fl^thematics library '^hite Hall 3 1924 073 853 339 The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924073853339 SOLID GEOMETEY. BY THE SAME AUTHOR. ELEMENTARY ALGEBRA, Sixth Edition. Revised and enlarged. Globe 8to. 4s. 6d. Key, 10s. 6d. For the Second Edition the whole book was thoroughly revised, and the early chapters remodelled and simplified, and chapters on Logarithms and Scales of Notation were added. The number of examples was also very greatly increased. A TREATISE ON ALGEBRA. Fifth Edition. Crown 8vo. 7s. 6d. Key, 10s. 6d. This book has now been made more complete by the addition of a Chapter on the Theory of Equations. AN ELEMENTARY TREATISE ON CONIC SECTIONS. Twelfth Edition. Crown 8to. 7s. 6d. Key, 10s. Gd. GEOMETRICAL CONICS. Crown 8vo. 65. LONDON: MACMILLAN AND CO. Ltd. AN ELEMENTAEY TREATISE ON SOLID GEOMETEY CHARLES SMITH, M.A. UASTEB OF SIDNEY SUSSEX COLLEQE, CAMBBIDQE. SIXTH EDITION. SonDon MACMILLAN AND CO., Limited NEW YOEK : THE MACMILLAN COMPANY 1897 All rights reserved. First Edition, 1884. Second Edition, 1886. Third Edition, 1891. Fourth EditioK, 1893. Fifth. Edition, 1895. Sixth Edition, 1897. CamfittSje: PBINTED BY J. & 0. P. OLAY, AT IHB UNIVEBSITY PRESS. PREFACE. The following work is intended as an introductory text- book on Solid Geometry, and I have endeavoured to present the elementary parts of the subject in as simple a manner as possible. Those who desire fuller information are referred to the more complete treatises of Dr Salmon and Dr Frost, to both of which I am largely indebted. I have discussed the different surfaces which can be represented by the general equation of the second degree at an earlier stage than is sometimes adopted. I think that this arrangement is for many reasons the most satisfactory, and I do not believe that beginners will find it diflScult. The examples have been principally taken from recent University and College Examination papers ; I have also included many interesting theorems of M. Chasles. I am indebted to several of my friends, particularly to Mr S. L. Loney, B.A., and to Mr R. H. Piggott, B.A., Scholars of Sidney Sussex College, for their kindness in looking over the proof sheets, and for valuable suggestions. CHARLES SMITH. SlDNEI SnSSEX CoiiliBGE, April, 1884. CONTENTS. CHAPTER I. Co-OBDINATES. PAGE Co-oidinates 1 Co-ordinates of a point which divides in a given ratio the line joining two given points 3 Distance between two points 4 Direction-cosines 5 Belation between direction-cosines .... . . 5 Projection on a straight line ... 6 Locus of an equation 7 Polar co-ordinates 8 CHAPTER II. The Plahb. An equation of the first degree represents a plane 9 Equation of a plane in the form ten- my -(-7jz=p . . . . 9 Equation of a plane in terms of the intercepts made on the axes . . 10 Equation of the plane through three given points 11 Equation of a plane through the line of intersection of two given planes 11 Conditions that three planes may have a common line of intersection . 11 Length of perpendicular from a given point on a given plane . . 12 Equations of a straight line 14 Equations of a straight line contain four independent constants . . 14 Symmetrical equations of a straight line Ifl VUl CONTENTS. Ectnations of the straight line through two given points Angle between two straight lines whose direotion-oosines are given Condition of perpendicularity of two straight lines Angle between two planes whose equations are given . Perpendicular distance of a given point from a given straight line Condition that two straight lines may intersect Shortest distance between two straight lines .... Projection on a plane Projection of a plane area on a plane Volume of a tetrahedron Equations of two straight lines in their simplest forms . Pour planes with a common line of intersection cut any straight line in a range of constant cross ratio ... Oblique axes Direction-ratios Eelation between direction-ratios Distance between two points in terms of their oblique co-ordinates Angle between two lines whose direction-ratios are given Volume of a tetrahedron in terms of three edges which meet point, and of the angles they make with one another Transformation of co-ordinates Examples on Chapter II PAOB 16 16 X7 18 19 19 20 22 23 24 25 26 26 27 28 28 28 29 34 CHAPTER III. SUBEAOES OF THE SeOOND DsaBED. Number of constants in the general equation of the second degree . 37 All plane sections of a surface of the second degree are conies . . 38 Tangent plane at any point of a couicoid 38 Polar plane of any point with respect to a oonicoid .... 89 Polar lines with respect to a coniooid 40 A chord of a oonicoid is cut harmonically by a point and its polar plane 40 Condition that a given plane may touch a conicoid .... 41 Equation of a plane which cuts a conicoid in a conic whose centre is given 43 Iiocus of middle points of a system of parallel chords of a conicoid . 44 Principal planes 44 CONTENTS. IX PAGE Parallel plane sections of a ooniooid are similar and similarly situated conies 45 Classification of coniooids 46 The ellipsoid .... 49 The hyperboloid of one sheet 50 The hyperboloid of two sheets .... . . 51 The cone 51 The asymptotic cone of a conicoid 52 The paraboloids 52 A paraboloid a limiting form of an ellipsoid or of an hyperboloid . . 54 Cylinders 54 The centre of a conicoid 56 Invariants .... 58 The discriminating cubic . . . . ... 59 Coniooids -with given equations . . 60 Condition for a cone 66 Conditions for a surface of revolution ... ... 66 Examples on Chapter III .... 67 CHAPTER IV. Coniooids eefekeed to thbib Axes. The sphere ... 69 The ellipsoid 71 Director-sphere of a central conicoid 72 Normals to a central conicoid 73 Diametral planes 74 Conjugate diameters 75 Eolations between the co-ordinates of the extremities of three conjugate diameters 75 Sum of squares of three conjugate diameters is constant ... 76 The paraUelopiped three of whose conterminous edges are conjugate semi-diameters is of constant volume 76 Equation of conicoid referred to conjugate diameters as axes . . 78 The paraboloids 80 Locus of intersection of three tangent planes which are at right angles 80 Normals to a paraboloid 81 Diametral planes of a paraboloid . . 81 X CONTENTS. FASE Oones , 83 Tangent plane at any point of a cone 83 Reciprocal cones 84 Beciprocal cones are co-axial 85 Condition that a cone may have three perpendicular generators . . 85 Condition that a cone may have three perpendicular tangent planes . 86 Equation of tangent cone from any point to a conicoid .... 86 Equation of enveloping cylinder . . 88 Examples on Chapter IV 90 CHAPTER V. Plane Seoiiokb oh Conicoids. Nature of a plane section found by projection 96 Axes and area of any central plane section of an ellipsoid or of an byperboloid 97 Area of any plane section of a central conicoid . . . . 98 Area of any plane section of a paraboloid 99 Area of any plane section of a cone 99 Directions of axes of any central section of a conicoid .... 101 Angle between the asymptotes of a plane section of a central conicoid . 101 Condition that a plane section may be a rectangular hyperbola . . 102 Condition that two straight hnes given by two equations may be at right angles 102 Conicoids which have one plane section in common have also another . 103 Circular sections 103 Two circular sections of opposite systems are on a sphere . . . 105 Circular sections of a paraboloid 105 Examples on Chapter Y 108 CHAPTER VI. Genebatino Lines of Coniooids. Euled surfaces defined 113 Distinction between developable and skew surfaces .... 113 Conditions that aU points of a given straight line may be on a surface . 113 The tangent plane to a conicoid at any point on a generating line contains the generating line II5 Any plane through a generating line of a conicoid touches the surface . 115 CONTENTS. XI PAGE Two generatins lines pass through every point of an hyperboloid of one sheet, or of an hyperbolic paraboloid 116 Two systems of generating lines . . 116 All straight lines which meet three fixed non-intersecting straight lines are generators of the same system of a eonicoid, and the three fixed lines are generators of the opposite system of the same eonicoid . 117 Condition that four non-intersecting straight lines may be generators of the same system of a eonicoid 117 The lines through the angular points of a tetrahedron perpendicular to the opposite faces are generators of the same system of a eonicoid 118 If a rectilineal hexagon be traced on a eonicoid, the three lines joining its opposite vertices meet in a point 118 Four fixed generators of a eonicoid of the same system out all generators of the opposite system in ranges of equal cross-ratio . . . 118 Angle between generators 119 Equations of generating lines through any point of an hyperboloid of one sheet ■ . . 120 Equations of the generating lines through any point of an hyperbolic paraboloid 122 Locus of the point of intersection of perpendicular generators . 124 Examples on Chapter VI 124 OHAPTEE. VII. Systems of Coniooids. Tangential Equations. Eecipbooation. All coniooids through eight given points have a common curve of intersection 128 Four cones pass through the intersections of two conicoids . . . 129 Self-polar tetrahedron 129 Coniooids which touch at two points . . .... 130 All conicoids through seven fixed points pass through another fixed point ............ 130 Eeotangular hyperboloids ... 131 Locus of centres of coniooids through seven given points . . 132 Tangential equations • • 133 Centre of eonicoid whose tangential equation is given . . . 134 Director-sphere of a eonicoid 135 Locus of centres of coniooids which touch eight given planes . . 136 Locus of centres of conicoids which touch seven given planes . . 137 XU CONTENTS. PAOB Director-spheres of coniooids which touch eight given planes, have a common radical plane 137 The director-spheres of all coniooids which touch six given planes are out orthogonally by the same sphere 137 Eeciprocation 137 The degree of a surface is the same as the class of its reciprocal . . 138 Eeciprooal of a curve is a developable surface 138 Examples of reciprocation 140 Examples on Chapter VII 141 CHAPTER VIII. CoNPOCAIi CONIOODJS. CoNOYOlIO CoNIOOIDS. FoOI OS CoNIOOIDS. Confocal coniooids defined 144 Focal conies. [See also 158] 145 Three coniooids of a confocal system pass through a point . . . 145 One oonieoid of a confocal system touches a plane .... 146 Two coniooids of a confocal system touch a line 146 Oonfocals out at right angles 147 The tangent planes through any line to the two oonfocals which it touches are at right angles 148 Axes of central section of a conicoid iu terms of axes of two oonfocals . 149 Corresponding points on conicoids 151 Locus of pole of a given plane with respect to a system of oonfocals . 152 Axes of enveloping cone of a conicoid 153 Ec[uation of enveloping oone in its simplest form 153 Locus of vertices of right oiroular enveloping cones .... 155 Concyclic conicoids 155 Beciprocal properties of confocal and concyclic conicoids . . .156 Foci of coniooids ^ . 156 Focal conios 158 Focal lines of oone 159 Examples on Chapter Vni 160 CHAPTER IX. QUADIUPLAKAB AND TbIBAHBDEAI, Co-OBDINATES. Definitions of Quadriplanar and of Tetrahedral Co-ordinates , . 164 Equation of plane 165 Length of perpendicular from a point on a plane 167 CONTENTS. Xlll Plane at infinity 167 Symmetrical equations of a straight line 168 General equation of the second degree in tetrahedral co-ordinates . 169 Equation of tangent plane and of polar plane 170 Co-ordinates of the centre 170 Diametral planes 171 Condition for a cone 171 Any two eoniooids have a common self-polar tetrahedron . . . 172 The circumscribing conicoid 172 The inscribed conicoid 172 The circumscribing sphere 173 Conditions for a sphere . . 173 Examples on Chapter IX 175 CHAPTER X. SUEFAOES IN GeNEBAIi. The tangent plane at any point of a surface 178 Inflexional tangents 179 The Indieatrix 180 Singular points of a surface 180 Envelope of a system of surfaces whose equations involve one arbitrary parajneter 181 Edge of regression of envelope 182 Envelope of a system of surfaces whose equations involve two arbitrary parameters 183 Functional and differential equations of conical snrfaces . . . 184 Functional and differential equations of cylindrical surfaces . . . 185 Conoidal surfaces 186 Differential equation of developable surfaces 188 Equation of developable surface which passes through two given curves 190 A conicoid will touch any skew surface at all points of a generating line 191 Lines of striction 191 Functional and difierential equations of surfaces of revolution . . 192 Examples on Chapter X. 194 XIV CONTENTS. CHAPTER XI. CUEVES. FAOB Equations of tangent at any point of a curve 197 Lines of greatest slope 198 Equation of osculating plane at any point of a curve .... 201 Equations of tlie principal normal 202 Eadius of curvature at any point of a curve 202 Direction-cosines of the binormal 203 Measure of torsion at any point of a curve 203 Condition that a, curve may be plane 204 Centre and radius of spherical curvature 206 Badius of curvature of the edge of regression of the polar developable . 207 Curvature and torsion of a helix ... .... 208 Examples on Chapter XI. 210 CHAPTER XII. CuBVATUBE or Sdeiaoes. Curvatures of normal sections of a surface 313 Principal radii of curvature . 214 Euler's Theorem . 214 Meunier's Theorem 215 Definition of lines of curvature 217 The normals to any surface at consecutive points of a line of curvature intersect 217 Differential equations of lines of curvature 217 Lines of curvature on a surface of revolution . . . . 218 Lines of curvature on a developable surface 218 Lines of curvature on a cone 219 If the curve of intersection of two surfaces is a line of curvature on both the surfaces cut at a constant angle 220 Dupin's Theorem . . 221 To find the principal radii of curvature at any point of a surface . . 222 TJmbilios 228 Prinoipalradiiof curvature of the surface 2=/ (a;, ^) .... 224 Gauss' measure of curvature 225 Geodesic lines 226 CONTENTS. XV PAGE Lines of curvature of a ooniooid are its curves of intersection with con- fooal conieoida 227 Curvature of any normal section of an ellipsoid 228 The rectangle contained by the diameter parallel to the tangent at any point of a line of crtrvature of a coniooid, and the perpendicular from the centre on the tangent plane at the point is constant . 228 The rectangle contained by the diameter parallel to the tangent at any point of a geodesic on a coniooid, and the perpendicular from the centre on the tangent plane, is constant 228 Properties of lines of curvature of coniooids analogous to properties of confocal conies 229 Examples on Chapter XU . . 230 Hisoellaneous Examples . . 237 SOLID GEOMETEY. CHAPTER I. CO-OEDINATES. 1. The position of a point in space is usually determined by referring it to three fixed planes. The point of inter- section of the planes is called the origin, the fixed planes are called the co-ordinate planes, and their lines of intersection the co-ordinate axes. The three co-ordinates of a point are its distances from each of the three co-ordinate planes, measured parallel to the lines of intersection of the other two. When the three co-ordinate planes, and therefore the three co-ordinate axes, are at right angles to each other, the axes are said to be rectangular. 2. The position of a point is completely determined when its co-ordinates are known. For, let YOZ, ZOX, XOY be the co-ordinate planes, and X'OX, Y'OY, Z'OZ be the axes, and let LP, MP, ^fP, be the co-ordinates of P. The planes MPN, NPL, LPM are paraUel respectively to YOZ, ZOX, XO Y; if therefore they meet the axes in Q, B, 8, as in the figure, we have a paraUelopiped of which OP is a diagonal; and, since parallel edges of a parallelepiped are equal, LP = OQ,MP= OB, and NP = OS. Hence, to find a point whose co-ordinates are given, we have only to take OQ, OB, 08 equal to the given co-ordinates, s. s. G. 1 2 CO-ORDINATES. and draw three planes through Q, R, S parallel respectively to the co-ordinate planes ; then the point of intersection of these planes will be the point required. Z M r S ^Y ^ ^ P XT' ^ T l^ ^ -T z' A If the co-ordinates of P parallel to OX, OY, 0^ respec- tively be a, i, c, then P is said to be the point (aj 6, c). 3. To determine the position of any point P it is not sufficient merely to know the absolute lengths of the liiies LP, MP, NP, we must also know the directions in which they are drawn. If lines drawn in one direction be con- sidered as positive, those drawn in the opposite directidn must be considered as negative. We shall consider that the directions OX, OY, OZ are positive. * The whole of space is divided by the co-ordinate planes into eight compartments, namely OXYZ, OX'YZ, OXY'Z, OXYZ, OXY'Z, OX'YZ, OXYZ, and OX'Y'Z. If P be any point in the first compartmeiit, there is a point in each of the other compartments whose absolute distances from the co-ordinate planes are equal to those of P ; and, if P be (a, h, c) the other points are (— a, &, c), (a, — 6, c), (as, 6, - c), (a, - 6, - c), (- a, b, - c), (- a, - b, c) and (- a,-b,-c) respectively. CO-ORDINATES. 4 To find the co-ordinates of the point which divides the straight line joining two given points in a given ratio. Let P, Q be the given points, and R the point which divides PQ in the given ratio m, : m^. Let Pbe (x^, y^, z^, Q be (x„ y^, z^), and R be {x, y, z). Draw PL, QM, PiV parallel to OZ meeting ZO Fin L, M, N. Then the points P, Q, B, L, M, N are clearly all in one plane, and a line through P parallel to LM will be in that plane, and will therefore meet QM, RN, in the points K, H suppose. ™ HB PR m. Then -r^ = dts = — r — • KQ PQ m^ + m^ But LP = z„MQ = a„ NR=z; z —z, m. Similarly When PQ is divided externally, m, is negative. 1—2 4 CO-OKDINATES, The most useful case is where the line PQ is bisected : the co-ordinates of the point of bisection are iC^i + a's). i(2/i + y,). i(«i + ^2)- The above results are true whatever the angles betweeri the co-ordinate axes may be. We shall in futwre consider the axes to he rectangular in all cases except when the contrary is expressly stated. 5. To express the distance between two points in terms of their co-ordinates. • Let Pbe the point {x^, y^, aj and Q the point {x^,y^, «,). Draw through P and Q planes parallel to the co-ordinate planes, forming a parallelopiped whose diagonal is PQ. z s: Let the edges PL, LK, KQ be parallel respectively to OX, Y, OZ. Then since PL is perpendicular to the plane QKL, the angle PLQ is a right angle, .•.PQ'^Pr + QL' = PB + LK' + KQ'. Now PL is the difference of the distances of P and Q from the plane YOZ, so that we have PL = x^ — x^, and similarly for LK and KQ. Hence PQ''^{x^-x,y+ {l/,-y,Y + {z,-z^y (i). The distance of P from the origin can be obtained from the above by putting aj, = Q,y^ = 0, z^ = 0. The result is (?P' = a:.»-l-2/,''-f-^,» (ii). CO-ORDINATES. 5 Ex. ]. Theco-ordinatesofthecentreofgrayityof thetrianglewhoseangnlar pomts are [x^, y^, is{), (x^, y^ g, (a;,, y^, z^) are J [x^+x^+x^, | (yi+y^+y^), and i(jSi + 22+«8). Ex. 2. Shew that the three lines joining the middle points of opposite edges of a tetrahedron meet in a point. Shew also that this point is on the line joining any angular point to the centre of gravity of the opposite face, and divides that line in the ratio of 3 : 1. Ex. 3. Find the locus of points which are equidistant from the points (:, 2, 3) and (8, 2, - 1). Am. x-2z=0. Ex. 4. Shew that the point (|, 0, |) is the centre of the sphere which passes through the four points (1, 2, 3), (3, 2, -!),(- 1, 1, 2) and (1, -1,-2). 6. Let a, ^, y be the angles which the line PQ makes with lines through P parallel to the axes of co-ordinates. Then, since in the figure to Art. 5 the angles FLQ, PMQ, PNQ are right angles, we have PQ cos a = PL, PQ cos ^ = PM, and PQ cos 'y=P]S'. Square and add, then P(^ {cos'a -t- cos^yg + cosV} = PL' + PM" + PF' = PQ". Hence cos'a -I- cos'/8 + cos'y = 1. The cosines of the angles which a straight line makes with the positive directions of the co-ordinate axes are called its direction-cosines, and we shall in future denote these cosines by the letters I, m, n. From the above we see that any three direction-cosines are connected by the relation P + m' + n' = l. If the direction-cosines of PQ be I, m, n, it is easily seen that those of QP will he — l,—m,—n; and it is immaterial whether we consider I, m, n, or the same quantities with all the signs changed, as direction-cosines. If we know that a, h, c are proportional to the direction- cosines of some line, we can at once find those direction- cosines. For we have - = ^ = - : hence each is equal to a b c ^ V(Z'-l-m'-Hw') . 1 , _ a „ V(a'+ 6'+c') ' '■®" V(a' -I- 6' -t- O ' • • VC^' + &' + <>') 6 CO-ORDINATES. Ex. The direction-ooBiues of a line are proportional to 3, - 4, 12, find their actual values. Ans. ^, —A) rf- 7. The projection of a point on any line is the point where the line is met by a plane through the point per- pendicular to the line. Thus, in the figure to Art. 2, Q, It, S are the projections of P on the lines OX, OY, OZ re- spectively. The projection of a straight line of limited length on another straight line is the length intercepted between the projections of its extremities. If we have any number of points P, Q, R, 8... whose projections on a straight line are p, q, r, s..., then the projections of PQ, QR, RS... on the line, Skre pq, qr, vs.... In estimating these projections we must consider the same direction as positive throughout, so that we shall always have pq-\-qr + rs = ps, that is the projection of PB on any line is equal to the algebraic sum of the pro- jections of PQ, QiJ and R8. This result may be stated in a more general form as follows : — The algebraic sum of the projections of any number of sides of a. polygon beginning at P and ending at Q is equal to the projection of PQ. 8. If we have any number of parallel straight lines, the projections of any other line PQ on them are the intercepts between planes through P and Q perpendicular to their directions. These intercepts are clearly all equal ; hence the projections of any line on a series of parallel straight lines are all equal. And, since the projection of a straight line on an intersecting straight line is found by multipljtng its length by the cosine of the angle between the lines, we have the following proposition : — The projection of a finite straight line on amy other straight line is equal to its length multiplied hy the cosine of the angle between the lines. 9. In the figure to Art. 2, let OQ = a,OR = h, OS=c. Then it is clear that x = a for all points on the plane PMQN, and that y = & for all points on the plane PNRL, CO-OEDINATES. 7 and that z = c for all points on the plane PL8M. Also along the line NP we have x = a, and y = h; and at the point P we have the three relations x = a,y = h,z = c. So that a plane is determined by one equation, a straight line by two equations, and a point by three equations. In general, any single equation of the form F{x,y, z) = 0, in which the variables are the co-ordinates of a point, represents a surface of some kind ; two equations represent a curve, and three equations represent one or more points. This we proceed to prove. 10. Let two of the variables be absent, so that the equation of the surface is of the form F {x) = 0. Then the equation is equivalent to (x — a) (x — b) (x — cj = 0, where a, b, c,... are the roots of F{x) = 0; hence all the points whose co-ordinates satisfy the equation F'(x) = are on one or other oithe plcmes x — a = 0, x — b = 0, x — c= 0, Let one of the variables be absent, so that the equation is of the form F {x, y) = 0. Let P be any point in the plane s = whose co-ordinates satisfy the equation F{x,y) = Q; then the co-ordinates of all points in the line through P parallel to the axis of z, are the same as those of P, so far as X and y are concerned; it therefore follows that all such points are on the surface. Hence the surface represented by the equation F (x, y) = is traced out by a line which is always parallel to the axis of z, and which moves along the curve in the plane z = defined by the equation F{x, y) = 0. Such a surface is called a cylindrical surface, or cylinder. Next let the equation of the surface be F(x, y, z) = 0. We have seen that all points for which x = a, and y = S lie on a straight line parallel to the axis of z. Hence, if in the equation F{x, y, z) = 0, we put x = a, and y = b, the roots of the resulting equation in z will give the points in which the locus is met by a line through (a, b, 0) parallel to the axis of 2. Since the number of roots is finite, the straight line will meet the locus in a finite number of points, and therefore the locus, which is the assemblage of all such points for different values of a and b, must be a surface and not a solid figure. 8 CO-OEDINATES. 11. The points whose co-ordinates satisfy two equations must be on both the surfaces which those equations represent and therefore the locus is the curve determined by the intersec- tion of the two surfaces. When three equations are given, we have sufficient equations to find the co-ordinates, although there may be more than one set of values, so that three equations represent one or more points. 12. The position of a point in space can be defined by other methods besides the one described in Art. 1. Another method is the following : an origin is taken, a fixed line OZ through 0, and a fixed plane XOZ. The position of a point P is completely determined when its distance from the fixed point 0, the angle ZOP, and the angle between the planes XOZ, and POZ are given. These co- ordinates are called Polar Co-ordinates, and are usually de- noted by the symbols r, 9 and 0, and the point is called the point (r, 6, (f>). If OX be perpendicular to OZ, and OF be perpendicular to the plane ZOX, we can express the rectangular co-ordinates of P in terms of its polar co-ordinates. Draw PN perpendicular to the plane XOY, and NM perpendicular to OX, and join ON. Then X = OM = OJV cos if) = OP sin cos ^ = rsin0 cos <^, y = JOr= 0-^sin = OP sin sin (f>=r sin sin 0, and z = NP = OP cos 0=rcos 0. We can also express the polar co-ordinates of any point in terms of the rectangular. The values are, r = ^/ (aP + f + z'),0 = tm-'^^^^^-±^, &ni = Un-'^ . X CHAPTER TL The Plane. 13. To shew that the surface represented hy the general equation of the first degree is a plane. The most general equation of the first degree is Ax+JBy + G2 + I) = 0. If («,, 2/,, z^) and (x^, y^, z^ be any two points on the locus, we have Ax^ + By, + Gz^ + D = 0, and Ax^ + By^ + Gz^ + D = Q. Multiply these in order by ^ — , and ' — and add; "^ ■' ■' m, + «ij, m^^ + m^ then we have j[ m,a;, + >»,a;, , ^ m^y, + m,y^ ^ ^ m^s, + m,z^ ^ j_^^ m^ + m^ ■" m^ + m^ m^+m^ This shews [Art. 4] that if the points {x^, y„ »,), {x^, y^, z^ be on the locus, any other point in the line joining them is also on the locus; this shews that the locus satisfies Euclid's definition of a plane. 14. To find the equation of a plane. Let p be the length of the perpendicular ON from the origin on the plane, and let I, m, n be the direction-cosines of 10 THE PLANE. the perpendicular. Let P be any point on the plane, and draw PL perpendicular on XOY, and LM perpendicular to OX. Then the projection of OP on OF is equal to the sum of the projections of OM, ML and LP on ON. Hence if P be {x, y, z), we have la>-\-my + nz=p (i), the required equation. By comparing the general equation of the first degree : with (i), we see that the direction-cosines of the normal to the plane given by the general equation of the first degree are proportional to A, B,C; and therefore [Art. 6] are equal to A B G ^/iA' + B' + G')' ^/{A' + B'+Gy sliA'+B^+Oy Also the perpendicular from the origin on the plane is equal to —D sl{A^ + ^-¥GY 15. To find where the plane whose equation is Ax + By-\-Gz-\-D = Q, « meets the axis of x we must put y=z = 0; hence if the intercept on the axis of x be a, we have Aa + 1> = 0. Similarly if the intercepts on the other axes are h and c we have Bh + D = 0, and Go + D = 0. Hence the equation of the plane is X y z - + ? + - = a c 1. This equation can easily be obtained independently. THE PLANE. 11 16. To find the equation of the plane through three given points. Let the three points be (a;,, y^, »,). [x^, y^, z^, (x^, y^ z^). The general equation of a plane is Ax + By + Gz-{-I) = 0. If the three given points are on this plane, we have Ax^ + By^ + Gz^ + D = Q, Ax^ + By^ + Cz^ + D = 0, and Ax^ + By^ + Cz^ + 2) = 0. Eliminating A, B, C, D from these four equations, we have for the required equation = 0. y . z , 1 Vk ^1. 1 y^. ^2> 1 2/8. ^8. 1 17. If i8'= and 8' = Q be the equations of two planes, 8—X 8' = will be the general equation of a plane through their intersection. For, since 8 and 8' are both of the first degree, so also is 8—X8'; and hence jS — \/8" = represents a plane. The plane passes through all points common to ^ = and 8' = 0; for if the co-ordinates of any point satisfy 8=0 and ;8" = 0, those co-ordinates wUl also satisfy 8 = \8'. Hence, since X is arbitrary, 8 — XS' = is the general equation of a plane through the intersection of the given planes. 18. To find the conditions that three planes may have a convmon line of intersection. Let the equations of the planes be ax + by + cz + d = (i), a'x + b'y-{-c's + d' = (ii), and a"x + h"y + c"z+d" = (iii). The equation of any plane through the line of intersection of (i) and (ii) is of the form (ax + by + CZ + d:)+X(a'x + b'y + c'z + d') = 0...(iv). 12 THE PLANE. If the three planes have a common line of intersection, we can, by properly choosing X, make (iv) represent the same plane as (iii). Hence corresponding coefficients must bo proportional, so that a + Xa' _ b + 'Kb' _ c + 'Kc' _ d + Xd' ~~^' b" ~ c" " d" • Put each fraction equal to — /t, then we have a + \a' + ixa" = 0, 6 + \6' + /i6"=0, c + Xc' + /*c"=0, and d + \d' + )u,d" = 0. Eliminating \ and /* we have the required conditions, namely a , b , c , d =0, a' , b' , c' , d' «" I." jt J" a , , c , a the notation indicating that each of the four determinants, ob- tained by omitting one of the vertical columns, is zero.* 19. We can shew, exactly as in Conies, Art. 26, that if Ax + 5^/ + Cs^ + Z) = be the equation of a plane, and x, •^j , z' be the co-ordinates of any point, then Ax' + Bif + Gz' + D will be positive for all points on one side of the plane, and negative for all points on the other side. 20. To find the perpendicular distance of a given point from a given plane. » Let the equation of the given plane be Ix + my + m=p (i), and let x', y' , s! be the co-ordinates of the given point P. The equation Ix + my ■¥nz=p' (ii) is the equation of a plane parallel to the given plane. It will pass through the point (a;', y', si) if ia;' + m2/'+ns'=p' (iii). * It is easy to shew that there are only two independent conditions, as is geometrically obvious, for if the planes have two points in common they must have a common line of intersection. THE PLANE. 13 Now if PL be the perpendicular from P on the plane (i), and ON, ON' the perpendiculars from the origin on the planes (i) and (ii) respectively, then will LP = NW = Ix' + my' + nz — p. Hence the length of the perpendicular from any point on the plane Ix + my + nz —p = is obtained by substituting the co-ordinates of the point in the expression Ix + my +ns — p. If the equation of the plane be Ax + By + Gs + D = 0, it may be written A B Q ^{A' + W + Cf'^ ^{A^-\-B' + G'')y^ >J{A^ + R + G'f ^^/(A'+B' + G') ' which is of the same form as (i) ; therefore the length of the perpendicular from (x, ^, z") on the plane is Aa^ ^By'+Gz ^-B Ex. 1. Find the equation of the plane through (2, 3, - 1) parallel to the plane 3a!- 4y+ 72=0. Am. 3x-ly-i-7z+lB=0. Ex. 2. Find the equation of the plane through the origin and through the intersection of the two planes fix — 3^ + 22 + 5 = and Sx - 5j/ - 2s - 7 = 0. Ans. 2Sx-2Sy+2z=0. Ex. 3. Shew that the three planes 2x+5y + 3z=0, x-y + 4z=2, and 7y-5z + i = intersect in a straight line. Ex. 4. Shewthatthefourplanes2ai— B^ + 22=0,a!+^-32=4,3a;-y+«=2, and 7a: - 5j/ + 6z = 1 meet in a point. Ex.5. Shew that the four points (0,-1,-1) (4, 5, 1), (3, 9, 4) and ( - 4, 4, 4,) lie on a plane. Ex 6. Are the points (4, 1^ 2) and (2, 3, - 1) on the same or on opposite sides of the plane 6ai - 7^ - 62 + 3 = ? Ex. 7. Shew that the two points (1, - 1, 3) and (3, 3, 3) are equidistant from the plane 5a! + 2y-7« + 9=0, and on opposite sides of it. Ex. 8. Find the equations of the planes which bisect the angles between the planes Ax+By + Cz+D=0, and A'x+B'y + C'z+D'=Q. Ax+By + Gz+B _ , A'x+Jffy+C'z+D 14 THE STRAIGHT LINE. Ex. 9. The locus of a point, whose distances from two given planes are in a constant latio, is a plane. Ex. 10. The locus of a point, which moves so that the sum of its distances from any number of fixed planes is constant, is a plane. 