or "AiTrORNTA S&N DIEGO F 3 1822 01264 8036 LIBRARY \ UNivjRscnr OF CALIFOivNlA SAN 01E6O presented to the UNIVERSITY LIBRARY UNIVERSITY OF CALIFORNIA SAN DIEGO by KARL nVK A 3 1822 01264 8036 SCIENCE AND ENGINEERING LIBRARY University of California, San Diego Please Note: This item is subject to RECALL after one week. DATE DUE Sis/ S7 MAR 4 mn SE 7 (Rev. 7/82) UCSD Libr. s^e ANALYTIC GEOMETRY OF SPACE BY VIRGIL SNYDER, Ph.D. (Gottingen) Professor of Mathematics at Cornell University C. H. SISAM, Ph.D. (Cornell) Assistant Professor of Mathematics at the University of Ilunois NEW YORK HENRY HOLT AND COMPANY Copyright, 1914, BY HENRY HOLT AND COMPANY April, 1924 PRINTED IN U. 8. A. PREFACE In this book, which is planned for an introductory course, the first eight chapters include the subjects usually treated in rectangular coordinates. They presuppose as much knowledge of algebra, geometry, and trigonometry as is contained in the major requirement of the College Entrance Examination Board, and as much plane analytic geometry as is contained in the better elementary textbooks. In this portion, proofs of theorems from more advanced subjects in algebra are supplied as needed. Among the features of this part are the development of linear systems of planes, plane coordinates, the concept of infinity, the treatment of imaginaries, and the distinction between centers and vertices of quadric surfaces. The study of this portion can be regarded as a first course, not demanding more than thirty or forty lessons. In Chapter IX tetrahedral coordinates are introduced by means of linear transformations, under which various invariant proper- ties are established. These coordinates are used throughout the next three chapters. The notation is so chosen that no ambigu- ity can arise between tetrahedral and rectangular systems. The selection of subject matter is such as to be of greatest service for further study of algebraic geometry. In Chapter XIII a more advanced knowledge of plane analytic geometry is presupposed, but the part involving Pliicker's num- bers may be omitted without disturbing the continuity of the subject. In the last chapter extensive use is made of the cal- culus, including the use of partial differentiation and of the element of arc. The second part will require about fifty lessons. CONTENTS CHAPTER I COORDINATES ARTICLE PAGE 1. Coordinates 1 2. Orthogonal projection 3 3. Direction cosines of a line 5 4. Distance between two points ........ 6 5. Angle between two directed lines ....... 7 6. Point dividing a segment in a given ratio 8 7. Polar coordinates 10 8. Cylindrical coordinates 10 9. Spherical coordinates 11 CHAPTER II PLANES AND LINES 10. Equation of a plane 12 11. Plane through three points ........ 13 12. Intercept form of the equation of a plane ..... 13 13. Normal form of the equation of a plane .14 14. Reduction of a linear equation to the normal form .... 15 15. Angle between two planes ........ 16 16. Distance to a point from a plane 17 17. Equations of a line .......... 19 18. Direction cosines of the line of intersection of two planes . . 19 19. Forms of the equations of a line 20 20. Parametric equations of a line 21 21. Angle which a line makes with a plane ...... 22 22. Distance from a point to a line 23 23. Distance between two non-intersec\'ing lines 24 24. System of planes through a line 25 25. Application to descriptive geometry 28 V VI CONTENTS ARTirLE PAGE 26. Bundles of planes 29 27. Plane coordinates 31 28. Equation of a point 32 29. Homogeneous coordinate of the point and of the plane ... 33 30. Equation of the plane and of the point in homogeneous coordinates 34 31. Equation of the origin. Coordinates of planes through the origin . 34 32. Plane at infinity 36 33. Lines at infinity .......... 35 34. Coordinate tetrahedron 35 35. System of four planes 36 CHAPTER III t TRANSFORMATION OF COORDINATES 36. Translation 38 37. Rotation 38 38. Rotation and reflection of axes 41 39. Euler's formulas for rotation of axes ...... 42 40. Degree unchanged by transformation of coordinates ... 42 CHAPTER IV TYPES OF SURFACES 41. Imaginary points, lines, and planes 42. Loci of equations 43. Cylindrical surfaces 44. Projecting cylinders 45. Plane sections of surfaces 46. Cones .... 47. Surfaces of revolution 44 46 47 47 48 49 50 CHAPTER V THE SPHERE 48. The equation of the sphere 52 49. The absolute 52 50. Tangent plane 65 51. Angle between two spheres 65 52. Spheres satisfying given conditions 66 63. Linear systems of spheres 67 54. Stereographic projection 59 CONTENTS Vll CHAPTER VI FORMS OF QUADRIC SURFACES ARTICLE PAGE 55. Definition of a quadric 63 56. The ellipsoid 63 57. The hyperboloid of one sheet 65 58. The hyperboloid of two sheets ....... 67 59. The imaginary ellipsoid ......... 68 60. The elliptic paraboloid 69 61. The hyperbolic paraboloid 70 62. The quadric cones .......... 71 63. The quadric cylinders 72 64. Summary 72 CHAPTER VII CLASSIFICATION OF QUADRIC SURFACES 65. Intersection of a quadric and a line 66. Diametral planes, center .... 67. Equation of a quadric referred to its center 68. Principal planes ...... 69. Reality of the roots of the discriminating cubic 70. Simplification of the equation of a quadric 71. Classification of quadric surfaces 72. Invariants under motion ..... 73. Proof that 7, t/, and D are invariant 74. Proof that A is invariant 75. Discussion of numerical equations . 74 75 77 78 79 80 €1 82 83 84 CHAPTER VIII SOME PROPERTIES OF QUADRIC SURFACES 76. Tangent lines and planes ........ 90 77. Normal forms of the equation of the tangent plane ... 91 78 Normal to a quadric 92 79. Rectilinear generators . . . . . . . . .93 80. Asymptotic cone .......... 95 81. Plane sections of quadrics 96 82. Circular sections 98 83. Real circles on types of quadrics 100 84. Confocal quadrics 104 85. Confocal quadrics through a point. Elliptic coordinates . . 105 86. Confocal quadrics tangent to a line ....... 107 87. Confocal quadrics in plane coordinates 108 Vlll CONTENTS CHAPTER IX TETRAHEDRAL COORDINATES ARTICLE PAOR 88. Definition of tetrahedral coordinates 109 89. Unit point 110 90. Equation of a plane. Plane coordinates . . . . .111 91. Equation of a point 112 92. Equations of a line 112 93. Duality 11.3 94. Parametric equations of a plane and of a point .... 114 95. Parametric equations of a line. Range of points. Pencils of planes 115 96. Transformation of point coordinates 117 97. Transformation of plane coordinates . . . . . .119 98. Projective transformations 120 99. Invariant points 121 100. Cross-ratio 121 CHAPTER X QUADRIC SURFACES IN TETRAHEDRAL COORDINATES 101. Form of equation ....... 102. Tangent lines and planes 103. Condition that the tangent plane is indeterminate . 104. The invariance of the discriminant 105. Lines on the quadric surface 106. Equation of a quadric in plane coordinates . 107. Polar planes 108. Harmonic property of conjugate points 109. Locus of points which lie on their own polar planes 110. Tangent cone ........ 111. Conjugate lines as to a quadric .... 112. Self-polar tetrahedron ...... 113. pjquation referred to a self-polar tetrahedron 114. Law of inertia 115. Rectilinear generators. Reguli .... 116. Hyperbolic coordinates. Parametric equations . 117. Projection of a quadric upon a plane 118. Equations of the projection ..... 119. Quadrics deterniineii by three non-intersecting lines 120. Transversals of four skew lines .... 121. The quatlric cone ....... 122. Projection of a quadric coue upon a plane 124 124 125 126 129 130 132 132 133 133 134 135 135 136 137 138 139 140 141 143 143 145 CONTENTS . IX CHAPTER XI LINEAR SYSTEMS OF QUADRICS ARTICLE 123. Pencil of quadrics 124. The \-discrirainant .... 125. Invariant factors ..... 126. The characteristic ..... 127. Pencil of quadrics having a common vertex 128. Classification of pencils of quadrics 129. Quadrics having a double plane in common 130. Quadrics having a line of vertices in common 131. Quadrics having a vertex in common 132. Quadrics having no vertex in common 133. Forms of pencils of quadrics . 134. Line conjugate to a point 135. Equation of the pencil in plane coordinates 136. Bundle of quadrics .... 137. Representation of the quadrics of a bundle by 138. Singular quadrics of the bundle 139. Intersection of the bundle by a plane 140. The vertex locus J .... 141. Polar theory in the bundle 142. Some special bundles .... 143. Webs of quadrics ..... 144. The Jacobian surface of a web 145. Correspondence with the planes of space 146. Web with six basis points 147. Linear sy.stems of rank r . . . 148. Linear systems of rank r in plane coordinates 149. Apolarity ...... 150. Linear sy.stems of apola,r quadrics . the points of a plane PAGE 147 147 148 150 161 161 161 151 152 156 163 165 166 167 168 168 169 170 171 173 175 175 177 177 180 181 181 186 CHAPTER XII TRANSFORMATIONS OF SPACE 161. Projective metric 188 152. Pole and polar as to the absolute ....... 188 153. Equations of motion 190 154. Classification of projective transformations 191 155. Standard forms of equations of projective transformations . . 196 156. Birational transformations . 196 157. Quadratic transformations ........ 198 158. Quadratic inversion ......... 201 159. Transformation by reciprocal radii 201 160. Cyclides 203 CONTENTS CHAPTER XIII CURVES AND SURFACES IN TETRAHEDRAL COORDINATES I. Algebraic Surfaces AKTICLE PAGE 161. Number of constants in the equation of a surface .... 206 162. Notation 207 163. Intersection of a line and a surface 207 164. Polar surfaces 208 165. Tangent lines and planes 209 166. Inflexional tangents 210 167. Double points 210 168. The first polar surface and tangent cone 211 169. Class of a surface. Equation in plane coordinates . . .212 170. The Hessian 213 171. The parabolic curve 214 172. The Steinerian 214 II. Algebraic Space Curves 173. Systems of equations defining a space curve . 174. Order of an algebraic curve 175. Projecting cones ....... 176. Monoidal representation ..... 177. Number of intersections of algebraic curves and surfaces 178. Parametric equations of rational curves 179. Tangent lines and developable surface of a curve . 180. Osculating planes. Equation in plane coordinates 181. Singular points, lines, and planes .... 182. The Cayley-Salmon formulas .... 183. Curves on non-singular quadric surfaces 184. Space cubic curves 185. Metric classification of space cubic curves 186. Classification of space quartic curves 187. Non-singular quartic curves of the first kind . 188. Rational quartics 216 216 217 219 221 222 224 224 226 226 228 230 234 235 238 242 CHAPTER XIV DIFFERENTIAL GEOMETRY I. Analytic Curves 189. Length of arc of a space curve 245 190. The moving trihedral 246 191. Curvature 248 CONTENTS XI ARTICLK 192. Torsion .... 193. The Frenet-Serret formulas 194. The osculating sphere 195. Minimal curves PAGF. 249 250 251 252 II. Analytic Surfaces 196. Parametric equations of a surface 197- Systems of curves on a surface 198. Tangent plane. Normal line 199. Differential of arc . 200. Minimal curves 201 . Angle between curves. Differential of surface 202. Radius of normal curvature. Meusnier's theorem 203. Asymptotic tangents. Asymptotic curves 204. Conjugate tangents 205. Principal radii of normal curvature 206. Lines of curvature . 207. The indicatrix Answers Index .... 254 255 255 257 258 259 259 261 261 262 263 266 269 287 ANALYTIC GEOMETRY OF SPACE CHAPTER I COORDINATES 1. Rectangular coordinates. The idea of rectangular coordinates as developed in plane analytic geometry may be extended to space in the following manner. Let there be given three mutually perpendicular planes (Fig. 1) XOY, YOZ, ZOX, intersecting at 0, the origin. These planes will be called coordinate planes. The planes ZOX, XOY intersect in X'OX, the X-axis; the planes XOY, YOZ intersect in Y'OY, the F-axis ; the planes YOZ, ZOX intersect in Z'OZ, the Z-axis. Dis- tances measured in the directions X'OX, Y'OY, Z'OZ, respectively, will be considered positive ; those measured in the opposite directions will be regarded as negative. The coordi- nates of any point P are its distances from the three coordinate planes. The distance from the plane YOZ is denoted by x, the distance from the plane ZOX is denoted by y, and the distance from the plane XOY is denoted by z. These three numbers X, y, z are spoken of as the x-, y-, z-coordinates of P, respect- ively. Any point P in space has three real coordinates. Con- versely, any three real numbers x, y, z, taken as x-, y-, and z- coordinates, respectively, determine a point P; for if we lay otf a distance OA = x on the X-axis, OB=y on the F-axis, OC = z on 1 COORDINATES [Chap. I. Fig. 2. the Z-axis, and draw planes through A, B, C parallel to the co- ordinate planes, these planes will intersect in a point P whose coordinates are x, y, and z. It will frequently be more convenient to determine the point P whose coordinates are x, y, and z, as follows : Lay off the distance OA = x on the X-axis (Fig, 2). From A lay off the distance AD = ?/ on a parallel to the F-axis. From D lay off the distance DP = 2 on a parallel to the Z-axis. The eight portions of space separated by the coordinate planes are called octants. If the coordinates of a point P are a, '^ b, c, the points in the remaining octants at the same absolute distances from the coordinate planes are (— a, b, c), (a, — b, c), (a, b, - c), (- a,-b, c), (- a, b, - c), (a, - b, - c), (- a, — 6,- c). Two points are symmetric with regard to a plane if the line joining tliem is perpendicular to the plane and the segment between them is bisected by the plane. They are symmetric with regard to a line if the line joining them is perpendicular to the given line and the segment between them is bisected by the line. They are symmetric with regard to a point if the segment be- tween them is bisected by the point. The problem of representing a ligure in space on a plane is considered in descriptive geometry, where it is solved in several ways by means of projections. In the figures appearing in this book a particular kind of parallel projection is used in which the X-axis and the Z-axis are represented by lines perpendicular to each other in the plane of the paper ; the F-axis is represented by a line making equal angles with the other two. Distances parallel to the X-axis or to the Z-axis are represented correctly to scale, but distances parallel to the F-axis will be foreshortened, the amount of which may be chosen to suit the particular drawing considered. It will usually be convenient for the student, in drawing figures on cross section paper, to take a unit on the y-axis 1/ V2 times as long as the unit on the other axes. Art. 2] ORTHOGONAL PROJECTIONS 3 EXERCISES 1. Plot the following points to scale, using cross section paper : (1, 1, 1), (2, 0, 3), (- 4, - 1, -4), (-3,-4, 1), (4, 4, - 1), (-7, 2, 3), (-1, 6, -6), (-4,2,8), (3, -4, -1), (2,1, -3), (-1,0,0), (4, -2, 2), (0, 0, 2), (0, -1, 0), (-3,0, 0), (0, 0, 0). 2. What is the locus of a point for which x = ? 3. What is the locus of a point for which x = 0, ?/ = ? 4. What is the locus of a point for which x = a, y = b? 5. Given a point {k, I, m), write the coordinates of the point symmetric with it as to the plane XOY; the plane ZOX; the X-axis; the F-axis; the origin. 2. Orthogonal projections. The orthogonal projection of a point oil a plane is the foot of the perpendicular from the point to the plane. The orthogonal projection on a plane of a segment PQ of a line* is the segment P'Q\ joining the projections P' and Q' oi P and Q on the plane. The orthogonal projection of a point on a line is the point in which the line is intersected by a plane which passes through the given point and is perpendicular to the given line. The or- thogonal projection of a segment PQ of a line Z on a second line /' is the segment P'Q' joining the projections P' and Q' of P and Q on I. For the purpose of measuring distances and angles, one direc- tion along a line will be regarded as positive and the opposite direction as negative. A segment PQ on a directed line is positive or negative according as Q is in the positive or nega- tive direction from P. From this definition it follows that PQ=-QP. The angle between two intersecting directed lines I and V will be defined as the smallest angle which has its sides extending in the positive directions along I and V. We shall, in general, make no convention as to whether this angle is to be considered positive or negative. The angle between two non-intersecting directed lines I and V will be defined as equal to the angle be- tween two intersecting lines m and m' having the same directions as I and V, respectively. * We shall use the word line throughout to mean a straight line. COORDINATES [Chap. I. Theorem I. The length of the projection of a segment of a directed line on a second directed line is equal to the length of the given segment midtiplied by the cosine of the angle between the lines. Let PQ (Figs. 3 a, 3 b) be the given segment on I and let P'Q' be its projection on V. Denote the angle between I and I' by 6. It is required to prove that P' Q' = PQ cos 0. Through P' draw a line I" having the same direction as I. The angle between V and I" is equal to 6. Let Q" be the point in Fig. 3 a. Fig. 3 6. which I" meets the plane through Q perpendicular to V. Then the angle P'Q'Q" is a right angle. Hence, by trigonometry, we have P'Q' = P'Q" cos e. But P'Q" = PQ. It follows that P'Q' = PQ cos $. It should be observed that it makes no difference in this theorem whether the segment PQ is positive or negative. The segment PQ = r will always be regarded as positive in defining Theorem II. The projection on a directed line I of a broken line made %ip of segments P^P^, P2P3, ••', Pn-\Pn of different lines is the sum of the projections on I of its parts, and is equal to the pro- jection on I of the straight line P^Pn- Arts. 2, 3] DIRECTION COSINES OF A LINE For, let P\, P'o, P'3, •••, P'„_i, P'„ be the projections of P^, P^, Ps, ••-, P„_i, P„, respectively. The sum of the projections is equal to P\P\, ; that is, P,P, + P,P, + ... + P'„_iP'„ = P,P,, But P'iP'„is the projection of PiP„. The theorem therefore follows. Corollary. If Pi, P2, •'•, P„-i «'"e the vertices of a polygon, the sum of the projections on any directed line I of the segments P^P^, PiP^i •••> Pn-\Pi formed by the sides of the 'polygon is zero. Since in this case P„ and Pi coincide, it follows that P\ and P'„ also coincide. The sum of the projections is consequently zero. EXERCISES 1. If is the origin and P any point in space, show that the projections of the segment OP upon the coordinate axes are equal to the coordinates of P. 2. If the coordinates of Pi are Xi, y^ ^i and of Po are x-j, 2/2, z-,, show that the projections of the segment PiP^ upon the coordinate axes are equal to ^•2 — a^i, 2^2 — 2/i> Zo — Z\, respectively. 3. If the lengths of the projections of PiP^ upon the axes are respectively 3, — 2, 7 and the coordinates of Pi are (- 4, 3, 2), find the coordinates of P2. 4. Find the distance from the origin to the point (4, 3, 12). 5. Find the distance from the origin to the point (a, h. c). 6. Find the cosines of the angles made with the axes by the line joining the origin to each of the following points. (1,2,0) (1,1,1) (-7,6,2) (0,2,4) (1,-4,2) {:>-,iJ,z) 3. Direction cosines of a line. kZ Let I be any directed line in space, and let V be a line through the origin which has the same direction. If «, fi, y (Fig. 4) are the angles which V makes with the coordinate axes, these are also, by definition (Art. 2), the angles which I makes with the axes. They are called the direction angles of I and their cosines are called direction cosines. The latter will be denoted by A, /x, v, respectively. ^ /,' ^J^ h^. r/ -^/s a 6 COORDINATES [Chap. I. ' Let P=(a, b, c) be any point on I' in the positive direction from the origin and let OP = r. Then, from trigonometry, we have a o b c \ = cos a = , /x = cos p = -, V = cos y = - • r r r Bnt r is the diagonal of a rectangular parallelepiped whose edges OA^a, OB = b, OG=c. Va^ -h62_,_c2 h Va' '■ + ¥ + (? c Hence, we obtain r = Va- + ^^ + cl In this equation, as in the formulas throughout the book, except when the contrary is stated, indicated roots are to be taken with the positive sign. By substituting this value of r in the above equations, we obtain \ = cos a = /x = cos (3 = v=C0Sy Va" + b^+c' By squaring each member of these equations and adding the results, we obtain ,00. X--^ + Ht--f- v-' = l, (1) hence we have the following theorem. Theorem. The sum of the squares of the direction cosines of a line is equal to unity. If Ai, fxi, vi and Ao, 1^2, v-^ are the direction cosines of two like directed lines, we have Xl = A2, fli = /Xo, Vi = V2. If the lines are oppositely directed, we have Ai = — Ao, /Ai = — fJ.2, Vi = — Vo. 4. Distance between two points. Let Pi =(x\, y^, z^), Pt = {x2,y2, Z2) be any two points in space. Denote the direction cosines of the Arts. 4, 5] ANGLE BETWEEN TWO DIRECTED LINES 7 line P1P2 (Fig. 5) by X, /a, v and the length of the segment P^Pz by d. The projection of the segment P^P., on each of the axes is equal to the sum of the projections of P,0 and OP., that is Xd = X, ~ x\, fxd = 2/2 — Vn vd = Zo — Zi. By squaring both members of these equations, adding, and extracting the square root, we obtain ^1 N^ N, Fig. 5. ^M^ -rX a = V(a?a - ^1)'^ + (:i/'2 - Vi)'^ + («2 - «i)2. (2) EXERCISES 1. Find the distance between (3, 4, — 2) and (— 5, 1, — 6). 2. Show that the points ( - 3, 2, - 7), (2, 2, - 3), and (- 3, 6, - 2) are vertices of an isosceles triangle. 3. Show that the points (4, 3, — 4), (- 2, 9, — 4), and (— 2, 3, 2) are vertices of an equilateral triangle. ■ 4. Express by an equation that the point (x, rj, z) is equidistant from (1, 1, 1) and (2, 3, 4). 5. Show that or^ + y'^ + z- = i \s the equation of a sphere whose center is the origin and whose radius is 2. 6. Find the direction cosines of the line P1P21 given : (a) Pi = (0, 0, 0), (h) P,= (l, 1, 1), ^(c) Pi = (l, -2,3), P2 = (2, 3, 5). P2 = (2, 2, 2). P2 = (4,2, -1). 7. What is known about the direction of a line if (a) cos a = ? (6) cos a — and cos /3 = ? (c) cos « = 1 ? 8. Show that the points (3, - 2, 7), (6, 4, — 2), and (5, 2, 1) are on a line. 9. Find the direction cosines of a line which makes equal angles with the coordinate axes. 5. Angle between two directed lines. Let li and l^ be two directed lines having the direction cosines X,, fXi, vi and A2, /X2, V2, respectively. It is required to find an expression for the cosine of the angle between l^ and I^. Through (Fig. 6) draw two COORDINATES [Chap. I. ^,' lines OPi and OP^ having the same di- rections as li and l^, respectively. Let OP2 = ^2 and let the coordinates of P^ be x^ = OM, 2/2 = MN, %<, = ArP2 X The projection of OP^ on OPi is equal to the sum of the projections of the ' Fig. 6. broken line OMNP. on OP^ (Art. 2). Hence OP^ cos = 03/ Ai + MN y.^ + ^^2 vi- < But OP2 = »'2J 03/= a;2 = r2A.2J ^^ = 2/2 = ^*2/>t2) -^-f* = ^2 = ^2>'2- Hence, we obtain or r^ cos ^ = rjXiAa + >'2/'ii/^2 + ^2»'i»'2) cos e = \i\2 -}- 1*1(1.2 + vivg. (3) The condition that the two given lines are perpendicular is that cos ^ = 0. Hence we have the following theorem : Theorem. Two lines l^ and Uwith direction cosines Aj, /xj, vi and X2, /Ao) V2, respectively, are perpendicular if '^l^a + t^lK-2 +''iv.2 = 0. (4) The square of the sine of 6 may be found from (1) and (3). Since sin^ ^ = 1 — cos^ 6, it follows that sin2 6 = (Ai^ + ix^ + v,2) {X^ +ix,^ + v.^) - (A1A2 + M1M2 + v,v,Y — (Ai)a2 — XofJ^xf + {lJ^iV2 — M2Vl)^ +(viA2 - V2Ai)2. <5. Point dividing a segment in a given ratio. Let Pj = (.t„ .Vi, z^) and P2 = (x2, 1/2, Z2) be two given points (Fig. 7). It is required to find the point P = (x, y, z) on the line P1P2 such that P,P : PP. = Wi : m.. Let A, iM, V be the direction cosines of (5) Fig 7. the line P1P2. Then (Art. 2, Th. I) we have P, P A = .T — Xi and PP2 \ = X2—x. Hence P^P A : PP2 \ = x — Xi : x^ — x^ m^ : wij. Art. 6] POINT DIVIDING A SEGMENT 9 On solving for x we obtain x = — ^-'— ! =, (6) mi + m2 c 1 1 „ m2!/i + mii/2 Similarly, y = ; . ~ mi + m^ It should be noticed that if vii and 7?i2 have the same sign, P^P and PPj a.re measured in the same direction so that Plies between Pj and Pj. If 7Jii and ?/i2 have opposite signs, P lies outside the segment PiPj. By giving mj and 7)12 suitable values, the coor- dinates of any point on the line P1P2 can be represented in this way. In particular, if P is the mid-point of the segment P1P2, vii = m^, so that the coordinates of the mid-point are _ a?! +X2 _ Vx + ?/2 „ _ ^^ +Z2 EXERCISES 1. Find the cosine of the angle between the two lines whose direction cosines are — ^, — ^, — ^ — - and — ;^, — ^::, — ^^- ^14 \/l4 Vli VSO VSO VSO 2. Find the direction cosines of each of the coordinate axes. 3. The direction cosines of a line are proportional to 4, — 3, 1. Find their values. 4. The direction cosines of two lines are proportional to 6, 2, — 1 and — 3, 1, — 5, respectively. Find the cosine of the angle between the lines. 5. Show that the lines whose direction cosines are proportional to 3, 6, 2 ; — 2, 3, — 6 ; — 6, 2, 3 are mutually perpendicular. 6. Show that the points (7, 3, 4), (1, 0, 6), (4, 5,-2) are the vertices of a right triangle. 7. Show that the points (3, 7, 2), (4, 3, 1), (1, 6, 3), (2, 2, 2) are the vertices of a parallelogram. 8. Find the coordinates of the intersection of the diagonals in the paral- lelogram of Ex. 7. 9. Show by two different methods that the three points (4, 13, 3), (3, 6, 4), (2, — 1, 5) are coUinear. 10 COORDINATES [Chap. I. 10. A line makes an angle of 75° with the A'-axis and 30° with the F-axis. How many positions may it have ? Find, for each position, the cosine of the angle it makes with the Z-axis. 11. Determine the coordinates of the intersection of the medians of the triangle witli vertices at (1, 2, 3), (2, 3, 1), (3, 1, 2). 12. Prove that the medians of any triangle meet in a point twice as far from each vertex as from the mid-point of the opposite side. This point is called the center of gravity of the triangle. " 13. Prove that the three straight lines joining the mid-points of oppo- site edges of any tetrahedron meet in a point, and are bisected by it. This point is called the renter of gravity of the tetrahedron. 14. Show that the lines joining each vertex of a tetrahedron to the point of intersection of the medians of the opposite face pass through the center of gravity. 15. Show that the lines joining the middle points of the sides of any quadrilateral form a parallelogram. 16. Show how the ratio mi : ??i2 (Art, 6) varies as P describes the line P1P2. 7. Polar Coordinates. Let OX, Y, OZ be a set of rectangular axes and P be any point in space. Let OP = p have the direc- AZ tion angles a, ^, y. The position of the line OP is determined by rt, (3, y and the position of P on the line is given by p, so that the position of the point P in space is fixed when p, a, ^, y are known. These quantities p, a, ft, y are called the polar coordinates Y ^^i"- «■ of P. As a, ft, y are direction angles, they are not independent, since by equation (1) cos^ a + cos^ ft -f cos^ y = 1. If the rectangular coordinates of P are x, y, z, then (Art. 3) X = p cos a, y — p cos ft, z = p cos y. 8. Cylindrical coordinates. A point is determined when its directed distance from a fixed plane and the polar coordinates of its orthogonal projection on that plane are known. These co- ordinates are called the cylindrical coordinates of a point. If the Arts. 8, 9] SPHERICAL COORDINATES 11 point P is referred to the rectangular axes X, y, z, and the fixed plane is taken as 2 = and the a;-axis for polar axis, we may write (Fig. 9) x= p cos 6, y = p sin 6, z = z, in which p, 0, z are the cylindrical coordi- nates of P. 9. Spherical coordinates. Let OX, Y, OZ, and P be chosen as in Art. 7, and let P be the orthogonal projection of P on the plane XOY. Draw OP. The position of P is defined by the distance p, the angle = ZOP wliich the line OP makes with the 2;-axis, and the angle 6 (measured by the angle XOP) which the plane through P and the 2;-axis makes witli the plane XOZ. The num- bers p, ^, 6 are called the spherical coordinates of P. The length p is called the radius vector, the angle <{> is called the co-latitude, and 6 is called the longitude. If P = (x, y, z), then, from the figure (Fig. 10), OP = p cos (90 - <^) = p sin <^. Hence x = p sin cf> cos 8, y = P sin <^ sin 6, FiG. 10. ' z = p cos <^. On solving these equations for p, ' m^ + m.2 m^ + mj The equation (1) is satisfied by the coordinates of P if nu{Ax, + By, -|- Cz, + D)-\- m,{Ax., + By., + Cz^ + Z)) = 0, but since the coordinates of I\ and P^ satisfy (1), we have Ax, + By, + Cz,-\-D = 0, Ax, 4- By, + Cz, + D = 0, hence the coordinates of P satisfy (1) for all values of m, and wij. 12 Arts. 11, 12] INTERCEPT FORM OF THE EQUATION 13 Finally, not all the points of space lie on the locus defined by (1), since the coordinates [ 0, 0, — ^ — ^ — ^ j do not satisfy (1). This completes the proof of the theorem. 11. Plane through three points. Let (ic,, y^, z^), (x^, n^, z^, (^3, ?/3, Zj) be the coordinates of three non-collinear points. The condition that these points all lie in the plane ^x + -B^ + Cz + Z> = is that their coordinates satisfy this equation, thus Ax^ + %i + C^i + i> = 0, Ax^ + By^_ + Cz2+ D=: 0, Ax, + By, + Cz, + D = 0. The condition that four numbers A, B, C, D (not all zero) exist which satisfy the above four simultaneous equations is X y z 1 .-c, ?/i Zi 1 = ^-^yL^.^.2..c.^Ml ^1 2/2 ^2 •^z Vz ^3 This is the required equation, for it is the equation of a plane, since it is of first degree in x, y, z (Art. 10). The plane passes through the given points, since the coordinates of each of the given points satisfy the equation. 12. Intercept form of the equation of a plane. If a plane inter- sects the X-, Y-, Z-axes in three points '^1, B, C, respectively, the segments OA, OB, and OC are called the intercepts of the plane. Let A, B, C all be distinct from the origin and let the lengths of the intercepts be a, b, c, so that A =(a, 0, 0), B =(0, b, 0), C = (0, 0, c). The equation (2) of the plane determined by these three points (Art. 11) may be reduced to ^ + f-f- = l. (3) a b c This equation is called the intercept form of the equation of a plane. 14 PLANES AND LINES [Chap. IL EXERCISES 1. Find the equation of the plane through the points (1, 2, 3), (3, 1, 2), (5, - 1, 3). •* 2. Find the e(iuation of the plane through the points (0, 0, 0), (1, 1, 1), (2, 2, - 2) . What are its intercepts ? 3. Prove that the four points (1, 2, 3), (2, 4, 1), (- 1, 0, 1), (0, 0, 5) lie in a plane. Find the equation of the plane. 4. Determine k so that the points (1, 2, - 1), (3, - 1, 2), (2, - 2, 3), (1, — 1, k) shall lie in a plane. ' 5. P'ind the point of intersection of the three planes, a: + ?/ + 2 = 6, 22-y+2a! = 0, x-2y + 33; = 4. 13. The normal form of the equation of a plane. Let ABC (Fig. 11) be any plane. Let OQ be drawn through the origin per- pendicular to the given plane and intersecting it at P'. Let the direction cosines of OQ, be A, fi, V and denote the length of the segment OP by^:*. Let P = {x, y, z) be any point in the given plane. The projec- tion of P on OQ is P' (Art.' 2). Draw OP and the broken line OMNP, made up of segments CM = X, MN = y, and NP = z, Fio. 11. parallel to the X-, Y-, and Z-axes, respectively. The projections of OP and OMNP on OQ are equal (Art. 2, Th. II). The projection of the broken line is \x + fxy + vz, the projection of OP is OP' ov jy, so that Xx + ixy + vz=2^. (4) This equation is satisfied by the coordinates of every point P in the given plane. It is not satisfied by the coordinates of any other point. For, if Pj is a point not lying in the given plane, it is similarly seen, since the projection of OPi on OQ is not equal to p, that the coordinates of Pi do not satisfy (4). Hence, (4) is the equation of the plane. It is called the normal form of the equation of the plane. The number p in this equa- tion is positive or negative, according as P' is in the positive or negative direction from on OQ. Art. 14] REDUCTION OF THE EQUATION 15 14. Reduction of the equation of a plane to the normal form. Let Ax + By+Cz + D = (5) be any equation of first degree with real coefficients. It is required to reduce this equation to the normal form. Let Q = (^4, B, G) be the point whose coordinates are the coefficients of x, y, z in this equation. The direction cosines of the directed line OQ are (Art. 3) X ^ B C ... Va^^+W+c^ V^2 + B' + C^ Va-" + ^ + (72 If we transpose the constant term of (5) to the other member of the equation, and divide both members by y/ A^ + B'^ + C^, we obtain A , B ■X + —Z -y ^ , ^ .= -^ • (7) The plane determined by (7) is identical with that determined by (5) since the coordinates of a point will satisfy (7) if, and only if, they satisfy (5). By subtituting from (6) in (7) and comparing with (4), we see that the locus of the equation is a plane perpen- dicular to OQ, and intersecting OQ at a point P' whose distance from is P= ^ ~^ =' (8) V^2 + B^ + C^ In these equations, the radical is to be taken with the positive sign. The coefficients of x, y, z are proportional to A, /a, v in such a way that the direction cosines of the normal to the plane are fixed when the signs of A, B, C are known. But the plane is not changed if its equation is multiplied by — 1, hence the position of the plane alone is not sufficient to determine the direction of the normal. In order to define a positive and a negative side of a plane we shall first prove the following theorem: Theorem. Tico points Pi, P^ are on the same side or on opposite sides of the plane Ax -\- By + Cz-\- D = 0, according as their coordi- nates make the first member of the equation of the plane have like or unlike signs. 16 PLANES AND LINES [Chap. IL For, let Pi=(xi, y^, z-^), Po = (x2, y-i, x^) be two points not lying on the plane. The point P = {x, y, z) in which the line PiPo inter- sects the plane is determined (Art. 6) by the values of mi, m^ which satisfy the equation my{Ax^ + By. -f- Cz^ + Z>) + m-lAxi + By^ + Cz^ + D) = 0. If Axi + 7?//i + Czi + D and Ax. + By. + Cz. + D have unlike signs, then m^ and iiu have the same sign, and the point P lies be- tween Pi and P.2. If Axi -f- £//i + Cz^ -f- Z> and yl.r. + By. -\- Cz^ + D have the same sign, then the numbers m^, m^ have opposite signs, hence the point P is not between Pj and Pg. When all the terms in the equation Ax + By + Cz-\-D = are transposed to the first member, a point (x^, _?/„ Zi) will be said to be on the positive side of the plane if Axi + By^ + Cz^ + Z) is a positive number; the point will be said to be on the negative side if this expression is a negative number. Finally, the point is on the plane if the expression vanishes. It should be observed that the equation must not be multiplied by — 1 after the positive and negative sides have been chosen. 15. Angle between two planes. The angle between two planes is equal to the angle between two dii-ected normals to the planes ; hence, by Arts. 5 and 14, we have at once the following theorem : Theorem. Tlie cosine of the angle 6 be- tween two planes Ax -{-By-\-Cz + D = 0, A'x + B'y + C'z + D' = is defined by the equation A A' +BB'-irCC' Fm. 12. COS 6 = z=:' (9) In particular, the condition that the planes are perpendicular is AA' 4- BB' + CC = 0. (10) Arts. 15, 16] DISTANCE TO A POINT FROM A PLAl The conditions that the planes ; ire parallel are (Art. 3) A B C A' B' C 17 (11) The equations (11) are satisfied whether the normals have the same direction or opposite directions. From the definition of the angle between two planes it follows that in the first case the two planes are parallel and in the second case they make an angle of 180 degrees with each other. We shall say, however, that the planes are parallel in each case. 16. Distance to a point from a plane. Let P = (x^, y^, z^) be a given point and Ax + B>/ + Cz + D = be the equation of a given plane. The distance to P from the plane is equal to the distance from the given plane to a plane through P parallel to it. The equation Ax + By+Cz- (Ax, + By, + Cz,) = represents a plane, since it is of first degree with real coefficients (Art. 10). It is parallel to the given plane by Eqs. (11). It passes through P since the coordinates of P satisfy the equation. When the equations of the planes are reduced to the normal form, they become, respectively, A , B C -D A , B — — x-\ y V^2 _^ ^ 4. (72 y-^42 + £2 ^ (72 I C .^ ^1-^1 + %i + Cz, _ ^W+W+~C^ VA^ + -B2 + 02 The second members of these two equations represent the dis- tances of the two planes from the origin, hence the distance from the first plane to the second, which is equal to the distance d to P from the given plane, is found by subtracting the former from the latter. 18 PLANES AND LINES [Chap. IL The result is ^ , n , ^ , r. ^ ^ Axt + By I + Czi+D _ .^2) Va'^ + B- + C^ The direction to P from the plane, along the normal, is positive or negative according as the expression in the numerator of the second member is positive or negative (Art. 14), that is, according as P is on the positive or negative side of the plane. EXERCISES / 1. Reduce the equation 3 x — 12 ?/ — 4 z — 26 = to the normal form. 2. Write the equation of a plane through the origin parallel to the plane X + 2 y = 6. 3. What is the distance from the plane 3x + 4y — z = 5 to the point (2,2,2)? 4. Find the distance between the parallel planes 2x — i/ + 32 = 4, -Ix-y + Zz + b =0. 5. Which of the points (4, 3, 1), (1, -4, 3), (3, 5, 2), (- 1, 2, -2), (5, 4, 6) are on the same side of the plane 5x — 2y — 32 = as the point (1, 6, - 3) ? 6. Find the coordinates of a point in each of the dihedral angles formed by the planes 3x + 2y + 5s-4 = 0, x-2?/-2; + 6 = 0. 7. Show that each of the planes 25 x + 39 ?/ + 8 2 — 43 = and 25 x — .39?/ -|- 112 2 + 113 = bisect a pair of vertical dihedral angles formed by the planes o x + 12 2 + 7 = and 3«/ — 42 — 6 = 0, Which plane bisects the angle in which the origin lies ? 8. Find the equation of the plane which bisects that angle formed by the planes 3x — 22/ + 2 — 4 = 0, 2x+2/ — 82 — 2 = 0, in which the point (1, 3, -2) lies. -; _ ' 9. Find the equations of the planes which bisect the dihedral angles formed by the planes AxX + Biy + dz + Di = 0, A-^x + B-iy + C-iZ + D-2 = 0, 10. Find the equation of the locus of a point whose distance from the origin-is equal to its distance from the plane 3x + 2/ — 22 = 11. 11. Write the equation of a plane whose distance from the point (0, 2, 1) is 3, and which is perpendicular to the radius vector of the point (2, — 1,-1). 12. Show that the planes 2x-j/ + 2 + 3 = 0, x-?/ + 42 = 0, Zx + y -22 + 8 = 0, 4x-2«/ + 22-5 = 0, 9x + 3«/-62 — 7=0, and lx-1 y + 28 2 — 6 = bound a parallelopiped. 13. Write the equation of a plane through (1, 2, — 1), parallel to the plane x — 2j/ — 2 = 0, and find its intercepts. Arts. 17, 18] DIRECTION COSINES OF THE LINE 19 14. Find the equation of the plane passing through the points (1, 2, 3), (2, — 3, 6) and perpendicular to the plane 4x + 2y + 3z = l. 15. Find the equation of the plane through the point (1, 3, 2) pei-pen- dicular to the planes 2x + Sy-iz = 2, ix-Sy-2z =:5. 16. Show that the planes x + 2y — z = 0, y + 1 z — 2 = 0, x — 2y — z — 4 = 0, X + Sy + z = -i, and Sx + Sy — z — 8 bound a quadrilateral pyramid. 17. Find the equation of the locus of a point which is 3 times as far from the plane 3x — 6y — 2z = as from the plane 2x — y + 2z = 9. 18. Determine the value of m such that the plane mx + 2y — Sz — 14i shall be 2 units from the origin. 19. Determine k from tlie condition that x — ky + Sz — 2 shall be perpen- dicular to 3 X + 4 y — 2 z = 5. 1 7. Equations of a line. Let A^x + B^y + C^z + D^ = and A2X -f- B^y + C2Z -\- 0-2 = he the equations of two non-parallel planes. The locus of the two equations considered as simultaneous is a line, namely, the line of intersection of the two planes (Art. 10). The simultaneous equations A,x + B,y + C,z + A = 0, A2X + Biy + C2Z + D2 = are called the equations of the line. The locus represented by the equations of two parallel planes, considered as simultaneous, will be considered later (Art. 33). 18. Direction cosines of the line of intersection of two planes. Let A, fx, V be the direction cosines of the line of intersection of the two planes ii = A,x + B,y -f- C,z -f Z>i = 0, L2 = -420; + B2y + C2Z -f A = 0- Since the line lies in the plane A = 0, it is perpendicular to the normal to the plane. Hence, (Arts. 5, 1^ \A, + fjiB, + vCi = 0. Similarly, XA. + fxBz + vC'j = 0. By solving these two equations for the ratios of A, /u,, v, we obtain ^ fjt. B,C2 - B2C, C,A2 - C2A, A.B. - A2B, (13) 20 PLANES AND LINES [Chap. IL The denominators in these expressions are, therefore, proportional to the direction cosines. In many problems, they may be used instead of the direction cosines themselves, but, in any case, the actual cosines may be determined by dividing these denominators by the square root of the sum of their squares. It should be observed that the equations of a line are not sufficient to deter- mine a positive direction on it. 19. Forms of the equations of a line. If A, /u,, v are the direction cosines of a line, and if P, ={xi, y^, Zj) is any point on it, the distance d from Pj to another point P = (x, y, z) on the line satis- fies the relations (Art. 4) Xd = X — Xi, fxd = y — yi, vd = z — z^. By eliminating d, we obtain the equations \ fJi V which are called the symmetric form of the equations of the line. Instead of the direction cosines themselves, it is frequently convenient to use, in these equations, three numbers a, b, c, pro- portional, respectively, to A, /a, v. The equations then become «i _ ?/ - .Vi _ ^ (15) a b c They may be reduced to the preceding form by dividing the de- nominator of each member by V«^ + ^^ + c^ (Art. 3). If the line (15) passes through the point P.;,={x2, y^, z^, the coordinates of P^ satisfy the equations, so that a b c On eliminating a, b, c between these equations and (15), we obtain x-xi ^ y-yi ^ z-Zi ^ ^^q-^ x^-xi yt-y, zi-zx These equations are called the two-point form of the equations of a line. Art. 20] PARAMETRIC EQUATIONS OF A LINE 21 20. Parametric equations of a line. Any point on a line may be defined in terms of a fixed point on it, the direction cosines of the line, and the distance d of the variable point from the fixed one. Thus, by Art. 4 x=:Xi + Xd, yz=y^^ fid, z = z^ + vd. (17) If A, /A, V are given and (x^, y^, z^) represents a fixed point, any point {x, y, z) on the line may be defined in terms of d. To every real value of d corresponds a point on the line, and conversely. These equations are called parametric equations of the line, the parameter being the distance. It is sometimes convenient to express the coordinates of a point in terms of a parameter k which is defined in terms of d by a linear fractional equation of the form y + 8k in which a, /3, y, 8 are constants satisfying the inequality «8 - ^y ^ 0. By substituting these values of d in (17) and simplifying, we obtain equations of the form in which a^, b^, etc., are constants. Equations (18) are called the parametric equations of the line in terms of the parameter k. It should be observed that the denominators in the second members of equations (18) are all alike. Each value of k for which a^ -\-b^K=^0 determines a definite point on the line. As «4 + b^K approaches zero, the distance of the corresponding point from the origin increases without limit. To the value deter- mined by tti 4- 64K = we shall say that there corresponds a unique point which we shall call the point at infinity on the line. EXERCISES 1. Fiud the points in which the following lines pierce the coordinate planes : (a) X +2y -Sz = 1, 3x-2y + 5z = 2. (b) x + Sy + bz = 0, 5x-Sy + z = 2. (c) x + 2y-5 = 0, 2x-Sy + 2z = T. 22 PLANES AND LINES [Chap. IL 2. Write the equations of the line x + y — 3 s = Cr, 2 x — y -\- 2z = 1 in the symmetric form, the two-point form, the parametric form. 3. Show that the lines 4a; + 2/ — 3^ = 0, 2x — y + 2z + Q — 0, and 8 x — y -\- z = 1, lOx + 2/ — •l^ + lrrO are parallel. 4. Write the equations of the line through (3, 7, 3) and (— 1, 5, 6). Determine its direction cosines. 5. Find the equation of the plane passing through the point (2, — 2, 0) and perpendicular to the line 2 = 3, ?/ = 2 .r — 4. 6. Find the value of k for which the lines ^^^ = ^-±-i = ^^^ and ^—^ 2 k k+l 3 3 y + 5 r + 2 T 1 = ^— ! — = — ! — are perpendicular. 1 ^•-2 7. Do the points (2, 4, 6), (4, 6, 2), (1, 3, 8) lie on a line ? 8. For what value of k are the points {k, — 3, 2), (2, — 2, 3), (fi, —1, 4) coUinear ? 9. Is there a value of k for which the points (k, 2, — 2), (2, — 2, A-), and (—2, 1, 3) are coUinear? 10. Show that the line' ^^—^ = ^-^ = ?-^ lies in the plane 2x+2y 3-14 -0 + 3 = 0. 11. In equations (18) show that, as k approaches infinity, the correspond- ing point approaches a definite point as a limit. Does this limiting point lie on the given line ? 21. Angle which a line makes with a plane. Given the plane Ax + By ^ Cz + D = and the line ^^^=^ = -"^^^^ = ^-^^^. a b c The angle which the line makes with the plane is the complement of the angle which it makes with the normal to the plane. The direction cosines of the normal to the plane are proportional to A, B, C and the direction cosines of the line are proportional' to a, b, c, hence the angle 6 between the plane and the line is de- termined (Art. 5) by the formula sin 6 = — — — ' (19) V^2 + B^+ C'2 Va2 + 62 + c2 Arts. 21, 22] DISTANCE FROM A POINT TO A LINE 23 EXERCISES 1. Show that the planes 2x -Sy + z + \ =0, 5x + z — l =0, ix + 9y — z — 5 = have a line in common, and find its direction cosines. 2. Write the equations of a line which passes through (5, 2, 6) and is parallel to the line 2 x — 3 z + y — 2 = 0, x + y + z + l=0. 3. Find the angle which the line x + y + 2z — 0, 2x— y + 2z — 1=0 makes with the plane Sx + 6z — 5y + l =0. 4. Find the equation of the plane through the point (2, — 2, 0) and peqiendicular to the line x + 2y — Sz = i, 2x — Sy + 4iZ = 0. 5. Find the equation of the plane determined by the parallel lines X + I _ y — 2 _ z X — .3 _ y + 4 _ z — I 3~2~r 3~2~1' 6. For what value of k will the two lines x + 2y — z + S = 0, Sx — y -{- 2z + l=0; 2 X— y + z— 2 = 0, x + y — z + k = intersect ? ^ 7. Find the equation of the plane through the points (1, — 1, 2) and (.3, 0, 1), parallel to the line x + y — z = 0, 2 x + y + z = 0. oci *i**i 1- X— 2 V + I z i X — 3 w + 4 z + 2 8. Show that the hues = ^-^ — = and = ^— J — = — ■ — 3 3-2 _ 1 3 2 intersect, and find the equation of the plane determined by them. 9. Find the equation of the plane through the point (a, b, c), parallel to each of the lines, "i^^ = ^^^ = l^lii ; "^^^^ = y^^^ = ^-^^^. 10. Find the equation of the plane through the origin and perpendicular to the line 'i x — y + i z -\- ii = 0, x + y — z = 0. 11. Find the value of k for which the lines ^' ~ '^ = -^-^ — = ^; 2 k k + l 5 ' X — I V + 5 z + 2 ,. , — - — = --^—^ — = are perpendicular. 3 1 k-2 ^ ^ 12. Find the values of k for which the planes kx — 5 y + (k + S^z + 3 = and (k — l)x + ky + z = are perpendicular. 13. Find the equations of the line through the point (2, 3, 4) which meets the I'-axis at right angles. 22. Distance from a point to a line. Given the line X — CCi _ 7/ — Pi _ Z — Zi X fJL V and the point P^ = (x^, y^, z^^ not lying on it. It is required to find the distance between the point and the line. 24 PLANES AND LINES [Chap. IL Fig. 13. Let Pi = (.Ti, y^, z,) (Fig. 13) be any point on the line ; let P be the foot of the perpendicular from Pj o^ the line ; the angle between the given line and •-j^ the line P1P2; let d be the length of the segment PiP^. We have (Fig. 13) P,P2 = P^P^^ sin^ = (r~- cP cos^ e. The direction cosines of the line P^P^ are '-- —, — —, d d from which (Art. 5) d d >j2 - yi _,_ ^. ^2 - zi d d Hence, d' cos^ e = {X, - x,f + (//, - y,y 4- (z. - z,y (20) 23. Distance between two non -intersecting lines. Given the two lines X- Xi _ y - y, ^ z - z, ^^^^^ x - x.^ ^ y - ?/, _ z - z^ Ai /*i Vl V2 which do not intersect. It is required to find the shortest dis- tance between them. Let A, ^, v be the direction cosines of the line on which the distance is measured. Since this line is per- pendicular to each of the given lines, we have, by Art. 5, Equations (4) and (5), AZ /* /i-lVo — V,/A2 V ±1 A1M2 ~ 1^1 A2 sin 6 where 6 is the angle between the given lines. The length d of the required perpendicular is equal to the projection on the common per- pendicular of the segment PP', ^ ^'°- ^^• and is equal to the projection of the broken line PMNP' (Fig. 14). Arts. 23, 24] SYSTEM OF PLANES THROUGH A LINE 25 or d = ± sin 6 Hi — 1/2 fJ-i H-2 Zi — e, I'l V2 EXERCISES 1. Find the distance from the oritiin to the line 1 (21) X- 1 _y-3_g-2 2 4 1 2. Find the distance from (1, 1, ]) to x + y + z = 0, Sx — 2y + 4z = 0. 3. Find tlie perpendicular distance from the point (— 2, 1, 3) to the line x + 2y-z + 'i-0,Hx — y + 2z + l-0. 4. What are the direction cosines of the line through the origin and the point of intersection of the lines x -\- 2 y — z + 3 = 0, ox — y + 2z + l = 0; 2x — 2y + 3z — 2=0,x-y-z + S = 0. 5. Determine the distance of the point (1, 1, 1) to the line a; = 0, y — and the direction cosines of the line on which it is measured. 6. Find the distance between the lines - = ^ = ~ ~ and ^ ~ - 2-2 1 4 ^y-3^z+l 2 - 1 ■ 7. Find the equations of the line along which the distance in Ex. 6 is measured. 8. Find the distance between the lines 2 x + y — z = 0, x— y -\-2z = 3 and x + 2y — 3z — 'i, 2x — 3?/ + 40 = 5. 9. Express the condition that the lines ^ ~ ^1 - ^ ~ ^i - ^ ~ ^1 ^ x - Xj h mi Hi h = y^ZLVl = IjlLll intersect. 