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 ELEMENTAEY TEEATISE 
 
 ON 
 
 SOLID GEOMETRY 
 
 BT 
 
 W. STEADMAN ALDIS, M.A. 
 
 TRINITY COLLEGE, CAMBRIDGE, 
 PROFESSOR OF MATHEMATICS IN THE UNIVERSITY COLLEGE, AUCKL-^ND, 
 
 NEW ZEALAND. 
 
 FOURTH EDITION, REVISED. 
 
 CAMBRIDGE: 
 DEIGHTOX, BELL, AND CO. 
 
 LONDON: GEOKGE BELL AND SONS. 
 
 1886 
 

 • • • •, • . 
 
 • • • • • • 
 
 ,•• ••••• • • 
 
 ... . ..'. '... ',.••.. . ... 
 
 
 PRINTED BY C. J. CLAY, M.A. & SONS, 
 AT THE UNIVERSITY PRESS. 
 
 PHYSI'^S f!EP' 
 
PREFACE TO THE FIRST EDITION. 
 
 The present work is intended as an introductory text-book 
 for the use of Students reading for the Mathematical 
 Tripos. Many of the higher applications of the subject 
 are therefore either omitted entirely or treated very briefly. 
 At the same time the Author believes that the book in- 
 cludes as much as the great majority of Cambridge Students 
 have time to master thoroughly, while those who are 
 desirous of making farther acquaintance with the subject 
 will perhaps find a work like the present not unsuitable as 
 an introduction to the more complete treatises of Salmon 
 and others. 
 
 The Author besfs to thank those of his friends who have 
 kindly assisted him by revising the manuscript and proof- 
 sheets, and will feel obliged to any one who will offer cor- 
 rections or improvements. 
 
 Examples, selected chiefly from recent College and Uni- 
 versity Examination Papers, will be found at the end of each 
 Chapter. 
 
 Cambridge, August, 1865. 
 
 665346 
 
VI PREFACES. 
 
 SECOND EDITION. 
 
 The present Edition has been revised and re-arranged 
 and somewhat enlarged. 
 
 Newcastle-ox-Tyne, September, 1873. 
 
 THIRD EDITION. 
 
 The Third Edition has been revised and farther en- 
 larged, chiefly by the addition of hints for the solution of 
 the Examples. 
 
 Newcastle-ok-Tt>'e, September, 1879. 
 
 FOURTH EDITION. 
 
 The Fourth Edition has been enlarged chiefly by an 
 additional Chapter on Vectors, and a few new Examples. 
 
 AuciiL.\ND, August, 188G. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 PAGE 
 
 Introductory Theorems. Co-ordinates. Direction-cosines. Projec- 
 tions. Polar Co-ordinates 1 
 
 CHAPTER II. 
 The Straight Line and Plane 18 
 
 CHAPTER III. 
 
 The Sphere. The Cone and Cylinder. The Ellipsoid. The Hyper- 
 
 boloids. The Paraboloids. Asymptotic Surfaces ... 40 
 
 CHAPTER IV. 
 Transformation of Co-ordinates 54 
 
 CHAPTER V. 
 On Generating lines and sections of Quadrics 63 
 
 CHAPTER VI. 
 
 Diametral Planes. Conjugate Diameters. Principal Planes . . 85 
 
 CHAPTER VII. 
 
 On the Surfaces represented by the General Equation of the Second 
 
 Degree 103 
 
viii CONTENTS. 
 
 CHAPTER YIIL 
 
 PAGE 
 
 Ou Tangent Lines and Planes. The Normal. Enveloping Cones and 
 
 Cylinders 1^7 
 
 CHAPTER IX. 
 
 On Curves in Space. The Tangent. Normal and Osculating Planes. 
 
 Line of Greatest Slope. Curvature of Curves .... 138 
 
 CHAPTER X. 
 On Envelopes, with one and two parameters. Reciprocal Polars . 163 
 
 CHAPTER XL 
 
 On Functional and Differential Equations of Families of Surfaces. 
 Conical, Cylindrical, and Conoidal Surfaces. Developable Sur- 
 faces. Surfaces of Revolution 172 
 
 CHAPTER XI L 
 On Foci and Confocal Quadrics 189 
 
 CHAPTER XIIL 
 
 On Curvature of Surfaces. Meunier's Theorem. Indicatrix. Prin- 
 cipal Sections. Lines of Curvature. Umbilici. Geodesic Lines 199 
 
 CHAPTER XIV. 
 
 On Vectors. Quaternions. Versors. Product of Vectors. Equation 
 
 of I'lanc and Sphere 229 
 
 Answers to the Examples 247 
 
I t •» - -. ., ,. 
 
 » » CI 
 
 SOLID GEOMETEY. 
 
 CHAPTER I. 
 
 INTRODUCTORY THEOREMS. ' 
 
 1. The position of a point in space is usually deter- 
 mined by referring it to three planes meeting in a point. 
 This point is called the origin, the three planes the co- 
 ordinate planes, and their three lines of intersection the 
 co-ordinate axes. The point of intersection of the three 
 planes is usually designated by the letter 0, and their lines 
 of intersection by the letters Ox, Oy, Oz. They are called 
 the axes of x, y, and z respectively, and the planes yOz, zOx, 
 xOy are called the planes of yz, zx, xy respectively. If the 
 three planes of yz, zx, xy, and consequently the three lines 
 Ox, Oy, Oz, are at right angles to each other, the co-ordinates 
 are said to be rectangular, and in all other cases oblique. We 
 shall generally make use of rectangular co-ordinates, but in 
 some cases the proofs and the results obtained will hold good 
 equally whether the axes be at right angles or not. 
 
 2. The position of any point P relatively to these three 
 planes is known, if its distance from each, measured parallel 
 to the intersection of the other two, be known. 
 
 For let PH, PK, PL be draw^n through P parallel to 
 Ox, Oy, Oz respectively to meet the planes of yz, zx, xy in 
 H, K, L ; and let a plane through PL, PK, w^hich by Euclid, xi. 
 15, is parallel to the plane of yz, meet Ox in M. Let also a 
 plane through PH, PL meet Oy in N, and a plane through 
 PH, PK meet Oz in R. Then if KR, KM be joined, KMOR 
 is obviously a parallelogram, and KR therefore equal to OM. 
 Similarly RKPH is a parallelogram, and KR equal to PH. 
 