21. The co-ordinates of any point on the line of intersection of two planes will satisfy the equation of each of the planes. Hence any two equations of the first degree represent a straight line. We can find the equations of a straight line in their simplest form in the following manner. Let PQ be the straight line, ^g' its projection on the plane XOY hy lines parallel to OZ. Then the co-ordinates a> and y of any point in PQ are the same as the co-ordinates x and y of its projection in ^q. Hence i£ la; + my = 1 be the equation of pq, the co-ordi- nates of any point od PQ will satisfy the equation lay+my=l. Similarly, if the equation of the projection of PQ on the plane FO^be ny+pz=l, the co-ordinates of any point on PQ will satisfy the equation ny+pz = l. Hence the equations of the line may be written la; + my = l, ny+pz=l. It should be noticed that the equations of a straight line contain four independent constants. The above equations are unsymmetrical and are not so useful as another form of the equations which we proceed to find. THE STRAIGHT LINE. 15 22. Let (a, ^, y) be any point A on a straight line, and (ps, y, z) any other point P on the line, at a distance r from («. A 7); and let I, m, n be the direction-cosines of the line. Draw through A and P planes parallel to the co-ordinate planes so as to make a parallelepiped, and let AL, LM, MP be edges of this parallelopiped parallel to the axes of x, y, z respectively. Then AL is the projection of AP on the axis of «; therefore T, oa — d a; — a = tr, or — i — = r. We have similarly y-^ = r, I z — y m n Hence the equations of the line are x—a y—^s—y I m n = r. = r. Ex. 1. To find in a symmetrieal form the equations of the line of inter- section of the planes hx-iy—\,Zy-f)Z='i. Hence the direction- if' «™ ^ 9t Z ^- The equations may be ■written —j-2=| = -5-5;. cosines are proportional to 4, 6, 3. The actual values of the direction- cosines are therefore f V^, iV2, tW2- Ex. 2. Find in a symmetrical form the equation of the line x-2y=5, 3x + y-1z=0. Am. ^(x-S)-y=z-i^. Ex. 3. Find the direction-cosines of the line whose equations are 12 3 x+y-z+l=Q, ix+y-2z + i=0. Ans. V14'V14'VW Ex. 4. Write down the equation of the straight line through the point (2, 3, 4) which is equally inclined to the axes. Ans. x-2=y-Z=z-i. 16 THE STEAIGHT LINE. 23. To find the equations of a straight line through two given points. Let the co-ordinates of the two given points AB be fljj, 2/,, a, and a;,, y,, z^; and let the co-ordinates of any poirit P on the line AB be so, y, z. Then the ratio of the projections of AP and AB on any axis is equal to AP : AB, Hence the equations of the line are «'2-«'i Vi-Vx ^a-^i" 24. To find, the angle between two straight lines whose direction-cosines are given. Let I, m, n and V, ni, n be the direction-cosines of the two lines, and let 6 be the angle between them. Let P,Q be any two points on the first line. Draw planes through P, Q parallel to the co-ordinate planes, and let PL, LM, MQ be edges of the parallelepiped so formed. Then the projection of PQ on the second line is equal to the sum of the projections of PL, LM, and MQ on that line. z r ">i f ■ ^ M k ^O Hence PQcoaO^PLA' + LM.m' + MQ.n'. But PL = l.PQ, LM=m.PQ, and MQ = n.PQ; THE STBAIGHT LINE, 17 therefore cos 0=U' + mm' + nn'. If the lines are at right angles we have W + mm' + nn' = 0. If i, M, iVare proportional to the direction-cosines of a line, the actual direction-cosines will be L M F Hence the angle between two lines whose direction-cosines are proportional to L, M, N and L', M', W respectively is _. LL' + MM' + NN' ^{r + M-' + N')^/{L" + M' + N")- The condition of perpendicularity is as before LL' + MM' + NN' =^ 0. Ex.1. Shew that the lines ^ = ^ = 3- and =- = -^ = ^ are at right angles. Ex. 2. Shew that the line ix=3y= -z is perpendicular to the line Sx=-y=-iz. SC li Z SB 1/ z Ex. 3. Find the angle between the lines - = f = - and -=-—=-. 1 1 U 0—45 Ans. cos-'i'jf. Ex.4. Shew that the lines 3x + 2y+z-5=0=x + y-2z-3, and Sa - 4y - 4z = = 7a! + lOy - 8z are at right angles. Ex. 5. Find the acute angle between the lines whose direction-cosines are Ex. 6. Shew that the straight lines whose direction-cosines are given by the equations 2l + 2m-n=0, and mn + nl+lm=0 are at right angles. Eliminating I, we have 2mn-{m+n){2m~n) = 0, or 2m'' - mn - m^ = 0. Hence, if the direction-cosines of the two lines be Zj, 7»i, raj and l^, in,^, »2' ^^ have^i— ^=-i. Similarly -^^s -1. Hence the condition l,}, + m,m„ mi"2 "i"2 Hi... +n^7i^=0is satisfied. Ex. 7. Find the angle between the two lines whose direction-cosines are given by the equations i-Hm-m =0, PH-m"- n' = 0. Am. W. Ex. 8. Find the equations of the straight lines which bisect the angles between the hnes y = - = - , and - = -^ = - . I m n I m n Let P, Q be two points, one on each line, such that OP—OQ^r. Then the co-ordinates of JP are Ir, mr, nr, and of Q are I'r, m'r, n'r; hence the co- ordinates of the middle point of PQ are J (J -h ?) r,i{m+ in') r,i(n+ n') r. Since S. S. G. 2 18 THE STRAIGHT LINE. the middle point is oa the bisector, the reciuired equations are Similarly the equations of the bisector of the z l+V m+m! n+n', supplementary angle are , — ,-. = — — ■. = , . 25. By the preceding Article cos 9= ir + mm + nn' ; therefore sin" ^ = 1 — {W + mm + nn')' = {P+m''+n''){r + m"' + n") — {W + mm' + nn'y ; therefore sin ^ = V { (mn' - m'n)' + (nl' - n'lf + Qm' - tmf}. 26. To find the angle between two planes whose equations are given. The angle between two planes is clearly equal to the angle between two lines perpendicular to them. Now we have seen [Art. 14] that the direction-cosines of the normal to the plane Ax+By+Cz + D = Q, are proportional to A, B, C. Hence by Article 24 the angle between the planes whose equations are Ax + By+ Cz +D = 0, A'a!+B'y+C's + D'=0, AA' + BB'+CC IS V (^» + B'+ C) V (A" + B" + G") ■ Ex. 1. Find the equation of the plane containing the line x+y+z^l, 2x+Sy + 4ji=5, and perpendicular to the plane x-y+z=0. Ans. x-^z+2=0. Ex. 2. At what angle do the planes a; + y + « = 4, a - 2^/ - z = 4 out ? Is the origin in the acute angle or in the obtuse? Is the point (1,-3, 1) in the acute angle or in the obtiise ? Ans. cos-iJ^2, acute, obtuse. Ex. 3. Find the equation of the plane through (1, 4, 3) perpendicular to the line of intersection of the planes 3x+4y + 7z+4:=0, and x-y + 2z+S=0; also of the plane through (3, 1, - 1) perpendicular to the Une of intersection of the planes 3a!+ 2/ -2 = 0, 5x-Sy + 2i=0. Am. 15x+y-7z + 2=0. Ans, a!+ll2/ + 14«=0. Ex. 4. Shew that the line ^= ?^ = 5. is parallel to the plane lx+my+nz+p=0 if l\ + mit + ni' =0, the axes being rectangular or oblique. THE STRAIGHT LINE. 19 27. To find the perpendicular distamce of a given point from a given straight line. Let the equations of the line be a;— a_ y — j3 _^ — 7 I m ~ n ' 0} Let (/ g, fi) be the given point P, and let PQ be the per- pendicular from P on the line. Let A be the point (a, /3, 7), and draw through A and P planes parallel to the co-ordinate planes so as to form a parallelopiped of which AL, LM, MP are edges parallel to the axes. Then AQ is the projection of AP on the given line, and is equal to the sum of the projections of AL, LM, and MP; therefore AQ = (/- a) i + (jr - /3) m + (A - 7) «. Hence Pq' = AP'-Aq' = C/"-«)=+(^-/8)"+(A-7)' -{llJ-a) + m{g-^) + n{h-rii\\ 28. To find the condition that two lines may intersect. Let the equations of the lines be I m, n i m n If the lines intersect they wUl lie on a plane ; andj since the plane passes through (a, /8, 7), we may take for its equation X(a;-a) + fi(i/-0) + v(z-y) = O (i). 2—2 ^ ^/r^A-- -^VV^' 20 THE STRAIGHT LINE. The point (a', 0, 7') is on the plane, hence we have X(a' -«) + /* (/3'-y3) + 1/(7' -7) = .(ii). Also, since the normal to the plane is perpendicular to both lines, we have , Xl +iim +vn =0 (iii), and \l' + iim' +vn' = (iv). Eliminating \, /*, v from the equations (ii), (iii) and (iv) we have the required condition, namely I , m , n m , n 0. If this condition be satisfied, by eliminating \, fi, v from (i), (iv), (iii), we find for the equation of the plane through the straight lines I , m , n I' m n = 0. If the equations of the lines be a^x + \y +c^z + ci, = 0, a^ + h^ + Cj^ + dj = 0, and a^ + i^+ c^ +d^ = 0, a^oa + h^y + c^z + 5j = 0, the condition of intersection of the lines is the condition that the four planes may have a common point, which is found at once by eliminating x, y, z. 29. To find the shortest distance between two straight lines whose equations are given. Let AKB and OLD be the given straight lines, and let KL be a line which is perpendicular to both. Then KL is the shortest distance between the given lines, for it is the projection of the line joining any other two points on the given lines \ Let the equations of the given lines be x—a _ y—b _ z—o _ m and '-a' _ y-V _ z-c' V m 1 We can find KL by the foUowing eonstruotion : — draw AE through A parallel to CD ; let AP be perpendicular to the plane EAB, and 1st the plane PAB cut CD in i j then if LK be drawn parallel to PA it will be the line re(iuired. THE STRAIGHT LINE. ^1 Let the equations of the line on which the shortest distance lies be w-j,^yj--£_g-y X fi V W- Since the line (i) meets the given lines, we have [Art. 28] • a-a, ^-b, y-c = (ii), and X, fi , V a— a, yS — 6', 7 — c' V , m' , n' = (iii). Since (i and ) is perpendicular to the given lines, we have Xi +/jim +vn =0, therefore \ _ /* _ " mn' — m'n nV — n'l Im — I'm ' Hence, from (ii) and (iii), we see that (a, ^, 7), which is- an arbitrary point on the shortest distance, is on the two planes and x — a, y — b, z — c I , m , n mn — m'n, nt — n'l, hn' — I'm x—a', y — V, « — c r , m' , n mn' — m'n, nl' — n'l, Im' — I'm, = 0. = 0. These planes therefore intersect in the line on which the shortest distance lies. We can find the length of the shortest distance from the fact that it is the projection of the line joining the points (a, b, c) and {a', b', c'). Now the projection of this line on the line whose direction-cosines are X, fi, v is (a - a')X + (6 — &') /" + (c - c') v. 22 THE STRAIGHT LINE. But as above mn' — m'n nl! — n'l Im' — I'm ' therefore each fraction is equal to 1 ^/{(mn' - m'n)' + {nl' - n'lf + Qm' - I'mY] ' Hence the length of the shortest distance is (a - a') {mn' - m'n) +{b- h'){nl' - n'l) + (c - c'){lm' - I'm) Vl(m»' - m'rif + {nV - n'l)' + {knf - I'mYi " Ex. 1. Find tlie perpendicular distance of an angnlar point of a cube from a diagonal which does not pftss through that angular point. Avs. a^jj. Ex. 2. How far is the point (4, 1, 1) from the line of intersection of /27 x+y+e=i, x-2y-z=4:J -Ans. /v/ J^- Ex. 3. Shew that the two lines x-2 = 2y-6=3z, ix-ll=iy- 13 = 3z meet in a point, and that the equation of the plane on which they lie ia 2a!-6^ + 3z + 14=0. Ex. 4. Find the equation of the plane through the point (a', /y, Y), and X — (t It ^ B z ^ V through the line whose equations are — =— = - — — = . x-a,y-p,z~y Arts, o'-o, jS'-jS.V-T I , m , n = 0. Ex. 5. The shortest distances between the diagonal of a rectangular parallelepiped and the edges which it does not meet are be ca ah yj(h^+cy v(o»+c«)' TjW+W)' where a, 6, c are the lengths of the edges. Ex. 6. Find the shortest distance between the straight lines 4(iB-l)=4(3/-2)=2-3, a,-nAy-mx=z=Q, 5m- 10 Am, Ex. 7. Determine the length of the shortest distance between the lines 4a;=8^=-z and 8 (a!-l)= -3^-2= -4« + 2. Find the equations of the straight line of which the shortest distance forms a part. Ana. ^, 30. If through any number of points, P, Q,R... lines be drawn either all through a fixed point, or all parallel to a fixed line ; and if these lines cut a fixed plane in the points PROJECTIONS. 23 P', Q',B! ..,; then P', Q', B! ... are called ilae pr(^ections of P, Q, M... on the plane. If the lines PP', QQ, MR... are all perpendicular to the fixed plane, the projection is said to be orthogonal. The orthogonal projection of a limited straight line on a plane is the line joining the projections of its extremities. It is easily seen that the projection of a line on a plane is equal to its length multiplied by the cosine of the angle between the line and the plane. 31. The orthogonal projection of any plane area on any other plane is found by multiplying the area by the cosine of the angle between the planes. Divide the given area into a very great number of rectangles by two sets of lines parallel and perpendicular to the line of intersection of the given plane and the plane of projection. Then, those lines which are parallel to the line of intersection are unaltered by projection, and those which are perpendicular are diminished in the ratio 1 : cos 0, where is the angle between the planes. Hence every rectangle, and therefore the sum of any number of rectangles, is diminished by projection in the ratio of 1 : cos^. But, when each of the rectangles is made indefinitely small, their sum is equal to the given area. Hence any area is diminished by projection in the ratio 1 : cos 0. 32. If we have more than one plane area, we must make some convention as to the sign of the projection, and we have the following definition : the algebraic pro- jection of any face of a polyhedron on a fixed plane is found by multiplying its area by the cosine of the angle between the normal to the fixed plane and the normal to the face, the normals to the faces being all drawn outwards or all drawn inwards. 33. Let A be the area of any plane surface ; l,m,n the direction-cosines of the normal to the plane ; A^, A^, A^ the projections of A on the co-ordinate planes. Then we have A^ = l.A, Ag = m.A, A, = n.J. 24 VOLUME OF TETRAHEDRON. Hence, since P + m* + n^ = 1, we have A^' + A J' + A; = A\ Also the projection of A on any other plane, the direction- cosines of whose normals are I', m', n', is A cos j and we have A cos 6 = (W + mm' + nn) A' = I'A^ + m'A^ + n'A,. Hence to find the projection of any plane area, or of the sum of any plane areas, on any given plane, we may first find the projections A^, A^, A, on the co-ordinate planes, and then take the sum of the projections of A„ A^, A, on the given plane. 34. To find the volvme of a tetrahedron in terms of the co-ordinates of its angular points. Let the co-ordinates of the angular points of the tetra- hedron ABGD be («„ y,, z^), (x^, y,, aj, (x^, y„ £■,), and (x^ y^, z^. The volume of a tetrahedron is one-third the area of the base multiplied by the height. Now the equation of the face BCD is oc. V: S > Ht< "2 > X., v.. Z., 1 = 0. "*> Vi' "*' The perpendicular p from A on this is found by sub- stituting the co-ordinates of A and dividing by the square root of the sum of the squares of the coefficients of x, y, and z. ^ Now the coefficients of x, y, z are y^> ^2' 1 > ajj. ^s. 1 > ^'s. 2/2. 1 2/,. ■^»> 1 «„ ^3, 1 «.> ^8. 1 VO ^4. 1 «4, Z^, 1 «4. Vv 1 respectively; and these coefficients are respectively equal to twice the area of the projection of BOB on the planes a; = 0, y = and a^ = 0. Hence the square root of the sum of the squares of the coefficients of x, y and' z is, by the preceding Article, equal to ItiBGI). TWO STRAIGHT LINES. 25 Therefore 2p . A5CZ) = S > S3> therefore volume of tetrahedron Z^y 1 a 1 ^2 > l HO. ^1. yi> z„ I «,. y,> z,. 1 ^'s. y,' ^,. 1 »4. y*' ^4. 1 35. The equations of two straight lines can be found in a very simple form by a proper choice of axeSi Let be the middle point of CC, the shortest distance between the two straight lines CD, CD'. Through draw OA, OB parallel to CD, CD', and let OX, OY bisect the angle AOB. Take OX, OY,OG for axes of co-ordinates; then, if AOB be 2a, the equations of OA, OB are y = a;tana z = 0, and y = — x tan a,z=^0. Hence the equations of the parallel lines CD, G'U are y = x tan a,z=C] and y = — x tan a, s = — c. When it is not of importance that the axes should be rectangular, we may take OA, OB, OG for axes: the equa- tions of CD, CD' will then be y = 0, ^ = c ; and x^O, z=—c. Also CG' may be any straight line which intersects CD and CD. 26 OBLIQUE AXES. 36. F = IV + mm' + nn' + (jnn' + m'n) cos \ + {nl -f n'h cos /* -I- (Jml + Im) cos V. 43. To find the volume of a tetrahedron in terms of three edges which meet im a point and of the angles they make with one another. Take the axes along the three edges, and let a, h, c be the lengths of the edges, and X, fi, v the angles they make with one another. Then Volume = J abc sin v cos 6, OBLIQUE AXES. 29 Let the direction-ratios of the normal to the plane XOY be I, m, n. Then from Art. 40 we have l + m cos v + n cos /* = 0, Z cos v + m + ra cos \ = 0, ^ cos /t + m cos \ + n = cos 0. Multiply by I, m, n and add, then, from (i) Art 41, M cos = 1. The elimination of I, m, n from the above equations gives = 0; 1 , cos V , cos /i , cos V , 1 , cos X , cos /t , cos \ , 1 , cos 6 0, 0, cos^, 1 therefore sin' v cos' ^ = I 1 , cos v , cos /* cosi;, 1, cos\ I cos /It , cos \ , 1 = 1 -cos''\-cosV-cos% + 2cos\cos/icos V. Hence the volume required = ^ abc V (1 - cos' \ — cos" fi — cos' v + 2 cos \ cos fi cos v). Tbansfoemation of Co-ordinates. 44. To change the origin of co-ordinates without changing the direction of the axes. Let j^ g, h be the co-ordinates of the new origin referred to the original axes. Let P be any point whose co-ordinates referred to the original axes are as, y, z, and referred to the new axes a;', y\ il. Let PL be parallel to the axis of x and let it meet YOZ'yh L, and T'OZ' in L'. 30 TRANSFORMATION OF CO-ORDINATES. Then therefore Similarly and LP = x,L'P = x x-af = LL'=f. y-y'=9, Hence, if in the equation of any surface we write x +f, y+g, e + h for X, y, z respectively, we obtain the equation referred to the point (/, g, h) as origin. 46. To change the direction of the axes without changing the origin, loth systems being rectangular. Let I , m^, w,; l^, m^, n^; and ?,, m,, w, be the direction- ifthe " " - -- cosines o: new axes referred to the old. Z JP Let P be any point whose co-ordinates in the two systems are x, y, z and x', y^, z'. » Draw PL perpendicular to the plane X' OY and LM per- pendicular to OX; then Oikf = »', ML = y', and LP=z. Since the projection of OP on OX is equal to the sum of the projections of OM, ML and LP, we have x=l^af + \y' + l^z'. Similarly y = m^(d +m^y' +m^ z', and z = n^x' ■{■n^y + n^ n! . TEANSFOEMATION OF CO-OEDINATES. 31 These are the formulae required. Since l^, m^, n^ ; l^, m^, n^; and l^, m^, n^ are direction- cosines, we have Z/ + m/ + < = l}. «/ + < + < = ! I Also, since OX', OY, OZ' are two and two at right angles, we have and l^^ + TOjTOj + mjTOj = J The six relations between the nine direction-cosines which we have found above are equivalent to the following : C +«/ +C =1,] < + m,= + m/=ia m^n^ + m^^ + mjji^ = 0, \ V. +^^2 +V3 =0. [- Z^m, + Z,m, +^37713=0,} This follows at once from the fact that l^, \, \; m,, m^, m,; and Jij,, n^, m, are the direction-cosines of OX, OY, 0^ referred to the rectangular axes OX', OY, OZ'. 46. Since l^l^ + m^OTa + WjWj = 0, and lJ,^ + m^m^ + n^n^ = Q, we have k _ «». _ w., Hence each fraction is equal to VIKtz., - m,/!,)' + (m,«3 - Wag" + {\m^ - l^m^f) ~ - TEANSFOEMATION OP CO-OEDINATES, Also Zj, m^, w, = ±(Zj' + m,= + = + l. 47. If in Art. 45 the new axes are oblique, we still have the relations a;= Z/ + l^' + I/, y = m^x + my + m/, z = n^x + my + Wj/. We can deduce the values of as', y', z' in terms of x, y, z: the results are m„ Mi, w» z„. ?» m„ »w. '"1 > '"2 I and two similar equations. 48. The degree of an equation is unaltered by any trans- formation of axes. From the preceding Articles we see that, however the axes may be changed, the new equation is obtained by sub- stituting for a;, y, z expressions of the form lx + my-\-nz-\-p. These expressions are of the first degree, and therefore if they replace x, y, and z in the equation, the degree of the equation will not be raised. Neither can the degree,of the equation be lowered; for, if it were, by returning to the original axes, and therefore to the original equation, the degree would be raised. 49. We shall conclude this chapter by the solution of some examples. (1) A line of eomtant length Jua its extremities on two fixed straight lines; shew that the locus of its middle point is an ellipse. If we take the axes of co-ordinates as in Art, 35, the equations of the lines will be y=mx, z=c; and y=-mx, z=-c. Let the co-ordinates of the EXAMPLES. 33 extremities of the line in any one of its possible positions be ajj, y^, z-^ and "'ai fe ^2 > ^iid let (a;, y, z) be the co-ordinates of the middle point of the line. Then, if 21 be the length of the line, we have But, since y.i=mxi and «i=c, and ^2= -mx^ z^=-c, we have yi - 2/j = m (iBi + ajj) = 2»w, Zi-z^=ic, and 22; = «i+«2=0. Hence the locus of the middle point is the ellipse whose equations are (2) ^ K?ie imves so as always to intersect three given straight lines, which are not all parallel to the same plane; find the equation of the siurface generated by the straight line. Draw through each of the Unas planes parallel to the other two; a parallelopiped is thus formed of which the given lines are edges. Take the centre of tiie parallelopiped for origin, and axes parallel to the edges, then the equations of the given lines are y = b, z= ~c; z—c, x= -a; and x=a, y=-b respectively. Let the equations of the moving line be x-a _y-p _z -y I m ~ n ' Since this meets each of the given lines we have 6-|8 -c-y c-y _ -a-a , a-a_ -h-p ——^ — , — — z , and ^—: — — . m n n I I m Hence, by multiplying corresponding members of the three equations, we see that (a, /3, 7), an arbitrary point on the moving line, is on the surface whose equation is {a-x){b-y){c-z) + {a+x){b+y){c + z)=0, |^ + i? + ^+l=0. be ca ao (3) TJie lines of intersection of corresponding planes of two homegraphic systems describe a surface of the second degree. We may take y=ina!, z=c, and y= -mx, 2= -c for the equations of the lines of intersection of the two systems of planes [see Art. 35.] Let the equations of corresponding planes of the two systems be y-mx+\(z-c)=0, and y+mx+W {z+c)=0. Since the systems are homographic there is one value of X' for every value of X, and one value of X for every value of X'; hence X, X' must be connected by a relation of the form XX'+.4X+BX' + C=0. S. S. G. 3 34 EXAMPLES ON CHAPTEK II. Substitute for \ and X', and we have y^-m?x'-A{z + c)(i/-mx)-B(z-c)(i/+mx) + C{z^-c^)-=0. Hence the line of interseotion of corresponding planes describes a surface of the second degree. Examples on Chapter II. 1. If P be a fixed point on a straight line through the origin equally inclined to the three axes of co-ordinates, any plane through P will intercept lengths on the co-ordinate axes the sum of whose reciprocals is constant. 2. Shew that the six planes, each passing through one edge of a tetrahedron and bisecting the opposite edge, meet in a point. 3. Through the middle point of every edge of a tetrahedron a plane is drawn perpendicular to the opposite edge ; shew that the six planes so drawn will meet in a point such that the centroid of the tetrahedron is midway between it and the centre of the circumscribing sphere. i. The equation of the plane through -= = — = - , and which m is perpendicular to the plane containing — = ^=- and -=j = is x{m-n) + y(n-l) + z(l-m) = 0. 5. Shew that the straight lines X _y _z X _ y _ z x _y _z a P y' aa b^ cy' I m~ n' will lie in one plane, if I „ , m , »w, ,»„ -ih-c) + -^{e~a) + -{a-b) = 0. b. Two systems of rectangular axes have the same origin; if a plane cut them at distances a, b, o, and a, b', c' from the origin, then i 1 L_i 1 1 EXAMPLES ON CHAPTER 11 35 7. Determine the locus of a point which moves so as always to be equally distant from two given straight lines. 8. Through two straight lines given in space two planes are drawn at right angles to one another ; find the locus of their line of intersection. 9. A line of constant length has its extremities on two given straight lines ; find the equation of the surface generated by it, and shew that any point in the line describes an ellipse. 10. Shew that the two straight lines represented by the equations ax + hy + cz = 0. i/z + ex + xy = wUl be perpendicular if 111. - + y + - = 0. a o c 11. Find the plane on which the area of the projection of the hexagon, formed by six edges of a cube which do not meet a given diagonal, is a maximum. 12. Prove that the four planes my + nz = (}, nz+lx = 0, Ix + my^O, lx + my + nz=p, form a tetrahedron whose volume is „, . 13. Find the surface generated by a straight line which is parallel to a fixed plane and meets two given straight lines. 14. A straight line meets two given straight lines and makes the same angle with both of them; find the surface wMch it lerates. 15. Any two finite straight lines are divided in the same ratio by a straight line ; find the equation of the surface which it generates. 16. A straight line always parallel tothe plane of yz passes through the curves x' + ^ = a', z = 0, and sa' = az, y = 0; prove that the equation of the surface generated is xy^ia^-azyia'-x"). 17. Three straight lines mutually at right angles meet in a point P, and two of them intersect the axes of x and y respec- tively, while the third passes through a fixed point (0, 0, c) on the axis of z. Shew that the equation of the locus of P is « + w + « :«=2fl 3—2 36 EXAMPLES ON CHAPTER II. 18. Find the surface generated by a straight line which meets y = mx, e = c; y = -mx, z = -c; and ^ + %^ = o', 03 = 0. 19. P, P are points on two fixed non-intersecting straight lines AB, A'B' such that the rectangle AP, A'P' is constant Find the surface generated by the line FF. 20. Find the condition that W + lif' + cii? + Idyz + Ih'fsx + ^c'xnj = may represent a pair of planes ; and supposing it satisfied, if 6 be the angle between the planes, prove that , . 2ja'' + h" + c"-ba-ca-ab tan Q = -^ J . a + b + c 21. Find the volume of the tetrahedron formed by planes whose equations are y + z = Q, a + a; = 0, x + y = 0, and x + y + z=l. 22. Find the volume of a tetrahedron, having given the equations of its plane faces. 23. Shew that the sum of the projections of the faces of a closed polyhedron on any plane is zero. 24. Find the co-ordinates of the centre of the sphere in- scribed in the tetrahedron formed by the planes whose equations are x = 0, y=0, z=:0 and x + y + z=l. t u h, b, f. V 9> f. c, V) u. V , w, d We will give special investigations in the two following cases which are of great importance: I. Let the equation of the surface be aaj' + fe/'f c^+d = 0. The tangent plane at any point {af, if, a') is cKc'aj + byfy + c^z + d = 0. Hence, comparing this equation with the given equation ha + my-\-'n,z + 'p = 0, ax _ hy' _oz _d I m n p' ^(aa!'^+by" + cz'^ + d) , wj: uu lis 1 we nave ,- = -^ = — = • Each fraction is equal to T^f^^!)' hence, since ax'^ + by'^ + cz'*-\- d = 0, the required condition of tangency is V - + a m bed II. Let the equation of the surface be ao(? + bf-{-2z = Q. The tangent plane at any point (as', y', a') is ax'ai + by'y + 3+^ = 0. Hence, comparing this equation with the given equation Ix + my -Vnz-irp^O, yg- CENTRE OP A PLANE SECTION. 43 , aa! hy' \ z' -^ , . . . we have -^ = ^- = - = - . Each fraction is equal to hence, since vw^ + ly'^ + 2/ = 0, the required condition of tangency is 58. K we find, as in Article 51, the quadratic equation giving the segments of a chord through (a, /3, 7) the roots of the equation will be equal and opposite, if ,dF ^ dF^ dF _ ... ^d^-^'"d^+'^^ = ^ w- In this case (a, ;S, 7) will be the middle point of the chord. Hence an infinite number of chords of the conicoid have the point (a, ^, 7) for their middle point. If we eliminate I, m, n between the equations of the chord and (i), we see that all such chords are in the plane whose equation is .(.-.)f+to-«f+{-,)f-0 (ii). Hence (a, /8, 7) is the centre of the conic in which (ii) meets the surface. This result should be compared with that obtained in Art. 52. Ex. 1. The locus of the centre of all plane sections of a conicoid which pass through. a fixed point is a conicoid. J7JT JTyl JTp The equation of the locus is {/- ») -j- + (jf - J') j- + (ft - z) t- =0, where /, g, h are theioo-ordiuates of the fixed point. Ex. 2. The locus of the centre of parallel sections of a conicoid is a straight line. 44. DIAMETEAL PLANES. The section whose centre ia (a, j3, y) is parallel to the given plane Zx + ma+7M!=0if dF dF dF da __dp _clry I TRi n Hence the locus is the straight line whose equations are \dF^\^^^ldF I dx~ m dy ~ n dz' The straight lines clearly all pass through the point of intersection of the dF dF dF . planes -^ = 3- = -^-=0. dx dy dz 59. To find the loom of the middle points of a system of parallel chords of a conicoid. As in the preceding Article, (a, /3, 7) will be the middle point of the chord whose direction-cosines are I, m, n, if jdF ^ dF ^ dF . Hence the locus of the middle points of all chords whose direction-cosines are I, m, n is the plane whose equation is • jdF , dF dF _ fix dy ■ dz Def The locus of the middle points of a system of parallel chords of a conicoid is called the diametral pkme. If the plane be perpendicular to the chords it bisects, it is called a principal plane. 60. To find the equations of the principal plames of a conicoid. The diametral plane of the chords whose direction- cosines are I, m, n is ,dF dF dF „ ^•j +m -r- + n -,- = 0, dx dy dz or, writing the equation in full, I (ax + hy+gs + u)+m (fix + by +fz + v) + n {gx +fy + cz+w)=0, or X (al + hni +gn) +y(hl + bm -l-/n) +z{gl +fm -H cw) ■\-ul + vm + wn = 0. c PRINCIPAL PLANES. 45 If this plane be perpendicular to the chords it bisects, we have ^ al + hm+gn _hl + bm+Jh _gl+fm+cn I m n ' Put A. for the common value of these fractions, then {a — X)l +hm +gn =0,1 hi. +{b-\)m+fn =0,| (i).. gl +fm + {0- - X) n = 0. Eliminating I, m, n we have a — X, h, h, b-X, ff> f. 9 / c — \ = 0, or \^-{a-irh + c)\^ + {bc + ca + ab-f-g''-h^)\ - {abc + 2fgh - af - 6/ - cV) = 0. This is a cubic equation for determining \ ; and when \ is determined, any two of the three equations (i) will give the corresponding values of I, m, n. Since one root of a cubic is always real, it follows that there is always one principal plane. Find the principal planes of the following surfaces : (i) x^+y'-z'+2yz + 22x-2xy=a?. (ii) lla?+10y''+6z''-8yz + 4zx-12xy=l. Ans. (i) x + y + z=0, x-y=Q, x+y-2z = 0. Ans. (ii) x+iy + 2z=0,ix+y-2z = 0,2x-2y + z=(i. 61. All parallel plane sections of a conicoid are similar and similarly situated conies. Change the axes of co-ordinates in such a way that the plane of sny may be one of the system of parallel planes ; and let the equation of the surface be, the general equation of the second degree. Let the equation of any one of the planes be s = ^. At all points of the section of the surface F(w, y, z)=0, by the 46 PARALLEL SECTIONS ARE SIMILAR. plane z = k both these relations are satisfied ; we therefore have oa?" + Sy + ck? + Ifyk + ighx + 2hxy + 2uiv + Ivy ^2wk + d=Q (i). Now the equation (i) represents a cylinder whose gene- rating lines are parallel to the axis of a, and which is cut by the plane = in the curve represented by (i). Since parallel sections of a cylinder are similar and simi- larly situated curves, the section of the surface F{ic, y,z) = hy z = k is similar to the conic represented by (i) and z = 0; and all such conies, for different values of k, are clearly similar and similarly situated : this proves the proposition. Classification op Conicoids. 62. We proceed to find the nature of the different surfaces whose equations are of the second degree ; and we will first shew that we can always change the directions of the axes of co-ordinates in such a way that the coefficients of yz, zx, and icy in the transformed equation are all zero. 63. We have seen [Art. 60] that there is at least one diametral plane which is perpendicular to the chords it bisects. Take this plane for the plane j? = in a new system of co- ordinates. The degree of the equation of the surface will not be aljered by the transformation ; hence the equation will be of the form aa? + hy^ + c^ + 2fyz + 2gzx + 2hxy + 2ua!+ 2vy + 2wz -1- d = 0. By supposition the plane z = Q bisects all chords parallel to the axis of z ; therefore if («', 1/, z) be any point on the surface, the point («', y', — z') will also be on the surface. From this we see at once i]\&if=g=w = 0. 2h Now turn the axes through an angle j^ tan"* =- , then [See Conies, Art. 167] the term involving xy will disappear. CLASSIFICATION OF CONICOIDS. 47 Hence we have reduced the equation to a form in which the terms yz, zx, and xy are all absent. 64. When the terms yz, zx, xy are all absent from the equation of a conicoid, it follows from Art. 60 that the co-ordi- nate planes are all parallel to priacipal planes. Hence by the preceding article, there are always three principal planes, which are two and two at right angles. This shews that all the roots of the cubic equation found in Art. 60 are real. For an algebraical proof of this important theorem see Todhunter's Theory of Equations. 65. We have seen that the general equation of the second degree can in all cases be reduced to the form Ax' + By^ + Gz'+'2.Ux->r'iVy->r2Wz + D = 0. I. Let A, B, be all finite. We can then write the equation -D = B'. 1 TI\^ f A[x + j)-^.B(y + B)' ^g(z + 1)' A^ B + G Hence, by a change of origin, we have Ax^ + By''-^- ■Cz" = 2)'. If D' be not zero we have A B G = 1, ich we can write in the form ^ + F + ? = ^ ^"^' x^ y' z' ^ /■o\ a^ + F-? = ^ (^^' (^ 'f Z^ - , V 48 CLASSIFICATION OF CONICOIDS. jy ly jy according as -j , -„ , yf are all positive, two positive and one negative, or one positive and two negative. [If all three are negative the surface is clearly imaginary.] If D be zero, we have Ao^ + Bf^-Cz^=^ (S). II. Let C, any one of the three coefficients A, B, G, be zero. Write the equation in the form then, if W be not zero, the equation can, by a change of origin, be reduced to Ax^-\-By^ + 'i.Wz = Q (e). If W be zero, we have the form Aic' + Bf + D' = Q (a or, if U be zero, the form Ao^ + By''=0 in). III. Let B, G, two of the three coefficients, be zero. We then have A(x + ^]+2ry+2Wz+D-j- = 0. IP Nowtake 2Vy+2Wz+D — -7- = ^ for the plane y = (S and the equation reduces to the form • x' = 2ky {&). If however V—W=0, the equation is equivalent to af' = k' (t). 