7712 «2 24. System of planes through a line. If ii = A^x + B^y + C^z + 7), = 0, L., = A2X + Boy + C.JX + Z>2 = ane the equations of two intersecting planes, the equation fcjLj + A:2Z/2 = is, for all real values of k^ and ^•2, the equation of a plane passing through the line Li = 0, Lo = 0. For, Jc^L^ + kjj^ = is always of the first degree with real coefficients, and is therefore the equation of a plane (Art. 10); this plane passes through the line ij = 0, 7^2 = 0, since the coordinates of every point on the line satisfy Z-i = and Xg = and consequently satisfy the equation 26 PLANES AND LINES [Chap. H. \Li + Tx-.L., = 0. Conversely, the equation of any plane passing through the line can be expressed in the form l\Li + k.^Lo = 0, since k^ and ^^, can be so chosen that the plane k^Li + A'2L2 = will contain any point in space. Since any plane through the given line is determined by the line and a point not lying on it, the theorem follows. To find the equations of the plane determined by the line L^ = 0, Z/2 = 0, and a point P^ not lying on it, let the coordinates of Pj be (xi, Pi, Zi). If Pi lies in the plane k^L^ + koL., = 0, its coordinates must satisfy the equation of the plane; thus k,(A,x, + B,>ji + C\z,+I),) + k,(A,Xi + B.^j, + C.,z^ + A) = 0. On eliminating A'^ and k^ between this equation and k^L^ + AvLj = 0, we obtain = {Aa, + B.^j, + a^i + A)(^4^^• + B,y + C,z + A) - {A,x^ + 5,^1 + C^z^ + A) {A^ + AV + C-Z + A), as the equation of the plane determined by the line A = 0, A = 0, and the point Pj. It will be convenient to write the above equation in the abbre- viated form A(a:'i)A('«-') — A(-^i) A(-t') = 0- The totality of planes passing through a line is called a pencil of planes. The number k^/'k^ which determines a plane of the pencil is called the parameter of the pencil. If, in the ecjuation A'l A + ^'2 A =^ 0, A'l and k., are given such values that the coefficient of x is equal to zero, the corresponding plane is perpendicular to the plane aj = 0. Since this plane contains the line, it intersects the plane cc = in the orthogonal projection of the line Lj = 0, A = 0. Similarly, if fci and k^ are given such values that the coefficient of y is equal to zero, the corresponding plane is perpendicular to the plane y = and will cut the plane y = in the projection of A = ^i A = on that plane ; if the coefficient of z is made to vanish, the plane will contain the projection of the given line upon the plane z =0. The three planes of the system k^L^ + k.L^^ obtained in this way are called the three projecting planes of the line T/j = 0, ij = on the coordinate planes. Art. 24] SYSTEM OF PLANES THROUGH A LINE 27 Since two distinct planes passing through a line are sufficient to determine the line, two projecting planes of a line may always be em- ployed to define the line. If the line is not parallel to the plane z = 0, its projecting planes on a; = and y = are distinct and the equations of the line may be reduced to the form (Fig. Fig. 15 15) X = mz + a, y = nz + h. (22) If the line is parallel to 2 = 0, the value of k for which the coeffi- cient of X is made to vanish will also reduce the coefficient of y to zero, so that the projecting planes on x" = and on ?/ = coincide. This projecting plane z =c and the projec- ting plane on z = may now be chosen to define the line. If the line is not "X parallel to the X-axis, the equations oi the line may be reduced to (Fig. 16) X = py -\- c, z = c. (23) Finally, if the line is parallel to the A"-axis, its equations may be reduced to (Fig. 17) y^b, z^ c. (24) If the planes L^ = 0, L, = are par- allel but distinct, so that Ao B., a Do' A "O Y X Fig. 17. then every equation of the form k^L^ -|- kj^^ = 0, except when k A H C --2=— =— i=— ij defines a plane parallel to the given ones. k\ A-y Bi Cz \ Conversely, the equation of any plane parallel to the given ones \can be written in the form k^Li + koL., = by so choosing k^ : k^ 28 PLANES AND LINES [Chap. II. tliat the plane will pass through a given point. In this case the system of planes k-^L^ + A;2L2 = is called a pencil of parallel planes. Two equations A = A^x + Bi!j + C,z + A = 0, Li^ A.^ + B.2y -\- G^z -\- Di = will represent the same plane when, and only when, the coefficients Ax, Bi, Ci, Di are respectively proportional to A2, B2, C2, D^; thus, when A2 B2 C2 A' These conditions may be expressed by saying that every deter- minant of order two formed by any square array in the system A, A Ci D, A2 A Q D., shall vanish. In this case multipliers ki, k^ can be found such that the equa- tion k^Li + kJjo = is identically satisfied. Conversely, if multipliers k^, ko can be found such that the pre- ceding identity is satisfied, then the equations Li = 0, L2 = define the same plane. EXERCISES '^ 1. Write the equation of a plane through the line 7 x + 2y ~ z — S = 0, 3x~3y + 2z — 5 = perpendicular to the plane 2x-\-y — 2z = 0. 2. What is the equation of the plane determined by the line 2x — Sy — z + 2 = 0, x-y + iz = S and the point (3, 2, — 2) ? / 3. Determine the equation of the plane passing through the line ^ Q y_L4 Z 7 X + 2 z = i, y — z = 8 and parallel to the line = ^— !^ — = . ^ ^ 112 4. Does the plane x + 2y — z + '4 = have more than one point in common with the line Sx — y + 2z+l=0, 2x — Sy + Sz-2 = 0? ^5. Determine the equations of the line through (1, 2, 3) intersecting the two lines x + 2 (/-3.j=0, >/— 4,j = 4 and 2x-y +3^ = 3. 3x + y -\-2z + 1 = 0. 25. Application in descriptive geometry. A line may be repre- sented by the three orthogonal projections of a segment of the line, each drawn to scale. Consider the X>^-plane (elevation, or verti- cal plane) as the plane of the paper, and the XF-plane as turned about the ,Y-axis until it coincides with the XZ-plane. The pro- Arts. 25, 26] BUNDLES OF PLANES 29 ^x jections iu the XF-plaue are thus drawn to scale on the same paper as projections on the XZ-plane, but points are distinguished by different symbols, as P', P^. q a Z The XF-plane is called the plan or horizontal plane. Finally, let the FZ-plane be turned about the Z^axis until it coincides with the XZ-plane, and let figures iu the new position be drawn to scale. This is called the end or profile plane. Thus, in the figure (Fig. 18), a segment PQ, wherein P=(7, 4, 8), Q = (13, 9, 12), may be indicated by the three segments P'Q', PiQi, PpQp- Example. Find the equations of the projecting planes of the line 2x + 32/ — 42 = 5, x — iy + 5z = 6. Here, Li = 2 x + S y — i z — ^, L2 = x — iy + 5z — 6, kiLi + k2L2 =(2 ki + k2)x + (3 ki - 4 A-^)?/ + (_ 4 A-i + 5 hi)z + ( - 5 fci - 6 A;2) = 0. If ki = —2 k\, the coefficient of x disappears ; thus the equation of the plane projecting the given line on the plane .r = is 11 ?/- 142 + 7 =0. 7. q If — =: -, the coefficient of y vanishes; the projecting plane on y = is ki 4 found to be 1 1 X 38. ko 4 Finally, if -^ = -, the projecting plane on 2=0 is found. ki b Its equation is 14 X — ?/ = 49. EXERCISES Find the equations of the projecting planes of each of the following lines : -'1. z + 2 y - 3 2 = 4, 2 x - 3 J/ + 4 2 = 5. 2 2x +y + z = 0, x — y + 2 z = S. 3. X + t/ + 2 = 4, X- y + 3z = 4. 4. Au- + Bxy + Ciz -f- Z)i = 0, .4oX + B-.y + C\z + Z>2 = 0. 26. Bundles of planes. The plane L^ = A-^x + B^y + C^z -\- D^ =0 will belong to the pencil determined by the planes A= Oj L2=0, assumed distinct, when three numbers k\, ko, k^, not all zero, can be found such that the equation k^L^ + k^Lo ■+■ k^L^ = is identi- 30 PpiNES AND LINES [Chap. II. cally satisfied for all values of x, y, z. This condition requires that the four equations \Ay + hoA., + ^s-^j = 0, k^B^ + k^B^ + Jc^B^ = 0, A^iCi + k-yC. + k^Cs = 0, k^Di + k^D. + ^^sA = are satisfied by three numbers k^, k^, k^, not all zero ; hence, that the four equa- tions I A,B,G, 1=0, I B,C,D, 1 = 0, 1 C,D,A, 1=0, \ D,A,B, are all satisfied, wherein we have written for brevity, AiB^C^i Ay A c, A. B. c. A B, c. etc. These simultaneous conditions may be expressed by saying that every determinant of order three formed by the elements contained in any square array in the system A, B, c; D, A, B, C, D, A3 B3 63 D3 shall vanish. Conversely, if these conditions are satisfied, then three con- stants ^'i, A'2, k^ can be found such that the equation k^Li + kzL^ -\- k^L^ = is identically satisfied, and the three planes L^ = 0, io = 0, -L3 = belong to the same pencil. Let L, = A,x + B,y + C',2 + D, = 0, L. = A^x + B.ai + C.^ + A = 0, L, = A,x + B,v + C,z + X>3 = be the equations of three planes not belonging to a pencil. If we solve these three equations for (x, y, z), we find for the coordinates of the point of intersection of the three planes, in case | ^diCoCsl IAAC3I Ul.ACsl \A,B.D, A,B,C3 AB^Cs AxBiCs (25) If \AiB.2C\\ =0, but not all the determinants in the numerators of (25) are zero, no set of values of x, y, z will satisfy all three ' equations. In this case, the line of intersection of any two of the planes is parallel to the third. For, if L^ = and L, = intersect, •» Arts. 26, 27] PLANE COORDINATES 31 the direction cosines of their Hue of intersection are proportional (Art. 18) to B,C^ - BoC„ C,A, - aA„ A^B, - A.B,. The condition that this line is parallel to the plane Ls = is (Art. 21) A,{B,a - B,C,) + B,(C,A, - G,A,) + C,{A,B, - A,B,) = 0, which is exactly the condition | A^B^C:^ j = 0. The proof for the other lines and planes is found in the same way. If at least one of the determinants | A^BoC^ \, \ D^BoCs \, | A^DoC^ \, and I A1B2D3 1 is not zero, the system of planes A-jZ/i + Jc.Jj., + k^L^ = is called a bundle. If \ABC\^ 0, all the planes of the bundle pass through the point (25), since the coordinates of this point satisfy the equation of every plane of the bundle. Conversely, the equation of every plane passing through the point (25) can be expressed in this form. This point is called the vertex of the bundle. If \ABC\ = 0, all the planes of the bundle are parallel to a fixed line (such as L^ = 0, L., = 0). In this case, the bundle is called a parallel bundle. 27. Plane coordinates. The equation of any plane not passing through the origin may be reduced to the form ux + vy -t- wz + 1=0. (26) When the equation is in this form, the position of the plane is fixed when the values of the coefficients «, v, w (not all zero) are known; and conversely, if the position of the plane (not passing through the origin) is known, the values of the coefficients are fixed. Since the numbers (a, v, iv) determine a plane definitely, just as (x, y, z) determine a point, we shall call the set of num- bers («, V, ic) the coordinates of the plane represented by equation (26). Thus, the plane (3, 5, 2) will be understood to mean the plane whose equation is 3 a; + 5 ?/ -f 2 2; + 1 = 0. Similarly, the equation of the plane (2, 0, — 1) is 2 ic — 2; + 1 = 0. If u, V, IV are different from zero, they are the negative recipro- cals of the intercepts of the plane (u, v, w) on the axes (Art. 12). 32 PLANES AND LINES [Chap. 1L If u = 0, the plane is parallel to the X-axis ; if u = 0, -y = 0, the plane is parallel to the XF-plane. The vanishing of the other coefficients may be interpreted in a similar way. 28, Equation of a point. If the point {x^, y^, z^ lies in the plane (26), the equation ux^ + vy^ + icz^ +1 = (27) must be satisfied. If x-^, y^, z^ are considered fixed and u, v, w variable, (27) is the condition that the plane (m, v, iv) passes through the point (a-„ y^, Zi). For this reason, equation (27) is called the equation of the point (a-j, yi, Zi) in plane coordinates. Thus, u-5v + 2iv-\-l =0 is the equation of the point (1, — 5, 2) ; similarly, 3u + IV + 1 =0 is the equation of the point (.3, 0, 1). If equation (27) is multiplied by any constant different from zero, the locus of the equation is unchanged. Hence, we have the following theorem : Theorem. TJie linear equation Au + Bv+Civ + D = (Z) ^ 0) is the equation of the point (—, — , ] in plane coordinates. Thus, u — 5v— 3^0 — 2 = is the equation of the point - 1 5 3^ 2 ' 2' 2^ The condition that the coordinates (w, v, w) of a plane satisfy two linear equations uxi + vyi -\- u'Zi 4-1=0, ^1X2 + vy., + wz., +1=0 is that the plane passes through the two points (x^, y^, z^ and {X2, 2/2) ^-i) and therefore through the line joining the two points. The two equations are called the equations of the line in plane coordinates. r>2A^^JUCX^./s~ryJU nru>4jt tCirY\jL . S- , *2 *% , 1 tf Arts. 28, 29] HOMOGENEOUS COORDINATES 33 EXERCISES 1. Plot the following planes and write their equations : (1, 2, i), (3, — I, 2. Find the volume of the tetrahedron bounded by the coordinate planes and the plane (— h ~ h ~ i)- 3. What are the coordinates of the planes whose equations are Tx + 6y-^z+l=0, x-6y + nz + o = 0, 9.r-4=0? 4. Find the angle which the plane (2, 0, 5) makes with the plane (-1, i2). ^ 5. Write the equations of the points (1, 1, 1), (2, - 1, ^,), (6, —2, 1). 6. What are the coordinates of the points whose equations are 2m-»-3w+1 = 0, ?t + 2 to -3=0, to -2 = 0? ♦^7. Find the direction cosines of the line 3?< — •o4-2mj + 1=0, u + ^ V + 2 w - I = 0. 8. What locus is determined by three simultaneous linear equations in (m, V, w) ? r^ ' *'9. Write the equation satisfied by the coordinates of the planes whose distance from the origin is 2. What is the locus of a plane which satisfies this condition ? 29. Homogeneous coordinates of the point and of the plane. It is sometimes convenient to express the coordinates x, y, z; of a point in terms of fonr numbers x', y', z' , t' by means of the equations x' v' z' — = X, ■^=y, — = z. A set of four numbers (x', y', z', t'), not all of which are zero, that satisfy these equations are said to be the homogeneous coordinates of a point. If the coordinates (x', y', z', t') are given, the point is uniquely determined (for the case t' = 0, compare Art. 32), but if (x, y, z) are given, only the ratios of the homogeneous coordinates are determined, since (x', y', z', t') and (kx', ky', kz', kt') define the same point, k being an arbitrary constant, different from zero. Similarly, if the coordinates of a plane are (w, v, iv), four num- bers (?t', v', w', s'), not all of which are zero, may be found such that u' v' w' — = u, - = v, —-10. s' s s' I^^n^z--^-^- ^^o^ 34 PLANES AND LINES [Chap. IL The set of numbers (?<', v', ?t'', s') are called the homogeneous coordi- nates of the plane. Where no ambiguity arises, the accents will be omitted from the homogeneous coordinates. 30. Equation of a plane and of a point in homogeneous coordinates. If, in the equation If, in the equation Ax + By-\-Cz-\- D = An + By + Cjo + i) = {D^O, and A, B, C are not all (D ^ 0, and A, B, C are not all zero) the homogeneous coordi- zero) the homogeneous coordi- nates of a point are substituted, nates of a plane are substituted, we obtain, after multiplying by we obtain, after multiplying by t, the equation of the plane in s, the eqiiation of the point in homogeneous coordinates homogeneous coordinates Ax + Bt/ -\-Cz + Dt = 0. All + Bv + Civ + Ds = 0. The homogeneous coordinates The homogeneous coordinates of this plane are (^1, B, C, D). of this point are (A, B, C, D). 31. Equation of the origin. Coordinates of planes through the origin. The necessary and sufficient condition that the plane whose equation is ?<.r + ^n -f- vz -\- st = shall pass through the origin is .s=0. We see then that s = is the equation of the origin, and that (u, v, iv, 0) are the homogeneous coordinates of a plane through the origin. Since s = 0, it follows from Art. 29 that the non-homogeneous coordinates of such a plane cease to exist. 32. The plane at infinity. Let {x, y, z, t) be the homogeneous coordinates of a point. If we assign fixed values (not all zero) to X. y, z and allow t to vary, the corresponding point will vary in such a way that, as ^ = 0, one or more of the non-homogeneous co- ordinates of the point increases without limit. If t = 0, the non- homogeneous coordinates cease to exist, but it is assumed that there still exists a corresponding point which is said to be at infinity. It is also assumed that two })oints at infinity coincide if, and only if, their homogeneous coordinates are proportional. The equation of the locus of the points at iniinity is ^ = 0. Since this equation is homogeneous of the first degree in x, y, z, t, it will be said that ^ = is the equation of a plane. This plane is called the plane at infinity. Arts. 33, 34] COORDINATE TETRAHEDRON 35 33. Lines at infinity. Any finite plane is said to intersect the plane at infinity in a line. This line is called, the infinitely dis- tant line in the plane. The equations of the infinitely distant line in the plane Ax + Bf/ + Cz + Dt = are Ax + By + Cz = 0,t = 0, Theorem. Tlie condition that two finite planes are ^mrallel is that they intersect the plane at infinity in the same line. If the planes are parallel, their equations may be written in the form (Art. 15) Ax + By+ Cz +Dt = 0, Ax + By+Cz + D't = 0. (28) It follows that they both pass through the line Ax->rBy+Cz=0, t = 0. (29) Conversely, the equations of any two finite planes through the line (29) may be written in the form (28). The planes are there- fore parallel. 34. Coordinate tetrahedron. The four planes whose equations in point coordinates are a; = 0, y = 0, 2 = 0, t = will be called the four coordinate planes in homogeneous coordi- nates. Since the planes do not all pass through a common point, they will be regarded as forming a tetrahedron, called the coordi- nate tetrahedron. The coordinates of the vertices of this tetra- hedron are (0,0,0,1), (0,0,1,0), (0,1,0,0), (1,0,0,0). The coordinates of the four faces in plane coordinates are (0,0,0,1), (0,0,1,0), (0,1,0,0), (1,0,0,0). The equations of the vertices are u = 0, v = 0, iv = 0, .s = 0. EXERCISES 1. Find the iKm-homogeneous coordinates of the following points and plane.s : ^ (h) 10 a; -3y-(- 15 = 0, (e) u + v-w-l=0, (c) x-2 = 0, (f) 2w+ 11 =0. >^ 2. Determine the coordinates of the infinitely distant point on the line Sx -j-2 >j + [,t = U, 2x— \{)z + At = 0. 36 PLANES AND LINES [Chap. IL r3. Show that if Li{u) =AiU + Biv + CiW + Dis = 0, and Z2(m) =^2« + iJo?? + C2W + D2S = are the equations of two points, the equation of any point on the joining line may be written in the form kiLi + k^Li = 0. t- ''4. Show that the planes X + 22/ + 72 — 3«=:0, x + 3?/+62 = 0, x + 4j/ + 52 — 2^ = determine a parallel bundle. Find the equation of the plane of the bundle through the points (2, — 1, 1, 1), (2, 5, 0, 1). 35. System of four planes. The condition that four given planes L, = A,x + B^y + C,z + D,t =% L2 = A.2X + B^y + C.2Z + Dot = 0, A = A,x + B,y + Qz + Dit = 0, L, = A,x + B,y + dz -{-D,t = all pass through a point\is that four numbers {x, y, z, t), not alfl zero, exist which satisfy the four simultaneous equations. The condition is, consequently, that the determinant A, B, a ^1 Ao B. a A A, B, A A is equal to zero. If this condition is not satisfied, the four planes are said to be independent. When the given planes are independ- ent, four numbers A,, k,, k^, k^ can always be found such that the equation A'lLi + A'oLa + A-jLj + kjj^ = shall represent any given plane. For, let ax -j- by -\- cz + d — be the equation of the given plane. The two equations will repre- sent the same plane if their coefficients are proportional, that is, if numbers Atj, k,, k^, k^, not all zero, can be found such that a = k^Ai + A;, A + hA + A-4.14, b = k,B, + k,B2 + k,B, + k,B„ c = A'lCi -|- kiCi -f ksC^ -H ^4^4, d = k,D^ + k,D, + k,D, + Jc^D,. Since the planes are independent, the determinant of the coeffi- cients in the second members of these equations is not zero, and the numbers A;,, k^, k^, ki can always be determined so as to satisfy these equations. Art. 35] SYSTEM OF FOUR PLANES 37 These results, together with those of Arts. 24, 26, may be ex- pressed as follows : The necessary and sufficient condition that a system of planes have no point in common is that the matrix* formed by their coefficients is of rank four ; the planes belong to a bundle when the matrix is of rank three ; the planes belong to a pencil when the matrix is of rank two ; finally, the planes all coincide when the matrix is of rank one. We shall use the ex- pression " rank of the system of planes " to mean the rank of the matrix of coefficients in the equations of the planes. \ \r EXERCISES ^' \jj^ 1. Determine the nature of the following systems of planes : ' l^a) 2x — 5y + z — 3t = 0, x + y + 'iz — 5t =0, x + Sy + 6z-t = 0. (ft) 3x + iy + 5z-5t = 0, 6x + 5y + 9z-l0t = 0, 3x + Sy + 5z -5« = 0, x—y + 2z = 0. (c) 2x-f4j/ = 0, Hx + ly + 2z = 0, Sx + iy - 2z + ?,t = 0, x = 0. (d) 2x + £>y + Sz-0, 7y-[,z + it = 0, x-y + iz = 8t. JL^^ ^-^. Show that the line x + ?jy — z + t = 0, 2x-y + 2z — St-0 lies in the plane 7 x + 1 y + z — 3t = 0. ^3. Determine the conditions that the planes X = cy + bz, y = ax + cz, z = bx + ay shall have just one common point ; a common line ; are identical. 4. Prove that the planes 2x — Sy — 7z = 0, 3 x — 14 y — 13 z = 0, 8x — 31?/ — 33 2 = have a line in common, and find its direction cosines. 5. Show that the planes 3x — 2y — t = 0, ix — 2z — 2 t = 0, 4x -\- 4y — b z =0 belong to a parallel bundle. * Any rectangular array of uumbers Ai 2?i C'l Di ... 3/] A.2 B.2 C'2 Di - Mi An B„ Cn Z)„ Mn is called a matrix. Associated with every matrix are other matrices obtained by suppressing one or more of the rows or one or more of the columns of the given matrix, or both ; in particular, associated with every square matrix, that is, one in which the number of rows is equal to the number of columns, is a de- terminant whose elements are the elements of the matrix. Conversely, associated with every determinant is a square matrix, formed by its elements. We shall use the word rank to define the order of the non-vanishing determinant of high- est order contained in any given matrix. The rank of tlie determinant is defined as the rank of the matrix formed by the elements of the determinant. CHAPTER III TRANSFORMATION OF COORDINATES The coordinates of a point, referred to two different systems of axes, are connected by certain relations which will now be determined. The process of changing from one system of axes to another is called a transformation of coordinates. 36. Translation. Let the coordinates of a point P with respect to a set of rectangular axes OX, OY, OZ be (.c, y, z) and with respect to a set of axes O'X', 0' Y', 0' Z', parallel respectively to the first set, be (.«', y', z'). If the coordinates of 0', referred to the axes OX, Y, OZ are {h, k, l) we have (Fig. 19) x = x' + h, y = y'-\-J,; z = z' + l. (1) For, the projection on OX of OP is equal to the sum of the pro- jections of 00' and O'P (Art. 2), but the projection of OP is x, of 00' is h, and of O'P is x'; hence x — x' + h. The other formulas are derived in a similar way. Since the new axes can be obtained from the old ones by moving the three coordinate planes parallel to the X-axis a distance h, then parallel to the y-axis a distance k, and parallel to the .Z-axis a distance I, without changing their directions, the transformation (1) is called a translation of axes. 37. Rotation. Let the coordinates of a point P, referred to a set of rectangular axes OX, OY, OZ, be x, y, z, and referred to another rectangular system OX', OY', OZ' having the same origin, be x', y', z'. Let x' = OL', y' = L'M', z' = M'PiFig. 20); and let the direction cosines of OX', referred to OX, O Y, OZ, be A,, fxi, vi ; those of OY' be Aj, fi.2, vo, and of OZ' be A3, yu.3, v^. 38 Art. 37] ROTATION 39 We shall show that X = Aix' + XoXi' + Ajz', y = ii.,x' + /xo?/' + M32', (2) 2 = vix' + v^y' + v^z'. For, the projection of OP (Fig. 20) on the axis OX is x. The sum of the projections of OL', L'M', and M'P is A,.r' + A.^^' + A32:'. That these two expres- sions are equal follows from Art. 2. The second and third equations are obtained in a similar way. The direction cosines of OX, Y, and OZ, with re- spect to the axes OX', OY', OZ' are Ai, A,, A3; /ii, /xo, fi^; vi) V2, V3, respectively. If Ave '^"' " ' project OP and 0L= x, LM = y, and MP = z on OX', OY', and OZ', we obtain (2') x' = X^x + [x{y + v^z, y' = X^x + fi.y + v^z, z' — X^x + fji^y + v^z. The systems of equations (2) and (2') are expressed in con venient form by means of the accompanying diagram. x< y' z' x K A3 A3 y H-i H-2 H-3 z Vl v. V3 Since Aj, /aj, v^ ; A2, /u,o, v., ; A3, //.j, V3 are the direction cosines of three mutually perpendicular lines, we have the six relations K^ + f^l^ + n^ = 1> A1A2 + /X1/U2 -f ViVo = 0, V -f- /tA2^ 4- V2^ = 1, A2A3 + /X21U.3 + V2V3 = 0, (3) K^ + H-3' + »'3^ = Ij A3A1 + /i,3/Ai -f J/3V1 = 0. 40 TRANSFORMATION OF COORDINATES [Chap. III. We have seen that Ai, A2, A3; /xj, ^u,.,, /^a^ v,, vo, v^ are also the di- rection cosines of three mutually perpendicular lines. It follows that ^1^ + A.2^ + A-3^ = Ij KH-I + ^^2/^2 + ^3/^3 = 0, P-i^ + /A2^+ M3^ = Ij /AlVi + /A2l'2 + H-3V3 = 0, (4) Vl^ + V2^ + V3^ = 1, ViAj + V2A2 + V3A3 = 0. It will next be shown that Aj = € (1X2V3 — V2IJ.3), A2 = e (/X3V1 — V3/A1), A3 = e (/AiV2 — V1/A2), /xj = e(v2A3 — A2V3), |Li2 = e(v3Ai — A3V1), ^3 = e(viA2 — AiVj), (5) vj = e (A2M3— fJ.2>^3), Vo = e (Ag^i — ^3 Aj), V3 = e (Aj/Xj — /XjAs), where e= ±1. From the first and third equations of the last column of (4) we obtain Ai ^ A2 ^ A3 fUV3 — V2/A3 /A3V1 — V3/I.1 /XiVo — I'lfJ.o If we denote the value of these fractions by e, solve for Ai, A2, and A3 and substitute in the first of equations (4), we obtain f^[(/tA2"3 — ^2^3)- + (/A3V, — V3/A1)' +(/tiV2 — Viflof] = 1. Since the lines OY' and OZ' are perpendicular, the coefficient of c2 is unity (Art. 5, Eq. (5)). It follows that e^ = 1 or e = ± 1. The first three of equations (5) are consequently true. The other equa- tions may be verified in a similar way. It can now be shown that Ai A2 A3 fJ-i P-2 ^3 V, V2 V3 = ±1. (6) For, expand the determinant by minors of the elements of the first row, and substitute for the cofactors of Aj, A2) A3 their values from (5), The value of the determinant reduces to It will be shown in the next Article that if e = 1, the system of axes 0-X'Y'Z' can be obtained by rotation- from 0-XYZ. If c = — 1, a rotation and reflection are necessary. Art. 38] ROTATION AND REFLECTION OF AXES 41 38. Rotation and reflection of axes. Having given three mutu- ally perpendicular directed lines, forming the trihedral angle 0-XYZ (Fig. 21), and three other mutually perpendicular directed lines through 0, forming the trihedral angle 0-X'Y'Z', we shall show that the trihedral angle 0-XYZ can be revolved in such a way that OX and OZ coincide in direction with OX' and OZ', respectively. OY will then coincide with OY' or will be di- rected oppositely to it. Let Xy be the line of intersection of the planes XOY and X'OY'. Denote the angle ZOZ' by 6, the angle XOX by , and Z' k Z the angle XOX' by ij/. Let the axes 0-XYZ be revolved as a rigid body about OZ through the angle , so that OX is revolved into the position OX. Denote the new position of OF by OY^, so that the angle YOY^ = (f>. The trihedral angle 0-XYZ is thus re- volved into 0-X^Y,Z. Now let 0-XY^Z be revolved about OX thi-ough an angle 6, so that OZ takes a position OZ', and OFj, a F^^- ^l- position OY2. Then the angle ZOZ' = angle YiOY2 = e. The trihedral angle O-X^Y^Z is thus brought into the position O-XY2Z'. Finally, let the trihedral angle in this last position be revolved about OZ' through an angle ij/, so that O^is revolved into OX'. By the same operation OFis revolved into a direction through perpendicular to OX' and to OZ'. It either coincides with OY' or is oppositely directed. In the first case the trihedral 0-XYZ has been rotated into the trihedral 0-X'Y'Z'. In the second case the rotation must be followed by changing the direc- tion of the F-axis. This latter operation is called reflection on the plane i/ =0. It cannot be accomplished by means of rotations. In case the trihedral 0-XYZ can be rotated into 0-X'Y'Z', the number « (Art. 37) is positive ; otherwise, it is negative. For, during a continuous rotation of the axes, the value of e (Eq. (6)) cannot change discontinuously. If, after the rotation, the trihe- drals coincide, we have, in that position, Ai = /Hj = V3 = 1 and the 42 TRANSFORMATION OP COORDINATES [Chap. III. other cosines are zero, so that (Eq. (G)) e = 1*. If, however, at the end of the rotation, Y and Y' are oppositely directed, Aj = V3 = 1, /A2 = — 1> and e = — 1. 39. Euler's formulas for rotation of axes. Let the coordinates of a point P referred to 0-XYZ be {x, y, z), referred to O-NY^Z be (xi, 2/1, ^i), referred to O-NY2Z be ix2, y-i, z^), and referred to 0-X' Y'Z' be {x', y', z'), (Fig. 21). In the first rotation, through the angle ^, z remains fixed. Hence, from plane analytic geometry, z = 2,, X = Xi cos — yi sin <^, y = x^ sin <^ + 2/1 cos cos ij/ cos ^) + z' sin sin ^. y = x' (sin <^ cos i/' + cos <^ sin ip cos 0)— ^(sin ^.sin \f/ — cos cos li' cos 6) — z' cos sin ^. z = x' sin i/^ sin 6 + y' cos i// sin ^ + z' cos ^. If 0-X' Y'Z' cannot be obtained from 0-XYZ hy rotation, the sign of y' should be changed. These formulas are known as Euler's formulas. 4-0. Degree of an equation unchanged by transformation of co- ordinates. If in an equation F{x, y, z) = the values of x, y, z are replaced by their values in any transformation of axes the degree of F cannot be made larger, since x, y, z are replaced by linear ex- pressions in x', y', z'. But the degree of the equation cannot be made smaller, since by returning to the original axes and to the original equation, it would be made larger, which was just seen to be impossible. Art. 40] EXERCISES 43 EXERCISES 1. Transform the equation x'^ — 3 yz + y- — 6 x + z = to parallel axes through the point (1, —1,2), 2. By means of equations (2) show that the expression x^ + y^ -\- z- is un- changed by rotation of the axes. Interpret geometrically. 3. Show that the lines x = ^ = ^; - = -^ = z ; - =y = -^— are mu- 4 22-1 '2^-3 tually perpendicular. "Write the equations of a transformation of coordinates to these lines as axes. *^4. Translate the axes in such a way as to remove the first degree terms from the equation x"^ -2y^ + Qz^ - \Gx — 4y — 24:Z + 37 = 0. ^. Show that the equation ax + by + cz + s = may be reduced to x = by a transformation of coordinates. ^' 6. Find the equation of the locus 11 x- + 10 y- + 6 z'^ — 8 yz + i zx — 12 xy — 12 = when lines through the origin whose direction cosines are ^, |, | ; h h ~ ^> ~ h h ~ i ^^® taken as new coordinate axes. 7. Show that if 0-X' Y'Z' can be obtained from 0-XYZ by rotation, and if OY can be made to coincide with OX by a revolution of 90 degrees, counterclockwise, as viewed from the positive end of the /^-axis, then OY' can be revolved into OX' by rotating counterclockwise through 90 degrees as viewed from the positive Z'-axis. 8. Derive from Ex. 7 a necessary and sufficient condition that 0-X Y'Z can be obtained from 0-X YZ by rotation. CHAPTER IV TYPES OF SURFACES 41. Imaginary points, lines, and planes. In solving problems that arise in analytic geometry, it frequently happens that the values of some of the quantities x, y, z which satisfy the given conditions are imaginary. Although we shall not be able to plot a point in the sense of Art. 1, when some or all of its coordinates are imaginary, it will nevertheless be convenient to refer to any triad of numbers x, y, z, real or imaginary, as the coordinates of a point. If all the coordinates are real, the point is real and is de- termined by its coordinates as in Art. 1 ; if some or all of the coordinates are imaginary or complex, the point will be said to be imaginary. Similarly, a set of plane coordinates u, v, w will de- fine a real plane if all the coordinates are real ; if some or all of the coordinates are imaginary, the plane will be said to be imaginary. A linear equation in x, y, z, with coefficients real or imaginary, will be said to define a plane, and a linear equation in ti, v, w, with coefficients real or imaginary, will be said to define a point. The equations of any two distinct planes, considered as simul- taneous, will be said to define a line. It follows that if (.x-j, y^, z{) and (.x-2, y-,,, 2.>) are any two points on the line, then the coordinates of any othei" point on the line can be written in the form A-jXi + A-jXg, etc. The line is also determined by the equations of any two distinct points on it. The line joining two imaginary points is real if it also contains two real points. If P =(a -\- ik, b -f il, c + im) is an imaginary point, the point P' =(a — ik, b — il, c — im), whose coordinates are the respective conjugates of those of P, is called the point conjugate to P. The line joining any two conjugate points is real ; tlius the equations of the line PP' are Ix — ky -{- bk — al = 0, (bm — d)x +{ck — am)y +(al — bk)z = 0. The line of intersec- 44 Art. 41] IMAGINARY POINTS, LINES, AND PLANES 45 tion of two imaginary planes is real if through it pass two distinct real planes. The line of intersection of two conjugate planes is rer' Fr^ii^ the preceding it follows that no imaginary line can con- tain more than one real point, and through an imaginary line cannot pass more than one real plane. If a plane passes through^ an imaginary point and not through its conjugate, the plane is Uttc^ imaginary. If a point lies in an imaginary plane and not in j its conjugate, the point is imaginary-. One advantage of using the form of statement suggested in this Article is that many theorems may be stated in more general form than would otherwise be possible. We may say, for example, that every line has two (distinct or coincident) points in common with any given sphere. With these assiimptions the preceding formulas will be applied to imaginary elements as well as to real ones. No attempt will be made to give to such f(n-mulas a geometric meaning when imagi- nary quantities are involved. In the following chapters, in all discussions in which it is necessary to distinguish between real and imaginary quantities, it will be assumed, unless the contrary is stated, that given points, lines, and planes, and the coefficients in the equations of given surfaces, are real. ' ' EXERCISES 1. Show that the point (2 + i, 1 + 3 i, i) lies on the plane x — 2y + 5 2=0. V2. Find the coordinates of the points of intersection of the line whose parametric equations are (Art. 20) x = 1 + ^^ d, y = — 2 + ^^ d, z = 5 — \f d, with the sphere x'^ + y'^ + z^ = 1. ' ^3. Show that the line of intersection of the planes x + ?^ = 0, (1 -j^ *)^^ a • 4. Find the coordinates of the point of intersection of the line through (3, 2, - 2) and (4, 0, 3) with the plane x + 3 y + (1 - 2 1)2^ + 1 = 0. ' 5. Find the equation of the plane determined by the points (5 -|- i, 2,-2 - 0, (4 + 2 f, -1 + 2 i, 0), (i, 1 + 2 (\ 1+3 0. 6. Determine the points in which the sphere (x - l)'^ + r/^ + (0 +2)2 = 1 intersects the X-axis. 46 TYPES OF SURFACES [Chap. IV. 42. Loci of equations. The loons defined by a single eqnation among the variables x, y, z is called a surface. A point P= (.x'l, yi, 2i) lies on the surface i^= if, and only if, the coor- dinates of P satisfy the equation of the surface. We have seen, for example, that the locus of a linear equation is a plane. More- over, the locus of the equation a;2 + ^2 ^_ ^2 ^ 1 is a sphere of radius unity with center at the origin. The locus of the real points on a surface may be composed of curves and points, or there may be no real points on the surface ; for example, the locus of the real points on the surface X^ + 7/2 = is the Z-axis ; the locus of real points on the surface •«' + y2 + 2^ = is the origin; the surface x'-\-if + z^ + l = has no real points. If the equation of a surface is multiplied by a constant different from zero, the resulting equation defines the same surface as be- fore; for, if P= is the equation of the surface and k a constant different from zero, the coordinates of a point P will satisfy the equation kF = if, and only if, they also satisfy the equation F=0. The locus of two simultaneous equations is the totality of the points whose coordinates satisfy both equations. If F{x, y, z)=0, fix, y, z) =0 are the equations of two surfaces, then the locus of the simultaneous equations ^=0, /=0 is the curve or curves in which these surfaces intersect. Every point on the curve of in- tersection may be imaginary. The locus of three simultaneous equations is the totality of the points whose coordinates satisfy the three simultaneous equations. /?■ EXERCISES 1. Find the equation of the locus of a point whose distance from the Z-axis is twice its distance from the JTF-plane. 2. Discuss the locus defined by the equation x^ + x'^ — yf-. "^ 3. Find the equation of the locus of a point the sum of the squares of whose distances from the points (1, 3, — 2), (6, — 4, 2) is 10. Arts. 43, 44] PROJECTING CYLINDERS 47 r 4. Find the equation of the locus of a point whicli is three times as far from the point (2, 6, 3) as from the point (4, — 2, 4). 5. Find the equations of the locus of a point wliich is 5 units from the XF-plane and 3 units from the point (3, 7, 1). '6. Find tlie equations of tlie locus of a point which is equidistant from the points (2, 3, 7), (3, -4, 6), (4, 3, -2). 7. Find the coordinates of the points in which the line x = — i, z = 2 in- tersects the cylinder y^ = -ix. 43. Cylindrical surfaces. It was seen in Art. 42 that the locus of a single equation F{x, y, z) = is a, surface. We shall now discuss the types of surfaces which arise when the form of this equation is restricted in certain ways. Theorem. If the equation of a surface involves only tivo of the coordinates x, y, z, the surface is a cylindrical surface ivhose generat- ing lines are ^mrallel to the axis ivhose coordinate does not appear in the equation. Let/(iK, ?/) = be an equation containing the variables x and y but not containing z. If we consider the two equations /(.c, y)=0, z = simultaneously, we have a plane curve f(x, y) = in the plane z = 0. It (xi, y-^, 0) is a point of this curve, /(o^i, ?/i) = 0. The coordinates of any poiut on the line x = x^, y = yi are of the form a^, y^, z. But these coordinates satisfy the equation f{xy, y^) = independently of z, hence every point of the line lies on the surface f(x, y) = 0. It is therem^e generated by a line moving par- allel to the Z-axis and always intersecting the curve /(.«, y) = in the XF-plane. The surface is consequently a cylindrical surface. In the same w^ay it is shown that {x, z) =0 is the equation of a cylindrical surface whose generating elements are parallel to the F-axis, and that F(y, z) ~0 is the equation of a cylindrical sur- face whose generating elements are parallel to the X-axis. 44. Projecting cylinders. A cylinder whose elements are per- pendicular to a given plane and intersect a given curve is called the projecting cylinder of the given curve on the given plane. The equation of the projecting cylinder of the curve of inter- section of two surfaces F(x, y, z) = 0, f(x, y, z) = on the plane 2 = is independent of z (Art. 43). The equations of this cylin- 48 TYPES OF SURFACES [Chap. IV. der may be obtained by eliminating z between the equations of the curve. If i^ and /are polynomials in z, the elimination may be effected in the following way, known as Sylvester's method of elimination. Since the coordinates of points on the curve satisfy i^=0 and /=0, they satisfy F= 0, 2i^= 0, z'F= 0, •", /= 0, zf= 0, ^y = 0, ..., simultaneously. If we consider these equations as linear equa- tions in the variables z, z"^, z^, —, and eliminate z and its powers, we obtain an equation R{x', y) = 0, which is the equation required. The following example will illustrate the method. Given the curve 2^ + 3 .T2 -f X + 2/ = 0, 2 ^2 + 3 2 + .f + ^2 ^ 0. The equation of its projecting cylinder on 2 = is found by elimi- nating 2 between the given equations and z^ -I- 3a-22 + {x-\-y)z = 0, 2 z' -\-3 z^ + (x 4- y^)z = 0. The result is 1 3x x+y 1 3 a; X + y 2 3 x + y^ 2 3 a; + ?/ which simplifies to (y- —2y — xy = 9 (1 — 2 a;) {xy'^ -\-x'^ — x — y). The equations of the projecting cylinders on a; = and on y = may be found in a similar manner. 45. Plane sections of surfaces. The equation of the projecting cylinder of the section of a surface F{x, y, z)=0 by a plane 2 = A; parallel to the XF-plane may be found by putting 2 = A: in the equation of the surface. The section of this cylinder F(x,y, k) = by the plane 2 = is parallel to the section by 2 = k. Since paral- lel sections of a cylinder, by planes perpendicular to the elements, are congruent, we have the following theorem : Theorem. If in the equation of a surface, loe put z = k and con- sider the result as the equation of a curve in the plane 2 = 0, this carve is congruent to the section of the surface by the plane z = k. = 0, Art. 46] CONES 49 46. Cones. A surface such that the line joining an arbitrar}^ point on the surface to a fixed point lies entirely on the surface is a cone. The fixed point is the vertex of the cone. Theorem. If the equation of a surface is homogeneous in x,y, z, the surface is a cone luith vertex at the origin. 'Letf{x, y, z)=0 be the equation of the surface. Let / be ho- mogeneous of degree n in {x, y, z), and let P^ =(xi, yi, Zi) be an arbitrary point on the surface, so that/(a;i, ?/i, Zi)= 0. The origin lies on the surface, since /(O, 0, 0) = 0. The coordinates of any point P on the line joining P^ to the origin are (Art. 6) x = kxi, yz=ky^, z^Jcz^, where k = r/i, + ma But the coordinates of P satisfy the equation, since f{^, y, 2) = /(A;.r„ %i, kz,) = k''f{x^, y„ z{)=0 for every value of k. Thus, every point of the line OPi lies on the surface, which is therefore a cone with the vertex at the origin. EXERCISES 1. Describe the loci represented by the following equations : (a) x2 + y2^4. ^ iL' + ^=l. ^ ^ 4 9 (b) y- = x. .. ^_L^= 1 ^^49' (c) j/ = sinx. v' (f) x(x-l){x-l){x-S) = 0. 2. Describe as fully as possible the locus of the equation 4 x- + i/^ = 25 z^. 3. Show that the section of the surface a;'^ + y- = 9 ^ by the plane 2 = 4 is a circle. Find the coordinates of its center and the length of its radius. \ 4. Find the equation of the projection upon the plane 2 = of the curve of intersection of the surfaces 2/2+1=0, {X? +?/2- 1)2 + 2 2/ = 0. * 5. Show that the section of the surface x'^z'^ + a?y'^ = r-z"^ by the plane z = k is an ellipse. Find its semi-axes. By giving k a series of values, de- termine the form of the surface. ^ 6. Show that if the equation of a surface is homogeneous in x — ^, y — k, z — I, tlie surface is a cone with vertex at (/i, k, I). 7. By using homogeneous coordinates, show tliat the cylinder /(x, y, t) =0 can be considered a cone with vertex at (0, 0, 1, 0). 50 TYPES OF SURFACES [Chap. IV. 47. Surfaces of revolution. The surface generated by revolving a plane curve about a line in its plane is called a surface of revo- lution. The fixed line is called the axis of revolution. Every point of the revolving curve describes a circle, whose plane is per- pendicular to the axis of revolution, whose center is on the axis and whose radius is the distance of the point from the axis. To determine the equation of the surface generated by revolving a given curve about a given axis, take the plane of the given curve for the X5^-plane and the axis of revolution for the X-axis. Let the equation of the given curve in 2 = be fix, y) = 0. Let Pi = (a*!, ?/i, 0), Fig. 22, be any point on the curve, so that f(xi, y^) = Fig. 22. and let P = (a-, ?/, z) be any point on the circle described by Pj. Since the plane of the circle is perpendicular to the X-axis, the equation of this plane is .k = ^,. The coordinates of the center C of the circle are C = (x^, 0, 0); and the radius CPi is y^. The distance from C to P is ,/, = V(^i - x,y + (y - Of +{z- oy = V2/2 + z^- On substituting , , ; a-i = X, ?/i = V.y- -f- z- in the equation /(.t'l, ?/i)=0 we obtain, as the condition that the point P lies on the surface, f{x, V?^+^)=0, which is the desired ecpmtion. In the same way it may be seen that the equation of the sur- face of revolution obtained by revolving the curve /(x, ?/)= about the F-axis is • /-or ? s n /( Vx^ ■+- z\ y) = 0. Art. 47] EXERCISES 51 EXERCISES 1. What is the equation of the surface generated by revolving the circle x2 + t/2 = 25 about the X-axis ? about the I'-axis ? y^l. Obtain the equation of the surface generated by revolving the line 2x + 3y = 15 about the X-axis. Show that the surface is a cone. Find its vertex. What is the equation of the section made by the plane x = ? Find the equation of tlie cone generated by revolving the line about the I'-axis. 3. Why is the resulting equation of the same degree as that of the gen- erating curve in Ex. 1, but twice the degi'ee of the given curve in Ex. 2? Formulate a general rule. • 4. What is the equation of the surface generated by revolving the line y = a about the X-axis ? about the i'-axis ? . 5. If the curve /(.r, tj) = crosses the x-axis at the point (xi, 0, 0), de- scribe the appearance of the surface /(./•, Vy'^ -f- z^) — near the point (xi, 0, 0). 6. Find the equation of the surface generated by revolving the following curves about the A'-a.-cis and about the F-axis. Draw a figure of each surface. («) T + -n- = l- ('■) y- = ^^- (^) 2/ = sinx. 4 9 (6) ^-f-'=l- (cO x2+(^- 1)2 = 4. if)y=e'. a- 0^ ^ / J J- X. ' . CHAPTER V THE SPHERE 48. The equation of the sphere. The equation of the sphere having its center at (a"o, y^, Zq) and radius r is (x - x,y +{y - y,y + {z- z,f = r\ (1) or ar' + 2/^ + 2- - 2 x^x -2yQy-2zoZ + Xq" + ?/o'^ + Zq^ -i~ = 0. Any equation of the form a{x'' + y^ + z'') + 2fx + 2gy + 2hz + k = 0, a^O (2) may be written in the form ..{J.(..^J.(..fJ=/l±^^if^^. (3) jf j2 _j_ ^2 _j_ ^2 _ (^j^. ^ Q^ |.j^^g ^g segi^^ \)y comparing with (1), to be a sphere with center at ( — — , —", ) and radius \ a a a) V /2 + ^2 ^ }C- - nk a If the expression under the radical sign vanishes, the center is the only real point lying on the sphere, which in this case has a zero radius, and is called a point sphere. If the expression under the radical is negative, no real point lies on the locus, which is called an imaginary sphere. 49. The absolute. We shall now prove the following theorem: Theorem I. All spheres intersect the plane at infinity in the same curve. In order to determine the intersection of the s])liere and the plane at infinity, we first write the equation of the sphere in homogeneous coordinates : a (x-2 + / + z"") +2fxt + 2 yyt + 2 hzt + kf^ = 0, a ^ 0. 52 Art. 49] THE ABSOLUTE 53 The equations of the curve of intersection of this sphere with the plane at infinity are t^O, a;- + / + 2' = 0. (4) Since these equations are independent of the coefficients a, f, g, h, k which appear in the equation of the sphere, the theorem follows. The curve determined by equations (4) is called the absolute. Since the homogeneous coordinates of a point cannot all be zero (Art. 29), there are no real points on the absolute. The equation of any surface of second degree which contains the absolute may be written in the form a {x^ + y- + z^) + (kx + hj-\- VIZ + nt) t = 0. It a ^ 0, this is the equation of a sphere (Art. 48). If a = 0, the locus of the equation is two planes of which at least one is ^ = 0. In the latter case also, we shall call the surface a sphere, since its equation is of the second degree and it passes through the abso- lute. When it is necessary to distinguish it from a proper sphere, it will be called a composite sphere. With this extended defini- tion, we have at once the following theorem : Theorem II. Every surface of the second degree which contains the absolute is a sphere. Any plane ux 4- vy 4- wz -\- st — 0, other than t = 0, intersects the absolute in two points whose coor- dinates may be found by solving the equation of the plane as simultaneous with the equations of the absolute. Any circle in this plane is the intersection of the plane with a sphere. Since the absolute lies on the sphere, the circle must pass through the two points in which its plane intersects the absolute. These two points are called the circular points in the plane. Evidently all the planes parallel to the given one will contain the same circular points. The reason for the designation circu- lar points is seen from the fact that any conic lying in any real transversal plane and passing through the circular points is a circle, as will now be shown. Since the equations of the absolute are not changed by displacement of the axes, it is no restriction 54 THE SPHERE [Chap. V. to take z = for the equation of the transversal plane. The coordinates of the points in which the plane 2=0 meets the curve ^ = 0, X' + y^ + z^ = are (1, i, 0, 0), (1, — i, 0, 0). A conic in the plane z = has an equation in homogeneous coordinates of the form Ax"- -f Bi/^ + 2 Hxy + 2 Gxt + 2 Fyt + Cf = 0. If the points (1, /, 0, 0), (1, — i, 0, 0) lie on this curve, A= B, 11=0. But these are exactly the conditions that the conic is a circle. Conversely, it follows at once that every circle in the plane z = passes through the two circular points in that plane. A conic in an imaginary plane will be defined as a circle if it passes through the circular points of the plane. If the two circular points in a plane coincide, the plane is said to be tangent to the absolute. Such a plane is called an isotropic plane. The condition that the plane ux + vt/ + ivz -j- st = is isotropic is found, by imposing the condition that its intersections with the absolute coincide, to be u" + V- + iv^ = 0. (5) This equation is the equation of the absolute in plane coordinates. EXERCISES 1. Write the equation of a sphere, given (a) center at (0, 0, 0) and radius r, (6) center at (— 1, 4, 2) and radius 6, (c) center at (2, 1, 5) and radius 4. 2. Determine the center and radius of each of the following spheres: (a) x^ + y'^ + z'^ + 7x + 2y + z + 5 = 0. lb) x2 + 2/2 -f ^2 + 2 X + 4 ?