 A. G. 1 
 
2 INTRODUCTOKY THEOREMS. 
 
 Hence PH is equal to OM, and similarly PL to OR, PK to 
 ON. If therefore we niqasure off from Ox, Oy, Oz, respec- 
 tively, ItjigthsfiMi Qlfy^QR equal to the given distances of 
 P from the, co-prdinate, planes, and through M, N, R draw 
 
 < e c « » 
 
 H 
 
 C ' 
 
 R 
 
 ] 
 
 K 
 
 
 f— 
 
 
 P 
 
 
 
 
 ^^ 
 
 
 
 
 
 • 
 
 
 
 
 / 
 
 
 
 / 
 
 M 
 
 X 
 
 N 
 
 planes parallel to those of yz, zx, xy, these planes will inter- 
 sect in P, the position of which is therefore determined. The 
 lengths PH, PK, PL, or OM, ON, OR, which are equal to 
 them, are called the co-ordinates of P, and are usually de- 
 noted by the letters x, y, z. 
 
 3. If the line xO be produced through to x , and from 
 Ox' we cut off a length OM' equal to OM, and repeat the 
 preceding construction, we obtain a point P' whose absolute 
 distances from the three co-ordinate planes are the same as 
 those of P. We must therefore have some convention to 
 enable us to distinguish between these two points. The 
 following is usually adopted. 
 
 The co-ordinates are considered positive if measured in 
 one direction along the axes from 0, and negative if measured 
 in the ojtposite. 
 
 The positive directions for the three axes are usually 
 taken to be those represented in the figure by Ox, Oy, Oz, 
 and tlie negative directions to be Ox, Oy, Oz. 
 
 It will be seen that the whole of space is divided by the 
 co-ortlinate i)lanes into eight compartments, and the signs of 
 
IXTKODUCTORY THEOREMS, 
 
 the co-ordinates of any point indicate in which of these com- 
 partments it is situated, while their absolute magnitudes 
 indicate its position in that compartment. Thus the co-ordi- 
 
 nates of a point whose absolute distances from the co-ordinate 
 planes are a, y8, 7 are represented bv (a, ^, 7), [—a, /?, 7), 
 (a, - A 7), {a, A - 7), (a, - A - t), (- «. A -7), (- a, - jB, 7), 
 (— OL, — P, — 7), according as the point lies in the compart- 
 ment Oxyz, Ox'yz, Oxy'z, Oxyz, Oxy'z, Ox'yz , Ox'y'z, Oxy'z, 
 respectively. 
 
 4. To find the distance of a point from the origin in terms 
 of its co-ordinates. 
 
 In this and Articles 5, 6 and 8 the co-ordinates are sup- 
 posed rectangular. 
 
 Let P be the point, x, y, z its co-ordinates. Through P 
 draw planes parallel to the co-ordinate planes and forming 
 with them a parallelepiped of which OP is the diagonal and 
 PL the edge through P parallel to Oz. 
 
 Join OP and OL. Then since PL is parallel to Oz which 
 is perpendicular to the plane of xy, PL is perpendicular to 
 the plane of xy, and therefore to the line OL which lies in 
 that plane. (Euclid, xi. Def. 3.) 
 
 Hence OP^ = OL' + PL'. 
 
 1—2 
 
INTRODUCTORY THEOREMS. 
 
 
 z 
 R 
 
 
 
 K 
 
 H 
 
 
 
 P 
 
 
 
 / 
 
 
 
 
 
 
 A 
 
 JK 
 
 \ 
 
 / 
 
 M 
 
 /N 
 
 (2), 
 
 But OU = OM' + MU ; 
 
 .-. OP^ = 0]\P + Mr + Pr = x" + y' + z" (1). 
 
 5. Let (X, ^, y be the angles between OP and the axes 
 of X, y, z respectively. Join PM. Then since Ox is perpen- 
 dicular to the plane PLM, it is perpendicular to PM. 
 
 Hence 
 
 OM = OP cos POM = OP cos a ; or a? = r cos a 1 
 
 Similarly, 
 
 ON = OP cos PON = OP cos 13; or?/ = rcos/S 
 
 OR = OP cos POR = OP cos 7 ; or ^r = 7^ cos 7 
 
 X, y, z being- the co-ordinates of P, and OP being denoted 
 by?'. 
 
 Squaring and adding, we get 
 
 Oil/= + ON'' 4- OR' = OP' (cos'^ a + cos' 13 + cos' 7), 
 or taking account of (1), 
 
 1 = cos' a -f- cos"' /S -f cos' 7 (3). 
 
 Tlie letters I, in, n are frequently used to denote cos a, j 
 cos)3, cos 7, which are called the direction-cosines of the line 
 OP. It is usual to denote by a, [3, 7 the angles which OP j 
 makes with the positive directions of the axes, in which case 
 the formula) (2) hold for all positions of the point P. 
 
INTRODUCTORY THEOREMS. 
 
 6. To find the distance between two points luJiose co-ordi- 
 nates are given. 
 
 Let P and Q be the two points ; x^, y^, z^\ x^, y^, z^ their 
 co-ordinates. Join PQ, and through P and Q draw planes 
 
 parallel to each of the co-ordinate planes, thus forming a 
 parallelepiped whose edges are parallel to the co-ordinate axes, 
 and are equal in length to x^ — x^, y^ — y^, z^ — z^, respectively. 
 
 As in Art. 4, we obtain 
 
 PQ^ = PH'' + HN^ + NQ^ 
 
 = (^,-^,y4-(y,-y,y + (^,-^,)^ (4). 
 
 We have also formulse similar to those of equation (2), 
 o(, /3, 7 being the angles between PQ and the lines drawn 
 through P parallel to the axes, viz. 
 
 PH — x^ —x^ = PQ cos a = lr '\ 
 
 PM=y^ — y^ = PQ cos /5 = mr> (5), 
 
 PL = ^2 ~ '^1 ^ ^Q' ^^^ 7 = nr J 
 
 where r represents the length of PQ, and I, 7n, n are the 
 
 direction-cosines of PQ. 
 
G 
 
 INTRODUCTORY THEOREMS. 
 
 7. To find the co-ordinates of a point which divides the 
 straight line joining tiuo given points in a given ratio. 
 
 Let P, Q be the two given points, and R the point in PQ 
 
 which divides PQ in the given ratio of n^ to n^. Let x^, y^, z^ 
 be the co-ordinates of P, ^'2, y^, z^ those of Q, x, y\ z those 
 of P. 
 
 Draw Pil/, P//, QK parallel to the axis of z to meet the 
 plane of xy in il/, Hy K. These points all lie in one straight 
 line, namely that in which a plane through PQ parallel to 
 the axis of z cuts the plane of xy. Draw PEF parallel to 
 MUK to meet RH in E and QK in F. 
 
 Then 
 Also 
 
 PM = z^, RII = z\ QK = 
 
 '2' 
 
 n. 
 