66. We now proceed to consider the nature of the surfaces whose equations are (a), (/3), (t) ; to one of which forms we have seen that the general equation is reducible. THE ELIJPSOID. 4D The surface whose equation is is called an ellipsoid. Let a, b, c be in descending order of magnitude; then {«, y, a) being any point on the surface, we have a a a a;" «' z' and _+^ + _^l. So that no point on the surface is at a distance from the origin greater than a, or less than c. The surface is therefore limited in every direction ; and, since all plane sections of a conicoid are conies, it follows that all plane sections of an ellipsoid are ellipses. The surface is clearly symmetrical about each of the co- ordinate planes. If r be the length of a semi-diameter whose direction- cosines are I, m, n, we have the relation 1 _f «^ 71? r" ~ a» "^ 6" "*■ c" • If two of the coefficients are equal, b and c suppose, the section by the plane x = 0, and therefore [Art. 61] by any plane parallel to a;=0, is a circle. Hence the surface is that formed by the revolution of the ellipse -J -f- T« = 1 about the axis of x. a The surface formed by the revolution of an ellipse about its major axis is called a prolate spheroid ; that formed by the revolution about the minor axis is called an oblate spheroid. li a=b = c the equation of the surface is as' + y" + ^' =5= a", which from Art. 5 represents a sphere. s. s. G. 4 50 THE HTPERBOLOID OF ONE SHEET. 67. The surface whose equation is ^^ + 1-1-1 is called an hyperholoid of one sheet. The intercepts on the axes of x and y are real, and those on the axis of z are imaginary. The surface is clearly symmetrical about each of the co- ordinate planes. The sections by the planes x = and y = are hyperbolas, and that by s = is an ellipse. The section by ^ = i is also an ellipse, the projection of x' v" A" which on z = is -5 + t« = 1 + -^, and the section becomes a" b" c" greater and greater as k becomes greater and greater. If a = d, the section of the surface by any plane parallel' to « = is a circle. Hence the surface is that formed by the THE HYPEEBOLOID OF TWO SHEETS. 51 revolution of the hyperbola —^ — ^ = 1 about its conjugate axis. The figure shews the nature of the surface. 68. The surface whose equation is a" b' c' ~ ' is called an hyperholoid of two sheets. The intercepts on the axis of x are real, those on the other two axes are imaginary. The sections by the planes y=0 and a = are hyper- bolas. The section by the plane a; = is imaginary. The parallel plane x = k does not meet the surface in real points unless k" > a*. If A' > a* the section is an ellipse the axes of which become greater and greater as k becomes greater and greater. The surface therefore consists of two detached portions as in the figure. If B = c, the section by any plane parallel to x=0 is a circle. Hence the surface is that formed by the revolution a;" «" of the hyperbola -i — ^ 1 about its transverse axis. 69. The surface whose equation is Ax' + By' + Cz'=0 is a cone. 4—2 52 THE CONE. A cone is a surface generated by straight lines which always pass through a fixed point, and which obey some other law. The lines are called generating lines, and the fixed point through which they pass is called the vertex of the cone. If the vertex of a cone be taken as origin, the equation of the surface is homogeneous. This follows at once from the consideration that if {x, y, z) be any point P on the surface, any other point (hx, hy, kz) on the line OP is also on the surface. Conversely any homogeneous equation represents a cone whose vertex is the origin of co-ordinates. For, if the values 00, y, z, satisfy a homogeneous equation, so also will kx, ky, kz, whatever the value of h may be. Hence the line through the origin and any point on the surface lies wholly on the surface. The general equation of a cone of the second degree, or quadric cone, referred to its vertex as origin is therefore cm;" + 6/ + ca" + ^yz + 2gzx + 2hxy = 0. 70. If r be the length of the semi-diameter of the surface asi? + hy^ ■\-cz' = 1, we have the relation -s=aP + bin' + cti'. r Hence the direction-cosines of the lines which meet the surface at an infinite distance satisfy the relation aP + brn^ + en" = 0. Such lines are therefore generating lines of the cone aiie' + by' + cz'' = 0. This cone is called the asymptotic cone of the surface. 71. The equation Ax" + By" + 2Wz = is equivalent to ne' y" a? y' 7- + -17 = 2^;, or J — ^ = 2s, according as the signs of A and B are alike or different. THE PARABOLOID. 53 The surface whose equation is I + 1'-"^"' is called an elliptic paraboloid. The sections by the planes a; = and y = are parabolas having a common axis, and whose concavities are in the same direction. The section by any plane parallel to z = is an ellipse if the plane be on the positive side of a = 0, and is imaginary if the plane be on the negative side of z = 0. Hence the surface is entirely on the positive side of the plane s = 0, and extends to an infinite distance. The surface whose equation is is called an hyperbolic paraboloid. The sections by the planes a; = and y = are parabolas which have a common axis, and whose concavities are in opposite directions. The surface is on both sides of the plane s = 0, and extends to an infinite distance in both directions. Si THE PARABOLOID. The section by the plane z = is the two straight lines a 2 given by the equation y - y = 0. The section by any plane parallel to z = is an hyperbola: on one side of the plane z = the real axis of the hyperbola is parallel to the axis of X, and on the other side the real axis is parallel to the axis of y. The figure shews the nature of the surface. 72. It is important to notice that the elliptic paraboloid is a limiting form of the ellipsoid, or of the hyperboloid of two sheets ; and that the hyperbolic paraboloid is a limiting form of the hyperboloid of one sheet. This can be shewn in the following manner. The equation of the ellipsoid referred to (— a, 0, 0) as T^ fJ^ Z^ 2iIC origin is -i + fs + -3 = 0. Now suppose that a, h, all b^ c" become infinite, while — , - remain finite and equal respec- a a v' z* tively to I and I' ; then, in the limit, we have j- + j, = 2a;, which is the equation of an elliptic paraboloid. The other cases can be proved in a similar manner. 73. The equation Aa^ + By''+D=0 represents a cylinder [Art. 10], being a hyperbolic cylinder if A and B have dif- ferent signs, and an elliptic cylinder if A and B have thp same sign. If the signs of A, B, D are all the same the surface is imaginary. ' The equation Aa? + By^ = Q represents two intersecting planes, which are imaginary or real according as the signs of A and B are alike or different. The equation s^ = Ihy represents a cylinder whose guiding curve is a parabola, and which is called a parabolic cylinder. The equation x^ = k represents the two parallel planes X = + s/h. EXAMPLES. 55 Ex. 1. The sum of the squares of the reciprocals of any three diameters of an ellipsoid which are mutually at right angles is constant. If j-j be the semi-diameter whose direction-oosinea are (Zj, mi, rij) we X Z ^ Tllrt^ tin '^^^® n — "S + w + a » ^^^ similarly for the other diameters. By addition , 111111 wehave^ + -, + -, = -, + -, + ^-,. Ex. 2. If three fixed points of a straight line are on, given planes which are at right angles to one another, shew that any other point in the line describes an ellipsoid. Let A, S, G be the points which are on the co-ordinate planes, and P (a;, y, z) be any other fixed point whose distances from A, B, G are a, 6, c. Then - = J, ^=m, and -=n, where I, m, n are the direction cosines of the a b c ^i yi ^i line. Hence the equation of the locus is -5 -H fr -i- -5=1. a' V c' Ex. 3, Find the equation of the cone whose vertex is at the centre of ai; ellipsoid and which passes through all the points of intersection of the ellipsoid and a given plane. x^ tfl z^ Let the equations of the ellipsoid and of the plane be -j -H ?j + -2=1, and lx + my+m=l. We have only' to make the equation of the ellipsoid homogeneous by means of the equation of the plane: the result is X^ w2 z^ For this equation being homogeneous represents a cone whose vertex is at the origin ; and it is clear that the plane cuts the cone and the ellipsoid in the same points. Ex. 4. Find the general equation of a cone of the second degree referred to three of its generators as axes of co-ordinates. The general equation of a quadric cone whose centre is at the origin is ax^ + by^ + cz^ + 2fyz + 2gzx + ihan) = 0. If the axis of a be a generating Une, then 3/=0, z=0 must satisfy the equation for all values of x ; this gives a=0. Similarly, if the axes of y and z be generating lines, 6=0 and c=0. Hence the most general form of the equation of a quadric cone referred to three generators as axes is fyz+gzx+hxy=0. Ex. 5. Find the equation of the cone whose vertex is at the centre of a given ellipsoid, and which goes through all points common to the ellipsoid and a concentric sphere. If the equations of the ellipsoid and sphere be ^ + ^ + -3=!, and j;3+^a+jj2=)-2 respectively; the equation of the cone will be 56 THE CENTBE. Ex. 6. Find the eguation of the cone whose vertex is the point (a, p, y) a? v^ and whose generating lines pass through the conic -^ + is=l> 2=0. Let any generator be -^;— = ^-^= — -. This meets s=0 where I m n ^=a-- 7, and y=p--y. Hence -, ^a-^^J +f, {P--y) =1. or -^{an-y^^+T^i^-ymY^n". Substitute for I, m, n from the equations of the line, equation. the line, and we have -i{az-yn:)''+j^{pz-yy'Y=(z-y)\ the required 74. »If the origin be the centre of the surface, it is the middle point of all chords passing through it; hence if (»,, 2/j, «j) be any point on the surface, the point (— aSj, — y^, — z^) will also be on the surface. Hence we have aiKj' + hi + C2i' + 2/y,^i + 2gz^a;^ + 2hxj/^ + 2ux^ + 2vy^ + 2waj + d=0, and ax^ + ly^ + cz^ + 2fy^z^ + 2gz^x^ + 2hxj)^ - 2ux^ - 2vy^ — 2wz^ + d = ; therefore ux^ + vy^ + wz^ = 0. Since this equation holds for all points (aSj, y,, «,) on the surface, we must have u, v, w aU zero. Hence, when the origin is the centre of a conicoid, the coefficients of x, y and z are all zero. 75. To find the co-ordinates of the centre of a conicoid. Let (f, 7), f) be the centre of the surface; then if we take (?f V' ?) for origin, the coefficients of x, y, and z in the trans- formed equation will all be zero. The transformed equation will be [Art. 44] + 2g{z+^){x + ^)+2h{x+^)(y^r,) + 2u{x + ^) + 2v{y + ri) + 2w{z-\-^ + d = 0. THE CENTRE. Hence the equations giving the centre are 57 and h^ + bv+f^+v = OA (i). Therefore ? h. 9> u h, /, V f> c. w -V a, 9> u h. f. V 9^ c , w a, h, u h, i, V 9> /. w -1 a. h, 9 h. h. f 9> /> G The equation of the conicoid when referred to the centre (?. V, as origin is ax^ + %" + c/ + 2fyz + 2gzx + 2hxy + <^' = (ii), whered' = ^(f »;,?). Multiply equations (i) in order by f , 77, ? and subtract the sum from F{i^, rj, f)j then we have d' = u^+vv + 'ui^+d (iii). From (i) and (iii) we have a, h, g, u =0; h, b, f, V 9. f' <^' «* u, V, w, d — d' therefore d' a, h, 9 = a, h, g. h, h, f h, h, f. 9' f. c 9, f> 0, U, V, w, u V w d .^iv). 58 THE CENTRE. The determinant on the right side of (iv) is called the discriminant of the function F {x, y, z), and is denoted by the symbol A. The determinant on the left side is the discriminant of the terms in F(x, y, z) which are of the second degree; it is also the minor of d in the determinant A, and, as in Art. 57, we shall denote it by D. Equation (iv) may therefore be written c-x But (ii) is the product of linear factors when \ is equal to a, P, or 7. Hence the coefficients a, yS, 7 are the three roots of the equation (iii). The equation when expanded is X»-V(a + 6+c) + \(a&+5e + ca-/'-/-A'') - (a6c + 2fgh - af -hg" - ch') = 0. This equation is called the discriminating cubic. It should be noticed that the equation is the same as that found in Art. 60. 79. We proceed to shew how to find the nature of a conicoid whose equation is given. » First write down the equations for finding the centre of the conicoid ; and from Art. 75 we see that there is a definite centre at a finite distance, unless the determinant a, h, g h, b, f 9> f> c = n IS zero. CONICOIDS WITH GIVEN EQUATIONS. 61 If Z) be not zero, change to parallel axes through the centre, and the equation becomes ckb' + by^ +0? + %fyz + 2gzx + 2hxy + d' = 0, where d' is found as in Art. 75. Now, keeping the origin fixed, change the axes in such a manner that the equation is reduced to the form Then, by Art. 78, a, ^, y will be the three roots of the dis- criminating cubic. [When the discriminating cubic cannot be solved, since its roots are all real [Art. 64], the number of positive and of negative roots can be found by Descartes' Rule of Signs.] Since Bd' = A, the last equation may be written in the form Daaf + i)/3/ + Dyz^ + A = 0. If the three quantities -r- , — r-, ~- are all negative, the surface is an ellipsoid ; if two of them are negative, the surface is a,n hyperboloid of one sheet; if one is negative, the surface is an hyperboloid of two sheets ; and if they are all positive, the surface is an imaginary ellipsoid. If A = 0, the surface is a cone. Ex. (i). llx'+10y''+6z^-8yz + izx-12xy + 72x-72y + 3&z + 150=O. X. ^ ^ J- ii. i dF dF dF . The equations for finding the centre are j— = -^ = -=-=0, or lis- 6y + 2z + 36=0, - 6x+10y-iii-3e=0, 2x~ iy + 6z + 18=0. Therefore the centre is ( - 2, 2, - 1). The equation referred to parallel axes through the centre will therefore be llx'^+10y'' + ez''-8yz+izx-12xy-12=0. [Art. 75 (iii).] The Discriminating Cubic is X* - 27\'' + 180\ - 324 = ; the roots of which are 3, 6, 18. Hence the equation represents the ellipsoid 3x^ + 6^" +182^=12, 3,2 „2 ^s or 4+"2+f-^- We can find the equations of the axes by using the formulae found in Art. 60. The direction-cosines of the axes are ^, |, |; |, |, -|; — 2 a _i 62 CONICOIDS WITH GIVEN EQUATIONS. Ex. (ii). x'' + 2y'+3z''-ixz-43ty + d=0. The Discriminating Cubic is \3-6\2 + 3X + 14=0. All the roots of the cubic are real ; hence, by Descartes' Eule of Signs, there are two positive roots and one negative root. The surface is therefore an hyperboloid of one sheet, an hyperboloid of two sheets, or a cone, according as d is negative, positive, or zero. 80. Next suppose that D = 0. Then the three planes [Art. 75 (i)] on which the centre lies will not intersect in a point at a finite distance from the origin, and we shall have three cases to consider according as the planes meet in a point at infinity, or have a common line of intersection, or are all. parallel to one another. These three cases we shall consider in the following Articles. It should be observed that when i> = one root of the discriminating cubic is zero. 81. The conditions that the planes whose equations are he + hy+fa + v=0, and gx-\'fy+cz + w=0,. may be parallel are These conditions may be written af=gh, bg=hf, ch=fg (i). Now these are the conditions that the terms of the second degree should be a perfect square ; and when this is thq^case it is obvious on inspection. When the terms of the second' degree are a perfect square, the general equation can be written in the form fgh f ^+ ^-+ Ij +2wc + 2vy + 2-wz + d-O'. (ii). If the plane uos + vy + wz = 0' is parallel to the plane f ff h ' - CONICOIDS WITH GIVEN EQUATIONS. 63 the equation (ii) will represent two parallel planes : the con- ditions for this are uf= vg=ivh (iii). If the conditions (iii) are not satisfied, the equation (ii) is of the form At/' +Ba) = 0; which represents a parabolic cylinder whose generating lines are parallel to y = 0, » = G. ^ Hence the general equation of the second degree repre- sents a parabolic cylinder whose generating lines are parallel to the line T. + - + r = 0. liio + yy + wz = Qiy J 9 h provided the conditions (i) are satisfied, and that (iii) are not satisfied. The latus-rectum of the principal parabolic section can be found by the same method as that employed in Conies, Art. 172. Ex. Find the nature of the conicoid whose equation is The equation is (2a! - ^ + izf + 2x-iy + 5z+X=0. This is equivalent to (2a!-3/ + 2«+X)»=x(4\-2)-j/(2X-4)+«(4X-5)-l. The planes 2a:-y + 22+X=0, and a;(4X-2)-j/(2X-4) + «(4X-5)-l = 0, will be perpendicular, if X=l. Hence the equation of the surface may be written (2a;-j; + 2z+l)2=2a; + 2^-z-l,. /•2a -y + 2z + l V_l 2x + 2y-z-l «"^ I, 3 ) ~3- 3 Hence, taking 2x-y+2z+l=0, and 2x + 2y-g-l=0 as the planes y=0 and a;=0 respectively, the equation of the surface will be Hence the latus-rectum of a principal parabolic section is = . 64 CONICOIDS WITH GIVEN EQUATIONS. 82. Next suppose that the three planes on which the centre lies are not all parallel, but that they have a common line of intersection. If -we take any point on the line of centres for origin, the equation will take the form ax^ + hy^ + cz^ + 2fyz + 2gzx + 2hoy + d' = 0. Then, keeping the origin fixed, by transformation of axes the equation will be reduced to the form aa?->r^y^-^d'=0 (i). One root of the discriminating cubic is zero, since i) = ; and the roots a, /3, are given by the equation X' - X'' (a + 6 + c) + X (6c + ca + ah -f - ^r' - A") = 0. If d' = 0, the surface represented by the equation (i) js two planes, real or imaginary. If d' be not zero, the surface is a cylinder. The conditions that the three planes ax-\-hy + gz + u=0, hx + by+fz + v =0, gx +fy + cz +w = 0, may have a common line of intersection, are given by a, h, g, u = 0, [Art. 18] h. b, /, V g, f, c, w u^v=w=i ) = 0. that is, Ez. Find the nature of the conicoid whose equation is 32a;'> + j^" + iz^ -16zx-6xy+%x-20y-8z + 103 = 0. The equations giving the centre are 32a!-4^-82 + 48=0, - ix+ y -10=0, and - 8a! +iz- 4=0. Hence there is a line of centres. Find one point on the line, for example (0, 10, 1), and change the origin to the point (0, 10, 1) : the equation will then become 32x^ +y^+ iz" - 16^3; - 8xy = 1. CONICOIDS WITH GIVEN EQUATIONS. 65 The Discriminating Cubic is \'-37X'+84:X=0. One root is zero, and ttie other two roots are positive ; hence the equation is an elliptic cylinder. The axis of the cylinder is the line of centres ; and its equations are X _ y-10 _ z-1 1~ i ~~2~" 83. If the planes on which the centre lies meet at a point at infinity, we proceed as follows. Since one root of the discriminating cuhic is zero, the equation can always be solved : let the roots be a, /3, 0. Find the directions of the principal axes of the surface, by means of the equations of Art. 60; and take axes parallel to these principal axes. The equation will then become aa;" + yS?/" + 2u'a! + 2v'y + %i)'z + d = 0, or, by a change of origin, aa!''+/32/" + 2wV = 0. Hence the surface is a paraboloid, the latera recta of its principal parabolic sections being and -3- . Ex. Find the nature of the surface whose equation is 32?- 6s^z-6za! - 7a;- 52/ + 62+3=0. The Discriminating Cubic is X'-3\2-18X=0; the roots of which are 6, -3,0. 1 1 —2 The direction-cosines of the principal axes are -j^ , —rz, -jr- ; Vo \/o VD -To » -To ) -Td ; ^iid "To > —m > "• Hence to find the equation referred to vo \/o vo y^ V^ axes parallel to the principal axes, we must substitute for X, y,s respectively. The equation will then become 6a;2-3j/»-V6a!-2^3j/-^22! + 3=0; or, by changing the origin ear" — 3^' - ^iz = 0. Thus the surface is a hyperbolic paraboloid, the latera recta of the principal parabolas being i,J2 and ^^2. s. s. a. 5 fab CONDITION FOR A CONE. 84. It follows from Art. 75 (ii) and (iv) that when D is not zero, the necessary and sufficient condition that the surface represented by the general equation of the second degree may be a cone is A = 0, When A = and also Z) = 0, then will U, Fand W be all zero*: hence [Arts. 81 and 82] the surface must be either a cylinder or two planes ; and cylinders and planes are limiting forms of cones. Conversely, when the surface re- presents a cylinder, or two planes, U, V, W and D are all zero, and therefore also A = 0. Hence A = is the necessary and sufficient condition that the surface represented by the general equation of the second degree may be a cone. 85. To find the conditions that the surface represenied hy the general equation of the second degree may he a surface of revolution. We require the condition that two of the roots of the dis- criminating cubic may be equal. In that case one' + hf+ c^ -I- Ifyz + 'igzx -|- ^hxy can be transformed into Hence as? + hf + c/ -1- Ifyz + ^zx -1- %hxy -\{a? -^f + ^)... (i), * This can be proved as follows : ^ We have uU+vV+wW+dD=^. And, since a determinant vanishes when two of its rows are identical, we have also aV+'hV+gW+uD=0, hTJ+hV+fW+vB=Q, and gX7+fV+cW+wD=0. Hence when A=0 and D=0, unless U, V, W are all zero, we can eliminate U, V, W from the first equation and any two of the others : we thus obtain three determinants which are all zero ; but these determinants are V, r, and W. SURFACE OF REVOLUTION. 67 can be transformed into ax'+ay^+V(c-\)=/Y V(c - X) V(a - \) =•*. I ^ (iii); V(a -X)V(6 ->.)■=> J /v Hence, iff, g, h be all finite, we have ^_^=6,_¥=,_f = x (iv),. the required conditions. Let h, any one of the three quantities f g, h, be zero ; then from (iii) we see that \ = a or \ =b, and therefore also 9 = or/=0. Suppose g = and h = 0; then \ = a, and the condition for a surface of revolution is (b-a)ic-a)=f (V). Examples on Chapter III. 1. Determine the nature of the surfaces represented by the following equations : (i) af-2y'+6»'+l2xs: + a' = 0. l^iA - I Jvju/ (ii) s(^ + y' + z' + 4:xy-2xz + 4:yz=l. .. ( (iii) aj' - 2xy — 2yz — 2zx = a'. ,, / '• (pr) S2s(?^y' + i!s'-\lozx-&xy=\. zM'j'-'^^- (v) Jx+Jy+Jz = Q. t^— . (vi) 2a;'~+5y* + «'-4x2/-2a;-42/-8 = 0. ^iJ^- ra-^ b—2 68 EXAMPLES ON CHAPTER III. 2. Find the nature of the surfaces represented by the following equations : (i) a;'+22/»-3«»-4y«+8aiB-12a!2/ + l=0. ^^ , ■ (ii) 2x' + 2y'-W-2yz-2zx-5xy-2x-2y + z=0. f^YJ^ (iii) 5x''-y' + z' + 6xz + ^y + 2x + iy + 6z = 8. « > (iv) 2a!' + Zy" + Syn + 2zx + hxy - 4^ + 8a - 32 = 0. i\ c^ Find the equations of the axes of (i), and the latera recta of the principal parabolas of (ii) and of (iii). ^.'>. . /—it -1— ' 3. Shew that the equation ' * ^ \^ y75 ' Ijj^/ a? + if + ^ + yz + zx + xy=\, represents an ellipsoid the squares of whose semi-axes are 2, 2, \. Shew also that the equation of its principal axis is k = ?/ = «. 4. Shew that, if the axes, supposed rectangular, be turned round the origin in any manner, m' + »' + vf will be unaltered. 5. Shew that, if three chords of a coniooid have the same middle point, they all lie in a plane, or intersect in the centre of the conicoid. 6. Through any point lines are drawn in fixed directions which meet a given conicoid in points P, P and Q, Q' respectively; shew that the rectangles OP, OP and OQ, OQ' are in a constant ratio. 7. If any three rectangular axes through a fixed point cut a given conicoid in P, P ; Q, Q' and R, R' ; then will PP' QQ" RR" OP. OP' "^ 0Q\ OQ" ■*" OR'. OR"' A 111 OP. OP OQ.OQ' OR . OR" be constant. CHAPTER IV. CoNicoiDS Eefeered to their Axes. 86. In the present chapter we shall investigate some properties of couicoids, obtained by taking the equations of the surfaces in the simplest forms to which they can be reduced. We shall begin by considering the Sphere. The Sphere. 87. The equation of the sphere whose centre is (a, b, c) and radius d is [Art. 5] (x-ay+(3/-by+{z-cy= ^i> 1 K + y2+^2' ^'a. 2/2' ^.' 1 «>S+ys+^>' <^^' 2/8' ■2^8. 1 "'i+Vi+^i' I"V ^4' ^4. 1 89. The equation of the tangent plane at any point {x', y', z') of the sphere whose equation is x^ + y^ + z^ = a" is xx' + yy' + zz = a' [Art. 52, Ex. 1]. This result can be obtained at once from the fact that the tangent plane at any point («', 2/'i z') on a sphere is perpendicular to the line joining {00, y', a') to the centre. This gives for the equation of the plane {X -x') x'+(y- t/) y' + {z- if) z' = 0, or xx' + yyf + ^/ = a^ The polar plane of any point (a/, 3/', nT) can be shewn, by 'the method of Art. 53, to be xx' + y'j/ + zz' = a". 90. It can be easily shewn, that if /S = be the equation of a sphere (where S is written for shortness instead of al' + f + z'+ 2Ax + 2By + 2Cz + D), and the co-ordinates of any point be substituted in 8, the result will be equal to the square of the tangent from that point to the sphere. Hence, \i 8=0, and /S' = be the equations of two spheres (in each of which the coefficient of a!" is unity), S^ = ;S'«is the locus of points, the tangents from which to the two spheres are equal. The surface whose equation ia 8— 8' = passes through all points common to the two spheres 8=0, and 8=0; for, if the co-ordinates of any point satisfy the equations 8 = and jSi' = 0, they will also satisfy the equation 8 — 8' = 0. NowjS— /S'=0 is of the first degree, and therefore represents a plane. The plane through the points of intersection of two spheres is called their radical plane. THE ELLIPSOID. 71 We have seen that the tangents drawn to two spheres from any point on their radical plane are equal. The radical planes of four given spheres meet in a point, viz. in the point given by 8^=8^ = 8^= 8^, where 8^ = 0, 8^ = 0, 8^ = 0, 8^=0 are the equations of the four spheres, in each of which the coefiScient of a^ is unity. This point is called the radical centre of the four spheres, Ex. 1. Find the equation of the sphere which has {x^, y^, Zj) and (jJa, ^2, Zj) for extremities of a diameter. If (a;, y, z) be any point on the sphere, the direction- cosines of the lines joining (k, y, z) to the two given points are proportional to x-x^, y-yi, 2-«i, and a!-a!j, y-y^z-z^. The condition of perpendicularity of these lines gives the required equation (x-x^(x-x^ + (y-y.^(y-y^ + (z-z^(z-z^ = 0. Ex. 2. The locus of a point, the sum of the squares of whose distances from any number of given points is constant, is a sphere. Ex. 3. A point moves so that the sum of the squares of its distances from the six faces of a cube is constant; shew that its locus is a sphere. Ex. 4. A,B axe two fixed points, and P moves so that P'A=nPB ; shew that the locus of P is a sphere. Shew also that all such spheres, for different values of n, have a common radical plane. Ex. 5. The distances of two points from the centre of a sphere are pro- portional to the distance of each ficom the polar of the other. Ex. 6. Shew that the spheres whose equations are a?+yi+z!'+2Ax + 2By + 2Cz+D=0, and x'+y'+z'' + 2ax + 2by + 2cz + d=0, cut one another at right angles, if 2Aa+2Bb + 2Ge-D-d=0. 91. We proceed to prove some properties of the ellipsoid; and we shall always suppose the equation of the surface to be «" y^ s^ , unless it is otherwise expressed. To obtain the properties of the hyperboloids we shall only have to make the necessary changes in the signs of V and c". 72 DIRECTOR-SPHERE. We have already seen [Art. 62] that the equation of the tangent plane at any point («', y', z') is axx; y]/ z^ . ,j. The length of the perpendicular from the origin on the tangent plane at the point (a/, ^ , /) is [Art. 20] given by the equation i,.<+C+$ J {aX + Vm^ + cX"). By squaring both sides of these equations and addlllg, we have in virtue of the relations between the direction-cosines of perpendicular lines a? + 'ff!? = a^ + V' + c\ The required locus is therefore a sphere. This sphere is called the director-sphere of the ellipsoid. 93. The normal to a surface at any point P is the straight line through P perpendicular to the tangent plane at P. NORMALS. 73 The normal to an ellipsoid at the point (td, y', z') is therefore x — x' _ y—y' _ z — z' JL~ £ ~ ^ ' a' 6» c" Since p«(^'+|; +$) = !, [Art. 91.] the direction-cosines of the normal are px 'Pi/ pz' 1?' w "?■• 94. If the normal at («', y', z') pass through the par- ticular point (/, g, h) we have f-^ . 9-1/ _^-!f JL JL jfl ' a' 6» c» Put each fraction equal to \, then aV , Par , , c% * = g . ^ , V = T. ^ and 2r = -= — - . Hence, since we have Since this equation for \ is of the sixth degree, it follows that there are six points the normals at which pass through a given point. Ez. 1. The normal at any point P of an ellipsoid meets a principal plane in G. Shew that the locns of the middle point of PG is an ellipsoid. Ex. 2. The normal at any point P of an ellipsoid meets the principal planes in (?„ G^, 63. Shew that PG^, PG^, PGg are in a constant ratio. Ex. 3. The normals to an ellipsoid at the points P, P" meet a principal plane in G, ^; shew that the plane which bisects PP' at right angles bisects GG'. 74 DIAMETEAL PLANES. Ex. 4. If P, Q be any two points on an ellipsoid, the plane through the centre and the line of intersection of the tangent planes at P, Q, will bisect PQ. Ex. 5. P, Q are any two points on an ellipsoid, and planes through the centre parallel to the tangent planes atP, Q cut the chord PQ in P', Q'. Shew that PP'=QQ'. 95. The line whose equations are « — a_y — y8_s— 7_ I ~ m n ' meets the surface where (oi+lrY (0+mrf . (y + nr)' _ a' ^ b" ^ c" ~ If (a, yS, 7) be the middle point of the chord, the two values of r given by the above equation must be equal arid opposite; therefore the coefficient of r is zero, so that we have «» ^ 6' "^ c« ~ Hence the middle points of all chords of the ellipsoid which are parallel to the line I m n are on the plane whose equation is This plane is called the diametral plane of the line I m n' The diametral plane of lines parallel to the diameter through the point («', y, z) on the surface is a^ + J« +c*"~" W, hence the diametral plane of any diameter is parallel to the tangent plane at the extremities of that diameter. CONJUGATE DIAMETERS. 75 The condition that the point {x', y", z") should be on the diametral plane (i) is a^x" i/y" a'y ^ The symmetry of this result shews that if a point Q be on the diametral plane of OP, then will P be on the diametral plane of OQ. Let OR be the line of intersection of the diametral planes of OP, OQ ; then, since the diametral planes of OP, OQ pass through OR, the diametral plane of OR will pass through P and through Q, and will therefore be the plane POQ, so that the plane through any two of the three lines OP, OQ, OR is diametral to the third. Three planes are said to be conjugate when each is dia- metral to the line of intersection of the other two, and three diameters are said to be conjugate when the plane of any two is diametral to the third. 96. If K, y„ »i), («„ y„ e,) and («,, y„ z,) be extremities of conjugate diameters, we have from Art. 95, „ + pS - U SB£!. ■ b' l"» + ^»+f!fs=0 .(i). Also, since the points are on the surface, + 6" + c' ,» ^ W T^ = 1 .(ii). 76 CONJUGATE DIAMETERS. Now from equations (ii) we see that a' y, b' c' a' b' c' ^^^ a' b' are direction-cosines of three straight lines, and from equations (i) we see that the straight lines are two and two at right angles. Hence, as in Art. 45, we have and «'iyi+^^y,+«><:ys=o\ .(iii), 3/i^i +y. ^.+2/8^8 = ^A + ^^,+z,x. = o] We have also from Art. 46. ^ h fs = 1, or a'l. y^ ^1 a' b' c a'a. y^. «2 5 h a' b' c a^g, ^8. «8 «3 y, a' 6-- c •Civ). = abc . .(V). From (iii) we see that the sum of the squares of the pro- jections of three conjugate semi-diameters of an ellipsoid on any one of its axes is constant. Also, by addition, we have, the sum of the squares of three conjiigate diameters of an ellipsoid is constant. ^ From (v) we see that the volume of the parallelopiped which has three conjugate semi-diameters of an ellipsoid for conterminous edges is constant. In the above the relations (iii) and (iv) were deduced from (i) and (ii) by geometrical considerations. They could however be deduced by the ordinary processes of algebra without any consideration of the geometrical meaning of the quantities, and hence the results are true for the hyper- boloids. CONJUGATE DIAMETEBS. 77 97. The two propositions (1) that the sum of the squares of three conjugate semi-diameters is constant, and (2) that the parallelopiped which has three conjugate semi-diameters for conterminous edges is of constant volume, are extremely important. We append other proofs of these propositions. Since in any conic the sum of the squares of two conjugate semi- diameters is constant, and also the parallelogram of which they are adjacent sides, it follows that in any conicoid no change is made either in the sum of the squares or in the volume of the parallelopiped, so long as we keep one of the three conjugate diameters fixed. We have therefore only to shew that we can pass from any system of conjugate diameters to the principal axes of the surface by a series of changes in each of which we keep one of the conjugate diameters fixed. This can be proved as follows : — let OP, OQ, OR, be any three conjugate semi-diameters, and let the plane Q OB cut a principal plane in the line OQ", and let OR' be in the plane QOR conjugate to Oi^; then OP, OQ', OR' are three conjugate semi-diameters. Again, let the plane POR' meet the principal plane in which Q' lies in the line OP', and let OR" be conjugate to OF' and in the plane POE ; then OF', OQ' and OR" are semi-conjugate diameters. But, since OR" is conjugate to OP" and to OQ', both of which are in a principal plane, it must be a principaJ diameter. Hence, finally, we have only to take the axes of the section Q'OF' to have the three principal diameters. 