/ - 6 2 + 14 = 0. i (c) 2(x'^ + y^ + z^)-x-2y + 5z + 6 = 0. (d) x^ + y^ + z^+fx^O. 3. Find the points of intersection of the absolute and the plane 2 x - y + 2 z + l^ t = 0. 4. Find the coordinates of the points of intersection of the line x =— 2 + I d, 2/ = 3 - f (?, = - 2 + i c? with the sphere x^ + y'^ + z- + 1=0. 5. Show that x^ -iry'^ + z^ = is the equation of a cone. 6. Find the distance of the point (1, 0, i) from the origin. / Arts. 49-51] THE ANGLE BETWEEN TWO SPHERES 55 7. Show that the radius of the circle in which ^ = 2 intersects the sphere - jJ^ ^ ^' '^ ^'-^ ^^ imaginary. iiVtui^ ) yv^^B. Prove that, if (xi, ij\, Zi) is any point exterior to the sphere (x — Xo)^ / ^^.v«-.«-ei^ + {y — 2/o)- + (2 — 2o)- - r'^ tlie expression (xi — Xq)- + (iji - 2/o)^ + {zi — ZoY \]j^it~^ — r^ is the square of the segment on a tangent from (xi, t/i, Zi) to the point of 'u > ^ contact on the sphere. /*^ 50. Tangent Plane. Let P = (.ri, ?/,, Zi) be any point on the sphere a(2;2 + y2 ^ ^2) ^ 2/a; + 2 gy + 2 liz + k = 0. The plane passing through P perpendicular to the line joining P to the center of the sphere is the tangent plane to the sphere at P. \i is required to find its equation. The coordinates of the center are ( — •-, — •-, )• The equations of the line joining \ a a a) the center to P are (Art. 19) X — a;, 11 — ?/, 2 — 2, - ^ - .Ti -'^-]}x ^x a a a The ecjuation of the plane passing through P and perpendicular to this line is Fig. 23. If we expand the first member of this equation and add to it a{Xi^ + 7/^'^ + Zi'^) + 2fXi + 2 gyi+2hz,+k, which is equal to zero since the point {x^, y^, z^) lies on the sphere, we obtain «('^-i^' + yiy + ^1^) + /(■« + .i-i) + giih + y) + K^^i ^z) + k = o, (6) which is the required equation of the tangent plane. 51. The angle between two spheres. The angle between two spheres at a point P^ on their curve of intersection is defined as equal to the angle between the tangent planes to the spheres at Py To determine the magnitude of tliis angle, let the coordinates of Pj be (xi, ?/i, 2i) and let the equations of the spheres be a(x-- + v/2 \- z-')+2fx -^2 gy + 2I1Z + k ^i), a'(x^ + 2/2 + Z-) + 2f'x -\-2g'y + 2h'z + k' = 0. 56 THE SPHERE [Chap. V. The equations of the tangent planes to these spheres at P^ are a{x^x + y,y + z,z) -\-f{x + x^) + g{y + Ih) + h{z + ^j) + A; = 0. a\x,x + y,y + z,z)+f\x + x,) + g\y + y,) + h'{z + ^i) + A;' = 0. Since the angle 9 between the spheres is equal to the angle between these planes, we have (Art. }5) ^ cog Q ^ L 9^%<4>iL itUiJ-^M^ y^ CA}<^^^ynjtXJ . {axi +f){a'xi +f) + (ayi + g)(a'yi + (j') + (azi+h){a'zi + h') ^(axi+fy + {a>ji + gy^+iazi + hyV{a'xi+f'y^ + {a'yi+g'y+{a'zi + h'y' Since (x■^, y^, z^) lies on both spheres, this relation reduces to 2 v/r + g^ + li" - akVf'^ + g"" + h''' - a'k' Since this expression is independent of the coordinates of P^, we have the following theorem : Theokem. Tiro sjyheres intersect at the same angle at all points of their carve of intersection. If ^ = 90 degrees, the spheres are said to be orthogonal. The condition that two spheres are orthogonal is 2 //•' + 2 (/r/' + 2 hh' - aJc' - a'k = 0. (8) 52. Spheres satisfying given conditions. The equation of a sphere is homogeneous in the five coefficients a, f, g, h, k. Hence the sphere may he made to satisfy four conditions, as, for example, to pass through four given points, or to intersect, four given spheres at given angles. If the given conditions are such that a = 0, the sphere is composite (Art. 49). EXERCISES 1. Prove that the point (— 3, 1, — 4) lies on the sphere x^ + y^ -]- z"^ + 6x + 24y-{-Sz = and write the equation of the tangent plane to the sphere at that point. 2. Find the angle of intersection of the spheres x^ -\- y- + z- + x + 6 y + 2 z + \) = 0, x:^ + y^ + z^ + 5 X + S z + i = 0. ^ 3. Find the equation of the sphere with its center at (1, 3, 3) and making an angle of 60 degrees with the sphere x"^ +■ ?/2 + z- = 4. • 4. Determine the equation of the sphere which passes through the points (0,0,0), (0,0,3), (0,2,0), (1,2, 1). Arts. 52, 53] LINEAR SYSTEMS OF SPHERES 57 5. Determine the equation of the sphere which passes through the points (1,3,2), (3,2, -5), (-1,2,3), (4,5,2). i^' 6. Write the equation of the sphere passing through the points (2, 2, — 1), (3, — 1, 4), (1, 3, —2) and orthogonal to the sphere ■X?- -\- y- -V z'^ — Z X -\- y + z = ^. *i^. Write the equation of the sphere inscribed in the tetrahedron x = 0, y = 0, 5x+12 2 + 3 = 0, 3x-12y + 42i=0. 53. Linear systems of spheres. Let S' = a' (x^ + y^ + z") + 2fx + 2 f/'y + 2 li'z + A;' = be the equation of two spheres. The equation A,6' + X^S' = 0, or (a.\i + a'\,){x'^ + y^ + z-^)+2 (f\, +/A,) x +2 (gX, + 9'X,)y + 2 (/iAi + /i'A,) z + kX, + k'X. = also represents a sphere for all values of X^ and A,. Every sphere of the system Ai*S + AoS" = (12) passes through the points of intersection of the spheres »S'=0, S' = 0, aS"' = 0. Every sphere of the system (12) determined by values of Aj, A2, A3 for which Aia + A,,o' +A3a" = is composed of two planes of which one is the plane at infinity and the other passes through the line 2 {a\f- af')x + 2 (a V/ - ag')y + 2 {a'h - ah')z + a'k - ak' = 0, (13) 2 (a"f-af")x + 2(a"(/ - ag ")y + 2{a"h - ali")z + a"k - ak" = 0. r a"$'etS' This line is called the radical axis of the system of spheres (12), By comparing equations (13) with (11) and the equation analo- gous to (11) for S" = 0, it may be shown that the radical axis is the locus of centers of the spheres which intersect all the spheres of the system (12) orthogonally. Now let S'" = a'" (.f2 + 7/2 + z^) + 2f"'x + 2g"'y + 2 h"'z + k'" = be the equation of a sphere whose center is not in the plane de- termined by the centers of /S' = 0, /S' = 0, ;S" = 0. The condition that a sphere of the system X,S + \oS' + X^S'' + XS"' = ^ is composite, is that Ai A2 A3 and A4 satisfy the relation Aitt -\- Ajtt' + Aatt" -f A4a"' = 0. Art? 53, 54] STEREOGRAPHIC PROJECTION 59 The sphere orthogonal to all the spheres of the system is in this case uniquely determined by equations analogous to (W). The center of this orthogonal sphere is called the radical center of the system. Through the radical center passes one plane of every composite sphere of the system. EXERCISES •7 1. Prove that the center of any sphere of the system \iS + X2 = 0. V£ti/^'y-y^'*^CU, + 2 r^Q V((7a;i + 2 ?-^)2+ ((7?/i + 2 rBf ^{C'x, + 2 r^')2+ (C'y, + 2 ri^')^ • By expanding this expression and making use of the fact that (.Tj, 2/i) lies on both circles, we may simplify the preceding equa- tion to AA' + BB' + CC ,.rr. cos<^ = — ~ — — =^=1 — • (17) V^2 + B'+ C VA'' + B" + C"2 From (16) and (17) we have cos 6 = cos . We may conse- quently choose the angles in such a way that 6 = 0. The segment joining the vertices on the X-axis is then known as the major axis ; that joining the vertices on the Y'-axis as the mean axis; that joining the vertices on the Z-axis as the minor axis. The section of the ellipsoid by the plane z = A; is an ellipse whose equations are C' 63 64 FORMS OF QUADRIC SURFACES [Chap. VI. The semi-axes of this ellipse are a*ji _ ^^ ^-Jl — — . As | A; | in- creases from to c, the axes of the ellipses of section decrease. If \Jc\=c, the ellipse reduces to a point. If j Zc j > c, the ellipse of section is imaginary, since its axes are imaginary. The real part of the surface j|l^_______^^ therefore lies en- Yv.,^ tirely between the planes z = c and z= — c. In the same man- ner, it is seen that the plane y = k' intersects the sur- face in a real ellipse if\k'\ 6. Finally, it is seen that the section x = k" is a real ellipse, a point, or an imaginary ellipse, according as | k" \ is less than, equal to, or greater than a. The ellipsoid, there- fore, lies entirely within the rectan- gular parallelepiped formed by the planes X = a, ?/= b, z =c; x= - a, y=-b, z= — c, and has one point on each of these '^^BBIll^^kis^'Ha^® planes (Fig 25). If a = b > c, the ellipsoid is a surface of revolution (Art. 47) obtained by revolving the ellipse x^ y^ _ M Fig. 25. about its minor axis. This surface is called an oblate spheroid. If a > 6 = c, the ellipsoid is the surface of revolution obtained Arts. 56, 57] THE HYPERBOLOID OF ONE SHEET 65 by revolving the same ellipse about its major axis. It is called a prolate spheroid. If a = b = c, the surface is a sphere. 57. The hyperboloid of one sheet. The surface represented by the equation „ „ „ ^" + ^ _ ?_" = 1 a^ 6^ c^ is called an hjrperboloid of one sheet. It is symmetric as to each of the coordinate planes, as to each of the coordinate axes, and as to the origin. The section of the surface by the plane z = k is an ellipse whose equations are This ellipse is real for every real value of k. The semi-axes are «v which are the smallest when A- = 0, and increase without limit as I k I increases. For no value of k does the ellipse reduce to a point. The plane y = k' intersects the surface in the hyperbola = 1, y = k: If \k' \ < b, the transverse axis of the hyperbola is the line 2 = 0, y = k', and the conjugate axis is x = 0,y = k'] the lengths of the semi-axes are a\ 1 , c\ 1 . As I A;' I increases from zero to b, the semi-axes decrease to zero. When j k' ] = b, the equation cannot be put in the above form, but becomes ~ = a 2 c^ and the hyperbola is composite ; it consists of the two lines "^ + ' = 0, y=b; "^-" = 0, y = b; a c a c 66 FORMS OF QUADRIC SURFACES [Chap. VI. when k' = — b, the hyperbola consists of the lines 0, y = -b. - + - = 0, y = -b; ---■■ a c a c These four lines lie entirely on the surface. If \k' \ > b, the transverse axis of the section is x — 0, y = k' and the conjugate axis is z = 0, y = k'. The lengths of the serai-axes are 62 ' \ Jf. They increase without limit as k' increases. The plane x = k" intersects the surface in the hyperbola y^ z^ bH\ cn\- A:"2 1, x=k'\ Of- J \ a" If I A;" I < a, the transverse axis of this hyperbola is 2 = 0, .-c = A:". The section on the plane x = a consists of the two lines _-^ "7'-0'-3 b 0, a; = a ; "^ = 0, a; = a. J -'rz: c be The section on the plane x = — a consists of the lines | + ? = 0, .r = -a; f-? = 0, x = -a. be be If I k" I > a, the line y = 0, x = k" is the transverse axis and 2=0, x = k" is the conjugate axis. As I k" I increases, the lengths of the semi-axes increase with- out limit. The form of the surface is indicated in Fi(A-, respectively. If A; < 0, the ellipse is imaginary. If A- = 0, the ellipse reduces to a point, the origin. As A' increases, the semi-axes of the ellipse increase without limit. The section of the paraboloid by the plane y = A' is the parabola For all values of k' these parabolas are congruent. As k' in- creases, the vertices recede from the plane y= along the parabola y-=2nz, x = 0. 62 V 70 FORMS OF QUADRIC SURFACES [Chap. VI. Fig. 28. li a — b, the paraboloid is tlie surface of revolution generated by revolving the parabola '— = 2 nz, y = about the Z-axis. 61. The hyperbolic parab- oloid. The surface defined by the equation The sections by the planes x = k" are the congruent parabolas r_9 = 'Z vz 7.M2 '^, x = k". Their vertices describe the pa- rabola — = 2 nz, y = o. The form of the surface is in- dicated by Fig. 28. is called an hyperbolic paraboloid. The surface is symmetric as to the planes x =0 and ?/ = 0, but not as to 2 = 0. As before, let it be assumed that n > 0. The plane z = k inter- sects the surface in the hyperbola ^^ - -^^^ =1 z = 7c a^2nk b^2nk ' If A; > 0, the line x — 0, z = k \s the transverse axis and y = 0, z = k is the conjugate axis. If k < 0, the axes are interchanged. The lengths of the semi-axes increase without limit as |A;| increases. "^, (I- J -J. Arts. 61, 62] THE QUADRIC CONES 71 When k = 0, the section of the paraboloid consists of the two lines ^ + ^ = 0,2 = 0; --^=0, z = 0. ah a b Fig. 29. The sections of the surface by the planes y = k' are the con- gruent parabolas V^lf'i a- 0- '.'.n The vertices of these parabolas describe the parabola ^ = - 2 H2r, a- = 0. The sections by the planes x = A'" are congruent parabolas whose vertices de- scribe the parabola ^jLtf^i* 62. The quadric cones. The cone (Art. 46) ^ 1. = 72 FORMS OF QUADRIC SURFACES [Chap. VI. is called the real quadric cone. Its vertex is at the origin. The section of the cone by the plane 2; = c is the ellipse The cone is therefore the locus of a line which passes through the origin and intersects this ellipse. If a = b, the surface is the right circular cone generated by re- volving the line - = , y = about the Z^axis. a c The equation o? b"^ & represents an imaginary quadric cone. There are no real points on it except the origin. 63. The quadric cylinders. The cylinders (Art. 43) whose equations are a^ b"^ ' «2 b"^ ^ a? b"^ are called elliptic, hyperbolic, imaginary, and parabolic cylinders, respectively, since the sections of them by the planes z = k are congruent ellipses, hyperbolas, imaginary ellipses, and parabolas, respectively. 64. Summary. The surfaces discussed will be enumerated again for reference. Ellipsoid. (Art. 56) x'y- z^ -, Hyperboloid of one sheet. (Art. 57) 3.2 y2 Z- _^ 'a? 1? ?~ ' Hyperboloid of two sheets. (Art. 58) x^ tf- I 2;^ _ 1. Imaginary ellipsoid. (Art. 59) ?, + -^ = 2'«- Elliptic paraboloid. (Art. 60) Art. 64] SUMMARY 73 ^ = 2 nz. Hyperbolic paraboloid. (Art. 61) a? b^ ^^t-t=,0. Keal quadric cone. (Art. 62) a^ b"^ c^ — + — -\ — = 0. Imaginary quadric cone. (Art. 62) a- b^ c^ — ± ^ = ± 1 ; y^ = 2px. Quadric cylinders. (Art. 63) a* &* EXERCISES Classify the following surfaces : 1. 4 22-6x2 + 2 2/2 = 3. 2. x-^ + 3 2/2 + 5 a; + 2 2/ + 7 = 0. 3. x2 + 3 2/2 + 4 X - 2 2 = 0. 4. 4 x2 + 4 2/2 - 3 ^2 = 0. 5. 2 22 _ x2 - 3 2/2 - 2 X - 12 2/ = 15. 6. x2 - 2 j/2 - 6 2/ - 6 2 = 0. 7. Find the equation of and classify the locus of a point which moves so that (a) the sum of its distances, {b) the difference of its distances from two fixed points is constant. Take the points (± a, 0, 0). 8. Find and classify the equation of the locus of a point which moves so that its distance from (a, 0, 0) bears a constant ratio to its distance (a) from the plane x = ; (6) from the Z-axis. 9. Show that the locus of a point whose distance from a fixed plane is al- ways equal to its distance from a fixed line perpendicular to the plane is a quadric cone. 10. A line moves in such a way that three points fixed on it remain in three fixed planes at right angles to each other. Show that any other point fixed on the line describes an ellipsoid. (Sug. Find the direction cosines of the line in terms of the coordinates of the point chosen, and substitute in formula (1), Art. 3.) CHAPTER VII CLASSIFICATION OF QUADRIC SURFACES 65. Intersection of a quadric and a line. The most general form of the equation of a quadric surface is (Art. 55) F{x, y, z) = ax^ + hy'^ + cz- + 2fyz + 2 gzx + 2 hxy + 2 te + 2 ?/i?/ + 2 //z 4- fZ = 0. (1) We shall suppose, unless the contrary is stated, that the coeffi- cients are all real, and that the coefficients of the second-degree terms are not all zero. To determine the points of intersection of a given line (Art. 20) ic = a-o + Ar, y = Po + f^r, z = z^ + vr (2) with the quadric (1), substitute tlie values of x, y, z from (2) in F{x, y, z) and arrange in powers of r. The result is Qf- + 2Rr + S = 0, (3) in which Q = a\2 + hfx^ + cv^ + 2/^v + 2 ^vX + 2 /^A/., (4) R = {axo + /i//o 4- gzo + A + (/i.i'o + f>yo +.fzo -\-m)ix + (gxo +/2/0 + cZq +n)v IfdF, , dF , dF = - A + /A H 1 2 \dxQ dyo Ozq S = F{Xo, yo, Zo). The roots in r of equation (3) are the distances from the point Pq = (^o> 2/o> ^o) 01^ the line (2) to the points in which this line intersects the quadric. If Q :^0, equation (3) is a quadratic in r. If Q = 0, but R and S are not both zero, (3) is still to be considered a quadratic, with one or more infinite roots. If Q = 72 = /S' = 0, (2) is satisfied for all values of r and the corresponding line (2) lies entirely on the quadric. We have, consequently, the following theorems : Theorem I. Every line ickkh does not lie on a given quadric surface has two {distinct of coincident) points in common loith the surjace. ,' 74 Arts. 65, 66] DIAMETRAL PLANES, CENTER 75 Theorem IL If a given line has more than two points in common with a given qnadric, it lies entirely on the quadric. For, if (3) is satisfied by more than two values of r, it is satis- fied for all values. 66. Diametral planes, center. Let P, and P2 be the points of intersection of the line (2) with the quadric. The segment P1P2 is called a chord of the quadric. Theorem I. T7ie locus of the middle point of a system ofp)arallel chords of a quadric is a plane. Let r, and r2 be the roots of (3) so that P^Pi = r^ and PqPo = r^. The condition that Pq ^s the middle point of the chord P1P2 is PoP, + P,P, = 0, or ri + r2 = 0. ^' Hence, from (4), we have JVAt+A^ so '{axo + hy^ + gz^ + 0'^ + (^'-^'o -|- ^J'h +.ho + m)fi + (.7-^0 + fUo + <^^o + n) V = 0'. '^ " ' (5) If, now, \, fx, V are constants, but x^, yg, Zq are allowed to vary, the line (2) describes a system of parallel lines. The locus of the middle points of the chords on these lines is given by (5). Since (5) is linear in Xq, y^, Zq, this locus is a plane. Such a plane is called a diametral plane. ^' /■ ."^ ^ ' r'' ^ X' Theorem II. All the diametral planes of a quadric have at least one (finite or infinite) jwint in common. For all values of A, ix, v the plane (5) passes through the inter- section of the planes ax + hy -\- gz + I = 0," hx+by+fz-\-m = 0,- (6) gx + fy -f cz -j- n = 0. In discussing the locus determined by (6), it will be convenient to put, for brevity, D = a h g a h I a g I h g I h b f , N = h b m , M= h f m , L = b f m 9 f c 9 .f n g c n fen (7) ^G 6 QUADRIC SURFACES [Chap. VII. If D =^ 0, the planes (6) intersect in a single finite point (Art. 26) L M N ,^. If this point (xq, y^, Zq) does not lie on the surface, it is called the center of the quadric. It is the middle point of every chord through it. If the point (xq, y^, Zq) does lie on the surface, it is called a vertex of the quadric. In either case the system of planes (5) is a bundle with vertex at f _L M _N\ \ D' D' D)' If Z) = 0, but L, M, N are not all zero, the planes (6) intersect in a single infinitely distant point, the homogeneous coordinates of which are found, by making (6) homogeneous and solving, to be (L, — M, N, 0). The system of planes (5) is a parallel bundle. The quadric is, in this case, said to be non-central. If the system of planes (6) is of rank two (Art. 35), the planes determine a line; the diametral planes (5) constitute a pencil of planes through the line. If this line is finite and does not lie on the quadric, it is called a line of centers ; if it is finite and does lie on the quadric, it is called a line of vertices. If the system is of rank one, the diametrical planes coincide. If each point of this plane does not lie on the quadric, it is called a plane of centers ; if every point of the plane lies on the quadric, it is called a plane of vertices. Example. Find the center of the quadric The equations (6) for determining the center are a: + 2?/ + 2;+l = 0, x + 2// + 2 + l = 0, x-|-2y — z — 1=0, from which x + 2i/ = 0, 2+1=0. This line is a line of centers unless d = — 1, in which case it is a line of vertices. EXERCISES 2 1. Find the coordinates of the points in which the line x=l + -r, 2 T y = — 2 r, 2=— 1+- intersects the quadric x^ + Sy"^ — 4:z'^ + 'iz~2 y— o o 6 ::=0. Arts. 66, 67] QUADRIC REFERRED TO ITS CENTER 77 Find which of the following quadrics have centers. Locate the center when it exists. 2. z^-2y'2 +6z^ + 12xz -U= 0. >^ 3. 2z^+y^ — z^ — 2xz + ixy + 4:yz + 2y-4z~i = 0. 4. xy + yz + zx — X + 2 y — z — 9 = 0. 5. 2 x^ + 5 y'^ + z" - 4 xy - 2 X - i y - 8 = 0. f^6. x^ — xz — yz — z = 0. #7. x^ + y^ + z' — 2yz + 2xz — 2xy — x + y - z =0. P e. x"^ + 4 y- + z^ - i yz — 2 xz + i xy + \0 x + ^ y - 7 z + 15 = 0. ■)Ikk 9. Show that any plane which passes through the center of a quadric is a diametral plane. 10. Let Pi and P^ be two points on an ellipsoid, and let be its center. Prove that if Pi is on the diametral plane of the system of chords parallel to OP2, then P2 is on the diametral plane of the system of chords parallel to OPi. 67. Equation of a quadric referred to its center. If a quadric has a center {xq, y^, Zq), its equation, referred to its center as origin, may be obtained in the following way : If we translate the origin to the center by putting x = x' + Xo, y = y'-\-yo, z = z' + z^, the equation F{x, y,z)= is transformed into ax" + by'' + cz'^ + 2fy'z' + 2 gz'x' + 2 hx'y' + 2{ax, + hy, + gZo+ l)x' + 2(/tXo + %o +fio + wi)^' + 2(gXo+fyo + cz^ + n)z' + S = wherein, as in Eq. (4), S = F(xq, yo, z^). Since (xq, y^, Zq) is the center, it follows from (6) that axo -\- hyo + gZo-\- I =0, hxo + byo +JZo + 7n = 0, (8) gxo + fyo + c'Zf, i- n =0, so that the coefficients of x', y', z' are zero, and the equation has the form (after dropping the accents) ax' + by' + cz' + 2fyz + 2 gzx+2hxy + S = 0.-^ (9) The function S = F{xq, yo, z^) may be written in the form S = F(xo, yo, Zo) = Xo{axo + hyo + gzo + I) + yo{hxo + byo -\-fzo + m) + Zo(gxo 4-./>/o+ c^o + n) + Ixo + myo + nzo + d. 78 QUADRIC SURFACES [Chap. VII. Hence, from (8) we have ^S = Ixo + m?/o + nzo + d. (10) By eliminating Xo, y^, 2;,, from (8) and (10) we obtain the relation h b f m I m n d-S = 0. This equation may be written in the form a h g h b f S = 9 f c h b f m 9 / f m c n ?i d Denote the right-hand member of this equation by A coefficient of S is D (Eq. 7). Hence DS = A, or, if D=^0, \:\\ The V.^- ,./v*^ D (11) If 2)^0 and A=0, it follows from (9) and (11) that the quadric is a cone (Art. 46). The vertex of the quadric is the vertex of the cone. v'l. If A = and S ^0, then Z> = 0. Since {Xq, iJq, Zq) was assumed to be a finite point, it follows that L = M= N= so that the surface has a line or plane of centers. If A = and S = D = 0, then from (9) the surface is com- posite. Every point common to the component planes is a vertex. The determinant A is called the discriminant of the given quadric. If A = 0, the (piadric is said to be singular. If A ^ 0, the quadric is non-singular. 68. Principal planes. A diametral plane which is perpendic- ular to the chords it bisects is called a principal plane. Theorem. If the coefficients in the equation of a quadric are real, and if the qnadric does not have the plane at in-finity as a com- ^ ponent, the quadric has at least one real, finite, principal plane. Arts. 68, 69] THE DISCRIMINATING CUBIC 79 The condition that the diametral plane (5) (ciA + hix + gv)x + {h\ + bfj.-\- fv)y + (fi'A +fiJ.-\- cv)z + IX + ?/iyu, + wv = is perpendicular to the chords it bisects is (Art. 14) aX + hfi + gv _ hX + ^i^ +> _ fifA +//a + cv ,^2\ A /A V If we denote the common value of these fractions by k, equa- tions (12) may be replaced by (a — A-)A + h/x -\- gv = 0, hx+{b-k)ti+fv=0, (13) gX+ffji-{-(c-k)v = 0. The condition that these equations in A, fx, v have a solution other ■ ■-'J-', O vv^-' '-- = 0, (14) than 0,0, is -^^--o vv- • - a — k h g h h-k f 9 f c - A; or, developed and arranged in powers of k, Jc^ -(a + 6 + c)k^+{ab + bc-^ca-f^- g"- - h'')k -D = 0, (15) where B has the same meaning as in (7). This equation is called the discriminating cubic of the quadric F{x, y, z) = 0. To each real root, different from zero, of the discriminating cubic corresponds, on account of (13), (12), and (5), a real finite principal plane. Our theorem will consequently be proved if we show that equation (15) has at least one real root different from zero. The proof will be given in the next article. . 69. Reality of the roots of the discriminating cubic. We shall first prove the following theorem : Theorem I. TJie roots of the discriminating cubic are all real. Let fcj be any root of (15) and let Aq, /xq, vq (not all zero) be values of A, fx, V that satisfy (13) when k = k^. If A; is a complex number, Aq, /hq, Vq may be complex. Let ^0=^1 + iX\, ixq = ixi + ifx\, vo = vi + iv\, where ?' = V — 1 and Aj, A'l, lx■^, fx\, vj, v\ are real. 80 QUADRIC SURFACES [Chap. VII. Substitute fcj and these values of X.^, /xq, vq for A;, A, fi, v in (13), multiply the resulting equations by Aj — i\\, fj.^ — iix\, vj — iv\, re- spectively, and add. The result is (A^2 + A V + /.i^ + tx.\' + V,' + v'r) k, = (Ai^ + A'l^) a + (/xi^ + ^Lt'i^) 6 + (vi^ + v'l^) c + 2 (/.iv: + /xVv'i)/ + 2 (viA, + v'A'O gr + 2(Ai/xi+AVi)/i- The coefficient of A:i is real and different from zero. The number in the other member of the equation is real. Hence k^ is real. Since A;i is any root of (15), the theorem follows. Theorem II. Not all the roots of the discriminating cubic are equal to zero. The condition that all the roots of (15) are zero is a-\-h + c = 0,ah + hc + ca-f- — g'^-h^=Q,D = 0. Square the first member of the first equation, and subtract twice the first member of the second from it. The result is a? + 62 _,_ ^2 ^ 2/2 + 2 ^2 _^ 2 7^2 = 0. Since these numbers are real, it follows that a = 6 = c = / = gr = /i = ; but if these conditions are satisfied, the equation of the quadric contains no term in the second degree in x, y, z, which is contrary to hypothesis (Art. 65). 70. Simplification of the equation of a quadric. Let the axes be transformed in such a way that a real, finite principal plane of the quadric F{x, y, z) =0 is taken as x = 0. Since the surface is now symmetric with respect to a; = (Art. 68), the coefficients of the terms of first degree in x must all be zero. Hence the equation has the form ax-2 + 6(/2 + cz"^ + 2fyz + 2my + 27iz + d = 0. Moreover, fiTtO, since otherwise x = would not be a principal plane (Art. 68). Now let the planes y = 0, 2 = be rotated about the X-axis 2 f through the angle 6 defined by tan 2 = — •—- • This rotation re- b — c . ^RTS. 70, 71] CLASSIFICATION OF QUADRICS 81 duces the coefficient of yz to zero, and the equation has the form "\ i. iL' a'x'-[-h'y- + c'z' + 2m'y-\-2n'z + d' = 0, (16) ■wherein a' ^ 0, but any of the other coefficients may be equal to zero. 71. Classification of quadric surfaces. Since the equation of a quadric can always be reduced to the form (16), a complete classi- fication can be made by considering the possible values of the co- V efficients. -- I. Let both h' and c' be different from zero. By translation of the axes in such a way that f 0, — — , — — j is the new origin, the equation reduces to If d" ^ 0, divide by d" and put a h c' the signs being so chosen that a, b, c are real. This gives the fol- lowing four types : t + t + t = i. Ellipsoid. (Art. 56) or Ir c- /p2 y2 2^ — \-- = 1. Hyperboloid one sheet. (Art. 57) a? \r '{x\ y' z') =f'{x', y\ z') - A:(x'== + y" + z"). If k has such a value that is the product of two linear factors in X, y, z, then, for the same value of k, the expression ' will be the product of two linear factors in x', y', z'. The condition that <^ is the product of two factors is that its discriminant vanishes, that is a — k h g h h-k / =0, g f c-k which, developed in powers of k, is exactly the equation of the dis- criminating cubic (Art. 68) 84 QUADRIC SURFACES [Chap. VIL Similarly, the condition that ' is the product of two linear fac- tors is Ji^-I'k^-\-J'k-D' = 0, where /', J', and D' are the expressions I, J, and D formed from the coefficients oif'(x', y', z'). These two equations have the same roots, hence the coefficients of like powers of k must be proportional. But the coefficient of k? is unity in each, hence, /' = 7, J' = J, D' = D, that is, I, J, D are invariants. From the theorem just proved the following is readily obtained : "' Theorem. When the axes are transformed in such a loay that the coefficients of xy, yz, and zx are all zero, the coefficients of x^, y^^ and z^ are the roots of the discriminating cubic. For, if the equation of the quadric has been reduced to a'x^ + by + c'z2 + 2rx + 2 m'y + 2 u'z + d' = 0, the discriminating cubic is A;3 _ (a' + 6' + c')k + {a'b' + b'c' + c'a.')k - a'b'c' = 0. The roots of this equation are a', b', and c'. This proves the proposition. From the theorem just proved, the following criteria immedi- ately follow : If two roots of the discriminating cubic are equal and different from zero, the quadric is a surface of revolution, and conversely. If all three roots of the discriminating cubic are equal and different from zero, the quadric is a sphere. If A T^ 0, and a root of the discriminating cubic is zero, the quadric is non-central. If two roots of the discriminating cubic are equal to zero, the terms of second degree in the equation of the quadric form a perfect square. 74. Proof that A is invariant. It will first be proved that A is invariant under rotation. The reasoning is similar to that in Art. 73. Let F(x, y, z)= ax"" + by"" + cz^ + 2fyz + 2 gzx + 2hxy + 2lx + 2 my + 2nz + d = Art. 74] PROOP^ THAT A IS INVARIANT 85 be the equation of the given quadric. Let this equation be trans- formed by a rotation into F{x\ y', z') = a'x'^ + b'y'^ + cY' + 2f'y'z' + 2 g'z'x' + 2 h'x'y' + 2 1'x' + 2 m'y' + 2 n'z' + d' = 0. This rotation transforms the expression $ {x, y, z) = F(x, y, z) - k (x' -{-y^ + z-' + l) into $'(ic', y', z') = F'{x', y', z') - k {x'^ + y"' + z"" + 1). The discriminants of $ and $' are, respectively, a — k h 9 h b-k f 9 f c — k I m n I m n d-k and a'-^■ h' g' V h' b'-k f m' g' f c'-k n' V m' n' d'-k The roots of the quartic equations in A; obtained by equating these discriminants to zero are equal ; since a value of k which makes = singular also makes 4>' = singular and conversely (Art. 67). Hence, since the coefficient of k* in each equation is unity, the constant terms are equal ; that is, A = A'. Hence, A is invariant under rotation. In order to prove that A is invariant under translation, let the axes be translated to parallel axes through (x^, y^, z^). The equa- tion of the quadric becomes (cf. Art. 67) F{x', y', z') = ax'^ -f by'^ + cz'^ + 2fx'y' + 2 gy'z' + 2 hz'x' + 2 (ax, + hy, + gz, + /) x' + 2 {hx, + by, +fz, + w) y' + 2 (gx, + /2/0 + cz, + n)z' + S = 0, where S = F{x,, y„ z^. The discriminant of F'(x', y', z') is a h g axo+hyo+gZ(, + l h b f hxo + byo + fzo + m 9 f c gxo+ fyo + czo+n axo+hyo+gzo+l hxo+byo+fzo+m gxo+fyo+czo+n S Multiply the first column by x„ the second by y,, the third by z,, and subtract their sum from the last column. In the resulting determinant, multiply the first row by x„ the second by y,, the third by z„ and subtract their sum from the last row. Finally divide the first row and column each by Xq, the second row and column each by y,, and the third row and column each by z,. 86 QUADRIC SURFACES [Chap. VII. The resulting determinant is A. Hence A' = A, so that A' is inva- riant under translation. Since A is invariant unaer both transla- tion and rotation, it is invariant under motion. 75. Discussion of numerical equations. In order to determine the form and position of a quadric with a given, numerical equa- tion, it is advisable to determine the standard form (Art. 71) to which the equation of the given quadric may be reduced, and the position in space of the coordinate axes for which the equation has this standard form. For this purpose the roots k^, ko, ks of the discriminating cubic and the value of the discriminant A should first be computed. A. If all the roots ki, k^, k^ are different from zero, the three principal planes may be determined as in Art. 68. If these planes are taken as coordinate planes, the equation reduces to (Art. 67, Eq. 11 ; Art. 73) ^•i^" + hy' + hz' + TT^ = 0. B. If one root k-^ is zero, two finite principal planes may be determined as before. Let these be taken as cc = and y = 0. At least one intei*section of the new Z-axis with the surface is at infinity. If this axis does not lie on the surface, and does meet the surface in one finite point, the axes. should be translated to this point as origin. The equation of the surface now has the form k\x''-\-k.^/^-^2n"z = 0. Since A = it follows that ^\ k. n n" k\k2 If the new Z-axis lies on the quadric, or if it has no finite point in common with it, any point on the new .Z-axis may be chosen for origin and the equation takes the form k^x" + k.^f + S=0, Art. 75] DISCUSSION OF NUMERICAL EQUATIONS 87 where (Art. 67) S = lXo+ viyo + nzo + d, and (xq, ijo, Zq) are the old coordinates of the new origin. C. If two roots of the discriminating cubic are zero, the terms of the second degree in the original equation form a, perfect square, so that the equation of the surface, referred to the original axes, is of the form ^^■'':: (ax + /3y + yzf + 2 Ix + 2 m?/ -f 2 nz + d = 0, or (ax + /3y+yz + Sf + 2(1 - a8)x + 2(m - /3B)y + 2(n - y8)z e^, +d-8' = 0. (17) If the planes ax + /3y -\-yz + 8 = 0, 2(1 - aS)x + 2(m - ^8)y + 2(n - y8)z + d - 8^ = are not parallel, we may choose 8 so that they are perpendicular. The first term of (17) is proportional to the square of the distance of the point (x, y, z) from the plane ax+ (iy + yz+8 = 0. . . The remaining terms of (17) are proportional to the distance to the second plane. If these planes, with the appropriate value of 8, are chosen as x = 0, y = 0, the equation reduces to («2 + ^2 + y--yf + 2V(? - a8f + (m - 138)'+ (n - y8y x = 0. If the two planes are parallel, 8 may be so chosen that l-a8 = 0, m - ^8 = 0, n - y8 = 0. The equation now becomes (u-^+fS^ + y^f + d -8^=0, wherein ax -\- /3y -\- yz -}- 8 — is the new y = 0. Example 1. Discuss the equation a;2 _ 2 j/2 ^_ 6 2> + 12 xz - 16 X - 4 y - 36 z + &2 = 0. Tlie equations determining tlie center are x + 6 z — S = 0, 2y + 2 = 0, 6 X -\- 6 z — 18 = 0, from which the coordinates of the center are (2,— 1, 1). The invariants are / = 5, J" = — 44, Z> = GO, A = 1800. Hence, the discriminating cubic is A;3 _ 5 ^.2 _ 44 ;t - 60 = 0. Its roots are ki = 10, k^ —— 2, kz = — 3. The transformed equation is 10 a;2 _ 2 2/2 _ 3 ^2 + 30 = 0. 88 QUADRIC SURFACES [Chap. VII. The direction cosines of the new axes through (2, — 1, 1) are found, as in Art. 68, by giving k the values 10, — 2, — 3, to be O Q S 2 ——, 0, -^; 0, 1, 0; — ^, 0, — =• V13 Vl3 Vl3 Vl3 The surface is an hyperboloid of one sheet. Example 2. Discuss the quadric 11 x2 + 10 2/2 + 6 22 - 8 J/.S + 4 2X - 12 xy + 72 X - 72 t/ + 36 2 + 150 = 0. The discriminating cubic is k^ - 27 k^ + 180 yfc - 324 = 0. Its roots are 3, 6, 18. A = ^ 3888. The surface is an ellipsoid. The equations for finding the center are 11 X - 6 y + 2 z + 36 = 0, - 6 x + 10 y - 4 2; - 36 = 0, 2 X - 4 ?/ + 6 .2 + 18 = 0. The coordinates of the center are (— 2, 2,-1). The direction cosines of the axes are 1 2 2. 21 _2. _22 1 3' 3' 5 ' 3' 3' 3 ' 3' 3' ^* The equation of the ellipsoid referred to its axes is 3x2 _^ (^yi + is^i- 12. Example 3. Discuss the quadric Sx^ -y2 + 2z^ + 6yz -4:ZX--2 ry - 14 x + 4 y + 20 2 + 21 = 0. The discrinainating cubic is ^-3_4i.-2_ 13^.^ 19 = 0. Its roots are approximately 1.2, 5.7, — 2.9. A = 0. The surface is a cone. The equations for finding the vertex are Sx-y-2z-7=0, - X - // + 3 2 + 2 = 0, - 2 x + 3 x + 2 z + 10 = 0. The coordinates of the vertex are (1, — 2, — 1). The direction cosines of the axes are approximately .8. .4, .5; .6, -4, -.7; 0, .6, - .4. The equation of the cone referred to its axes is approximately 1.2 x2 + 5.7 2/2 _ 2.9 22 = 0. Example 4. Discuss the quadric 4 x^ + y"^ + z^ - 2 yz + i xz — i xy — 8 X + i z + 1 = 0. This equation may be written in the form (2x-2/ + z + 5)2= (8 + 4 5)x-2 52/-(4-25)2-7 +52. . If 5 = — 1, the planes 2Xf-y +z-l -0 and 4x + 2y — 6z — 6--0 are perpendicular. If we take these planes as ?/' = and x' = 0, the equation of the surface reduces to 6 2/2 = V56 x. The surface is a parabolic cylinder. Art. 75] DISCUSSION OF NUMERICAL EQUATIONS 89 EXERCISES Discuss the quadrics: 1. 3 a;2 + 2 2/2 + 22 _ 4 xy - 4 yz + 2 = 0. 2. x^-y^ + 2z^-2yz + 4iXZ + Axy-2x-4:y-l=0. ''§. 6 y2 + 8 2^ + 6 yg + 6 xz + 2 xy + 2 X + 4 y - 2 z - 1 = 0. 4. 4x2 + y2-8z2 + 8yz-4xz + 4xy-8x-4y + 4z + 4 = 0. 6. 3x2 + 2y2 + 2z2-4y3-2zx + 2xy-6x + 2y + 2z-12 = 0. '6. 6x2-2z2-6yz-6x3-2xy + 2x + 4y + 2z = 0. 7. 4x2 + 4y2 + 22-4yz-4xz + 8xy-6y + 6z-3 = 0, /^ 8. x'' - yz + xz - xy + X + y + 2 z - 2 = 0. ■■>' 9. 3 y2 + 6 yz + 6 xy - 2 X + 2 z + 4 = 0. 10. 3 x" + 3 y2 + z2 + 2 yz + 2 xz - 2 xy - 7 X + y + 6 z - 7 = 0. 11. 3 x2 - 5 y2 + 15 z2 - 22 yz + 14 xz - 14 xy + 2 X - 10 y + 6 z - 5 = 0. 12. x2 - y2 - 2 z2 - 4 yz + 2 xy - 2 y + 2 z = 0. 13. x" - 6 yz + 3 zx + 2 xy + X - 13 z = 0. 14. x2 - 2 y2 + z2 - 4 zx - 12 xy + 4 y + 4 z - 9 = 0. 16. x2 + 2 y2 + 2 z2 + 2 xy - 2 X - 4 y - 4 z = 0. 16. 3 x2 + y2 + z2 + yz - 3 zx - 2 xy + 2 X + 4 y + 2 z = 0. 17. For what values of c is the surface 5 x2 + 3 y2 + cz2 + 2 xz + 15 = a surface of revolution? 18. Determine d in such a way that x2 + y'^ + 5z2 + 2ya + 4x2-4xy + 2x + 2y + d = is a coue. CHAPTER VIII SOME PROPERTIES OF QUADRIC SURFACES 76. Tangent lines and- planes. If the two points of intersection of a line and a quadric coincide at a point Pq, the line is called a tangent line and Pq the point of tangency. If the surface is sin- gular, it is supposed in this definition that Pq is not a vertex. Theorem. TJie locus of the lines tangent to the quadric at Pq is a plane. Let the equation of the quadric be F{x, y, z) = ax^ + by'^ + cz^ + ^fy^ + 2 gzx + 2 hxy + 2lx + 2 my + 2 7iz -j- d — 0, (1) and let the equation of any line through P „ = (-^'o ^o ^o) ^^ (Art. 20) X = Xo + Xr, y = y,-}- fxr, z = Zq + vi: (2) Since Pq lies on the quadric, F(xq, y^, Zq) = 0. Hence, one root of equation (3), Chapter VII, which determines the intersections of the line (2) with the quadric (1), is zero. The condition that a second root is zero is P = 0, or A {axQ + hy^ -f- (/^^ + /) + /a (JiXa + hy^ + Jzq + m) + v{gxQ + /Vo -+- ''^0 + »0= 0- (3) If we substitute in (.3) the values of A, p., v from (2), we obtain (x — Xo)(axo + hyo + gz^ + l) + (y - yo)(hXo + byo + J'Zq + m) + {z- Zo) (gxa + fyo + cz^ + 7i) = 0, (4) which must be satisfied by the coordinates of every point of every line tangent to the quadric at Pq. Conversely, if (x, y, z) is any point distinct from Pq, whose coordinates satisfy (4), the line de- termined by (ic, y, z) and Pq is tangent to the surface at Pq. Since (4) is of the first degree in (.t, y, z), it is the equation of a plane. This plane is called the tangent plane at Pq. 90 Arts. 76, 77] TANGENT PLANE NORMAL FORM 91 The equation (4) of the tangent plane may be simplified. Mul- tiply out, transpose the constant terms to the second member, and add Ixq + m>jQ -f tiZq + d to each member of the equation. The second member is F(Xa, r/^, Zq), which is equal to zero, since Pq lies on the quadric. The equation of the tangent plane thus reduces to the form axxo + byi/o + czZq + /{yz^ + zy^) + f/ {zx^ + -rzc) +■ h (-^V/o + V^o) + l{x + .i-o) + m (y + .vo) + 7^ (z + z,)-{-d = 0. (5) * This equation is easily remembered. It may be obtained from the equation of the quadric by replacing x^, y'^, z^ by xx^^ yy^, zZq ; 2 yz, 2 zx, 2 xy by yz^ + zyo, zx^ + xz^, xy^ + yx^ ■ and 2x,2y,2zhy ^ + 3*0) y + ?/o) 2! 4- ^0, respectively. 77. Normal forms of the equation of the tangent plane. The equa- >'^ion of the tangent plane to the central quadric ax'' + by"" + cz^ = 1 (6) at the point (x^, y^, Zq) on it is axxQ + byyo + czZq = 1, Let the normal form of the equation of this plane (Art. 13) be A.T -\- fj.y + vz = p, (7) so that A u , V - = aXo, ^ = bye, - = czq. P P P Since (xg, Ih, Zq) lies on the quadric, we have from which «.V + ^Jlfo^ + t'^o' = 1, X2 „2 „2 ^4-^^- + ^=^^ (8) a b c Conversely, if this equation is satisfied, the plane (7) is tangent to the quadric (6). By substituting the value of p from (8) in (7), we have Xx + f,y + rz = J^ + i + -, ^ a b c which is called the normal form of the equation of the tangent plane to the central quadric (6). 92 PROPERTIES OF QUADRIC SURFACES [Chap. VIII. It follows from (8) that the necessary and sufficient condition that the plane ux -{- vy + wz = 1 is tangent to the quadric (6) is that t + t + }!^==l. VlV' (9) a b c This equation is called the equation of the quadric (6) in plane coordinates. Again, if ax^ + bif=2 nz (10) is the equation of a paraboloid (Arts. 60 and 61), it is proved in a similar way that the normal form of the equation of the tangent plane to the paraboloid is X. + ,y + v. = -i(f + f) (11) and that the condition that the plane iix -{- vy -\-wz = 1 is tangent to the paraboloid is 'iV - + — = 0. (12) a b n Equation (12) is the equation of the paraboloid in plane coordinates. 78. Normal to a quadric. The line through a point P(, on a quadric, perpendicular to the tangent plane at P^, is called the normal to the surface at Pq. It follows from equation (4) that the equations of the normal at Po to the quadric F{x, y, z) = are X — Xn _ y — lk _ 2: — Zq aXf^ + hyf^ + gZf^ + I hxo + byo + fz^ + m gxo + fyo + cz^ + n (13) EXERCISES 1. Show that the point (1, - 2, 1) Hes on the quadric x^ — y^ + z^ + 4 yz + 2 zx + xy — X + y + z + 12 = 0. Write the equations of the tangent plane and the normal line at this point. 2. Show that the equation of the tangent plane to a sphere, as derived in Art. 76, agrees with the equation obtained in Art. 50. Arts. 78. 79] RECTILINEAR GENERATORS 93 3. Prove that the normals to a central quadric ar? -\- by^ + cz- = 1, at all points on it, in a plane parallel to a principal plane, meet two fixed lines, one in each of the other two principal planes. 4. Prove that, if all the normals to the central quadric ax^ + hif -}- cz^ = l intersect the X-axis, the quadric is a surface of revolution about the JT-axis. 5. Prove that the tangent plane at any point of the quadric cone ax2 + 6^2 -f. 0^2 = passes through the vertex. 6. Prove that the locus of the point of intersection of three mutually per- pendicular tangent planes to the central quadric ax^ -f- hy'^ ■\- cz^ =\ is the concentric sphere z^ + y^ + z'^ = --\ 1 This sphere is called the director a b c sphere of the given central quadric. 7. Prove that through any point in space pass six normals to a given central quadric, and five normals to a given paraboloid. 79. Rectilinear generators. The equation of the hyperboloid of one sheet or or also X' y^ ^ — 1 tie form a cj\a CJ \ b^ iH a c b 1 + ^' ?_?' b a c a c - >. i-.y ^_? b a c (14) (15) Let the value of each member in (14) be denoted by ^, so that by clearing of fractions we have For each value of ^, these equations define a line. Every point on such a line lies on the surface, since its coordinates satisfy 94 PROPERTIES OF QUADRIC SURFACES [Chap. VIII. (14). Moreover, through each point of the surface passes a line of the system (16) since the coordinates of each point on the sur- face satisfy (14) and consequently satisfy (16). The system of lines (16), in which | is the parameter, is called a regulus of lines on the hyperboloid. Any line of the regulus is called a generator. Similarly, by equating each member of (15) to yj, we obtain the system of lines whose equations are in which -q is the parameter. This system of lines constitutes a second regulus lying on the surface. The two reguli will be called the ^ regulus and the rj regulus, respectively. Through every point P of the surface passes one, and but one, generator belonging to each regulus. Moreover, any plane that contains a generator of one regulus contains a generator of the other regulus also. The equation of any plane through a genei'ator of the ^ regulus, for example, may be written in tlie form (Art. 24) a c \ bj \a c Since this equation may also be written in the form it follows that this plane also passes through a generator of the rj regulus. Every such plane is tangent to the surface at the point of intersection P of the generators in it, since every line in the plane through P has its two intersections with the surface coincident at P. Example. The equations of the reguli on the hyperboloid 4 ' 9 4- ^ — t I 1 - + ^ _ ^2 = 1 X o + 'S and |+, = ,(,_!), 1+| = , (!-»)• The point (2, 6, 2) lies on the surface. The values of | and tj which Arts. 79, 80] ASYMPTOTIC CONE 95 determine the generators through this point are | = 1, tj =— 3. Hence, the equations of these generators are ? + ^ = l + 2/, i_y = ?_^, and ? + 0=-3fl-2/V i + ^=:_3f?-^V 2 3' 3 2 ' 2 V 3^ 3 V2 / The equation of the plane determined by these lines is This is the equation of the tangent plane at (2, 6, 2) (Art. 76). It is similarly seen that the equation a? ¥ of the hyperbolic paraboloid may also be written in the forms ah 1 (. and Hence, on this surface also, there is a ^regulus and an yj regains The generators of the ^ regains are parallel to the fixed plane — " = ; those of the -q regains, to the fixed plane " + -^ = 0. a b a b By writing the above equations in homogeneous coordinates, it is seen that the line - -f ^ = 0, f = in the plane at infinity belongs a b to the $ regulus ; and the line ' — ^ = 0, < = to the n regains. a b Hence the plane at infinity is tangent to the paraboloid. The hyperboloid of one sheet and the hyperbolic paraboloid are sometimes called ruled quadrics, since the reguli on them are real. It will be shown (Art. 115), that on every non-singular quadric there are two reguli ; bat, on all the quadrics except these two, the reguli are imaginary. 80. Asymptotic cone. The cone whose vertex is the center of a given central quadric, and which contains the curve in which 2nz ' X y a b X _y a b 1 2nz x^y a b 96 PROPERTIES OF QUADRIC SURFACES [Chap. VIII. the quadric intersects the plane at infinity, is called the asymp- totic cone of the given quadric. If the equation of the quadric is the equation of its asymptotic cone is ax^ + by"^ + cz^ = 0. For, this equation is the equation of a cone with vertex at the center (0, 0, 0, 1) of the given quadric (Art. 46). Its curve of intersection with the plane at infinity coincides with the curve of intersection of the given surface with that plane. EXERCISES (1. Show that the quadric xy = z \s ruled. Find the equations of its gen- erators. 2. Show that x^ — 2 z^ + by — x + 9iz = is Si ruled quadric. 3. Prove that, for all values of k, the line x + \ = ky — — {k + \)z lies on the surface yz + zx + xy + y + z = 0. 4. Prove that (y + rnz) (x + nz) — z represents an hyperbolic paraboloid which contains the X-axis and the F-axis. 5. Show that every generator of the asymptotic cone of a central quadric is tangent to the surface at infinity. From this property derive a definition of an asymptotic cone. 6. Show that every generator of the asymptotic cone of an hyperboloid of one sheet is parallel to a generator of each regulus on the surface. 81. Plane sections of quadrics. Theorem I. The section of a quadric by a finite plane, lohich is not a component of the surface, is a conic. For, let TT be any given finite plane, and let the axes be chosen so that the equation of this plane is 2 = 0. Let the equation of the quadric, referred to this system of axes, be ax^+by^ + cz''+2fyz+2gzx+2hxy+2lx+2my-^2nz + d=0. (17) If, when 2 = 0, (17) vanishes identically, the given quadric is Abt. 80] PLANE SECTIONS OF QUADRICS 97 composite and z = is one component ; otherwise, the locus defined in the XF-plane by putting 2; = in (17) is a conic. Theorem II. Tlie sections of a quadric by a system of parallel planes are similar conies and similarly placed. Let the axes be chosen so that the equations of the given sys- tem of parallel planes is 2; = A;, and let (17) be the equation of the given quadric. The equation of the projecting cylinder of the section by the plane z = k is ax^ + '2hxy+by^ + 2(l+ gk)x + 2 (?n +fk)y + ck^ + 2nk + d = 0. The curves in which these cylinders intersect z = 0, and conse- quently (Art. 45) the curves of which they are the projections, are similar and similarly placed, since the coefficients of x'^, xy, and y"^ in the above equation are independent of k* The equations of the section of the surface by the plane at infinity are found by making (17) homogeneous in .-r, y, z, t and put- ting t= 0. They are ax^ -\- by^ + cz"^ + 2 fyz +2 gzx -\- 2 hxy =0,t = 0. The locus of these equations is called the infinitely distant conic of the quadric. This conic consists of two lines if the first mem- ber of the first equation is the product of two linear factors. The condition for factorability is D = 0. 'iOfi EXERCISES lid the semi-axes of the ellipse in which the plane z = 1 intersects the quadric x^ + i y"^ — 3 z^ + i yz — 2 x — 4 y = I. 2.; Show that the planes z = k intersect the quadric 2 x'^ — y"^ + 3 z^ + 4 oiz^ 2 yz + i X + 2 y = in hyperbolas. Find the equations of the locus of the-cgnters of these hyperbolas. A ( 3. Show that the curve of intersection of the sphere x"^ + y"^ + z"^ = r^ and f\ ^ — '' 3.2 ,/2 g2 the ellipsoid — f- ^ + — = 1 lies on the cone a2 b^ c^ a-z r^j ^\fy2 r"-) W r^ Find the values of r for which this cone is composite. Show that each com- ponent of the composite cones intersects the ellipsoid in a circle. * Cf. Salmon, " Conic Sections," 6th edition, p. 222. 