The weight of the system, found by the same process as before, is |K(n-l)(ii- 2) \*\' + \m (to - 1) (to - 2) /*>' + (w-l)(n-2)(TO-2)(X,^'+i^V)+(»»-l)(w-2)(w-2)(X^ / u,'+^ i! \') ■+ (to - 1) n (n - 1) (n - 2) (Xk'a. -f JA.V) -I- (n — 1) to (to — 1) (to — 2) (fifia + $fi?a.') + 2 ( m ~ 2) ( n ~ 2) {2tow — to — n) (X/i'a + \'/j,a + \/j,a!) + {\n (n - 1) (n - 2) to (to - 2) + £« (« - 1) (n - 2)} (a 2 X/ 4 2aa'X) + {£m (to - 1) (to - 2) m (m - 2) + \m (to - 1) (to - 2)} (a 8 /*' + 2aa» + Ito (to — 1) (to — 2) n [n — 1) (« — 2) a*a'. 12. The next system we discuss is that formed by the system of conditions that the three equations at' + It 1 + &c. = 0, a't m + b't"- 1 + &c. = 0, a"f + h"f* + &c. = may have a common factor. The system may be expressed ON THE ORDEE OF SYSTEMS OF EQUATIONS. 501 by the three equations obtained by eliminating t in turn be- tween every pair of these equations, a system equivalent to two conditions. The order of the system may be found by eliminating from the equations, x, y which enter implicitly into a, 6, c, when the order of the resulting equation in t determines the order of the system. Let us suppose that a, a, a 1 ' are homogeneous functions in x ) Vi z i °f the degrees X, /a, v respectively, that b, b\ b" are of the degrees X — 1, /* — 1, v — 1, &c. ; then if we take the reciprocal of t for the linear unit, the equations denote surfaces of the X 1 , fi" 1 ) v tb degrees on each of which the point xyz is a multiple point of the orders X — I, fi — m, v - re, respectively. The number of their other points of intersection is therefore X/jlv — (X — I) [fi — m) [v — «), or lfA.v + mvX + nXfjj — Xmn — (inl— vim + linn. But this is evidently the same as the order of the equation obtained by eliminating x, y, between the equations. If the order of 5, &', c, c', &c. had been X + a, /jl + a, X + 2a, ft + 2a, &c. ; then the order of the system would have been lfx,v + mvX + nX/j, + ) + Q. This result may be stated as follows, in a way which leads us at once to foresee the answer to some other questions that may be proposed as to the order of systems of these equations. In the case we are considering, the entire number of columns, counting the relation columns, is one more than the number of rows; and the order of the system is that given by the rule of Art. 5, if we give a negative sign to the orders in the relation columns. In like manner, when the number of columns, counting the relation columns, is equal to the number of rows, the system by Mr. Cayley's theorem, represents a determinant whose order is that which we should obtain by calculating the order of the entire system considered as a determinant, the orders in the relation columns being taken negatively. And so no doubt if the entire number of columns exceeded the number of rows by two, the order of the system would be found by the same modification from the rule of Art. 7. ON THE OKDER OF SYSTEMS OF EQUATIONS, 505 1 7. Let us now apply the rule just arrived at to the problem proposed in Art. 14. We suppose that in the equations of the three curves the coefficients of the highest powers of x, viz. a; ! , x m 7 x n are of the orders X, fi, v ; those of x'' 1 ^, « M a are of the orders X + a, X + a', and so on, the orders of the coefficients increasing by a for every power of y, and by a! for every power of s. Then the terms in the first column consist first of J (m + n — 1) {m + n — 2) terms whose orders are X ; X - a, X — a' ; X — 2a, X — a — a', X - 2a', &c. ; secondly, of \{n\l — 1) {n + l — 2) terms whose orders are /x, /u. — a, /t — a', &c, and thirdly of £(Z+m — 1) (l + m- 2) similar terms in v. These may be taken for the numbers a, /3, 7, &c. of Art. 5. The numhjers a, h, c, &c. of that article are 0, a, a' : 2a, a + a', 2a', &c, there being in all |(Z + m + n - 1) (Z + m + n — 2) such terms. Lastly, the numbers A, B, G, &c. of the last article are found to consist of \{l — 1) (I— 2) terms, /t+v, /i+v -a, /u+v— a'; together with |(w — 1) (m — 2) and ^ (n — 1) (n— 2) correspond- ing terms in v + X and X + /*. In calculating I have found it convenient to throw the formula of the last article into the shape where s 2 denotes the sum of the squares of the terms a, b, c, &c. Also if 4> (X) = AT + M* + Cf + DI + E, it is convenient to take notice that (l+m+n)+(T)+(p(m)^(j)(n)-^(l^m)-(n+T} = YlAlmn (I + m + n) + &BlX 4- {(nZ-1) (Hm-l)-i(n-l)(n-2)}X/* + mrih [Imn - Z + 1 - \ (m + n)} (a. + a) + nl/J. [Imn - m + 1 - | (« + Z)} (a + a') + Zmv {?»