98. It is known that any two conjugate diameters of a conic will both meet the curve in real points when it is an ellipse; that one will meet the curve in imaginary points when it is an hyperbola ; and that both will meet the curve in imaginary points when it is an imaginary ellipse. Hence, by transforming as in the preceding Article, we see that three conjugate diameters of a conicoid will all meet the surface in real points when it is an ellipsoid ; that one will meet the surface in imaginary points when it is an hyper- 78 CONJUGATE DIAMETERS. boloid of one sheet ; and that two will meet the surface in imaginary points when it is an hyperboloid of two sheets. 99. To find the equation of an ellipsoid referred to three conjugate diameters as axes. Since the origin is unaltered we substitute for x, y and z expressions of the form Ix + my + nz in order to obtain the transformed equation [Art. 47]. The equation of the ellipsoid will therefore be of the form Ao^ + Bf + C/ + 'LFyz + 2GZW + 2Hxy = 1. By supposition the plane ai = bisects all chords parallel to the axis of x. Therefore if («„ y^, z^ be any point on the surface, (— a;,, y„ z^ will also be on the surface. Hence Gz^x^ + -5«,yi = for aU points on the surface : this requires that(? = S=0. Similarly, since the plane ^ = bisects all chords parallel to the axis of y, we have H= F=0. Hence the equation of the surface is 2 2 2 X y * _ i or ^«+F' + 7'~ ' where a', b', c' are the lengths of the semi-diameters, 100. We may obtain the relations between conjugate diameters of central conicoids by the following method : — The expression -r" 77" z" is transformed, by taking for axes three conjugate diameters which make angles a, 13, 7 with one another, into the expression x" ifl 1^ -75 + p2 + -^ + X («' + 2/' + «° 4- 2yz cos a. + Izx cos j8 + 2xycoS'y). The two expressions will therefore both split up into linear factors for the same values of X. Hence the roots of the cubica & + ^)(^' + '^)& + '^) = ^' CONJUGATE DIAMETEES. 79 and ^ + x. X. cos 7 , \C0Sy8 \ COS 7 , y2 + X , Xcosa \cos/S, \ cos a , ^i+x = are equal to one another. Hence, by comparing coefficients in the two equations, we have a» + 6« + c^ = a"' + 6'» + c'» (i), hY + cV + a'V = 6'V» sin^a + c^d^ sin=/3 +a%"&\ii' f , c, n I, TO, n, or Ar + Bm' +Gn' + 2 Fmn +2Gnl+ 2Hlm = 0, where A, B, C, &c. are the minors of a, b, c, &c. in the deter- minant a, h. 9 h, h, f 9, /. 108. If through the vertex of a given cone lines be drawn perpendicular to its tangent planes, these lines generate another cone called the reciprocal cone. The line through the origin perpendicular to the plane lx + my + ns = 0, is f = ^ = i . RECIPROCAL CONK. 85 Hence, from the result of the last article, the reciprocal of the cone as^ + hy^ + 6^ + 2/2/2 + 2gzx + 2hooy = 0, is Ax* + Bf + Ge* + 2Fyz + 20zx + IHxy = 0. Since the minors of A, B, C, &c. in the determinant A, H, H, B, F G, F, are proportional to a, b, c, &c., we see that the relation be- tween the two cones is a reciprocal one. As a particular case of the above, the reciprocal of the cone aa^ + by' + c^ = 0,is-+f + - = 0. " a b c From this we see at once that a cone and its reciprocal are co-axial. 109. To find the condition that a cone may have three perpendicular generators. Let the equation of the cone be aa? + bf + cz* + 'ifyz + Igzx + 2hxy = (i). If the cone have three perpendicular generators, and we take these for axes of co-ordinates, the equation will [Art. 73, Ex. 4] take the form Ayz-\-Bzx+Gxy=Q .: (ii). Since the sum of the co-efficients of a?', y* and z^ is an in- variant [Art. 79] and in (ii) the sum is zero ; therefore the sum must be zero in (i) also. Therefore a necessary condition is a + b + c=0 (iii). If the condition (iii) is satisfied there are an infinite number of sets of three perpendicular generators. For take any generator for the axis of x; then by supposition any point on the line y — 0,z = Q is on the surface ; therefore the 86 CONE WITH THREE PERPENDICULAR GENERATORS. co-eflScient of a? is zero, so that the transformed equation is of the form %'+ C!?+2fys+ 2gzx+ 'ihxy = (iv); and since the sum of the co-efficients of tc", i/^, z' is an in- variant, -sve have 6 -I- c = 0. Now the section of (iv) by the plane a; = is the two straight lines bf+cz' +2/^2=0; and these are at right angles, since h +c = 0, 110. If a cone have three perpendicular tangent planes, the reciprocal cone will have three perpendicular generators. Hence the necessary and sufficient condition that the cone ax' + hf + cz^ + Ifyz + 2gzx + 2kxy = 0, may have three perpendicular tangent planes is ^+5-1- (7 = 0. Ex. 1. CP, CQ, GK are three central radii of an ellipsoid which are mutually at right angles to one another ; shew that the plane PQi2 touches a sphere. Let the equation of the plane PQiJ be fci!+m2/+7i«=y. The equation of the cone whose vertex is the origin, and which passes through the intersection of the plane and the ellipsoid ^^ + |' + f!=l, is^-' + g +?;= {^f^±JI^^\\ By supposition the cone has three perpendicular generators; therefore 1 i: i-i Ex. 2. Any two sets of rectangular axes which meet in a point form six generators of a cone of the second degree. • Ex. 3. Shew that any two sets of perpendicular planes which meet in a point all touch a cone of the second degree. 111. To find ^e equation of the tangent cone from any point to an ellipsoid. Let the equation of the ellipsoid be a c TANGENT CONE. 87 Let the co-ordinates of any two points P, Q be x', y', z' and a;", y", /' respectively. The co-ordinates of a point which divides PQ in the ratio m : n are wa;' -H mal' ny' + my" m' + tnz" m + n ' m + n ' m + n If this point be on the ellipsoid, we have (nx + mcd'y {ny -f m«/")' [nz -f- »n/')' , .„ or If the line P Q cut the surface in coincident points, the above eqiiation, considered as a quadratic in — , must have equal roots ; the condition for this is U+F + 7-VU+V + ^-O _/«« «2/ zz _ Y ~ V a" 6' c' / ■ Hence, if the point P («', y', z) be fixed, the co-ordinates of any point Q, on any tangent line from P to the ellipsoid, must satisfy the equation U* + 6» + c' -^JU J' c" ; Hence (i) is the required equation of the tangent cone from {x', y', z^) to the ellipsoid. 112. If we suppose the point («', y', z') to move to an infinite distance, the cone will become a cylinder whose generating lines are parallel to the line from the centre ot the ellipsoid to the point {x', }/, /). 88 ENVELOPING CYLINDER. Hence, if in the equation of the enveloping cone we put a;' = h',y' = mr, z' = nr, and then make r infinitely great, we shall obtain the equation of the enveloping cylinder whose generating lines are parallel to X _y _z I m n' Substituting Ir, mr, nr for x', y', z' respectively in the equation of the enveloping cone we have (^ f z' \fP m^ m" 1\ \a 6 c r) Hence, when r is infinite, W + t' + c* V V 6" cV U'^ V ^ cV 113. The equation of the enveloping cylinder can be found, independently of the enveloping cone, in the following manner. The equations of the straight line which is drawn through any point [x , y, z') parallel to as _ y _z I m n' co — x v — v z — z are — = — = - — ^- = = r. I m n The straight line will meet the ellipsoid in two joints whose distances from {x', if, a') are given by the equation The straight line will therefore touch the surface, if («l\y\z'* ^(V m' n\ (lx'my'nz'\' EXAMPLES. 89 Hence the co-ordinates of any point, which is on a tangent line parallel to - = ^ = - I m n' satisfy the equation which is the required equation of the enveloping cylinder. Ex. (i). To find the condition that the enveloping cone may have three perpendicular generators. The equation of the enveloping cone whose vertex is (a;*, y', s!) is If this have three perpendicular generators the sum of the coefficients of a', y\ and «» must be equal to zero [Art. 109]. Hence {of, y, «"), the vertex of the cone, is on the surface Ex. (ii). Shew that any two enveloping cones of an ellipsoid intersect in plane curves. The equations of the cones whose vertices are (a/, /, 2') and (aj", 1/', «") are respectively. The surface whose equation is passes through their common points, and clearly is two planes. Ex. (iii). Find the equation of the enveloping cone of the paraboloid 03!" + 6^" + 22=0. Ant. (oa? + Ji/" + 22) [asfl +62/''' + 22') = {aastl +byy'+z+z')K Ex. (iv). Find the locus of a point from which three perpendicular tangent lines can be drawn to the paraboloid aa;" + ii^'' + 22=0. Ans. ab( a? z' 8 = 0, and whose vertex is any point of the hyperbola -5 — j-, — r-, = 1, y = 0, is a right circular cone. 22. A cone, whose equation referred to its principal axes, is is thrust into an elliptic hole whose equation is -5 + fa = 1 J shew that when the cone tits the hole its vertex must lie on the ellipsoid IB' y' ,/l 1\ , 23. In a cone any system of three conjugate diameters meets any plane section in the angular points of a triangle self polar with respect to that section. 24. The enveloping cones which have as vertices two points on the same diameter of a conicoid intersect in two parallel planes between whose distances from the centre that of the tangent plane at the end of the diameter is a mean proportional. What is the corresponding proposition for a paraboloid 1 25. Shew tliat any two enveloping cones intersect in plane curves; and that when the planes are at right angles to one another, the product of the perpendiculars on one of the planes of contact from the centre of the ellipsoid and the vertex of the corresponding cone, is equal to the product of such perpendiculars on the other plane of contact. 26. If a line through a fixed point be such that its con- jugate line with respect to a conicoid is perpendicular to it, shew that the line ia a generating line of a quadric cone. 27. The locus of the feet of the perpendiculars let fall from points on a given diameter of a conicoid on the polar planes of those points is a rectangular hyperbola. EXAMPLES ON CHAPTER IV. 93 28. Prove that the surfaces K_" y^ _ 2« ^' 3/* _ 2« x" y' 2a will have a common tangent plane if V, a.% a,' =0. 29. Prove that an ellipsoid of semi-axes a, 6, c and a concen- tric sphere of radius ^ are so related that an in- ^6 V + c V + d?V definite number of octahedrons can be inscribed in the ellipsoid, and at the same time circumscribed to the sphere, the diagonals of the octahedrons intersecting at right angles in the centre. 9 g s 30. Pind the locus of the centre of sections of -s + ?j +.-, = 1 c? 6' ^ x' tf s' which touch -^ + |^ + -15 = 1. 31. Planes are drawn through a given line so a,s to cut an ellipsoid; shew that the centres of the sections so formed all lie on a conic. 32. Find the locus of the centres of sections of an ellipsoid by planes which are at a constant distance from the centre. 33. Shew that the plane sections of an ellipsoid which have their centres on a fixed straight line are pai-allel to another straight line, and touch a parabolic cylinder. 34. The locus of the line of intersection of two perpendicular tangent planes to ax' + by' + C!^ = is a{b + c)os' + b{o + a)y' + c{a + h)i^ = 0. 35. The points on a conicoid the normals at which intersect the normal at a fixed point all lie on a cone of the second degree whose vertex is the fixed point. 36. Normals are drawn to a conicoid at points where it is met by a cone which has the axes of the conicoid for three of its generating lines; shew that all the normals intersect a fixed diameter of the conicoid. 94 EXAMPLES ON CHAPTER IV. 37. Shew that the six normals which can be drawn from any point to an ellipsoid lie on a cone of the second degree, three of whose generating lines are parallel to the axes of the ellipsoid. 38. Find the equations of the right circular cylinders which circumscribe an ellipsoid. 39. If a right circular cone has three generating lines mutually at right angles, the semi-vertical angle is tan" '^2. 40. If one of the principal axes of a cone which stands on a given base be always parallel to a given right line, the locus of the vertex is an equilateral hyperbola or a right line according as the base is a central conic or a parabola. 41. The axis of the right circular cone, vertex at the origin, which passes through the three lines, whose direction-cosines are (?,, m,, Mj), (If, m^, Wj), (l^, m^, raj is normal to the plane = 0. 0, 1, 1. 1 <», K, ^.. h y. m^, »»,. m, m. w„ M,. »•. 42. The equations of the axes of the four cones of revolution which can be described touching the co-ordinate planes are of siu^a sin=^" smy a, j8, y being the angles YOZ, ZOX, and XOZ respectively. 43. Prove that four right cones may be described, passing through three given straight lines intersecting in the same point, and that if 2a, 2/8, 2y be the mutual inclinations of the straight lines, the equations of the cones referred to the straight lines as co-ordinate axes will be sinV sin'^fi sin"-/ . sin'a coa'B cos'v „ -+ — -+ '- = 0, -f ^-1- ^ = 0, y z X y z X cos'a X Bin B cos- + — !i+ y z 1 = 0,- cosa X cos'/S sin^y y « = 0. EXAMPLES ON CHAPTER IV. 95 44. Shew that, if P, Q, R be extremities of three conjugate diameters of a conicoid, the conic in which the plane PQR cuts the surface contains an infinite number of sets of three conjugate extremities, which are at the angular points of maximum triangles inscribed in the couic PQR. 45. Shew that, if the feet of three of the six normals drawn from any point to an ellipsoid lie on the plane Ix + my + nz + p = 0, the feet of the other three will be on the plane aa; 6v CIS 1 „ I m n p the equation of the ellipsoid being aa^ + hy' + Ga? = \. 46. Prove that the locus of a point with which as a centre of conical projection, a given conic on a given plane may be projeotpd into a circle on another given plane, is a plane conic. 47. If C be the centre of a conicoid, and P (Q) denote the perpendicular from P on the polar plane of Q ; then will P{Q) G{Q) Q{P)-G(Py 48. The locus of a point such that the sum of the squares of its normal distances from a given ellipsoid is constant, is a co-axial ellipsoid. 49. If a line cut two similar and co-axial ellipsoids in P, P" ; Q, Q'; prove that the tangent plane to the former at P, P", meet those to the latter at Q or Qi in pairs of parallel lines equi- distant respectively from Q or Q', 50. A chord of a qxiadric is intersected by the normal at a given point of the surface, the product tJf the tangents of the angles subtended at the point by the two segments of the chord being invaiiable. Prove that, being the given point and P, P" the intersections of the normal -witli two such chords in perpendi- cular normal planes, the sum of the reciprocals of OP, OP", is invariable. CHAPTER V. Plane Sections of Conicoids. 114 We have seen [Art. 51] that all plane sections of a conicoid are conies, and also [Art. 61] that all parallel sections are similar conies. Since ellipses, parabolas, and hyperbolas are orthogonally projected into ellipses, parabolas, and hyperbolas respectively, we can find whether the curve of intersection of a conicoid and a plane is an ellipse, parabola, or hyperbola, by finding the equation of the pro- jection of the section on one of the co-ordinate planes. For example, to find the nature of plane sections of a paraboloid. The plane Ix + my + nz + p = cuts the paraboloid ax* + bi^+2z = 0,m a curve through which the cylinder a (my + m +pf + Wf -^ 2Vz = passes. The plane a; = 0, which is perpendicular to the generating lines of the cylinder, cuts it in the conic whose equations are a = 0, a {my ■+nz + pf + Wi^ -f %l^z = ; and this conic is the projection of the section on the plane x=0. If M = 0, the projection will be a parabola ; but, if n be not zero, the projection will be an ellipse or hyperbola accord- ing as m^{am*+W) - a^m^r^ is positive or negative, or ahl'n^ positive or negative, that is, according as the surface is an elliptic or hyperbolic paraboloid. AREA OF CENTRAL SECTION. 97 Hence all sections of a paraboloid which are parallel to the axis of the surface are parabolas ; all other sections of an elliptic paraboloid are ellipses, and of a hyperbolic paraboloid are hyperbolas. Ex. 1. Find the condition that the section of ax^ + by" + cz''=lhy the plane la>+my + nis+p = may be a parabola. P m' n" . Ans. - + -r H — =0. a c Ex. 2. Shew that any tangent plane to the asymptotic cone of a conicoid meets the conicoid in two parallel straight lines. 115. To find the axes and area of any central plane section of an ellipsoid. Let the equation of the ellipsoid be a;" 2/" / , U2- -I = 1 „» -I- J2 + p. J-. and let the equation of the plane be Ix + my + nz = (i). Every semi-diameter of the surface whose length is r is a generating line of the cone whose equation is [p. 55, Ex. 5] «-(i-^.) + ^(^i) + «-(i-^)-o (ii). This cone will, for all values of r, be cut by the plane in two straight lines which lie along equal diameters of the section ; and, when r is equal to either semi-axis of the section, these equal diameters will coincide. That is, the plane (i) _ will touch the cone (ii) when r is equal to either semi-axis of the section of the ellipsoid by the plane. The condition of tangency gives r m' , n' a' r" S From (iii) we see that .(iii). abc abc .(i^), where r^, r^ are the semi-axes of the section, and p is the perpendicular on the parallel tangent plane. s. s. G. '^ 98 PLANE SECTIONS. From (iv) we see that the area of the section is equal to iraho 116. To find the area of any plane section of an ellipsoid. Take for co-ordinate planes three conjugate planes of which ^: = is parallel to the given plane; then the equations of the surface and of the given plane will be respectively of the forms a? v" / ^7-2 + ^2+^2=1. and0 = A. The cylinder whose equation is ^,2+^2+^-2 1, passes through the curve of intersection of the surface and the plane ; and the area of the section of this cylinder hj z = k\s irab smi/l 1 — tjI , V being the angle X07. The area of the section of the ellipsoid by ^^ = is ira'b' sin v. Hence, if A be the required area, and A^ be the area of the parallel central section, we have A = A,{1-^). Now the tangent plane at (0, 0, c') is » = c' ; therefore the perpendicular distances of the given plane and of the parallel tangent plane from the centre are in the ratio of k : c. Hence A = aJi-^1) (i), where p and p„ are the perpendicular distances of the given plane and of the parallel tangent plane from the centre. This gives the relation between the area of any section and of the parallel central section ; and we have found, in Art. 115, the area of any central section. PLANE SECTIONS. 99 Hence the area of the section of the ellipsoid whose equation, referred to its principal axes, is a» + 6= + c" ~ ^' made by the plane whose equation is lx+ my + nz =p, is -r-7-. V(a irahv. r ■+ bW (l P' + cV j • For A = irahc [Art. 115], V(a=i' + 6W + cW) and p:- = aT + 6W + cV [Art. 91]. Ex. 1. To find the area of the section of a paraboloid by any plane. Let the equation of the paraboloid be aa? + hy^ + ^=(i, and let the equa- tion of the section be te+mj+n2+p = 0. The projection of the section on the plane «=0 is the conic o ax^ + 6j/' — (te + m?/ +y) = 0, The area of the projection is and therefore [Art. 31] the area of the section is T i'" »»^ n \ 7= -- +T-+2pret. ifijab'-a b -^ ) Ex. 2. To find the area of the section of the cone — + ^ + - = by the plane Ix+my+m^p. H-i yl ni The area of the section "^ Ti + ^ + •t = 1 by the given plane is ^(kaP+klm''+kcn^ ( kaP + kbm'+hcn'y If we put ft = the surface becomes the cone. The required area is therefore irp^ Jcibc (aP + bm^+cn?)^ Ex. 3. If central plane sections of an eUipsoid be of constant area, their planes touch a cone of the second degree. 7-2 100 PLANE SECTIONS. Let the area be ~- , and let the equation of one of the planes be d Then we have iraie Tobe IT' {a? - d^ P + {V- d^) m" + (c" - d') n' = 0. This shews that the plane lx+my+nz=0 always touches the cone 2^" a'-d'^h'>-cP c'^-d^ " 117. We can find, by the method of Art. 115, the area of a central plane section of the surface whose equation is oar' + 6^" + cz^ + 2fys + 2gzx + 2hxy = 1. For the semi-diameters of length r are generating lines of the cone whose equation is {a-'^x' + (h-\^f + {c-'^^ + 2fyz + 2gzx + 2hxy = 0. When r is equal to either semi-axis of the section of the surface by the plane Ix -f- my + m=0, the plane will be a tangent plane of the cone. The condition of tangency gives, for the determination of the semi-axes, the equation 1 h, 9- h. r /. 9' /> 1 I I, m, n, = 0. This result may also be obtained by finding the maxi- mum value of x^ + y^ + z^ = r', subject to the conditions aac' + by^ + cz' + 2fyz + 2gzx + 2hosy = 1, and la! + my + nz = 0. AXES OF CENTRAL SECTIONS. 101 118. To find the directions of the axes of any central section of a conicoid. Let the equation of the surface be aa^ + bf + els' + 2fyz + 2gzx + 2hoBy = 1, and let the equation of the plane be Ix + my + nz = 0. Then, if P be any point on an axis of the section, the line joining P to the centre of the section will be perpendicular to the polar line of P in the plane of the section. Hence, if P be {^, ij, f), and if the direction-cosines of the polar line be X,, /jl, v, we have ■K^ + firf+v^=Q (i). Also, since the polar line is on both the planes x{a^-^}wi+g^) + y{h^+hr,+fi) + z{g^+f7) + c^ = l, and Ix + my + nz=0, it is perpendicular to the normals to those planes ; hence \(a^-^hr,+g^) + ,j,{h^+hn+f^) + v{g^+f'n + cK) = Q...(ii), and Xl -\- fi.m + vn = Q (iii). Eliminating \, fi, v from the equations (i), (ii), (iii), we have 0. t V, K a^ + hv+g^, h^+br}+f^, g^+fy + c? I, m, n Hence the required axes are the lines in which the given plane cuts the cone whose equation is «>, y, z ax + hy + gz, hx+by+ fz, gx+fy + cz I, m, n = 0. 119. To find the angle between the asymptotes of any plane section of a conicoid. Let 6 be the angle between the asymptotes of the plane section, and let the semi-axes of the section be a, /S. 102 CONDITION FOR RECTANGULAR HYPERBOLA. Then tan f = V^Hf i^ ; 2i a. This gives the required angle, since we have found, in the preceding articles, the axes of any plane section. Ex, 1. Find the angle between the asymptotes of the section of ax''' + by^ + c^=l by the plane lx + my + 'nz=0. The semi-axes are the roots of the equation '^-^ *-jr2 «--2 therefore tan' B ^__47^V _ \a b e/ ■ (V + r,^)" {P{b + c) + m'{c + a) + n'' (a + b)]^ ' Ex. 2. To find the condition that the section of the conicoid aii? + by'' + cz'' + 2/yz + 2gzx+2hxy=l by the plane h!+my + nz=0 may be a rectangular hyperbola. The square of tiie reciprocal of the semi-diameter whose direction-cosines are X, /t, v is given by ^ = aX" + bii'> + ei'^+ ifiiv + 2gv\ + 2h\n. Take any three perpendicular diameters; then we have by addition 111 M "^2 '^S Now, if *•,, rj be the lengths of any two perpendicular semi-diameters of a rectangular hyperbola, r^^ -i- r/ = 0. Hence for any semi-diameter of the conicoid which is perpendicular to the plane of a section which is a rectangular hyperbola, we have -^=a + b + c. The required condition is therefore al^+bm!' + en'+2fm,n+2gnl + 2hlm=a+b + e = {a + b + c)(P + m^ + n''). Ex. 3. Shew that the two lines given by the equations ax' + by^ + cz^=0, lx+my+nz=Q will be at right angles, if P{b + c)+m''{c + a)+n!'(a + b)=0. The lines are the asymptotes of the section of the conicoid aac' + by'+ez' = l by the plane Ix+my + m=0. CIRCULAR SECTIONS. 103 120. // two conicaids have one plane section in common all their oth^r points of intersection lie on another plane. Let the equations of the common plane section be ax' + by^ -H 2hxif + 2va! + 2vy + c = 0, z = Q. The most general equations of two conicoids which pass through this conic are aos^ + by^ + 2hxy + 2ux + 2vy + o + z{la!+ my + nz +p) = 0, and ax' + by' + 2hxy + 2ux + 2vy + c + z (J'x + m'y + n'z + p) = 0. It is clear that all points which are on both surfaces, and for which z is not zero, are on the plane given by the equation Ix + my + nz +p = Z'a; + m'y + n'z+p' ; this proves the proposition. Circular Sections. 121. To find the circular sectiors of an ellipsoid. Since parallel sections are similar, we need only consider the sections through the centre. Now all the semi-diameters of the ellipsoid which are of length r are generating lines of the cone whose equation is If there be a circular section of radius r, an infinite number of generating lines of the cone will lie on the plane of the section; hence the cone must be two planes. This will only be the case when r is equal to a, or b, or c. If r = a, the two planes pass through the axis of oo, their equation being 2'1p-^)+ (ii). Now AP, BP are the generators through P; hence from (ii), 9 + } cos <}> — cos9 , si n {9 + ^) cos — sin ^ sm ^ sm 4> or to a sin {0 + (/>), — b cos (9 + ^), c; hence the equations of AP are 01 — a cos 9 sec _y — bsm0 sec _2 — c tan ) ~ —b cos (9 + ) c Similarly the equations of BP are as —a cos sec ^ _y— h sin sec (p _z — c tan (^ a sin (9 — ) — b con (9 — cf)) — c 122 GENERATING LINES OF A PARABOLOID. Cor. The equations of the generators, through the point on the principal elliptic section whose eccentric angle is 6, are OB— a cos 9 _y — bsm6 _ z a sin ^ — 6 cos ~ c ' These equations may also be obtained as follows : The line whose equations are w — a cos _y — b sin 6 _z _ I on n ^ will meet the surface, where (g cos e + Iry , (6 sin 6> + mrf n'r^ _ , ffl» ''" b' e" ~ Hence, in order that the straight line may be a generating line, we must have u"'^ b" c'~ ' , I cos 6 m sin . and 1 T — =0. Whence The equations .of the generators are therefore a; — acos0_y—bsm0_z ^ asinB —bcos ~ ~ c' 139. To find the eqimtions of the generating lines through any point of a hyperbolic paraboloid. Let the equation of the paraboloid be a' h^-^""- GENERATING LINES OF A PARABOLOID. 123 Let the equations of any line be x — T. _y — fi z— 7_ I m ~ n ~ ' The points of intersection of the line and the surface are given by the equation Hence, in order that the straight line may be a generating Jine, we must have ^ m^ . ... ^-"6^ = ^ W' „.-^-« = (11), and __^_27 = (iii). The equation (iii) is satisfied if (a, jS, 7) be any point on the surface ; from (i) we have - = + -r- ; and, substituting in (ii), a we obtain Z _ m _ n a~ lb" a_§' Hence the equations of the two generating lines through tlie point (a, jS, 7) are a;-a_ y-^ _z-y a ~ ±b ~^-i ^ '' It is clear from the above that any generator of the paraboloid is parallel to one or other of the two planes a 124 GENERATING LINES OF A PARABOLOID. Ex. 1. Shew that the projections of the generating lines of an hyper- boloid on its principal planes are tangents to the principal sections. The tangent plane at any point P on a principal section is perpendicular to that section. Hence the projection on the principal plane of any Hne in the tangent plane at P is the tangent line which is in the principal plane. This proves the proposition, since the generating lines through P are in the tangent plane at P. Ex. 2. Find the locus of the point of intersection of perpendicular generators of an hyperboloid of one sheet. If the generating lines at any point P are at right angles, the parallel central section is a rectangular hyperbola, and therefore the sum of the squares of its axes is zero. But the sum of the squares of three conjugate semi-diameters of the hyperboloid is constant and equal to a'' + 6* - e*. Hence OP*=o2 + 62_(.a; so that the points are all on a sphere. This is the result we should obtain by putting tan S = a> in the result of Art. 136, We could also find the locus by using the equations of Art. 138; Ex. 3. .Find the angle between the generating lines at any point of a hyperbolic paraboloid. The result is obtained at once from equations (iv),. Art, 139;. The gene- rators are at right angles, if aS_62+4-^!=0, orif 27+o'-62=0. Thus generators which are at right angles meet on the plane 2 = i (6' - a') . Ex. 4. A luie moves so as always to intersect three given straight lines which are all parallel to the same plane : shew that it generates a hyperbolic paraboloid. Ex. 5. A line moves so as always to intersect two given straight lines and to be parallel to a, given plane : shew that it generates a hyperbohc paraboloid. Ex. 6. AB and CD are two finite non-intersecting straight lines ; shew that the lines which divide AB and CD in the same ratio are generators of ' one system of a hyperbolic paraboloid, and that the lines which divide AC and BD in the same ratio are generators of the opposite system of the same paraboloid. « Examples on Chaptek VI. 1. A straight line revolves about a fixed straight line, find the surface generated. 2. If four non-intersecting straight lines be given, shew that the four hjrperboloids which can be described, one through each set of three, all pass through two other straight lines. EXAMPLES ON CHAPTEB VI. 125 3. Find the equation of the conicoid, three of •whose generat- ing lines are !» = 0, y = a ; y = 0,z = a; z = 0, x = a. Shew that it is a surface of revolution, and find the eccentricity of its meridian section. 4. Find all the straight lines which can be drawn entirely coinciding (i) with the surface y" - «" = 3a'x ; and (ii) with the surface y* — z* = ia'x. 5. Normals are drawn to an hyperboloid of one sheet at every point through which the generators are at right angles ; prove that the points, in which the normals intersect any one of the principal planes, lie in an ellipse. 6. Given any three lines, and a fourth line touching the hyperboloid through the three lines, then will each one of the four lines touch the hyperboloid through the other three lines. 7. A line is drawn through the centre of ax' + bf,y', z') and (a/', y", /') be any two points. The equations of the polar planes of these points wiU be axa^+byy'+czi! + d=0 and a^i^' + iyy" + czz" + d=0. The conicoid X (ax^ +hy^+ci^+d)-{axx' + byy' +czz'+ d) {axa^' + byy" + czz" + d) = is the general equation of a conicoid through the two plane sections. The conicoid will pass through (x', y, si) if \ be such that X (oj;" +hy"' + cz'' +d)- (aaj'^ + by'^ + cz'^ + d) (oaV + by'y" + czV + d) = 0, orif \=ax'x" + by'y"+cz'z"+d. The symmetry of this result shews that the conicoid will likewise pass through (x", y", z"). Ex. 3. Through the curve of intersection of a sphere and an ellipsoid four quadric cones can be drawn; and if diameters of the ellipsoid be drawn parallel to the generators of one of the cones the diameters are all equal. Also the continued product of the four values of such diameters is equal to the continued product of the axes of the ellipsoid and of the diameter of the sphere. Let the equations of the ellipsoid and of the sphere be and {x-a)<' + {y-p)'+{z-y)'=r\ The general equation of a conicoid through the curve of intersection is x(^I + |-2 + J-l-) + (»-''f + (2/-«H(^-7)=-'-==0 (i). 9—2 132 EXAMPLES, This conicoid will be a cone, if the co-ordinates of the centre satisfy the equations (1 + ^)2^-^=0. (l + |).-7=0. and -ax-py-ye + a' + p' + y'-r^-\ = 0. Eliminating x, y, z we have If, for any particular value of X, the conicoid given by (i) is a cone, the equation of the cone, when referred to its vertex, takes the form and therefore the direction-cosines of any diameter which is parallel to one of the generating lines of the cone, satisfy the equation P m? rfi _ 1 Hence the square of the semi-diameter is constant and equal to - X. Hence also the continued product of the squares of the four values of the semi-diameters is equal to the product of the four roots of the equation (ii) ; and the product of the roots is easily seen to be a^lfcV. Ex. 4. The locus of the centres of all conicoids which past through seven given points is a cubic sm/rface, which passes through the middle point of the line joining any pair of the seven given points. Let 5i=0, 8^=0, 5g=0 be any three conicoids through the s|ven given points ; then the general equation of the conicoids is Si-t-XSs-l-/4Sa=0. The equations for the centre are dx dx dx dy dy '^ dy dz dz dz TANGENTIAL EQUATIONS. 133 Hence the equation of the locus of the centres, for difierent values of \ i 11, ia Si' dx ' dx =0, dy ' dy dS, dz ' dz ' dSs dz which is a cuhic surface, since -r-i &c. are of the first degree. ax Now, to have the centre of a conicoid given, is equivalent to having three conditions given ; hence a conicoid which has a given centre can be made to pass through any six points. Hence, if A, B be any two of the seven given points, one conicoid whose centre is the middle point of AB will pass through A and through the remaining five points ; and a conicoid whose centre is the middle point of AB, and which goes through A, must also go through B. Thus the middle point of AB is a point on the locus of centres ; and so also is the middle point of the line joining any other pair of the given points. [Messenger of MatJiematics, vol. xni. p. 145, and xiv. p. 97.] Tangential Equations. 145, If the equation of a plane be fa; + my + n^: + 1 = 0, then the position of the plane is determined if I, m, n are known, and by changing the values of I, m and n the equation noay be made to represent any plane whatever. The quantities I, m, and n which thus define the position of a plane are called the co-ordinates of the plane. These co- ordinates, when their signs are changed, are the reciprocals of the intercepts on the axes. If the co-ordinates of a plane be connected by any relation, the plane will envelope a surface; and the equation which expresses the relation is called the tangential equation of the surface. 