98 PROPERTIES OF QUADRIC SURFACES [Chap. VIII. 82. Circular sections. We shall prove the following theorem : Theorem I. Through each real, finite point in space pass six planes which intersect a given non-composite, non-spherical quadric in circles. If this quadric is not a surface of revolution nor a para- bolic cylinder, these six p)lanes are distinct; two are real and four are imaginary. If the quadric is a surface of revohdion or a para- bolic cylinder, four of the planes are real and coincident and two are imaginary. Two proofs will be given, based on different principles. Proof I. Since parallel sections of a quadric are similar, it will suffice if we prove this theorem for planes through the origin. The planes through any other point, parallel to the planes of the circular sections through the origin, also intersect the quadric in circles. Let the axes be chosen in such a way that the equation of the quadric is (Art. 70) k.x'' -f- W + ^-3^' + 2 Ix + 2 my ^2nz + d = 0, (18) where k^, k^, k^ are the roots of the discriminating cubic (Art. 73). The condition that a plane intersects this quadric in a circle is that its conies of intersection with the given quadric and with a sphere coincide. The curve of intersection of the quadric (18) with the sphere k(x~-\-y'^-^z'')-\-2lx-h2 7ny-\-2nz-^d = (19) coincides with the intersection of either of these surfaces with the cone (^•l - k) x-" + {k^ - k)y^ + (k, - k) z" = 0. This cone is composite if the first member of its equation is factorable, that is, if k is equal to k^, k.,, or k^. It follows that each of the six planes VA:i - k^ x=± Vits —kiy -Vki — ^2 X = ± VA-2 — k^z Vfcj — ki y = ± VA,*! — k^ z intersects the quadric (18) in a conic which lies on the sphere (19) and is consequently a circle. Art. 82] CIRCULAR SECTIONS 99 If A;, > k2 > k^, the six planes are distinct. The planes VA;i — fcj a; = ± VA-'j — fcj 2 are real. The others are imaginary. If ki = k.2^k3, the last four planes coincide with z = 0. The other two are imaginary. If ki = ^2 ^ 0, the quadric (18) is a surface of revolution (Art. 73). If k^ = k., = 0, it is a parabolic cylinder (Art. To). If the equation of the surface is in the form (17), and k^, k^, k^ are the roots of its discriminating cubic, it follows from the dis- cussion in Article 73, that the equations of the planes of the circular sections through the origin are ax2 + bf + cz"" + 2fyz + 2 gzx + 2 hxy - k, (x"" + y^ + z") = 0, ax'^ + by- + cz- + 2fyz + 2 gzx + 2 hxy — k.-, (x-^ + y2-\-z'^) = 0, ax2 + 6^2 ^ cz"" + 2fyz + 2 gzx + 2 hxy - ^-3 (a^ + ?/' + 2') = 0. Proof II. It was shown (Art. 49) that a plane section of a quadric is a circle if it passes through the circular points of its plane. The conic in which the quadric meets the plane at infinity has four points of intersection with the absolute. Any plane other than the plane at infinity which passes through two of these points will meet the quadric in a conic through the circular points of the plane ; hence the section is a circle. The coordinates of the points of intersection may be found by making the equations ax- + by"^ + cz^ + 2fyz + 2 gzx + 2 hxy = 0, x^ + ?/2 + z* = simultaneous. Since both equations have real coefficients and the second is satisfied by no real values of the variables, it follows that the four points Pj, P.,, P3, P^ consist of two pairs of conjugate imaginary points, or of one pair counted twice. In the first case, let Pj, P^ be one pair of conjugate points, and P3, P4 the other. The lines P^P., P^Pi are real (Art. 41), while the lines P1P3, P2P4, P1P4, P2P3 are imaginary. The pairs of lines PiPo, P3P4; P1P3, P2P4; PiP4> P^Pz constitute composite conies passing through all four of the points Pj, P2, P3, P,. In the second case, let P., = P4 and Pj = P3. The lines P1P2 and P3P4 coincide, and the lines PiP3, P2P4 are tangents to both curves, which have double contact with each other at these points. 100 PROPERTIES OF QUADRIC SURFACES [Chap. VIII. In either case the equations of the lines /*,Pt can be found as follows. Through the points of intersection of (17) and the abso- lute passes a system of conies ax' + b>f^cz-+2fiiz^2 gzx + 2 hxi/ - k{x--\-y^+z^)=0, t=0. (19') A conic of this system will consist of two straight lines through the four points of intersection if its equation is factorable, that is, a —k h g ' h b-k / =0; 9 f c-k, thus k must be a root of the discriminating cubic (Art. 73). Let A;,, ki, ks be the roots of this equation. The equations of the pairs of lines are then ax.2 4. jjy2 ^ ez2 + 2 ./}/2 + 2 gz.v + 2 hxy - k, {£■ + f ^-7?) = ^, t= 0, (20) with similar expressions for ki and k^. From Art. 4t it follows that for one of the roots k^ the two factors of the first member of the quadratic equation (20) are real, but the factors for each of the others are imaginary when the roots k^ are all distinct. If ?«, V are the two linear factors of (20), then the line u = 0, ^ = will pass through one pair of points and v = 0, < = will pass through the other. A plane of the pencil u +2?< = will cut the quadric in a circle. Since a plane is determined by a line and a point not on the line, the theorem follows. In case two roots of the discriminating cubic are equal and different from zero, the quadric is one of revolution; the two conies in the plane at infinity now have double contact. If fcj > k^ > A'3, the planes determined by the second root are real. 83. Real circles on types of quadrics. The above results will now be applied to the consideration of the real planes of circular section for the standard forms of the equation of the quadric (Chap. VI). (a) For the ellipsoid ^' + ^ + ^ = 1, a^ 6^ (? the roots of the discriminating cubic are 1/a*, 1/6'', 1/c*. Art. 83] REAL CIRCLES ON TYPES OF QUADRICS 101 Let a > 6 > c > 0. Since parallel sections of the surface are similar, it follows that the equations of the real planes of circular section are cy/a''-¥x±a^b''-c^z-{-d = 0, (21) where d is a real parameter. The circle in which a plane (21) intersects the ellipsoid is real if the plane intersects the ellipsoid in real points, that is, if it is not more distant from the center than the tangent planes parallel to it. The condition for this is (Arts. 76 and 16) | d ] £. ac^a? — fl If I d I > ac Va^ — c^ the circles are imaginary. If 1^1= ac Va^ — c", the circles are point circles. The four planes determined by these two values of d are the tangent planes to the ellipsoid at the points -, 0, ±cyA- ' ^a^ — c Each of these points is called an umbilic. The two systems of planes (21) are also the real planes of circu- lar section of the imaginary cone ^4.^ + 5^=0, a? b^ c^ ' and of the imaginary ellipsoid ^! 4. 2/' > ?" ^ _ 1 a^ b^ c" (b) The equations of the real planes of circular section of the hyperboloids of one and two sheets and of the real cone a"" b'' c2 ' a" 62 c2 ' where a > 6 > 0, are found to be c Va2 -b^y ±b Va^ + e" z + d = 0. 102 PROPERTIES OF QUADRIC SURFACES [Chap. VIII. On the hyperboloid of one sheet and the real cone, the radii of the circles are real for all values of d. On the hyperboloid of two sheets, the circles are real only if \d\ >bc ^ b^ + c^. The coordi- nates of the umbilics on the hyperboloid of two sheets are (c) The real planes of circular section of the elliptic paraboloid '^ + l = 2nz, a>6>0, n>0 and the real or imaginary elliptic cylinders -, + f =±1, a>6>0 are determined by ± V«2 - b' y + bz + d = 0. On the real elliptic cylinder, the circles are real, and on the imaginary cylinder they are imaginary, for all values of d. On the elliptic paraboloid, the circles are real if d<-^(a^ — b"^). The coordinates of the umbilics on the elliptic paraboloid are n 0, ±bn^(e--b-, ^{a'-b') (d) For the hyperbolic paraboloid 2nzt x^_y^_r, a" b' and the hyperbolic cylinder ^2 ^ y2 ^ ^^ a" b' the equations of the planes of the circular sections are bx ± ay -\- dt = 0. The circles in these planes are all composite. For, the planes bx + ay -\- dt = Art. 83] REAL CIRCLES ON TYPES OF QUADRICS 103 intersect these surfaces in the fixed infinitely distant line hx + ay — 0, t = and in a rectilinear generator which varies with d. Similarly, the planes hx — ay -\-dt = intersect them in the line hx — ay = 0, t = and in a variable generator. Also on the parabolic cylinder a;2 = 2 myt the real circles are all composite, since the planes x = dt intersect the surface in the fixed line x = t= 0, and in a variable generator. We have, therefore, the following theorem : Theorem II. On the hyjjerhoh'c 2^("'<^l>oloid, the hyperholic cylinder, and the j^'^i'f'ci'hoUc cylinder, the real circular sections are composite. The components of each circle are an infinitely distant line and a rectilinear generator ivhich i)itersects it. ^ EXERCISES f 1.: Find the equations of the real circular sectipns of the surface 4 '^^ 2 2/2 + z^ + 3yz + X2 = 1. ^>H^^ '^ "^ "^^ P*^ -kCit^-^'i "^^ J*) -' « «> 2., Find the equations of the real circular sections of the surface 2 x^+ 5 2/2 + 3 5;- + 4 .r2/ = 1. i' 3. Find the radius of a circular section through the origin in Ex. 2. .-^.! Find the equations of the real planes through (1. —3, 2) which in- tetsfect the ellipsoid 2 x- + y'^ + iz'^ = 1 in circles. 5. Find the conditions which must be satisfied by the coefficients of the equation F{x, y, z) = oi a, quadric if the planes z = k intersect it In circles. 6.' Show that the centers of the circles in Ex. 5 lie on a line. Find the equations of this line. //y. Find the second .system of real planes cutting circles from the quadric Kn Ex. 5. 8. Find the conditions which must be satisfied by the coefficients if the plane Ax + By + Cz -{- D = intersects the quadric F{x, y, z) = in circles. 9. Find the coordinates of the center and the radius of the circle in which the plane x =2 z + 5 intersects the cone 3 x- + 2y'^ — 2 z^ = 0. ^tp ±-i 45^^ROPERTIES OF QUADRIC SURFACES [Chap. VIII. V 10. Show that, for all values of \, the equation of the planes of the cir- cular sections of the quadrics (a + X)x2 + (6 4- X)2/2 + (c + \)z2 = 1 are the same. The quadrics of this system are said to be concyclic. 84* Confocal quadrics. The system of surfaces represented by the equation + -^— + -^^— = 1, a>b>c>0, (22) in which k is a parameter, is called a system of confocal quadrics. The sections of the quadrics of the system by the principal planes x = 0, y = 0, z = are confocal conies. If A; > — c^, the surface (22) is an ellipsoid ; if — c- >k> — b"^, the surface is an hyperboloid of one sheet; if — 6- > fc > — a^, the surface is an hyperboloid of two sheets ; if — a^ > k, the surface is an imaginary ellipsoid. If A:> — c^, but approaches — c^ as a limit, the minor axis of the ellipsoid approaches zero as a limit, and the ellipsoid ap- proaches as a limit the part of the XF-plane within the ellipse -^— + -^^ = 1. (23) If — e^>A; > — b"^, the surface is an hyperboloid of one sheet. As k approaches — c-, the surface approaches the part of the XF-plane exterior to the ellipse (23). As A; approaches — 6^, the surface approaches that part of the XZ-plane which contains the origin and is bounded by the hyperbola = 1. (24) a^ — b"^ b^ — c^ If — ¥ >k> — o}, the surface is an hyperboloid of two sheets. As k approaches — 6^, the hyperboloid approaches that part of the plane y = which does not contain the origin. As k approaches — ci^, the real part of the surface approaches the plane a;=0, counted twice. The ellipse (23) in the XF-plane and the hyperbola (24) in the XZ-plane are called the focal conies of the system (22). Arts. 84, 85] ELLIPTIC COORDINATES 105 The vertices of the focal ellipse are (-t-Va^-cS 0, 0). The foci are (±Va^^^^ 0, 0). On the focal hyperbola the vertices are ( ± Va^ — 6^ 0, 0) and the foci are (± Va^ — c^, 0, 0). Hence, on the focal conies, the ver- tices of each are the foci of the other. 85. Confocal quadrics through a point. Elliptic coordinates. Theorem I. Three confocal quadrics pass through every point P in space. If P is real, one of these quadrics is an ellipsoid, one an hyperholoid of one sheet, and the third an hyperholoid of two sheets. If P = (Xi, yi, Zi) lies on a quadric of the system (22), the param- eter k satisfies the equation {j£ + a''){k + b'){k + c^) - x^(k -f ¥)(k + c^) - y,\k + c%k + a'-) - z,\k + a^k -f h^)= O.sr^i -1^X25) Since this is a cubic equation in k, and each of its roots determines . 9- a quadric of the system through P, there are three quadrics of eT *^ the system (22) which pass through P. ^'jJc Let P be real. **«> If ^' = -f- 00, the first member of (25) becomes positive. -Mj h 1 / If A; = — c^, it is — 2;i2(— c2-|-a2)(— c^ 4- &2)^-vvhich is negative. =V^'^i If k = — b"^, it is — yi'^( — ¥ + c'^){— ¥ + a-), wliich is positive. ^ -f C~ ^^< Itk = — a^, it is — x^% — a^ + ?/)( — a^ + c^), which is negative. ^ ■} ' ~* Hence the roots of (25) are real. One is greater than — c^, one lies between — c^ and —b^, and the third between — 6^ and — a^- Denote these roots by ki, k^, k^. Hence, we have ki >-c'>k,>-¥>k,>- a\ Then, of the three quadrics f 1,9! I 7. ' _9 1 7. ' a2 -\-k, b'' + k, c2 -1- fci CC^ iP" 2*^ 106 PROPERTIES OF QUADRIC SURFACES rj [Chap. VIII. ^^ which pass through P, the first is an ellipsoid, the second an hy- berboloid of one sheet, and the third an hyperboloid of two sheets. Theorem II. TJie three quadrics of a confocal system which pass through P intersect each other at right angles. For, the equations of the tangent planes to the first two quad- rics (26) are ^ ^ a^ + K + 4- 62 + A;i + + (? + fci a^ + k. ' 5- + A-., ' c'^ + k^ These planes are at right angles if = 1, = 1. V n t S 'S^ <'.'? + Vi + ■ (a^ + ;ci)(a2 + k,) {1/ + k,){b' + k.^ (c^ + k,){c'' + k,) = That this condition is satis- fied is seen by substituting the Jj-*] coordinates of P in (26), sub*!*" tracting the second equation? from thefirst, and removing the*^' factor A'2— A'l, which was seen to be different from zero. The proof for the other pairs may be obtained in the same way. The three roots k^, k^, A3 of equation (25) are called the el- liptic coordinates of the point P. To find the expressions for the rectangular coordinates of P,in terms of the elliptic coordinates, we substitute the coordinates (.Ti, ?/i, 2,) of P in (26) and solve for Xi^, yx, %{-. The result is ,.,_ («^ + A-0(a^ + A,)(a^ + A3) ' (a2-62)(a2-c2) ' {WJrK^{W^-k^{W^k,^ ^' (^2 _ (j2)(52 _ c2) ' ,^ (c'' + A0(c^ + A,)(c^-j-A-3) ' (c2-a2)(c2-62) (27) Arts. 85, 86] QUADRICS TANGENT TO A LINE 107 It is seen at once from these equations that Ji\, fcj, and ^'3 are the elliptical coordinates, not only of P, but also of the points sym- metric with P as to the coordinate planes, axes, and origin. 86. Confocal quadrics tangent to a line. Theore3i I. An.i/ line touches tivo quadrics of a confocal system. The points of intersection of a given line with a quadric of the system (22) are determined by the equation (Art. 65) oL^ U^aaa^ ■*"* 3*< witC *> 14/ i'v Sam** a" -\-k ¥ + k c- + kj \a'' + k b^ + k c^~ + k W + k b^ + k c-' + k J ^ The condition that this line is tangent is -^ y j^-a..*:^ w^ vJ^^^ a^j^kh-'^kc'^k) ^^"^ + v;;^ + -;r^ -^F^ + t/^ + ^t^-I =0. When expanded and simplified, this equation reduces to y? -f [(62 + c2) A^ + {c? + «') /x^ + («' + /'^) v^ - (a^oi^ - y,\Y — (z/o«' — ^Jq/a)" — (2;o^- — -I'd")'] ^ + [ft-c^A^ + c^aV + a-lP-v^ — i?'^l>' - .'/o^)c' - (.Vol' - 2;oa)/j- — (^oA. - .Tov)a2] = 0. Since this equation is quadratic in k, the theorem follows. Theorem II. If tico confocal quadrics touch a line, the tangent 'A. planes at the points of contact are at right angles. fl Let A"i and A'o be the parameters of the quadrics, and let ^ P'={x', y', z'), P" = {x", y", z") be the points of tangency of the \^^line with the given quadrics. The equations of the tangent '.J planes at P' and P" are (Art. 76), respectively, jj ;j» x'x y'y ^'^ =\ ^"^ 1 ?/"-^^ 1 ^'''^ —\ ,^ a^+A-i h'^-^ki^ c'-^-k^ ' a2 + A-o &2 ^ A:., c2 + A;2 These planes are at right angles, if ^'*^" + ^NL + ^^ = 0. (28) (a2 + A'0(a2 + A-o) (6^ + a-,)(62 + A%) (c^ + A-i)(c2 + A-,) I l08 properties of QUADRIC surfaces [Chap. VIII. Since the line through P' and P" is tangent to both quadrics, it lies in the tangent planes at both points. Hence P* and P" lie in both planes, so that ,.U" ,.'-.," /v'-v" ^'t" ii'^/" -^'-v" 1 J u.^ , yy" , z-z" ^^ x'xr yy z'z" ^^ /,j> I*! V ,.■> , 1. ' -LO , 1. ' .9 I 7. ' „9 1 7« '" t9 1 7. ' ,.01 1 7. a2+A:j 62+^"j c2 + ;fci ' o? + k^ b' + k. c'+h ' J^<' » xjj subtracting one of these equations from the other, it is seen';^^ ^ J that (28) is satisfied. The planes are therefore at right angles. , ^/ 87. Confocal quadrics in plane coordinates. The equation of tl^ ^/ system (22) in homogeneous plane coordinates (Art. 77) is /v^"'^ *^ ahr + 6V + c'^iv^ - s^ + k(if + w^ + w^) = 0. ■^y * V^ v* Since this equation is of the first degree^i k, we have the follow- ^* ing theorem:6jc^ (^''+ A.;!^,V^. • - • V U;-^^ M^^'+'^V^iJl Theorem. An arbitrary plane («i, Vj, ?f'i, s,)t^s tangent to one and onlij one quadric of a confocal system. - _ ^^t^-^^JUXc Lc^*-n/^ "^ The (imaginary) planes whose homogeneous coordinates satisfy the two equaUons^frv^i_JQj_^^^^^^ r * J "a^M^ + 6^y^ + c^zo' — s^ = 0, w'^ + V" + w^ = are exceptional. They touch all the quadrics of the system. Hence, all the quadrics of a confocal system touch all the planes common to the quadric k = and the absolute. EXERCISES *» 1. Prove that the difference of the squares of the perpendicular from the center on two parallel tangent planes to two given confocal quadrics is con- stant. This may be used as a definition of confocal quadrics. ^2. Prove that the locus of the point of intersection of three mutually per- pendicular planes, each of which touches one of three given confocal quadrics, is a sphere. 3. Write the equation of a quadric of the system (22) in elliptic coordi- nates. Derive from (27) a set of parametric equations of tliis quadric, using elliptic coordinates as parameters. r, \ *( 4. Discuss the system of confocal paraboloids y i V^ s?a ^^^^ a^ + k b-^ + k 5. Discuss the confocal cones + — ^ — = 2iiz + kn^. i^vA + -^ — + _^=0. a^ + k b'^ + k c- + k x + y + z — t = in tetrahedral coordinates. r^ 5. Write the equation of the surface XiXo + x^Xi = in rectangular I ^j coordinates. ♦^ ■ . — ^ 6. Solve Exs. 1 and 2 when the point whose rectangular coordinates are ^Q (3, 1, — 2, 2) is taken as unit point. S^ 7. Why may not anoint lying in a face of the tetrahedron of reference ( be taken as unit point ./to i ^^O^.^,^,^^^,^^^ ^e> , f 90. Equation of a plane. Plane coordinates. From the equation /[^ Kx -f vy -f- wz -\- sf = (5) y» of a plane in homogeneous rectangular coordinates, the corre- ^ spouding equations in tetrahedral coordinates can be found by "5(5^ solving equations (3) for x, y, z, t and substituting in (5). The ^^ resulting equation is linear and homogeneous in Xi, x.2, x^, x^ of :!^ the form ^ Wj-Tl + U^X^ + UsXs + U^Xi = 0, (6) J* . . * with constant coefficients ^^■^, ti^, 113, W4. Conversely, any equation T of the form (6) defines a plane. For, if x^, x^, x^, X4 are replaced 4 by their values from (3), the resulting equation is I* ux + vy-^ wz -f- si = 0, 112 TETRAHEDRAL COORDINATES Chap. IX. wherein xi, = A^i^ + Am^ + A^u^ -\- -44M4, V = A«i + B^iL, + B^u^ + 54W4, ty = Ci^i + (72?t2 + (73H3 + (74?/4, (7) The coefficients u^, Uo, u^, u^ in (6) are called the tetrahedral coordinates of the plane (compare Arts. 27 and 29). It follows from equations (7) and (2) that, if u^, U2, x(s, '«4 (not all zero) are given, the plane is definitely determined, and that, if the plane is given, its tetrahedral coordinates (wj, U2, ih, it^ are fixed except for an arbitrary multiplier, different from zero. 91. Equation of a point. Let {x^, x^, x^, x^) be the coordinates of a given point. The condition that a plane whose coordinates are (u^, xu, 7/3, xi^) passes through the given point is, from (6) Wl^i + Xl..^2 + ^'3^'3 + "4»"4 = 0. (8) This equation, which is satisfied only by the coordinates of the planes which pass through the given point, is called the equation of the point (xj, x^, x^, x^) in plane coordinates (cf. Art. 28). It should be noticed that, in the equation (6) of a plane, (?fj, U2, M3, W4) are constants and (x^, Xo, x^, Xi) are variables. In the equation (8) of a point (x^, x.,, Xg, X4) are constants and (wj, U2, u-i, W4) are variables. 92. Equations of a line. The locus of the points whose coordi- nates satisfy two simultaneous linear equations U'\X^ + ?/"2iC2 + U'^X^ + ^"4X4 =0 ^ ^ is a line (Art. 17). The two simultaneous equations are called the equations of the line in point coordinates. Similarly, the locus of the planes whose coordinates satisfy two simultaneous linear equations X jMi + X 2?^2 "T ^ 3^*3 "I "^ 4^*4 ^^ ") /-I A\ JC'>, + CC"2W2 + •'C'V'3 + a^'>4 = ^ is a line (Art. 28). These two simultaneous equations are called the equations of the line in plane coordinates. Arts. 91-93] DUALITY 113 EXERCISES 1. Write the equations and the coordinates of the vertices and of the faces of the coordinate tetrahedron. 2. Write the equations in point and in plane coordinates of the edges of the coordinate tetrahedron. 3. Find the equations of the folio wing points: (1, 1, 1, 1), (3, — 5,7, — 1), (_ 1, 6, -4, 2), (7, 2, 4, 6). ,.r 4. Write the coordinates of the following planes : ^\ + y-i. + X3 + a;4 = 0, 7 .Ti — X2 — 3 .rs ^ X4 = 0, a;i + 9 j-2 — 5 X3 — 2 X4 = 0. 5. Write the equations of the line Xi + 0:2 = 0, X3 — 7 3:4 = in plane coordinates. ScG. Write the equations of two points on the line. 6. Find the coordinates of the point of intersection of the planes (1, 2, 7, 3), (1, 3, 6, 0), (1, 4, 5, 2). 93. Duality. We have seeu that any four numbers x^, X2, x^, x^, not all zero, are the coordinates of a point and that any four num- bers ?i.stii>ct points (.)•') hy tiro giren distinct planes («') and {x"), every determinant of and {a" ), every determinant of order three in the matrix order three in the tnatrix Jby JU-~} iCQ it. J Xi A it/ o '^ 3 '^ I cV 1 •1/9 "^ 3 *^ is equal to zero. Foi', the points (a;), {x'), {x") and any fourth point (x'") are coplanar. Their coordinates consequently satisfy (11). Since (11) is satisfied for all values of x"\, x"\, x"\, x"\, it follows that the coefficient of each of these variables is equal to zero, that is, that all the determi- nants'of order three in (15) are equal to zero. M, Wo "3 V (15) n\ u'. "'3 u' u'\ u'\ u'\ u is eqiu d to zero. (16) For, the planes (w), {u'), («") and any fourth plane ('<"') are concurrent. Their coordinates consequently satisfy (12). Since (12) is satisfied for all values of u"\, u"\, u"\, u"\, it follows that the coefficient of each of these variables is equal to zero, that is, that all the determi- nants of order three in (16) are equal to zero. 116 TETRAHEDRAL COORDINATES [Chap. IX. Conversely, if the determi- nants of order three in (15) are all equal to zero, the points (x), {x'), and {x") are collinear, since they are coplanar with any fourth point (x'") what- ever. It follows from the above theorem that there exist three numbers j), l^, l^, not all zero, such that Conversely, if the determi- nants of order three in (16) are all equal to zero, the planes (w), («'), and {u") are collinear, since they have a point in com- mon with any fourth plane {u'") whatever. It follows from the above theorem that there exist three numbers p, l^, l^, not all zero, such that px, = l,x\+lix'\, ?-=l,2,3,4. (17) pn^ = l,u\+ku" ., /=1,2,3,4. (18) In particular, we have p ^ 0, since otherwise the coordinates of the points {x') and (x") would be proportional so that the points would coincide. Equations (17) are called the parametric equations of the line determined by {x') and {x"). The coefficients l^ and U are called the homogeneous param- eters of the points on the line. The system of points (17) is said to form a range of points. The equation of the points of this range is found, by substituting from (17) in the equation Sm.cc^ = of a point, to be U'^x\u- -I- l^'^x'^Ui = 0. In particular, we have p ^ 0, since otherwise the coordinates of the planes (w') and («") would be proportional so that the planes would coincide. Equations (18) are called the parametric equations of the line determined by (w') and {u"). The coefficients l^ and I2 are called the homogeneous param- eters of the planes through the line. The system of planes (18) is said to form a pencil of planes (Art. 24). The equation of the planes of this pencil is found, by substituting from (18) in the equation Sw-x. = of a plane, to be l{^u\Xi + k'Zu'^Xi = 0. Arts. 95, 96] TRANSFORMATION 117 EXERCISES 1. Prove the following theorems analytically. State and prove their duals, (a) A line and a point not on it determine a plane. (6) If a line has two points in common with a plane, it lies in the plane. (c) If two lines have a point in common, they determine a plane. (d) If three planes have two points in common, they determine a line. 2. Write the parametric equations of the plane determined by the points (1, 7, - 1, 3), (2,5, 4, 1), (10, - 1, -3, - 5). Find the coordinates of this plane. 3. Write the parametric equations of the point determined by the planes (- 5, 3, 4, 1), (7, - 5, 8, 2), (6, - 4, — 3, 1). Find the coordinates of this point. 4. Write the equation, in plane coordinates, of the field of points in the plane xi + 2 X2 — Xs — xt = 0. ScG. First find the coordinates of three points in the plane. 5. Find the parametric equations of the pencil of planes which pass through the two points Ui — 5 W2 + 3 Ms — M4 = 0, 7 Mi + 2 Mo — Us — M4 = 0. 6. Prove that the points (1, 2, - 3, - 1), (3, -2, 5, - 2), (1, -6, 11, 0) are coUinear. Find the parametric equations of the line determined by these points and the equation in plane coordinates of the range of points on this line. 96. Transformation of point coordinates. Let (Xi, x.,, x^, x^) be the coordinates of a point referred to a given system of tetra- hedral coordinates, so that X- =aiiX + a^.2y + a.i^z + aj, « = 1, 2, 3, 4, (19) in which the determinant of the coefficients * Let the coordinates of the same point, referred to a second S3^stem of tetrahedral coordinates, be a;'. = a\yx + a\.jj + a'.^z + a' J, i = 1, 2, 3, 4, (20) in which A' = I a'li a'22 a'33 a'44 1 ^ 0. * The sj'mbol ] a-^ a-^ a^ 044 1 will be used for brevity to denote «ii f'i2 ai3 "14 the determinant ^21 ^22 "23 ^'24 Osi ^82 Osa 034 041 043 043 044 118 TETRAHEDRAL COORDINATES [Chap. IX. It is required to determine the equation connecting the two sets of coordinates (x^, x.,, x^, x^ and (x\, x'^, x\, x'^). For this purpose solve equations (20) for x, y, z, t. The results are A'x = %A\,x\, A'y = :^A\,x'„ A'z = 2^1 ',30;'^, A't = %A\,x'„ in which A'^^ is the cofactor of a'-^. in the determinant A'. Sub- stitute these values of x, y, z, t in (19) and simplify. The result is of the form x^ = «ii.i-'i + «,„x-'. + «i3.»'3 + wi^a/^, X2 = (it.2iX y -j- «22^ 2 ~r ^23*'^' 3 "I" ^''24'*^ 4J X^ = CC^iX y -\- (t^'yX 2 + «33-'*'' 3 + '<34't' 4, .^4 = a^yx\ + a^rx". + aax\ + a^^x'^, wherein. ^'«,, = a,,A\, + a,2.1',2 + «i3^1'A.3 + nu^^'u, The determinant T^ is called the determinant of the transformation (21). This deter- minant is different from zero, for if we substitute in it the values of the a-i^ from (22), we have at once * f 1 (21) i, k = l,2, 3, 4. (22) «n «12 «13 «1 «21 «22 «L>3 «; «31 «32 «33 «^ «41 «42 «43 « ^1'"' Y 4' A' ^i 11 ^±22 ^13 A' I — A^^ * The product of two determinants of order four A = I (111 «2-2 "33 «44 I and /? = | ^u 622 633 644 I is also a determiuaut of order four C'=lcii e.22 C33 C44I, in which Cik = a.i&ii + ai^bki + CHsbkz + a,-46A-4, £, A; = 1, 2, 3, 4. This theorem can easily be verified by substituting these values of ak in C and ex- pressing V as the sum of determinants, every element of each being the product of an element of A and an element of B. Of the sixty-four determinants in the sum, forty vanish identically, having all the elements of one column proportional to the elements of another. Each of the remaining twenty-four determinants has .B as a factor. When the factor B is removed, the resulting expression is the expansion of the determinant A. t The determinant | A'n A'^^ A'^ A'^ \ whose elements are the cofactors of the elements of A' is equal to A'^, as is seen immediately by multiplying it by A' by the preceding rule, and simplifying the result. Arts. 96, 97] TRANSFORMATION 119 Since T=f^O, the system (21) can be solved for x\, x\, x\, x\ in terms of X i, x 2, x 3, x ^. The results are TX\ = /3uXi + Al^'2 + ^31-^-3 + (3iiXi, Tx\ = p,.x, + /SooXo -I- ^30X3 + ^4,X4, Tx\ = ^,3.7-1 -f /3,3X-., 4- 1833^^3 + ^843^^4, Tx\ = p,,x, + /3,^, + l3,iX, + /?«.T„ (23) in which ^^^ is the cofactor of a.^ in the determinant T. The transformations (2l) and (23) are said to be inverse to each other. 97. Transformation of plane coordinates. Let UiXi + ii^Xo + U3X3 + u^Xi = (24) be the equation of a given plane, referred to tlie system of tetra- hedral coordinates determined by (19). Let the equation of the same plane, referred to the system (20), be I, \x\ 4- ti'ox', + u ',x', + H ,x\ = 0. (25) If, in (24), we replace a;,, x^, x^, x^ by their values from (21), we obtain, after rearranging the terms, (ail?'i + It-zi^U + a^i'(3 + «4l"4)-l''l +(«i>"l + «_•_•": + «lj"3 + "-i;>'A)-'>^'-2 + («13"l + «J3"2 + «,!3"3 + ^<43"4)-''''3 + («14"l + «-:i"2 + «34«3 + a^,,,).i-',= 0. (26) Since equations (25) and (26) are the equations of the same plane, their coefficients are proporticmal, hence l>t'\ = «i."i 4- (W^ + «'."3 4- «u"47 '■ = 1, 2, 3, 4, (27) where j^^O is a factor of i)ruportionality. If we solve equations (27) for u^, V.-,, j/j, v^, we have crn, = p^,n\ + ^^,n\ + ^.3.^-3 + ^,,.^'4, i = 1, 2, 3, 4, (28) in which o- t^ and the p^,^ have the same meaning as in (23). Since, when x^, x^, x^, x^ are subjected to a transformation (21), M„ T<2' "3) "^h ai"6 subjected simultaneously to the transformation (28), the systems of variables (x) and (ii) are called contragredient. 120 TETRAHEDRAL COORDINATES [Chap. IX. EXERCISES 1. Prove that the four planes determined by equating to zero the second members of equations (23) are the faces of the coordinate tetrahedron of the system (x'l, x'2, x'3, x'4). 2. State and prove the dual of the theorem in Ex. 1 for the second mem- bers of equations (27). 3. By means of equations (21) and (23) find the coordinates in each sys- tem of the unit point of the other system. 4. Determine the equations of a transformation of coordinates in which the only change is that a different point is chosen as unit point. 98. Projective transformations. Equations (21) were derived as the equations connecting tlie coordinates of a given arbitrary point referred to two systems of tetraliedral coordinates. We shall now give these equations another interpretation, entirely distinct from the preceding one, but ec^ually important. Let there be given a system of equations (21) with determinant T not equal to zero. Let P' be a given point and let its coordi- nates, in a given system of tetrahedral coordinates, be {x\, x'2, x\, x'^. By substituting the coordinates of P' in the second members of (21), we determine four numbers a^i, x,, Xg, x^, which we consider as the coordinates (in the same system of coordinates as those of P') of a second point P. To each point P' in space corresponds, in this way, one and only one point P. Moreover, when the coordi- nates of P are given, the coordinates of P' are fixed by (23), so that to each point P corresponds one and only one point P'. It is useful to think of the point P' as actually changed into P by the transformation (21) so that, by means of (21), the points of space change their positions. A transformation determined by a system of equations of the type (21), with determinant T not equal to zero, is called a pro- jective transformation. The projective transformation (23) is called the inverse of (21). If, by (21), P' is transformed into P, then, by (23), P is transformed into P'. By (21), the points of the plane («') are transformed into the points of the plane (») determined by (28). Equations (28) are called the equations of the transformation (21) in plane coor- dinates. Arts. 98-100] CROSS RATIO 121 99. Invariant points. The points which remain fixed when operated on by a given projective transformation (21) are called the invariant points of the transformation. To determine these points, put x, = p.«'i in (21). The condition on p in order that the resulting equations («ii —P) ^\ + «i2^"'2 + a^zx\ + a^iX\ = 0, Ojix'i + {0.21— P)x' 2 + a^^x'; + a^iX\ = 0, ttaiic'i + ttsox', + («33 — 1^) ^"'3 + a34^'4 = 0, a^^x'i + aiox\ + a43x'3 + {a^^ —p) x'4 = have a set of solutions (not all zero) in common is that D{p) = an-p «12 «13 «14 «21 «22-i> «23 «24 "31 «32 «33-P «34 a,. «42 "43 «44 = 0. (30) P Let Pi be a root of D{p)= 0. If 7)1 is substituted forp in (29), the points (x') whose coordinates satisfy the resulting equations are ifi yarra nt points of the given transformation. If D{pi) is of rank three, equations (29) determine a single invariant point when p = Pi (Art. 85). If D(2h) is of rank two, equations (29) detennine a line when 2^ — P\- Each point of this line is an invariant point of the transformation. If D(pi) is of rank one, equations (29) determine a plane of invariant points when p=p^. If all the elements of D(p^) are zero, every point in space remains fixed. In this last case, the transformation is called the identical transformation. 100. Cross ratio. The cross ratio of four numbers k^, k^, k^, k^ is defined by the equation __ ki — ^2 _ ^3 — ^2 ^ /Cj — /t^ fC^ — K^ The cross ratio of four collinear points Pj, P2, P3, P^, or of four collinear planes ttj, ttj, tts, ^4, is equal to the cross ratio of the ratios of their homogeneous parameters (equations (17) or (18)). If the parameters of the given points or planes are, respectively, li, l^'i l\, l\; I" I, V\\ r"i, V'i, it follows that their cross ratio is ^ ^ Ul\ - l,V, . V\ l\ - i \ y\ ir\-i,v".'i"d"\-v\v\' 122 TETRAHEDRAL COORDINATES [Chap. IX. If o- = — 1, the four given points or planes are said to be harmonic. An important property of the cross ratio is stated in the follow- ing theorem : Theorem. The cross ratio of four x>oints (or planes) is equal to the cross ratio of any four points (or 2ilanes) into which they can be projected. In the projective transformation (21), let the points (x') and (x") of equation (17) be projected into (y') and (y"), respectively. It follows that the point of the range (17) whose parameters are /i and h is projected into a point (?/) of the range determined by (?/') and (y") such that yi = i,y'i + i2y"i, ^• = l, 2, 3,4. Since the parameters of the points are unchanged, the cross ratio is unchanged. Similarly for a set of four planes through a line. Conversely, two ranges of points, or pencils of planes, are pro- jective if the cross ratio of any four elements in the first is the same as that of the corresponding elements in the second. ^. EXERCISES ' 1. Let .4^(1, 0, 0, 0), 7^=(0, 1, 0, 0), C=(0, 0, 1, 0), Z)=(0, 0, 0, 1), E={1, 1, 1, 1). Find the equations of a projective transformation which interchanges these points as indicated, determine the roots of D( p) — 0, and find the configuration of the invariant elements when (a) J. is transformed into A, B into i?, C into C, D into E^ E into D. (6) A is transformed into B, B into ^, C into D, Z> into C, E into E. (c) A is transformed into 5, B into C, C into ^4, Z) into Z>, E into E. (d) A is transformed into B, B into C, C into Z>, B into £", E into A. 2. Show that a projective transformation can be found that will transform five given points A, B, C, D, E. no four of which are in one plane, into five given points ^4', B', C", D', E' , respectively, no four of which lie in one plane. Show that the transformation is then uniquely fixed. ^ 3. A non-identical projective transformation that coincides with its own inverse is called an involution. Find the condition that the transformation (21) is an involution. ^' 4. Show that the tran.sformations X\ — x'l, Xo = x'2, X3 = ± x'3, X4 = — x'4 are involutions. Find the invariant points in each case. Art. 100] CROSS RATIO 123 5. If P, P' are any two distinct corresponding points in either involution of Ex. 4, prove the following statements : (a) The line PP' contains two distinct invariant points 31, 31'. (/>) The points {PP'313I') are harmonic. 6. Find the invariant points of the transformation Xi = x'2, x.^ = x'3, 3:3 = x'4, Xi = x'l. Show that the points of space are arranged in sets of four which are interchanged among themselves. '7., Interpret the equations (Art. 36) of a translation of axes as the etpiations of a projective transformation. Find the invariant elements. 8. Interpret the equations (Art. 37) of a rotation of axes as the equations of a projective transformation. Show how this transformation can be effected. 9. Find the cross ratio of the four points on the line (17) whose param- eters are (0, 1), (1, 1), (1, 5), (4, 3). CHAPTER X QUADRIC SURFACES IN TETRAHEDRAL COORDINATES 101. Form of equation. Since the equation F(x, y, z, t) = may be transformed into an equation in tetrahedral coordinates by means of equation (3) of Art. 88, it follows that the equation of a quadric surface in tetrahedral coordinates is of the form + 2 a^iX^x^ + 2 a23.r2.r3 + 2 a^iX^Xi + 2 Us^x^Xi = 0. a^^ = a^^. (1) Conversely, any equation of this form will represent a quadric surface, since by replacing each x^ by its value from (3), Art. 88, the resulting equation F (x, y, z, t) = is of the form discussed in Chapters VI, VIT, and VIII. 102. Tangent lines and planes. Let (x) and (y) be any two points in space. The coordinates of any point (z) on the line joining (x) to (y) are of the form (Art. 95) z^ = \x^+l.y,, 1=1,2,3,4. (2) If (2:) lies on the quadric A = 0, then \'A(x) + 2Xf,A(x,y) + ,.'A{y) = 0, (3) wherein A(x, y) = A(y, x) = (a„?/i + a^,y., + a3i?/3 + a,iy,)xi + (a2l2/l+ a222/2 + «322/3 + «422/4)^2 + («3uVl + (^3^2+ ttsS^/s + «342/4)a'3 + (a4l2/l + a42.V2 + 0432/3 + a442/4)^'4 = 9 2^ ^ 2/. = ."^ A ^^ •^'•- ^'^^ If (y) lies on J. = 0, then A(y) = and one root of (3) is X = 0. If (y) is so chosen that both roots of (3) are A = 0, we must have A(x, y) = 0. If {x) is regarded as variable, and A{x, y) is not identically zero, the equation A{x, y) = defines a plane. The line joining any point in this plane to the fixed point (y) on the quadric A touches the surface at the point (y) (Art. 76). The line is a tangent line and the plane A(x, y) = is a. tangent plane to ^ = at (y). 124 Arts. 101-103] INDETERMINATE TANGENT PLANE 125 EXERCISES ' 1. Find the equation of the tangent plane to x^ + Xi^ + X3- — a^Xi'' = at the_point (0, 0, a, 1). 2. Show that equation (4) vanishes identically if A = axi^ + bx2^ + cxs^ = and (y) = (0, 0, 0, 1). 3. Determine the coordinates of the points in which the line Xi + 2 X2 4- 354 = 0, ^3 — 2 X4 = meets the surface Xi^ — xiX2 + a;2X3 + 4 ^3^= 0. 4. Show that the line Xi = 0, a;i — 3 X2 = touches the surface 0-4- — 3 xr + bxo'^ -\- Xi{xi + 5 X2) + a;3X4 = 0. 103. Condition that the tangent plane is indeterminate. If equation (4) is satisfied identically, the coefficient of each x, must vanish. Thus we have the four equations «Uyi + 0212/2 4- Clsilh + «4l2/4 = 0, «12yi + «222/2 + 032^3 + «42y4 = 0, /gx ai3^1 + «23?/2 + «33?/3 + «43?/4 = 0, «142/l + «242/2 + «342/3 + ^uVi = ^^ If these equations are multiplied by yi, y2, Vz, y^i respectively, and the products added, the result is ^(?/)=0, hence if the coordinates of a point {y) satisfy all the equations (o), the point lies on the surface ^ = 0. From (3) it follows that the line joining any point in space to a point {y) satisfying equations (5) will meet the surface ^4=0 in two coincident points at (.?/). If {x) is any other point on the surface A. so that A{x) = 0, it follows from (3) that every point on the line joining (x) to (?/) lies on the surface. The surface A is in this case singular and (y) is a vertex (Arts. 66 and 67). Conversely, if A{x) = is singular, with a vertex at (?/), the two intersections with the surface of the line joining {y) to any point in space coincide at {y). The coefficient ^(.r, y) is identi- cally zero and the coordinates of {y) satisfy (5). Since these co- ordinates are not all zero, it follows that the determinant A = ttji a, «13 ai4 (^23 «24 033 n^i «« a^ (6) 126 QUADRIC SURFACES [Chap. X. vanishes. Conversely, if A = 0, then four nmnbeis y^, j/o, Vz, Vi can be found such that the four equations (5) ai-e satistied. The point iy) lies on A{;x) = and in the plane A{x, y)=0. The line joining (y) to any point {x) will have two coincident points in common with A(x) = at (y) ; that is, (y) is a vertex of the quadric A. We thus have the following theorem : Theorem. The necessary and sufficient condition that a quadric surface is singidar is that the determinant A vanishes. The determinant A is called the discriminant of the quadric A. If it does not vanish, the quadric will be called non-singular. Unless the contrary is stated, it will be assumed throughout this chapter that the surface is non-singular. 104. The invariance of the discriminant. In Chapter VII cer- tain invariants under motion were considered. We shall now prove the following theorem which will include that of Art. 74 as a particular case. Theorem I. If the equation of a quadric surface is subjected to a linear transformation (Art. 96), the discriminant of the transformed equation is equal to the product of the discriminant of the original equation and the square of the determinant of the transformation. A 4 Let A{x) = 2] 2 ^^ik^i^k = be the equation of a given quadric, and let a;. = ai]X\ -f a^^x'^ + a^^'s + oLh^'a, i = 1, 2, 3, 4 define a linear transformation of non-vanishing determinant T. If these values of a;,- are substituted in A(x), the equation becomes 1=1 A=l in which 4 4 Art. 104] INVARIANCE OF DISCRIMINANT If we now put 127 rik = X^' Im^^mky it follows that 'a-=2"''^*'^- If we form the discriminant A' of A'(x'), we may write «ll'*ll + «2l'*21 + «3l'"31 + "4l''41 «u''l2 + (h\'''22 + «3l''32 + «4l''42 «12^11 + «22'*21 + «32''31 + «42'*41 «12»'l2 + «22^'22 + «32'"32 + «42''42 A' = This determinant may be expressed as the product of two deter- minants T and R (Art. 96, footnote), thus «11 «12 «13 «14 ^•ll '•l2 ''-13 r "21 «22 «23 «24 »*21 5'22 '-23 r "ai «32 "33 «34 '31 ^-32 ^*33 r «41 «42 «43 «44 ni *-42 '•43 1 the columns in the first factor being associated with the columns in the second to form the elements of the rows in the product. Similarly, the second factor may be expressed in the form «ll«n + «12«21 + ai3f<31 + «14«41 «21«U + «22a21 + «23«31 + ^hi<^i\ «U«12 + «12«22 + <'l3«32 + «14«42 «21«12 + «22«22 + «23«32 + «24«42 which is the product of A by T, the elements of the rows in the first factor being multiplied by the elements of the columns in the second, hence . , _ ji^k On account of this relation, the discriminant is said to be a rela- tive invariant under linear transformation of tetrahedral coordi- nates. ISIoreover, the following theorem will now be proved. TriKMFtEAr IT. Any sth minor of ^' may he expressed as a linear fiiiiciinii nf i]i(>_ stli tuinors of ^. 128 QUADRIC SURFACES [Chap. X. The method of proof will be sufficiently indicated by consider- ation of the minor This determinant, when written in full, «l2'''ll + «22''21 + «32'*31 + «42'"41 '*u''l2 "1" ^^21^22 ~l~ '''si^Sa ~r C'4l'*42 «12''l2 + «22'*22 + "32^32 + a42''42 may be expressed as the sum of sixteen determinants, four of which vanish identically. The remaining ones may be arranged in pairs, by combining the determinant formed by the ith term of the first column and the kth term in the second with that formed by the kth. term in the first column and the ith in the second. Every such pair is equivalent to the product of a second minor of A and a second minor of T. If / = 2, k= 3, for example, we have CC21T21 tt31^'32 ^22^21 ^32^*32 In this way it is seen that every second minor of A' is a linear function of the second minors of the determinant R, the coeffi- cients not containing yv^. 4 By replacing each ?-.^ by its value ^^''w""''; ^"^^ repeating the m=l same process, it may be seen that each second minor of R may be expressed as a linear function of the second minors of A, the coefficients not containing any a,^.. The same reasoning may be applied to the first minors of A'. This proves the proposition. As a corollary we have the further proposition : Theorem III. Tlie rank of the discriminant of the equation of a quadric surface is riot changed by any linear transformation with non-vanishing determinant. + «32'' 31 "21*'22 31 '*22^22 = «22 «31 «32 '■21^32 - «21 «32 «31 «22 021 «31 »*21 »-31 «22 «32 5 '22 ^•32 Arts. 104, 105] LINES ON THE QUADRIC SURFACE 129 For, it follows from Th. II that the rank of A' is not greater than that of A. Neither can it be less, since by the inverse trans- formation the minors of A may be expressed linearly in terms of those of A'. We may now conclude : if the discriminant A is of rank four, the quadric A(x) = is non-singular (Art. 103). If A is of rank three, ^ = is a non-composite cone, for if we take its vertex (Art. 103) as the vertex (0, 0, 0, 1) of the tetrahedron of refer- ence, the equation A = reduces to a^Xi^ + cu.X2^ + 033X3^ + 2 a^nX^x^ 4- 2 ai3cria;3 + 2 a23X2Xs = 0. The line joining any point on the surface to (0, 0, 0, 1) lies on the surface, which is therefore a cone (Art. 46). Since by hypothesis A is of rank three, we have I «lia22«33 ! ^ ^, hence the cone is non-composite. If A is of rank two, the quadric is composite, for if we take two vertices as (0, 0, 0, 1) and (0, 0, 1, 0), the equation reduces to aiiX'i^ + «22-V + 2 ai2XiX2 = 0, which is factorable. Since by hypothesis A is of rank two, o„a22 — du is not zero, hence the two components do not coincide. If A is of rank one, the equation may be reduced to the form .rj2 = 0, which represents a plane counted twice. 105. Lines on the quadric surface. Theorem. TTie section of a quadric surface made by any of its tangent planes consists of two lines passing through the point of tangency. For, let (?/) be any point on a quadric surface ^ = 0, and (z) any point on the tangent plane at (y), so that A(y) = 0, A(y, z) = 0. If (2) is on the curve of intersection of A(x) = 0, A{x, y) = 0, then A{z) = and (3) is identically satisfied, hence every point of the line joining (y) to (2) lies on the surface. Since the sec- tion of a quadric made by any plane is a conic (Art. 81) and one component of this conic is the line joining (y) to (2), the residual component in the tangent plane is also a straight line. 130 QUADRIC SURFACES [Chap. X. The second line also passes through (y), since every line lying in the tangent plane and passing through (y) has two coincident points of intersection with the surface at (y). , 106. Equation of a quadric in plane coordinates. Let the plane u^x^ + UnX^ + n^x^ + v^Xi = (7) be tangent to the given quadric ^1, and let (y) be its point of tangency. Since A(x, y) = is also the equation of the tangent plane at (?/), the equation 2?/,;*;^ = must differ from A{x, y)=0 by a constant factor k (Art. 24), hence (hlVl + a2l2/2 + «3l2/3 + fl'41.V4 = kUu ^ «122/l + «222/2 + a32.V3 + «42i/4 = ^'^2, (g^) auVi + (hslh + «332/3 + ttisy* = ^«3> «14^1 + ^24^2 + «34.V3 + «44.V4 = ^"4- Moreover, since (?/) lies in the tangent plane, we have ^^i2/i + ^'22/2 + n^lh + "4?/4 = ^- (9) On eliminating y^, y^, y^, y^ and k between (8) and (9), we obtain as a necessary condition that the plane (?<) shall be tangent to the surface, $(?<) = «11 «21 «31 «12 «22 «32 "13 "23 «33 ai4 «24 «34 u. U^ U. (10) Conversely, if the coordinates of a plane (it) satisfy (10), and if also A ^ 0, then the plane is tangent to the quadric A = 0. For, if (10) is satisfied, five numbers y^, y.,, yz, y^, k, not all zero, can be found which satisfy (8) and (9). In particular, k^O, for otherwise, since ^=^0, it would follow from (8) that y^ = y^ = y^ = y^ = 0, contrary to the hypotheses. Since t*,, ?/,, u^, u^ are not all zero, it follows from (8) that y^, y., y^, y^ are not all zero, and hence are the coordinates of a point. By solving (8) for u^, U2, Us, u^ and substituting in (9), we obtain ^l(//)=0, hence the point (//) lies on the quadric A. From (4) and (7) it follows that the plane (7) is tangent to A at the point (y). Art. 106] EXERCISES 131 The equation ^ (») = is of the second degree in ?fj, U2, u^, n^. It is the equation of the quadric in plane coordinates. By duality it follows that any equation of the second degree in plane coordinates, whose discriminant is not zero, is the equa- tion of a quadric surface in plane coordinates. If A is of rank three, so that A = is the equation of a cone, the equation (^(a) =0 reduces to C^k-u^f^ 0, 2fci"i = being the equation of the vertex of the cone. If A is of rank less than three, ^{u) = vanishes identically. The equation (?<) = was in fact derived simply by imposing the condition that the section of the quadric by the plane (u) should be composite. EXERCISES 1. If the equation ^(.r) = cuntaius but three variables, show that it represents a singular quadric. 2. Calculate the discriminant of Xi- + 'j-{' + X2'^ — x^^ = 0. 3. Show that the di.scriuiinaiit of (?6) = contains the discriminant of ^(x) = as a factor. 4. Given A{x) = axi'- + bx-^^ + cxi^ + dx^^ = 0, determine the form of the equation $(m) = 0. 5. When the equation *(«) = is given, show how to obtain the equation A{x) = 0. 6. Given A (x) = axi^ + bx-r + 2 cxai = 0, find (?<) = 0. 8. Given 4>(?<) = Ui^ — 2 uiUo + u-2^ + 2 ti^uz + 2 xiiin — 2 xi^u-i — 2 u^u^ + M3- + »4^ + 2 u-iUi = 0, find A{x) = and interpret geometrically. •* '"i" * 9. Find the two lines lying in the tangent plane Xi = to the quadric X\X-2 + xi- — xi^ — 0. 10. \yrite the equation of a quadric passing through each vertex of the tetrahedron of reference. 11. \Yrite the equation of a quadric touching each of the coordinate planes (use dual of method of Ex. 10). 12. Write the equation of a quadric which touches each edge of the tetra- hedron of reference. 13. What locus is represented by the equation 'LatkUiUk — when the dis- criminant is of rank three ? of rank two ? of rank one ? 14. Show that through any line two planes can be drawn tangent to a given non-singular quadric. 132 QUADRIC SURFACES [Chap. X. 107. Polar planes. When the coordinates Zj, Z2, z^, Zi of any point (z) in space are substituted in A(;x, z) = 0, the resulting equation defines a plane called the polar plane of (z) as to the quadric A. Let (y) be any point in the polar plane of (2), so that A(y, z) = 0. Since the expression A(y, z) = A(z, y) is symmetric in the two sets of coordinates yi, y2, y^, 2/4 and z^, z^, z^, z^, it follows that (z) lies in the polar plane of (?/). Hence we have the following theorem : Theorem. If the j)oiut (y) lies on the polar plane of (z), then (2) lies on the polar plane of (y). Any two points (?/), (z), each of which lies on the polar plane of the other, are called conjugate points as to the quadric A(x) = 0. Dually, any two planes are said to be conjugate if each passes through the pole of the other. 