«« - « + 1 - § (Z + m) (a + a') + £Zmn(Z»m-Z-m-w+2)(a'' ! +0 + £Zw2n(2Z?rtn-Z-m-ft+l)aa'. 506 ON THE OEDEE OP SYSTEMS OF EQUATIONS. If the order of all the terms in the first equation be X, in the second /i, in the third v, we have only to make a and a! = in the preceding formula. In this case supposing X = fi = v ■= 1, and l = m = n, the order becomes \n [n — 1) (ri 2 + n — 1). 18. The preceding formulae enable us to determine the order (B) in the coefficients that a curve should have two double points, and the order ( G) that it should have a cusp. In either of these cases, the three polar curves Z7„ U a U s have two common points, either distinct or coincident. Writing then n — 1 for n in the last formulas, we have 2(5+(7) = 9(ji-1) (n-2) (V-m-1). But either from a formula, given note p. 179, or from p. 417, we can infer 2B+ 3 C= 3 (n - l) 2 {3 (n - If - 1} - Sn [n - 1) = 3 (»- 1) (« - 2) (3rc 2 - 3« + 1). Hence we have (7= 12 (n - 1) (w - 2), B= f (n - 1) (n - 2) (Sw 2 - 3m - 1). Mr. Cayley had arrived at these numbers by a different process in a Memoir communicated to the Cambridge Philosophical Society, but not yet published. ( 507 )« APPENDIX V. ON THE PROPERTIES OF SYSTEMS OP SURFACES. 1. In this appendix we shall give a brief account of methods by which M. Chasles has investigated properties of systems of plane curves (Comptes Rendus, 1864, t. lviii.), and which have been applied to surfaces by M. De Jonquieres. It is simpler to illustrate the method, by first giving as specimens a few of M. Chasles's theorems for plane curves. Let us suppose that we are given one less than the number of conditions necessary to determine a curve of the n" order, then the curves satisfying these conditions form the system whose properties have been studied by M. Chasles. The curve would be completely determined, if in addition a point on the curve or if one of its tangents were given. Let the number of solutions of the problem in these two cases be /* and v ; that is to say, let it be supposed that /j, curves of the system pass through any given point, and that v of them touch any given line: then these two numbers are called the characteristics of the system. (1) The locus of the poles of a fixed line, with respect to the curves of the system, is a curve of the degree v. For every point of the locus which lies on the line itself is the point of contact of a curve of the system touching the line. There can, by hypothesis, only be v such points. Since then the locus is one which meets a given line in v points, it is a curve of the V th order. As a particular case of this theorem, we can deter- mine the order of the locus of centres of conies satisfying any four conditions. (2) Reciprocally, the envelope of the polars of a fixed point, with respect to the curves of the system, is a curve whose class is p. 508 ON THE PROPERTIES OF SYSTEMS OF SURFACES. (3) The locus of the points of contact of tangents drawn from a fixed "point to all the curves of the system, is a curve of the degree /j. + v. Consider any line drawn through the fixed point ; its v points of contact with curves of the system are points on the locus ; and the fixed point itself is a multiple point, being a point on the locus for each of the ft curves passing through the point. No other points of the locus can lie on the line ; and the locus is therefore a curve of the degree ft+v. 2. If there be on a right line two series of mutually corte- sponding points ; such, that to any point of the first system corre- spond m points of the second, and that to any point of the second correspond n points of the first, then there will be m + n points of either system which coincide with points corresponding to them in the other. This theorem is proved in the case where m = n = 1 , [Conies, Arts. 331, 340); and the proof in general is the same as that given in the place referred to. Let. x, x be the dis- tances measured from any fixed origin of two mutually corre- sponding points, then since if x is given there may be m values of x, and if x is given there may be n values of x, these distances must be connected by a relation of the m th degree in x' and of the n" 1 in x, as for instance (ax' m + bx""' 1 ■+ &c.) x" + &c. = 0. If then we put x = x', we obtain an equation of the degree m + n to determine the a: of a point which coincides with its conjugate. The following example shows the use that may be made of this principle. (4) The locus of a point whose polar with regard to a fixed curve of the m degree, coincides with