146. If the tangential equation of a surface be of the n'" degree, then n tangent planes can be drawn to the surface through any straight line. For, let the straight line be given by the equations ax+by + cz+ 1=0, a'x + b'y + c'z + l=0; then the co-ordinates of any plane through the line will be % — , — t and T — — . If these co-ordinates be sub- l + \ ' 1+X 1-l-X. 134 CENTRE OF CONICOID. stituted in the given tangential equation, we shall obtain an equation of the n^ degree for the determination of X, which proves the proposition. Def. A surface is said to be of the w"" class when n tangent planes can be drawn to it through an arbitrary straight line. 147. We have shewn in Art. 57 that the plane Ix + my + n2 + 1=0 will touch the conicoid whose equation is aar'4- 62/°+ c/ + 2fyz + 2gzx + Ihxy + 2ux + 2vy + 2wz +d - 0, if Al^ + Sm" + Gn^ + IFrnn + 2Qnl + 'i.Elm + 2111+ 2Vm+2,Wn + D = 0, where A, B, G... are the co-factors of a, b, c... in the dis- criminant. Hence the tangential equation of a conicoid is of the second degree. Conversely every surface whose tangential equation is of the second degree is a conicoid. 148. Since the tangential equation of a conicoid is of the second degree, which in its most general form contains nine constants, it follows that a conicoid can be made to satisfy nine conditions and no more ; and in particular a conicoid can be made to touch nine given planes. 149. To find the Gartesian co-ordinates of the centum of the conicoid given by the general tangential equation of the second degree. The two tangent planes to the conicoid which are parallel to the plane a; = are those for which m = n = 0. The values of I are therefore given by the equation aP + 2ul H- d = 0. Now the centre of the surface is on the plane midway between these ; and hence the centre is on the plane <» = -;. d DIRECTOR-SPHERE. 135 Similarly the centre is on the planes y = -^ , and ^r = u V u d' d' d' Hence the required co-ordinates are ^ , ^ , -r . [See Art. 76.] 150. We may take the equation of the moving plane to be Ix -I- my ■^nz+p = Q; and the plane will envelope a surface if I, m, n, p be connected by a homogeneous equation ; for any homogeneous equation in I, m, n, p would be equivalent to an equation between the constants - , — . - . P p ' p If we take loo + my + nz +p = for the equation of the plane, we may suppose I, m, n to be the direction-cosines of the normal to the plane. 151. To find the director-sphere of a conicoid whose tamgehtial equation is given. If we eliminate p between the equation of the surface and the equation Ix + my + nz + p = 0, we shall obtain a relation between the direction-cosines of any tangent plane which passes through the particular point (x, y, z). The relation will be al^ + tm' + cw" -f d (fa; -)- TTjy -t- nzf + ^fmn -1- 'ignl -t- 2AZm — 2 (ul +vm + wn)(lx + my -I- nz) = 0. If (p, y, z) be a point on the director-sphere, three per- pendicular tangent planes will pass through it ; the above relation must therefore be satisfied by the direction-cosines of each of three perpendicular planes. Hence, by addition, we have a + h + o- 2ux — Ivy - 2wz + d(oi^ + y' + z") = 0, which is the required equation of the director-sphere. 152. li 8=0 and S' = Ohe the tangential equations of any two conicoids which touch eight given planes, then the equation 8 + X/S' = will be of the second degree, and will therefore be the tangential equation of a conicoid; and it is clear that the conicoid 8 + \S' = will touch the common 136 CONICOIDS WHICH TOUCH SEVEN PLANES. tangent planes of /Sf = and S' = 0, for if the co-ordinates of any plane satisfy the equations 8=0 and 8' = 0, they will also satisfy the equation 8 + X8' = 0. Also, by giving a suitable value to \, the conicoid 8 + \8' = can be made to touch any ninth plane : it will therefore represent any coni- coid touching the eight given planes. 153. If 8^ = 0, /Sij = 0, /Sfg = be the tangential equations of any three conicoids which touch seven given planes ; then the conicoid whose tangential equation is 8^ + X8^ + fi8, = will touch each of the seven given planes, for if the co- ordinates of any plane satisfy the three equations 8^ = 0, /Sj = and 8^ = 0, it will also satisfy the equation 8, + XS^ + fi8^ = 0. Also, by giving suitable values to X and fi, the conicoid S, + X8, + ^l8, = can be made to touch any two other planes ; hence 8, + X8, + ^l8, = is the most general equation of a conicoid which touches the seven given planes. Similarly, if 8^ = 0, 8^ = 0, 8^ = and (Sf, = be the tangential equations of any four conicoids which touch six given planes, 8^ + X8^ ■+ fj,Sg + v8^ = will be the general tangential equation of the conicoids which touch those six planes. Ex. 1. The centres of all conicoids which touch eight given planesare on a straight line. If S=0 and S'=0 be the equations of any two conicoids which touch the eight given planes, then S + \S'=0 will be the general equation of a conicoid touching them. The centre of the conicoid is given by _u+\u' _ v + \v' _ w + 'Kw ' EHminating X we obtain the equation of the centre locus, namely da>-u _ dy-v dz-w ^ d'x -u'~ d'y - «' ~ d's - w' ' hence the locus is a straight line. EXAMPLES. 137 Ex. 2. The centres of all conicoids which touch seven given planes are on a plane. If S=0, S'=0, S"=0 be the equations of three conicoids which touch the seven given planes, then the general equation of a ooniooid which touches the planes will be S + \S'+/iS"=0. Ex. 3. The director-spheres of all conicoids which have eight common tangent planes have a common radical plane. The director-sphere of the ooniooid S + XS'=0 is a+i-¥c-iux-2vy-2wz + d(x^+y'' + z^) +\ {a' + b' + c' -2u'x-iify-2w'z + d' (x^ + y^ + !i^))=Q. Ex. 4. The director-spheres of all conicoids which touch six given planes are cut orthogonally by the same sphere. [P. Serret's Theorem.] If Ci=0, C/„=0, Gg=0 and 04=0 be the equations of any four conicoids which touch the six planes; then the general equation of the conicoids will be Ci + XCa+;iiCs + i'C4=0. Now from Art. 151 we see that the equation of the director-sphere of a ooniooid is Unear in a, b, c, &o. It therefore foUows that, if 5i=0, ^2=0, Ss=0 and 8^=0 be the equations of the director-spheres of the conicoids Ci=0, Oj=0, Cg=0 and 04=0 respectively, the equation of the director- sphere of Ci+\G^+ /tCj + vCi=0 will be Si -I- XSj + 1183 + 98^ = 0. Now from the condition that two spheres may out orthogonally [Art. 90, Ex. 6], it foUows that a sphere can always be found which wUl cut four given spheres orthogonally; and it also follows that the sphere which cuts orthogonally the four spheres Si=0, S^^O, 8^—0 and 8^=0, wiU cut orthogonally any sphere whose equation is Si + \8^-i-)iS3 + vSi=0. This proves the proposition. Ex. 5. The locus of the centres of conicoids which touch six planes, and have the sum of the squares of their axes given, is a sphere. [Mention's Theorem.] By Ex. 4 all the director-spheres of the conicoids are out orthogonally by the same sphere; and the director- spheres have a constant radius. Hence their centres, which are the centres of the conicoids, are on a sphere con- centric with this orthogonal sphere. Reciprocation. 154. If we have any system of points and planes in space, and we take the polar planes of those points and the poles of the planes, with respect to a fixed conicoid G, we obtain another system of planes and points which is called 138 EECIPEOCATION. the polar reciprocal of the former with respect to the aimMary conicoid G. When a point in one system and a plane in the reciprocal are pole and polar plane with respect to the auxiliary conicoid G, we shall say that they correspond to one another. If in one system we have a surface • 8, the planes which correspond to the different points of S will all touch some surface S'. Let the planes corresponding to any number of points P, Q, R... on a plane section of S meet in T; then T is the pole of the plane PQR with respect to C, that is the plane PQR corresponds to T. Now, if the plane PQR move up to and ultimately coincide with the tangent plane at P, the corresponding tangent planes to S' will ultimately coincide with one another, and their point of intersection T will ultimately be on the surface 8'. So that a tangent plane to the surface 8 corresponds to a point on the surface iSi', just as a tangent plane to 8' corresponds to a point on 8. Hence we see that 8 is generated from 8' exactly as 8' is from 8. 155. To a line L in one system corresponds the line L' in the reciprocal system which is the polar line of L with respect to the auxiliary conicoid. If any line L cut the surface 8 in any number of points P, Q, R... we shall have tangent planes to 8' corresponding to the points P, Q, R..., and these tangent planes will all pass through a line, viz. through the polar line of L with respect to the auxiliary conicoid. Hence, as many tangent planes to /Sf can be drawn through a straight line as there are points on 8 lying on a straight line. That is to siy the class [Art. 146] of 8' is equal to the degree of 8. Reciprocally the degree of 8' is equal to the class of 8. In particular, if /S be a conicoid it is of the second degree and of the second class ; hence 8' is of the second class and of the second degree, and is therefore also a conicoid. 156. The reciprocal of a point which is common to two surfaces is a plane which touches both the reciprocaJ surfaces. RECIPROCATION. 139 If two surfaces have a common curve of intersection, they have an infinite number of common points ; the reciprocal surfaces therefore have an infinite number of common tangent planes. These common tangent planes form a surface : and, since the line of intersection of any two consecutive- planes is on the surface, it is a ruled surface, the generating lines being the lines of intersection of consecutive planes. Any one of the planes contains two consecutive generating lines, so that two consecutive generators must intersect ; hence the surface is a developable surface. If all the points of the curve lie on a plane, all the tangent planes to the developable pass through a point; the developable must therefore be a cone. Hence the reciprocal of a plane curve is a cone. It follows by reciprocation from Art. 144, that all coni- coids which touch seven fixed planes will touch an associated eighth plane. It also foUows from Art. 140 that all conicoids which touch eight given planes have an infinite number of common tangent planes, provided that the eight given planes do not form an associated system. 157. The reciprocation is usually taken with respect to a sphere, and since the nature of the reciprocal surface is in- dependent of the radius of the sphere, we only require to know the centre of the sphere, which is called the origin of reciprocation. The line joining the centre of a sphere to any point is perpendicular to the polar plane of the point. Hence, if P, Q be any two points, the angle between the polar planes of these points with respect to a sphere is equal to the angle that PQ subtends at the centre of the sphere. 158. If any conicoid be reciprocated with respect to a point 0, the points on the reciprocal surface which corre- spond to the tangent planes through to the original surface must be at an infinite distance. 140 EECIPEOCATION. Hence the generating lines of the asymptotic cone of the reciprocal surface are perpendicular to the tangent planes of the enveloping cone from to the original surface. In particular, if the point be on the director-sphere of the original surface, that is if three of the tangent planes from be at right angles, the asymptotic cone of the reciprocal surface will have three generating lines at right angles. Corresponding to a point at infinity on the original surface we have a tangent plane through to the reciprocal surface. Hence the tangent cone from the origin to the reciprocal surface has its tangent planes perpendicular to the generating lines of the asymptotic cone of the original surface. In particular, if the asymptotic cone of the original surface have three perpendicular generating lines, three of the tangent planes from to the reciprocal surface will be at right angles, so that is a point on the director-sphere of the reciprocal conicoid, 159. As an example of reciprocation take the theorem : — "If two of the conicoids which pass through eight given points are rectangular hyperboloids, they will all be rect- angular hyperboloids." If this be reciprocated with respect to any point we obtain the following, "If the director- spheres of two of the conicoids which touch eight given planes pass through a point 0, the director-spheres of all the conicoids will pass through 0." Hence " the director-spheres of all conicoids which touch eight given planes have % com- mon radical plane." As another example of reciprocation take the theorem : — " A straight line is drawn to cut the faces of a tetrahedron ABGD which are opposite to the angles A, B, G, D in a, b, c and d respectively. Shew that the spheres described on the straight lines Aa, Bh, Co, and Dd as diameters have a common radical axis." Let be a point of intersection of the spheres whose diameters are Aa, Bb and Gc. If we reciprocate with RECIPROCATION. 141 respect to we shall obtain another tetrahedron whose faces and angular points correspond respectively to the angular points and faces of the original tetrahedron. Corre- sponding to the four points a, b, c, d which are on a straight line, we shall have four planes with a common line of inter- section; and, since a, b, c, d are on the faces of the original tetrahedron, the corresponding planes will pass through the angular points of the reciprocal tetrahedron ; also since the angles AOa, BOb, COo are right angles, the three pairs of planes corresponding respectively to a and ^, to 6 and B, and to c and G will be at right angles ; this shews that the line of intersection of the planes corresponding to a, b, c, d will meet three of the perpendiculars of the reciprocal tetrahedron. But we know [Ait. 135, Ex. 4], that every line which meets three of the perpendiculars of a tetrahedron, meets the remaining perpendicular; and hence the planes corresponding to d and D are at right angles, which shews that the angle dOD is a right angle. Hence is also on the sphere whose diameter is Dd. Ex. 1. The reciprocal of a sphere with respect to any point is a ooniooid of revolution. Ex. 2. Find the reciprocal of aa;' + 6y'' + C2^=l with respect to the sphere ai'+y' + z^=l. Am. - + |!+-=1. " a b e Ex. 3. Shew that the reciprocal of a ruled surface is a ruled surface. Ex. i. Shew that if two conicoids have one common enveloping cone they also have another. [The reciprocal of Art. 120.] Ex. 5. Either of the two surfaces ax^ + by^= ±2^ is self reciprocal with respect to the other. Examples on Chapter VII. 1. When three conicoids pass through the same conic, the planes of their other conies of intersection pass through the same line. 2. Shew that, if the curve of intersection of two conicoids cross itself, the conicoids wUl touch at the point of crossing ; and that if the curve of intersection cross itself twice, it will consist of two conies. 142 EXAMPLES ON CHAPTER VII. 3. Shew that three paraboloids will pass through the curve of intersection of any two oonicoids. 4. Shew that a surface of revolution will go through the intersection of any two coniooids whose axes are parallel. 5. If a conicoid have double contact with a sphere, the square of the tangent to the sphere from any point on the conicoid is in a constant ratio to the product of the distances of that point from the planes of intersection. 6. Any two oonicoids which have a common enveloping cone intersect in plane curves. 7. Shew that the polar lines of a fixed line, with respect to a system of oonicoids through eight given points, generate an hyper- boloid of one sheet. 8. Shew that the polar planes of a fixed point, with respect to a system of oonicoids through seven given points, pass through a fixed point. 9. Shew that the poles of a fixed plane, with respect to e, system of oonicoids which touch seven given planes, lie on a fixed plane. 10. The polar planes of a point with respect to two given oonicoids are at right angles ; shew that the locus of the point is another conicoid. 11. All oonicoids through the intersection of a sphere and a given conicoid, have their principal planes, and also their cyclic planes, in fixed directions. 12. If be any point on a conicoid, and lines be drawn through parallel to equal diameters of the conicoid, thes» lines will meet the surface on a sphere whose centre is on the normal at 0. 13. If be the centre of any conicoid through the intersec- tion of a sphere and a given conicoid, the line joining to the centre of *the sphere is perpendicular to the polar plane of with respect to the given conicoid. 14. Shew that, in a system of oonicoids which have a common curve of intersection, the diametral planes of parallel diameters have a common line of intersection. EXAMPLES ON CHAPTER VH. 143 15. If a system of conicoids be drawn through the inter- seotion. of a given conicoid and a sphere whose centre is 0, the normals to them from form a cone of the second degree, and their feet are on a curve of the third order which is the locus of the centres of aU the surfaces. 16. If any point on a given diameter of an ellipsoid, be joined to every point of a given plane section of the surface, the cone so formed will meet the surface in another plane section, whose envelope will be a hyperbolic cylinder. 17. A cone is described with its vertex at a fixed point, and one axis parallel to an axis of a given quadric, and the cone cuts the quadric in plane curves; shew that these planes envelope a parabolic cylinder whose directrix-plane passes through the fixed point. 18. If two spheres be inscribed in any conicoid of revolution, any common tangent plane of the spheres wUl cut the conicoid in a conic having its points of contact for foci. 19. If the line joining the point of intersection of three, out of six given planes, to the point of intersection of the other three, be called a diagonal ; shew that the ten spheres described on the diagonals have the same radical centre, and the same orthogonal sphere. 20. The circumscribing sphere of a tetrahedron which is self polar with respect to a conicoid cuts the director-sphere of the conicoid orthogonally. CHAPTER VIII. oonfocal conicoids. concyclic conicoids. Foci of Conicoids. 160. Conicoids whose principal sections are confoeal conies are called confoeal conicoids. The general equation of a system of confoeal conicoids is Suppose a, b, c to be in descending order of magnitude. If X is positive, the surface is an ellipsoid, and the principal axes of the surface will increase as \ increases, and their ratio wiU tend more and more to equality as \ is increased more and more ; so that a sphere of infinite radius is a limiting form of one of the confocals. If \ is negative and less than c* the surface is an ellipsoid ; but the ellipsoid becomes flatter and flatter as \ approaches the value — c'. Hence the elliptic disc whose equations are is a limiting form of one of the confocals. If \ is between — c° and — 6' the surface is an hyperboloid of one sheet. When \ is very nearly equal to — c°, the hyperboloid is very nearly coincident with that part of the plane z=0 which is exterior to the ellipse -5 5 + j^ — j =1. CONFOCAL CONICOIDS. 145 When \ is very nearly equal to — 6', the hyperholoid is very nearly coincident with that part of the plane y — O which contains the centre and is bounded by the hyperbola P "*"«»_ ^2 ~ ^• If \ is between — V and — a", the surface is an hyper- holoid of two sheets. When \ is very nearly equal to — b'\ the hyperholoid is very nearly coincident with that part of the plane y = which does not contain the centre and is off' z" bounded by the hyperbola ,_,, + -^ — j-^ = 1. When \ is between — a" and — oo the surface is imaginary. The two conies 'd'-c' b'-c"' ' which we have seen are the boundaries of limiting forms of confocal conicoids, are called focal conies, one being the focal ellipse, and the other the focal hyperbola. 161. Three conicoids, confocal with a given central conicoid, mil pass through a given point ; and one of the three is an ellipsoid, one an hyperholoid of one sheet, and one an hyper- boloid of two sheets. Let the equation of the given conicoid bo ^+^V^=i Any conicoid confocal to this is x- y -^ v + P^r + -« — r=i (i)- a - A, — \ c — X This will pass through the particular point (/ g, h) if /' (6' - X) (c' - X) + / (c» - X) (a^ - X) + A''(a^-X)(6''-X)-(a=-X)(6'''-X)(c'-X) = (ii). S. S. G. 10 146 CONFOCAL CONICOIDS. If we substitute for \ the values a", 6°, c", and — oo in succession, the left side of the equation (ii) will be +, —,+,—; hence there are three real roots of the equation, namely one between a" and ¥, one between b' and c", and one between c' and — 00 . When X is between c' and — oo , all the coefiScients in (i) are positive, and the surface is an ellipsoid ; when \ is between c" and 6', one of the coefficients is negative, and the surface is an hyperboloid of one sheet ; and when X is between 6'' and a' two of the coefScients are negative, and the surface is an hyperboloid of two sheets. 162. One conicoid of a given confooal system will touch any plane. Let the equation of the plane be Ix + my + nz = p. The plane will touch the conicoid if (a' + \) l'+{b^+X) m' + (c' + X)n' =p\ which gives one, and only one, value of X. Hence one con- focal will touch the given plane. 163. Two conicoids of a confocal system will touch any straight line. Let the straight line be the line of intersection of the planes lai + my+ nz +p = 0, I'x + m'y + n'z +p' = 0» Any plane through the straight line will be Q+ kl')x+ {m + km')y + (n + kn')z + (p + kp') = 0. This plane will touch the conicoid a,' + X^b'+X^c' + X~ ' if (a" + X)(l + kiy + Qf + \) (m + kmj + (c' + X) (« + knf = (p + kpj. CONFOCAL CONICOIDS. 147 Now, if the given line be a tangent line of the conicoid, the two tangent planes through it will coincide. Hence the roots of the above equation in k must be equal. The condition for this gives the following equation for finding \ {(a" + X)P + (V + \) m" + (c' + A,) n" - /} {(a' + X) r + (b" + \) m'" + (c" + \) n" - p"} = {(a' + X) W + (6= + \) mm' + {(f + \) nn' - ppf. Since the equation is of the second degree, there are two confocals which touch the given line. 164. Two confocal conicoids cut one another at right angles at all their common points. Let the equations of the conicoids be «' y" ^ •, a' + X 6' + X c' + X and let (x'y'z') be a common point ; then the co- ordinates x, y, z will satisfy both the above equations. Hence, by subtraction we have ».'s «'" a"* ^ I " I -0 {K\ . a«(a" + X)^6^(6'' + X)^c*(c' + X) " ^ ^• Now the equations of the tangent planes at the common point {oiiy'si) are The condition (i) shews that these tangent planes are at right angles. 10—2 148 CONFOCAL CONICOIDS. 165. If a straight line touch two confocal conicoids, the tangent planes at the points of contact mil be at right angles. Let (x'y'e), (x"y"g") be the points of contact, and let the conicoids be x' f z" a« + \ + i« + \ + c'' + \ ' , a!° , y' , z' _., """^ ^+V + F+V + c" + X,'~-^' The tangent planes will be at right angles if {a'+x){a^+-s:y{y+x){b^+\')^{c'+X){c'+\') "■••'>''■ But, since the line joining the two points is a tangent line to both conicoids, each point must be in the tangent plane at the other. Hence -'"'" I y'y" , ^'^" _i a" + \ 6" + \ c' + \ ■ x'af' y'y" and r3-rT7 + j#:^ + By subtraction we see that the condition (i) is satisfied. Ex. 1. The difference of the squares of the perpendiculars from the centre on any two parallel tangent planes to two given confocal conicoids. is constant. \iPi-'Pi=\i-\.'\ Ex. 2. The locus of the point of intersection of three planes mutually at right angles, each of which touches one of three given coufocals, is a sphere. {See Art. 92.] % Ex. 3. The locus of the umbilici of a system of confocal ellipsoids is the focal hyperbola. [The umbilici aie given by Ex. 4. K two concentric and co-axial conicoids out one another everywhere at right angles they must be confocal. Ex. 6. Pi Q are two points, one on each of two confocal conicoids, and the tangent planes at P, Q meet in the line JRS; shew that, if the plane through SS and the centre bisect the line PQ, the tangent planes at P and Q must be at right angles to one another. CONFOCAL CONICOIDS. 149 Ex. 6. Shew that two confocal paraboloids out everywhere at right angles. [The general equation of confocal paraboloids is ,-^ + — — =2z + \.'\ l+\ m+\ ' 166. We have seen that three coaicoids confocal with a given conicoid will pass through any point P, the parameters of the confocals being the three values of \ given by the equation x' f z" aJ'+X^h' + X^ c' + X ' where x, y, z are the co-ordinates of P. If the roots of the above equation be \, \, \, it is easy to shew that with similar values for y^ and ^. '■ Hence the absolute values of the co-ordinates of any point can be expressed in terms of the parameters of the conicoids which meet in that point, and are confocal with a given conicoid. 167. The parwmeters of the two confocals through any point P of a conicoid are equal to the squares of the axes of the central section of the conicoid which is parallel to the tangent plane at P ; and the normals at P to the confocals are parallel to the axes of that section. Let (x', y', z') be any point P on the conicoid whose equation is ^V^^+i^i- a^'^b'^c' ' then, if P be on the confocal whose parameter is \, we have and therefore yl-^^^=o (i). a'ia"- X) ^ 6' {V- X) ^ c" (c' - X) 150 CONFOCAL CONICOIDS. The equation of the central section parallel to the tangent plane at P is ■^"•■"i^ + c'-'""- Hence the equation giving the squares of the axes of the section is f! y!! i! a" r-» ¥ r" c' r" ..'2 1-^. + ..: „n =o (ii). Comparing (i) and (ii), we see that the squares of the axes of the section are the two values of \. The equations of the diameter which is parallel to the normal at P to one of the confocals are X y^ z «' ~ y' / a'-X V-\ c'-X The length of the diameter will be equal to 2\/\ if it be one of the generating lines of the cone the condition that this may be the case is x'" (I \\ , y^ (\ 1\ , /' (1 V\_^ and it is clear from (i) that this condition is satisfied. Hence an axis of the central section is parallel to the normal to one of the confocals through P, and the square of the length of the semi-axis is equal to the parameter of that confocal. CONFOCAL CONICOIDS. 151 CoE. If diameters of a conicoid be drawn parallel to the normals to a confocal at all points of their curve of inter- section, such diameters ■will be of constant length. 168. Two points (x, y, z), {^, rj, ^), one on each of two co-axial conicoids whose equations are x" «' z' , «" y^ z^ , respectively, are said to correspond when 5 = !, f=land? = -^. a a. p c y . In order that real points on one conicoid may correspond to real points on the other, the two surfaces must be of the same nature, and must be similarly placed. It follows at once from the equations (i), Art. 96, that if on one of the conicoids three points be taken which are ex- tremities of conjugate diameters, the three corresponding points on the other conicoid will be at extremities of con- jugate diameters. 169. The distance between two points, one on each of two confocal ellipsoids, is equal to the distance between the two corresponding points. Let («!, y,, z^, {x^, y^, z^ be the two points on one conicoid, and (^j, •qy Q, (l^, Vi> O the corresponding points on the other conicoid. Then ^'-fi. ^=^5, ^ = ^; a a b Id c y 1 «'S!_?2 y2_V3 ^^_^1 ^°^ a~a' "6-/8' c~7" We have to prove that " ("f.->.)'-(i'.-f^o'-ef.-?'0" 152 CONFOCAL CONICOIDS. , 2 or which is clearly the case, since the conicoids are confocal, and 170. The locus of the poles of a given plcme with respect to a system of confocal conicoids is a straight line. Let the equation of the confocals be a^' _f_,_^_ a'-x'^ b^-X'^ c'-x"" ' and let the equation of the given plane be Ix + Tuy + nz = 1. The equation of the polar plane of the point {x', y, /) is ip^' I yy' I zz' ^ Comparing this equation with the equation of the given plane, we have = ''> 72 = «i, and -i — ^.- = n ; d'-X "• b'-x ' c'-X therefore ^ _ a» = ^ _ 6« = ?! _ c'. t m n Hence the locus of the poles is the straight line«?hose equations are X — aH _y — Vm _z—' + X>2/' + (c" + -K)z^ = ic\ It is clear that the reciprocal conicoids have the same cyclic planes for all values of X. Hence a system of confocal conicoids reciprocates into a system of concentric concyclic conicoids. loG FOCI OF CONICOIDS. 174. The following are examples of reciprocal properties of confocal and concentric concyclic conicoids. Three conoyolios touch any plane, namely an ellipsoid, an hyperboloid of one sheet, and an hyperboloid of two sheets ; also the lines from the centre to the points of contact of the plane are at right angles. Two concyclics touch a straight line, and the lines from the centre to the points of contact are at right Three confocals pass through any point, namely an eUipsoid, an hyperboloid of one sheet, and an hyperboloid of two sheets; also the tangent planes at the point to the three surfaces are at right angles. Two confocals touch a straight line, and the tangent planes at the points of contact are at right angles. One conicoid of a confocal system touches any plane. One conicoid of a concyclic system passes through any point. The locus of the pole of a given plane with respect to a system of confocals is a straight line. The principal planes of a cone enveloping a conicoid are the tangent planes to the confocals through its vertex. The envelope of the polar plane of a given point with respect to a system of concyclics is a straight line. The axes of a cone whose vertex is at the centre of a conicoid and base any plane section, are the lines from the centre to the points of contact of the plane with the concyclics which touch it. Foci of Conicoids. 175. There are two definitions of a conicoid which corre- spond to the focus and directrix definition of a conic. One definition, due to Mac CuUagh, is as follows : — A conicoid is the locus of a point which moves so ''that its distmce from a fixed point, called the focus, is in a constant ratio to its distance {measured parallel to a fixed plane) from, a fixed straight line called the directrix.' Let the origin be the focus, and the plane z — O the fixed plane. Also let the equations of the directrix be ^-f^y-g^z-h I m n ' FOCI OF CONICOIDS. 157 Let CO , y, z' be the co-ordinates of any point P on the locus, and let a plane through P parallel to « = meet the directrix in M, then M is |/+^ (/ -h), g + "^{z- h), z\ . Now OP' = e" . PM", e being the constant ratio. Hence the equation of the locus of (x', y', /) is a^+y'+z'=e'' \.-f-liz-h)]\{y-,-^iz-h^}' ..(i). The locus is therefore a conicoid, and is such that sections parallel to z = are circles. If the axes be changed in any manner (i) will always be of the form {a;-ay+(y-l3y+{z-yY-A = 0, where A is the sum of two squares, or is the product of two imaginary factors. We can therefore find the foci of any given conicoid whose equation is (S = 0, from the consideration that 8 -X{{x-ay + (y - ^y + (z - 7)'} will be the product of imaginary linear factors if (a, /Q, 7) be a focus, provided a suitable value be given to X. 176. The other definition of a conicoid, due to Salmon, is as follows : — A conicoid is the locus of a point the square of whose distance from a fixed point, called a focus, varies as the pro- duct of its distances from two fixed planes. The equation of the locus is clearly of the form {x-ay+{y-^y-\-{z-riy=¥{lx+my + nz+p){l'x+m'y+n'z+p'). We can find the foci of any conicoid according to this definition by the consideration that s-\{{x-ay + {y-^y+{z-iy} will be the product of real linear factors if (ot, yS, 7) be a focus, provided a suitable value be given to \. 158 FOCAL CONICS. 177. To find the foci of the conicoid whose equation is We have seen in Articles 175 and 176 that (a, ^, y) is a focus when ax'+bf + cz" -l-X{(jv-ay + (y - ^y + i^ -ryf) (i) is the product of linear factors. Hence \ must be equal to a, or b, or c. Let \ = a, then (i) becomes (6 - a) y'+ (c - a) /+ 2aaai + 2a^y+ 2ayz - a (a»+ /S'+y') -1, or J ab^^ acf + 2aaa>—aoL—, — 1. — a c — a Hence, in order that (i) may be the product of linear factors, we must have a = 0, and -^+-:^=i. i_i^i_i b a c a Similarly, if = b, we have /S = and i_i + i_i '■' a b c b and, if \ = c, we have 7 = 0, and .^+-^=1 1_1^1_1 • a b There are therefore three conies, one in each principal plane, on which the foci lie. FOCAL LINES OF A. CONE. 159 178. If the surface be an ellipsoid whose semiaxes are a, h, c, the conies on which the foci lie are ^.37^+^-^ = 1,^ = (i), ^^d F^=+7^==l' ^ = ("i)- Since a, h, c are in descending order of magnitude (i) is an ellipse, (ii) is an hyperbola, and (iii) is imaginary. These conies are called the focal conies; and, as we have seen in Art. 160, they are the boundaries of limiting forms of confocal conicoids. 179. The focal conies of the cone aoe' + by' + cz'' = can be deduced from the above, or found in a similar manner. The conies become b a e a c b a b a? «" and z = Q, ? — T + ri"^^" a c b c One of the focal conies of a cone is therefore a pair of real straight lines which are called the focal lines ; the other focal conies are pairs of imaginary straight lines, which we may consider as point- ellipses. Ex. 1. Two cones which have the same focal lines cut one another at right angles. Ex. 2. Shew that the enveloping cones from any point to a system of confocals have the same focal lines. Ex. 3. Shew that the focal conies of a paraboloid are two parabolas. 