108. Harmonic property of conjugate points. We shall prove the following theorem. Theorem. Any two conjugate j^oints (x), (y) and the two points in which the line joining them intersects the quadric constitute a set of harmonic jioints. The coordinates of the points (z) in which the line joining the conjugate points (x), (y) as to the quadric A are obtained by putting Zi = \x- + /A?/- and substituting these values in A{z) = 0. The values of the ratio X : fx are roots of the equation (Art. 102) A2.4(x) + 2 XfxA(x, y) + fi?A(y) = 0. Since A{x, y) = 0, if one root is Aj : /xi, the other is — Ai : /aj. The coordinates of the points (x), (y) and the two points of intersec- tion are therefore of the form ^i, Vi, K^i + Mi, K^\ - Mi, ?■ = 1, 2, 3, 4, hence, the four points ai'e harmonic (Art, 100). Dually, any two conjugate planes («), {v) and the two tangent planes to the quadric through their line of intersection determine a set of harmonic planes. Arts. 107-110] TANGENT CONE 133 109. Locus of points which lie on their own polar planes. The condition that a point (y) lies on its own polar plane A{x, y) =0 as to A(x) = is A{y, y) = A{y) = 0, that is, that the point lies on the quadric. We therefore have the theorem : Theorem. The locus of points ichich lie on their polar planes as to a quadric A(x) = is the quadric itself. Since when (y) is a point on A(x) = 0, A{x, y) = is the equa- tion of the tangent plane to A(x) =0 at {y), it follows that the polar plane of any point on the surface is the tangent plane at that point. A point which lies on its own polar plane will be said to be self-conjugate. Dually, a plane which passes through its own pole will be said to be self-conjugate. 110. Tangent cone. If from a point (y) not on the quadric A all the tangent lines to the surface are drawn, these lines define a cone, called the tangent cone to ^1 from (//). Theorem. Tlie tangent cone to a quadric from any point not on the surface is a quadric cone. Let (x) be any point in space. The coordinates of the points (z) in which the line joining (x) to (y) meets the quadric A are of the form 2;. = A.i\ -|- fxy^, in which A : fi are roots of the quadric equation \'A{x) + 2 \^A(x, y) + /.2.4(v/) = 0. The two points of intersection will be coincident if [A{x, y)J = A{x)A{y). (11) If now {y) is fixed and (x) is any point on the surface defined by (11), then the line joining {x) to {y) will be tangent to A = 0. Since this equation is of the second degree in x, the theorem follows. The curve of intersection of the tangent cone from (?/) and the quadric is found by considering (11) and A{x) = simultaneous. The intersection is evidently defined by \_A{x,y)J=0, A{x) = 0. 134 QUADRIC SURFACES [Chap. X. This locus is the conic of intersection of the quadric and the polar plane of the point (t/), counted twice. If (y) is a point on the surface, then ^l(^) = and the tangent cone reduces to the tangent plane to ^4 = at {y), counted twice. 111. Conjugate lines as to a quadric. We shall now prove the following theorem. Theorem. The polar j^layie of every jwinf of the line joining any two given points (y), (z) jiasses through the line of intersection of the polar planes of {y) and (z). The polar planes of (y) and of (z) are A{x, y) = and A(x, z) = 0. The coordinates of any point of the line joining (jj) and (z) are of the form Xy--j-iJiZ-; and the polar plane of this point is A{x, Xy -\- jxz) = 0. Since this equation is linear in Xy^ + /x2,, it may be rewritten in the form XA{;x, y) + fxA{x, z) = 0, which proves the theorem. From Art. 107 it follows that the polar plane of every point of the second line passes through the first. Two such lines are called conjugate as to the (paadric. If from P, any point on the quadric, the transversal to any pair of conjugate lines is drawn, it will meet the quadric again in the harmonic conjugate of P as to the points of intersection with the conjugate lines, since its inter- sections with these lines are conjugate points (Arts. 107, 108). EXERCISES 1. Determine the equation of tlie polar plane of (1, 1, 1, 1) as to the quadric .ii"- + »;2" + *'3^ + ^i" = 0. 2. Find the equation of the line conjugate to a:i = 0, a;2 = as to the quadric s'l'^ + X2^ + X3- + Xi^ = 0. 3. Show that any four points on a line have tlie same cro.ss ratio as their four polar planes. 4. Find the tangent cone to 3:1X2 — x^Xt = from the point (1, 2, 1, 3). 5. If a line meets a ([uadric in P and Q, show that the tangent planes at P and Q meet in the conjugate of the line. Arts. 111-113] SELF-POLAR TETRAHEDRON 135 6. Show that t]\e quadrics xr + xr + x-i^ — kXi~ = 0, xr + Xo- + x.r — Ix^^ = are such that the pohxr plane of (0, 0, 0, 1) is tlie same for both. Inter- pret this fact geometrically. 7. Write the equation of a quadric containing the line Xi = 0, X2 = 0. How many conditions does this impose upon the equation ? 8. Write the equation of a quadric containing the line Xi = 0, x^ = and the line X3 = 0, Xi = 0. 9. Show that through any three lines, no two of which intersect, passes one and but one quadric. 112. Self-polar tetrahedron. Associated with every tetrahedron P1P2F3P4 is a tetrahedron 7ri7r27r37r4 formed by the polar planes of its vertices, ttj of P^, w^, of P^, ttj of P3, and -n-^ of P^. Conversely, it follows from Art. 107 that the plane P^P^P^ is the polar plane of the point ttittottj, etc. Two tetrahedra P^P^P^P^, Tr^ir^rr^-i, such that the faces of each are the polar planes of the vertices of the other as to a given quadric, are called polar reciprocal tetrahedra. If the two tetra- hedra coincide, so that the plane ttj is identical with the plane P^P^Pi, etc., the tetrahedron is called a self-polar tetrahedron. To determine a self-polar tetrahedron choose any point Pi not on A{x) and determine its polar plane tt). In this polar plane choose any point P, not on A{x) and determine its polar plane tt^. This plane passes through P^ (Art. 107). On the line of inter- section of 7rj7r2 choose a third point P3 not on A{x) and determine its polar plane ^3. The plane tt^ passes through Pi and P.,. Finally, let P4 be the point of intersection of ttittotts. The polar plane 774 of P4 passes through points P^P^P^. Hence the tetra- hedron P^P^P^P^ = TTi 772773774 is a self-polar tetrahedron. 113. Equation of a quadric referred to a self -polar tetrahedron. Theorem. Tlie necessarj/ and sufficient condition that the equa- tion of a quadric contains only tlie squares of the coordinates is that a self polar tetrahedron is chosen as tetrahedron of reference. If the tetrahedron of reference is a self-polar tetrahedron, the polar plane of the vertex (0, 0, 0, 1) is x^ = 0. But the equation ^(^> 2/) = <^ of the polar plane of (0, 0, 0, 1) is a4ia;i + a42a;2 + a43.'r3 + ««-'^*4 = 0, hence a^ = a^ = O43 = 0. Since the polar plane of 136 QUADRIC SURFACES [Chap. X. (0, 0, 1, 0) is x^ = 0, it follows further that O13 = 033 = 0, and since the polar plane of (0, 1, 0, 0) is X2 = 0, that a^z = 0. But if these conditions are all satisfied, then the polar plane of (1, 0, 0, 0) is Xj = 0, and the equation of the quadric has the form Conversely, if the equation of a quadric has the form OtllX'j -|- 0!22'^2 I" '^33'^3 1 fl'44'^4 ^ "j the tetrahedron of reference is a self-polar tetrahedron. Since A^O, the coefficients a^j are all different from zero. If the coefficients in the equation of a quadric are real numbers, it follows from equation (4) that the polar plane of a real point is a real plane, hence from Art. 88 the equation of the quadric can be reduced to the form 2a„.'c'^ = by a real transformation of coordinates, that is, one in which all the coefficients in the equa- tions of transformation are real numbers. By a suitable choice of a real unit point the equation of the quadric may further be reduced to the form Xj^ + X2^ ± x^ ± x^ = 0. 114. Law of inertia. The equation of a quadric having real coefficients may thus be reduced by a real transformation to one of the three forms (a) Xi^ + xi + x^^ + x^ = 0, (6) xy-" + x,-" + x,"" - x-" = 0, (c) .Xi^ + .r2- — x^^ — x^^ = 0. Theorem. TTie equation of any real non-singxdar quadric may he reducedbya real transformation to one and only one of the types (a), (6), (c). A quadric of type (a) contains no real points, as the sum of the squares of four real numbers can be zero only when all the num- bers are zero. If the equation is of type (6), the surface contains real points, but no real lines, for a real line lying on the surface ■would cut every real plane in a real point, but the section of (b) by 3^4 = is the conic x^^ + x^^ + .T3'^ = 0, which contains no real Arts. 113-115] RECTILINEAR GENERATORS 137 points. If the equation of a quadric can be reduced to type (c), the surface contains real points and real lines. The line x^ — X3 — 0, X2 — x'4 = 0, for example, lies on the surface. Any real plane through it will intersect the quadric in this line and another real line. If the equation of a quadric can be reduced to one of those forms by a real transformation, it can evidently not be reduced to either of the others, since real lines and real points remain real lines and real points. The theorem of this Article is known as the law of inertia of quadric surfaces. It states that the numerical difference between the number of positive terms and the number of negative terms is a constant for any particular equation independently of what real transformation is employed. By a transformation which may involve imaginary coefficients the equation of any quadric may be reduced to the form 2x--^ = 0. For this purpose it is necessary only to replace x- by — l^z in the equation Sa^.x,^ = of Art. 113. 115. Rectilinear generators. Reguli. If in the equation 2j;^.2 = 0, the transformation Xi = a; J -{- X 2} X2 = l\X 1 — ^2)} ■^i^^ ^{.^ 3 I" -^ 4)9 ^4 ^^ i*^ 3 '''4/ is made, it is seen that the equation of any quadric can also be written in the form If the quadric is of type (c), its equation can be reduced to (12) by a real transformation. In the other cases the transformation is imaginary. The line of intersection of the planes ""1*^1 ~" '^2*''^3j /CjiC^ ^^ ^2*^2 \ *^/ lies on the quadric for every value of k^ : ^^, since the coordinates of any point (y) on (13) are seen by eliminating k^ : k^ to satisfy (12). Conversely, if the coordinates of any point (.?/) on the quadric are substituted in (13), a value of k^ : ko is determined such that the corresponding line (13) lies on the quadric and passes through [y). 138 QUADRIC SURFACES [Chap. X. No two lines of the system (l.'>) intersect, for if k^Xi = A^-jXa, k^x^ = knXo, and k\Xi = ^''2^2J k\x^ = ^'2-^3 are the two lines, the con- dition that they intersect is k, fco k\ k', k, A-, k'. k\ — — (kiK 2 — "-'2"^ 1 ) — '-'• But this condition is not satisfied unless k^ : k, = A;'i : A;',, that is, unless the two lines coincide, hence : Theorem. Througk each point on the quadric (12) jjcisses one and bwt one line of the sytitem (13), lying entirehj on the surface. A system of lines having this property is called a regulus (Art. 79). In the same way it is shown in the system of lines l^x, = l.,x„ 1^X3 = 1x2 (14) is a regulus lying on the same quadric (12). Those two reguli will be called the A:-regulus and the Z-regulus, respectively. It was seen that no two lines of the same regulus intersect. It will now be shown that every line of each regulus intersects every line of the other. Let 7. . _ 7. . ^ . _ 1. A'j.i'j — A'2.^'3, KiX^ — /l2^2 be a line of the A'-regulus and (j.l'l = 1 0X4, tj2?3 = I2X2 be a line of the Z-regulus. The condition that these lines intersect ^^^^^^ k, A-2 ko k, li l, /. ^1 But this equation is satisfied identically ; hence the lines intersect for all values of k^ : k^ and Zj : l^. 110. Hyperbolic coordinates. Parametric equations. Each value of the ratio k^ : k., uniquely determines a line of the A;-regulus ; each value of li : L uniquely determines a line of the /-regulus. These two lines intersect; their point of intersection lies on the quadric; = 0. Arts. 115-117] PROJECTION UPON A PLANE 139 through this point passes no other line of either regains. Thus, a pair of values k^ : A:, and Zj : I2 fixes a point on the surface. Conversely, any point on the surface fixes the line of each system passing through it, and consequently a pair of values of Jc^ : k^ and ^1 : ^2- These two numbers are called hyperbolic coordinates of the point. From equations (13), (14) the relations between the coordinates x^, X2, X3, x^ of a point on the surface and the hyperbolic coordi- nates ki : k.,, li : I2 are These equations are called the parametric equations of the quadric (12). Since the equation of any non-singular quadric can be reduced to the form (12) by a suitable choice of tetrahedron of reference, it follows that the general form of the parametric equation of a quadric surface, referred to any system of tetra- hedral coordinates, may be written in the form ic. = ciijcl, + «-2i^\h + (hi^J'i + ^ji^'j^jj * = 1, 2, 3, 4. 117. Projection of a quadric upon a plane. Given a quadric surface A and a plane tt. If each point Pof A is connected with a fixed point on A but not on tt, the line OP will intersect w in a point P', called the image of P. Conversely, if any point P' in tt is given, the point P of which it is the image is the residual point in which OP intersects A. If P describes a locus on A, P' will describe a locus on tt, and conversely. This process is called the projection of A upon tt. Through pass two generators ,7, and g^ of A, one of each regulus. These lines intersect tt in points 0^, 0,, which are singidar elements in the projection, since any point of ^i has Oi for its image, and any point of g.^ has 0^ for its image. The tan- gent plane to A at contains the lines ^i, r/o) hence it inter- sects the plane tt in the line 0^0,. Any point P' of O^Oi will be the image of 0. The line O/Jj will be called a singular line. The tangent lines to A at form a pencil in the tangent plane; any line of this pencil is fixed if its point of intersection Avith O1O2 is known. If a curve on A passes through 0, the point in which its tangent cuts 0^0^ will be said to be the image of the 140 QUADRIC SURFACES [Chap. X. point on that curve. The generators of the regulus to which gr, belongs all intersect g^ ; each, with 0, determines a plane passing through g^, and the intersections of these planes with tt is a pencil of lines passing through Oo. Similarly for the other regu- lus and Oj. The two reguli on A have for images the pencils of lines in ir with vertices at Oj, O2. 118. Equations of the projection. Let 0, Oj, 0^ be three vertices of the tetrahedron of reference ; take for fourth vertex the point of contact 0' of the other tangent plane through O1O2. If = (0, 0, 0, 1), 0, = (0, 0, 1, 0), 0' = (1, 0, 0, 0), 0, = (0, 1, 0, 0), the equation of the surface may be written Let ^1, 4) 4 be the coordinates of a point in the image plane, re- ferred to the triangle of intersection of o^j = 0, Xj = 0, ccj = and the image plane tt or ^a^x- =0. Any point of the line joining (0, 0, 0, 1) to (?/i, 2/25 Vz, Vi) on A will have coordinates of the form Tcy^, ky^, ky^, A'l/^ + A, wherein k'^a^y^ + a^X = for the point in which the line pierces the plane tt. Moreover, since $i = kyi (i = 1, 2, 3) and yiy^ — y^^ = 0, ,, y^ih 7, v?t3 Hence, a point (?/) on A and its image (|) in tt are connected by the equations Plh = ^l^ plh = Ii4. pVz = 44> Plh = 44- (16) If 4 = 0, tlien ^1 = 0, 2/2 = 0, 2/3 = 0, so that any point of the line 0;02 corresponds to 0. If ^1 = and ^ = 0, all the ?/■ vanish, but if we allow a point to approach Oi in tt along the line ^ — t^ = 0, then the corresponding point on A is pyx=T%\ py2 = rt'^S Plh = r^2^z, Pl/i=i2^3, from which the factor ^ can be removed. If now ^ is made to vanish, the point on A is defined by Z/i = *N 2/2 = 0, ?/3 - T.V4 = 0. Arts. 117-119] QUADRIC THROUGH THREE LINES 141 Thus, to the point Oy correspond all the points of the generator gi, but in such manner that to a direction |i — t^2 = through Oj corresponds a definite point (0, 0, t, 1) on g^. To the line li — T^2 = as a whole corresponds the line ?/i - ry. = 0, ^3 - ry^ = 0, that is, a generator of the regulus g^. A plane section cut from A by the plane 2«iic, = has for image in tt the conic whose equa- tion is It passes through Oy, Oz- EXERCISES 1. Prove that if the image curve C is a conic not passing through Oi nor O2, then the curve C on J. has a double 'point at 0, intersects each generator of each reguhis in two points, and is met by an arbitrary plane in four points. 2. If C is a conic through Oi but not O2, then C passes through 0, inter- sects each generator gi in two points and each generator g2 in one point ; it is met by a plane in three points. 3. By means of equations (16), show that C of Ex. 1 lies on another quadric surface, and find its equation. 4. By means of equations (16), show that C of Ex. 2 lies on another quadric, having a line in common with A. Find the equation of the surface and the equations of the line common to both. 119. Quadric determined by three non-intersecting lines. Let the equations of three straight lines I, I', I", no two of which inter- sect, be respectively 2m.x.. = 0, ^ViXi = ; 2w>, = 0, ^v\x, = ; ^u",x, = 0, ^v'\x, = 0. It is required to find the locus of lines intersecting I, I', I". Let (y) be a point on l" so that 2»",7/,. =0, 20. =0. (17) The equation of the plane determined by (_?/) and I is 2M,.?/i2v,a;. — 2",ic.2'y,?/, = 0, (18) and of the plane determined by (y) and I' is ■%a\y,^v',Xi - •$u\xr$o',y, = 0. (19) 142 QUADRIC SURFACES [Chap. X. The planes (18) and (19) intersect in a line which intersects I, I', I". Moreover, the equations of every line which intersects the given lines may be written in this form. If we eliminate ?/i, 1/2, ?/3, 2/4 from (17), (18), (10), we obtain a necessary condition that a point (x) lies on such a line. The equation is u^{vx) — t\{ux) U2(vx) — V2{vx) U3(vx)—V3{iix) u^(vx)~i\(vx) u\(v'x) — v\(u'x) — — u'\ u\ u'\ u'\ " '"" v'\ v'' v V = 0, (20) wherein (iix) is written for 2m(m) = (2A\».)2 = (Art. 106). If k\^0, the section of A=0 by the plane x\ = is a conic whose equation in plane coordinates is obtained by equating to zero the first minor of $(m) correspond- ing to a^i- The first minor of any element a-^ of the principal diagonal equated to zero, together with ^(y() = 0, will, if l\ ^ 0, define the given cone. 122. Projection of a quadric cone upon a plane. Given a point on a cone K, but not at its vertex. To project the cone from upon a plane ir not passing through 0, connect every point Pon ^ with 0. The point P' in which OP cuts tt is called tlie ju'ojcc- tion of P upon tt. Let g be the generator of /i" through 0, and 0' the point in which g pierces tt. Let I be the line of intersection of TT and the tangent plane along g. The point on K corre- sponds to any point of /, and to 0' in tt correspond all the points of g. With these exceptions there is one-to-one correspondence between the points of tt and of K. A curve defined on either will uniquely determine a curve on the other. Let /i be defined by XiX^ — x.f=0, v by x^=0, and O=(0, 0, 1, 0). If P' = (ii, $2i ^1 ^4)? the coordinates of P = {Xi, x^, x^, x^ are seen, as in Art. 108, to be connected with those of P' by the equations 146 QUADRIC SURFACES [Chap. X. EXERCISES 1. Show that 4 xi2 + 6 ri.ra + 8 x-r + 9 Xg^ + 12 X3X4 +4 3:42 = represents a cone. Find the coordinates of its vertex. 2. Find a value of k such that the equation x{- — 5 x\X2 + 6 X2^ + 4 x^ — TcXi'X'i + a'4"'^ = represents a cone. 3. Write tlie equations of tlie cone of Ex. 1 in plane coordinates. 4. In equations (24), replace m by x,- and interpret the resulting equations. 5. Prove that if the two lines of intersection of a quadric and a tangent plane coincide, the surface is a cone. 6. What locus on the cone K has for its projection in tt a conic : (a) not passing through 0' ? (6) passing through 0', not touching I ? (c) touching I at O' ? 7. State some properties of the projection upon tt of a curve on K which passes k times through 0, has A;' branches at the vertex, and intersects g in n additional points. CHAPTER XI LINEAR SYSTEMS OF QUADRICS In this chapter we shall discuss the equation of a quadric sur- face under the assumption that the coefficients are linear functions of one or more parameters. 123. Pencil of quadrics. If A = 2a.,.avx-^ = 0, B= ^b^XiX,^ = are the equations of two distinct quadric surfaces, the system .1 - A5 = 2 (a,, - Xb,,) x,x, = 0, (1) in which X is the parameter, is called a pencil of quadrics. Every point which lies on both the given quadrics lies on every quadric of the pencil, for if the coordinates of a point satisfy the equations A = (), B = Q, they also satisfy the equation A — XB = for every value of X. Through any point in space not lying on the intersection of A = 0, B = passes one and but one quadric of the pencil. If (y) is the given point, its coordinates must satisfy the equation (1), hence A(y)-XB{y) = 0. If this value of X is substituted in (1), we obtain the equation Aiy)B-Biy)A = of the quadric of the pencil (1) through the point (y). 124. The X-discriminant. The condition that a quadric A — XB = of the pencil (1) is singular is that its discriminant vanishes, that is, aji — A.&,i a,2 — A6i2 Clfj2 — X0i2 CI22 — XU22 ai3 — AO13 a23 — A623 Uu — Xbu a24 — A524 147 I a.i. — A6,. I = ai3 — ^&i3 «14 — ^K C<23 — AO23 024 - -^^24 ^33 — •^^sa «34 — ^^34 "34 — A63, Ou - A644 = 0. (2) 148 LINEAR SYSTEMS OF QUADRICS [Chap. XI. This deterraiuant will be called the X-discriminant. If it is iden- tically zero, the pencil (1) will be called a singular pencil. If the pencil is not singular, equation (2) may be written in the form AX* + 4 ©A^ 4- 6 4>A2 + 4 ®'A -h A' = 0. (3) If A ^ 0, this equation is of the fourth degree in A. If A = 0, the equation will still be considered to be of the fourth degree, with one or more infinite roots. It follows at once from equation (3) that in any non-singular pencil of quadrics there are four distinct or coincident singular quadrics. If in (3), A is put equal to zero. A' results. But from (2), this is the discriminant of ^ = 0. Similarly, A is the discriminant of B = 0. Let jS,^ be the cofactor of 6,^. in A. From (2) and (3) we obtain - 40 = ciuAi + «22/^22 + ••• + (isAi- If = 0, yl = is said to be apolar to S = 0. Similarly, if 0'= 0, B= is said to be apolar to ^1 = 0. A geometric interpretation of this property will be given later (Art. 149). 125. Invariant factors. If the equations of the quadrics of a non-singular pencil are transformed by a linear substitution such that ^ = is transformed into A' = and B = into B' = 0, then A — XB = becomes A' — \B' = 0. Moreover, if T is the determinant of the transformation of coordinates, then (Art. 104) \a\,-kb',,\ = T'\a,,-\b,,\. From this formula we have at once Theorem I. If (A— Ai)*o is a factor of | a.^ — Ai.t |, it is also a factor of I a',,. — A^'.^ | and conversely. Hence the numerical value and multiplicity of every root of the A-discriminant is invariant under any linear transformation of coordinates. Moreover, by a proof similar to that of Theorem II, Art. 104, we obtain the following theorem : Theorem II. Every sth minor of the transformed X-discriminant is a linear function of the sth minors of the original X-discriminant and conversely. From the two theorems I and II we obtain at once Arts. 124, 125] INVARIANT FACTORS 149 Theorem III. If (A — A,)* is a factor of all the sth minors of |fl.^_X5.^|^ then it is also a factor of all the sth minors of I a'i* — -^^'i* I ^'^^ conversely. Let (A — A,)*o be a factor of the A-discriminant, (X. — Ai/i of all its first minors, (A — Ai)*2 of all its second minors, etc., k^ being the highest exponent of the power of (A — Aj) that divides all the sth minors, and k, being the first exponent of the set that is zero. Let also Li = A'o — A"i, Lo = A;i — k^, •••, -i/, = a;,._i. (4) From Theorem III we have : Theorem IV. The expressions (A-AOS (A-AOS •••, (A-Ai)^^ are independent of the choice of the tetrahedron of reference. These expressions are called invariant factors or elementary divisors to the base A — A, of the A-discriminant. We shall next prove the following theorem : Theorem V. The expimcnt of t^ack invariant factor is at least unity. ^^^ I f'.. - A^, I = (A - AO*^'i^(A), wliere -F(A) is not divisible by (A — Ai). Then ,/ ~ I ",. - A/.,, 1 = (A - X,r^-\f{\), where /(A) is not divisible by (A — Ai). But the derivative of I tt,t — A/>.j. I with respec-t to A may be expressed as a linear function of the first minors,* and is conse(]uently divisible by (A — Ai)*i at least. * If the elements of a determinant \ahcd\ are functions of a variable, it follows from the definition of a derivative that the derivative of the determinant as to the variable may be expressed as the sum of determinants of the form \u'hcd\ -\-\ah'c(l\ + \ abc' d \ -\- \ uhcd' \, in which a\ is the derivative of Oj, etc. If these determinants are expanded in terms of the columns which contain the derivative, it follows that the derivative of the given determinant is expressible as a linear function of its lirst minors. 150 LINEAR SYSTEMS OF QUADRICS [Chap. XI. Hence 7^7 1 r \ i The proof in the other cases may be obtained in a similar way. 120. The characteristic. It is now desirable to have a symbol to indicate the arrangement of the roots in a given A.-discriminant. There may be one, two, three, or four distinct roots. If k^ = 1 for any root Aj, then L^ = l, and no other L^ appears for that factor. If /Cq = 2, then L^ may be 1 or 2, according as the same factor is contained in all the first minors or not. If all the exponents L, associated with the same root are enclosed in parentheses {L^, L2, •••), and all the sets for all the bases in brackets, the config- uration is completely defined. This symbol is called the charac- teristic of the pencil (1). E.g., suppose and that X — X^ is also a factor of all the first minors, but that X — Xi is not. The characteristic is [2(11)]. If A — Aj is also a factor of all first minors so that Lx = l, Lo = l to the base A — Aj, the symbol has the form [(11)(11)]. From (4) it is seen that T^i + Zo -|- ••• -{- L^ = A'^, that is, that the sum of the exponents for any one root is equal to the multiplicity of that root. Since the sum of the multiplicities of all the roots is equal to four, we have the following theorem : Theorem. The sum of the exponents in the characteristic is always equal to four. 1. Express the minor EXERCISES \h': of I a',i — ^ft'itl in terms I a'23 — X6'23 «'33 of the second minors of la.-jt — Xft,*!. 2. Find the invariant factors and characteristic of each of the following forms : (a) (c) 1- \ X 0X0 ; i.b) X 1 X 1 1 X ' X X 1 0X0 X X (d) X 1 X 1 ' 1 X 0X10 1 - \ Arts, 125-130] QUADRICS WITH LINE OF VERTICES 151 127. Pencil of quadrics having a common vertex. If the A-dis- criminant is identically zero, the discussion in Arts. 124-126 does not apply. In case all the quadrics have a common vertex, we may proceed as follows. If the common vertex is taken as (0, 0, 0, 1), the variable Xi will not appear in the equation. We then form the A,-discriminant of order three of the equation in a'l, X2, X3. If this discriminant is not identically zero, we deter- mine its invariant factors and a characteristic such that the sum of the exponents is three. Similarly, if the quadrics have a line of vertices in common, we form the A.-discriininant of order two, and a corresponding charac- teristic ; if the quadrics have a plane of vertices in common, the A-discriminant is of order one. 128. Classification of pencils of quadrics. The principles de- veloped in the preceding Articles will now be employed to classify pencils of quadrics and to reduce their equations to the simplest forms. When the equation of the pencil is given, the charac- teristic is uniquely determined. It will be assumed that for any given pencil A — \B = 0, the A-discriminant has been calculated and the form of its characteristic obtained. For convenience, the cases in which A = and B = coincide will be included in the classification, although in this case A — \B = does not constitute a pencil as defined in Art. 123. Since any two distinct quadrics of a pencil are suflBcient to define the pencil, we shall always suppose that the quadric B = is so chosen that the A-discriminant has no infinite roots. 129. Quadrics having a double plane in common. By taking the plane for x^ = 0, the equation reduces to A-iX^ — AXy ^ U, A = \,x,\ B = x,\ and the characteristic is [1]. 130. Quadrics having a line of vertices in common. Let x^ = 0, X2 = be the equations of the line of vertices. Every quadric consists of a pair of planes passing through this line, and the equation of the pencil has the form A — XB = ajiXi^ -f 2 ai2XiX2 -f a22X2^ — A(6ii^i^ + 2 bi2XiX2 + 622^2^ = 0- 152 LINEAR SYSTEMS OF QUADRICS [Chap. XI. Three cases appear : (a) The A-discriminant has two distinct roots Aj, Xj- (b) The A-discriminant has a double root Ai, but not every first minor vanishes for A = A^ (c) The A-discriminant is of rank zero for A = Aj. In case (a), ^4 — AjB is a square and A — AjB is another square. Let the tetrahedron of reference be so chosen that A-XiB = .r/, ^ - A2B = Xi\ If we solve these equations for ^4 and B, we may, after a suitable change of unit point, write A, B in the form A = Aia;,^ + X^x.^ = 0, B = x''-\- x^. In case {b) we have the relation (rtH&22 - a^lKY = 4 (aii&n - Ol2^'ll)(«12^22 - Ct22&12)> which is the condition that A = 0, 5 = have a common factor. By calling this common factor ic,, and the other factor of B = (which is by hypothesis distinct from the first) 2 x^, we may put A-\^B = Xy^, B = 2x^X2. Solving for A, B, we have In case (c), we have ^ — A,B = 0, hence we may write at once Ki^'i' + ^2') - A(a;,2 + x/) = 0. The invariant factors are A — Aj, A — Aj. In this case we have then the following types : [11 ] A = Ai-Ti^ 4- x^x^^ B^xi^^ x^, [2] A = 2 AiX'iO^a + x^, B = 2 XiX2, [(11)J A = X,{x,' + X,'), B = X,' + x,\ 131. Quadrics having a vertex in common. Let the common vertex be taken so that the equation of the pencil contains only three variables, .t,, a;,, x^. It will first be assumed that the A-dis- criminant is not identically zero. Suppose |a,.^ — A^i, 1 = has at least one simple root Aj. The expression A — XiB is the product of two distinct linear factors, hence the quadric A — X^B = consists of two distinct planes, Arts. 130, 131] SINGLE C0M2>10N VERTEX 153 both passing through the point (0, 0, 0, 1). Let the line of inter- section of the planes be taken for Xj = 0, rcj = ^> so that the ex- pression A — \iB does not contain x^. It follows that «ii - Kbn = 0, a,2 - Ai^i2 = 0, 013 - -^-^13 = 0- By means of those relations a^, ajj, a^^ can be eliminated from the X-discriminant. The result may be written in the form I Oik - bik I = 5u(Xi - X) fei2(X, - X) fci3(Xi - X) bn{\i — A) O22 — X022 O23 — X623 bu(Ki - X) 0^3 - X623 033 - X633 Since Xi was assumed to be a simple root of | a-^ — Xb^^. \, it follows that 6,1 ^ 0. The equation of the pencil now has the form — \{bnxi^ + 2 bi^XoXi + 2 bi^x^Xj + 622^2^ + 2 623a-2iC3 + 633X3^)= 0. If we make the substitution yi = x, + -i?-^- — '-^, 2/2 = x„ y, = Xs, then replace t/i, 1/21 Vs by x^, x^, x^, the equation of the pencil takes the form Xi^i^ + <^(.T2, .T3) - \{x,^ +f{x.^, 2:3))= 0, in which (x2, x^ and f{Xo, 2-3) are homogeneous quadratic func- tions of X2, x^. The above transformation may be interpreted geometrically as follows : Since 61, ^t 0, the quadric JB = does not pass through the point (1, 0, 0, 0). The polar plane of the point (1, 0, 0, 0) as to B is consequently not a tangent plane to B at this point. The transformation makes this polar plane the new x^, changes the unit point, and leaves x^ = 0, x^ = unchanged. The expression <^(a;2, x^) — A/(a;2, x^) may now be classified ac- cording to the method of Art. 130, and the associated functions of Xi, X2, X3 are obtained by adding X^x^^ to (l>{x.,, x^), x^ to/(.T2. Xg). Next suppose that |a,i — X&.^l =0 has no simple root. It has, then, a triple root which we shall denote by Xj. If X— Aj is not a factor of all the first minors, the quadric ^ — XiB = consists 154 LINEAR SYSTEMS OF QUADRICS [Chap. XI. of two distinct planes. Let the tetrahedron of reference be chosen in such a way that these two planes are taken as x^ = 0, X3 = 0, so that the equation of the quadric has the form A — XiB = 2 (a23 — Xib23)X2X3 = 0, wherein 033 — A16.23 t^ 0, but «ii — Ai&„ = 0, Oo., — XA2 = 0, ajs — A1633 = 0, ai2 — A/>,2 = 0, «i3 - ^1^13 = 0, and I dik - Xbifc I = 6u(Xi - X) 6i2(X, - X) b,3(X, - X) buiXl - X) 622(X - Xl) 023 - X623 ftlo(Xi - X) 023 - Xfe23 b33(Xl - X) Since (X — Ai)' is a factor of this determinant and a^^ — Xib^s ^ 0, it follows that ftn = 0, and 613^12 = 0, that is, either &13 = or &j,, = 0. Since it is simply a matter of notation which factor is made to vanish, let &i3=0. Then 612 =^0, since 1 0,-^ — A&j^ | ^ 0. Geometrically, this means that the plane ajj = touches B = along the line X2 = 0,X3 = 0. The plane x^ = intersects the cone £ = in the line X2 = 0, 0^3 = and in one other line. By a further change of coordinates, if necessary, the tangent plane to B = along this second line may be taken for x^ = 0. We then have but since A = XiB + 2 (01,3 - Ai&23)a^2a^3, we may, by a suitable choice of unit point, write the equation of the pencil in the form A-XB = Ai(2 xix^ + ^-32) + 2 XVT3 - A(2 x^x., + x^") = 0. If A — Ai is also a factor of all the first minors of the A-dis- criminant, but not of all its second minors, A — X^B is a square and represents a plane counted twice. This pla:ne may be chosen for Xj = so that A —XiB = (a.22 — Xj)n-^ x.2^. Since (A — Aj)' is a factor of the A-discriminant, we must also have ?>n^>33 - ^'13- = 0. Art. 131] SINGLE COMMON VERTEX 155 Geometrically, this condition expresses that Xj = is a tangent plane to the cone B = 0. We may now write Hence, by a suitable choice of unit point, the equation of tlie pencil may be reduced to A,(2 x^, + x,') + X.? - A(2 x,x, + xi) = 0. If A — Ai is also a factor of all the second minors of | a^^ — \h.^ j , the equation of i? = is a multii)le of that of ^4 = and the equa- tion of the pencil may be written in the form We have thus far supposed, in this Article, that the A-discrimi- nant did not vanish identically. It may happen that the deter- minant I a -J. — A^,^. ! is identically zero even though the quadrics of the pencil do not have a line of vertices in common. In this case every quadric of the pencil consists of a pair of planes. Let A = <)){Xi, .i'o), B=f(x.,, X3). Since jai^. — A^.-^l is identically zero, it follows that f'll(&22^33 - ^23') = 0, 633(«U^'22 " ai2') = 0, and hence that Ou = 0, ^^33 = 0, as otherwise the quadrics would have a line of vertices in common, contrary to hypothesis. By an obvious change of coordinates, we may write the equa- tion of the pencil in the form 2 XjXj — A2 x^x^ = 0. This is called the singular case in three variables. Its characteristic will be denoted by the symbol \S\. Collecting all the preceding results of the present Article, we have the following types of pencils of quadrics with a common vertex. [Ill] \,x,- -f- X,x,^ + A3.T32 x-^ + x.^ + CC3' [21] AiO-i^ + 2 Aoa-2-^3 + x^^ x^^ + 2 XoX^ [1(11)] AjXi^ + A,(.^./ + x,') x,^ + x,^ + x,^ [3] A,(2 x,x, + x,^) + 2 .T,a-3 2 x,x,+x,^ [(21)] A,(2 x,x, + x,^) + xi 2 x,x, -f x,^ [(111)] X.ixC- ^ x,^ + x,^) .ri^ + .r^^ + a-3* 156 LINEAR SYSTEMS OF QUADRICS [Chap. XI. EXERCISES 1. Determine the invariant factors for each pencil in the above table. 2. Determine the nature of the locus ^ = 0, B = for each pencil in the above table. 3. Find the invariant factors and the characteristic of the pencils of quadric cones defined by (a) ^ = 3 a;r -f 9 Xi"^ -f- 4 3:2X3 — 2 xiXs — 6 xiXi — 0, B = o x{^ + 8 Xo2 - 2 xs^ — 6 XiiCs — 14 XiXg = 0. (b) ^ = 5 Xi2 + 3 X2'^ + 2 X32 + 4 X2X3 — 2 X1X3 + 2 3:1X2 = 0, B = 9 xr - X22 + X32 - 4 x.:X3 + 14 X1X3 + 42 X1X2 = 0. (c) A = 5 x.^ - 5 X22 + X32 + 6 X2X3 + 10 X1X3 - 4 X1X2 = 0, B = 10 Xi2 + 2 X22 + 10 x^- 10 X2X3 + 24 X1X3 - 16 X1X2 = 0. {d) A = 2 xi2 + 2 X2^ - 2 X2X3 - 2 X1X3 = 0, B = Xi2-f 3 X.22 + X32 — 4 X2X3 - 2 X1X3 = 0. 4. Find the form of the intersection of vl = 0, jB = in each of the pencils of Ex. 3. 5. Write the equations of each of the pencils in Ex. 3 in the reduced form. 132. Quadrics having no vertex in common. As in the preced- ing case, we shall suppose, except when the contrary is stated? that ] ttik — ^&,)fc I is not identically zero. If (A — X{) is a simple factor of the A.-discriminant, then A — X^B = is the equation of a cone. By choosing its vertex as (1, 0, 0, 0) and proceeding exactly as in Art. 131, the equation may be reduced to the form AiXi^ + (fi(x2, X3, X4) - X(a:i2 +/(.C2, 0^3, Xi))= 0. By this process the variable x^ has been separated and the func- tions {x2, .T3, X4), f(x2, x^, x^) can be reduced by the methods of Art. 131, not including the singular case. The only new cases that arise are those in which the roots of I a,4 — A&.jt I = are equal in pairs or in which all four are equal. Consider first the case in wliich there are two distinct double roots Ai and A25 neither of which is a root of all the first minors of the X-discriminant. The quadrics A — X^B = 0, ^ — X2B = are cones having distinct vertices. Let the vertex of the first be taken as (0, 0, 0, 1) and that of the second as (0, 0, 1, 0). The equation of the former does not contain x^. Hence, we have «i4 — -^1^14 = 0, a24 — A1624 = 0, a34 - A1634 = 0, a^ — A,?>44 = 0. Arts. 131, 132] NO VERTEX IN COMMON 157 When those values of a^i are substituted in | a^^— A6,j | = 0, A. — Aj is seen to be a factor. The condition that (A — Ai)^ is a factor is that either b^ = or that A — Aj is a factor of the minor cor- responding to a^ — A644. But in the latter case A — Aj is a factor of all the first minors, contrary to hypothesis, hence 644 = 0. Proceeding in the same way with the factor A— A2, it is seen that and also that 633 = 0. Hence the vertices of both cones lie on the quadric B= 0. Let the tangent plane to 5 = at (0, 0, 0, 1) be taken as cc, = 0, and the tangent plane to 5 = at (0, 0, 1, 0) be taken as x^ = 0. Since B = is non-singular, 613 in the trans- formed equation does not vanish, hence the plane a^j = intersects the cone A — \iB = in the line x^ =3^2=^ and in another line. Let the tangent plane along this second line be taken as CC3 = 0; that is, make the transformation yi = ^i, 2/2 = ^2, 2 &,3(A2 - Ai)y3 = («!! - Kbn)Xl + 2 (ai2 - \A2)X2 + 2 &i3(A2 - Ai).T3, 2/4 = ^i- The equation of the cone has now the form A-\iB= (((02 - Kb.„)x.^ + 2(ai3 - kAs)^^'''.^ = 0. Similarly, the plane .<•, = () intersects the cone ^ — A2B = in the line x^ = 0, x^ — and in another line. Make a further trans- formation by choosing the tangent plane to -4 — A2B = along this line for the new x^, thus Vl = ^1, Ih = ^2, Ik = ^3, 2 624(^1 — ■^2)2/4 =(«12 — '^2^12)-^'l +(«22 — A2&22)^'2 + 2 624(^1 " •^)-'^'4- The equation of the second cone now has the form A — X^B = ((In — X.bid^i^ + 2(a24 — X^hd^^^i = 0. By a suitable choice of unit point the equation of the pencil may be reduced to . Ai(xi2 -f 2 X.X,) + A2(.r.r + 2 x.x^) - Afa^i^ -f o-J 4-2 x^x^ -\- 2 x.x^) = 0. If the invariant factors are (A — A,), (A — Aj), (A — ^y, the quad- ric A — Ai-B = is a pair of distinct planes and as before A — A2B =0 is a cone having its vertex on the quadric B = 0. Let the line of 158 LINEAR SYSTEMS OF QUADRICS [Chap. XI. intersection of the two planes of ^ — X^B = be taken as x^ = 0, x^ = 0, and let the vertex of ^ — AgB = be at (0, 0, 1, 0) as before. Since this vertex lies on A — X^B = and on B = 0,\t lies on every quadric of the pencil, in particular, therefore, on A — XiB = 0. Thus, one of the planes of the pair constituting ^ — A]B = is the plane x^ = 0. The other may be taken as a^g = so that A-XyB = (a34 - Xib^i)XsXi = 0. The plane x^ = is not tangent to ^ — X^B = 0, since otherwise the discriminant |a,jt — Xb^^l would vanish identically. Hence we may choose for x^ = 0, and X2 = any pair of planes conjugate to each other and each conjugate to Xi = as to the cone A —X2B = 0. The equation of the cone A — X^B = is now A — X^B^ (a„ — Xa&iOa^i^ + (0^22 — KK^^t^ + («44 — Kbi^x^ = 0. From these two equations we may reduce the equation of the pen- cil to the form 2 X^x,x, + X,(x,' + x^'-j- X,') - A(2 x,x, + .t^^ + x^' + .^•/) = 0. If (A— A2) is also a factor of all the first minors, so that the in- variant factors are (A — Ai), (A — Ai), (A — A2), (A — A2), the quadrics A — Ai-B = and A — A2B = both consist of non-coincident planes. These four planes do not all pass through a common point, since in that case all the quadrics of the pencil would have a common vertex at that point, contrary to the hypothesis. We may conse- quently take A-X,B = (r<33 - Xfi,:,)x^^ + (a44 - X,b^,)xi^ = 0, A — X2B = (ftii — XJ)n)Xi^ + (a22 — Xj32->)X2 = 0. By a suitable choice of unit point the equation of the pencil as- sumes the form Ai(.r,^ + .^-2') + X^x,^ + .r/) - X{x,^ -f- x.,' + a'a" + x,^) = 0. The remaining cases to consider are those in which | ^hi a44 - A644 Since (A — A,)* is a factor and b^^ =^ 0, it follows that H(A-AO^ = I — A6.,. «34 - A634 The section of the pencil of quadrics A — XB = by the plane X.J = is the pencil of composite conies 033X3^ -f a^Xi^ + 2 a^^x^Xi — A(633a;32 4- b^^x^^ 4- 2 b^^x^x^) = 0, x.2 = 0. The characteristic of this pencil of composite conies is [2]; it con- sists (Art. 130) of pairs of lines through (1, 0, 0, 0) all of which have one line g in common. The plane iC2 = cuts the cone A — Ai-B = in the line g counted twice, and g is defined by one of the factors of 633a;3^+ 2 b^^x^x^+b^^x^-, since it is common to all the conies of the pencil. The tangent plane 0:2 = to J5 = therefore con- tains the line g and another line gf. Through the line g', which passes through the vertex of the cone A—kiB=0, can be drawn two tangent planes to the cone. One of them is X2 = 0. Choose the other for x^ = 0. The plane x^ = will touch the cone A — AjB — along a line g". The plane containing the two generators g, g" of the cone is next chosen as Xi = 0. The equation of the cone A — AjB = now has the form A — AiB = 2(a23 — Ai623)^'2'''^3 + («44 — Ai644~)a;/ = 0. The plane 0-3 = contains the generator g' ot B = 0, hence it is tangent to 5 = 0, and intersects B = in a line gr, of the other regulus. The plane x^—0 contains the generator ^ of B = 0, hence meets the surface in another line _f/2. The lines g, g' are of opposite systems, hence ^1, g^_ belong to different reguli and inter- sect. The plane of g^, g^ may be taken as the plane iCj = 0. The quadric 5 = now has the equation 5 = 2 bxtX^x^_ -f 2 634a;3a-4 = 0. 160 LINEAR SYSTEMS OF QUADRICS [Chap. XI. By means of this equation and the equation of the cone A— \iB = it is seen that the equation of the pencil may be reduced, by a suitable choice of unit point, to Xi(2 X1X2 + 2 x^x^) + 2 x^s + x^^ — A(2 x^x^ + 2 x^x^) = 0. Now suppose A. — Aj is also a factor of all the first minors, but not of all the second minors. The surface A — X^B = consists of a pair of planes which may be taken for Xs = and ^4 = 0, so that A- XiB= 2(034 - Khd^i^i = 0, and A — \B = 2(034 — '^i^34)^V^4 + (K — ^) B. If the A-discriminant is calculated and the factor (A— Aj)^ re- moved, it is seen that in order for | a -^ — Xbi^ \ to have the further factor (A — Ai)^ the expression 611622 — ^12^ must vanish. Hence ^u^i'^ + 2 6120:1.^2 + 6223^2^ either vanishes identically, or is a square of a linear expression. In the first case, 611 = 0, 612 = 0, 6,2 = 0, so that the line x^ = 0, x^ = lies on the quadric B = 0. The plane ^3 = passes through this line and intersects B = in a second line g'. Similarly, CC4 = intersects i? = in x^ — and in another line g". Another tangent plane through g' may be taken as x., = 0, and the plane of g" and the second line in x^ = as x'l = 0. The equation oi B = is and the equation of the pencil may be reduced to the form Ai(2 x^Xs + 2 a-2a;4) + 2 a;3a;4 — A (2 x^x^ + 2 x'2a.*4) = 0. In case 6iia;i^ + 2 6i2X'iJ*2 + 6222^2^ is a square, not identically zero, the line x^ = 0, .^4 = touches 5 = but does not lie on it. Let the point of tangency be taken as (0, 1, 0, 0) so that 612 = 0, 622 = 0. If we now remove the factor (A — Ai)' from the A-dis- criminant and then put A equal to Ai, the result is 603624(034— A1634). This expression is equal to zero, since (A — A,)* is a factor of the A-discriminant. But O34 — Ai634^0, as otherwise A would be identical with B ; hence either 603 = or 624 = 0. Let the nota- tion be such that 694= 0. Then the section of the quadric B = by the plane aja = consists of two lines through (0, 1, 0, 0). Art. 132] NO VERTEX IN COMMON 161 Let L be the harmonic conjugate of the line a^j = 0, 0:4 = with regard to these two lines, and let P be any point on the conic rr4 = 0, ^ = 0. If the plane determined by P and L is chosen for Xi = and the tangent plane to B = at P is taken for x^ = 0, the equation of 5 = becomes B = biixi^ + 2 b23X2X3 + hi^x^ = 0, and the equation of the pencil has the form Xi(.ri2 -^x^^ + 2 x,x,) + 2 .j'3.r, - \ (.^i^ + x,' + 2 x,x,) = 0. Now suppose that A — A, is a factor of all the second minors, but not of all the third minors, so that A — \iB — is a plane counted twice. Let this plane be taken as x^ = 0. vl-AiB = (a«-A,6«).vr = 0. By substituting these values in the A-discriminant, it is seen that the determinant l^n'^oi^ssl must also vanish if A — Aj is to be a fourfold root. This means tliat the section of the quadric J5 = by the plane x^=0 consists of two lines, hence that cc^ = is a tangent plane to B =0. Let planes through these two lines be taken as x^ = 0, x.^ = 0. The remaining generators in Xi = and in ;», = belong to opposite reguli and therefore intersect. The plane determined by them is now to be taken as x^ = 0. The equation of B = is 2 bioX^Xo + 2 b^^x^x^ = 0, hence the equation of the pencil may be reduced to the form A, (2 x,x, + 2 x,x,) + x^ - A(2 x^x. + 2 x^x,) = 0. If finally A — Ai is a factor of all the third minors, the two equations ^4 = 0, B = differ only by a constant factor. If B = is reduced to the sum of squares by referring it to any self-polar tetrahedron, the equation of the pencil becomes \{x^ + x,J 4- x^ + x^-) — A x^ + X.} +^^32+ x{) = 0. Thus far it has been assumed that the A-discriminant did not iden- tically vanish. Now suppose | a -j — A6,^| = so that all the quad- rics of the pencil are singular. By hypothesis they do not have a common vertex. In the singular pencil two distinct composite quadrics cannot exist, for, if ^ = 0, B = were composite, we could choose A — 2 x^x^, -B = 2 X2,Xi, since the quadrics of the pencil 162 LINEAR SYSTEMS OF QUADRICS [Chap. XI. do not have a common vertex. But the A-discriminant of the pencil A — XB — is not identically zero, contrary to hypothesis, hence the pencil does not contain two distinct composite quadrics. The quadrics ^ = 0, B = may therefore be chosen as cones. Let the vertex of ^ = be taken as (0, 0, 0, 1) and the vertex of 5 = as (1,0, 0, 0). Let g, g' be generators of ^ = 0, B = which intersect, but such that the tangent planes along each of them does not pass through the vertex of the other cone. The plane g, g' can be taken as x^ = 0, the tangent plane to ^ = along g as a-j = 0, and the tangent plane to B = along g' as x^ = 0. The equations of the singular quadrics ^ = 0, B = are now of the form B = 633.T3- + 2 biiX^i + 2 634X30:4 + 6440:42 = 0, and the X-discriminant is I a,-. — Xh, I = Since this expression vanishes identically, the coefficient of each power of X must be equal to zero. These conditions are a^ = 0, 644 = 0, ai2634 — 624ai3 = 0. The last condition expresses that the planes ai2.T2 4- a^jX^ = and 624.r2 + 634.^3 = are coincident. By transforming the equation of this plane to x^ = 0, the equation of the pencil reduces to 2 XiXo + ax^^ — A(2 0:20:4 + 0:3^^) = 0. This case is called the singular case in four variables. The char- acteristic will be denoted by the symbol [ \3\ 1]. The determination of the invariant factors and the form of the characteristic for each of the above pencils is left as an exercise for the student. The properties of the curve of intersection will be developed in Chapter XITI, but in each case the curve is described in the following table for reference. The table includes only those forms which do not have common double point. ttu ai2 «13 ^12 -A624 «13 f'33 — '^633 -X634 - A624 ■^34 -A644 Arts. 132, 133] FORMS OF PENCILS OF QUADRICS 163 133. Forms of pencils of quadrics. Simplified Formb of A and B Character- ■ ISTIC [1111] [112] [11(11)] [13] [1(21)] [1(111)] [22] [2(11)] B = X,' + x.^ + .x-a^ + .^•4' A = Ai^'i^ + Aa-^a^ + 2 X^x^x^ + x^ B = x^^ + x.y^ + 2 x^Xi A = Aia',2 + A^a-.^^ + X.ix-' + x,^) B = X,' + x,^ + x-' + x,^ A = AiXi^ H- A2(2 x-o.-Tj + x,^) + 2 x,x, ^ = AiXi^ + A2(2 x,x, + a-,2) + x,"^ B = x,^ + 2 oj^ajg + .r/ A = AjOJi^ + A^CiCo^ + X32 + X42) £ = x^"^ + a-/ + x^^ + a;^^ A2 (.1-/ + 2 aJiXa) A = Ai(a-i2 + a;,^ + x^^) + 2 AoiKga'^ [(11)(11)] A =K{x,' + x.^) + \,(x,^ + X,') B=x,' + x,' + x-' + x,- Curve op Intersection OF ^= AND ^=0 A general space quartic of the first species. A nodal quai'- tic. Two conies which intersect at two distinct points. A cuspidal quartic. Two conies which touch each other. A conic counted twice. At each point of this conic the quadrics are tangent. A generator and a space cubic. The generator and the cubic intersect in distinct points. Two intersect- ing generators, and a conic which intersects each generator. The three points of in- tersection are dis- tinct. Four generators which intersect at four points. 