 RE PR 
 
 QF PQ 7^ + n/ 
 
 z — z 
 
 or 
 
 1 
 
 n. 
 
 z — z 
 2 1 
 
 n, + n^ 
 
 whence 
 
 n^z^-^7i^z^ 
 
 X/ — 
 
 n^ + n, 
 
 Siniihirly it may be shewn that 
 
 i\ + n^ 
 
 y 
 
 n, + n^ 
 
INTRODUCTORY THEOREMS. 
 
 If R be the middle point of PQ, n^ — n^, and we have 
 
 5j^ ._ 2/1 + 3/2 
 
 ^ ~ 2 ' ^ ~ 2 
 
 /_^i+^2 
 
 8. To ^?icZ ^/le angle between Uuo straight lines whose 
 direction-cosines are given. 
 
 Since by Euclid, xi. 10, the angle between any two straight 
 lines is equal to that between any other two respectively 
 parallel to them, we need only consider the case of two lines 
 through the origin. 
 
 Let OP, OQ be the two lines ; I, m, n the direction-cosines 
 
 of OP; V, m, n those of OQ. Let x^, y^, z^, be the co-ordi- 
 nates of P any point in OP ; x^, y^, z^ those of Q any point 
 in OQ. 
 
 Then by Art. (6) 
 
 -p«'^=K-^ir+(2/2-2/j+(^2-^j' 
 
 = < + y" + < + < + y' + ^1' - 2 {x,x^ + y,y, + ^x^J- 
 
 But by Art. (4) 
 
 x:' + y: + z,'=OP\ 
 
 x:+y:^z:=oQ\ 
 
8 INTRODUCTORY THEOREMS. 
 
 And by Art. (5) 
 
 x^=OP.l, y^ = OP.m, z^ = OP.n, 
 oc^=OQ. r, y^ = OQ. m, z^=OQ. n ; 
 
 and .-. x^x^ + y^y^ + z^z^ = 0P .OQ {W + mm + nn). 
 
 Hence PQ' = OP' + OQ' - 20P . OQ {W -f mm + nn'). 
 
 But by Trigonometry we have from the triangle OPQ 
 
 PQ' = OP' + 0Q'-20P.0Q. cos POQ. 
 Comparing these two expressions for PQ', we get 
 
 cos POQ = ir + mm' +7in (6). 
 
 The formula3 (1), (3), (4) and (6) are of very frequent use, 
 and should be carefully remembered by the student. 
 
 From (G) we can deduce 
 sin'^ POQ = 1 - (W + mm + nnf 
 
 = {mil - VI nY + (/zZ' - nlf + (hi - l'm)\ 
 
 9. If from the ends of a straight line PQ of limited 
 length there be drawn perpendiculars on a fixed plane and 
 the feet of these perpendiculars be joined by a straight line, 
 the joining line is called the projection of PQ on the plane. 
 Thus in the figure to Art. (6) if the edges LP, QN of the 
 parallelepiped PKQM be produced to meet the plane of xy 
 in y^ and F, EF is the projection oi PQ on the plane of xy, and 
 is equal and parallel to PN. Also 
 
 PN=PQcosQPN. 
 
 But QPN is ec[ual to the angle which PQ makes with the 
 plane of xy. Hence we derive the theorem : 
 
 The projection of a straigJit line of limited length on a 
 (jiven pUnie is equal to tJie length of the line multiplied by the 
 cosine of the angle between the line and plane. 
 
 If from the angular points of a triangle we draw perpen- 
 diculars on a fixed plane the feet of these perpendiculars form 
 a triangle whicli is called the projection of the given triangle 
 on the fixed plane. 
 
INTRODUCTORY THEOREMS. 
 
 Let PQR be the triangle. Through R draw a plane RHS 
 parallel to the plane on which the projection is made. 
 
 Let PQ produced meet this j)lane in >S'. Draw PK per- 
 
 pendicular to RS, and PH and QL perpendicular to the 
 plane RHS. Then RHL is equal to the projection of PQR on 
 the given plane. 
 
 Join HK, then it follows, from Euclid, xi. 8, that RS is 
 perpendicular to the plane PKH and therefore to KH. 
 Hence PKH is the angle between the plane PQR and the 
 plane on which the projection is made. Let this angle 
 be e. 
 
 Then 
 
 Similarly 
 Therefore 
 
 ^HRB = iSR . HK 
 
 = iSR.PK cose 
 = APSR . cos e. 
 
 ARSL = AQSR . cos 6. 
 
 ARHL = APQR . cos 6. 
 
 Hence the lirojection of a triangular area on a fixed j^lctne 
 is equal to the area of the triangle multiplied hy the cosine of 
 the angle between the plane of the triangle and the fixed 
 plane. 
 
 The proposition can easily be extended to any plane area. 
 
10 
 
 INTRODUCTORY THEOREMS. 
 
 10. If again from P and Q we draw perpendiculars on 
 some fixed line, the portion of the second line intercepted 
 between the feet of these perpendiculars is called the iwojec- 
 tion of PQ on the fixed line, and the following theorem holds : 
 
 The projectio7i of a straiglit line of limited length on a 
 second straight line, is equal to the lengtli of the first line mid- 
 tiplied hy the cosine of the angle between the two lines ; under- 
 standing by the angle between two lines which do not meet, the 
 angle between any two lines parallel to them luhich do meet. 
 
 This theorem is proved as follows : 
 
 Let PQ be the line of limited length, and AB the line on 
 wliich it is to be projected. Through P draw PR parallel, 
 and PA perpendicular to AB. Through Q draw a plane 
 perpendicular to AB meeting AB in B, and PR in R. Join 
 QR, RB, BQ. Then AB is the projection of PQ, for AB is 
 perpendicular to QB which lies in the plane QBR. Then 
 
 since PR is parallel to AB, which is perpendicular to the 
 plane RBQ, J\R is also perpendicular to this plane and there- 
 fore perpendicular to QR and RB. Hence PRBA is a paral- 
 lelogram, and therefore AB = PR. But PR = PQ cos QPR, 
 since 1*R(^ is a right angle. 
 Therefore 
 
 AB = PQ cos QPR, 
 
 the theorem required. 
 
 11. If we take any two points, P, Q, and draw from P 
 in any direction a straight line PR of any length, from R 
 a straight line RS, and join ^Q ; and from P, R, S and Q 
 
INTKODUCTORY THEOREMS. 
 