its polar with respect to a curve of the given system, is a curve of the degree ft [m — 1) + v. Take any right line, let A and a be two points on it, such that the polar of A with respect to the fixed curve is the same as the polar of a with regard to one of the curves of the system ; then the problem is to determine in how many cases A can coincide with a. Now first if A be given, its polar with respect to the fixed curve is also given ; the locus of the poles of this line with respect to curves of the system is a curve, which, by theorem (1), meets the assumed line in v points. There are therefore v positions of a corresponding to any position of A. ON THE PROPERTIES OP SYSTEMS OF SURFACES. 509 On the other hand if a be given, its'polars with respect to curves of the system, by theorem (2), envelope a curve of the //" class. But the polars of the points on a right line, with respect to a curve of the m tb degree, envelope a curve of the [m — I)'" class. The two curves then may have p (m - 1) common tangents ; and these may accordingly be fi (m — 1) points A corresponding to any point a. There will therefore be on any right line A 4 [m — 1) + v points A coinciding with a ; and this will there- fore be the degree of the locus of such coincident points. The points where this locus meets the fixed curve, will be points of contact of that curve with curves of the system. The number therefore of curves of the system which can touch a fixed curve of degree m is fim (m - 1) + vm ; or more generally /in -f vm, where n is the class of the curve. 3. Now, in like manner, let it be supposed that we are given one less than the number of conditions necessary to determine a surface of the n ,b order ; the surfaces satisfying these conditions form a system whose characteristics are /i, v, p ; where fi is the number of surfaces of the system which can be drawn through any point, v is the number which can touch any plane, and p the number which can touch any line. It is obvious that the sections of the system of surfaces by any plane form a system of curves whose characteristics are /i, p ; and the tangent cones drawn from any point form a system whose characteristics are p, v. Several of the following theorems answer to theorems already proved for curves. (1) The locus of the poles of a fixed plane with regard to surfaces of the system, is a curve of double curvature of the order v. The locus is a curve, since the plane itself can only be met by the locus in a finite number of points, v. Taking the plane at infinity, we find as a particular case of the above, the locus of the centre of a quadric satisfying eight conditions. Thus when eight points are given, the locus is a curve of the third order : when eight planes, it is a right line. (2) The envelope of the polar planes of a fixed point, with regard to all the surfaces of the system, is a developable of the class /i. 510 ON THE PROPERTIES OF SYSTEMS OF SURFACES. (3) The locus of the poles with regard to surfaces of the system, of all the planes which pass through a fixed right line, is a surface of the degree p. There are evidently p, and only p points of the locus, which lie on the assumed line. The theorem may other- wise be stated thus ; understanding by the polar curve of a line with respect to a surface, the curve common to the first polars of all the points of the line ; then, the polar curves of a fixed line with regard to all the surfaces of the system lie on a surface of the degree p. (4) Reciprocally, The polar planes of all the points of a line, with respect to surfaces of the system, envelope a surface of the class p. (5) The locus of the points of contact of lines drawn from a fixed point to surfaces of the system, is a surface of the order fi + p, having the fixed point as a multiple point of order p. This is proved as for curves. The problem may otherwise be stated, " To find the locus of a point such that the tangent plane at that point to one of the surfaces of the system which pass through it, shall pass through a fixed point." Hence we may infer the locus of points where a given plane is cut orthogonally by surfaces of the system. It is the curve in which the plane is cut by the locus surface p + p, answering to the point at infinity on a perpendicular to the given plane. (6) The locus of points of contact, with surfaces of the system, of planes passing through a fixed line, is a curve of the order v + p meeting the fixed line in p points. This also may be stated as the locus of points, the