164 LINEAR SYSTEMS OF QUADRICS [Chap. XI. CllARAO- TEKISTIC [^] Simplified Fokms of A AND B A = Xi{2 X1X2 + 2 x^Xi) -f 2 x^, + x,^ Curve of Intf.rsection op ^ = AND ^=0 A generator and a space cubic. The generator touches the cubic. [(22)] [(31)] A B. A B [(211)] A: B. [(1111)] A B [13|1] A B Xi(2 XyX^ -\- 2 x^Xi) + 2 x^x^ "T -^ ^3'*'4 Ai(^ ^'1^2 ~f" -^ X^X^j -\- Xi x,^ + x,^ + X,' + x,^ 2 .rox^ + .Ts^ Three generators, one counted twice. This generator intersects each of the others. Two intersecting gener- ators and a conic which touches the plane of the generators at their point of inter- section. Two intersecting gener- ators each counted twice. The quadrics touch at each point of each generator. The quadrics coincide. A conic and a generator counted twice. The vertices of the cones all lie on this gen- erator. EXERCISES 1. Derive the invariant factors of each of the above systems of quadrics. 2. Find the equations of each conic and eacli rectilinear generator of in- tersection of the quadrics of the above pencils. 3. Determine the invariant factors; find the equations of the curve of intersection, and write the equations in the reduced form of the pencils determined by A - Xi2 — a-u2 4. 2 X32 + 2 x^ + 5 XiX.^ = 0, L' = 3 a*]- — x-r + xi^ — 8 xi^ — 2 xiXo — 2 x^Xi = 0. (a) AuTs. 133, 134] LINE CONJUGATE TO A POINT 165 .^s A = x{^ + Xi^ + 4 3:32 + X42 + 4 xiX-2 + <3 X2X3 + 4 XiX^ = 0, B = X2'^ + SX3^ + Xi^ + 2XiX3 + 2x.iX3^0. ,. A = 3 xr - X2^ - 2 X32 + 2 X42 + 2 xiXg — 4!ciX3 = 0, B = i xr' - X22 + 2 X32 + 3 X42 + 2 XiX2 + 2 xix^ + 4 X3X4 = 0. /^N ^ = 3 Xr + 2 X22 — Xs'^ — X42 + 4 X1X2 — 2 X3X4 = 0, i> = 3 Xi'-' — ^2^ — X32 — X42 + X1X2 — 2 X3X4 — 3 X2X4 — 3 X1X4 = 0. 4. To what type does a pencil of concentric spheres belong ? A pencil of tangent spheres ? 134. Line conjugate to a point. The equation of the polar plane of a point (y) with respect to any quadric of the pencil (1) is As A varies, this system defines a pencil of planes (Art. 24). The axis of the pencil, namely the line is said to be conjugate to the point (y) as to the pencil of quadrics. Let (y) describe a line, two points of which are (/) and (y"). It is required to find the locus of the conjugate line. Since y. = H-iy'i + M"i, i = 1, ^, 3, 4 (Art. 95), the line conjugate to (?^'is, by definition, H-i'^aiky'i^k + H-2^aiky"iXk = 0, ' ixyV),kU\^k + H"i^b,,y'\x, = 0. As (y) describes the line joining {y') to (y") the ratio fx^ : ^uo takes all possible values. If between these equations (x.^ : fi., is elimi- nated, the resulting equation defines the quadric surface %a,^\x, . 26,,y",a;, - %a,,y",x, ■ ^j,^\x, = 0. (5) From the method of development it follows (Art. 119) that all the lines of the system belong to one regulus (Art. 115). The polar planes, with respect to a given quadric of the pencil, of two fixed points {y'), (y") on the given line intersect in the line 2a.y.a;, - \%h,^\x, = 0, 2a,^",ar, - X^b,,y",x, = 0. If between these equations A is eliminated, the resulting equation defines the same quadric (5). From Art. 115 it follows that this second system of lines constitutes the other regulus on the surface. (Ill — A^u ((12 — A612 «13 - A&13 «14-A6i4 Wl <^i2 — •^^^la ({22 — A622 (123 - A623 Cl'24 "~~ '^^24 M2 «13 - 'V_>i3 (123 — AO23 '^^33 — AO33 «34 - A634 ^*3 «14 — -^^14 O24 — A&24 (^34 — Atl34 044 - Af'44 «4 Ml %U U, Ui IGG LINEAR SYSTEMS OP QUADRICS [Chap. XI. 135. Equation of the pencil in plane coordinates. Let A —XB= be the erjuation of a iion-singular pencil of quadrics. The equation = (6) expresses the condition that the section of a quadric of the pencil by a plane (w) is composite (Art. 106). For a given value Aj of A, (6) is the equation of the quadric ^ — Aii5 = in plane coordi- nates, if it is non-singular. If ^ — AiJ3 = is a cone, (6) is the equation of its vertex counted twice. If A — AjB = is composite, (6) vanishes identically. Equation (6) is called the equation of the pencil in plane coor- dinates. Arranged in powers of A, it is of the form $i(m)A' + 3 ^,{u)X^ -t- 3 *2(")^ + ^2('0 = 0. (7) If $i(rf) ^ 0, the equation is of the third degree in A. When (7) is not identically zero, it will be said to be a cubic in any case, even if it has one or more infinite roots. Hence we have the fol- lowing theorem : Theorem. Every plane intersects three distinct or coincident quadrics of a non-singtdar j}encil in composite conies. The coefficient of each power of A in (7) is homogeneous and of the second degree in Wi, U2, n^, v^ (if it is not identically zero), hence, when equated to zero, it defines a quadric in plane coordinates. Since the pencil is non-singular, we may, without loss of general- ity, assume that the quadrics ^=0, andi? = are non-singular (Art. 128). The equation ^.^(n) = is seen, by putting A = in (6), to be the equation of ^ = in plane coordinates. An analo- gous statement holds for i(?/) = and B = 0. The geometric meaning of the other coefficients will be discussed later (Art. 149). Arts. 135, 136] BUNDLE OF QUADRICS 167 EXERCISES 1. Write the equation in plane coordinates of the pencil of quadrics x{^ - xr + :c3^ + oXi^ — 6 xiXi + i xsXi — X(2 X2X4 4 xr + xo^ + xs^) = 0. 2. Determine the equations of the three quadrics of the pencil of Ex. 1 which touch the plane 3:4 = 0. 3. Determine equation (7) for the pencil a(2 X1X2 + 2 a;3X4) + x{^ — X(2 Xix.y + 2 X3X4) = 0. Show that (7) vanishes identically for each of the planes Xi= 0, X3 = 0, X4 = 0, and interpret the fact geometrically. 136. Bundle of quadrics. If ^1 = '^a^i^x^x^ = 0, B = Vj^^x^x^ = 0, C = %Ci^x-x^ = () are three given quadrics which do not belong to the same pencil, the system defined by the equation Ai^ + A,B + A3 6' = 0, (8) in which Xy, A,? ^3 are parameters, is called a bundle of quadrics. The three given quadrics ^ = 0, J3 = 0, C=0 intersect in at least eight distinct or coincident points,* through each of which pass all the quadrics of the bundle. These eight points cannot be taken at random, for in order that a quadric shall pass through eight given points, the coordinates of each point must satisfy its equation, thus giving rise to eight linear homogeneous equations among the coefficients in the equation of the quadric. If the eight given points are chosen arbitrarily, these eight equations are independent and the system of quadrics determined by them is a pencil. It is seen that seven given arbitrarily chosen points determine a bundle of quadrics passing through them. Since all the quadrics of the bundle have at least one fixed eighth point in common, we have the following theorem : Theorem I. All the quadric surjaces ivhich pass through seven independent 2^0 i) its in space pass through a fixed eighth point. * Three algebraic surfaces whose equations are of degrees m, ?i, p, respectively, intersect in at least mnp distinct or coincident points. If they have more than mnp points in common, then they have one or more curves in common. For a proof of this theorem see Salmon: Lessons Introductory to Modern Higher Algebra, Arts. 73, 78. We shall assume the truth of this theorem. 168 LINEAR SYSTEMS OF QUADRICS [Chap. XI. These points are called eight associated points. If the coordi- nates of any fixed arbitrarily chosen point {y) are substituted in (8), the condition that {y) lies on the cpiadric furnishes one linear relation among the A^. Hence through {y) pass all the quadrics of a pencil and therefore a proper or composite quartic curve lying on every quadric of the pencil. This quartic curve passes through the eight associated points of the bundle. If (?/) is chosen on the line joining any two of the eight asso- ciated points, every quadric of the pencil passing through it will contain the whole line, since each quadric of the pencil contains three points on the line (Art. 65, Th. II). The residual intersec- tion is a proper or composite cubic curve passing through the other six of the associated points and cutting the given line in two points. 137. Representation of the quadrics of a bundle by points of a plane. Let Ai, \o, A3 be regarded as the coordinates of a point in a plane, which we shall call the A-plane. To each point of the A- plane corresponds a definite set of values of the ratios Aj : A2 : A3 and hence a definite quadric of the bundle (1) and conversely, so that the quadrics of the bundle and the points of the A-plane are in one to one correspondence. To the points of any straight line in the A-plane correspond the quadrics of a pencil contained in the bundle. The line wdll be said to correspond to the pencil. Since any two lines intersect in a point, it follows that any two pencils of quadrics contained in the bundle have one quadric in common. 138. Singular quadrics of the bundle. Those values of Aj, A2, A3 which satisfy the equation lAia,. + X2&a + V.J = (9) will define singular quadrics of the bundle. Unless special rela- tions exist among the coefficients a^^, b^^, c^^, none of these cones will be composite, for in that case all of the first minors of (9) must vanish, thus giving rise to three independent conditions among the A,, A2, A3, which are not satisfied for arbitrary values of the coefficients. It follows further that, under the same conditions, no two cones contained in the bundle have the same vertex. For, if /r= 0, L = were two cones having the same vertex, then every Arts. 136-139] PLANE SECTION OF A BUNDLE 169 cone of the pencil Ai7v'+ A2L = would have this point for a ver- tex. By choosing this point as vertex (0, 0, 0, 1) of the tetrahe- dron of reference, the pencil could be expressed in terras of the three variables x^, Xo, x^. Tlie discriminant of this pencil equated to zero would be a cubic in Ai : Aj whose roots define composite cones which were shown above not to exist for arbitrary values of ^iki ^iki ^ik- It follows from (9) that the points in the A-plane determined by values of A], Ao, A3 which define cones of the bundle of (8) lie on a quartic curve C4. Every point of this curve defines a cone of the bundle, and conversely. Each cone has a vertex, and it was just shown that no two cones have the same vertex. We have therefore the following theorem : Theorem. The vertices of the cones in a general bundle describe a space curve J. The points of J are in one to one correspondence with the points of the curve G^ in the X-plane. The four points in which any line in the A-plane intersects C4 correspond to the four singular quadrics of the pencil which cor- responds to the line. If P is any point on the quartic curve, the tangent line to C4 at P defines a pencil of quadrics in which one singular quadric is counted twice ; if the residual points of inter- section of the tangent line and C4 are distinct from each other and from the point of contact, the characteristic of the pencil is [211]. All the quadrics of the pencil pass through the vertex of the cone corresponding to the point of contact. 139. Intersection of the bundle by a plane. If the quadrics of the bundle (8) are not all singular, the equation = 0, (10) •wherein s.^ = X^a^t. -f XJI\^ + X^c-^, is called the equation of the bundle in plane coordinates. If the coordinates of a given plane (a) are substantiated in (10), the resulting equation, if it does not vanish identically, is homogeneous of degree three in Ai, A2, A3 and ^11 S12 ■Sl3 ^U Wl §12 6'o2 •*''23 S,, «2 Sl3 ^23 ■''33 •%4 ^h Su S24 ^34 S44 U, "1 ^'2 «3 "4 170 LINEAR SYSTEMS OF QUADRICS [Chap. XI. is consequently the equation of a cubic curve C3 in the A-plane. Equation (10) is the condition that the section of the quadric (Ai, A2, A3) by the plane (») shall be composite. Every such com- posite conic in the plane («) has at least one double point. It will now be shown that the locus of the point of tangency to (?t) of the quadrics of the bundle which are touched by (11) is a cubic curve. The equation of any plane (w) may be reduced to x*4 = by a suitable choice of coordinates. Let Ai, A2, A3 be any set of values of Aj, A2, A3 which satisfy (10) when we have replaced «i, Wj, M3, each, by zero and M4 by 1. The section of the quadric Ai^l + A2-B + A3C=0 by the plane a;4 = is a composite conic having at least one double point (^y^, y^, 2/3, 0). The coordinates of (jj) must satisfy the relations K^a,,y, + ~X.^h,^, + AaSc.,,?/, = 0, for i = 1, 2, 3. If from these three equations Aj, Ao, A3 are eliminated, the result is the equation of the locus of the point of contact (?/). Since the re- sulting equation is of degree three in the homogeneous variables .Vu .%) .V31 the locus is a cubic curve. It is called the Jacobian of the net of conies in the given plane. 140. The vertex locus /. The order of a space curve is defined as the number of its (real and imaginary) intersections with a given plane. We shall now prove the following theorem : Theorem. The vertex locus J of a general bundle is of order six. For, the condition that the vertex of a cone of the bundle lies in a given plane (») is that the corresponding point in the A-plane lies on each of the curves (9) and (10). The theorem will follow if it is shown that these curves have contact of just the first order at each of the common points so that their twelve intersections coincide in pairs. Let the given plane be taken as x^ = 0. The equation of a cone of the bundle having its vertex in this plane Dan be reduced to X,' + x-' + .r/ = 0, and that of the bundle to the form X,A -f A^B + A3(.«-o' + .^3^ + .V) = 0. Arts. 139-141] POLAR THEORY IN A BUNDLE 171 The point in the A-plane corresponding to the cone is (0, 0, 1). It lies on C^^X) and on C3(X). It is to be shown that Ci(X), Cs{\) have the same tangent at (0, 0, 1), but that they do not have con- tact of higher than the first order. In (9) put c^^ = % = C44 = 1 and all the other c^^ = 0, and develop in powers of A3. The re- sult may be written in the form (aiiAi + 511X2) V + n U ^14 <^44 0, wherein ,^ = a^^Xi + b.^X^ = ki- Similarly in (10) put Wi = ^2 = ^3 = 0, mJ = 1, c.jj. = 0, and develop in powers of A3. The result is (auXi + bnX^)Xs'' + <^12 <^22 + <^13 <^13 <^33 Xs + 0. These curves both pass through the point (0, 0, 1) and have the same tangent anAi + &11X2 = at that point. By making the two equations simultaneous, it is seen that they do not have contact of order higher than the first unless anXi -\- 611X2 is a factor of 011<^44 - 4>\i^, which is not the case unless particular relations exist among the coefficients a.t, ?>,.. 141. Polar theory in a bundle. Theorem. T7ie polar planes of a point (y) toith regard to all the guadrics of a bundle pass through a fixed point {y'). For, the polar plane of the point (?/) with regard to a quadric of the bundle Aj^ + A2B + A3C = has the equation X^^a^t^x-y^ + X^'^bii.x^y^ + X^tcufc-y^ = 0. For all values of A,, Ao, A3 this plane passes through the point (?/') of intersection of the three planes SffifT^y, = 0, ^b,,x,y, = 0, 2c,iX,?/4 = 0. (11) From the theorem that if the polar plane of (y) passes through (y'), then the polar plane of (?/') passes through (y), it follows that all the points in space are arranged in pairs of points (y), (y') 172 LINEAR SYSTEMS OF QUADRICS [Chap. XI. conjugate as to every quadric of the bundle. Since the coordinates t/i, y^, yz, 2/4 a^wd v'u y'2) l/zi y\ appear symmetrically in the equations 2a.,/.2/A. = 0, ^h,,y\y, = 0, 2c.y,?/, = defining the correspondence between {y) and {y'), the correspond- once is called involutorial. By solving the equations defining the correspondence for y\, y'2, y'z, y\ we obtain 2a2t.V* 2a3,j/^ ^o.^yk o■y^ ^Kyk ^^Akyk and similar expressions for y'^, y\. y\. If we denote the second members of the respective equations by ^i{y), then replace both 2/j and y\ by x^ and x\, respectively, the equations defining the involution may be written in the form a.<. = «/,,(.!;), px, = ,{x'). (12) If {y) describes a plane S'ti.i'j = 0, the equation of the locus of (?/') may be obtained by eliminating the coordinates of {y) from (11) and the equation 2' -^2) + ^4(^1) ^2) = 0, in which <^2) ^3; ^4 do not contain A3. Hence the point Ai = 0, Aj = is a double point on Ci{\)= ; it corresponds to the quadric a;iiC2 = 0. The points of Ci{X) are now in one to one correspond- ence with the curve of order five, forming one part of J, and the double point is associated with the whole line x^^ = 0, 0^2 = 0. Similarly, buudles of quadrics may be constructed having eight associated double points lying on two, three, four, five, or six pairs of planes. In the last case the equation of the bundle may be written in the form AiC^i^ - ^4') + K{^2'- - 3^4') + K{x,^ - re/) = 0. The eight associated points are (±1, ±1, ±1, 1). The curve J consists of the six edges of a tetrahedron and C4(A) is composed of the four sides of a quadrilateral. Its equation is AiA2A3(Ai + A2 + A3)=0. In this case the equations (12) of the involution (1/), (2/') have the simple form y\ = -, 1 = 1,2,3,4, in which o- is constant. Bundles of quadrics exist having a common curve and one or more distinct common points. The spheres through two fixed points furnish an example. EXERCISES 1. Show that (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (1, 1, 1, 1), (1, 1', -1, -1), (1, -1, 1, -1), (1, -1, -1, 1) are eight associated points. Arts. 142-144] THE JACOBIAN SURFACE OF A WEB 175 2. Prove that if P is a given point and I a given line through it, there is one and only one quadric of the bundle to which I is tangent at P. 3. Determine the characteristic of the pencil of quadrics in a general bundle corresponding to : (o) A tangent to Ci(\). (6) A double tangent to Ci(\). (c) An inflexional tangent to Ci{\). 4. What is the general condition under which C4(X) may have a double point ? 5. Determine the nature of the bundle Xi(a:i2 - x.xz) + Mx^^ + x^^ + X3' - 4 X42) +X3(xi2 - 3:32) = and of the involution of corresponding points (y), (?/'). 6. If three quadrics have a common self-polar tetrahedron, the twenty-four tangent planes at their eight intersections all touch a quadric. 7. Write the equation of a bundle of quadrics passing through two given skew lines and a given point. 8. If four of the eight common tangent planes of three quadrics meet in a point, the other four all meet in a point. 9. Show that the cubic curve, image of an arbitrary line, intersects the locus of vertices J in 8 points. 10. Show that the surface B of Art. 141 contains J" as a threefold curve. 143. "Webs of quadrics. If J. = 2o,ta;,.T^ = 0, B = '^bn^x^x^ = 0, C = 2c,fc.r,a-4 = 0, i) = Sc^i^cc.a^^ = are four quadrics not belonging to the same bundle, the linear system X,A + X^B + X3C+\,D = (14) is called a web of quadrics. Through any point in space pass all the quadrics of a bundle belonging to the web, through any two independent points a pencil, and through any three independent points, a single quadric of the web. 144. The Jacobian surface of a web. The polar planes of a point {y) with regard to the quadrics of a web form a linear system ^i^^ik^iVk + K'^^\k^^yk + AaSCi^a^i?/, 4- XiMik^iVk = 0. (15) If the point (?/) is chosen arbitrarily, this plane may, by giving Aj, A2, X3, X4 suitable values, be made to coincide with any plane in 176 LINEAR SYSTEMS OF QUADRICS [Chap. XI space, unless there are particular relations among the coefficients a^k, 6,-^., c.jt, dij. Thus an arbitrary plane is the polar plane of (y) with regard to some quadric of the web. There exists a locus of points (y) whose polar planes with regard to all the quadrics of a web pass through a fixed point (?/'). This locus is called the Jacobian of the web. Since the equations connecting (y) and (?/') are symmetrical, it follows that (y') also lies on the Jacobian. A pair of points (?/), (y') such that all the polar planes of each pass through the other are called conjugate points on the Jacobian. To determine the equation of the Jacobian, we impose the con- dition that the four polar planes of (y) pass through a point. K.= It. The result is 2au?/i 2a2i?/i S^aJ/i 2a4i2/i :s&ii//i 2?>2i.Vi ^b,i!/i ^KVi 2c,i?/,. Scjii'/i '^c,,y, ^CAiVi 2rfu-,'/i Srfoi?/; '^^hiVi 2t?4i?/i = 0. (16) The condition that a point (,;/) is the vertex of a cone contained in the web is that its coordinates satisfy the equations K%a,,y, + X,-%h,,y, + AaSc.,^/, + X,-^d,,y, =0, k= 1, 2, 3, 4 (17) for some values of A„ A2, A.3, A4. By eliminating A,, A2, A3, A4 we obtain equation (16). This gives the theorem : Theorem I. The Jacobian surface is the locus of the vertices of the cones contained in the web of quadrics. If from equations (17) we eliminate yi, y^, 2/3, 2/4 we obtain nx)- ^u tn tn tu t2X ^22 t2»t3*t'^ a^i «1 1 2 3 4 1 0^2 02 1 3 4 1'' Xj "a 1 ^4123 X, 04 1 of planes (123), (45G), are composite quadrics of the web. The line of vertices of each pair lies on 7t4 = 0. The surface K^ = also contains the fifteen lines joining the basis points by twos, since through any point of such a line five lines can be drawn to the six basis points, and a quadric cone of the web is fixed by these five lines. If the basis points are taken for vertices of the tetrahedron of reference, the unit point, and the point (a^, a^, a^, at), the equation of K^ = is found to be = 0. This surface is known as the Weddle surface.* If in (17) the values of y^, y^, y^, y^ are eliminated, the resulting equation A(A) = of degree four in the A^ will define those values for which the equation XiA + X2B-\-X^C-{-X4D = is a cone of the web. The vertex of this cone is a point (^) = (^'). Let A,, X.,, A3, A4 be considered as the tetrahedral coordinates of a plane. To each plane (A) corresponds a quadric of the web (14) and con- versely. A linear equation with given coefficients aXi + bXn + cA3+(/A4 = determines a point in the A-space (Art. 91). By making this equation and (14) simultaneous, we define a bundle whose basis points are the points (x) whose coordinates satisfy the equations abed Of the eight associated points so determined, the given points 1, 2, 3, 4, 5, 6 are six. Either of the remaining points P = ($), P' = ($') will uniquely determine the other and also uniquely determine the point (a, b, c, d) in the A-space. The equation aXi + bXi + CA3 + dXi = thus defines a one to two correspondence between the points of the A-space and the points P and P. For * First discussed in the Cambridge and Dublin Mathematical Journal, Vol. 5 (1850), p. 69. 180 LINEAR SYSTEMS OF QUADRICS [Chap. XL points of A", P and P' coincide. The locus of the corresponding point (a, h, c, d) is called the Kummer surface.* We have thus proved the following theorem : Theorem IV. Tlie j^oints of the Weddle surface and the points of the Kummer surface are in one to one correspondence. EXERCISES 1. Show that all the quadrics having a common self-polar tetrahedron form a web. 2. Determine the Jacobian of the web of Ex. 1. 3. Determine under what conditions the Jacobian of a web will have a plane as component. 4. Find the Jacobian of the web defined by the spheres passing through the origin x = 0, ?/ = 0, z = 0. 5. Show that the Jacobian of a web having two basis lines is inde- terminate. 6. Discuss the involution of conjugate points (y), (y') for the web of Ex.4. 7. Show that the spheres cutting a given sphere orthogonally define a web. 8. Show that the equation of the quadric determined by the lines joining the points (1, 0, 0, 0), (ai, aj, «3, ^4); (0, 1, 0, 0), (0, 0, 1, 0); (1, 1, 1, 1), (0, 0, 0, 1) is x^Xl{a2 — az)+ (a^T^ — 02X3) + Xi(aiXi — aiX2) = 0. 147. Linear systems of rank r. The linear system of quadrics A.i^i + X,Fo + ■'■-\-KF, = 0, (19) wherein is said to be of rank r, if the matrix (20) "11 "22 "33 • "34 n "-2) "22 "33 • . a <2) U.34 "11 a.,2<'' "33 U34 * First discussed by E. E. Kummer in the Monatsberichte der k. preussischen Akademie der Wissensehaften, Berlin, 1863. Arts. 146-149] APOLARITY 181 is of rank r, that is, if there does not exist a set of values of Aj, A2, •••, Xrf ^lot all zero, such that the expression k,F, + X,F^ + :- + X,F, is identically zero. All the quadrics in space form a linear sys- tem of rank ten, since the equation of any quadric may be ex- pressed linearly in terms of the ten quadrics, x^, x^, •••, x^Xi for which the matrix (20) is of rank ten. All the quadrics in space whose coefficients satisfy 10 — r independent homogeneous linear equations form a linear system of rank r. For, if ^b-^x^x^ = is the equation of any quadric whose coefficients satisfy the given conditions, then all the co- efficients 6j^ can be expressed linearly in terms of the coefficients of r quadrics belonging to the system. Thus bi, = Aia.,<" + Vf.,"' + - + KaJ'\ h k = 1, 2, 3, 4, (21) wherein ^ ,,> ^ „ . ^ are fixed quadrics belonging to the system. Conversely, 10 — r independent homogeneous linear conditions may be found which are satisfied by the coefficients in the equa- tions of the quadrics F, = 0, i^2 = 0, -•, F, = 0, and consequently by the coefficients in the equations of all the quadrics of the linear system (19) of rank r. 148. Linear systems of rank r in plane coordinates. The system of quadrics AA + A.2<^2 + ••• +A,$, = 0, wherein $, = 2/3./"". "« is called a linear system of rank r in plane coordinates if there does not exist a set of values Ai, A2, — , A^ for which the given equation is satisfied identically. These systems may be discussed in the same manner as that considered in the preceding article. 149. Apolarity. Let F= Sa-^x.x-^ = be the equation of a quadric in point coordinates and ^ ~ S/?,;,?/,?^^. = be the equation of a quadric in plane coordinates. If the equation 2a,i^,, ~ a„/3„ + a,,f3,, + a,,l3,, + a,,(3» + 2 a^^As + 2 a,,l3,, + 2 a,,^,, + 2 cu,(3,, + 2 ao,A4 + 2 a,,/3,, = (22) 182 LINEAR SYSTEMS OF QUADRICS [Chap. XI. is satisfied by the coefficients in the eqnations of the two quadrics, F=0 is said to be apolar to = 0, and $ = is said to be apolar to F = 0. It should be noticed that in this definition the equa- tion F=0 is given in point coordinates, and that of = in plane coordinates. It should also be noticed that if i^=0 and 4> = are two given apolar quadrics, and if 2«i4.?/,% = is the equation of i^ = in plane coordinates, and '!S,bi;^x-x^ = is the equation of ^ = in point coordinates, then it does not necessarily follow that Sa.At = ^ because SttaiSit — " 0- In order to show the significance of the condition (22) of apolarity, we shall prove the following theorem : Theorem I. TJie expression a^^/Sik is a relative invariant. Let the coordinates of space be subjected to the linear trans- formation »,. = ttax'i + Ui^x'z + a^^^x'i + a^x'^, i = 1, 2, 3, 4 of determinant T^O. The coordinates of the planes of space undergo the transformation (Art. 97) M. = Ai^n\ + yl.ou'z + As^'s + Ai^i'i, i = 1, 2, 3, 4. The equation F(x)= goes into '^^a'^^x'-x'^ = 0, wherein (Art. 104) and $ = is transformed in ^(i'-f.u-u\ = 0, wherein The proof of the theorem consists in showing (Art. 104) that In the first member, replace a',^, fi\^ by their values from the above equations, and collect the coefficients of any term tti^/Si^ in the result. We find hence which proves the proposition. The vanishing of this relative invariant may be interpreted geometrically by means of the following theorem : Art. 149] APOLARITY 183 Theorem II. If F = 0, ^ = are apolar quadrics, there exists a tetrahedron self-polar as = and inscribed in F = 0. This theorem should be replaced by others in the following exceptional cases in which no such tetrahedron exists. (a) If 2^=0 is a plane counted twice. In this case (22) is the condition that the coordinates in this plane satisfy $ = 0. (b) If $ = is the equation of the tangent planes to a proper conic C and ii F = intersects the plane of C = in a line counted twice, (22) is the condition that this line touches C. We shall consider first the special cases (a) and (b). Let F = (uiXi + »2'^2 + "3^3 + ^^^^^y^ Then a^^ = M,?/fc and (22) reduces at once to $ = 0. In case (6), let the plane of C be taken as ^4 = and the line of intersection of F =0 with X4 = be taken as x^ = x^ = 0. Then $ = fin< + PlM.^ + /833"3' + 2 /3i27(,«2 + 2 ^23^2^3 + 2 P,,U,U, = (i, and F = a^^x^- -\- 2 a^iXyX^ + 2 a^^x.x^ + 2 a^^x^Xi + 2 a^^x^ = 0, where a^ ^ 0. Hence (22) reduces to ^^ = 0, that is, to the con- dition that Xj = .i'4 = touches C. To prove Theorem II, excluding cases (a) and (6), we must consider various cases. First suppose 4> = is non-singular. Choose a point Pj on F = 0, not on the intersection F = 0, ^ — 0, and find its polar plane tt, as to 4> = 0. In tti take a point Pg ^^^ F = 0, not on 4> = 0, and find its polar plane tt.^ as to ^ = 0. On the line ttittj choose a point P^ on F = 0, not on $ = 0, and find its polar plane ttj. If the point of intersection of ttj, ttj, ttj is called P4, then P1P2P3P4 = TTiTToTTzTr^ is taken for the tetrahedron of reference; we may, by proper choice of the unit plane, reduce the equation of $ = to ^^i^ + n.,^ + 11 ^^ -\- ii^- = 0. Equation (22) now has the form On -f «22 + ^33 + f'« = ^- Since three of the vertices P,, P2, P3 were chosen on P = 0, three coefficients a^, = 0, hence the fourth must also vanish, which proves the proposition for this case. It should be observed that if P = 0, * = define the same 184 LINEAR SYSTEMS OF QUADRICS [Chap. XI. quadvic, equation (22) cannot be satisfied since tlieir equations may be reduced simultaneously to F = x^^ + x.^ + xi + 0^4^ = 0, 4> = u^ + W2' + ^i + n^ = 0. Now let $ = be the equation of the tangent planes to a proper conic c. Take the plane of O as x^ = 0, so that )8h = )8,, = /334 = 1844 = 0. If 2^=0 is composite and x^ is one component, equation (22) is identically satisfied. In this case we may take three vertices of a triangle in x^^^ self-polar as to the conic C and any point on i^ = not on .T4 = as vertices of a tetrahedron self-polar to $ = and inscribed in jP = 0. If jp' = consists of ^'4 = counted twice, (22) expresses the condition that the plane belongs to $ = 0, whether $ = is singular or not. This is the exceptional case (a). If CC4 = is not a component of /^ = 0, (22) has the form ttiiiSn + atSit + 033/833 + 2 a,,^i.3 -f 2 «,3;8i3 + 2 a.^S1^= 0, which is the condition that the section C" of i^ = by the plane 0:4 = is apolar to C. It follows by the theorem for apolar conies analogous to Theorem II that a triangle exists which is inscribed in C" and is self-polar to C. A tetrahedron having the vertices of this tri- angle for three of its vertices and a fourth vertex on i^ = but not on 0^4 = satisfies the condition of the theorem (dual of Th. I, Art. 121). If = is the equation of two distinct points, (22) expresses the condition that these points are conjugate as to ii^= 0. This is also the condition that a tetrahedron exists which is inscribed in F= and is self-polar to = 0. If 4> = is the equation of a point counted twice, (22) expresses that the point lies on i^= 0. This is the dual of the exceptional case (a). In each of the above cases, the teti-ahedron which satisfies the conditions of the theorem can be chosen in an infinite number of ways, hence we have the following theorem. Theorem III. If one, tetrahedron exists ivhich is inscribed in F= and is self-polar as to $ = 0, the^i an infinite number of such f£trahedra exist. Art. 149] APOLARITY 185 By duality we have the following theorems : Theorem IV. //' i^ = 0, $ = are apolar quadrics, there exists a tetrahedron self-polar as to F = and circumscribed = 0. Theorem V. If one tetrahedron exists which is circumscribed to $ = and is self-polar as to F = 0, then an infinite number of such tetrahedra exist. Moreover, both the exceptional cases of Theorem II have an immediate dual interpretation: they will not be considered further. With the aid of these results we can now give an interpretation to the vanishing of the coefficients and ©' of equation (3), Art. 124, and of %{u), %{n) of equation (7), Art. 135. If 5 = in (1) is non-singular, let its equation in plane coordinates be 2;8i^.tt,i<^. = 0. Since /S.-^ is the first minor of 6-^ in the discriminant of B = 0, it follows at once from equation (3) that ©' = 'S.ai^^i^- Hence 0' = is the condition that ^ = is apolar to B = 0. If 5 = is a cone, it is similarly seen that 0' = is the condition that the vertex of the cone B = lies on ^ = 0. If 5 = is composite, 0' is iden- tically zero, independently of A, since the discriminant of 2? = is of rank two, hence all the coefficients ^^^ vanish. An analogous discussion holds for = 0. The surface *i(w) = (Art. 135) may be defined as the envelope of a plane which intersects ^4 = in a conic which is apolar to the conic in which it intersects B = 0. For particular singular quadrics this definition will not always apply. Let an arbitrary plane of *i(w) = be taken as x^ = 0. It fol- lows from equation (7) that I «11&22&33 1 + 1 ^ll«22&33 | + | hAi^hz \ = 0. (23) Let the sections of A = 0, B = by x^ = he C, C, respectively. If C" is not composite, it is seen by writing the equation of C in line coordinates that (23) is the condition that C is apolar to C. If C is a pair of distinct lines, (23) is the condition that their point of intersection lies on C. If C is a line counted twice, (23) is satisfied identically for all values of a^^, since all the first minors of the discriminant of C" vanish. An analogous discussion holds for *2(^) = ^- 186 LINEAR SYSTEMS OF QUADRICS [Chap. XI. 150. Linear systems of apolar quadrics. Since equation (22) is linear in the coefficients of i^= 0, from Art. 147 we may state the following theorem : Theorem 1. All the quadrics apolar to a given quadric form a linear system of rank nine. Conversely, since the coefficients of the equations of all the quadrics of a linear system of rank nine satisfy a linear condition which may be written in the form of equation (27), we have the further theorem : Theorem 11. All the quadrics of any linear system of rank nine are apolar to a fixed quadric. From the condition that a plane counted twice is apolar to a quadric (Art. 149), it follows that this fixed quadric is the envelope of the double planes of the given linear system. If a quadric F —0 is apolar to each of r quadrics 4.1 = 2A.,<»M,% = 0, $, = 2y8,,<^>.^,% = 0, ..., , = :^/3J'-\r,, = 0, the coefficients in its equation satisfy the r conditions It follows that if a quadric is apolar to each of the given quadrics, it is apolar to all the quadrics of the linear system The conditions that this linear system is of rank r are equivalent to the conditions that the corresponding equations (24) are inde- dendent. Hence : Theorem 111. All the quadrics apolar to the quadrics of a linear syste^n of rank r in plane coordinates form a linear system of rank 10 — r in point coordinates and dually. EXERCISES 1. Find the equation of the quadric in plane coordinates to which all the quadrics through a point are apolar. Art. 150] LINEAR SYSTEMS OF APOLAR QUADRICS 187 2. How many double planes are there in a general linear system of rank seven in point coordinates ? 3. Show that all the pairs of points in a linear system of rank six in plane coordinates lie on a quartic surface. 4. Show that all the spheres in space form a linear system and find its rank. 5. Find the system apolar to the system in Ex. 4. 6. Show that a system of confocal quadrics (Art. 84) is a linear system of rank two in plane coordinates. Detei-mine the characteristic and the singular quadrics of the system (Art. 133). 7. Show that, if the matrix (20) is of rank r' < r, the system of quadrics (19) is a linear system of rank r'. CHAPTER XII TRANSFORMATIONS OF SPACE 151. Projective metric. In order to characterize a transfor- mation of motion, either translation, or rotation, or both, or a trans- formation involving motion and reflection, as a special case of a projective transformation, it will first be shown under what cir- cumstances orthogonality is preserved when a new system of coordinates is chosen. If the new axes can be obtained from the old ones by motion and reflection, the plane t = must evidently remain fixed, and the expression x"^ + 1/"^ + z^, which defines the square of the dis- tance from the point (0, 0, 0, 1) to the point (x, y, z, 1), must be transformed into itself or into {x — atf + (y — hty + (^ — cty, according as the point (0, 0, 0, 1) remains fixed or is transformed into the point (a, h, c, 1). It will be shown that, conversely, any linear transformation having this property is a motion or a motion and a reflection. 152. Pole and polar as to the absolute. We shall first point out the following relation between the direction cosines of a line and the coordinates of the point in which it pierces the plane at infinity. Theorem I. T7ie homogeneous coordinates of the point in ivhich a line meets the plane at infinity are proportional to the direction cosines of the line. The equations of a line through the given finite point (xq, y^, Zq, t^ and having the direction cosines (A, [x, v) are t{ kX 2 ^ 02)3^1 4" tt22'''2 1 ^"23*^3 I '^24'''4> xwv kX 3 = 0^31 a^i + <^32'*'2 "T <^ 33*^3 "T C34'^4> kX 4 = C41-'^l "T ^42^*2 I '^43'^3 I tt44^4' 192 TRANSFORMATIONS OF SPACE [Chap. XII. We shall now consider the problem of classifying the existing types of such transformations and of reducing their equations to the simplest form. The invariant points of the transformation (7) are determined by those values of k which satisfy the equation D(k) ,-k «12 «13 «14 «21 Ctoo A- a23 024 «31 «32 a33 — k «34 041 «42 «43 a. ^-k = 0. (8) The classification will depend fundamentally on the invariant factors (Art. 125) of this determinant. In equation (7), (x) and {x') are regarded as different points, referred to the same system of coordinates. In order to simplify the equations, we shall refer both points to a new system of coordinates. To do this both (x) and (x') are to be operated upon by the same transformation We shall use the symbols {x), (y) to indicate coordinates of the same point, referred to two different systems of coordinates, while equations between (.r) and {x') or between (y) and (y') will define a projective transformation between two different points, referred to the same system of coordinates. Let fcj be a root of D{k) = 0. The four equations («ii — ki)x^ + a^.^^ + ais^s + a^Xi = 0, aoi-Ti +(022 — ''^■1)^2 + 023^3 + ^ii^i = 0> Osi^'l + a32^"l +(033 — ^"l)-^'3 + 034-»*4 = 0, 041X1 + 042X2 4- 043^3 +(«44 — K)Xi = are therefore consistent and determine at least one point invariant under the transformation. Let 2/3.A. = {i = 2, 3, 4) be the equations of three planes passing through this invariant point but not belonging to the same pencil, and let 2/Su.T, = 4=1 Art. 154] PROJECTIVE TRANSFORMATIONS 193 be the equation of any plane not passing through the invariant point. If now we put y, = ^I3,,x„ i = 1, 2, 3, 4, and solve the equations for the x-, Xi = 2yit?/,„ and put also x\ = ^yiky'k, *=i then substitute these valaes in the members of (7), the new equa- tions, when solved for y\, will be of the form y'l = Kyi + binJ/2 + ^132/3 + &i4y« y\ = 622^2 + ^232/3 + ^242/4. y'3 = Kyi + Kyz + hiy» y\= bi.y2-\-bi^y^ + biiyi. Without changing the vertex (1, 0, 0, 0), the planes 3/2 = 0, 7/3 = 0, 2/4 = may be replaced by others by repeating this same process on the last three equations; in this way we may replace the coefficients 632, ^42 by ; by a further application to the variables 2/3> 2/4 we may replace 643 by 0. Referred to the system of coordinates just found, the equations of the projective transformation (7) are X I = fC^Xi -f- C12X2 + Ci3^3 + ^'l4''^4> (9) x',= C22X2 T" C23''C3 + C24X4, x\ = ^333^3 + C34X4, x\ = CaaXa, in which C22, C33, C44 are all roots of D (k) = 0. Equations (9) represent the form to which the equations of any projective transformation may be reduced. The further simplifi- cation depends upon the values of the coefficients, that is, upon the characteristic (Art. 127) of D{k). If c^i^ and C33 ^ C44, make the further transformation ^1 ^ 2/ij ^2 == 2/2J ^3 = ^3 H ' > ^4 ^ y*- ^44 ^33 On making this substitution we reduce the equations of (9) to a form in which the coefficient C34 is replaced by zero. 194 TRANSFORMATIONS OF SPACE [Chap. XII. In any case, if i <,k and c^^ ^ c^.^, we may always remove the term c,^ by replacing x\. by x^ -\ — ^''^'^^' in both members of the equation. If c„ = c^^. and c^^ ^0, by a change of unit point, o.^ may be replaced by unity ; thus, if C33 = C44 and C34 4^ 0, by writing 034X4 = 1/4, we obtain the equations ^'3 = 6332/3 + ?/4, .V'4 = C332/4. These two types of transformations will reduce the equations to their simplest form in every case in which D{k) = has no root of multiplicity greater than two. If D(k) = has one simple root k^ and a triple root k^, the pre- ceding method can be applied to reduce the equations of the transformation to , X 1 = K^Xi, X 2 = "'2*''2 i~ ^23*^3 I ^24'''4J X 2= KqX^ -f" '^34*''4> X 4 ^^ K^X^t In case a24 = 0, the preceding method can be applied again ; thus, if a34^0, a^ 9^0, each may be replaced by unity; if coeflBcients a23, a24, a34 are zero, the transformation is already expressed in its simplest form. If a24 = 0, either or both of the coefficients 023 and a34, if not zero, may be replaced by unity by a transformation of the type just discussed. If O24 ^ 0, a34 ^ 0, replace x^ by the substitution ■ «->4?/3 ^34 In the transformed equation, the new a24 is zero. In the same way, if 0*24 ^ 0, ^,34 = 0, but aog =^ 0, put «23 and the same result will be accomplished. Finally, if a24=?!=0, but O34 = 0, 023 = ^) P"t ^1 = 1/1, x., = y3, X3 = y2, x^ = y^ (10) in both members of the equation. Now a24 = 0, and the complete reduction can be made as before. Arts. 154, 155] FORMS OF TRANSFORMATIONS 195 If D{k)=0 has a fourfold root k^, equations (9) reduce to '^1*^2 1 ^^23*^3 1 ^^24*^4? X 2 — iK 3 = x\ = rC^X^. By transformations analogous to those in the preceding case, the coefficients cii^, a^i, and 024 may be reduced to zero, and the coeffi- cients ai2, «23> 3'i^d a^i to zero or to unity. This completes the problem of reduction. The determination of the locus of the invariant points and the characteristic of D(k) in the various cases is left as an exercise for the student. The re- sults are collected in the following table. projective transformations. Locus OF InVAKIANT I'uINTS Four distinct points. Two distinct, two co- incident points. Two distinct points and a line. x^ One distinct, three coincident points. A point and a line. A point and a plane. C Two pairs of coinci- dent points. Two coincident points and a line. Two lines. ^ Four coincident points. 155. Standard forms of equations of CHARACrERISTI.- EurATUiNs [1111] X 1 =: K^Xi, Jj •) ^— /I'Q'^S ^ 3 ^^ "^3**'3) tJC ^ A/^X^ [112] X J ^ '*'l'^'l> *fc — A. 2**^2 .7; 3 := /CjXj -|- ^4, ii' 4 ^ A'3X'4 [11(11)] »!/ 1 — ^ /i. 1 iC/ 1 • X 2 — ■ fi/^^^'Z ^ 3 ^= "^3'^3) X 4 =: ^'3a•4 [13] X 1 = fCiXi, X 2 — ''-'2**^2 ' iC 3 = K2X^ -p iC^, X 4 ^^ tt'i'^A [1(21)] X , = kiXi, X 2 ■— /toXo a* 3 ^ /Cj^'s 4" •'^4) X 4 — /i'2**^4 [1(111)] X 1 = ft'i^i. X 2 — "'2 2 X I =: fC^X^^ X^ 2 ~~ ^2*^4 [22] X J — A/jU/j -p *^23 X 2 — KyOutj a; 3 = K^^ -\- x^, JO 4 — /C2X4 [2(11)] X 1 = k^Xi -\- X2, X 2 """" A/ 1X0 a; 3 = k.^x^y •C 4 — ^-2X4 [(11)(11)] X 1 = k^Xi, •t/ 2 ^~ *^\*'^2 X 3 = K2X3y *C 4 — h-^JCA [4] x\ = A'la-i -f X2, X 2 = A.'iX*2 + a. 3 = a'jiCs -\- x*4, X 4 -^ /tj^X4 196 TRANSFORMATIONS Oharactbristic Equatio N8 [(22)] X 1 = KiXi + X2, X 2 ^^^ nTj^2 x\ = k,X3 + Xi, X 4 = li^iX^ [(13)] X i^ fCiX^, X 2 — '*^i^2 ~i I iC 3 = riiX^ -\- X^y X 4 =^ /tj.?/^ [(112)] X 1 =: fC^Xi, *C 9 A/ J »*/2 a; 3 = /C1X3 + x^, X ^ — rtJjU/j [(1111)] X 1 = fCiXi, iC 2 — 'i-i'^2 iC 3 = tCiX^f EXERCISES SPACE [Chap. XII, Locus OF Invariant Points A line. A lina A plane. All points of space ; the identical trans- formation. 1. In type [1111] obtain the necessary and sufficient condition that the transformation obtained by applying the given transformation p times is the identity. 2. In [1(111)] show that the line joining any point P to its image P' always passes through the invariant point. 3. In Ex. 2, let O be the invariant point, and let a line PP' intersect the invariant plane in 3L Show that the cross ratio of OMPP' is constant. This transformation is called perspectivity. If the points OMPP' are har- monic, it is called central involution. 4. In [(11) (11)] show that the line joining any point P to its image P' meets both invariant lines, and that the cross ratio of P, P' and these points of intersection is constant. 5. Discuss the duals of the types of transformations of Art. 155. 156. Birational transformations. Besides the projective trans- formations, we have already met (Arts. 141, 146) with certain non-linear transformations in which corresponding to an arbitrary- point (x) is a definite point (x') and conversely. These are all particular illustrations of a class of ti*ansformations which will now be considered. Let x\ = i{x„ x^, x^, X,), i = 1, 2, 3, 4 (11) be four rational integral functions of Xi, a;,, X3, x^, all of the same degree. When Xi, x., x^, x^ are given, the values of x\, x\_, x:^, a^i are uniquely determined, hence corresponding to a point (a;) is a Art. 156] BIRATIONAL TRANSFORMATIONS 197 definite point (x'). If the equations (11) can be solved rationally for Xi, X2, X3, x^ in terms of x\, x'o, x\, x' ^ X, = U^\, x'„ x\, x\), i = 1, 2, 3, 4, (12) in which all the functions 1//^ are of the same degree, then to a point {x') also corresponds a definite point {x). In this case the transformation defined by (11) is called birational ; that defined by (12) is called the inverse of that defined by (11). When the point (.«') describes the plane 2«',«'i = 0, the corre- sponding point {x) describes the surface w'i<^i(^) + w'22(^) + u'3^(x) = 0. (13) This surface will be said to correspond to the plane (/<■'). If the m', are thought of as parameters, we may say : corresponding to all the planes of space are the surfaces of a web defined by (13). In the same way it is seen that, corresponding to the planes "Siii-x- =0 of the system (.«), are the surfaces of the web u,il,,(x') + ti.^lx') + n,i(;,(x') + u,^,(x') = 0. (14) Three planes («') which do not belong to a pencil have one and only one point in common, henoe three surfaces of the web (13), which do not belong to a pencil, determine a unique point (x) common to them all, whose coordinates are functions of the coor- dinates of («'). This fact shows that in the case of non-linear transformations the web defined by (13) cannot be a linear combination of arbi- trary surfaces of given degree. For if the <^- are non-linear, any three of them intersect in more than one point, but it was just seen that of the points of intersection there is just one point whose coordinates depend upon the particular surfaces of the web chosen. The remaining intersections are common to all the sur- faces of the web. They are called the fundamental points of the system (x) in the tranformation (11). When the coordinates of a fundamental point are substituted in (9), the coordinates of the corresponding point (x') all vanish. For the fundamental points the correspondence is not one to one. The fundamental points of (x') are the common basis points of the surfaces *pi(x') = 0. 198 TRANSFORMATIONS OF SPACE [Chap. XII. 157. Quadratic transformations. We have seen (Art. 98) that if the 4>i are linear functions, the transformation (11) is projective, and that no point is common to all four planes i are quadratic. We shall consider the case in which all the quadrics of the web have a conic c in common. Let the equations of the given conic be 2i/.a5.- = 0, f{x) = 0. Any quadric of the system 2?/.a;.(AiaJi + ^^x^ + X3X3 + \^x^) + \5f(x) = will pass through this conic. Among the quadrics of this system those passing through an arbitrary point P define a web. Any two quadrics Hi = 0, H.^^O of this web intersect in a space curve consisting of the conic c and a second conic c' which passes through P. The planes of c and of c' constitute a composite quadric belonging to the pencil determined by i/i = and H = 0, and the conies c, c' lie on every quadric of the pencil. Hence c, c' intersect in two points, as otherwise the line of intersection of the two planes would have at least three points on every quadric of the pencil, which is impossible. Any third quadric H^ = of the web but not of the pencil determined by H^ = 0, H^ = passes through c and P. The plane of c' intersects H^^O in a conic c" passing through P and the two points common to c, c' and in just one other point. The posi- tion of this fourth point of intersection depends on the choice of the bundle H^ = 0, //a = (^ H^ = 0. We have thus proved that the web of quadrics defined by a conic and a point P has the neces- sary property mentioned in Art. 156 possessed by the web deter- mined by a birational transformation. Let the equations of the conic c be x^ = 0, e^x^ + 62^2" + ^3^3* = 0. If P is not on the plane x^ = 0, it may be chosen as vertex (0, 0, 0, 1) of the tetrahedron of reference. The equation of the web has the form Art. 157] QUADRATIC TRANSFORMATIONS 199 In analogy with equation (11) we may now put ic 1 ^ ^iX^y a; 2 = X2X^, x ^ := XyC^, x ^^ ^i-^i r 62-'K2 -r 63X3 . \^0) The most general form of the transformation of this type may be obtained by replacing the x'^ by any linear functions of them with non-vanishing determinant. In the derivation of equations (12) it makes no difference whether the conic c is proper or composite, hence three cases arise, according as ej = 62 = ^s = 1 or ^j = ej = 1, 63 = or e^ = 1, 62 = ^3 = 0. The equations are Cb 1 ^^ •t/iX^^ Jb o — JUo^iy **^ 3 — **^3*^4 ^ 4 •"" **^i "t" *^2 1" 3 * \ / CC 1 — — **^i"-'45 *^ — ' **^2 4) 3 — •^3*^4 *^ 4 — ~ **^l 1* ^^2 • \ ) CC 1 — — "^1*^41 *^ 2 — *^2 4' 3 ^"~ **^3*^4 *^ 4 — *^l • \ / Now let P approach a poiut K on the conic c. If c is com- posite, suppose its factors are distinct and that K lies on only one of them. In the limits the line KP is tangent to all the quad- rics of the web determined by c and P. But the tangent to c at K is also tangent to all these quadrics at K. Hence the plane of these two lines is a common tangent plane to all the quadrics of the web at K= P. Let P be taken as (1, 0, 0, 0), the common tangent plane at P as X2 = 0, and let the equations of the conic be reduced to x^ = 0, iCia;2 + ex^^ = 0. The equation of the web has the form KiX^Xi + X2X3X4 + Agx/ + \i{xiX2 + 6X3^) = 0. The two cases, according as e = 1 or e = 0, give rise to the transformations X J = ^20^4, X 2 — ™ •^3^4, *^ 3 ^— «^4 , •C 4 ^ ^\*^2 "I" *^3 > V / • 1 "^ X^Xa* X 2 —' *^3**'4, X 3 — fcCj ■ *C 4 — XyX^ \ / of this type. Finally, let c be composite and let the point K which P ap- proaches lie on both components of c. Since all the quadrics thi'ough c have in this case the plane of c for common tangent plane at K, the point P must approach c in such a way that the line KP approaches the plane of c as a limiting position. The conies in which the quadrics of the web are intersected by any plane through Pand A" have two points in common at K and one 200 TRANSFORMATIONS OF SPACE [Chap. XII. at P. Hence in the limit, all these conies must have three inter- sections coincident at K = P. Let the equations of c be Xi = 0, x^ + ex^ = 0, and the coordi- nates of P be (1, 0, 0, 0). The equations of the system of rank five of quadrics through c is XiX^Xi + \2X.Xi + XsX^Xi + XiXi^ + X^(xi^ + 6X3^) = 0. The section of this system by any plane through P, different from .T4 = 0, will consist of a system of conies touching each other at P. The required web belongs to this system and satisfies the condi- tion that its section by any plane through P other than a'4 = is a system of conies having three intersections coincident at (1, 0, 0, 0). The equations of the section by the plane .T3 = are Ai^jS/^ -f- A2'1^2'^4 ~r A4.T4 -|- A5X2 ^ "5 '^3 ~^ ^• All these conies touch each other at P. Let \\, A'2, A'4, A'5 be the parameters of one conic, and Aj, A2, A4, A5 of another contained in this system. The equations of the lines from (1, 0, 0, 0) to the two remaining intersections of these two conic are (AiA'a - A^A'O-bf +(AiA'2 - A2A'i)avr4 +(AiA'4 - K^'i)^*' = 0. One of these remaining points is also at P if AjA'j — A5A', = 0. Hence all the quadrics of the web satisfy a relation of the form A5 -f A;Ai = 0. It is no restriction to put A; = 1. It can now be shown that the conies cut from the quadrics of the web A5 + Ai = by any plane aiX^ 4- aoXj -f O3.V3 = through P have three coinci- dent points in common at P. The equation of the web is Ai.r.,.r4 + X.,XyXi + X:^Xi^ + Xi(x2^ + ex^ - 0^10:4) = 0. The two birational transformations defined by webs of quadrics of this type are X 1 = X2.T4, .1; 2 = •'^3"^4» '^ 3 ^^ '^*4 > 3J 4 = .