 11 
 
 draw perpendiculars PA, RG, SD, QB on AB ; AC, CD and 
 BB will be the projections of PR, RS and SQ on AB ; and 
 as long as A, C, D, B fall in the order represented in the 
 figure, the arithmetic sum of these projections is equal to 
 AB, the projection of PQ. The same would be true if we 
 had taken any number of lines between P and Q. If how- 
 ever C fall to the right of D, or C or D fall to the right of 
 
 A C 
 
 D B 
 
 B or to the left of A, this will be no longer the case. We 
 may agree to consider the projection of a line to be equal 
 to its length multiplied by the cosine of the angle which 
 it makes with the second line, those angles being always 
 taken which are formed by the successive lines PR, RS, SQ 
 with AB towards the same part. Thus if D come to the 
 left of C, the angle between RS and AB will be obtuse, and 
 the projection of RS will be negative. And since 
 
 AC-CD + DB = AB, 
 
 we still have the theorem that " the algebraical sum of the 
 projections on a given line, of a series of lines by which lue 
 
 A DC 
 
12 
 
 INTRODUCTORY THEOREMS. 
 
 pasfi from one point to a second, is equal to the j^'^ojection 
 on the same line, of the straight line joining the tiuo points. 
 
 This statement may be illustrated thus. Suppose a point 
 to move from P to Q along PR, RS, SQ, and from each 
 of its successive positions imagine a perpendicular let fall 
 on AB. As the point moves along PR, the foot of this 
 perpendicular will move along AB from A towards B, or 
 in the opposite direction, according as the angle between 
 PR and AB is acute or obtuse, and the length traversed 
 by it along AB in the projection of PR, and is positive if it 
 travels from A towards B, and negative if in the opposite 
 direction. It is clear that as the moving point passes from 
 P to Q, the foot of the perpendicular will pass from A to B, 
 and hence AB which is the projection of PQ will also be the 
 algebraical sum of the distances travelled by the foot of the 
 perpendicular, or of the projections of PR, RS, SQ. The 
 same theorem will be obviously true if instead of three lines 
 we have any number. By the angle between PR and AB 
 is meant the angle which would be formed if from any point 
 were drawn lines in the directions of PR and AB. Thus 
 the angle between PR and AB is the supplement of that 
 between RP and AB. 
 
 12. By means of the result of the last Article, another 
 proof of the formula (6) of Art. 8 can be obtained. 
 
INTRODUCTORY THEOREMS. 13 
 
 If, in the figure of that Article, QN be drawn parallel 
 to the axis of z to meet the plane of oci/ in N, and jS3I drawn 
 parallel to Oy to meet Ox in M, it follows that the pro- 
 jection of OQ on OP is equal to the sum of the projections 
 of OM, J\m and A^Q on OP, that is, if 6 be the angle POQ, 
 and I, m, n ; V, m', n be the direction-cosines of OP and OQ 
 respectively, 
 
 OQ cos e = OM. I + MN . m + NQ. n 
 
 = OQ . r . I + OQ . m . m + OQ . n .n; 
 
 . ' . cos 6 = ir + mm + nn. 
 
 >^ 13. To find the distance of a j^oint from the origin luhen 
 the co-ordinates are oblique. 
 
 The formulae of Arts. 4, 5, 6 and 8 were obtained on the 
 supposition of rectangular co-ordinates. Let Ox, Oy, Oz be 
 oblique axes, and P any point. Through P draw planes 
 parallel to the co-ordinate planes to meet the axes in M, 
 N, R ; and join OP. The ratios of OM, OJSf and OR to OP 
 
 1 
 
 will be clearly the same whatever be the position of P, pro- 
 vided it lie in the same straight line through 0. These 
 ratios are called the direction-ratios of the line OP, and are 
 usually denoted by the letters I, in, n. We then get formulae- 
 corresponding to those of Alt. (5), 
 
 x = l.OP, y = m. OP, z = n. OP. 
 
14 INTRODUCTORY THEOREMS. 
 
 Again, let X, yi, v be the angles between {Oy, Oz), (Oz, Ox), 
 (Ox, Oij). Then we have, if PL be the edge of the paral- 
 lelepiped through P parallel to Oz, 
 
 or = OM^ + MU - 20 M . ML cos OML 
 = x^ -\-y'^ + 2xy cos v. 
 
 And OP" = OU + PL"" - 20L . PL cos OLP. 
 
 But the projection of OL on OR is equal to the sum of 
 the projections of 021 and ML on OR, or by Art. 9, 
 
 OL cos ROL = 03/ cos /^ + i/i^ cos \ = - OL cos OXP ; 
 
 and therefore 
 
 OP^ = a^ + y^ + 2:'^ + 2yz cos X + 2ja; cos fi + 2^^ cos i/. 
 
 Conibining this with the formulae x = l. OP, y =m. OP, 
 z = n . OP, we get 
 
 j \ = l"^ + m^ + n^ -\- 2?nn cos X + 2nl cos //, + 2???i cos i^. . .(1), 
 
 the relation which holds between the direction-ratios of any 
 straight line. 
 
 In the same manner we could shew that the distance be- 
 tween two points x^y y^, z^; x^, y,^, z^ is 
 
 {x^ - x./ -h (y, - y/ + (z^ - zj +2(y^- y^) {z^ - z^) cos X 
 
 H- 2 {z^ — z.) (x^ - x^) cos fi + 2 (x^ - x^) (y^ - y^ cos v. 
 
 And as in (8) that the cosine of the angle between two lines 
 whose direction-ratios are I, m, n ; I', m, n is 
 
 IV 4- mill + mi -f- {mil -\- mn) cos X 
 I -}- [nV + nl) cos fi + {Im + I'm) cos v...{2). 
 
 ^5" 14. The volume of the parallelepiped of which OP is the 
 
 diagonal is evidently equal to the product of the area of the 
 
 1 parallelogram OMLN into the perpendicular from R on the 
 
 \ ])lane of xy. If 6 be the angle between OR and a line per- 
 
 ' l)cndicular to the plane of xy, this volume would equal 
 
 OM . ON sin V X OR cos d 
 
 1 = xyz . sin v . cos 0. 
 
 IBut if /', m\ n be the direction-ratios of the line through 
 jH.'rpcndicular to the plane of xy, since it is perpendicular 
 
INTRODUCTOKY THEOREMS. 15 
 
 to Ox and Oy whose direction-ratios are (1, 0, 0), (0, 1, 0) 
 respectively, we have, by formula (2) of the last Article, 
 
 r + m cos V -\-n cos//. = (1), 
 
 r cos V + m^ -\- n cos \ = (2). 
 
 And since it makes an angle 6 with Oz whose direction-ratios 
 are (0, 0, 1) we have 
 
 n -\- r cos fjb + m'cosX = cos 6 (3). 
 