tangent planes at which to surfaces of the system passing through it, contain a given line. (7) The locus of a point such that its polar plane with regard to a given surface of degree ra, and the tangent plane at that point to one of the surfaces of the system passing through it intersect in a line which meets a fixed right line is a surface of the degree tn/i + p. The locus evidently meets the fixed line in the p points where it touches the surfaces of the system, and in the m points where it meets the fixed surface, these last being multiple points on the locus of the order p. (8) If in the preceding case the line of intersection is to lie in a ON THE PROPERTIES OP SYSTEMS OP SURFACES. 511 given plane, the locus will be a curve of the order m (m—1 ) fi+mp+v. The v points where the fixed plane is touched by surfaces of the system are points on the locus ; and also the points where the section of the fixed surface by the fixed plane is touched by the sections of the surfaces of the system. But (Art. 2) the number of these last points is fim (m — 1) + mp. The locus just considered meets the fixed surface in m {m (m - 1) fi + mp -f v} points. But it is plain that these must either be the fim (m — 1) + mp points just mentioned ; or else points where surfaces of the system touch the fixed surface. Subtracting then from the total number, the number just written, we find that — (9) The number of surfaces of the system which touch a fixed surface is fim [m — l) 2 + pin (m — 1 ) + vm ; or more generally if n be the clas3 of the surface, and r the order of the tangent cone from any point, the number is /in + rp + vm. We can hence determine the number of surfaces of the form \U+/iV which can touch a given surface. For if U and V are of the degree m, these surfaces form a system for which /t=l, v = 3(m— l) z , p = 2(m— 1). If then n be the degree of the touched surface, the value is n[n- If + 2n [n - 1) (m - 1) + 3« [m - If, the same value as that given, p. 439. This conclusion may otherwise be arrived at by the following process. 4. If there be two systems of mutually corresponding points in a plane, such that to any point of the first system correspond m of the second, that to any point of the second correspond n of the first, and that any right line contains r pairs of corresponding points ; then the number of points of either system which coincide with points corresponding to them is m-\-n + r. Let us suppose that the co- ordinates of two corresponding points xy, x'y, are connected by a relation of the degrees /i, /*' in xy, x'y' respectively ; and by another relation of the degrees v, v ; then if x'y be given, there are evidently fiv values of xy, hence n = pv. In like manner m = /i'v. If we eliminate x, y, between the two equa- tions, and an arbitrary equation ax + by + c = 0, we obtain a result 512 ON THE PROPERTIES OF SYSTEMS OF SURFACES. of the degree fiv + fi'v in x'y ; showing that if one point describe a right line, the other will describe a curve of the degree /iv + ytt'v, which will of course intersect the right line in the same number of points, hence r — p.v + /jfv. But if we suppose x and y' respectively equal to x and y, we have (/«, + //) (v + v) values of x and y ; a number obviously equal to tn + n + r. 5. Let us proceed now, as in theorem (4) for curves, to inves- tigate the nature of the locus of points, whose polar planes with respect to surfaces of the system coincide with their polars with respect to a fixed surface ; and let us examine how many points of this locus can lie in an assumed plane. Let there be two points A and a in the plane, such that the polar plane of A with respect to the fixed surface coincides with the polar plane of a with respect to surfaces of the system. Now first if A be given, its polar plane with regard to the fixed surface is given : and the poles of that plane with respect to surfaces of the system lie, by theorem (1), on a curve of the order v. This curve will meet the assumed plane in the points a which corre- spond to A y whose number therefore is v. On the other hand if a be given, its polar planes with respect to surfaces of the system envelope, by theorem (2), a developable whose class is p,; but the polar planes of the points of the given plane with regard to the fixed surface envelope a surface whose class is (to — 1)'' ;* this surface and the developable have common p, (m — l) z tangent planes, which will be the number of points A corresponding to a. Lastly, let A describe a right