160 EXAMPLES ON CHAPTEH VIII. 180. The focal lines of a cone are perpendicular to the cyclic planes of the reciprocal cone. The equations of any two reciprocal cones referred to their axes are aa!'-\-by' + cz' = 0, and ^'+^ + ^ = 0. The cyclic pknes are [Art. 121] (a-6)^'+(c-6)^' = 0, andg-J)^'+g-J)^' = 0. The focal lines are by the last article of / a;" z^ a b c b , It is therefore clear that the focal lin!es of one cone are perpendicular to the cyclic planes of the other. Examples on Chapter VIII. 1. Three confocal conicoids meet in a point, and a central plane of each is drawn parallel to its tangent plane at that point. Prove that, one of the three sections will be an ellipse, one an hyperbola, and one imaginary. 2. Plane sections of an ellipsoid envelope a confocal ; shew that their centres lie on a surface of the fourth degree. 3. P, Q are two points on a generator of a hyperboloifl; P', Q the corresponding points on a confocal hyperboloid. Shew that P<^ is a generator of the latter, and that PQ = PQ'. 4. Shew that the points on a system of confocals which are such that the normals are parallel to a given line are on a rect- angular hyperbola. 5. If three lines at right angles to one another touch a conicoid, the plane through the points of contact will envelope a confocal. EXAMPLES ON CHAPTER Vlir. 161 6. If three of the generating lines of the enveloping cone of a paraboloid be mutually at right angles, shew that the vertex will be on a paraboloid, and that the polar plane of the vertex wiU always touch another paraboloid. 7. If through a given straight line tangent planes be drawn to a system of confocals, the corresponding normals generate a hyperbolic paraboloid. 8. Shew that the locus of the polar of a given line with respect to a system of confocals is a hyperbolic paraboloid one of whose asymptotic planes is perpendicular to the given line. 9. Planes are drawn all passing through a fixed straight line and each touching one of a set of confocal ellipsoids; find the locus of their points of contact. 10. At a given point the tangent planes to the three coni- coids which pass through 0, and are confocal with a given conicoid, are drawn ; shew that these tangent planes and the polar plane of form a tetrahedron which is self-conjugate with respect to the given conicoid. 11. Through a straight line in one of the principal planes tangent planes are drawn to a series of confocal ellipsoids ; prove that the points of contact lie on a plane, and that the normals at these points pass through a fixed point. If a plane be drawn cutting the three principal planes, and through each of the lines of section tangent planes be drawn to the series of conicoids, prove that the three planes which are the loci of the points of contact intersect in a straight line which is perpendicular to the cutting plane, and passes through the three fixed points in which the three series of normals intersect. 12. Any tangent plane to a cone makes equal angles with the planes through the line of contact and the focal lines. 13. If through a tangent at any point of a conicoid two tangent planes be drawn to a focal conic, these two planes wUl be equally inclined to the tangent plane at 0. 14. The focal lines of the enveloping cone of a conicoid are the generating lines of the confocal hyperboloid of one sheet which passes through its vertex. S. S. G. 11 162 EXAMPLES ON CHAPTER VIII. 15. Any section, of a cone which is normal at P to a focal line, has P for one focus. 16. If a section of an ellipsoid be normal to a focal conic at P, then P -will be a focus of the section. 17. The product of the distances of any point P on a focal conic of an ellipsoid, from two tangent planes to the surface which are parallel to one another and to the tangent at P to the focal conic, is constant for all positions of P. 18. From whatever point in space the two focal conies are viewed they appear to cut at right angles. Hence shew that the focal conies project into confocals on any plane. 19. If two confocal surfaces be viewed from any point, their apparent contours seem to cut at right angles. 20. If two cylinders with parallel generators circumscribe confocal surfaces their sections by a plane perpendicular to the generators are confocal conies. 21. The centres of the sections of a series of confocal conicoids by a given plane lie on a straight line. 22. Shew that those tangent lines to an ellipsoid from an external point whose length is a maximum or minimum are normals at their respective points of contact to confocals drawn through those points : and further, that the locus of these maximum and minimum lines to a series of ellipsoids confocal with the original one is a eone of the second degree. 23. A straight line meets a quadric in two points P, Q so that the normals at P and Q intersect : prove that PQ meets any confocal quadrio in points, the normals at which intersect, and that if PQ pass through a fixed point it lies on a quadric cone. 24. If from any point normals are drawn to a system of confocals (1) these normals form a cone of the second degree, (2) the tangent planes at the feet of the normals form a developable of the fourth degree. Consider the case of being in one of the principal planes. EXAMPLES ON CHAPTER VIII. 163 25. Th.e envelope of the polar plane of a fixed point with respect to a system of confocal quadrics is a developable surface. Prove this, and shew that the developable surface touches the six tangent planes to any one of the confocals at the points where the normals to that confocal through the fixed point meet that confocal. 26. Prove that the developable which is- the envelope of the polar planes of a fixed point P with respect to a system of confocal quadrics, meet Q the polar plane of P with respect to one of the confocals in a line, whose polar line with respect to the same confocal is perpendicular to Q ; and that these polar lines generate the quadric cone six of whose generators- are the normals at P to the three confocals through P, and the three lines through P parallel to their axes. 27. Prove that if a model of a hyperboloid of one sheet be constructed of rods representing the generating lines, jointed at the points of crossing; then if the model be deformed it wUl assume the form of a conibcal hyperboloid, and prove that the trajectory of a point on the model will be orthogonal to the system of confocal hyperboloids. 28. The two quadrics 2ayz+'2,hzx + 2cxy=\ and 2a'yz + Wzx + ^c'xy=t can be placed so as to be confocal if ahc a'h'c' _ ^ a'bV a%"c" ^T¥T? "^ a"+b"+o' ~ ' {a' + b' + cy ^ (»"■+ 5"= + c'")' ~ "• 29. Two ellipsoids, two hyperboloids of one sheet; and two hyperboloids of two sheets belong, to the same confocal system; shew that of the 256 straight lines joining a point of intersection of three surfaces to a point of intersection of the other three, there are 8 sets of 32 equal lines, the lines of each set agreeing either in crossing or in not crossing each of the principal planes. 30. A variable conicoid has double contact with each of three fixed confocals ; shew that it has a fixed director-sphere. 11—2 CHAPTER IS. QtTADEIPLANAR AND TeTEAHEDEAL Co-OEDINATES. 181. In the quadriplanar system of co-ordinates, four planes, which form a tetrahedron, are taken as planes of reference, and the co-ordinates of any point are its perpen- dicular distances from the four planes. The perpendiculars are considered positive when they are drawn in the same direction as the perpendiculars from the opposite angular points of the tetrahedron. Since the perpendicular distances of a point from any three planes are sufficient to determine its position, there must be some relation connecting the four perpen- diculars on the planes of reference. Let A, B, G, D be the angular points of the tetrahedron, and a, h, c, d be the areas of the faces opposite respectively to A, B, G, B; then, if a, ^8, 7, 8 be the co-ordinates (Jf any point, the relation will be where V is the volume of the tetrahedron ABGD. This is evidently true for any point P within the tetrahedron, since the sum of the tetrahedra BGDP, GDAP, BABP, ABGP is the tetrahedron ABGD ; and, regard being had to the signs of the perpendiculars, it can be easily seen to be universally true. TETEAHEDRA.L CO-ORDINATES. 165 182. The tetrahedral co-ordinates a, /8, 7, S of any point P are the ratios of the tetrahedra BGBP, GJDAF, DABP, ABGP to the tetrahedron of reference ABGD. The relation between the co-ordinates is easily seen to be a-t-)SH-7-l-S=l. It is generally immaterial whether we use quadriplanar or tetrahedral co-ordinates, but the latter system has some advantages, and in what follows we shall always suppose the co-ordinates to be tetrahedral unless the contrary is stated. We shall also suppose that the equations are homogeneous, for they can clearly always be made so by means of the relation a-l-/S-f- 74-8=1. When the equations are homogeneous we can use instead of the actual co-ordinates any quantities proportional to them. 183. The co-ordinates of the point which divides the line joining (a^, /S,, 7,, SJ and (a^, /S^, 7,, SJ in the ratio \ : fi. are easily seen to be X-|-/i ' \ + /t ' X,-|-/A ' \ + n 184. The general equation of the first degree represents a plane. The general equation of the first degree is la. + myS + ny+pB = 0. We may shew that this represents a plane by the method of Art. 13. Since the equation la. + m/3 + wy + p8 = contains three independent constants it is the most general form of the equation of a plane. The equation of the plane through the three points («i. A. 7i> ^1). («3. ^2. 72. K)' («s. /^s. 78. K) is :0. a, /8. 7, S «.. -8.. 7i. s. «2. ^2. 72. ^ «3. ^3. 7a. ^ 166 TETRiHEDRAL CO-ORDINATES. 185. To shew that the perpendioidars from the angular points of the tetrahedron of reference on the plane whose equation is la. + m^+wy +pB = are proportional to I, m, n,p. Let L, M, N, P be the perpendiculars on the plane from the angular points A, B, G, D respectively; the perpendicu- lars being estimated in the same direction. Let the plane meet the edge AB in K ; then at K we have 7 = 0, 8 = Q and la. + mB = 0.: therefore — = — -, .. m, I Now L:M::AK: BK. But AK : AB :: AGBK : ACBB :: B •■ I; simUarly KB : AB :: KBGB : ABGB :: a : 1; .-. L:M::AK:-KB::B--a"l-m; .'. -T= — , and similarly each = — = — . I m ■' n p 186. The lengths of tbe perpendiculars on a plane from the vertices of the tetrahedron of reference may be called the tangential co-ordinates of the plane; and, from the preceding article, the equation of the plane whose tangential co-ordinates are I, m, n, p is h + m/3 + nr/ +pS = 0. The co-ordinates of all planes which pass through the point whose tetrahedral co-ordinates are a,, /3„ y^, Sj, are connected by the relation la^ + m^^ + ny^ + pS^ = 0. Hence the tangential equation of a point is of the first degree. * 187. The equation of any plane through the intersection of the two planes whose equations are la + TO/S +ny+pS = 0, and I'a. + m'0 + n'y + p'S = 0, is {I + W) a + (to -I- \mO ^ + {n + \n')y + {p + Xp') S = 0. Hence the tangential co-ordinates of any plane through the line of intersection of the two planes whose co-ordinates are I, m, n, p and I', to', n', p' are proportional to I + XI', TO -I- \m', n + Xn', p + Xp'. TETRAHEDEAL CO-ORDINATES. 167 188. To find the perpendicular distance of a point from, a plane. Let the equation of the plane be k+?MyS + W7+pS = (i), and let its equation referred to any three perpendicular axes be Ax + By-^Gz + D = (ii). We know that the perpendicular distance of any point from the plane (ii) is proportional to the result obtained by substituting the co-ordinates of the point in the left-hand member of the equation. Hence the perpendicular distance of any point from (i) is proportional to the result obtained by substituting the co-ordinates in the expression la. + m/3 + ?i7 +ph. Hence, if I, m, n, p be eqiial to the lengths of the perpendiculars from the angular points of the tetrahedron of reference, the perpendicular distance of any other point («', ^S', y, B') will be lot' + mfi' ■+ ny +pB'. 189. If a plane be at an infinite distance from the angular points of the tetrahedron of reference, the perpen- diculars upon it from those points are all equal. Hence the equation of the plane at infinity is a+^+y + B=0. This result may also be obtained in the following manner. Let kix, k^, hy, kS be the co-ordinates of any point; then the invariable relation gives ka -{■ k^ + ky + kS = 1, or a-{-(S4-'y4-S=j^. If therefore k become infinitely great, we have in the limit a. + ^ + y + S = 0. This is the relation which is satisfied by finite quantities that are proportional to the co-ordinates of any infinitely distant point. 190. Let a,, /3,,7,, S, be the co-ordinates of any point P, and a, /S, 7, S the co-ordinates of a point Q. Also let 6^, 6.^, 6^, 6^ 168 TETEAHEDRAL CO-ORDINATES. be respectively the angles between the line PQ and the perpendiculars from the angular points A, B, G, D of the fundamental tetrahedron on the opposite faces. Then, a, b, c, d being the areas of the faces opposite to A, B, G, D respectively, we have a- o, = ^ a. PQ cos ^„ ^ -/3, = ^6.PQcos0j, 7 — 7, = ^ c.PQ cos^g,, and S — 8, = ^ d . FQ cos d^. The equations of the straight line through P, whose direction-angles are ^j„ ff^., 6^., d^, are therefore a-a , ^ /3-/3. ^ 7-7i ^ ^-^i ^i^._ a cos ^j 6 cos 6^ c cos 0^ d cos 6^ * Since the sum of the projections of the four faces of the tetrahedron on a plane perpendicular to PQ is zero, we have a cos ^j, + b cos 0^.+ c cos 0s + d cos 0^ = 0, or, putting I, m, n, p instead of acos^j, bcos0^, ocosd^, d cos 0^ respectively, l + m + n + p = 0. Ex. 1. Find the conditions that thiee planes may have a common line of intersection. Ex. 2. Find the conditions that two planes may be parallel. Ex. 3. Find the equation of a plane through a given point parallel to a given plane. [Any plane parallel to la + m^+ny+pS=0, is la+mp+ny+pS + \{a + p+y+S)=0. Hence the parallel plane through (a', ^, y, S') is la. + mp+ny+pB=iJ,a'+m^'+WY'+pS'){a+p+y + S).'\ y Ex. 4. The equations of the four planes each of which passes through a vertex of the tetrahedron of reference and is parallel to the opposite face are (3+7 + 8=0, y + S+a = 0, 8 + a+^=0, and o + j3+7=0. Ex. 5. Find the condition that four given points may lie on a plane- Ex. 6. Find the condition that four given planes may meet in a point. Ex. 7. The equations of the four planes each of which bisects three of the edges of a tetrahedron are o=/3 + 7+8, /3=7 + B + o, 7=S + o+/3, and S=o+/3+7. TETEAHEDBAL CO-ORDINATES. 169 Ex. 8. Shew that the lines joining the middle points of opposite edges of a tetrahedron meet in a point. Ex. 9. Find the equations of the four lines through A, B, G, B respec- tively parallel to the line whose equations are }a+7Hj3+m7+j)S=0, l'a.+m'^+n'y+p'S = 0. Ex. 10. A plane outs the edges of a tetrahedron iu six points, and six other points are taken, one on each edge, so that |eaoh edge is divided harmonically : shew that the six planes ea(£ of which passes through one of the six latter points and through the edge opposite to it, will meet in a pomt. Ex. 11. Lines AOa, BOi, COc, DOd through the angular points of a tetrahedron meet the opposite faces in a, b, c, d. Shew that the four lines of intersection of the planes BCD, icd; CDA, cda; DAB, dab ; and ABC, abc lie on a plane. [If be (a', ;8',. y'y «') the equation of bed is hence the line of intersection of BCD, bed is on the piano a p 7 o Ex. 12. If two tetrahedra be such that the straight lines joining corresponding angular points meet in a point, then will the four lines of intersection of corresponding faces lie on a plane. 191. We shall write the general eq^uation of the second degree in tetrahedral co-ordinates in the form ga" + r-jS' + s-f + 1^ + 2//3y + 2g-/a. + 'ihiP + 2MaS + 2wyS8 + 2W7S = 0. The left side of the equation will be denoted by F^oi, 13, 7, S). 192. To find the points where a given straight line cuts the surface represented by the general equation of the second degree in tetrahedral co-ordinates. Let the equations of the straight line be I m n p 170 TETRAHEDRAL CO-ORDINATES. To find the points common to this line and the surface, we have the equation F (a, + Ip, ^, + mp, % + np, S, +pp) = 0, or F{a,, P,,r,^,KHp{l^ + m^+n-^+p^J + p^F{l,m,n,p) = 0. Since there are two values of p, the surface is a conicoid. 193. To find the equation of a tangent plane at any point of a conicoid. If (hj, /8j, 7,, Sj) be a point on the surface, one root of the equation found in the preceding article will be zero. Two roots will be zero, if ,dF dF dF dF ^ The line will in that case be a tangent line to the surface. Substituting for I, mt, n,p from the equations of the straight line, we obtain the equation of the tangent plane, namely But, since the equation F(a, 0, y, S) =0 is homogeneous, dF^^ dF ^ dF^^ dF . therefore the equation of the tangent plane at the point («!. A. 7,. S,) is dF r, dF dF ^dF „ » 194. It can be shewn by the method of Art. 53, that the equation of the polar plane of any point (a^, yS^, 7^, S^) is dF^^dF^ dF^^dF . ''d^^^w^''^^^dsr^■ 195. To find the co-ordinates of the centre of the conicoid. The polar plane of the centre is the plane at infinity, whose equation isa + j8-t-7+S = 0. TETEAHEDEAL CO-OEDINATES. 171 Hence, if (oj, fi^, y,, SJ be the centre of the comcoid, we must have dF^dF_dF_dF da, di8i~dy, ~d^' 196. The diametral plane of a system of parallel chords of the conicoid can be found from Art. 192. The equation of the plane is jdF^ dF , dF , dF ^ Since l+m. + n+p = [Art. 190], it follows that all the diametral planes pass through the centre, that is through the point for which dF^dF^dF^dF dx d0 dy dB' 197. To find the condition that a given plane may touch the conicoid. The condition that the plane lu + m/S + ny + pB = may touch the conicoid can be found as in Art. 57. The result is QP + Bm" + (Sw= + Tp" + 2Fmn + IGnl + 2Hlm + 2Ulp + 2Vmp + 2Wnp = 0, where Q, R, S &c. are the co-factors of q, r, s &c. in the dis- criminant. 198. To find the condition that the surface represented by the general equation of the second degree may be a cone. The polar planes of the angular points of the fundamental tetrahedron with respect to a cone meet in a point, namely in the vertex of the cone. The equations of the polar planes are q% + h0 + gy + uS = O, ha + r0 + fy+ vB = 0, goL+f0+ S7-(-wS = 0, and UT. + t;/3 +W7 + th = Q. 172 TETRAHEDRAL CO-ORDINATES. The required condition is therefore q, h, g, u =0. h, r, f, V 9, f, s, w U, V, w, t 199. To shew that any two conicoids have a common self- polar tetrahedron. We can shew, as in Art. 142, that four cones can pass through the intersection of any two conicoids, and that the vertices of the four cones are the angular points of a tetrahe- dron self-polar with respect tO' any conicoid through the curve of intersection of the given conicoids. The equation of a conicoid, when referred to a self-polar tetrahedron, takes the form For, since a = is the polar plane of the point (1, 0, 0, 0), we have h=g = u = 0; and similarly /=« = w = 0. 200. To find the general equation of a conicoid circum- scribing the tetrahedron of reference. If we substitute the co-ordinates of the angular points of the tetrahedron of reference in the general equation of the second degree, we have the conditions q = r = s = t = 0. Hence the general equation of a conicoid circumscribing the tetrahedron of reference is f^y + gj^ + fi^^ + wS + v^S + wyB = 0. « 201. To find the general equation of a conicoid which touches the faces of the tetrahedron of reference. The planes a=0, /8 = 0, 7 = and 8 = will touch the conicoid given by the general equation of the second degree if Q = 0,B = 0,8 = 0&nAT=0. [Art. 197.] Hence conicoids which are inscribed in the tetrahedron of reference are given by the general equation, with the con- ditions Q= R=S = T=0. TETSAHEDEAL 00-ORDINATES. 173 Ex. 1. Find the equation of a ooniooid which oireumscribes the tetra- hedron of reference, and is such that the tangent planes at the angular points are parallel to the opposite faces. Ans. ^+ya+ap + aS+p3 + yS=0. Ex. 2. Find the equation of the conicoid which touches each of the faces of the fundamental tetrahedron at its centre of gravity. Am. w'+^ + y'' + S'-Py-ya-ap-ad-^5-yS = 0. 202. To find the equation of the sphere which circum- scribes the tetrahedron of reference. The general equation of a circumscribing conicoid is /^7 + S'ya + ^2/8 + waS + v^h + wyS = 0. If the conicoid be the circumscribing sphere, the section by S = will be the circle circumscribing the triangle ABC. Now the triangular co-ordinates of any point in the plane S = 0, referred to the triangle ABG, are clearly the same as the tetrahedral co-ordinates of that point, referred to the tetrahedron ABGD. Hence, when we put S = in the equa- tion of the conicoid, we shall obtain an equation of the same form as the triangular equation of the circle circumscribing ABG. Hence, comparing the equations f^r^ + gycL+ha^ = 0, and 5C''/37 + CJlV + -45'«/3 = 0, we obtain W^GA'^AW By considering the sections made by the other faces of the tetrahedron, we obtain the equation of the circumscribing sphere in the form BC^y + GA'ycc + AB'a^ + AB'oiB + BB'^B + GD^yS = 0. 203. To find the conditions that the general equation of the second degree may represent a sphere. Since the terms of the second degree in the equations of aU spheres, referred to rectangular axes, are the same; if (S = be the equation of any one sphere, the equation of any other sphere can be written in the form /S + Za + my8 + ny -^-ph = 0, or, in the homogeneous form, 8+{la + m^ + nr/+pB) (a + yS + 7-f S) = 0. 174 TETBAHEDBAL CO-OEDINATES. If this be the same conicoid as that given by the general equation of the second degree, S = being the equation of the circumscribing sphere found in Art. 202, we must have, for some value of X, 'kq = l, Xr = m, Xs = n, Xt =p ; also 2Xf=BC' + m + n, and five similar equations. f ^ s 2f Hence the required conditions are that — p^„ •' should be equal to the five similar expressions. The conditions for a sphere may also be obtained by means of the equation found in Art. 192 j or in the following manner. To find the points, P,, P^ suppose, where the edge BG meets the conicoid given by the general equation of the second degree, we must put a = 0> 8 = 0-, and we obtain rfi' + sy^ + 2/y37 = ; we have also jS + 7 = 1 ; .-. r-/3= + s (1 - /3)» + 2/^ (1 - ^), = 0, and, if the roots be /8„ yS,., we have ^'^'' " r + s - 2/' Now ;3A = — ^^; hence, if the conicoid be a sphere, and if t^, i^, f„ t^ be the lengths of the tangents from the points A, B, G, D respectively, we have * r + s — 2/ 8 BG" ~t^'- By considering the edges GB, GA we have similarly s + t — 2w _ q + s — 2g s GD' ~ GA'' ~t;- Hence, as above, the required conditions axe that — dTts should be equal to the similar expressionsi EXAMPLES ON CHAPTER IX. 175 Examples on Chaptee IX. 1. Shew that, if qa? + r^ + gy' + iS" = be a paraboloid, it will touch the eight planes a=fcj8±7±8 = 0^ 2. The locus of the pole of a givea plane with respect to a system of conicoids which touch eight fi:xed planes is a straight line. 3. The polar planes of a given point, with respect to a system of conicoids which pass through eight given points, all pass through a straight line. 4. If two pairs of the opposite edges of a tetrahedron are each tO' each at right angles to one another, the remaining pair will be at right angles. Shew also that in this case the middle points of the six edges lie on a sphere. 5. Shew that an ellipsoid may be described so as to touch each edge of any tetrahedron in its middle point. 6. If six points are taken one on each edge of a tetrahedron such that the three lines joining the points on opposite edges meet in a point, then will a conicoid touch the edges at those points. 7. If two conicoids touch the edges of a tetrahedron, the twelve points of contact are on another conicoid. 8. If a conicoid touch the edges of a tetrahedron, the lines joining the angular points of the tetrahedron and of the polar tetrahedron will meet in a point. 9i Shew that any two conicoids, and the polar reciprocal of each with respect to the other have a common self- polar tetrahedron. 10. A series of conicoids C^, U^, U^... are such that U^^^ and ZT _, are polar reciprocals with respect to iT. ; shew that U^^^ and U _, are also polar reciprocals with respect to U^. 11. The rectangles under opposite edges of a tetrahedron are the same whichever pair is taken ; prove that the straight lines joining its comers to the corners of the polar tetrahedron with respect to the circumscribed sphere will meet in a point. 176 EXAMPLES ON CHAPTER IX. 12. If four of the eight common tangeut planes of three conicoids meet in a point, the other four will also meet in a point. 13. A plane moves so that the sum of the squares of its distances from two of the angles of a tetrahedron is equal to the sum of the squares of its distances from the other two ; prove that its envelope is a hyperbolic paraboloid cutting the faces of the tetrahedron in hyperbolas each having its asymptotes passing through two of the angles of the tetrahedron. 14. If ABGD be a tetrahedron, self-conjugate with respect to a paraboloid, and DA, DB, DG meet the surface in A^, B^, C, respectively ; shew that DC]', DA, AA, BB, ■^ccj 15. If a tetrahedron have a self-conjugate sphere, and if its radius be R, prove that =-=5 = S „■« — 5- > where s is the sum of the squares of the edges of one face, and iS the sum of the squares of all the edges. 16. Shew that the locus of the centres of all conicoids which circumscribe a quadrilateral is a straight line. 17. The locus of the pole of a fixed plane with respect to the conicoids which circumscribe a quadrilateral is a straight line. 1 8. The polar plane of a fixed point with respect to any conicoid which circumscribes a given quadrilateral passes through a fixed line. 19. The sides of a twisted quadrilateral touch a conicoid; shew that the four points of contact lie on a plane. 20. A system of conicoids circumscribes a quadrilaterM : shew (1) that one conicoid of the system will pass through a given point, (2) that two of the conicoids will touch a given line, (3) that one conicoid will touch a given plane. Shew also that the conicoids are cut in involution by any straight line ; also that the pairs of tangent planes through any liae are in involution. 21. If three conicoids have a common self -polar tetrahedron, the twenty-four tangent planes at their eight common points touch a conicoid, and the twenty-four points of contact of their eight common tangent planes lie on another conicoid. EXAMPLES ON CHAPTER IX. 177 22. Nine conicoids have a common self-polar tetrahedron; shew that the eight points of intersection of any three, the eight points of intersection of any other three, and the eight points of intersection of the remaiaing three are aU on a conicoid. 23. The sphere which circumscribes a tetrahedron self-polar with respect to a conicoid cuts the director-sphere orthogonally. 24. The feet of the perpendiculars from any point of the surface - + s -^ ^-^ = 0, on the faces of the fundamental tetra- a ;8 y 8 hedron lie in a plane, a, h, c, d being proportional to the volumes of the tetrahedron formed by the centre of the inscribed sphere and the feet of the perpendiculars from it on any three of the faces, and the co-ordinates being quadriplanar. 25. The middle points of the twenty-eight lines which join two and two the centres of the eight Spheres inscribed in any tetra- hedron are on a cubic surface which contains the edges of the tetra- hedron. Shew also that the feet of the perpendiculars from any point of the cubic surface on the faces of the tetrahedron lie on a plane. 26. The six edges of a tetrahedron are tangents to a conicoid. The plane through the three points of contact of the three edges which meet in the same vertex meet the face opposite to that vertex in a straight line : shew that the four such lines are gene- rators of the same system of an hyperboloid. 27. When a tetrahedron is inscribed in a surface of the second degree, the tangent planes at its vertices meet the opposite faces in four lines which are generators of an hyperboloid. 28. The lines which join the vertices of a tetrahedron to the points of contact of any inscribed conicoid with the opposite faces are generators of an hyperboloid. 29. The lines which join the angular points of a tetrahedron to the angular points of the polar tetrahedron are generators of the same system of a conicoid. 30. Cones are described whose vertices are the vertices of a tetrahedron and bases the intersection of a conicoid with the oppo- site faces. The other planes of intersection of the cones and conicoid are produced to intersect the corresponding faces of the tetrahedron. Prove that the four lines of intersection are genera- ting lines, of the same system, of a hyperboloid. S. S. G. 12 CHAPTER X. Surfaces in General. 204. We shall in the present Chapter discuss some properties of surfaces of higher degree than the second. 205. Let F(!c, y,z)=Q be the equation of any surface. To find the points of intersection of the surface and the straight line whose equations are x— x _y — y' _z — if _ I m n we have the equation Fix' + Ir, y' + mr, z' + nr) = 0, or dF . dF . dF ^,, , ,, (,dF dF dF\ +o(^i+-i'+"i'y^+ =Y^- If the equation of the surface be of the rfi^ degree, the equation (i) will be of the «.* degree. Hence a straight line will meet a surface of the rfi^ degree in n points, and any plane will cut the surface in a curve of the n*'' degree. 206. To find the equation of the tangent plane at any point of a surface. If (x', y, a!) be a point on F (x, y, z) = 0, one root of the equation for r, found in the preceding article, will be zero. INFLEXIONAL TANGENTS, 179 Two roots will be zero if I, m, n satisfy the relation jdF , dF , dF ^ The line will in that case be a tangent line to the surface ; and the locus of all the tangent lines is found by eliminating I, m, n by means of the equations of the straight line. We thus obtain the required equation of the tangent plane , ,.dF . ,.dF , ,, dF ^ If the equation of the surface be z — /(«, y) = 0, it is easy to deduce from the above, or to shew independently, that the equation of the tangent plane at («', y, /) is 207. The two real or imaginary lines whose direction- cosines satisfy both the relations , dF . dF ^ dF „ meet the surface in three coincident points. Hence at any point of a surface two real or imaginary tangent lines meet the surface in three coincident points. These are called the inflexional tangents. 208. The tangent plane at any point of a surface will meet the surface in a curve of the n^ degree; and, since every line which is in the tangent plane, and which passes through its point of contact, meets the surface, and therefore the curve of intersection, in two points, it follows that the point of contact is a singular point in the curve of inter- section. When the inflexional tangents are imaginary, the point is a conjugate point on the curve of intersection. When the inflexional tangents are real, two branches of the curve of 12—2 180 INDICATEIX. intersection pass through the point of contact; and these branches coincide when the inflexional tangents are coin- cident. 209. The section of any surface by a plane parallel and indefinitely near the tangent plane at any point is a conic. Let any point on a surface be taken for origin, and let the tangent plane at the point be the plane z = 0. Let the equation of the surface be z =f{x, y) ; then, since z = is the tangent plane at the origin, we have z = ao^ + ^hxy + hy^ + higher powers of the variables. Hence, if we only consider points so near the origin that we may neglect the third and higher powers of the co-ordinates, the section of the given surface by the plane ^: = ^, is the same as the section of the conicoid whose equa- tion is z = ai^ ■\-hy^ ^-^hxy, by the plane z = h; the section is therefore a conic. The conic in which a surface is cut by a plane parallel and indefinitely near the tangent plane at any point, is called the indicatrix at the point ; and points on a surface are said to be elliptic, parabolic, or hyperbolic, according as the in- dicatrix is an ellipse, parabola, or hyperbola. 210. If, at the point (x', y, z') on the surface F{x, y, z) = 0, we have dx'd'i/ dz'~^' every straight line through the point (m', i/, z) will meet the surface in two coincident points. Such a point is called a singular point on the surface. All straight lines whose direction-cosines satisfy the relation dy' will meet the surface in three coincident points and are ENVELOPES. 181 called tangent lines. Eliminating I, m, n, by means of the equations of the line, we obtain the locus of all the tangent lines, viz. the cone whose equation is When the tangent lines at any point of a surface form a cone, the point is called a conical point; and when all the tangent lines lie in one or other of two planes, the point is called a nodal point. Ex. 1. Find the equation of the tangent plane at any point of the surface x'+y^+z'^=a^; and shew that the sum of the squares of the inter- cepts on the axes, made by a tangent plane, is constant. Ex. 2. Prove that the tetrahedron formed by the co-ordinate planes, and any tangent plane of the surface xyz=a?, is of constant volume. Ex. 3. Find the oo-ordinates of the conical points on the surface xyz - a (x^+y^ + z^) + ia?