 From these, since by formula (1) of the last Article 
 
 r Q! + m cos V + n cos fju) + m (m + n cos \ + l' cos v) 
 + n (n + t cos /M + mf cos X) = P + m'^ + n'^ 
 
 * + 2m V cos X + 2nr cos //, + 21' mf cos v = l, 
 
 we have n cosO = 1 (4). 
 
 And from (1) and (2) we have 
 
 I m n 
 
 cos /t — COS X cos V COS X — cos fJL COS V cos^ z^ — 1 
 
 cos^ , 
 
 = — 2T-^ -i — ; 2 s ^ =r ov (3), 
 
 cos X + cos yU, + COS V — Z COS X COS fJb COS l* — 1 " ^ ^ 
 
 whence we get 
 
 cos^ 6 sin^ V = 1 — cos^ X — cos^ //, — cos'^ z^ + 2 cos X cos /z cos z/. 
 And the volume of the parallelepiped becomes 
 
 xyz Jl — cos'^ X — cos''' /a — cos" v + 2 cos X . cos /^ . cos v. 
 
 The volume of the tetrahedron cut off from the co-ordi- 
 nate axes by a plane through R, M, N, is evidently one-sixth 
 of the above expression. - 
 
 15. The position of a point in space is sometimes de- 
 termined by means of polar co-ordinates. Thus if Ox, Oy, 
 Oz be rectangular axes and P any point, the position of P is 
 clearly determined if we know OP the distance of P from 
 the origin ; the angle POz which OP makes with a fixed 
 line the axis of z ; and thirdly, the angle between the plane 
 through OP and Oz and some fixed plane through Oz, as the 
 
16 
 
 INTRODUCTORY THEOREMS. 
 
 plane of zx. These are called the polar co-ordinates of P 
 and are usually denoted by the letters r, 6, (p. They are 
 connected with the rectangular co-ordinates of P referred to 
 the axes Ox, Oy, Oz by very simple relations which can 
 be obtained thus. Draw FN parallel to Oz to meet the 
 l)lane of xy in iV, and NM parallel to Oy to meet Ox in M. 
 Join ON. 
 
 Then 
 X = 0M= ON cos cj) = OP sin 6 cos <f) = r sin 6 cos 0, 
 y = MN = ON sin </> = OP sin ^ sin <^ = r sin 6 sin (^, 
 z = PN= OP cos ^ = 7- cos 6>, 
 
 from which we can obtain the equivalent system 
 
 r^^x^^ y' 4- z^, 
 
 tan 6 = -^^^ ^ , 
 
 7/ 
 
 tan = * ; 
 wliich give ?•, 6, </> in terms of x, y, z. 
 
( 17 ) 
 
 EXAMPLES. CHAPTER I. 
 
 1. Find the distances between each pair of the points 
 ' whose co-ordinates are (1, 2, 3), (2, 3, 4), (3, 4, 5) respectively. 
 
 2. Prove that the triangle formed by joining the three 
 points whose co-ordinates are (1, 2, 3), (2, 3, 1), (3, 1, 2) 
 respectively is an equilateral triangle. 
 
 3. The direction-cosines of a straight line are propor- 
 tional to 1, 2, 3 ; find their values. 
 
 4. The direction-cosines of a straight line are propor- 
 tional to 2, 3 and 6 ; find their values. Find also the angle 
 between this line and that in question (3). 
 
 5. Find the angle between two straight lines whose 
 direction-cosines are proportional to 1, 2, 3 and (5, —4, 1) 
 respectively. 
 
 6. A, B, G are three points on the axes of sc, y, z 
 respectively ; if OA = a, OB =h, 00 = c, find the co-ordi- 
 nates of the middle points of AB, BO and GA respectively. 
 
 7. In the last question find the co-ordinates of the 
 centre of gravity of the triangle ABO and the distances of 
 this point from A, B, respectively. 
 
 8. Shew that if D, E be the middle points of BO, CA in 
 the last question, J)JS = J BO. 
 
 9. Find the distance between two points in terms of their 
 polar co-ordinates. 
 
 10. The co-ordinates of a point are (JS, 1, 2 ^3) ; find 
 its polar co-ordinates. 
 
 11. The polar co-ordinates of a point are (4, ^ , ^j ; 
 find its rectangular co-ordinates. 
 
 A. G. 2 
 
^ ^ CHAPTER 11. 
 
 THE STRAIGHT LIXE AND PLANE. 
 
 16. Before proceeding to find the equations of the 
 straight line and plane, we must examine the nature of the 
 locus represented by an equation of the form 
 
 F{x,y,z) = ^ (1). 
 
 Solving with respect to z we obtain 
 
 .z=f{x, y), 
 
 where z may have one or more values for each set of values 
 of X and y. Hence if we take any point in the plane of xy 
 whose co-ordinates are a, h we get one or more values of z, 
 that is, the straight line drawn through the point (a, h) 
 parallel to the axis of z will meet the locus in one or more 
 definite points. Hence the equation (1) must represent 
 a surface and not a solid figure. 
 
 Two equations 
 
 F, (x, y, z) = 0, 
 
 F, (^, y, z) = 0, 
 
 considered as simultaneous will be satisfied by the co-ordi- 
 nates of all the points of intersection of the two surfaces 
 
 ^x ('^'> 2/> ^) = ^> 
 
 K (^> y> ^) = o> 
 
 that is, will represent a line. 
 
 The simplest line with which we are acquainted is the 
 straight line, and the simplest surface the plane. It would 
 perhaps be more logical to find the equation of the plane 
 first, and then, since any two planes intersect in a straight 
 
 J 
 
 ] 
 
THE STRAIGHT LINE AND PLANE. 19 
 
 line, the equations of two planes considered as simultaneous 
 would represent a straight line. The equations of a straight 
 line can however be obtained most simply mthout reference 
 to that of a plane, and we shall therefore invert the ap- 
 parently natural order. 
 
 17. To find the equations of a straight line. 
 
 Let Z, m, n be the direction-cosines of the straight line, 
 a, fi, 7 the co-ordinates of some fixed point in it, and x, y, z 
 those of any other point in it. Also let r be the distance 
 between these points. Then by Art. (6) we have 
 
 a? — a = It, y — (B = mr, ^ — 7 = nr, 
 
 X— OL y — ^ Z — rj 
 or — f- =^ — -= -=-r (1). 
 