^2 -f- .^3 XyC^. yj ) X ] = ■^2"^4» '^ 2 ^^ •^i'^i: *^ 3 ^ •''4 J iK 4 = 3?2 X^X^. \^yj The inverse transformations of forms (a) ••• (g) are also quadratic. For this reason transformations of this type are called quadratic- quadratic. Arts. 158, 159] RECIPROCAL RADII 201 158. Quadratic inversion. A geometric inpthod of constructing some of the preceding types of birational transformations will now be considered. Given a quadric A and a point 0. Let P be any point in space, and P' the point in which the polar plane of P as to A cuts the line OP. The transformation defined by hav- ' ing P' correspond to P is called quadratic inversion. If does not lie on the quadric ^ = 0, let = (0, 0, 0, 1) and let the equa- tion of ^1 = be If P = (yi, y^, 2/3, y^, the coordinates of P' are y\ = yiyt, y't = yiyi, y'z = M4, y\ = euVi^ + 622/2^ + e^x^^ which include forms (a), (6), (c). If lies on A, we may take A = x^- + e,x,^ - x,x, = 0, = (0, 0, 0, 1). The coordinates of P' in this case are functions of y^, y^, y^, y^ defined b}'' (/) and (r/). The quadratic-quadratic transformations (a), (b), (c), (/), (g) can therefore be generated in this manner. 159. Transformation by reciprocal radii. If, for the quadric ^ = (Art. 158) we take the sphere a;2 ^y^ + z^ = m"" (16) and for the center of this sphere, the equations of the trans- formation assume the form x' = k'^xt, y' = k^yt, z' = kht, if = x^ + y^ -^ x". (17) On account of the relation OP ' OP' = ¥ (18) existing between the segments from to any pair of correspond- ing points P, P', it is called the transformation by reciprocal radii. Any plane not passing through goes into a sphere passing through and the circle in which the given plane meets the sphere (16), which is called the sphere of inversion. The fundamental elements are the center 0, the plane at infinity, and the asymptotic cone of the sphere of inversion. 202 TRANoFORMATIONS OP SPACE [Chap. XII. A plane ax + by -\- cz -\- dt = not passing through the origin (d =^ 0) is transformed into a sphere ak^xt + bkhjt + ckht + d(.«2 ^y'i+z'^)=0 passing through the origin. A plane passing through the origin is transformed into a com- posite sphere consisting of the given plane and the plane at in- finity. We shall say that planes through the origin are trans- formed into themselves. A sphere a(a;- + y'' + z"^) -\- 2 fxt -f 2 gyt -f 2 hzt + mf = (19) not passing through the origin (in -^^ 0) is transformed into the sphere wi(a;2 + 2/2 + z-") + 2fk\xt + 2 gk'^yt + 2 hk^zt -^ ak' = 0. (20) The factor x^ + y"^ + z"^ can be removed from the transformed equation. A sphere passing through the origin {in = 0) is trans- formed into a composite sphere consisting of a plane and the plane at infinity. If any surface passes through the origin, its image is seen to be composite, one factor being the plane at infinity. The plane at infinity is the image of the center 0, which is a fundamental point. In particular, the sphere (19) will go into itself if m = ak^ ; but this is exactly the condition that the sphere (19) is orthogonal to the sphere of inversion, hence we may say : Theorem I. Tlie sphe7-es tchich are orthogonal to the sphere of in- version go into themselces when transformed by reciprocal radii. We shall now prove the following theorem : Theorem II. Angles are preserved ivhen transformed by recipro- cal radii. Let AyX + B{y + C^z + D^t = 0, A^x -f B^y + G^z + 0^1 = be any two planes. The angle 6 at which they intersect is de- fined by the formula (Art. 15) QQg Q _ ^1 ^ 2 + A -S 2 + C] C2 ^ /o\\ -JIaJTW+^WaFVW+W) Arts. 159, 160] CYCLIDES 203 These planes go into the spheres Z)i(x2 + ?/2 -f z2) + A^k'^xt + BJi-yt + CJiht = 0, i>2(x2 + ?/2 + z2) + A.Ji'-xt + JSjA;^^^ 4- Cpzt = 0. Since the angle of intersection of two spheres is the same for every point of their curve of intersection (Art. 51) and both spheres pass through 0, we may determine the angle at which the spheres intersect by obtaining the angle between the tangent planes at 0. These tangent planes are A^x + B^y + CjZ = 0, A^x + B^y + C^z = 0, hence the angle between them is defined by (21). Since the angle of intersection of any two surfaces at a point lying on both is de- fined as the angle between their tangent planes at this common point, the proposition is proved. 160. Cyclides. Since lines are transformed by reciprocal radii into circles passing through 0, a ruled surface will be transformed into a surface containing an infinite number of circles. A quadric has two systems of lines, hence its transform will contain tw^o sys- tems of circles, and every circle of each system will pass through 0. Moreover, the quadric contains six systems of circular sections lying on the planes of six parallel pencils (Art. 82). Hence the transform will also contain six additional systems of circles, not passing through 0, but so situated that each system lies on a pencil of spheres passing through 0. By rotating the axes (Art. 37), we may reduce (Art. 70) the equation of any quadric not passing through to the form ^ ax" + hy"" + c2^ + i^ + 2 Ixt -f 2 myt -f 2 nzt = (22) without changing the form of the equation of the sphere of inver- sion. By transforming this surface by reciprocal radii, we obtain (a;2 + t/2 + 22)2 + 2 k^x"" + ?/ + z2)(/.i- + my + nz)t + k\aj^ + by^ + cz'^)f = 0. This surface is called the nodal cyclide ; it contains the absolute as a double curve and has a double point at the point 0.* * A point P on a surface is called a double point or node when every line through P raeets the surface in two coincident points at P. A curve on a surface is called a double curve when every point of the curve is a double point of the surface. 204 TRANSFORMATIONS OF SPACE [Chap. XII. If the given quadric is a cone with vertex at P, its image will have a double point at and another at P'. The circles which are the images of the generators of the cone pass through and P'. The equation of the cone may be taken as a{x- fty -\-b{y- gty + c(z - htf = (23) and the equation of the transform is (a/2 _,_ ^g2 _,_ e/i2)(a;2 4- ^2 _,_ ^2^2 _ q m{x^ +if +z'){afx + bgy + chz) + k\ax' 4- &/ + cz'y = 0. This surface has a node at the origin and at the transform (/) 9^ h, p-\- rf -\- 7*2) of the vertex of the cone (23). It is called a binodal cyclide. If, in equation (22), 6 = c, so that the given quadric is a surface of revolution, the transformed equation may be written in the form [ic2+ ^2 _,_ ^2 _|_ ^tQ,-^ + my + nz)t + ^ IH'^'y + (a - h)k*xH'^ - k4lx 4- my + nz + ^ k''t\H'' = 0. It has a node at and at the points in which the line a; = 0, 2lx-{-2 my + 2 nz + k'^ht = intersects the sphere a;^ + 3/^ + 2^ + 2 Ixt + 2 myt -f 2 nzt + bkH- = 0. It is called the trinodal cyclide. Finally, if the cone (21) is one of revolution, the resulting cyclide has four nodes, and is called a cyclide of Dupin. If the center of inversion is within the cone, so that no real tangent planes can be drawn to the cone through the line OP, the surface is called a spindle cyclide; if the center is outside the cone, the resulting surface is called a horn cyclide. The generating circles of a cone of revolution intersect the recti- linear generators at right angles. Since both the lines and the circles are transformed into circles and angles are preserved by the transformation, we have the following theorem: Theorem III. Through each point of a cyclide of Diqjin pass two circles lying entirely on the surface. Tliese circles meet each other at right angles. Art. 160] CYCLIDES 205 A particular case of the spindle cyclide is obtained by taking the axis of the cone through the center of inversion. The result- ing cyclide is in this case a surface of revolution. It may be generated by revolving a circle about one of its secants. If the points of intersection of the circle and the secant are imaginary, the cyclide is called the ring cyclide. It has the form of an anchor ring. In this case all the nodes of the cyclide are imaginary. EXERCISES 1. If A consists of a pair of non-parallel planes and is taken on one of them, show that the quadratic inversion reduces to the linear transformation defined in Art. 155, Ex. 8 as central involution. 2. Obtain the transform of the ellipsoid ?! + m! + ^ = 1 a?- y- €^ with regard to the sphere x~ + ;/'- ■\- z' — 1. How many systems of circles lie on the resulting surface ? Show that four minimal lines pass through and lie on the surface. 3. Show that the transform of the paraboloid ax^ + hxp- = 2 ^ by reciprocal radii is a cubic surface. How many systems of circles lie on this surface ? How many straight lines ? 4. Discuss the transform of a quadric cone by reciprocal radii when the center of the sphere of inversion lies on the surface but is not at the vertex. 5. Show that a surface of degree n passing k times through the center of inversion is transformed by reciprocal radii into a surface of degree 2(n — A;), 2, >- having the absolute as an {n — ^•)-fold curve. 6. Show that the center of an arbitrary sphere is not transformed into the center of the transformed sphere by reciprocal radii. 7. Given the transformation X'l = {Xx — Xz)X-2, x'o=(Xi — X2).r:i, X'3 = (Xl — X2).r4, X'i = X2X3. Find the equations of the inverse transformation and discuss the basis points in (x). 8. Given the transformation x'l = X1X2, X'o = X2X3, X':? = .r^Xi, X'i = X4(Xi -I- X-2 + X3). Find the equations of the inverse transformation. Discuss the basis points in the web of quadrics XiXiXo + X2X2X;i + \3X3X1 + X4X4(xi + X2 + X3) = 0. CHAPTER XIII CURVES AND SURFACES IN TETRAHEDRAL COORDINATES I. Algebraic Surfaces 161. Number of constants in the equation of a surface. The locus of the equation f(^) = 2 ,o^,g, aa^Y6«'i"'»2 V^/ = 0, (1) «! p! y! 6! wherein a, /S, y, 8 are positive integers (or zero) satisfying the equation « + /5+ y + 8 = n, is called an algebraic surface of degree n. If the equation is arranged in powers of one of the variables, as .^4, thus n^," + u.x,'^-^ + . . . + w„ = 0, (2) in which u- is a homogeneous polynomial of degree i in x^, x^, x^, the number of constants in the equation can be readily calculated. For we may write n. == (^o^V + <^ia-3'-^ H + cf>i, 4>k being a homogeneous polynomial in x^, x^^ of degree A; and con- sequently containing A; + 1 constants. The number of constants in XI- is therefore 1 I o , , n ;^^ (^ + l)(r + 2) _(^ + 2)!. 2 i!2! This number of constants in u^ is now to be summed for all inte- gral values of i from to n. By induction the sum is readily found to be ^ 2^ 3j_4 (n + l)(n+2) _ (n + 3)! 22 2 n!3! which is the number of homogeneous coefficients in the equation of the surface. The number of independent conditions which the surface can satisfy is one less than this or (n + 3) ! _ ^ ^ n^ + 6 w' + 11 n w!3! ~ 6 206 Arts. 161-163] INTERSECTION WITH A LINE 207 162. Notation. It will be convenieut to introduce the follow- ing symbols : A f(x) = v ^M + v ^^ + » ^-f^ + v M^- Vv ^^ A//(x-) = V — — ■ yi^y-^yiUi d\f(x) wherein 1 ^ r ^n and a, b, c, d are positive integers (or zero), satisfying the condition a-f-6 + c + d = r, EXERCISES V Let /(.x) = aioooXi* + ao4ooX2* + aooioX3* + aQoo4Xi* + 6 02200X1^X2^+600220X2^X3^ + 6 ao202XrX4^ + 6 aoo22.'*;3-^4'^ + G a202o.i;i''^-^3"^ + 6 a2002Xi'Xi^. 1. Find A//(x) for r = 1, 2, 3, 4. 2. Show that A, [A//(.r)] = A//(x). 3. Show that ^A,V(^) = Ax/(y). 4. Show that Ay-f(x) = Ax^fiy). 5. Show that A,f(x) =4/(x) ; AxVC^:) = 12/(x) ; A^VW = 24/(x). 163. Intersection of a line and a surface. If (y), (x) are any two points in space, the coordinates of any point (z) on the line joining them are of the form z^ = \y^ + fjiX^ (Art. 95). If (z) lies on f[x) — 0, then f{\y + \x.x) = 0. By Taylor's theorem for the expansion of a function of four variables, we have, since A/+*/(?/) = for all positive integral values of A;, Siky + /.X-) = A"/(^) + A-VA^/Cv) + - ^A//(^) + -+-^ r! n + ^-^ A//(^) + • . • + ;^ A/ fiy) = 0. (4) This equation may also be written in the form f{\y + iix)= iL-fix) + /x" iAA,/(.r) + /• ! w : + Lr^ A//(x) + - + ^ A,"/(x) = 0, (6) which is equivalent to the preceding one. 208 CURVES AND SURFACES [Chap. XIII. If these equations are identically satisfied, the line joining (y) to (x) lies entirely on the surface. If they are not identically satisfied, they are homogeneous, of degree n in X, fx and conse- quently determine n intersections of the line and the surface. If we define the order of a surface as the number of points in which it is intersected by a line, we have the following theorem. Theokem. The order of a surface is the degree of its equation in point coordinates. 164. Polar surfaces. In (5) let the point (y) be fixed but let (x) vary in such a way that the equation a;/(x) = o (6) is satisfied. This equation defines a surface of order n — r called the rth polar surface of the fixed point (y) with regard to the given surface /(.!•) = 0. When r = n — l, the surface (6) is a plane. It is called the polar plane of the point (y) as to f(x) = ; when r = n — 2, the resulting quadric defined by (6) is called the polar quadric, etc. In the identities (4) and (5) the coeflicients of like powers of A, fi are equal, that is, i- a; fix) = ^ A— /(y)- r ! (n — r) ! From this identity we have the following theorem : Theorem I. If (x) lies on the rth jjolar of (y), then (y) lies on the (n — r)th polar of (x). If in (4), the two points (y), (x) are coincident, then fi\x + ^) = (A + ^yf{x) = X-fix) + A-VA^/Cx) + •.. \n—r,,r By expanding (A + fi)" by the binomial theorem and equating coefficients of like powers of A, /a in the preceding identity, we obtain A/f(x)=--^f(x), (w — ?•) ! Arts. 164, 165] TANGENT LINES AND PLANES 209 which is called the generalized Euler theorem for homogeneous functions. From this identity we hav^e the following theorem : Theorem II. The locus of a jwint which lies on any one and therefore on all its own polar surfaces is the given surface f(x) — 0. From the definition of ^Jf{x) (Art. 162) it follows that if ^ <'■<"' A//(.r) = A/[A-/(aO]. From this identity we have the theorem : Theorem III. Tlie sth polar surface of the (r — s)th polar siir- face of (y) with respect to f(x) = coincides with the rth polar sur- face of (y). EXERCISES 1. Determine the coordinates of tiie points in which the line Joining (1, 0, 0, 0) to (0, 0, 0, 1) intersects the surface Xi'^ + 2 X^^ — X3^ — 4 Xi^ + 4 Xi2X4 — XiX4''^ + 6 X2'^X3 — 6 X1X2X.3 = 0. 2. Determine a so that two intersections of the line joining (0, 1, 0, 0) to (0, 0, 1, 0) with the surface Xl* + X2* + X-i* + Xi* 4- 0X2^X3 + 2 (a - 1)X22X,32 + 4 X2X33 + 6 X1X2X3X4 = coincide. 3. Show that any line through (1, 0, 0, 0) has two of its intersections with the surface 3 XzW + Xi* + 6 Xi2X2- + 12 X22X42 + 4 Xirs"^ + 24 Xi,r2X3X4 = coincident at (1, 0, 0, 0). 4. Prove the theorems of Art. 164 for the surface of Ex. 3 by actual differentiation. 165. Tangent lines and planes. A line is said to touch a sur- face at a point P on it if two of its intersections with the surface coincide at P. In equation (4) let (?/) now be a fixed point on the given surface so th.a.t fy) = 0. One root of (4) is now /x = 0, and one of the intersections (x) coincides with (y). The condition that a second intersection of the line (y)(x) coin- cides wdth (y) is that fj} is a factor of (4), that is, that (x) is a point in the plane dyi 5//2 a^/3 dy^ All the lines which touch f(x) = at (?/) lie in the plane (7) and every line through (y) in this plane is a tangent line. This plane 210 CURVES AND SURFACES [Chap. XIII. is called the tangent plane of (>/). The plane (7) is also the polar plane of (ij) ; hence we have the theorem : Theokkm. The polar plane of a point P on the surface is the tangent plane to the surface at P. From Art. 164, Theorem III it also follows that the tangent plane to /(.r) = at a point (y) on it is also the tangent plane at (y) to all the polar surfaces of {y) with regard tof(x) = 0. 166. Inflexional tangents. A line is said to have contact of the second order with a surface at any point P on it if three of its intersections with the surface coincide at P. Let (?/) be a given point on the surface, so that/(?/)=0. The condition that the line {y){z) has contact of the second order at (y) is that fx^ is a factor of (4), that is, that (2:) lies on the tangent plane and on the polar quadric of (?/). Hence (2;) lies on the intersection of A,/(z/)=0, A//(^/) = 0. Since A^/(?/) = is the tangent plane of the quadric A//(y)=0 at (y), the conic of intersection of the taiigent plane and polar quadric consists of two lines, each of which has contact of the second order with _/(,r) =0 at the point (?/). These two lines are called the inflexional tangents to the surface at the point P. The section of the surface by an arbitrary plane through either of these lines has an inflexion at (?/), the given line being the inflex- ional tangent. 167. Double points. A point P is said to be a double point or node on a surface if every line through the point has two inter- sections with the surface coincident at P. If (y) is a double point on f(x) = 0, equation (4) has /x^ as factor for all positions of (2), that is, ^J{y) = is an identity in z^, z.,, z^, z^. It follows that if (?/) is a double point, its coordinates satisfy the four equations M2()=0, M^ = o, M^ = o, M^=0. (8) Conversely, if these conditions are satisfied, it follows, since nfijj) = ^yfQl), that equation (4) has the double root /x^ = and Arts. 165-168] FIRST POLAR SURFACE 211 {y) is a double point. Hence the necessary and sufficient condi- tion that/(x') = has a double point at {y) is that the coordinates of {y) satisfy equations (8). Unless the contrary assumption is stated, it will be assumed that/(ic) = has no double points. EXERCISES 1. Impose the condition that the point (0, 0, 0, 1) lies on the surface f{x) = and find the equation of the tangent plane to the surface at that point. 2. Determine the condition that the surface of Ex. 1 has a double point at (0, 0, 0, 1). 3. Show that the point (1, 1, 1, 1) lies on the surface of Ex. 1, Art. 164, and determine the equation of the tangent plane at that point. 4. Find the equations of the inflexional tangents of the surface of Ex. 1, Art. 164, at the point (1, 1, 1, 1). 5. Show that the lines through a double point on a surface f{x) = which have three intersections with the surface coincident at the double point form a quadric cone. 6. Show that there are six lines through a double point on a surface f(x) = which have four points of intersection with the surface coincident at the double point. 7. Prove that the curve of section of a surface by any tangent plane has a double point at the point of tangency, and the inflexional tangents are the tangents at the double point. 168. The first polar surface and tangent cone. If in equation (7), the coordinates Xi, x.;,, x., x^ are regarded as fixed, and ?/i, ?/2, 2/3, y^ as variable, the locus of the equation is the first polar of the point (x). Theorem I. Tlie first polar surface of any point in space passes through all the double j^oints of the given surface. For, if f(x)=0 has one or more double points, the coordinates of each must satisfy the system of equations (8) and also (7). Theorem II. Tlie joints of tangency of the tangent planes to the surface from a point {x) lie on the curve of intersection of the given surface and the first polar of{x). For, if (y) is the point of tangency of a tangent plane to f(x)=0 which passes through the given point (x), the coordi- 212 CURVES AND SURFACES [Chap. XIII. nates of (y) satisfy /(//)= and A^/'(^) = 0, Conversely, if (y) is a non-multiple point on this curve, it follows that the tangent plane at {y) passes through the given point (ic). Since the line joining (x) to any point (y) on the curve defined in Theorem II lies in the tangent plane at (y), it follows that it is a tangent line. The locus of these lines is a cone called the tangent cone from (x) to the surface f(x) = 0. To obtain the equation of this cone we think of (x) as fixed in (4) and impose the condition on (y) that two of the roots of the equation in A : /x shall be coincident. Hence we have the following theorem : Theorem III. The equation of the tangent cone from any point (x) is obtained by equating the discriminant of (4) to zero. 169. Class of a surface. Equation in plane coordinates. A point (x) lies on tlie surface /(a?) = if its coordinates satisfy the equa- tion of the surface. Similarly, a plane (u) touches the surface if its coordinates satisfy a certain relation, called the equation of the surface in plane coordinates. The class of a surface is the dual of its order ; it is defined as the number of tangent planes to the surface that pass through an arbitrary line and is equal to the degree of the equation of the surface in plane coordinates. Theorem. The class of an algebraic surface of order n, having 8 double points and no other singularities, is n(7i — ly — 2 8. Let /(.^•) = be of order n, and let Pi=(y), P.^={z) be two points on an arbitrary line I. The point of tangency of every tangent plane to/(ic) = that passes through I lies on the surface /(.c) = 0, on the polar of {y) and on the polar of (2;), so that its coordinates satisfy the equations /(.r)=0, A,/(x)=0, A,/(a;) = 0. These surfaces are of orders n, n — 1, n — 1, respectively, and have n(n — 1)^ points in common ; \i f{x)=: has no double points, each of these points is a point of tangency of a plane through the line I, tangent to the given surface. If /(a:)=0 has a double point, Aj^/(x)=0 and A,/(a7) = 0, both pass through it, hence the number of remaining sections is reduced by two. If the plane u^x^^ + u^x^ -{- n^x^ + u^x^ = is tangent to f{x)= Arts. 168-170] THE HESSIAN 213 at (.y), then this plane and that defined by equation (7) must be identical, hence Moreover, (y) lies in the given plane and also on the given sur- face, hence ^^^^^ _^ ^^^^^ ^ ^^^^ _^ ^^^^ ^ 0^ ^^^^^ ^ 0. (10) If between (9) and (10) the coordinates of (y) are eliminated, the resulting equation will be the equation of the given surface in plane coordinates. If /(a;) = has double points, the resulting equation will be composite in such a way that the equation of each double point appears as a double factor. EXERCISES 1. Determine the equation of the tangent cone to the surface Xi^ + X2^ + Xs^ + Xi^ = from the point (1, 0, 0, 0). 2. Write the equation of the surface of Ex. 1 in plane coordinates. 3. Write the equation of the surface a;ia:23;3 + a:ia;3a;4 + XiXoXt + XiX^Xt = in plane coordinates. 4. Write the equation of the surface Xi'^x^ — X2-X4 — in plane coordinates. 170. The Hessian. The locus of the points of space whose polar quadrics are cones is called the Hessian of the given sur- face f{x) = 0. The equation of the polar quadric of a point (x) may be written in the form y^'my^y, = 0, (11) ^ ax. 5.1V in which y^, y^, y^, y^ are the current coordinates. This quadric will be a cone if its discriminant vanishes (Art. 103), hence if we put for brevity ^ gy^^ ax-ax^ the equation of the Hessian may be written in the form /u /12 /l3 J\i TT fl2 J 22 /23 X24 ./l3 /23 JZ3 /34 7l4 ./24 /34 744 It is of order 4 (n — 2). = 0. (12) 214 CURVES AND SURFACES [Chap. XIII. It will now be shown that the Hessian may also be defined as the locus of double points on first polar surfaces of the given surface. The equation of the first polar of (y) as to/(a;) = is If this surface has a double point, the coordinates of the double point make each of its first partial derivatives vanish, by (8), thus ax^ axiox.^ dx^^dx^ dx^dXi yi^+2/.|^+ 2/3 ^ + 2/4^=0, (13) 0X16X2 OX^ 0X20X3 0X20Xi Qlf Q2f Q2f Q2f OXyOX:^ OX2OX3 0X3^ OX^OX^ oxiox^ dXiOX^ ox^ox^ oxv The condition that these equations in ^j, ?/2> 2/3) 2/4 ^^^ consistent is that their determinant is equal to zero, that is, that (x) lies on the Hessian. 171. The parabolic curve. The curve of intersection of the given surface with its Hessian is called the parabolic curve on the surface. Theorem. At any point of the parabolic curve the two inflexional tangents to the surface coincide. For, let (x) be a point on the parabolic curve. Since (x) lies on the Hessian, its polar quadric is a cone. This cone passes through (x) (Art. 164). The inflexional tangents are the lines which the cone has in common with its tangent plane at (x) (Art. 166). These lines coincide (Art. 121). 172. The Steinerian. It was just seen that the polar quadric of any point on the Hessian is a cone. Let (.c) be a point on H, and (y) the vertex of its polar quadric cone. As (x) describes H, (y) also describes a surface, called the Steinerian of f(x) = 0. The polar quadric of (;c) is given by equation (11). If (y) is the ver- tex of the cone, its coordinates satisfy (13). The equation of the Arts. 170-173] ALGEBRAIC SPACE CURVES 215 Steiuerian jnay be obtained by eliminating x^, x^, x^, x^ from these four equations (13). As the equations (13) were obtained by im- posing the condition that the first polar of (?/) has a double point, we may also define the Steinerian as the locus of a point whose first polar surface has a double point (lying on the Hessian). EXERCISES 1. Prove that the Hessian and the Steinerian of a cubic surface coincide. 2. Prove that if a point of the Hessian coincides with its corresponding point on the Steinerian, it is a double point of the given surface, and con- versely. 3. Determine the equation of the Hessian of the surface ai-Ti^ + a-ix-^ + asa-s' + a^x.^ + OsCx] + .ro + X3 + X4)^ = 0. 4. Determine the order of the Steinerian of a general surface of order n. II. Algebraic Space Curves 173. Systems of equations defining a space curve. A curve which forms the complete or partial intersection of two algebraic surfaces is called an algebraic curve ; if the curve does not lie in a plane, it is called a space curve. If a given curve C forms the complete intersection of two sur- faces F^ =0, i^2 = ^1 so that the points of C, and no other points, lie on both surfaces, then the equations of these surfaces, consid- ered as simultaneous, will be called the equations of the given curve. If the intersection of the surfaces i^i = and i^j = is composite, and G is one component, the equations 2^^ = 0, i^2 = are satisfied not only by the points of C but also by the points of the residual curve. If a surface i^3 = through G can be found which has no points of intersection with the residual curve except those on C, the simultaneous equations F^ = 0, i^2 = 0, i^3 = are satisfied only by the points of G and are called the equations of the curve. If the surfaces F^ = 0, i^^ = 0, i^3 = through G have one or more points in common which do not lie on C, then a fourth sur- face i^j = can be found through G which does not contain these residual points, but may intersect the residual curve of J^i = 0, 7^2 = in other points not on 2^3 = ; in this case the simultaneous 216 CURVES AND SURFACES [Chap. XIII. equations F, = 0, i^2 = 0, i^s = 0, i^4 = are called the equations of the curve. In this way a system of equations can be found which are simultaneously satisfied by points of C and by no others. As an illustration, consider the composite intersection of the quadric surfaces It consists of a space curve and the line X2 = 0, x^ = 0. The surface XiX^ — XzX^ = also contains the space curve since it contains every point common to the quadrics except points on the line X2 = 0, ^3 = 0. These three surfaces are sufficient to define the curve. The surface XiXi(xi — x^)—X2^-{-X2X^Xi=zO also contains the given curve. It does not, however, with the two giveu surfaces constitute a system whose equations define the given curve. All three equations are satisfied, not only by the coordinates of the points of the curve, but by the coordinates of the point (1, 0, 0, 1) which does not lie on the curve, since it does not lie on the surface XiX^ — x^x^ = 0. The sur- face x-^x^x^ + x^ — ;r./ — xi = passes througli the curve but not through the point (1, 0, 0, 1). The curve is therefore completely defined by regarding the four equations XiXi (Xi — x^) — xi + x^^Xi = 0, X1CC4 (a^i + X4) — xi — x^ = as simultaneous. 1 74. Order of an algebraic curve. Let F^ = 0, F^- = be two surfaces of orders /*, /x', respectively, and let C be their (proper or composite) curve of intersection. Any plane that does not con- tain C (or a component of it) intersects C in fxfjj points. For, any such plane intersects F^ = in a curve of order /*, and inter- sects i^ - in a curve of order jx. These coplanar curves have precisely jx^jJ points in common.* It can in fact be shown that every algebraic curve, whether defined as the complete intersection of two surfaces or not, is intersected by any two planes, neither of which contains the curve or a component of it, in the same number of points.t We * See, e.g., Fine: College Algebra (1905), p. 519. t Halphen: Jour, de I'ecole polyteclinique. Vol. 52 (1882), p. 10. Arts. 173-175] PROJECTING CONES 217 shall assume, without proof, the truth of this statement. The number of points in which an arbitrary plane intersects an alge- braic curve is called the order of the curve (Art. 140). 175. Projecting cones. If every point of a space curve is joined by a line to a fixed point P in space, a cone is defined, called the projecting cone of the curve from the point P. If the point P lies at infinity, the projecting cone from P is a cylinder (Art. 44). Except in metrical cases to be discussed later we shall make no distinction between cylinders and cones. For an arbitrary point P an arbitrary generator of the project- ing cone intersects the curve in only one point. It may happen, however, for particular positions of the point P, that every generator meets the curve in two or more points. If in this case P does not lie on the curve or if P lies on the curve and every generator through P intersects the curve in two or more points distinct from P, the curve is called a conical curve. Let P be a point not on the curve, such that an arbitrary generator of the projecting cone from P meets the curve in just one point. The order of the projecting cone is the number of generators in an arbitrary plane through its vertex. Each gener- ator contains one point on the curve, hence the order of the pro- jecting cone is equal to the order of the curve. If P is on the curve, the order of the projecting cone is one less than the order of the curve. Theorem. To find the equation of the 2^'>'ojecting cone of the simple or composite curve defined by the complete intersection of two surfaces, from a vertex of the tetrahedron of reference, eliminate be- tween the equations the variable which does not vanish at that vertex. Let the equations of the given surfaces be P^ = and F^. = and let it be required to project the curve of intersection of these surfaces from the point (0, 0, 0, 1). Let (y) be any point of space. The coordinates of any point (x) on the line joining (0, 0, 0, 1) to (y) are of the form X, = A?/i, a'2 = Xy2, x^ = Xy^, x^ = Xy^ + o-. 218 CURVES AND SURFACES [Chap. XIII. The points in which this line intersects F^ = 0, F^. = are de- fined by F^,{x) = F^{Xy„ Xy„ Xy^, Xy^ + a) = >^''Ffy„ y„ y„ y,+ f) = 0, ^ ^^ (14) F^.{x) = F^-{Xyi, Xy^, Xy^, Xy^ + a) = Xt^'FJy„ y.„ 2/3, 2/4+H=0, respectively. The condition that the line intersects both surfaces in the same point is that these equations have a common root in -, hence the equation of the projecting cone is obtained by elimi- A nating - between these two equations (cf. Art. 44). If - is elimi- A A nated from (14), y^ is also eliminated and the resulting surface is identical with that obtained by eliminating x^ between the equa- tions of the given surfaces. If the curve of intersection is composite, the projecting cone is composite, one component belonging to each component curve. A method for determining the projecting cone from any point P in space may be deduced by similar reasoning, but the process is not quite so simple. EXERCISES 1. Show that the intersection of the surfaces 3:1X2 — X3X1 + X^'^ — X2X3 = 0, XiXs'^ — X1X2X4 + X2(X42 — X2X3) = is composite. 2. Represent each component curve of Ex. 1 completely by two or more equations. 3. Find the equation of the projecting cone of the curve Xi'^ + X32 + X42 + 2 X1X4 = 0, X42 + 2 X2X4 - Xi^ + 2 X32 = from the point (0, 0, 0, 1). 4. Find the equation of the projecting cone of the curve Xi2 + 4 X32 - X42 = 0, Xi2 _ 2 X22 + 2 X32 - 3 .r4- = from the point (0, 0, 0, 1). 5. Find the equation of the projecting cone of the curve xi"^ + X2'^ + X32 4- xt^ = 0, aixi2 + a2X:2 -f- 03X32 + a^Xi^ = from the point (0, 0, 0, 1), Arts. 175, 176] MONOIDAL REPRESENTATION 219 6. Show by means of elimination that, if (0, 0, 0, 1) does not lie on the curve Ffj^= 0, i^^-^O, the order of the projecting cone from (0, 0, 0, 1) is ix/x', provided the curve is not conical from (0, 0, 0, 1). 7. Find the equation of the projecting cone of the curve Xi^ + 2 X2^ — Xs^ — 0, Xi' — 0*2^:3 + Xi^ = from the point (1, 1, 1, 1). 176. Monoidal representation. If a non-composite space curve (7„ of order m is defined as the complete or partial intersection of two surfaces F^ = 0, F^. = 0, other surfaces on which the curve lies can be obtained from the given ones by algebraic processes. Among such surfaces we have already discussed the projecting cone from a given point P. We shall now show how to obtain the equation of a surface which contains C„ and has at P a point of multiplicity one less than the order of the surface. Such a surface is called a monoid. In determining the equation of a monoid through C„, we shall assume that neither the complete intersection of P^ = and F^. = nor any component of it is a conical curve from P. We shall also assume that P does not lie on this curve of intersection. Let P be chosen as (0, 0, 0, 1) and let the equations F^ = 0, F^. = be arranged in powers of x^ (Art. 161). F^ = u^,^ + Kior/"' + ••• + w^ = 0, F^- = iVV' + ^iaJ4''"' + ••• +v^- = 0, wherein ?^,-, v,- are homogeneous functions of x^, x^, x-^ of degree i. Let the notation be so chosen that /x' ^ fx. The equation v^/->'F^ - u,F^. = contains x^ to at most the power /a' — 1. The surface represented by it passes through the curve C„, since the equation is satisfied by the coordinates of every point which satisfy F^ — and F^. = 0. The equation V .F -u F . = MM MM is divisible by x^. If this factor is removed, the resulting equation is of degree at most /x' — 1 in .^4, and determines a surface which passes through C'„. If either of these equations contains x^ to the first but to no 220 CURVES AND SURFACES [Chap. XIII. higher power, the surface determined by it is of the type required. If not, the two equations cannot both be independent of x^ nor can they coincide, since in that case the curve F^ = 0, F^- = would be conical from (0, 0, 0, 1). By applying this same process to the two equations just ob- tained, we may obtain two new ones which contain x^ to at most the power fi — 2. Continuing in this way with successive partial elimination, we obtain finally an equation of the form M= x^(f}„_y{Xi, X2, Xs) - <^„(a^i, X2, Xs) = 0, in which <^„_i and (^„ are homogeneous functions, not identically zero, of degree n — 1 andn, respectively, in x^, X2, Xy The surface M = is of order n and has an (it — l)-fold point at (0, 0, 0, 1). It is consequently a monoid. The surface <^„ = is a cone ; it is called the superior cone of the monoid. If ?i > 1, <^„_i = is the equation of another cone, called the inferior cone of the monoid. Let/„(.Ti, X2, .T3) = be the equation of the projecting cone from (0,0,0,1). The equations are said to constitute a monoidal representation of the curve C„. The advantage of this representation is that the residual inter- section, if any, of the two surfaces M = 0, f„ = consists of lines common to the three cones /„ = 0, <^„, =0, 0„ = O. For, let P be a point common to /„ = 0, M— 0, but not lying on C„, nor at the vertex (0, 0, 0, 1). The generator of /„ = passing through P intersects C„ in some point P'. Since this generator has P, P' and n — 1 points at (0, 0, 0, 1) in common with M = 0, it lies entirely on the monoid (Art. 163). For every point of this line, that is, independently of the value of x^, the equation Xi(f)„_i — cf)^ = must be satisfied ; hence the given generator lies on the inferior cone and on the superior cone. It follows at once from the above discussion that if any genera- tor of /„ = intersects C„ in two points P, Q, it lies entirely on the monoid and forms a part of the residual intersection. Such a line is called a double generator of the projecting cone, since, in Arts. 176, 177] NUMBER OF INTERSECTIONS 221 tracing the curve, the generator takes the position determined by Pon C„ and also the position, coincident with the first, determined by Q. Every such line counts for two intersections of M= and .4 = 0. Each of these bisecants of the curve is said to determine an apparent double point of C„ from (0, 0, 0, 1) ; the curve appears from (0, 0, 0, 1) to have a double point on each of these lines. It can be proved* and will here be assumed that the number of apparent double points of a given curve is the same for every point not lying on it, except the vertices of the cones, if any, on which Cn is a conical curve. This number will be denoted by h. We shall now show that if a point P which does not lie on C„, nor on any line that intersects C„ in more than two points, nor at the vertex of a cone (if any) of bisecants to C„„ is chosen for the vertex, then the order of the monoid from P is at least half the order of C„. The complete intersection of the projecting cone/„ = and the monoid a;4<^„ i — „ = is a curve of order mn. The curve C^ is one component of order m, and the h bisecants of C^ through (0, 0, 0, 1) together form a component of order 2h. If the num- ber of residual intersecting lines is denoted by k, then mn — m — 2 h = Jc, k ^ 0. The h bisecants of C„ and the k residual lines are components of the intersection of <^„_i =0, <^„ = 0. Hence from which and 7i[u — l) = h-}-k =7n()i — 1)— h, {m — n){n — l) = h^ - (n — 1), > 7n n — v ' which proves the proposition. 177. Number of intersections of algebraic curves and surfaces. Theorem. Any surface of order jj. lohich does not contain a given non-composite curve of order m intersects it in mfx j)oints. *N()etlier: Ziir Grumllegiing der Tlv^orie der alfjebraischeii Raumkurven, Ab- handlungen der k. preussisclien Akademie der Wissensehaften fiir 1S82. 222 CURVES AND SURFACES [Chap. XIII. Let C„ be the given curve and i^^^ = be the equation of the given surface. Choose (0, 0, 0, 1) not on F^ = 0, and let the monoidal equations of C„ be /„ = 0, Xi^_i — <^^ = 0. The com- plete intersection of /„ = 0, .t^4<^„_i — <^„ = consists of (7„ and of m{n — l) lines through (0, 0, 0, 1). As F^=0 does not pass through (0, 0, 0, 1), it cannot contain any of these lines. Hence Ffj, = 0, /„ = 0, M^ = have no common component. They con- sequently intersect in mri/j. points. Of these, mfx{n — 1) points are where the residual lines intersect F^= 0. The remaining m/x points lie on C„. If C„ has vi/x. + 1 points on F^ = 0, it lies on the sur- face, for the three surfaces /^ = 0, M^ = 0, F^ = have now mnfx + 1 points in common, and therefore all contain a common curve. Since the lines do not lie on ^^^ = 0, and/„ = 0, 3/"^ = have no other component curve except C„, it follows that C„ must lie on F^ = 0. EXERCISES 1. Show that a plane or any proper quadric is a monoid. 2. Write the equation of a monoid of order three. 3. Show that the only curve of order one is a line. 4. Show that the only irreducible curve of order two is a conic. 5. Show that a composite curve of order two exists which does not lie in a plane. How many apparent double points has this curve ? 6. Show that a bundle of quadrics pass through a proper space cubic curve. 7. Write a monoidal representation of a space cubic curve. 8. Show that every irreducible curve of order four lies on a quadric surface. 9. Discuss the statements of Exs. 6 and 8 for the case of composite cubics and composite quartics. 178. Parametric equations of rational curves. Since a space curve is defined as the complete or partial intersection of two surfaces, the coordinates of its points are functions of a single variable. The expressions for the coordinates of a point as func- tions of a single variable may not be rational. A curve which possesses the property that all its coordinates can be expressed as rational functions of a single variable is called a rational curve. "By definition the equations of such a curve can be written para- metrically in the form ^i =.W) = tt.o^"' + (I J""'' + - + «•>., i = 1, 2, 3, 4. Arts. 177, 178] PARAMETRIC EQUATIONS 223 Since the variables x^ are homogeneous, it is no restriction to suppose that the polynomials /^O liave no common factor. To every value of t corresponds a unique point {x) on the curve, but it may happen that more than one value of t will define the same point (x) on the curve. If, for example, the functions fi{t) can be expressed in the form in which F^ are homogeneous rational functions, of the same order, of the two polynomials {t), i}/{t), then f-(t) will define the same point for every value of t that satisfies the equation where s is given. In this case the coordinates of the points on the curve are rational functions of s. Conversely, it will now be shown that if to each point (x) of the curve correspond 7i values of t{n^ 1), then t may be replaced by a new variable, in terms of which the correspondence between it and the point (x) on the curve is one to one. Let fi, t2, •••, tn all correspond to the same point (x). The expressions Mt)Mh)-Mti)Mt) i, k - 1, 2, 3, 4 vanish for t = ti, tz, •••, t„, hence they have a common factor of order n, whose coefficients contain ti, o{h)t^ + 0i(«i)«"-i + ••• + n{h)- If ti is replaced by ti, the expression must have the same factor, hence the function ^ (m-l)(m-2) _^^_^^^^^^(n-l)(n-2)_^^^^^^^^ 183. Curves on non-singular quadric surfaces. It has been seen (Art. 115) that the equation of any non-singular quadric surface may be reduced to the form ^'l^2 — •''^3^4 = 0> (1^) and that through each point of the surface passes a generator of each regulus of the two systems x^ — Xx^ = 0, a'a — Xx2 = 0, (18) a-3 — ixx^ = 0, X2 — fxx^ = 0. (18') The coordinates of the point of intersection of the generator A= constant with the generator /ia = constant are (Art. 115) pXi ^ A, px^ = [X,. px^ = Xjx, px^ = 1. (19) Consider the locus of the points whose parameters A, p. satisfy a given equation /(A, /x) = 0, algebraic, and of degree A;, in A and of degree Jc2 in p.. The curve /(A, p.)=0 meets an arbitrary generator p. = constant in ki points, and an arbitrary generator A = constant in k^ points. It will be designated by the symbol [Aij, k2^. The order of the curve is k^ 4- A^j, since the plane determined by any two generators of different reguli meets the curve in k^ + Atj points on these two lines, and nowhere else. By replacing A, p. in /(A, p.) = by their values, we see from (17), (18), (18') that the curve is the intersection of the two surfaces \Xi xj Arts. 182, 183] CURVES ON QUADRIC SURFACES 229 The second is a monoid of order two (Art. 176) and the first is a cone with vertex at (0,0, 1, 0), a(2 — l)-fold point on the monoid. Thus, these equations constitute a particular monoidal representar tion of the curve. The equations of the image (Art. 118) of the given curve on the plane Za = are /A^^^O, .^3 = 0. Va-4 xj The two generators to the quadric through the vertex of the cone /=0 meet the plane in the points (1, 0, 0, 0), (0, 1, 0, 0). The former is a Avfold point on the plane curve, and the latter a ^■l-fold point. Theorem I. Tvoo curves of symbols [k^^, k^'], [k\, A-'.,] on the same non-singular quadric intersect i)i kjc'., + k2k\ x^oints. Let C, C" be the given curves of symbols [A^i, A-.,], [k\, A'2], re- spectively, and let the equation of the quadric be reduced to the form (17) in such a way that the point (0, 0, 1, 0) does not lie on either curve, and that the generators .Ti = 0, x^ = i)\ .1-2 = 0, x^ — i) through (0, 0, 1, 0) do not pass through a point of intersection of the given curves. Project the curves from (0, 0, 1, 0). Their images on x^ = are of orders k^ + k^, k\ + k\, respectively ; they intersect in {k^ + k^{k\ + A'j) points. Of these points, kji\ coin- cide at (0, 1, 0, 0) and kjk'^^ at (1, 0, 0, 0). They are the projections of the points in which the curves meet the generators passing through (0, 0, 1, 0), the vertex of the projecting cone, and are therefore apparent, not actual, intersections of the space curves. The remaining (ki ~p koj(k I -\- tC 2) — iCifC J — rCoK 2 ^^ fC^fC 2 ~r" '12"' 1 intersections of the plane curves are projections of tlie actual in- tersections of the space curves, hence the theorem is pioved. Theorem II. Tlie number of ap^jarent double points of a carve of symbol [Aj, ^2] on a quadric is h = ^{k,^ + k2'-k,-k2). Through an arbitrary point on the surface pass only two lines which meet the curve in more than one point, namely, the two generators passing through (). The generator fx = constant 230 CURVES AND SURFACES [Chap. XIII. through meets the curve in A', points, consequently counts for -~{k^~l) bisecants through 0. Similarly, the generator A = constant, which passes through (0, 0, 1, 0), meets the curve k in A'2 points and counts for -~ {k^ — 1) bisecants. The number of api^arent double points is the sum of these two numbers. 184. Space cubic curves.* Theorem I. Tliroiigh any six given points in space, no four of which lie in a plane, can he passed one and only one cubic curve. Let Pj, •••, Pq be the given points. The five lines connecting P^ to each of the remaining points uniquely determine a quadric cone having Pi as vertex. Similarly, the lines joining Pj to each of the other points define a quadric cone having P, as vertex. These two cones intersect in a composite curve of order four, one component of which is the line P1P2, since it lies on both cones. The residual is a curve of order three. This curve cannot be com- posite, for if it were, at least one component would have to be a straight line common to both cones. But that would require that the cones touch each other along P1P2, which would count for two. The residual intersection would in that case be a conic passing through P3, •••, Po- But this is impossible as it was assumed that the points P3, —, P« do not lie in a plane. No other cubic curve can be passed through the given points, for every such curve would have seven intersections with the two cones (the vertex counting for two). Hence it would lie on their curve of intersection, wliich is impossible, since the complete intersection is of order four. Theorem II. A space cubic curve lies on all the qnadrics of a bundle. For, let Pj, •••, P- be seven given points of the curve. Every quadric through these points has 2-3 + 1 points in common with the curve and consequently contains the curve (Art. 177). But through the given points pass all the quadrics of a bundle (Art. 136), which proves the theorem. Not all the quadrics of this bundle can be singular, for if so, at * Unless otherwise stated, it will be assumed in the following discussion that the curve is non-composite. Arts. 183, 184] SPACE CUBIC CURVES 231 least one of them would be composite (Art. 131) and still contain the curve. This is impossible, since the given curve is not a plane curve. The symbol (Art. 183) of a space cubic curve on a non-singular quadric is [2, IJ or [1, 2], since such symbols as [0, 3] and [3, 0] simply define three generators belonging to the same regulus. The forms of /(A, /x) corresponding to these symbols are (a,X^ + 2 a,\ + a,),M + (b,X +2b,\ + b,) = 0, (20) (aV^ + 2 aV + a',) A + (?