=0; and shew that the tangent cones at the conical points are right circular. [The conical points are (2a, 2a, 2a,) {2a, -2o, -2a,) (-2a, 2a, -2a) and ( - 2a, - 2a, 2a). The tangent cone at the first point is x^+f + z^-2yz-2zx-'ixy=Q.] Envelopes. 211. To find the locus of the ultimate intersections of a series of surfaces, whose equations involve one arbitrary para/meter. Let the equation of one of the surfaces be F {x, y, s, a) = Q, where a is the parameter. 182 ENVELOPES. A consecutive surface is given by the equation F{a}, y,z, a+Sa)=0, or Fi^x, y, z, a) + ^i?'(a!, y, z, a) Sa + = 0. Hence, when Ba is made indefinitely small, we have for the ultimate intersection of the two surfaces the curve given by the equations F(a!, y, z, a) = 0, and ^ F {oo, y, z, a) = 0. The required envelope is found by eliminating a from these equations. The curve in which any surface is met by the consecutive surface is called the characteristic of the envelope. Every characteristic will meet the next in one or more points, and the locus of these points is called the edge of regression or cmpidal edge of the envelope. 212. To find the equations of the edge of regression of the envelope. The equations of the characteristic corresponding to the surface F(x, y, z,a)=0 are F(a!, y, z,a) = and ^ F {ao, y, z, a) = 0. The equations of the next consecutive characteristic are therefore Fix, y,z,a+ Sa) = and -r- F {x, y, z, a + Sa) «»= 0, or F+^ha + ... = 0, and^+^8a + = 0. da da da Hence at any point of the edge of regression we must have T. (.dF . A^F f^ F = 0, ■y- = 0, and -r-» = 0. da da' The equations of the edge are found by eliminating a from the above equations. ENVELOPES. 183 213. The envelope of a system of surfaces, whose equation involves only one parameter, will touch each of the surfaces along a curve. Let A, B, Ghe three consecutive surfaces of the system ; and let PQ be the curve of intersection of the surfaces A and B, and P'Q' the curve of intersection of the surfaces B and G. Then the curves PQ and P'Q' are ultimately on the envelope. Let E be any point on the curve PQ ; and let 8, T be two points, very near the point R, one on the curve PQ, and the other on P'Q'. Then the plane RST will in the limiting position be the tangent plane at It both to the surface B and to the envelope ; and hence the envelope touches the surface B, and similarly every other surface of the system, along a curve. 214. To find the envelope of a series of surfaces whose equations involve two arbitrary parameters. Let the equation of any surface of the system be F(x,y,z, a,b) = 0, where a, b are the parameters. A consecutive surface of the system is Fix, y, z,a + ha,b + Sb) = 0, Jjp JET or F(a;, y, z, a,b) + Sa -^ + Sb -^ + = 0. Hence, when Sa and Bb are made indefinitely small, we must have at a point of ultimate intersection F=0,andBa^+Sb~ = 0, or, since Sa and Sb are independent, „ ^ dF . jdF ^ F=Q, J- = 0, and jt = da db Hence the curve of intersection of F with any surface consecutive to it goes through the point which satisfies the 184 FAMILIES OF SURFACES, equations The required envelope is found by eliminating a and b from the above equations. 216. To shew that the envelope of a series of surfaces, whose equations involve two arbitrary parameters, touches each surface of the series. Let the curves of intersection of the surface F with consecutive surfaces of the system pass through the point P ; then P is a point on the envelope. Let F^, F^ be any two surfaces consecutive to F, and let Q, R be the points on the envelope which correspond to these surfaces. Then all surfaces consecutive to F^, and therefore the surface F, will pass through Q ; similarly the surface P will pass through R. Hence, in the limit, the envelope and the surface F have the three points P, Q, R, which are indefinitely near to one another, in common ; they therefore have a common tangent plane. Hence the envelope touches the surface F, and simi- larly for any other surface. Bz. 1. Find the envelope of the plane which formB with the co-ordinate planes a tetrahedron of constant volume. Am. x^z= constant. Ez. 2. Find the envelope of a plane such that the sum of the squares of its intercepts on the axes is constant. Ans. x''+y^+i^=eonata,nt. Ez. 3. Find the equations of the edge of regression of the envelope of the y\a,ne X Bva.d -y COS 8= aB-cz, Am. x'+y^=a!', y=xta,n — . Families of Surfaces. 216. To find the general functional and differential equa- tions of conical surfaces. The equation of any cone, when referred to its vertex as origin, is homogeneous ; and is therefore of the form ^(;. !)-»• CONICAL SURFACES. 185 Hence the equation of any cone whose vertex is at the point (a, /8, 7) is of the form j^(£z^. ^ = (i). This is the required functional equation. The tangent plane at any point of a cone passes through the vertex of the cone. Hence, if the equation F {x, y,z) = represent a cone whose vertex is (a, /S, 7), we have , sdF . o,dF^, ,dF - .... (^-«)^+(y-^)^+(«-7)^=0 (XI), which is the required differential equation. 217. To find the general functional and differential equa- tions of cylindrical surfaces. A cylinder is the surface generated by a straight line which is always parallel to a given straight line, and which obeys some other law. Let the equations of the fixed straight line be X _ y _z I m n' The equations of any parallel line are ai-a_ y-P _z . I ~ m n ^'' the two constants a and /8 being arbitrary. Now, in order that the line (i) may generate a surface, there must be some relation between the constants a and /8. Let this relation be expressed by the equation a =/(/8); then, we have from (i) I .[ m \ or F{nx — Iz, ny — mz) =0 (ii), which is the required functional equation. 186 CONOIDAL SURFACES. The tangent plane at any point of a cylinder is parallel to the axis of the cylinder. Hence, if the equation F{a;, y, «) = represent a cylinder, whose axis is parallel to the line OS _ y _is I m,~ n* we have t-; — |-m-f- + w-r- = 0, dx dy dz which is the required differential equation. 218. To find the general fumctional and differential equa- tions ofconoidal surfaces, Def. a conoidal surface is a surface generated by the motion of a straight line which always meets a fixed straight line, is parallel to a fixed plane, and obeys some other law. The surface is called a right conoid when the fixed plane is perpendicular to the fixed line. Let the fixed straight line be the line of intersection of the planes Ix + my + nz+p =0, l'a! + m'y + n'z +p' = ; and let the fixed plane, to which the moving line is to be parallel, be Xa; + (ly + vs = 0. The equations of any line which satisfies the given conditions are lai + my + m +p + A (I'x + m'y + n'z +p') = 0, and Xas + fiy + vz + B = 0. In order that the straight line may generate a siirface, there must be some relation between the constants A and B. Let this relation be expressed by the equation A =/(B) ; then we have Ix + my + nz+p .. . ... -=; r 7 — ^-' = f (Kaa + iiy + vz) (i), lx + m'y-\-nz+p •' ^ ^fu^ i \ /> the required functional equation. If we take two of the co-ordinate planes through the fixed straight line, and the third co-ordinate plane parallel to the DEVELOPABLE SURFACES. 187 fixed plane, the above equation reduces to the simple form ^=/(-) (ii)- The differential equation of conoidal surfaces which corresponds to the functional equation (ii), can be readily shewn to be dx " dy ' The differential equation may also be obtained as follows. The generator through any point is a tangent line to the surface ; and the condition that X y ' may be on the plane dF dF ^ Ex. 1. Shew that xyz=sc {x^ - y^ represents a oonoidal surface. Ex. 3. Find the equation of the right conoid whose axis is the axis of z, and whose generators pass through the circle x=a, y^+z^=b''. Am. ahf+xh^^Vx^. Ex. 3. Find the equation of the right conoid whose axis is the axis of z, and whose generators pass through the curve given hy the equations x=aaos'nji,y=aBmnz. Arts. ^=a;tan7iz. Ex. 4. Shew that the only conoid of the second degree is a hyperbolic paraboloid. 219. Cones, cylinders and conoids are special forms of ruled surfaces. There are two distinct classes of ruled surfaces, namely those on which consecutive generators inter- sect, and those on which consecutive generators do not intersect ; these are called developable and skew surfaces respectively. We proceed to consider some properties of developable and skew surfaces. 188 DEVELOPABLE SURFACES. 220. Suppose we have any number of generating lines of a developable surface, that is any number of straight lines such that each intersects the next consecutive. Then, the plane containing the first two lines can be turned about the second line until it coincides with the plane containing the second and third lines ; this plane can then be turned about the third line until it coincides with the plane through the third and fourth lines; and so on. In this way the whole surface can be developed into one plane without tearing. 221. The tangent plane at any point of a ruled surface must contain the generator through the point [Art. 129]. If the surface be a skew surface, the tangent plane will be different at different points of the same generator ; but, if the surface be a developable surface, the tangent plane will be the same at all the different points of a given generator, for the tangent plane is the limiting position of the plane through the given generator and the next consecutive generator. Since any tangent plane to a developable surface touches the surface at all points of a straight line, it follows from Art. 213, that a developable surface is the envelope of a plane whose equation contains only one variable parameter. 222. To find the general differential equation of develop- able surfaces. The tangent plane at any point of a developable surface meets the surface in two consecutive generating lines which are the two inflexional tangents at the point. ' Hence, at any point of a developable surface, the two lines given by the equations l^+m^+n — = dx dy dz ' must coincide. DEVELOPABLE SURFACES. 189 The condition that this may be the case is d'F d'F d^F dF = das' ' dxdy' dxdz' dx d'F dJ'F d'F dF dxdy' df • dydz' dy d'F d'F d'F dF dxdz ' dydz' dz' ' dz dF dF dF dx ' dy ' dz ' .(i). This is the required differential equation. The differential equation may also be obtained from the property, proved in the last Article, that a developable surface is the envelope of a plane whose equation involves only (me parameter. For, the general equation of the tangent plane of a surface at the point {x, y, z) is ,^/ df ^-'=^^-'"^Tx-^^''-yHy- Hence, if the surface is a developable surface, there must be some relation connecting -^ and -^ ; that is, connecting T- and -r- \ we therefore have dx ay Therefore and da? d'z dx \dyl ^F'i^' d\ dyj ' dxdy ' dxdy \dy) ' dy'' Hence ^ c^z _ / d'z Y da? ' df ~ \dxdy) ' which is equivalent to (i). 190 DEVELOPABLE SURFACES. 223. We can find the equation of the developable surface which pafeses through two given curves, in the follow- ing manner. The plane through any two consecutive gene- rating lines of the surface will pass through two consecutive points on each of the given curves ; hence the tangent plane to the required developable surface will toiich each of the given curves. Now the equation of a plane in its most general form contains three arbitrary constants, and the conditions of tangency of the two given curves will enable us to express any two of these constants in terms of the third, and the equation of the plane will thus be found in a form involving only one arbitrary parameter. The developable surface is then obtained as the envelope of the moving plane. Ex. Find the equation of tlie developable surface whose generating lines pass through the two curves y^=iax, z=0 and x^=iay, 2=e; and shew that its edge of regression is given by the equations cx^-3ayz=0=cy^-3a,x(c-z). Let one of the tangent planes of the developable be lx + my+nz+l=0. The plane touches the first curve, if lx+my + l=0 touches y''~4:ax=0; that is, if l=.airfi. The plane touches the second curve, if lx + 'my+nc + l=Q touches x'^=iay; that is, if m(rec + l)=oP. Hence, the equation of the tangent plane of the developable is found in the form am?x + my + {a?m^ -1) - + \=0 (i). The surface is therefore given by the elimination of m between (i), and iamx + y + B =0 M}i). For points on the edge of regression we have also ax + 3 =0 (m). c ^ ' From (ii) and (iii) we have m= - — ; and therefore, from (iii), cx'^Sayz, This is the equation of one surface through the edge of regression. We obtain another surface through the edge by substituting m= - — in (i); the result is y'z^ifl {c-z), and at all points common to the surfaces (!a?'= and y'''z=(^{c-z), we must have cy'=3ax (c-z). SKEW SURFACES. 191 224. To shew that a conicoid can be draum which will touch any skew surface along a generating line. Let AB, A'B , A"E' be three consecutive generators of any skew surface. Then, [Art. 134], a conicoid will have these three lines as generators of one system, and any line which intersects the three given lines will be a generator of the opposite system of the same conicoid. Through any point Q on AE draw the line PQR to intersect the lines AB and A"B!'. Then this line passes through three con- secutive points of the given surface, and is therefore a tangent line to the surface. Hence the plane through A'B' and PQR touches both the given surface and the conicoid. Hence the conicoid touches the given surface at all points of the line A'R. By means of the above theorem many properties of a ruled conicoid may be shewn to be true of all skew surfaces. 225. To find the lines ofstriction of any skew surface. Def. The locus of the point on a generator of a ruled surface where it is met by the shortest distance between it and the next consecutive generator, is called the line of striction of the surface. If we know the equations of any generating line, we can at once find the direction of the shortest distance between it and the next consecutive generator, and this shortest distance is a tangent line of the surface. Hence, in order to find the point on the line of striction, which corresponds to any particular generator, we have only to write down the con- dition that the normal at a point on the generator may be perpendicular to the shortest distance between the given generator and the next consecutive. Ex. 1. To find the lines of stiiction of the hyperboloid x' y" z' . — + 2 = 1. o" J» c" The direotion-cosines of a generator, and of the next consecutive generator, are proportional respectively to a sin S, -b cos 6, c, and a sin {e + dS), -b cos (9 + dS), c. 192 SKEW SURFACES. Hence the direotion-cosines of the shortest distance are proportional to - be sin ff, ca cos 6, ab. Now, if {x, y, z) be the point where the shortest distance meets the con- secutive generators, the normal at (x, y, z) must be perpendicular to the given generator, and also to the shortest distance. We therefore have - sin e - r cos 9 - -=0, a b c and ■^sin9-^cos9 + -.=0. or 0" c"* Eliminating $, we get for the lines of striotion the intersection of the surface and the quartic Ex. 2. To find the lines of striction of the paraboloid whose equation is x' y^ All the generating lines of one system are parallel to the plane ^-1=0 (i). a b ' The shortest distance between two consecutive generators of this system will therefore be perpendicular to the plane (i). Hence, at a point on the corresponding line of striction, the normal to the surface is parallel to (i). The equations of the normal at {x, y, z) are l-g v-y f-'g X = y = -1 , Hence one line of striction is the intersection of the surface and the plane ao^b' "■ Similarly, the line of striction of the generators which are parallel to the plane - + |=0 is the parabola in which the plane -^-^=0 cuts the surface. [See a paper by Prof. Larmor, Quarterly Journal of Mathematics, Vol. XIX. page 381.] 226. To find the general functional and differential equa- tions of surfaces of revolution. Let the equations of the axis of revolution be x—a _y—h _z—o I m n ' SURFACES OF REVOLUTION. 193 The equations of a section of the surface by a plane perpendicular to the axis are of the form {x-aY + {y-h)'+{z-cf = r\ and Ix + my + nz=p. Hence, since there must be some relation between r* and p, the required functional equation is (x - af + {y- by + (z- c)' =f{lx +my + nz). The normal at every point of a surface of revolution intersects the axis. The equations of the normal at the point («', 2/', z') of the surface Fije, y, z) = are x — x' _ y — y' _ z — z dF ~ dF~ dF ' dx dy' dz By writing down the condition that the normal may in- tersect the axis, we see that at every point of the surface, dF dF^ dF =0; dx' dy' dz x — a, y — b, z — c I, m, n this is the dififerential equation of surfaces of revolution. Note. In the above, and also in Articles 216 and 217, we have obtained the functional equation and the diffe- rential equation by independent methods. The differential equation could however in each case be obtained from the functional equation; this we leave as an exercise for the student. For fuller treatment of Families of Surfaces the student is referred to Salmon's Solid Geometry, Chapter xiii. a s. G. 13 194 EXAMPLES ON CHAPTER X. Examples on Chapter X. 1. Prove that a surface of the fourth degree can be described to pass through all the edges of a parallelopiped, and that if it pass through the centre it also passes through the diagonals of the figure. 2. Shew that at any point on the axis of « there are two tangent planes to the surface os^ = a? (c' — z'). 3. Find the developable surface which passes through a parabola and the circle described in a perpendicular plane on the latus rectum as diameter. 4. Find the equation of the developable surface which contains the two curves y = 4aa;, «=0; and (y-6)' = 4c«, a; = 0; and shew that its cuspidal edge lies on the surface (ax + hy + cz)' = SaAx (y + b). 5. The developable surface which passes through the two circles whose equations are x' + ^ = a',z = 0, and as' + ^ = c", y = 0, passes also through the rectangular hyperbola whose equations are « - y' = -= 5 and a; = 0. 6. Prove that the surface has two conical points, and two singular tangent planes. * 7. Explain what is meant by a nodal line on a surface, and find the conditions for such a line on the surface ^ {x, y, «) = 0. There is a nodal line on the surface z {a? + y') + 2axy = ; find it. 8. Give a general explanation of the form of the surface z (x' + y') = 2kxy. Shew that every tangent plane meets the surface in an eUipse whose projection on a plane perpendicular to the nodal line is a circle. EXAMPLES ON CHAPTER X. 195 9. Examine the general form of the surface scyz - a'x — h'y — c'e + 2a5c = 0, and shew that it has a conical point. Shew also that each of the planes passing through the conical point and a pair of the inter- sections with the axes touches the surface along a straight line. 10. If a ruled surface be such that at any point of it a straight line can be drawn lying wholly on the surface and intersecting the axis of «, then at every point of the surface .(Fz . d'z .d'z . 11. Shew that the surface whose equation is determined by the elimination of between the equations X cos + y sin 6 = a, X sin 6 — y cos = — {o6 — e), is a developable surface, and find its edge of regression. 12. What family of surfaces is represented by the equation I ? Describe the form of the surface whose equation is '-*($): sin"' - = w tSa:;^ . If ra = 2, prove that through any point an infinite number oiJ)lanes can be drawn, each of which shall cut the surface in a conic section. 13. At a point on the surface (cc - y) «' + aa; (a + o) = there is in general only one generator, but at certain points there are two, which are at right angles. 14. Any tangent plane to the surface a{a? + y')+xyz = meets it again in a conic whose projection on the plane of xy is a rectangular hyperbola. 15. Shew that tangent planes at points on a generator of the surface ym^—c^z = cut a; = in parallel straight lines. 16. Prove that the equation as" + y° + «° - "ixyz = o° represents a surface of revolution, and find the equation of the generating <;urve. 17. From any point perpendiculars are drawn to the generators of the surface 8(a!' + 3/*)-27»ajy = 0; shew that the feet of the perpendiculars lie upon a plane ellipse. 13—2 196 EXAMPLES ON CHAPTER X. 18. Shew that all the normals to a skew surface, at points on a generator, lie on a hyperbolic paraboloid whose vertex is at the point where the generator meets the shortest distance between it and the next. 19. A generator PQ of the surface xyz - A (as' + y") = meets the axis of z in P. Prove that the tangent plane at Q meets the surface in a hyperbola passing through P, and that as Q moves along the generator the tangent at P to the hyperbola generates a plane. 20. Prove that all tangent planes to an anchor-ring which pass through the centre of the ring cut the surface in two circles. Also if a surface be generated by the revolution of any conic section about an axis in its own plane, prove that a double tangent plane cuts the surface in two conic sections. 21. Prove that a flexible inextensible surface in the form of a hyperboloid of revolution of one sheet, out open along a generator, may be bent so that the circle in the principal plane becomes the axis, and the generators the generating lines of a conoid of uniform pitch inclined to the axis at a constant angle. 22. Prove that every cubic surface has twenty-seven lines and forty-five triple tangent planes real or imaginary, and that every cubic surface which has a double line is a ruled surface. Discuss some properties of the surface whose equation is y' + x^z + yzw = 0. 23. Four tangent planes to any skew surface which are drawn through the same generator have their cross-ratio equal to that of their four points of contact. 24. Any plane thi'ough a generator of a skew surface is a tangent plane. at some point P and a normal plane at some point P; shew also that there is a point on the generator such that the rectangle OP. OP" is constant for all planes through it 25. Shew that the wave-surface, whose equation is aV by c'z' a^ + if' + z'-a' af + y' + s/'-b' a:* -h j/' 4- «* - c' has four conical points, and four singular tangent planes, = 0, CHAPTER XI. Curves. 227. We have already seen that any two equations will represent a curve. By means of the two equations of the curve, we can, theoretically at any rate, express the three co-ordinates of any point as functions of a single variable ; we may, for example, suppose the three co-ordinates of any point of a curve expressed as functions of the length of the arc measured along the curve from some fixed point. 228. To find the equations of the tangent at any point of a curve. Let X, y, z be the co-ordinates of any point P on the curve, and let x-^hx,y->rhj,z-\- Bz be the co-ordinates of an adjacent point Q. Then, if Ss be the length of the arc PQ, we have, since the arc is ultimately equal to the chord, Bx^ + Sy' + Bz'^Ss'; ■■• ®'-&h&'- Also, since the direction -cosines of the ' chord PQ are proportional to Bx, By, Bz, and the tangent coincides with the ultimate position of the chord, the direction-cosines of the tangent are equal to dx dy dz ds ' ds ' ds so that the required equations of the tangent at {x, y, z) are g-a; ri-y _ }^-z dx dy dz ' ds ds ds 198 TANGENT TO A CURVE. If the curve be the curve of intersection of the two surfaces Fix, y,s) = and Q (x, y, z) = 0, the tangent line at any point is the line of intersection of the tangent planes of the two surfaces at that point. Hence the equations of the tangent at any point (x, y, z) are ,, .dF . .dF ^.^ \dF f. ^^-"'^Tx-^^^-y^Ty + ^^-'^Tz = ^' tf \dG , f \dG , n. sdO „ (f-^)^ + ('?-2/)^+(?-^);^ = 0. 229. To find on a given surface a curve such that the tangent line at amy point makes a maooimum angle with a given plane. It is clear that the tangent line to such a curve at any point is in the tangent plane to the surface at that point, and is perpendicular to the line of intersection of the tangent plane and the given plane. Let the equation of the given plane be Ix + my + nz = 0. Then the direction-cosines of the line of intersection of the given plane and the tangent plane at any point (x, y, z) of the surface F {x, y, z) = 0, are proportional to dF dF dF ,dF ,dF dF dz dy dx dz ay dx The direction-cosines of the tangent to the curve are dx ^ dz ' ds' ds' ds' Hence we have ds \ dz dy) ds \ dx dz J dy the required differential equation ds \ dy dx) ' CURVES. 199 If the given plane be the plane ^ = 0, the differential equation of a line of greatest slope •will be dFdy dFdx_ dx ds dy ds Ex. Find the lines of greatest slope to the plane « = on the right conoid whose equation is x=yf (z). The differential equation of the projection on z = of a line of greatest slope is xdx+ydy = 0. Hence the projections of the lines of greatest slope on the plane 2=0 are circles. 230. Definitions. If A, B, G be three points on a curve, the limiting position of the plane ABG, when A, G are supposed to move up to and ultimately to coincide with B, is called the osculating plane at B. The circle ABG in its limiting position is called the circle of curvature at B, the radius of the circle is the radius of curvature, and its centre the centre of curvature at B. The normals to a curve at any point are all in the plane through the point perpendicular to the tangent to the curve : this plane is called the normal plane at the point. The normal which is in the osculating plane at any point of a curve is called the principal normal. The normal which is perpendicular to the osculating plane is called the binormal. The surface which is the envelope of all the normal planes of a curve is called the polar developable. The angle between the osculating planes at any two points P, Q of a curve is called the whole torsion of the arc PQ. The limiting value of the ratio of the whole torsion to the arc is called the torsion at a point. The radius of the circle whose curvature is equal to the torsion of the curve at any point, is called the radius of torsion at that point, and is represented by a. The radius of the sphere which passes through four consecutive points of a curve is called the radius of spherical curvature. Note. In what follows we shall have frequent occasion 200 CURVES. to employ differential coefficients with respect to the arc ; and we shall for shortness write id, a!', x'" &c. instead of dm cfa; T-j &c. 231. In the annexed figure A, B, G, D, E, F... are sup- posed to be consecutive points of a curve, and p, q, r... are the middle points of the chords AB, BG, GD..., Planes are THE OSCULATING PLANE. 201 drawn through p, q,r... perpendicular to the chords AB, BG, CD..., and LP, MQP, NMQ... are the lines of intersection of the planes through p and q, q and r, r and s, . . . . The lines pL, qL are in the plane ABG, and perpendicular respectively to AB and BG; the lines qM, rM are in the plane BGB, and perpendicular respectively to BG, GD. Then, in the limit, when the chords AB, BG, GD... become indefinitely small the planes ABG, BGD,... become osculating planes of the curve; the planes pLP, qMQ,... become normal planes of the curve ; the points L, M, N be- come centres of curvature of the curve ; the lines LP, MQP, NRQ... become generating lines of the polar surface, and are called polar lines; and the points P, Q, R... become con- secutive points on the edge of regression of the polar surface. All points on the plane pLP are equidistant from A and B, all points on the plane qMP are equidistant from B and G, and all points on the plane rMP are equidistant from G and D ; therefore a sphere with P for centre will pass through A, B, G, D; hence the edge of regression of the polar surface is the locus of the centre of spherical curvature. 232. To find the equation of the osculating plane at any point of a curve. Let P, Q, R be three consecutive points on the curve such that PQ = QR = Ss ; and let s be the length of the arc measured from some fixed point up to Q. Then, if the co-ordinates of Q be x, y, z, those of P, for which the arc is s — hs, will be, if we neglect powers of 8s above the second, x-co'hs+'^h^, y-y'Ss + ^Ss', z-zSs + jS^; and the co-ordinates of R will be found by changing the sign of Ss. The equation of any plane through Q is of the form i(?-«)+Jf(77-y) + i^(?-a) = 0. 202 THE PRINCIPAL NORMAL. If this plane pass through the points P and jB, we must have Lx +My' +N^ =0, and, eliminating L, M, N, we have the required equation of the osculating plane, namely ^-a!,r)-y,^-z =0. «', y, z 233. To find the equations of the principal normal, and, the curvature, at any point of a curve. Let P, Q, B be three points on a curve such that Then, if V be the middle point of PB, QF is in the plane PQB ; and, since the chords PQ and QB only differ by cubes of Ss, QVia ultimately perpen(£cular to PB, and is therefore the principal normal at Q. Then, the co-ordinates of P, Q, B being as in the last Article, the co-ordinates of V are x" v" z" a,+ '^Ss\ y + ^Bs\ z + ^Ss\ Hence the equations of QF are ^~as _ v-y ^-z «>" ~ 2/" ~~i^ ^^'' Again, the circle PQB, in its limiting position, ig» the circle of curvature. Hence, if p be the radius of curvature, we have in the limit ^P-QV But QV' = -^ (a/" + y"' + z"% and PQ = Ss; .■.^ = x"' + y"' + z"\ THE BINOEMAL. 203 Hence, the direction-cosines of the principal normal, which from (i) are proportional to a;", y", z", are equal to px", py" and pz". The co-ordinates of the centre of curvature are easily seen to be w+p'x",y + pY, z + pV. 234. To find the direction-cosines of the binormal. The binormal is perpendicular to the osculating plane. Hence, if I, m, n be the direction-cosines of the binormal, we have from Art. 232 I _ ""^ _ " y'z" - sfy" ~ i^x" - x'z" ~ x'y" - y'x' ' But {y'z" - z'y'J + {z'x" - x'z'J + {x'f - y'x'J = (x" + y'" + z") (x'" + y'" + z'") - (x'x" + y'y" + //')' 1 since x" + y'^ + z'" = 1, and therefore x'x" + y'y" + z'z" = 0. Hence the required direction-cosines are p (y'z" - z'y"), p (z'x" - x'^'), p {x'f - ^x"). 235. To find the measure of torsion at any point of a curve. Let I, m, n be the direction-cosines of the normal to the osculating plane at P ; and let l + Bl,m + Sm, n + Sn be the direction-cosines of the normal to the osculating plane at Q, where FQ = Ss. Then, if St be the angle between the osculating planes, we have sin" St = (mSn — nSm)" + (nU — IBn)' + (IBm — mSlf. 204 MEASXTEE OF TOESION. Hence, in the limit, we have /drV / dn dm\^ I dl jdnV , /, dm dlV [ds) =rTs-''-dF) +v'd-s-^d^) +[^ib-'^ds) ' or, 4 = (™w' - m'ny + {nV - n'lf + {Im' - I'mf (i). Now l = p{y'if'-zy"); and similarly for m' and n'. Hence mn' - m'n = p" (/x" - afz") [xf - y'x'") - p' {z'x'" - x'z'") [x'f - y'x") = /) V ^' . .y'. z' x", y"> z" x'", y"\ z'" We can find similar expressions for nV — n'l, and for Im' — I'm ; and substituting in (i), we have pV «! , y , z „»/ ^jf „tf o , y , z I 'I .'II jn 236. To find the condition that a curve may he a plane cwrve. Let X, y, z be the co-ordinates of any point P on the curve, expressed in terms of the arc measured from a fixed point up to P ; and let Q be the point at a distance a- measured along the curve from P. Then the co-ordinaflfes of Q will be x + trx' + ^x" + j^x"' + , y+<^y' + ^y" + %y"'+ , z+)x" + {n-y)y"+{^-z)z" = x'^ + y" + z" = l (ii), and (? - x) c^" + {'n-y) y'" +(?-«) /" = 0. . .(iii), since x'x" + y'l/' + z'z" = 0. 238 In the figure to Art. 231, we have p=pL = qL, p + Sp = qM = rM, and BT = LqM=LPM. If K be the point of intersection of MQP and qKL, we have to the second order, Mq = Kq, and KP = LP ; .'. LK = Sp, and LP=-^- = t^ ultimately (i). Also MEASURE OF TORSION. 207 •(ii), where R is the radius of spherical curvature. Projecting the sides of the triangle KLP on the axis of X, we have, if I, m, n be the direction-cosines of the binormal, therefore ultimately px" = ^.^ = ^^. d/r dp as dr ' or px" = (7l' (iii). Since I = p (yV - z'y") [Art. 234] we have from (iii) px" = sec' a. We therefore have x"= - cos 8 cos^a r= -Ein9 , z"—0; and also Hence and r = -J sin e ooss a,y"'=--^ cos e cos' a, «'" = 0. •°"'=ro^' 1 _ oos^ a - sin e cos o, cos fl cos o, sir - - cos 9 cos' a, — sinecoB^o, -jSineoos^a, - -^ cos fl cos' a, = —. cos' o sin a ; sin a cos a It should be noticed that the principal normals all intersect perpendicularly the axis of the cylinder. This is seen at once by writing down the eauatious 01 the prmcipal normal at 9, namely a-gcos9 _ y - g sin g _z-ad tan a cos 8 ~ sin 8 ' EXAMPLES. 209 Ex. 2. To find the equations of the principal normal, and of the osculating plane at any point of the curve given by the equations x=iacos^e,y = ia sin^ $, z — do cos 29. We have x'= - 12a cos^ 9 sin S . 6', y' = 12a sin^ fl cos S . S', «'=-6cBin2e. 6'. Square and add, then 1 = 6 ^(a^ + c^) sin 29 . 6'. The equations of the principal normal are therefore x-ia cos' B _y -ia sin' 9 _ 2 - 3c cos 25 sin 9 ~ cos 9 ~ The equation of the osculating plane is -4acoB*9, 3/-4asin3 9, ^-3ccos29 =0. - a cos 9, a sin 9, - c sin 9, cos 9, Ex. 3. To find to the third order the co-ordinates of any point of a curve in terms of the arc, when the axes of co-ordinates are the tangent, the principal normal, and the binormal at the point from which the arc is measured. Let OX, OY, OZ be the tangent, principal normal, and binormal at the point of a curve. Let x, y, z be the co-ordinates of a point at a distance s from 0. and let - and - be the curvature and torsion of the curve at 0. P " Them, at the origin, x' = l, y'=0, z^=0; also pa;" = 0, p2/"=l, !!"