 These are the symmetrical equations of a straight line. 
 li A,B,G \)Q any quantities which are proportional to I, m, n, 
 we can replace these equations by 
 
 x~ a _y — ^ _z— y 
 
 B G 
 
 (2), 
 
 but these fractions are no longer equal to r. Conversely any 
 equations of the form (2) represent a straight line whose 
 direction-cosines are proportional to A, B, G. The values of 
 these direction-cosines can be found ; for supposing them to 
 be I, m, 71, we have 
 
 I _in _n _ jJP + m' -{-n^ _ 
 
 A B G JA' + B' + G" JA' + B' + G'' 
 The equations (2) can be also written thus : 
 
 y^A^'^K^-AV^ 
 
 G ( G 
 
 G I G 
 
 Or writing 
 
 B ^ B G G 
 
 'j = m, fi--a=p, -j = n, y--Ta = q, 
 
 2—2 
 
20 THE STRAIGHT LIXE AND PLANE. 
 
 2' = ""' + H (3), 
 
 z = nx -\- q) 
 
 which are the simplest forms of the equations of a straight 
 line, and useful in many cases. The student is however ad- 
 vised especially to attend to the forms (1) and (2). 
 
 The equations in (3) are those of planes drawn through 
 the line parallel to the axes of 2 and y respectively, the inter- 
 sections of which with the planes of xy and zx are the pro- 
 jections of the given line on those planes. (Art. 19.) 
 
 18. To find the equations of a straight line passing 
 through two given points. 
 
 Let a, y8, 7 ; a, /3', y be the co-ordinates of the two given 
 points. 
 
 By the last article the equations of any straight line 
 through (a, yS, 7) can be written in the form 
 
 x—a_y—^_z—y 
 
 I m n 
 
 (i). 
 
 But if the line also pass through the point (a', ^, 7) we 
 must have 
 
 a^^ff-l^j^^-y 
 
 t m n 
 
 Dividing each member of (1) by the corresponding mem- 
 ber of (2); we get as the equations required 
 
 x—a _y—^_z—y 
 
 i'^~/3^^~7^7" 
 
 19. To find the equation of a plane. 
 
 Let OD be drawn perpendicular on the plane from the 
 origin, and let the length of OD be p, and Z, m, n its direc- 
 tion-cosines. Let P be any point in the plane. Then since 
 OD is perpendicular to the plane it is perpendicular to PD. 
 Hence OD is the projection of OP on OD. 
 
 Draw PM parallel to Oz to meet the plane of xy in M, 
 and MN ])arallel to Oy to meet Ox in N. Then the projec-' 
 tion of OP on OD is the sum of the projections of ON, NM 
 
THE STRAIGHT LINE AND PLANE. 
 
 21 
 
 and MP on OD. But these are Ix, my, nz, respectively, and 
 the projection of OP on OD is p. Hence 
 
 Ix -\- my ■\-nz—p (1) ; 
 
 a relation which is satisfied by the co-ordinates of any point 
 in the plane, and therefore the equation of the plane. 
 
 If the plane is perpendicular to one of the co-ordinate 
 planes, as for instance that of xy, OD will lie in that plane, 
 and we have ?i = 0. Hence the equation in that case be- 
 comes 
 
 Ix + my =p (2), 
 
 and does not contain the variable 2. 
 
 If the plane is perpendicular to two of the co-ordinate 
 planes, as those of xy and zx, 1 = 1, m = 0, n = 0, and the 
 equation becomes 
 
 x = p (3). 
 
 These results are geometrically evident. 
 
 20. To find the equation of the plane in terms of its in- 
 tercepts on the axes. 
 
 This can be deduced from the equation (1) in the last 
 
 ^lrticle, but may also be obtained independently thus. 
 Let the plane cut the axes m A, B, G; and let any plane 
 parallel to that of yz cut the co-ordinate planes of zx, xy in 
 
 n 
 
 ^1 
 
22 
 
 THE STRAIGHT LINE AND PLANE. 
 
 the lines RN, NQ, and the given plane in RQ. Let P be 
 any point in RQ and therefore any point in the plane. Then 
 by Euclid, xi. 16, the lines RN, NQ, QR are parallel to the 
 lines CO, OB and BG, respectively. Draw PM parallel to 
 RJSr to meet QN in if. 
 
 Let 0]Sr=w, NM=y, MP = z, OA=a, OB = b, OG = c. 
 
 Then by similar triangles 
 
 Also 
 
 Hence 
 
 PM 
 RN 
 RN 
 
 CO 
 
 MQ^ NM 
 NQ NQ ' 
 
 PM RN 
 
 X 
 
 RN CO 
 
 PM MN 
 ' CO'^ BO 
 
 AN_NQ 
 AO ~ BO' 
 AN_NM NQ 
 AO NQ ^ BO' 
 A^N _ _0N 
 A0~ AO' 
 
 X y z ^ 
 or - + ^-+ =1, 
 
 a c 
 
 .(4). 
 
 21. All these forms of the equation of the plane are in- 
 cluded in the form 
 
 Ax + By + Cz = D (5). 
 
 Conversely we can shew that any equation of the form 
 (5) represents a plane. 
 
THE STRAIGHT LINE AND PLANE. 23 
 
 For let a, ^, y; a, ^', 7 be the co-ordinates of any two 
 points in the locus represented by (5). The equations of the 
 straight line joining these two points are 
 
 x-a _y-^ _z-y ' ,.v 
 
 / — /3' a ' \yj' 
 
 a —a p —p 7 — 7 
 
 But since (a, /3, 7), (a', /3', 7') lie on (5) we have 
 
 AoL + B^ + Gy = D, 
 Aoi + BjS' + Cy = D. 
 
 Subtracting, A (a-a') +B{^ - /S') + (7 - y) = 0. 
 And therefore by (6) 
 
 A (oj-a) + 5 (2/ - /3) + (^-7) = 0, 
 Avhere cc, y, z are the co-ordinates of any point in the line (6), 
 or Ax^By \Gz = AoL-\-B^-\-Gy = D. 
 
 Hence Xy y, z, the co-ordinates of any point in the line 
 (6), satisfy the equation of the locus. That is, if any two 
 points be taken in the locus of (5) and be joined by a straight 
 line, this straight line lies wholly in that locus. Therefore 
 the surface represented by (5) is a plane. 
 
 An equation of the form 
 
 Ax + By= D 
 
 represents a plane perpendicular to the plane of xy, and an 
 equation of the form 
 
 Ax = D 
 
 represents a plane perpendicular to the axis of x (Art. 19). 
 These are particular cases of (5), and may be obtained from 
 it by making first G to vanish, and secondly both B and G 
 to vanish. 
 
 22. To find the distance from the origin of the point 
 at which the plane (5) cuts the axis of x we must put y = 
 
 and z = 0. We thus obtain Ax = D 01c x = -7-\ or if this 
 
 A ' 
 
 distance be called a, -j = ct. Similarly ^ = h, ^ = c ; and 
 
 substituting for A, B, G in (5) we get 
 
24f THE STRAIGHT LINE AND PLANE. 
 