line, then its polar planes with respect to the fixed surface envelope a developable of the class to — 1; but with respect to the fixed surface, by theorem (3), envelope a surface of the class p. There may therefore be p (m—1) planes whose poles on either hypothesis lie on the assumed line. Hence, by Art. 4, the number of points A, which coincide with points a is p, (m — l) 2 + p (to — 1) + v. The locus then of points whose polar planes with respect to the system, and with respect to a fixed surface coincide, will be a curve of * It was mentioned (p. 417) that if the equation of a plane contain two parameters in the degree n, its envelope will be of the class n 2 . ON THE PROPERTIES OF SYSTEMS OF SURFACES. 513 the degree just written, and which will meet the fixed surface m the points where it can be touched by surfaces of the system. 6. We add a few more theorems given by De Jonquieres. (10) The locus of a point such that the line joining it to a fixed point, and the tangent plane at it to one of the surfaces of the system which pass through it, meet the plane of a juiced curve in a point and line which are pole and polar with respect to that curve, is a curve of the degree ftm [m — 1) . + pm + v. This is proved as theorem (8). Let the fixed curve be the imaginary circle at infinity ; and the theorem becomes, the locus of the feet of the normals drawn from a fixed point to the surfaces of the system is a curve of the degree 2p + 2p + v. (11) If there be a system of curves, whose characteristics are fij, v, the locus of a point such that its polar with regard to a fixed curve of degree m, and the tangent at it to one of the curves of the system which pas3 through it, cut a given finite line harmonically, is a curve whose degree is m/j, + v. Consider in how many points the given line meets the locus, and evidently its v points of contact with curves of the system are points on the locus. But, reasoning as in Art. 3, we find that there will be m points on the line, whose polars with respect to the fixed curve divide the given line harmonically. And since these are points on the locus for each of the /j, curves which pass through them ; the degree of the locus is m/j, + v. Taking for the finite line the line joining the two imaginary circular points at infinity, it follows that there are m (m/n + v) curves of the system which cut a given curve orthogonally. De Jonquieres finds that in like manner the locus of a point such that its polar plane with regard to a fixed surface, and the tangent plane at that point to one of the surfaces of the system meet the plane of a fixed conic in two lines conjugate with respect to the conic, is a surface of the order in/j, + p. And consequently that a surface of this order meets the fixed surface in points where it is cut orthogo- nally by surfaces of the system. (12) If from each of two fixed points Q, Q' tangents be drawn to a system of curves of the w" 1 class, the locus of the 514 ON THE PROPERTIES OF SYSTEMS OP SURFACES. intersections of the tangents of one system with those of the other is a curve of the order v (2w — ] ). For consider any curve touching the line QQ\ then one point of the locus will be the point of contact, and n — 1 of the others will coincide with each of the points Q, Q. And since there may be v such curves, each of the points Q, Q', is a multiple point of the order (n — 1) v, and the line QQ' meets the locus in v(2w — 1) points. Let the points QQ be the two circular points at infinity, and it follows that the locus of foci of curves of the system is a curve of degree v (2k — 1). If we investigate, in like manner, the locus of the intersection of cones drawn to a system of surfaces from two fixed points QQ', it is evident from what has been said, that any plane through QQ meets the locus in a curve whose order is p (2n — 1 ) ; but the line QQ' is a multiple line of degree p, being common to both cones in every case where the line QQ' touches a surface of the system. The order of the locus therefore is 2np ; and accordingly, Ap is the order of the locus of foci of sections of a system of quadries by planes parallel to a fixed plane.* * Chasles has given the theorem that if there be a system of conies whose characteristics are p., y, then 2y — p. conies of the system reduce to a pair of lines, and 2/i — j/ to a pah- of points. It immediately follows hence, as Cremona has remarked, that if there be a system of quadries, whose characteristics are ft, v, p, of which a- reduce to cones and