=0. We have, at any point of the curve, !t'a;"+2/y'+0V'=O. Difierentiating, we have \ + x'x"'-i-y'y"'+i'z'" = '(i). P Also, by differentiating 1 we have at any point , S. S. G. ^=x"x"'-i-y"y"'-\-z"i"' (ii). 14 210 EXAMPLES ON CHAPTER XI. Also we know that 1 a;', 3,'. ,^ x", y", «" «'", J,"', z"' From (i), (ii), (iii) we see that at the origin Hence, by Maclaurin'a Theorem, we have to the third ordor 8' s2 s» dp «8 .(iii). Examples on Chapter XI. 1. Find the equation of the surface generated by the principal normals of a helix. 2. Find the osculating plane at any point of the curve x = a cos 6 + h sin 6, y = a sin Q + h cos 6, z = c sin 2$. 3. Find the equations of the principal normal at any point of the curve x' + i/'^a!', az = sif-y'. 4. A point moves on an ellipsoid so that its direction of motion always passes through the perpendicular from the centre of the ellipsoid on the tangent plane at any point; shew that the curve traced out by the point is given by the intersection of the ellipsoid with the surface a!'"""^""' !!;'"" = constant, ' I, m, n being inversely proportional to the squares of the semi- axes of the ellipsoid. 5. A curve is traced on a right cone so as to cut all the generating lines at the same angle; shew that its projection on the plane of the base is an equiangular spiral. 6. Shew that any curve has an infinite number of evolutes which lie on its polar developable. Shew also that the locus of the centre of principal curvature is not an evolute. EXAMPLES ON CHAPTER XI. 211 7. If a circular helix be drawn passing through four con- secutive points of a curve in space, prove that when the four points ultimately coincide the radius of the helix equals /"^ , and its slope is tan"' - . 8. Shew that if the osculating plane at every point of a curve pass through a fixed point, the curve will be plane. Hence prove that the curves of intersection of the surfaces whose a* equations are a;* + «/' + «° = a", and x* + y* + z'^ = -^ are circles of radius a. 9. Prove that the helix is the only curve whose radius of circular curvature and radius of torsion are both constant. 10. A curve is drawn on the cylinder whose equation is cutting all the generators at an angle a ; shew that its radius of curvature at any point is p cosec^ a, where p is the radius of curvature of the principal elliptic section through the poitit. 11. If a curve in space is defined by the equations a; = 2a cos t, y=2a sin t, z = ht', prove that the radius of circular curvature is equal to 2 /r ( a' + hH'Y ■) 2 /( [ a' + b". a\/\a' + b' + bH'i' 12. In any curve if Ji be the radius of spherical curvature, o the radius of absolute curvature and - the tortuosity at any point (x, y, «), then , ([d?x\^ (dW /(fzV) . ^ E' 13. If the tangent and the normal to the osculating plane at any point of a curve make angles a, ^ with any fixed line in space, sin a da = -5 + n eos*-, where g is the distance, p a a ' measured along the are, of the point from a certain fixed point of the curve. CHAPTER XII. Curvature of Surfaces. 241. We have already seen, in Art. 209, that the section of any surface, by a plane parallel to and indefinitely near the tangent plane at any point on the surface, is a conic, ■which is called the Indicatrix, and whose centre is on the normal at 0. 242. Let any section of the surface, drawn through the normal OV, cut the indicatrix in the diameter QVQ', and let p be the radius of curvature at of the section. Then we have, in the limit, 2p.0V=QV^. Hence, for different normal sections through 0, the radius of curvature varies as the square of the diameter of the indicatrix through which the section passes. 243. Since the sum of the squares of the reciprocals of any two perpendicular semi-diameters of a conic is constant, it follows from the last article that the sum of the reciprocals of the radii of curvature of any two perpendicular normal sections through a given point of a surface is con- stant. 244. Since the semi-diameter of a conic has a maximum and a minimum value, it follows from Art. 242 that the radius of curvature of a normal section through any point of a surface has a maximum and a minimum value, the corre- sponding sections being those which pass through the axes of the indicatrix. 214 CURVATURE OF SURFACES. The maximum and minimum radii of curvature are called the principal radii of curvature, and the corresponding normal sections are called the principal sections. The locus of the centres of principal curvature at all points of a given surface is called its surface of centres. 245. If the axes of a: and y be taken in the direction of the axes of the indicatrix the equation of the surface will he, when the terms of the third and higher orders are neglected, 2z = ax' + bif. Let Pj,, P2 1>6 the principal radii of curvature, that is the radii of curvature of the sections made by the planes y = 0, w=-0 respectively ;. then it is clear that p, = -, andpj = r ■ Hence the equation of the surface will be 2. = ^ + 2i\ Pi P» The semi-diameter of the indicatrix which makes an angle 6 with the axis of x is given by 2z cos"^ sin'^ — = + • Pi P^ If p be the radius of curvature of the corresponding section, we have r' = 2pz. „ 1 cos'^ sin=6' Hence - = 1 . P Pi Pa The results of Articles 243, 244 and 245 are due to Euler. * 246., When the indicatrix at any point of a surface is an ellipse, the sign of the radius of curvature is the same for all sections; this shews that the concavity of all sections is turned in the same direction, so that the surface, in the neighbourhood of the point, is entirely on one side of the tangent plane. The surface in this case is said to be Synclastic at the point. When the indicatrix is an hyperbola, the sign of the radius of curvature is sometimes positive and sometimes mettniee's theorem. 215 negative, shewing that the concavity of some sections is turned in opposite directions to that of others. The surface in this case is said to be Anticlastic at the point. The radius of curvature of a section which passes through an asymptote of the indicatrix is infinite ; hence the asymptotes divide the sections whose concavity is turned one way from those whose concavity is turned the other way. In the figure of Art. 71, the concavities of the sections by the planes a; = and y = Q are turned in opposite direc- tions ; and the normal sections through the two generating lines at are the sections of zero curvature. When the indicatrix is a parabola, that is to say is two parallel straight lines, which become ultimately coincident, one of the principal radii of curvature is infinite ; and, if p^ be the finite radius of principal curvature, the curvature of any other normal section is given by the formula - = . 247. To find the radius of curvature of any oblique section of a surface. Let any oblique section through the point of a surface cut the indicatrix in the line RKK, and let the normal section through the same tangent line cut the indicatrix in the line Q VQ' parallel to RKR'. Let K, V be the middle points of RR', QQ' respectively, and let p, p„ be the radii of curvature of the sections ROR', QOQ' respectively. Then we have, in the limit, 2p.0K = Rtr, and 2p,.0V = QV\ But OV, and therefore VK, is small compared with QV; hence RR' and QQ; are ultimately equal. Also 0V= OK cos 6, where is the angle between the planes ROR' and QOQ'. 216 LINES OF CURVATURE. Hence we have ultimately, p or „ or P — Po ^°^ ^• This is called Meunier's Theorem. 24-8. From Meunier's Theorem, and the theorem of Art. 245, it follows that if two surfaces touch one another, and have the same radii of principal curvature at the point of contact, then all sections through that point have the same curvature. 249. The following proof of Meunier's Theorem is due to Dr Besant. Let OT be any tangent line at the point of a surface, and let P be a point contiguous to on the normal section through OT, and Q a point contiguous to on an oblique section through OT. Then a sphere can be described to touch OT at 0, and to pass through P and Q; and the sections of this sphere by the planes TOQ,, TOP are ultimately the circles of curvature at of the sections of the surface by those planes. Hence, as Meunier's Theorem is obviously true for a sphere, it is true for the surface. Ex. 1. Find the principal radii of curvature at the origin of the surface iz = %x^-5xy-6y^. Am. ^„ -^^. Ex. 2. Find the radius of principal curvature at any point of the curve of intersection of two surfaces. « Let p be the required radius of curvature at any point P. Let the surfaces intersect at an angle a, and let 6, a-0 he the angles between the principal normal of the curve of intersection, and the normals to the two surfaces. Let pi, p^ be the radii of curvature of normal sections of the two surfacni; through the tangent line at P. Then, by Meunier's Theorem, p=PjCOse, a,ndip=p^cos(a.-6). Hence, eliminating B, we have sin'a _ 1 1 2ooso LINES OF CURVATURE. 217 250. Def. a line of cuniature on any surface is a curve sucli that the tangent line to it at any point is a tangent line to one of the principal sections of the surface at that point. 251. The normals to any surface at consecutive points of one of its lines of curvature intersect. Let P be an extremity of an axis of the indicatrix which corresponds to the point of a surface, then 0, P are consecutive points on a line of curvature. Let V be the centre of the indicatrix, then OV will be the normal to the surface at 0. The tangent line at P to the indicatrix is perpendicular to the normal to the surface at P ; it is also perpendicular to OV; and, since P is an extremity of an axis of the indicatrix, the tangent line is perpendicular to PV. Hence OV, PV, and the normal at P are in a plane, and therefore the normals at and P will intersect. Conversely, if the normals at P and intersect, the tan- gent line at P to the indicatrix will be perpendicular to the plane which contains the normals at and P ; therefore the tangent line will be perpendicular to PV, and hence PF is an axis of the indicatrix. 252. To find the differential equations of the lines of curvature on any surface. Let F{x, y, z) = be the equation of the surface. Then the equations of the normal at any point {x, y, z) are ^ — x Tf — y ^— z 1F^~dF dF • dec dy dz The normal at the consecutive point {x -^dx, y + dy, z + dz) is ^-x — dx _ -q—y — dy ^—z-dz dF .(dF\~dF (dF\~W~7fdF\. ^ + '^fej Ty+^Uy) Tz + ^W) 218 LINES OF CUEVATURE. The condition of intersection of the two normals gives the equation = 0...(i). dx, dy, dz dF dF dF das' dy' dz d(^^\ '^UyJ' a(a + X) ' 6(6 + X)' c{c+\) The elimination of a + X, 6 + X, c + A. from (iii), (iv); (v) gives = (vi), X a ' y b' z c dx. dy. dz dco dy dz a' 6' c which is the differential equation of the curve of mtersection of (i) and any one of its confocals ; and it is easy to see, by compaxing with (i), Art. 252, that (vi) is the differential equation of a line of curvature. ^ 15—2 228 LINES OF CUEVATUBE OF CONICOIDS. 265. The radius of curvature of any normal section of a central conicoid may be found as follows. The radius of curvature of any central section of a coni- coid through a point P is, by a well-known formula, equal to — , where d is the semi-diameter parallel to the tangent at P, and ^ is the perpendicular from the centre on the tangent at P. Hence, by Meunier's Theorem, the radius of curvature of any normal section of a conicoid through the point P is equal to — , where jj, is the perpendicular from the centre on the tangent plane at P, and d is the semi-diameter parallel to the tangent line at P ; for the cosine of the angle between the normal section and the central section is — . JP 266. At any point of a line of curvature of a central conicoid, the rectangle contained by the diameter parallel to the tangent at that point and the perpendicular from the centre on the tangent plane at the point is constant. Let p be the perpendicular from the centre on the tangent plane at any point P of a given line of curvature, and let a, yS be the semi-axes of the central section parallel to the tangent plane at P. Then, one of the axes, a suppose, is parallel to the tangent at P to the line of curvature, and the other axis is of constant length for all points on the line of curvature [Art. 167, Cor.]. Hence, since ^a/3 is constant, it follows that px is constant throughout the line of curvature. 267. At any point of a geodesic on a central coniiioid, the recta/ngle contained by the diameter parallel to the tangent at that point and the perpendicular from the centre on the tangent plane at the point is constant. The differential equations of a geodesic on the conicoid a^ + bf + c/ = 1 are d^ ^ d^z d£_^d£^d£ aas by ~ cz ' or GEODESICS ON CONICOIDS. 229 '--t-'--x (i). ax by cz ^ ' We have to prove that pr is constant, where p = aa;" + 62,'= + c^'» (ii), and 1 = aV + &y + cV (iii). Differentiating oa;' + hy^ + c^" = 1 twice with respect to s, we have aa!^ + hy"' + cz'^ + axte' + hyy" + czz' = (iv). From (i) we have aoox" Jr byyf' + czz" p' . .... A, = ■ — a a , 72 2 . — 5-5— = — S . "0™ (ill) and (iv). ., ^ ax'x" + by'y" + cz'/' r' ds . .... .,.,., Also A, =—2 — , , , ? , , — a — -= 1 J , from (11) and (111). a XX + byy + c zz \ dp ^ ' ^ ^ „ Idr Idp . Hence - -r- H r- = Oj r ds p ds and therefore pr is constant. Ex. 1. The constant pr is the same for all geodesies which pass through an umbUic. This follows from the fact that the central section parallel to the tangent plane at an umbUio is a circle, and therefore the semi-diameter parallel to the tangent to any geodesic through an umbilic is of constant length, Ex. 2. The constant^ has the same value for all geodesies which touch the same line of curvature. At the point of contact of the line of curvature and a geodesic which touches it, both p and r are the same for the line of curvature and for the geodesic. Ex. 3. Two geodesies which touch the same line of curvature make equal angles with the iSies of curvature through their point of intersection. From Ex. 2, the semi-diameters parallel to the tangents to the two geodesies, at their point of intersection P, are equal to one another, and are therefore equally inclined to the axes of the central section which is parallel to the tangent plane at P. But the axes of the central section are parallel to the tangents to the lines of curvature through P: this proves the proposition. 230 EXAMPLES ON CHAPTER XII. Ex. 4. Two geodesies which pass through umbilica make equal angles with the lines of ourvatuie through their point of intersection. Ex. 5. Any geodesic through an umbilic will pass through the opposite umbilic. Ex. 6. The locus of a point which moves so that the sum, or the differ- ence, of its geodesic distances from two adjacent umbilics is constant, is a line of curvature. Ex. 7. All geodesies which join two opposite umbilics are of constant length. Ex. 8. The point of intersection of two geodesic tangents to a given line of ourvatuie, which intersect at right angles, is on a sphere. Let t-j, r^ be the semi-diameters parallel to the tangents to the geodesies at P, their point of intersection. Then, since the geodesies cut at right angles, where a and |3 are the semi-axes of the central section parallel to the tangent plane at P. But, if ^ be the perpendicular on the tangent plane at P, then p»'i=yr2=constant, from Ex. 2. Hence, since pajS is constant, and also o=+/32 + 0P=, it follows that OP is constant. Ex. 9. The point of intersection of two geodesic tangents, one to each of two given lines of curvature, wMch cut at right angles, is on a sphere. Examples on Chapter XII. 1. A surface is formed by the revolution of a parabola about its directrix ; shevsr that the principal curvatures at any point are in a constant ratio. 2. If p, p' be the principal radii of curvature of any point of an ellipsoid on the line of its intersection v^ith a given concentric sphere, prove that the expression ^ "\ will be invariable. p + p 3. If Ui+u^ + u^+ w„ = be the equation to a surface vrhere u^ is a homogeneous function of x, y, %, of the rth degree, then M, -f- Mj + w, (Sb -I- my + nz) = will be the general equation of surfaces of the second order having the same curvature at the origin. EXAMPLES ON CHAPTER XII. 231 . ^' . '^.^^ normal at each point of a principal section of an ellipsoid is intersected by the normal at a consecutive point not on the principal section ; shew that the locus of the point of inter- section is an ellipse having four (real or imaginary) contacts with the evolute of the principal section. 5. In the surface y cos — x sin - = 0, a a the principal radii of curvature at (x, y, z) are ± - — - — --. CO 6. Shew that the umbilici of the surface lie on a sphere whose centre is the origin and whose radius is , J abo equal to -= — = . ab + oo + ca 7. The centres of curvature of plane sections of a surface at any point lie on the surface 8. Prove that the line which separates the synclastic from the anticlastic parts of a surface is a line of curvature, and that along it the inflexional tangents coincide. 9. The projections of the lines of curvature of an ellipsoid on the cyclic planes, by lines parallel to the greatest axis of the surface, ar'e confocal conies. 10. If one of the lines of curvature on a developable surface lies on a sphere all the other lines of curvature, other than the rectilineal ones, lie on concentric spheres. 11. A plane curve is wrapped upon a developable surface. If p is the radius of curvature of the plane curve at any point, p the corresponding radius of circular curvature of the curve upon the surface, B the corresponding principal radius of curvature of the surface, and i^ the angle at which the curve intersects the , , . sin^rf. 1 1 generator of the surface, -=5- = -7^ - -a . 232 EXAMPLES ON CHAPTER XII. 12. If one system of lines of curvature of a surface are ciicles, the surface is the envelope of a sphere whose centre moves on a given curve. 13. If a geodesic line is either a b'ne of curvature or a plane curve it is both ; but a plane line of curvature is not necessarily geodesic. Shew that if one series of the lines of curvature is geodesic they are all repetitions of the same plane curve. 14. Shew that if the normal to a surface always passes through a given curve, one set of the lines of curvature are circles; and that those normals which pass through a given point on the curve are generating lines of a right cone whose axis is the tangent at that point. Hence shew that if the normal always passes through two curves, these curves must be conies in planes at right angles, the foci of one being the vertices of the other. 15. Find the differential equation of the projection on the plane xy of each family of lines of curvature of the surface which is the envelope of a sphere whose centre lies on the parabola x^ + ^ay = 0, s = 0, and which passes through the origin. 16. Shew that the principal curvatures at any point of a surface are given by the equation dl 1 dl dx P' dy dm dm 1 dx » dy + - P dn dn dx J 'dy dl dz dm dz dn dz = 0, P I, m, n are the direction-cosines of the normal ht the where point. , 17. The tangent planes to the surface of centres at the two points where any normal meets it are at right angles. 18. Shew that the point for which x = y = z is &tx umbilic of x'" + y" + z'"=a'", and the radius of curvature there is nj- 1 (3)' EXAMPLES ON CHAPTER XII. 233 19. In a hyperbolic paraboloid, of -wrhich the principal parar bolas are equal, the algebraic sum of the distances of all points of the same line of curvature from two fixed rectilinear generators is constant. 20. Along the normal at a point P of an ellipsoid is measured FQ of a length inversely proportional to the perpendicular from the centre on the tangent plane at P; prove that the locus of Q is another ellipsoid, and that the envelope of all such ellipsoids is the " surface of centres," that is the locus of the centres of principal curvature. 21. Shew that the specific curvature at any point of the surface xi/z = dbc varies as the fourth power of the perpendicular from the origin on the tangent plane at the point, and that at an umbilicus it is |- (abcy^- 22. If a surface have one principal radius of curvature con- stant it is the envelope of a sphere of constant radius. 23. Find the umbilici of the surface — + ^ + - = ii;', and a e X v z shew that at the umbilicus - = ^ = - the directions of the three a c lines of curvature are given by the equations dx dy dy dz . dz dx . , — = -v- , -i^ = — and — = ^— respectively. abbe c a '' 24. If two geodesies be drawn on an ellipsoid from any point to two fixed points, the sine of the angle between them varies as the perpendicular on the tangent plane at the point. 25; Shew that on a surface of revolution, the distance of any point of a geodesic from the axis varies as the cosecant of the angle between the geodesic and the meridian. 26. If a geodesic line be drawn on a developable surface and cut any generating line of the surface at an angle xj; and at a distance t from the edge of regression measured along the generator, prove that d^ , , * -— + cot \lr . t = p, ay/ 234 EXAMPLES ON CHAPTER XH. where p is the radius of curvature of the edge of regression at the point where the generator touches it. 27. Shew that the tangent to a geodesic or line of curvature on a quadric always touches a geodesic or line of curvature respectively on a confocal quadric. 28. Shew that the reciprocals of the radii of curvature and torsion of a curve drawn on a developable surface are sin" 6 , sin 6 cos 6 da and p cos a p ds where p is the principal radius of curvature of the surface at the point, the angle the tangent line to the curve makes with the generator through the point, and a the angle between the normal to the surface and the principal normal of the curve. If a geodesic on a developable surface be a plane curve it must be one of the generators or else the surface must be a cylinder. 29. If - and - be the curvature and tortuosity at any point of a geodesic drawn on a surface, and — , — be the principal curvatures of the surface at that point, shew that <^ \Pi P/ XPa P/ 30. Through a given generator of a hyperboloid of one sheet, draw a variable plane ; this will touch the surface at some point A on the generator and will contain the normal to the surface at another point S. Shew that the sum of the square root* of the measures of curvature of the surface at A and £ is constant for all planes through this generator. Hence shew that the same proposition is true for any skew surface. 31. If sr be the pitch of the screw by which any generator of a skew surface twists into its consecutive position, shew that ■a^ + pp = 0, where p, p' are the principal radii of curvature at the point where the shortest distance between the two consecutive generators meets them. EXAMPLES ON CHAPTER XII. 235 32. If a geodesic he drawn on an ellipsoid from an umbilicus to an extremity of the mean axis, prove that its radius of torsion at the latter point is where a, b, c are the semi-axes of the ellipsoid arranged in descending order of magnitude. 33. If from any point on a surface a number of geodesic lines be drawn in all directions, shew (1) that those which have the greatest and least torsion bisect the angles betweea the principal sections, and (2) that the radius of torsion of any line, making an angle 6 with a principal section, is given by the equation. ^ = ( ) sm cos 6, -« \P, Pi/ where p^, p, are the radii of curvature of the principal sections. 34. Find the equation to the surface which is the locus of the central circular sections of a series of confocal ellipsoids. Prove that this surface cuts all the ellipsoids orthogonally, and that the orthogonal trajectories of the circles, drawn upon the surface, are lines of curvature upon two hyperboloids confocal with the ellipsoids. 35. If a cone of revolution circumscribe an ellipsoid, prove that the plane of contact divides the ellipsoid into two portions whose total curvatures are 27r (1 + sin a) and 27r (1 - sin a), where 2a is the vertical angle of the cone. 36. If any cylinder circumscribes an ellipsoid it divides it into portions whose integral curvatures are equal. 37. The measure of curvature at any point of the surface ^ 5=lis {c' + ry where r is the length of the generator through the point cut oflf by the plane g = 0. 38. Prove that, if radii be drawn to a sphere parallel to the priiicipal normals at every point of a closed curve of continuous 236 EXAMPLES ON CHAPTER XII. curvature, the locus of their extremities divides the surface of the sphere into two equal parts. Hence shew that the total curvature of a geodesic triangle on any surface is equal to the excess of its angles over two- right angles. 39. Define the radius of geodesic curvature of a curve drawn upon a surface, and shew that at any point it is equal to B cot <^, where H is the radius of curvature of the normal section contain- ing the tangent to the given curve, and <^ is the inclination of the osculating plane to that section. 40. If a surface roll on a second surface without rotation about the common normal, and the trace on one surface is a geodesic, the trace on the other surface is a geodesic. Hence prove that Gauss's measure of curvature is constant for all areas enclosed by geodesies. MISCELLANEOUS EXAMPLES. 1. The inclinations to the horizon of two lines which are at right angles to one another are o, j8, the lines being on a plane in- clined to the horizon at an angle 6; shew that sin" 6 = sin' a + sin"/3. 2. Shew that the volume of the tetrahedron of which a pair of opposite edges is formed by lengths r, r' on the straight lines whose equations are x — a y — h z — c . x — a' _y-h' _z-c' I i^ n .. ^, m' n a — a', 6-6', c-c' . I , m , n I' , m' , n' 3. A parallelogram of paper is creased along its shorter diagonal, and the two halves are folded so as to make an angle with each other : find the distance between the extremities of the longer diagonal, and prove that it is equal to the shorter, if sin' - = cot a cot )8, where a and ^ are the angles the sides make with the shorter diagonal. 4. The ends of a straight line lie on two fixed planes which are at right angles to one another, and the straight line subtends a right angle at each of two given points: shew that the locuis of its middle point is a plane. 5. The equations of three straight lines are y -» = 1, x = Q; z-x= 1, y==0; and x-y = l, z = 0; prove that the locus of all straight lines which intersect the three lines is x'' + y'' + si'-2yz-2zx-2xy= 1 238 MISCELLANEOUS EXAMPLES. 6. Three fixed lines are cut by any other line in the points A, jB, G, and D is the point on the line ABC such that {ABCD} is harmonic : shew that the locus of /> is a straight line. 7. A point moves so that its perpendicular distances from two given lines are in a constant ratio : shew that its locus is an hyperboloid; 8. A straight line slides upon two fixed straight lines in such a way that the part intercepted subtends a right angle at a fixed point : shew that the line generates a conicoid. 9. A sphere touches the six edges of a tetrahedron : shew that the three lines joining pairs of opposite points of contact will meet ia a point. 10. A straight line moves in such a manner that each of four fixed points on the line is always on a given plane j shew that any other fixed point on the line describes a plane ellipse. 11. Any three points P, Q, R, and the polar planes of those points with reference to any conicoid are taken. PQi, PR. are the perpendiculars from P on the polar planes of Q and R respec- tively; QR^, QP^ are the perpendiculars from Q on the polar planes of R and P respectively; and RP,, RQ^ are the perpen- diculars from R on the polar planes of P and Q respectively. Shew that PQ^ . QR^ . RP^= PR^ . QP^. RQ^, 12. Shew that, if the equation aal' + hy' + cs? + 2fyz + 2gzx + 2hxy = 0; represent two planes, the planes which bisect the angles between them are given by the equation » « , y , z ax + hy + gz, hx + hy+fz, gx+fy + cz _l _J. 1 a/-gh ' hg-hf ch-fg 13. Shew that, if the equation ax" + hy' + cz' + ifyz + 2gzx + 2?ixy = 0, = 0. MISCELLAJSTEOUS EXAMPLES. 239 represent two planes, the product of the peipendiculars on the planes from the point («, y, z) is aa;' + 6y' + eg' + Ifyz, + ^gisK + ^hxy (a + 6 + o)' + 4(/''-6c) +4(^''-ca) +4(A'-a6) * 14. li 17= (ahcdlmnpqr) {scyzwY = is the equation of a cone, shew that the co-ordinates of the vertex satisfy the equations du d£ dn_ da _ db _ _Sl _ da db dl where A is the discriminant. 15. Shew that, if the equation ax' + by' + cz' + 2fyz + 2gzx + 2hxy + 2ux + 2vy + Iwz +d = Q, represent a paraboloid of revolution, c = b =i=a. Shew also that if c = 6 + a, the equations of the axis of the paraboloid wUl be cz + w = Q, (px + u) J a + {cy + v) Jb= 0. 16. Shew that the three principal planes of the surface aa^ + by' + cs* + 2fyz + 2gzx + 2hxy = 1 are given by the equations ax +hy + gz, hx + by +Jz , gx +fy + cz =0, Ax + Hy+Gz, Hx + By + Fz, Gx + Fy + Gz X , y , z where A, B,. C... are the minors of a, b, c in the determinant a, h, 9 k, 6, f 9> /, c 17. If »' he any semi-axis of the conicoid aaf + bi/' + c^ + 2fy» + 2gzx + 2hxy = 1, prove that the values of r will be given by gh ¥ fg J -=1. ^A-a/+ti ¥-^9 + h f9-oh + - =(' 240 MISCELLANEOUS EXAMPLES. 18. The ellipse 6 V + aV - aV = 0, f8=0 is a plane section of a cone whose equation, referred to its principal axes, is Pyx' + yoy' + oyS*' = 0. Shew that the vertex of the cone is on the curve K + y' + g'-a'-feY ^ f a'6'-6V-ffly-(a° + 5'K y I a + /3 + y J I ^y + ya + aj8 J (a'h'z'Y 19. Shew that the conicoid ax" + by' + cz'' + d=0 is its own polar reciprocal with respect to any one of the conicoids =fc ok" ± hy'^&^'^d = 0. 20. Find the locus of the centre of the sphere which passes through two circular sections of a conicoid which are of opposite systems and whose planes are equidistant from the centre. 21. Prove that the foci of sections of an ellipsoid made by a series of parallel planes lie on an ellipse, 22. Shew that the perpendicular from the centre on the tangent plane at any point of — -^ = = 1 is , , where r " ^ ■' ^ a' c' Ja'+7^ is the length of a generator through the point cut off by the plane ot xy. 23. The six lines AB', B'O, OA', A'B, BC, C'A are six gene^ rators of the hyperboloid aaf + by" + cs* = 1, and AB', B'G, CA', are respectively parallel to A'B, BC, C'A ; shew that, if the parallelepiped of which the six generators are edges be completed, the corners which are not on the hyperboloid will be on ' ax' + by" + cs"+3 = 0. 24. Shew that at any point the rate per unit of length of generator at which the normal to the hyperboloid —■ 1 = 1 twists round a generator as we move along it is -5 — -, , where r is the distance, measured along the generator, of the point from the plane of xy. MISCELLANEOUS EXAMPLES. 241 25. ABCDQ is a twisted polygon all -whose angles are right angles ; AB, OD lying on fixed straight lines. Shew that if A, B, G, D be any points on their respective lines, the locus of P or Q is an hyperboloid of one sheet. 26. If I be the latus-rectum of a parabola, and l^, l^, l^ the latera recta of its orthogonal projections upon a rectangular system of co-ordinate planes makiag angles a, /3 and y respectively with the plane of the original parabola, then 2 cos^a cos^jS cos^y P ?/ z^t l^^ 27. If the sis points on a conicoid, normals at which meet in a point, are joined in pairs by three lines, prove that whatever set of joining lines is taken the sum of the squares of the semi- diameters parallel to them is constant. 28. A conicoid whose centre is D touches the three planes YOZ, ZOX, XOY ia A, B, G respectively : shew that the lines through A, B, G parallel respectively to OX, OY, OZ, and the line OD are four generators of an hyperboloid of one sheet. 29. Three perpendicular tangent planes are drawn, one to each of three confocal conicoids : shew that the normals at the points of contact of the planes, and the line joining their point of intersection to the centre of the conicoids are generators of an hyperboloid of one sheet. 30. If any line through a fixed point meet any number of fixed planes in the points A, B, G , and on the Une a point X be taken such that jr^ "^ TTS ''' 7)S '*' OG*'"' ^^^^ *^** *^® ^""^ of X will be a plane. 31. If any line through a fixed point meet any given sur- face in the points A, B, G, D..., and X be taken such that -UX = m^m'--5G'-m^-' *^'" ^^^ *^" ^'^ of X be a plane. S. S. G. 16 242 MISCELLANEOUS EXAMPLES. 32. Two straight lines drawn in fixed directions through any point meet a given surface in. the points A, B, C, D... and A', B, G', D'...; shew that q^> q^' , QO' . OD' ... '® *=°''^**'^*- 33. Prove that the pedal of a helix with regard to any point on its axis is a curve lying on a hyperboloid of one sheet ; and that, if the pitch of the helix be Jir, this curve will cut perpen- dicularly all the generators of one system of the hyperboloid. 34. A curve is drawn on a sphere of radius a cutting all the meridians at a constant angle ; shew (i) that the foot of the per- pendicular from the centre of the sphere upon the osculating plane is the centre of curvature ; (2) that if p, a- be the radii of curva- ture and torsion