 Dx By Dz ^ 
 a c 
 
 X y z ^ 
 or - + ^ + - =1, 4 
 
 a c ^ 
 
 the equation found in Art. 20. 
 
 23. By Art. 19 it appears that every plane can be repre- 
 sented by an equation of the form 
 
 Ix + my -\-nz =p, 
 
 where I, m, n are the direction-cosines, and p the length, of 
 the perpendicular from the origin on the plane. But 
 
 Ax + By + Cz = D 
 
 represents a plane. Hence if these represent the same plane, 
 we have 
 
 I _m _ n _ p 
 
 Also r + m^ -\-n^ = ly 
 A 
 
 .'. 1 = 
 
 m 
 
 n = 
 
 and p = 
 
 JA'+B'+C' 
 B 
 
 jA' + B'-\-C'' 
 
 G 
 JA' + B'+C' 
 
 D 
 
 JA^+WVC'' 
 
 Thus the direction-cosines of the perpendicular from the 
 origin on the plane 
 
 Ax + By-^Gz = D 
 
 are proportional io A, B, G, and the length of the perpendi- ( 
 
 cular is 
 
 JA' + B'+G'' 
 
 24. The angle between any two planes whose equa- 
 tions are 
 
 Ax -\- By -^ Gz =^ D, 
 
 A'x^By-\-G'z==D\ 
 
THE STKATGHT LINE AND PLANE. 25 
 
 is the same as the angle between the perpendiculars on them 
 from the origin. But the direction-cosines of these perpen- 
 diculars are (Art. 23) 
 
 A ^B G 
 
 JA' + B'+C' JA'+B'+C' JA^ + B' + C 
 
 A' E a 
 
 JA'^^B"'+G"' JA'' + B"+C"' JA" + B" + G'^' 
 
 and the cosine of the angle between the planes is therefore 
 equal to 
 
 AA' + BB' + GG 
 
 J A' -{-B'+G' J A" + B" + G" * 
 
 The condition that the two planes should be at right 
 angles is therefore 
 
 AA' 4- BB' + GG' = 0. 
 
 The conditions that they should be parallel may be 
 obtained by equating the cosine of the angle between them 
 to unity. It will be found that this leads to the con- 
 ditions 
 
 A^_B^_0^ 
 A'~B'~G" 
 
 These may be also obtained independently from the con- 
 sideration that the direction-cosines of the perpendicular on 
 the one plane are proportional to A, B, G, and those of 
 the perpendicular on the other to A', B\ G' \ and if the 
 planes be parallel, and consequently the perpendiculars from 
 the origin on them coincident, we must have A, B, G pro- 
 portional to A', B', G', or 
 
 A^_B^_G 
 A'~ B'~ G" 
 
 25. The equation of a plane through a point {a, jS, <y) 
 parallel to the plane 
 
 Ax-{-By-]-Gz = D (1) 
 
 is easily seen to be 
 
 A{x-a)-{-B(y-^) + G{z-y)=0, 
 
 or Aa) + By + Gz = AoL-\~B^ + Gy (2). 
 
26 THE STRAIGHT LINE AND PLANE. 
 
 For this equation does represent a plane parallel to (1) by 
 the last article, and it is satisfied by the values 
 
 Now the length of the perpendicular from the origin on 
 the plane (1) 
 
 B 
 
 and the length of the perpendicular from the origin on the 
 plane (2) is similarly 
 
 ^ Aa -^ BI3 ■{- Cry 
 
 jA'-hB'-^-C ' 
 The difference of these, or 
 
 (Aa+B^ + Cy)~D 
 
 is the length of the perpendicular from the point (a, y8, 7) on 
 the plane (1). 
 
 If we take the equation of the plane in the form 
 
 Ix + my -\-nz —p = Oy 
 the numerical value of the length of the perpendicular from 
 any point {x, y, z) on this plane is 
 
 ± {Ix + my + nz —p). 
 It is easily seen that the expression 
 
 Ix + my + nz — p 
 
 is positive if the point {x, y, z) is on the opposite side of the ! 
 plane from the origin, and negative when the point (x, y, z) 
 is on the same side of the plane as the origin. If the ex- 
 pression be denoted by a, the length of the perpendicular 
 from any point on the plane 
 
 a = 
 
 is -I- a or - a, according as the point and the origin are on 
 the same or opposite sides of the plane. 
 
 2G. If we take four planes forming a tetrahedron whose 
 ccjuations are 
 
 a = 0, y8 = 0, 7 = 0, 8 = 0, 
 
THE STRAIGHT LINE AND PLANE. 27 
 
 all expressed in the form 
 
 Ix + my + nz —p = 0, 
 
 any other plane may be represented by the equation 
 
 la. + m/3 + ny + qB = 0. 
 
 For this represents some plane, being of the first degree in 
 X, y, z, and since it contains three arbitrary constants, namely, 
 the ratios of three of the quantities I, m, n, q to the fourth, 
 it may be made to satisfy three conditions, and may therefore 
 be made to represent any plane. 
 
 This method of representing planes may be developed in 
 a similar manner to that used for straight lines in Plane Co- 
 ordinate Geometry (Todhunter's Conic Sections, Chap. iv.). 
 Thus the equations of the two planes bisecting the angles 
 between the planes a = 0, /3 = 0, will be 
 
 a-l3 = and a+/3 = 0, 
 
 the former bisecting that angle within which the origin lies, 
 and the latter the supplementary angle. 
 
 Any equation which is not homogeneous in a, /3, y, 8, can 
 be rendered so by means of the relation , 
 
 Aa-\-Bfi + Gy-^DS = -SV, \ 
 
 where V is the volume of the tetrahedron, and A, B, G, D 
 the areas of its faces. This equation merely states that the 
 algebraic sum of the four tetrahedra whose vertices are at the 
 point (a, /5, 7, h) is equal to the fundamental tetrahedron. , 
 
 27. If a straight line 
 
 x-a _ y-P ^ z-y 
 A ~ B G ^ ' 
 
 is parallel or perpendicular to a plane 
 
 A'x + Fy-^G'z^D (2), 
 
 it is perpendicular or parallel respectively to the perpen- 
 dicular on that plane, whose direction- cosines are proportional 
 to A', B\ G'. 
 
 The condition that (1) may be parallel to (2) is therefore 
 
 AA' + BB'^-GG'=0, 
 
28 
 
 THE STRAIGHT LINE AND PLANE. 
 
 and the conditions that (1) may be perpendicular to (2) are 
 
 A_B^G 
 A'~ B'~ C" 
 
 28. It is often requisite to know the length of the per- 
 pendicular on a given straight line from a given point. 
 
 Let the equations of the straight line be 
 
 x-a _ y-(B ^z-^ 
 A ~ B G~ ^ ^' 
 
 and let a', jS', y' be the co-ordinates of the given point. 
 The equation of any plane through (a', /S', 7') is 
 
 \(x-a') + fi(y-^') + p(z-y') = (2). 
 