> V + '-' '^> + '^'2) = 0. (20') Conversely, every irreducible equation of this form will define a cubic curve on the quadric. Since these equations have six homogeneous coefficients, five in- dependent linear conditions are sufficient to determine a curve of either system. Hence through any five points on a given non- singular quadric can be drawn two cubics, one of each symbol. Some of these cubics may be composite. From the formula of Art. 183 it follows that on a given non- singular quadric two cubics having the same symbol intersect in four points, while two cubics having different symbols intersect in five points. Theorem III. Every space cubic curve is rational. Let the parametric equations of a non-singular quadric through the given cubic be reduced to the form (19). The equations of the curve in A, (x are of the form (20) or (20'). In (20), let X=t, solve for /x in terms of t, and substitute the values of A and of /x in terms of t in (19). The resulting equations reduce to the form X, = a^^f + a.^f' + a J -f a-,, i = 1, 2, 3, 4. (21) These are the parametric equations of the curve (Art. 178). Since the curve is by hypothesis of order three, to each value of t cor- responds a definite point on the curve, and conversely. Since the cubic (21) does not lie in a plane, the determinant I a-j. I ^ 0. The parametric equations, referred to the tetrahedron defined by Xi = a^ox'i + aiix'2 + a.ax'j + a^^x^, i = 1, 2, 3, 4, are, after dropping the primes, x^ = ^, 0^2 = f", X3 =t, X4 = 1. (22) 232 CURVES AND SURFACES [Chap. XIIL From (22), the intersections of the curve with the plane '2,u-x- = are defined by the roots of the equation ?/l«3 -f M2«2 4- Ust + »4 = 0. (23) The condition that the pla.ne is an osculating plane is that the roots of (23) are all equal (Art. 180). It follows that the coor- dinates of the osculating plane at the point whose parameter is t may be expressed in the form Mj =1, 7(2 = — 3 t, u^ — 3 t', Ui = — f. These equations are called the parametric equations of the cubic curve in plane coordinates. The condition that the osculating plane at the point whose parameter is t passes through a given point (//) in space is that t is a root of the equation lUf -3y,t' + 3y,t-y,=0. (24) Since this equation is a cubic in t, it follows that the cubic curve is of class three. We shall now pfove the following theorem: Theorem IV. TJie points of contact of the three osculating jyJanes to a cubic curve throv/jh an arbitrarij point P lie in a plane passing through P. Let %a-Xi = be the plane passing through the points of oscu- lation of the three y^lanes passing through any given point P = (?/). The parameters of the points of osculation of the three osculating planes through {y) are the roots of (24). The roots of (24) must also satisfy the equation ttii? -\- a^'^ + aji + a^ = 0, hence '■^ Ik - V\ From these conditions it follows that 2a,.V, = 0, so that (?/) lies in the plane of the points of osculation. By the method of Art. 179 the equation of the developable sur- face of the cubic curve is found to be (X^X^ X^pt'^j 4(^X2 X'^X'i^jyX^ 3^2X4^= U. Art. 184] SPACE CUBIC CURVES 233 This is also the condition that equation (24) has two equal roots. From this equation it follows that the rank of the cubic curve is four (Art. 179). It was stated without proof in Art. 133 that the basis curve of a pencil of quadrics of characteristic [22] is a cubic and a bi- secant ; it was also stated that the basis curve of a pencil of char- acteristic [4] is a cubic curve and a tangent to it. We shall now prove these statements. It was shown in Art. 132 that the [22] pencil of quadrics is defined by the two surfaces x^^ + 2 x^x^ = 0, x^^ + 2 x^x-i = 0. These quadrics intersect in the line x^^O, Xo = ^ and the space cubic whose parametric equations can be found by putting Xi = 1, Xi = 2t in the equations of the surfaces, in the form It intersects the line Xi = 0, x., = in the two points (0, 0, 0, 1), (0,0,1,0). Similarly, it was seen that a pencil of characteristic [4] is defined by the surfaces The basis curve of this pencil consists of the cubic and of the line 0:2 = 0, x^ = which touches it at (1, 0, 0, 0). If in the parametric equation (20) of a cubic we replace X by — , and /x by -^, we determine as the projecting cone from x^ x^ (0, 0, 1, 0) a cubic cone with a double generator. It follows that the projecting cone of the cubic is intersected by a plane in a nodal or cuspidal plane cubic curve. We shall now prove the converse theorem. Theorem V. Any nodal or cuspidal plane cubic curve is the projection of a space cubic. Let the plane of the cubic be taken as x^ = 0, and the node or cusp at (0, 1, 0, 0). The equation of the curve is of the form XtittffCi + 2 a^x^x^ + anX^) + b^x^Xi + 2 b^x^x^ + b.,x^^ = 0. 234 CURVES AND SURFACES [Chap. XIII. By dividing this equation by x^ and replacing x, : x^ by \, x^ : x^ by IX, we obtain equation (20) of a space cubic curve of which the given curve is the projection. Theorem VI. Amj plane nodal cubic curve has three points of inflexion lying on a line. If a space cubic is projected from any point {y) upon a plane, the osculating planes from {y) will be cut by the plane of projec- tion in the inflexional tangents of the image curve and the points of osculation will project into the points of inflexion. From the theorem that the points of osculation lie in a plane through (?/) it follows that the points of inflexion of the plane cubic lie on a line. EXERCISES 1. Show that any space cubic curve and a line which touches it or inter- sects it twice form tlie basis curve of a pencil of quadrics. 2. Show that a composite cubic curve exists, through which only one quadric surface can pass. 3. Prove that the osculating planes to a cubic curve at its three points of intersection with a given plane (w) intersect at a point in (?(). 4. Show tliat if a cubic curve has an actual double point or a trisecant it must lie in a plane. 5. Obtain all the Cayley-Salmon numbers for the proper space cubics. 6. Where rau.st the vertex of the projecting cone be taken, in order that the plane projection of a proper space cubic shall have a cusp ? 7. Show that the projection of a space cubic upon a plane from a point on the curve is a conic. 8. Show that the cubic curve through the six basis points of a web of quadrics determined by six basis points lies entirely on the Weddle surface (Art. 146). 9. Show that a cubic through any six of eight associated points (Art. 136) will have the line joining the other two for bisecant (or tangent). 185. Metric classification of space cubic curves. The space cubic curves are metrically classified according to the form of their intersection with the plane at infinity. If the three intersections are real and distinct, the curve is called a cubical hyperbola. It has three rectilinear asymptotes and lies on three cylinders all of which are hyperbolic. If the points at infinity are all real and two are coincident, the curve is called a cubical hyperbolic Arts. 185, 186] SPACE QUARTIC CURVES 235 parabola. It has one asymptote, and lies on one parabolic cylin- der and on one hyperbolic cylinder. If all three of the points of intersection are coincident, the plane at infinity is an osculating plane. The curve is called a cubical parabola. It has no recti- linear asymptote and lies on a parabolic cylinder. Finally, two of the points of intersection may be imaginary. The curve is now called a cubical ellipse. It has one rectilinear asymptote and lies on one elliptic cylinder. An interesting particular case of the cubical ellipse is the curve called the horopter curve on account of its part in the theory of physiological optics. If one looks with both eyes at a point P in space, the eyes are turned so that the two images fall on corresponding points of the retinae. The locus of the points in space whose images fall on correspond- ing points is a horopter curve through the point P. 186. Classification of space quartic curves.* Theorem I. Every space q^iartic curve lies on at least one quad- He surface. For, through any nine points on the curve a quadric surface can be passed. This surface must contain the curve, since it has 2 X 4 -f 1 points in common with it (Art. 177). If a quartic curve lies on two different quadrics A = 0, B= 0, it is called a quartic of the first kind. A quartic of the first kind is the basis curve of a pencil A — XB = of quadrics. Not all the quadrics of this pencil are singular, since in every singular pencil are some composite quadrics. Composite quadrics are im- possible in this case, since the curve does not lie in a plane. The symbol of the curve on any non-singular quadric on which it lies is [2, 2], since each generator of one quadric will intersect the other quadric defining the curve in two points. A quartic having the symbol [1, 3] cannot lie on two different quadrics, nor can it lie on a quadric cone, since every generator would have to cut the curve in the same number of points. The [1, 3] curve is called a quartic of the second kind. It follows from Arts. 132 and 184 that except in the cases of the characteristics [1111], [112], [13], the basis curve of a pencil * See footnote of Art. 184. 236 CURVES AND SURFACES [Chap. XIII. of quadrics is composite. It will now be shown that in these three cases the basis curve is not composite, that in the case [1111] the basis curve has no double point, that in the case [112] it has a node, and that in the case [13] it has a cusp or stationary point (Art. 181). That the basis curve is not composite may be seen as follows: If it were, one component would have to be a line or a conic. It cannot be a line, for the line would have to lie on every quadric of the pencil, hence pass through the vertex of every cone contained in the pencil. From the equations of the pencils having these characteristics (Art. 133) it is seen that in each case there is at least one cone whose vertex does not lie on the basis curve. Moreover, one component cannot be a conic, for the quadric of the pencil determined by an arbitrary point P in the plane of the conic would contain the plane of the conic, and hence be composite ; but pencils having these characteristics have no composite quadrics. It will now be shown that the basis curve of the pencil [1111] has no actual node or cusp. It will be called the non-singular quartic curve of the first kind. Suppose the basis curve had a node at 0. The projecting cone to the curve from is of order two. The quadric of the pencil through an arbitrary point P on the projecting cone contains the line OP, since it has three points in common with it. This quadric and the cone must coincide, since they have a quartic curve and a straight line in common. Hence the cone would belong to the pencil, but this is impossible, since no cone of the pencil [1111] has its ver- tex on the basis curve. From the equation of the pencil of characteristic [112] it follows that the vertex (0, 0, 0, 1) of the cone (A, - X,)x,^ + {K - K)^-? + a^s' = of the pencil lies on the basis curve. This point is an actual double point on the curve, since every plane through it has two points of intersection with the curve coincident at that point. All the quadrics of the pencil touch the plane x-^ = at (0, 0, 0, 1); every plane through either of the distinct lines (A,, — A3)a;i'^ + (A2 — \:i)x.^ = 0, in which x^ — intersects the cone has three in- tersections with the curve coincident at (0, 0, 0, 1). These two lines are tangents at the node. Art. 186] SPACE QUARTIC CURVES 237 Finally, the vertex of the cone of the [13] pencil is a double point on the basis curve. The tan- gent lines Xi = 0, x^ = coincide. The double point is a cusp. The parametric equation of a quartic of symbol [2, 2] has the form (ta^ + 2 OiX + «o)^2 + 2(&o\' + 2 6iX+&2V+ CoA2+2 c,\+c, = 0. (25) The quartic defined by (25) is the intersection of the quadric X1X2 — x^x^ = (Art. 1 83) and the quadric aoO^s^ + 2 aiXs-Tj -f aj-r^^ + 2 boX^x^ + 4 b^x^Xz + 2 62^2^4 + (^0^1^ + 2 010:1X4 ' +c,x,' = 0. (25') If the quartic of intersection has a double point or cusp, we may take the double point as (0, 0, 0, 1), and a cone with vertex at that point for one of the quadrics passing through it. The parametric equation (25) now has the form (2 a,X + a,)ix? + 2(b,\' + 2 b,\),M + CoX^ = 0. (26) If in (26) we put X = fxt, solve for t, and put the values of ft and \ = yd in equations (19), we obtain a set of parametric equations of the singular quartic curve of the first kind, of the form x, = a,o^ + a,,e + a,/ + a J + a,^, i = 1, 2, 3, 4 ; (27) hence the nodal and cuspidal quartics are rational. A quartic of the second kind can be expressed parametrically in terms of the parameter which appears to the third degree in its parametric equation, hence the quartics of the second kind are also rational. Rational curves will be discussed later (Art. 188). Theorem II. Through a quartic curve of the second kind and any two of its trisecants can be passed a non-composite cubic surface. For, through nineteen points in space a cubic surface can be passed (Art. 161). Choose thirteen on the quartic curve, one on the trisecant g, one on the trisecant g', not on the curve, and four others in space, not in a plane nor on the quadric on which the quartic lies. The quartic curve and the lines g and g' must lie on the non-composite cubic surface determined by these nineteen 238 CURVES AND SURFACES [Chap. XIII. points as well as on the quadric containing the regulus of tri- secants, hence together they form the complete intersection of the cubic and the quadric. 187. Non-singular quartic curves of the first kind. Two quartic curves of the first kind lying on the same quadric intersect in eight points (Art. 183) ; these points ai-e eight associated points defining a bundle (x\rt. 136), since they lie on three distinct quadrics not having a curve in common. The number of apparent double points of a non-singular quartic Ci of the first kind is two. For each bisecant of C^ through an arbitrary point P is a generator of the quadric of the pencil hav- ing C4 for basis curve which passes through P. Conversely, each generator of every quadric through C4 is a bisecant. Of the Cayley-Salmon numbers we now have m = 4, h = 2, /8 = 0, H= 0. It also follows from the definition that G = v = to = 0, hence from the formulas of Art. 182 we have m = 4, n = 12, r = 8, H=0, h = 2, G = 0, g = 38, a = 16, (3 = 0, v = 0, 0) = 0, x = 16, y = 8, J) = 1. Theorem I. Through any bisecant of a non-singular space quartic curve of the first kind can be drawn four tangent planes to the curve, besides those having their poiyit of contact on the given bisecayit. Let the given bisecant be taken as .t, = 0, x^ = and the quadric of the pencil containing it as x^Xi — x^x^ = 0. Let another quadric of the pencil be determined by (25'). Any plane of the pencil Xi = mx^ intersects C4 in two points on .x, = 0, x^ = and in two other points determined by the roots of the quadratic equation in X2 '■ x^ x^ {a^ni^ + 2 a{tn + a,) + 2 XM^ {bfpn"^ -\- 2 b^m + 62) -f x^ (cqWi^ + 2 Ci?«, -\- C2) = 0. The planes determined by values of m which make the roots of this equation equal are tangent planes. The condition on m is 4(6om2+2 6im + &2)^-4(aom2+2 a^m + a.^{cam''-\-2 c{)n-\-c^ = Q. (28) Since this equation is of the fourth degree, the theorem is established. Art. 187] QUARTICS OF THE FIRST KIND 239 Theorem II. An arbitrary tangent to a non-singular quartic of the first kind intersects four other tangents at points not on the curve This is a particular case of Theorem I, since a tangent is a bisecant whose points of intersection with the curve coincide. Theorem III. The cross ratio of the four tangent planes through any bisecant is the same number for every bisecant of the curve. Two cases are to be considered, according as the two given bisecants intersect on C^ or not. Let g, g' be two bisecants through a point P on C^, but not lying on the same quadric of the pencil. Let the equation of the quadric of the pencil through C4 which contains g be reduced to the form x^x^ — x^Xi = Q in such a way that the equations of g are x^ = 0, 054 = and the points of intersection of g' with C4 are (0, 0, 1, 0) (0, 0, 0, 1). In (25') we now have a^ = 0, c, = 0, and also in (28). The points of inter- section not on g' of a plane x^ = nx^ and C4 are determined by the roots of the equation 2 {c^n- + b.m) x.^ + (c^^n- + 4 61/* -f- a,) x^x^ + 2 {b^n + a^) x^^ = 0. The parameters rii, Wji '>hj ^h 0^ the four tangent planes are roots of the equation (CqW^ + 4 b^n + aof — 16 {bf^n + a^) {c{n^ + &2'0 = 0. The cross ratio of the four roots of this equation is equal to the cross ratio of the roots of (28) (when Uq = 03 = 0), since the two equations can be shown to have the same invariants.* To prove the theorem when g, g' intersect at P on C4 and lie on the same quadric through C4, consider any third bisecant g" of C4 through P. The cross ratios on g and on g' are each equal to that on g". This completes the proof of the first case. To prove the theorem when the two bisecants do not intersect on Ci, consider a third bisecant connecting a point of intersection on the first with a point of intersection on the second. The cross ratio on each of the given lines is equal to that on the transversal. This cross ratio is called the modulus of the quartic curve. * Burnside and Panton : Theory of Equations, 3d edition, p. 148, Ex. 16. It will be found that I and J have the same values for each equation. 240 CURVES AND SURFACES [Chap. XIII. The projecting cone of C\ from a point on it is a cubic cone. The section of this cone made by a plane not passing through the vertex is a cubic curve. Conversely, any plane cubic curve is the projection of a space quartic curve of the first kind. Consider the cubic curve in the plane x^ = 0. It is no restriction to choose the triangle of reference with the two vertices (1, 0, 0, 0), (0, 1, 0, 0) on the curve. The most general cubic equation in x^, x^, Xi, but lacking the terms x^, x.^, may be written in the form 2 a^x.^Xi + a^x^x^ + 2 h^^Xi + 4 b^x^x^x^ + 2 h^x^x^ -\- c^x^Xi + 2 c^x^x^ + c^x^ = 0. But this is exactly the result of projecting (Art. 175) from the point (0, 0, 1, 0) the curve (25) for the case ciq = 0, that is, when the quartic curve passes through (0, 0, 1, 0). From Theorem III it now follows that the cross ratio of the four tangents to any non-singular cubic curve from a point on it, not counting the tangent at the point, is constant. It was seen that every non-singular quartic lies on four quadric cones whose vertices (Art. 133) are the vertices of the tetrahedron self-polar as to the pencil of quadric surfaces on which the curve lies (Art. 112). Let t, t' be two distinct tangents of C^ which intersect in a point P. The plane tt determined by t, t' touches C^ in the points of contact T, T of t, t', respectively. The following properties will now be proved : (1) The line I = TT' is a generator of a quadric cone on which Ci lies. (2) The plane tt is a tangent plane to this cone along I. (3) The point P lies in the face of the self-polar tetrahedron opposite to the vertex through which I passes. The plane n cuts the pencil of quadric surfaces on which C4 lies in a pencil of conies touching each other at T and T. One conic of this pencil consists of the line I counted twice, hence Z is a generator of a cone of the pencil and tt is its tangent plane. More- over, I is the polar line of P as to the pencil of conies, hence the vertex of the cone and the point P are conjugate points. Thus P lies in that face of the self-polar tetrahedron which is opposite the vertex of the cone. If TT approaches a stationary plane (Art 181), then T, T, P Art. 187] QUARTICS OF THE FIRST KIND 241 approach coincidence, and the tangents t, t' both approach I. This occurs at every point in which C4 intersects the faces of the self- polar tetrahedron. We have thus the following theorem : Theorem V. Tlie points of contact of the sixteen stationary planes (a = 16) of a non-singular qaartic curve of the first kind lie in the faces of the common self-polar tetrahedron. The planes he- longing to the points in each face pass through the opposite vertex. Referred to the self-polar tetrahedron, the equations of the quartic are (Art. 133) The equation of the developable was derived in Art. 179. The section of the developable surface by the plane Xi = is the quartic curve (o,^ = a ■ — a^), Ct24Cli3 Xi X^ -\- CI34I12 "^1 ■^2 1 Ct23(,Cll2^34 l" ^IZ^^Uj'^i X3 = U counted twice. It is a double curve on the developable. It is the locus in the plane 0:4 = of the points of intersection of tan- gents to C4. A similar locus lies in each of the other faces of the self-polar tetrahedron. Since the Cayley-Salmon number x is 16, the entire locus of intersecting tangents to C^ is these four curves. Since the points of intersection of G^ with the faces of the self-polar tetrahedron are the points of contact of the sixteen stationary planes, the coordinates of these points are (±Va^, ±Va3i, ±Vai2, 0), (±Va^, ± Va4i, 0, ±Va|^), (±Va34, 0, ±Va^, ±Vai3), (0, ± Va34, ±Va42, ±Va^). EXERCISES 1. Find the locus of a point P such that the two bisecants to d from P coincide. 2. How many generators of each quadric through d are tangent to the curve ? 3. By the method of Art. 180 find the equations of the stationary planes. 4. Show that any plane containing three points of contact of stationary planes will pass through a fourth. How many distinct planes of this kind are there ? 5. Find the locus of a point P such that the plane projection of d from P will be a quartic curve with one double point and one cusp ; two cusps. 242 CURVES AND SURFACES [Chap. XIII. 188. Rational quartics. The parametric equations of any rational quartic may be written in the form x^ — a^^} + 4 a-i^ + 6 a^J? + 4 0,3^ + a -4, i = 1, 2, 3, 4. The parameters of the points of intersection of the curve with any plane "^u-x^ = are the roots of the equation i^2«,a,o + 4 t^'^u-aa + 6 i^Su.a.-o + 4 f^UiOis + '^u-a^ = 0. Let ti, <2, is, U be the roots of this equation. From the formulas expressing the coefficients in terms of the roots we have at once (,, = 0, ( (24) A 9 — , H-2 — ) / Vi V2 = -) cr are called the Frenet-Serret formulas. The first three follow at once by replacing X, /x, v and Aj, /u,i, vi by their values from (9) and (14). To derive the last three, differentiate the identities V + fJ^2 + vi = 1, AAo + fllX.2 -(- VV2 = with respect to s and substitute for A', jx, v their values from (24) which we have just established. The results are AjA'j -f tHfJ^'i + V2v'2 = 0, AA'2 + /X/x'j + vv'2 = 0. From these equations we obtain, after simplifying by means of (19), A'2 = SAi, /x'2 = 8/xi, v'2 = Svi, Arts. 193, 194] THE OSCULATING SPHERE 251 8 being a factor of proportionality. To determine its value, sub- stitute these values of A'2, yx^, v'l in (23). Since \^ •\- \x.^ + ^i^ = 1, we find 8 = -. The last three equations of (24) are thus established. To find the values A'l, differentiate the identity \ = /^jv — vjyu. (19) and substitute for /x', /, /a',, v'2 their values from (24). By (19) the result reduces to the form A'i = — f- + -?j. The values of /u,'i, v\ are found in the same way. 194. The osculating sphere. The sphere which has contact of the third order with a curve at a point P is called the osculating sphere of the curve at P. To determine the center and radius of the osculating sphere at P^{x, y, z), denote the coordinates of the center by (0:2, y^, ^2) ^"^^ the radius by R. The equation of the sphere is (X - 0^2)^ + ( r- 2/2)2+ (^ _ ^^y ^ ^2, This equation must be satisfied by the coordinates defined by (4) to terras in (As)' inclusive. From these conditions we obtain the following equations {x - x,y + (y - y,y + (z -z,y = r^, {x - x^)x' + {y- y^)y' + (z - z.y = 0, {X - x,)x" + (y - y,)y" +(z- z,)z" + 1 = 0, ^^""^ (x - x.y" +{y- yy" + (z - z,)z"' = 0. By solving the last three equations for x — x^, y — y^,^ — 2^2 ^.nd simplifying by means of (21), (22\ and (24) we find X^ = X-\- p\i — p'crXo, 2/2 = .'/ + pP-i — p'fJ'H-iJ Zo = Z + pvi— p'crv2- (26) If we substitute these values of a-,, ^2' ^2 ^^ the first of equations (25) and simplify, we obtain R' = p^ + (t'p'\ (27) Theorem. TTie condition that n space curve lies on a sphere is p + (t{(t'p' + a-p") = 0. If a given curve lies on a sphere, the sphere is the osculating sphere at all points of the curve so that x^, y^, z^ and R are con- 252 DIFFERENTIAL GEOMETRY [Chap. XIV. stants. Conversely, if these quantities are constants, the curve lies on a sphere. To determine the condition that the coordinates of the center are constant, differentiate equations (26) and simplify by means of (24). Since Aj, fj^, v^ are not all zero, the condition is p + o-(cr'/3' + up") = 0. By differentiating (27) we see that R is also constant if this equa- tion is satisfied. This proves the proposition. 195. Minimal curves. We have thus far excluded from discus- sion those curves (Art. 189) x=f,{u), y=f^{u), z=f,{u), for which Such curves we called minimal curves. A few of their properties will now be derived. From (28) we may write dx , . di/ _ . dz du dii du J .(dx .dy\ dz and t{ — -— 1—^]=—- \du dtij du in terms of a parameter t. From these equations we deduce dx dy dz du _ du _ du 2 2 If we denote the value of these fractions by <^(w), solve for — , — , — and integrate, assuming that <^(m) is integrable, we da da du find that the equations of a minimal curve may be written in the form x = - C{1- t'^)(fi{u)du, y= f ("(1 -f t~)cf>{u)du, z = Ct(fi{u)du^ 2J 2J J ^29) in which i is a constant or a function of n. If t is constant, the locus (29) is a line. For, let k be defined by fc = | cf>(u)du. Art. 195] MINIMAL CURVES 253 lu terms of k we obtain X = -^— k + a?!, y = ^(1 + t'^)k + y^, z = tk + z^, wherein x-^, ?/,, z^^ are constants of integration. The locus of the point {x, y, z) is the minimal line through the point {x^, y^, z^ x—Xi y-yx z-Zx \-P i (l + O t 2 2 The equation of the locus of the minimal lines through any point (Xi, ?/,, Zi) is found by squaring the terms of these equations and adding numerators and denominators, respectively, to be the cone {X.-X,Y+ y-y,Y^{z-Z,Y=^, having its vertex at (xi, ?/i, z^ and passing through the absolute. This is identical with the equation of the point sphere (Art. 48). If t is not constant, but a function of ?t, we may take t as the parameter. Let u = \\i{t.), and let ^{iC)(ln ^= <^{^{tj)\\i^ {pjdt be re- placed by F{t)dt. Equations (29) have the form x = ^^{l-t')F{t)dt, y='^j{l + t')F{t)dt, z=JtF{t)dL (30) Jjet f{t) be defined by-^ = i^(^). By integrating equations (30) by parts we have y=^s^^m-u'J^+im+y,, (31) dt^ dt ^i> yi> ^\ being constants. The equations of any non-rectilinear minimal curve may be expressed in this form. EXERCISES 1. The curve X = a cos 0, y — a sin cj), z = a(j> is called a circular helix. Find the parametric equations of the curve in terms of the length of arc. 254 DIFFERENTIAL GEOMETRY [Chap. XIV. 2. At an arbitrary point of the helix of Ex. 1 find the direction cosines of the tangent, principal normal, and binormal. Also find the values of p and (T. 3. Find the radius of the osculating sphere at an arbitrary point of the space cubic x = t , y = t^, z — t^. 4. Show that the equations of a curve, referred to the moving trihedral of a point P on it, may be written in the form + z=- — + o p(r 2p \ds)6p^ s being the length of arc from P. 5. Discuss the equations (31) of a minimum curve in each of the follow- ing cases: C'^) /(O ^ quadratic function of t. (6) f(t) a cubic function of t. II. Analytic Surfaces 196. Parametric equations of a surface. The locus of a point (x, y, z) whose coordinates are analytic real functions of two in- dependent real variables u, v x=f^{u,v), y=f2{u,v), z=fs{u,v), (32) such that not every determinant of order two in the matrix M ^ ^3 du du du df\ df, % do dv dv (33) is identically zero, is called an analytic surface. The locus de- fined by those values of ri, v for which the matrix (33) is of rank less than two is called the Jacobian of the surface. Points on the Jacobian will be excluded in the following discussion. The reason for the restriction (33) is illustrated by the follow- ing example. Example. Consider the locus X = u + V, y =(u + vy, z =(u + vy. For any given value t, any pair of values ?«, v which satisfy the equation u -{- V = t define the point (t, t'-, «'). The locus of the equations is a space cubic curve. In this example the matrix (33) is of rank one. Arts. 196-198] TANGENT PLANE. NORMAL LINE 255 The necessary and sufficient condition that u, v enter /i,/2,/3 in such a way that x, y, z can be expressed as functions of one vari- able is that the matrix is of rank less than two. 197. Systems of curves on a surface. If in (32) u is given a constant value, the resulting equations define a curve on the sur- face. If u is given different values, the corresponding curve describes a system of curves on the surface. Similarly, we may determine a system of curves v = const. The two systems of curves n = const., v = const, are called the parametric curves for the given equations of the surface ; the variables u, v are called the curvilinear coordinates on the surface. Any equation of the form {u, v) = c (34) determines, for a given value of c, a curve on the surface. The parametric equations of the curve may be obtained by solving (34) for one of the variables and substituting its value in terms of the other in (32). If we now give to c different values, equation (34) determines a system of curves on the surface. If (ii, v) = c, \p{u, V) = c' are two distinct systems of curves on the surface, such that dcf) dip dcf> 5i/' p> du dv dv du by putting <^(?*, v) = u', \p (n, v) = v' and solving for u, v we may express x, y, z in terms of u', v'. This process is called the trans- formation of curvilinear coordinates. 198. Tangent plane. Normal line. The tangent plane to a sur- face at a point P on it is the plane determined by the tangents at P to the curves on the surface through P. The equations of the tangent lines to the curves u = const, and V = const, at P = (x, y, z) = {u, v) are (Art. 190) X-x Y-y Z-z dx dv dz_ dv X-x Y-y Z-z dx du dji du dz du 256 DIFFERENTIAL GEOMETRY [Chap. XIV. The plane of these two lines is X-x T-y Z-z dx dy dz dti du du dx dy dz dv dv dv = 0. (35) Let V = («) be the equation of any other curve on the surface through (u, v). The equations of its tangent lines are X-x Y-y Z-z X dx I dx dcj) du dv du dy dy d4> du dv du dz dz d(l> du dv du This line lies in the plane (35) independently of the form of (u), hence (35) is the equation of the tangent plane. The normal is the line perpendicular to the tangent plane at the point of tangency. Its equations are Y-v Z-z X — X dy dz dz dy dz dx _ dx dz du dv du dv du do du dv dx dy du dv dy dx du dv We shall denote the direction cosines of the normal by X, fx., v. Their values are r _ 1 fdy dz _ dz dy'^ D\dudv dudVj wherein '3?/ dz D' = du dv dz dy^ du dv _ _ 1 /dz dx dx dz D\du dv dudv _ _ 1 fdx dy _ dy dx D \du dv du dv dz dx dx dz -^^ M-^~- dudv dudv 4- dx dy du dv dy dx\^ du dv) (36) (37) If D — 0, the tangent plane (35) is isotropic (Art. 152), and the formula for determining the direction cosines of the normal fails. We shall limit our discussion to the case in which D^O. The equation of the tangent plane may be written in the form X{X-x) + ]J.{Y-y)^v{Z-z)=0. A.RT. 199] DIFFERENTIAL OF ARC 257 199. Differential of arc. Let =-l, 6 7i-3» + tr + 3 = 0, 6M-2» + ir+l=0. -1 (2, -1, -3); (— , 0, — 0, 0, 7. — — , — . V3' VS' V3 9. 4(m'^ + v2 + w-)= 1. A sphere. ^» 272 ANSWERS Page 35. Art. 34 (c)(^,o,o). (/)(«. ".n)- 2. (-10,15,-2,0). 4. 7x + 9j/ + 542-59« = 0. Page 37. Art. 35 1. (a) Parallel bundle. Rank 3. (b) Rank 4. (c) Rank 4. (d) Parallel bundle. Rank 3. -1 c b 3. The determinant -69 5 4. a — 1 c 6 a - 1 - 19 is of rank 3 ; of rank 2 ; of rank 1. \/3867 \/3867 \/3867 Page 43. Art. 40 1, x2-3 2/2 + 2/2-4x-8y + 42 + 4 = 0. ^21 V6 VU ^ix' y' ^ z' V21 V6 vTi' z ^ ^x' y' Zz' ~ V21 V6 Vli 4. New equation is x^ — 2 j/^ + 6 2^ = 49. Translation is x = x' + 8, ?/ = y' — 1, 2 = 2' + 2. 6. 3 x2 + 6 y2 + 18 ^2 = 12. Page 45. Art. 41 - /28±6i -6±8i 5T24i\ ^ /17-4t ) - (^- H^' -*'•)■ 13 ' 13 ' 13 ; " V 6 6. (13 + 9i)x+(3 + 4i)2/ + (16-7i> = 23 + 64 i. 6. (1 ± iV-T, 0, 0). Page 46. Art. 42 1. x2 + j/2 = 4 z^. 3. x2 + y'^ + 2^ - 7 X + .V + 30 = 0. 4. 8(x2 + y2 ^. ^2) _ 68 X + 48 2/ - 66 2 + 275 = 0. ANSWERS 273 5. « = 5, (x-3)2+(2/-7)2 + (;j-l)2 = 9. 6. 2 a; - 14 y - 2 3 + 1 = 0, 4 X - 18 + 33 = 0. 7. (- 4, ± 4 i, 2). Page 49. Art. 46 8. Center at (0,0, 4); radius 6. 4. x"^ - y"^ = 1. 6. r, — • Page 51. Art. 47 1. a;2 + 2/2 + ^2 = 26. 2. 9(y2 + z^) = (15 - 2 a;)*. Vertex (Y, 0, 0) ; z = 0, 9(2/2 + ^2) = 225 ; 4(x2 + z^) = 9(5 - y)^. 4. 2/2 + 22 = (,2 ; y = a. 6. (a)?! + l^4-^=l; ?-% 1^ + ^' = 1. ^^499 ' 494 ^ ' a2 62 fc2 ' a2 1-1 a2 (c) 2/2 + 22 =: 8 x ; 2/* = 64(x2 + z^). (d) (x2 + 2/2 + ^2 _ 5)2 = 16 - 4 x2 ; x2 + (y - 1)2 + 22 = 4. (e) 2/2 + ■^2 = sin2 x ; ?/ = sin V'x2 4- 22. (/) y2 + 22 = e2' ; 2/ = e-^x^'. Page 54. Art. 49 1. (o) x2 + 2/2 + z2 := ^2. (6) x2 + 2/2 + 22 + 2 x - 8 y - 4 2 = 16. (c) x2 + y2 + 22 - 4 x - 2 y - 10 2 + 14 = 0. (a) Center f- I, -1, --\ radius ^^. ^ ' V 2 ' 2/ 2 (6) Center (-1, - 2, 3) ; radius 0. (c) Center fl, 1, ^V radius ^' ^^^ ^ ^ U 2 4 r lius 2 4 (d) Center ( _ /, 0, O^j; radius 8. (-4±3i, 2±6i, 5, 0). . / 2±2iV2 1 T 2 t V2 _2±\/2i\ I 3 ' 3 ' 3 } Page 56. Art. 52 1. y = 1. 2. Arc cos . The spheres have no real point in common. 8. x2 + y2 + 22 _ 2 X - 6 y - 6 2 + 10 = and x2 + y2 + 22 - 2 x - 6 y -62-6=0, 274 ANSWERS 4. x^ + y^ + z^ + x-2y-Sz = 0. 5. 2 X — o y + z + 5 = 0. The sphere is composite. 6. 10(x2 + y2 + 2-2) ^-jix -6Sy -8[)z - 185 = 0. 7. 4232(a;'-! + y-^ + z^) - 276 a; + 27G y + 1032 ^ + 225 = 0. Page 69. Art. 59 1. Center (1, 1, -2); semi-axes^, — , — • ^ ' ' ^' 2 3 2 2. Sphere ; center [ 2, '-, — 5j; radii V'2U5 2 3. y = 0, 2 x^ = 3 z^ + 5 z + 7 . Rotated about the Z-axis. 4. X = 1, y = z ; x = 1, y =— z; x — — 1, y = z; x = — I, y =— z. 5. (a) Ellipsoid, (ft) Hyperboloid of two sheets, (c) Hyperboloid of one sheet. (cZ) Hyperboloid of revolution of one sheet, (e) Ellipsoid. (/) Imaginary ellipsoid. Page 73. Art. 64 1. Hyperboloid of one sheet. 2. Imaginary cylinder. 3. Elliptic paraboloid. 4. Real cone. 5. Hyperboloid of two sheets. 6. Hyperbolic paraboloid. 7. — + ^ "*" ^ = 1. 8. (a) (1 - r-2)a;2 + y^ + z-i - 2 ax + d^ = 0. (b) (1 - r2)x2 + (1 - r2)j/2 + z^-2ax + a^ = 0. Page 76. Art. 66 ^ / -8±n/I09 5 T V109 - 17 ± VlO!) \ g fO 0^ ' \ 3 ' 3 ' (J j • V . 7 > 3. (-1,1, -A)- 4. (-1, 2,-1). 5.(1,1,0). 6. Vertex (0, -1,0). 7. Plane of centers 2(x - y + «)- 1 = 0. 8. Non-central. Page 89. Art. 75 1. Hyperboloid of two .sheets. Center (0, 0, 0). Direction cosines of axes 2 _ 2 1.2 1 2.1 2 2 fl r2 -U 9 »;2 .^i 4_ 2 — 2. Hyperboloid of one sheet. Center (1, |, — f). Direction cosines of axes 2 + 2-\/5 VS-l 5+\/5 . 2-2V5 -\/5 -1 2V15 + 4V6 2Vl5+4v'5 2Vl5 + 4\/5 2Vl5-4\/5 2Vl5-4V5 5_V5 . -3 4 2 5 + V5^2 , •''> - v^^' ^2 _ 3 ^2 = 10 2V15-4V5' ^29 V29 v/29 22 3 cosines ANSWERS 275 3. Hyperboloid of one sheet. Center ( — , -pr- , -pr- 1 • Direction cosines of axes, 7n'^'^'^'^';ll';;|5' ^'^- 11-^ + 4 2/^-^^ = A. 2 1 4. Real cone. Vertex, (1, 0, 0). Direction cosines of axes, -7= > — 7=» -12 11-25 V5 V5 Ve Ve a/6 V30 V30 V30 ^ 6. Elliptic paraboloid. New origin, ( - , — , — 1 . Direction ^ 2 1 -1.1 -1 l.n-1-1 of axes, — F ) -7= > —7= ) -7= ) —7= > —7= > "> -7= > —7= • 6 x2 + 3 2/2 = Ve Ve Ve V3 V3 V3 V2 V2 -^ y (4 - 1 22 \ ^ ' -7- > 57 1 . Direction 1-5 3 1 2 3-3^1 cosines oi axes, / — > ~7= » , — > , — > , — » , — : > , — > "> , — • V35 V35 V35 Vl4 Vl4 Vl4 VlO VlO 2 x2 - 5 2/2 + 7 z2 + — =0. 35 7. Parabolic cylinder. New origin on2x + 22/-2z-l=0. 2x - y 2—1222 2 — 1 + 2 z - 2 = 0. Direction cosines of axes, - > -r- > - j ^ > ^ > ^ J — - , cy c} *j O *J O fj (j O 3 > - • 5 2/2 + 6 X = 0. 8. Two real intersecting planes. Line of vertices, x + z - 1 = 0, X - 2/ + 2 = 0. Direction cosines of axes, —7^1 ^^ > —7=^; 0, ~7^ > —p^', _1 _1 1 021, 'V6 V6 V6' ' \/2 V2' ZJ:^ ij _L_. 3x^ l_y2 V3' Vs' ^3' 2 ~ 2 " 9. Hyperbolic paraboloid. New origin, (1, 0, - 1). Direction cosines of axes, -^ , _?^ , _L ; J_ , III , J_ ; ^ ,0, — . fi j-2 _ q >/2 - 2^ z. ^6' Ve' Ve' V3' V3' V3 V2' V2 ^ "^ ^ ^ - 10. Elliptic paraboloid. New origin, (0, - 1, 1). Direction cosines of 1 1 1.1-1^1 1-2 axes, -7= ' -7= , -7= , —^ ,—=., U ;-—,—=,-—= . 3 x2 + 4 2/^ = V3 V3 V3 V2 V2 V6 Ve V6 -r « sVe z. 11. Hyperbolic cylinder. Line of centers 3x-72/ + 7z + l=0, or 2x-22/ + 4z + l=0. 7X + 52/ + 112 + 5 = 0. X + 32/+ 2 + 1=0. 1—12 1 3 1 — 7 Direction cosines of axes, —7= > -7^ . — ;=^ ! — f= » —7^:^ . —7=^ J -7= , 14 ' Ve Ve Ve \/n Vu Vn Vee Vl' V^- 24x2-112/2-5 = 0. 276 ANSWERS 12. Hyperbolic paraboloid. Origin ( — , — , ^— )• Direction cosines \72 72 i2 I f 1 V7-2 V7-3 -1 of axes — — , • — - , — - ; — — , V28-10\/7 V28- lOvT V28 - 10 V? V28 + 10 V7 V7 + 2 V7 + 3 1 -1 1 V28+IOVT V28+IOV7 ^ >/3 V3 (_ 1 + V7)a;'^ -(1 + ^7)2/2 = 4 V3^. 13. Hyperboloid of one sheet. Center (^, ^- , — V Direction cosines of axes .21, - .65, .69; .91, .41, .10 ; .36, - .64, - .68. 3.09 a;2 + 1.59 y2 _ 3.67 ^2 ^ as, (2 3 — 26\ — , - , ) . Direction cosines 15 5 15 / of axes -.77, .56, .28; .14, -.31, .94; .63, .76, .13. 6.17 x2 + . 712,2 -6.8822=^^9. 15. Ellipsoid. Center (0, 1, 1). Direction cosines of axes 2 1+^^ .0= -^ . -^+^L ,0:0.o,i. V'l0+2V5 V1O + 2V5 V1O-2V5 V1O-2V6 — x^ -\ y^ + 2 z^ — i. 2 2 (—9—7 \ , , — 6 1. Direction cosines of axes .83, - .33, - .44 ; .26, .95, - .22 ; .49, .07, .87. 4.20 a;2 + .59 y^ + .20 z^ = ^. 17. I, 5±2i. 18. ^=^. ^' 28 Page 92. Art. 78 1. X+IO2/-32 +22 = 0, ^^^zJ: = 1+l = ?^zl. " 1 10-3 Page 96. Art. 80 1. y -i, z = kx,; x--n, z = -r)y. Page 97. Art. 81 1. V5, :^. 2. a; + z + 1 = 0, 2/ + 2 - 1 = 0. 3. a, 6, c. ANSWERS 277 Page 103. Art. 83 1. x + y — z = dandx- y + 2z = p. 2. x — {2 ± y/6)y = d. 3. -^. 4. y + S±\/2{z-2)-0. 5. a = 6, h=0. V3 6. ax + fir.2; + Z = 0, ay + fz + m = 0. 7. 2 g'x + 2/y 4-(c - a)^ = d. 8. (6 -i-)^- +(«-^')-S2 -2/4^5 = 0, (c - i-)-S2 + C?> - A-) C2 - 2 /J5C' = 0, (a - A;) C2 + (c - A)^^ _ 2 ^C^ = 0, A" being a root of the discriminating cubic. 9. (-1,0, -3). ^. Page 108. Art. 87 8. ki = cons, i = 1, 2, 3, 4. For parametric equations, substitute this value of*,-, inEqs. (27). Page 111. Art. 89 1. (-6x4-61/- 12 «, x + 2y-2z + t, Qx + Sy + iz + it, —x + Sy -z-2t), (-12, 1, 4, -2), (12, -2, -20, 1), (18, -6, -16, 1), (12,-3, -28,1), (3,2,1,2). 2. (-373, 179, 92, 283), (-500, 181, 145, 344), (-153, 61, 38, 107), (-37x1-96x2-9x3 + 156x4, 11 Xi + 24 X2 - 3 X3 - 60 X4, 8x1 + 48x2 - 6 xa - 36 X4, 31 Xi + 60 X2 + 3 X3 - 108 X4). 3. 15 X + 5 J/ + 11 + 16 e = 0. 4. 197 xi + 468 X2 + 57 X3 - 792 X4 = 0. 5. 6x'^-15y^ + 2 z^ + 3 yz- zx- 3 xy + n xt + 9 yt- 6 zt + 10 t^ = 0. 6. (22 X - 22 2/ + 44 «, 12 x + 24 j/ - 24 + 12 «, 33 x + 33 y + 22 2; + 22 1, 66 x-198 2/ + 66^+132 0, (22, 6, 11, 66), (22, 12, 55, 33), (33, 36, 44, 33), (22, 18, 77, 33), (22, -48, -11, 264), (-97121, 36427, 22804, 66851), (296167, - 115487, - 64346, - 205981), (- 185625, 71181, 42570, 128403), (814 xi - 6912 X2 - 297 X3 - 61776 X4, - 242 xi + 1728 X2 - 99 X3 + 23760 X4, - 176 xi + 3456 X2 - 198 X3 + 14256 X4, -682 Xi+4320 X2+99 X3 + 42768 X4). Page 113. Art. 92 1. Vertices : ?(i = 0, (1, 0, 0, 0) ; M2 = 0, (0, 1, 0, 0) ; M3 = 0, (0, 0, 1, 0); Ui = 0, (0, 0, 0, 1). Faces : Xi = 0, (1, 0, 0, 0); X2 = 0, (0, 1, 0, 0); X3 = 0, (0, 0, 1, 0); X4 = 0, (0, 0, 0, 1). 2. Xi = 0, X2 = ; M3 = 0, M4 = 0. Xi = 0, X3 = ; ?<2 = 0, M4 = 0. iCi = 0, X4 r= ; U2 — 0, W3 = 0. 278 ANSWERS X2 = 0, Xa = ; Ml = 0, Hi = 0. X2 = 0, a;4 = ; Ml = 0, u^ = 0. Xs = 0, a;4 = ; Ml = 0, M2 = 0. 3. Ill + ^'2 + ^(3 + W4 = 0, 3 Ml — 5 M2 + 7 Ms — W4 = 0, — Ml + 6 2t2 — 4 M3 + 2 M4 = 0, 7 Ml + 2 M2 + 4 M3 + 6 M4 = 0. 4. (1, 1, 1, 1), (7, -1,-3, 1), (1, 9, - 6, 2). 5. Ml - M2 = 0, 7 Ms + U4 = 0. 6. (- 9, 1, 1, 0). Page 117. Art. 95 2. pxi = li + 2 12 + 10 h, px2 = lh + 5h- h, (176, - 175, 40, 363). pX3=—h + 4:l2-S h, px4 = 3li + h— ^h. 3. pui -—bh + lh + Gh, pu2 = 3h-5h-^h, (21,32,1,5). pMs = 4 Zi + 3 ^2 - 3 Zs, J9M4 = ?1 + 2 Z2 + h- 5. ^Mi = Zi + 7 I2, pu2 = — 5 Zi + 2 ^2, puz = 3 Zi — Z2, pUi =— h — h. 6. px\ = Zi + 3 Z2, pa;2 = 2 Zi — 2 Z2, px^ = — 3 Zi + 5 Z2, pXi=—h-2 h. Zi(Mi + 2 M2 — 3 Ms — U4) + Z2(3 Ml — 2 ?f2 + 6 7<3 — 2 M4) = 0. Page 120. Art. 97 3. («11 + «12 + «13 + «14, "21 + «22 + «2S + «24, Csi + «32 + "33 + a34, a41 + ^42 + «43 + «44). (/3ll + )321 + ftl + i34l, j3l2 + 1822 + /332 + /342, /3l3 + ^23 + ft3 + ^43, /3l4 + ^24 + ^4 + /344)- 4. xi = A;iXi', X2 = kiXi', Xs = k^x^', X4 = k4X4'. Page 122. Art. 100 1. (a) xi = Xi' — 3:4', a-2 = X2' — X4', Xs = xs' — X4', X4=— X4'. D(p) = (l + p)(l — py. Invariant points are (1, 1, 1, 2) and all the points of X4 = 0. (6)Xi = X2', X2=Xi', X3 = X4', X4 = Xs'. D(p) = (p^-iy. Every point on each of the lines Xi + X2 = 0, Xs + X4 = ; xi — X2 = 0, X3 — X4 = 0. (C) Xi=X3', X2 = Xl', X3 = X2', X4 = X4'. D{p) = {I — p^ip"^ + p + 1). The points (1, w, w^, 0), (1, u"^, w, 0), w^ _ j^ and every point of the line Xl = X2 = Xs. (d) Xl = — X4', X2 = Xl' — X4', Xs = X2' — X4', X4 = Xs' — X4'. D(p)=p*+p^ + p''+p + i. (d, i + 0, -e»(i + e), -6^), 6^ = 1, e^i. 3. "'* = cons. i,k = l, 2, 3, 4. ANSWERS 279 4. In case X3 = :cs\ the point (0, 0, 0, 1) and all the points of the plane 3-4=0. In case Xs = — xz', every point of each of the lines xi = 0, a-2 = ; 3-3 = 0, Xi = 0. 6. (1, 1, 1, 1), (1, - 1, 1, - 1), (1, i, -1, _ I), (1, - 1, - 1, i). 7. All the points in the plane at infinity. 9. ^'^. Page 125. Art. 102 1. X3 — aXi = 0. 3. (2 ± 2 iy/l2f, - 3 T iVl2f, 8, 4). Page 131. Art. 106 2. A=- 1. abed' 6. bcui^ + cath'^ 4- 2 abuzUi = 0. 7. A = ^. *(?«) = M2^ — U1U2 + U1U3 — U2U3 — iiiiu + M2W4 — 2 M3W4 = 0. 8. ^(x)=0. 9. Xi = 0, X3 — 3:4 = and Xi = 0, Xs + X4 = 0. 10. an =0. 1= 1, 2, 3, 4. 12. a,fc2 = a^i . a^j, /, k - 1, 2, 3, 4. 13. A conic ; two distinct points ; two coincident points. Page 134. Art. Ill 1. xi + a;2 + xs + a;4 = 0. 2. X3 = 0, X4 = 0. 4. (2 Xi + X2 - 3 X3 — X4)2 + 4(XiX2 - X3X4) = 0. 7. XiSWiX,- — XiZvXi = 0. Three. 8. 013X1X3 + ai4XiX4 + 023X2X3 + a24X.X4 = 0. Page 141. Art. 118 3. aiixi^ + 022^^2- + «33^"3'^ + 2 ai2XiX2 + 2 023X2X3 + 2 013X1X3 = 0. 4. oiiXi^ + 022X2* + 2 012X1X2 + 2 023a-2a;3 + 2 013X1X3 = 0. Xi =0, X2 = 0. Page 143. Art. 120 1. 8 xi2 + xo2-5 X32-2 X42 + 9 X1X2 + 5 X1X3 + 18 X1X4 + 13 X2X4 -7 X3X4=0. 2. i(xi + X3) - (X2 + X4) = 0, i(xi + X4) - (X2 - X3) = 0, and i(xi + X3) + (X2 + X4) = 0, iXxi + X4) + (a-2 - X3) = 0. 5. Equations of faces 20^4X4 =0, i = 1, 2, 3, 4. 280 ANSWERS Page 146. Art. 122 (0,0,2, -3). 2. k = ±4. 72 Mi^ + 36 U2^ + 23 H3- - 54 Mi?t2 = ; 2 ?t3 - 3 «4 = 0. (a) A quartic curve with double point at 0. (b) A cubic curve passing through 0. (c) A plane section of K, not passing through O. 3. Page 150. Art. 126 2 \<^ik — ^^ifc an — Xbii (a) X - 1, X2 X, [1(21)]. (6) M [4]. «il «.3 aA2 aw "ml "mS aj2 «!3 (C) X2, X2 ((Z) X3, X- Art. 131 Page 156. [111]. Four distinct lines. [21]. Two distinct and two coincident lines. [1(11)]. Two pairs of coincident lines. [3]. Three coincident lines and one distinct line. [(21)]. Four coincident lines. [(111)]. A quadric cone. {3}. A plane and a line. (a) X-1, X-J, X-^ (6) X + ^, X - i, X - i. (c) (X-l)2, X-^. (d) (X-l)3. 4. (a) Four distinct lines. (6) Two pairs of coincident lines. (c) Two distinct and two coincident lines. (d) Three coincident lines and one distinct line. [111]. [1(11)]- [21]. [3]. («) a:i- + ^ + ^ - X (xi^ + 3:2^ + X32) :0. (b) xr 41 + '^'^^' + '"'«") - X(a;i2 + X22 + 3:3^) = 0. [(22)]. [31]. (c) ^ + 2 X2X3 + X2^ - X(xi2 + 2 X2X3) = 0. (d) 2 X1X2 + X32 + 2 X2X3 - X(2 X1X2 + X32) = 0. Page 164. Art. 133 2. [11(11)]. V Xi - X3 X1 ± VXa - X2X2 = 0, Xi2 + X22 + X32 + X42 = 0. [1(21)]. V\2 - XiXi ± X3 = 0, Xi2 + 2 a:2.r3 + X42 = 0. [1(111)]. Xi=0, X22 + X32 + X4'^ = 0. ANSWERS 281 [22]. n =0, a-2 = 0. [2(11)]. Xi + ix3 = 0, 3:4=0 ; xi— 1X2=0, X4=0; X3=0, 3ri2+a;2- + 3-42=0. [(11) (11)]. Xi + 1X2 = 0, Xs + 1X4 = ; Xi + 1X2 = 0, X3 — 1x4 = ; Xi — 1x2 = 0, X3 + JX4 = ; xi — ix2 = 0, Xg — 1X4 = 0. [4]. X2 = 0, X4 = 0. [(22)]. X2 = 0, X3 = ; xi = 0, X4 = 0; X3 = 0, X4 = 0; the last one counted twice. [(31)]. xi + JX4 = 0, X3 = 0; xi— (>4=0, X3=0 ; X4=0, Xi'- + 2x2X3=0. [(211)]. Xi = 0, .r3 = ; Xi = 0, X4 = 0. [{3}1] . X2 = X3 = ; xi - nxi = 0, 2 x^^Xi + X3- = 0. 3. (a) (X - i)2, (\ - ^y. xi - X2 = 0, xs + 2 X4 = ; Xi - X2 - V3(X3 + 2 X4)= 0, V^(xi + X2) + 2 X3 + Xi = ; Xl — X2 + V3(X3 + 2X4) = 0. V o(Xi + X2)— 2 X3 — X4 = 0. X1X3 + X2X4 + 2 X3X4 — X(2 X1X3 + 2 X2X4) = 0. (b) X — 1, X — 1, (X + 1)2. xi + X3 = 0, xi — X2 + iXi = ; Xi + X3 = 0, .71 — X2 — 1x4 = ; a:i + X3 + 4 X2 = 0, x'-4 + (xi — X2)2 — 24 X2'- — 16 X2X3 = 0. xi^ + X2^ + X42 — 2 X3X4 — X(xi2 + X22 + X42 + 2 X3X4) = 0. (c) X + 3, X - 1, X - 1, X - 1. xi + 2 X3 + X4 = 0, 5 xi2 - xr + 6 Xs* + 4 X1X3 + 2 X1X2 = 0. - 3 xi^ + xr + X32 + xr — X(xi2 + X2- + X32 + x^^) - 0. (d) X — 1, X — 1, X-. xi + X2 = 0, Xi + ,T3 + X4 = ; xi + X2 = 0, xi — X3 — X4 = ; X2 + X4 = 0, 3 x\- + X22 — X32 + 4 X1X2 + 2 X2X3 = 0. Xi'- + X22 + X4'- — X(xi2 + X22 + X42 + 2 X3X4) = 0. 4. [1(111)]. [2(11)]. Page 167. Art. 135 1. (X3 + 2 X2 + 4 X + l)wi2 + (3 X2 + 7 X - 10)K2=' + (X^' + 2 X2 + 9 X +Q)ui^ + (X2- l)(\_l)M42_6(X2_\)„i„2+ 12(X^+ 1)2hiH3 + Q{\'^ - l)uxU4 + 4(X2- x)?<2?<3 + 2 x(x - i)2m2M4 - 4(x2 - i') mm = 0. 2. 2 X22 — 3 X42 + 6 X1X4 + 2 X2X4 — 4 X3X4 = 0, twice. 2 xi2 + 2 X32 + 3 X42 — 6 X1X4 + 2 X2X4 + 4 X3X4 = 0. 3. 2(miM2 + M3M4)^^ + ("2^ — 6 a?«i«2 — 6 auzUi)X- + (6 a?uiU2 + 6 a?UiU4 — 2 au2^)\ + a2M22 — 2 ahtiti2 — 2 ahisUi = 0, Page 174. Art. 142 3. (a) [211]. (5) [22]. (c) [31]. 4. All the quadrics of the bundle toucli a fixed line at a fixed point. 5. The quadrics touch x^ = 0, X2 — 2 X4 = at (0, 2, 0, 1), and X3 = 0, X2 + 2x4 = at (0, 2, 0, —1); they have four basis points in the plane X2 — X3 = 0, at the points 282 ANSWERS (2, 2, 2, V3), (2, 2, 2, -V>]), (-2, 2, 2, \/3), (-2, 2, 2, -VS). y'l = -1 yrii4, y'o = -^ 2/12/4 (^ 2/3 - ^2), 2/'3 = 4 ?/i(/32/4, /y'4 = 2/1 (- 2/;!'' + '^ 2/22/3 — 2/2^). 7. Xi(Ma;)(Mx')[(«"'x)(M"x")-(?t"x)(?("'x')] + \2{u"x){u"x')\_{u'x){ux')-{nx){u'x')^ + \zl{u'x){u"'x){ux'){u'ix')-{ux){%i"x){u'x'){ti"'x')'\^0. (For notation, see Art. 119.) Page 180. Art. 146 2. yvyiy^yi = 0. 3. The plane counted twice is a quadric of the web. 4. (x^ + y- + Z')fi = 0. 6. Any point on x^ + y'^ -{- z^ = is conjugate to any point on « = 0. Page 187. Art. 150 1. (SMix'i)^ = 0. 2. 8. 4. 5. 5. Xi(?t'-^ — W-) + \2(i)- — r«-) + Xs^n; + \iVio + XjMj?j = 0. 6. [1111]. (a2 _ c2)m-; + (/)•■! _ c-^)v2 = s'^ ; (a^ — 6-)?|2 _ (ft2 _ c2)^2 — s2 ; (a^ - 62) m2 + (a2 _ c2)i,2 ^ s2 = ; y2 -f ■j;2 ^ ^(;2 _ 0. Page 196. Art. 155 1. kiP = kiP = JC3P = kiP. Page 205. Art. 160 2. (x2 + 2/2 + ^2)2 = ^^^ + g + g^f2. Eigllt. 3. 2 0(x2 + y- + Z-) = (ax- + by-)t. Eight. Fifteen. 7. xi = .r2'(.)-i' + .r4')(-^2' + :*-4')' ^2 = a;2'a-4'(xi' + X4'), X3 = X2'Xi'{X2' + Xi'), X4 — Xz'Xi'{Xi' + ^i')- (1, 0, 0, 0), (0,1,0,0) ; the line xi = 0, a-o = 0. Touch at (0, 0, 0, 1). 8. Xi = Xi'Xs'iXl'Xi' + Xi'Xs' + Xs'Xl'), X2 = X\'X2'{Xi'X2' + X2'X3' + Xz'Xx'), Xz = X2'X3'{Xi'X2' + X^'Xz' + X^'Xl'), X4 - Xi'xJ Xz'Xi' . (1, 0, 0, 0), (0, 1, 0, 0), (1, 0, 0, 0). Four coincident at (0, 0, 0, 1). Page 207. Art. 162 1. A„/(a;) = 4 2/i(«40ooa;i-' + 3 02200X112- + 3 a202oa;i«3^ + 3 a20023'iX42) + 4 2/2(3 ffl2200'>'l'^-'*"2 + «0400a'2^ + 3 ao22oa;2*'3" + 3 ao202X2X42) + 4 2/3(3 a2020''l23-3 + 3 nQ22QXrXZ + n'0040^3' + 3 rt00225C3^4^) + 4 J/4 (3 a2002.''l''.r4 + 3 ao202a'2'^3'4 + 3 ao022X32.r4 + ffl0O04a;4^) • ANSWERS 283 ^U^fip^) = 12 ?/l-(«4000-'>'l- + «2200^'2- + «2020a'3"^ + «2002a^4-) + 12 y2'-^(ffl2200a;i'-^ + (?0400-^2- + a0220a^3- + «0202:''4-) + 12 2/3-(rt202oa'i- + no22QXr' + «oo40»"3- + «oo22a:4-) + 12 2/4-(«2002A"l- + a0202a-2'-^ + «0022X3^ + «0004-^"4"'^) + 48 tjiyiarmXiXi + 48 yxijsaim'XiXi + 48 ?/i?/4«2002a^ia^4 + 48 y^yiaxt'i-ioXiXz + 48 yiyiUma-i^iXi + 48 (/3.V4«oo22a:3^4- ^uV(^) = 24(?/i3a4oooXi + y2^«040oa:^2 + 2/3^«oo4oa;3 + 2/4''rtooo4a;4 + 3 ?/i-*/2a220o3'2 + 3 yiy2-n22ooXi + 3 yi'-2/3«202oa'3 + 3 ?/l?/3-a2020-*'l + 3 .'/l-l/4«2002-T4 + 3 2/l2/4-«2002a;i + 3 ?/2-'2/3ao22oa;3 + 3 y2yrao22oX2 + 3 !/22/y4ao202.*'4 + 3 y22/4-«0202a^2 + 3 2/3-J/4«oo22a-4 + 3 2/32/4-aoo22a;3) • A„V(a;) = 24/(2/). Page 209. Art. 164 1. (1, 0, 0, 1), (1, 0, 0, - 1), (4, 0, 0, - 1). 2 4 2359 + 1 31 ■v/17 2359- 131 Vl7 ' 376 ' 376 Page 211. Art. 167 1. (tOOOn = 0) aiWn-lXl + flolO n-lX2 4" «n01 h-1-^3 = 0- 2. osooon = 0, aioon-1 = 0) rtoiOrt-i = 0, aonin-i = 0. 3. 2(3:1-3-3) +5(a;2-X4) =0. 4. 2(a;i-a;3)+ 5(a;2-a-4) =0, 4 xi + 32 X2 - 36 3-4 ± Vl042(a;2 -0:4) = 0. Page 213. Art. 169 1. (X2^ + 0:3^ + X4^y- = 0. 3 ,,^1 ^ „^i + ,,3^ ^ ^J ^ 0. 2. ?tl^ + U2^ + 1(3^ + Ui^ = 0. 4. Mi-?(3 + U2'^Ui = 0. Page 215. Art. 172 3. J- + -i- 4- A_ + Jl_ + 1 =^0. 4. 4(n-2)3. aixi a2X2 azxz 04X4 ai{x\ + 0^2 + X3 + X4) Page 218. Art. 175 2. X\ = 0, 3:42 — X2%z = ; X2 — X3 = 0, X2 — X4 = ; X1X2 — X1X3 + X42 — X2a;3 = 0, xi^ — X2a-4 — X2'-^ + X2X3 = 0, x^ + a;2a;3 + 2 X2X4 + X3X4 — xiX2 = 0. 3. (Xl2 + X22 - X32)2 - 4(Xi - X2) (Xi3 + X-? + X^Xa^ - 2 X1X32) = 0. 4. x^ H- X22 + 5 X32 = 0. 5. (ai — a4)xr + («2 — «4)x22 + {a% — ai^X'^ — 0. 7. (xi2- 2 X22+X32 + 2x42 -2x2X3)2 - 2(xi _ 3 X2 + 2 X4) [(xi2 + X42 - .T0X3) (2 n + 4 X2 - 2 X3) - (xi2 + 2 X22 - X32) (2 xi - X2 - .T3 + 2 X4) ] = 0. 284 ANSWERS Page 225. Art. 180 1. .Tl = t(t- - 1), X2 = f- - 1, a-3 = (<■- - 1 )2, X4 = t. 2. (4 TsXi — xiX2y^ — HX2^ + 2 a:ia:3)(a-i'- + 2 X2a-4) = 0. 3. 12(ui^ - U3^){U2^ — M4^)— l2(UiU2 — If3«4)- + (Mi2 + 2 Mo^ - Ms'^ - M4^)2 = 0. 36(Mo2 _ M32) (M22 - M42) (mi2 4- 2 tiz^ - W32 - M42) + 18(?