 If this plane be perpendicular to (1) we have 
 
 \ _/jb _v 
 
 A^B~G' 
 
 and its equation becomes 
 
 A{x-a')-\-B(ij-0')-\-C{z-y) = O (3). 
 
 The point where this plane meets the line (1) is evidently i 
 the foot of the perpendicular from (a', y8', 7') on (I). 
 
 Let then P be the point {a, (3, 7), F the point (a , 13', 7 ), 
 
 P' 
 
 a 
 
 JS 
 
 and Q the foot of the perpendicular from P' on the line (1); 
 therefore PQ is the perpendicular from P on the plane (3), 
 
 ] 
 
THE STRAIGHT LINE AND PLANE. 29 
 
 and we have PQ= Ja^+B'^ + C^ ' 
 
 and P'Q^ = PF^-PQ^ by the right-angled triangle P'QP', 
 .-. P'Q^ = (a - a!f + (/S - ^')' + (7 " iJ 
 
 A' + B' + C" 
 
 29. I\? yl?^cZ ^/le conditions that a straight line may lie 
 wholly in a given jjlane. 
 
 "^-'^/-'-^ « 
 
 be the equations of the line, 
 
 A'x + B\j-\-C'2 = D (2) 
 
 the equation of the plane. 
 
 Put each of the fractions in (1) equal to k. 
 
 Therefore 
 
 a) = a+ Ak, y = ^ + Bk, z = y + Gk, 
 
 and if the line (1) lies wholly in (2), these values of x, y, z 
 must satisfy (2) whatever be the value of k. Hence the 
 equation 
 
 J.'a + B^ + (7'ry - D + ^AA' + BB + CC) k = 0, 
 
 must be satisfied independently of k. This gives us the two 
 conditions 
 
 A'a + B'l3-{-G'y-D = 0, 
 
 A A' + BB' + (7(7' = 0. 
 
 The first of these equations denotes that the point (a, ft 7) 
 lies in the plane (2), and the second that the angle between 
 the line (1) and the perpendicular on the plane (2) is a 
 right angle. These are evidently necessary and sufficient 
 conditions. 
 
 
 30. To find the shortest distarice between tiuo straight 
 lines whose equations are given. 
 
 We must first prove that the shortest line between two 
 1 1 given straight lines is perpendicular to each of them. 
 
30 
 
 THE STRAIGHT LINE AND PLANE. 
 
 Let BC, ^D be the two straight lines, and AB a line 
 perpendicular to each of them. Then AB is clearly shorter 
 than the line joining A with any other point of BC, and 
 also than the line joining B with any other point of AD. 
 Let P be any point in AD, and Q any point in BC. Then j 
 
 P 
 
 D 
 
 PA and QB are both perpendicular to AB, and therefore AB 
 is the projection of PQ on AB, and is equal to the length 
 of PQ multiplied by the cosine of the angle between them, 
 and is therefore less than PQ, since the cosine of any angle 
 is less than unity. 
 
 — =^_^.i- a). 
 
 x-a _y-^' _ z-r^' 
 
 A' B a ^ ^' 
 
 be the equations of the two straight lines. Let the equation 
 of any plane through (1) be 
 
 P(^-a) + a(y-/3) + -K(2-7) = (3). 
 
 Then we have, since (3) contains (1), 
 
 PA + QB + RC=0 (4). 
 
 And if we take the plane through (1) to be also parallel 
 to (2), we have 
 
 PA'+QB'+RC' = (5). 
 
 From (4) and (;")) we have 
 
 P _ Q ^ R 
 BC'-HC CA'-G'A AB'-A'B' 
 
THE STEAIGHT LINE AND PLANE. SllT 
 
 The equation of a plane through (1) parallel to (2) is 
 therefore i I 
 
 {BC-B'C){x-a)+{GA'-C'A)(y-l3)+{AB'-A'B)(z-ry)=0{6). 
 Similarly the equation of a plane through (2) parallel to 
 
 {Ba-B'C)(w-oL')+{CA'-C'A)(7j-l3')+{AB'-A'B){z-Y)=0 (7). 
 The length of the perpendicular from the origin on (6) is 
 (BC - FG) a + {GA' - C'A) jS + (AB - A'B) y 
 J(BC' -B'Cy + iGA' -G'Ay + {AB' -A'B)' ' 
 and the length of the perpendicular on (7) is 
 
 ( BG' - B'G) d + {GA' - G'A) ^' + jAB' - A'B) y' 
 J{BG' - B'Gf + {GA' - G'Af + {AB - A'Bf 
 The difference of these, or 
 {BG'- BG) {a-a!) + {GA'- G'A) i/S-jS') + {AB- A'B) (7-7 ) 
 J(BG'- BG)' + (GA'- G'Af -f {AB - A'Bf 
 
 is clearly the perpendicular distance between the two given 
 lines. 
 
 The equations of the line AB can be obtained by finding 
 the equations of two planes, one of which contains the straight 
 line BG and is perpendicular to the plane (6), and the other 
 contains the line AD and is perpendicular to the same plane. 
 Each of these planes evidently contains the straight line AB, 
 and their equations considered as simultaneous determine the 
 line. The requisite conditions for the two planes will be 
 found in Articles 24 and 29. 
 
 31. To find the condition that two straight lines whose 
 equations are given may intersect. 
 
 Let the equations of the straight lines be 
 
 x-cL _ y-^ ^ z-y 
 A'B G ^^' 
 
 A' " B ~ G' ^ ^• 
 
 d 
 
32 
 
 THE STRAIGHT LINE AND PLANE. 
 
 Then if they intersect, a plane can be made to pass through 
 both of them. Let this plane be 
 
 Since this contains the line (1) we have, by Art. 29, 
 
 Pa + Q^-]-Ry = D (3), 
 
 PA +QB + RC=0 (4). 
 
 And since it contains the line (2) we have 
 
 Pot'+ Qff + Ry' = I) (5), 
 
 PA' + QB' + Ra = (6). 
 
 From (3) and (5) we have 
 
 P {a -a') + Q(/3 - ^') +R {y-'y') = (7). 
 
 And eliminating P, Q, R from (4), (6) and (7) we get with 
 the usual notation of determinants, 
 
 ABC 
 
 A' E C 
 
 a /3-0 7-7 
 
 a 
 
 = 0, 
 
 or 
 
 (0L-a)(BC'-B'C)-\-{^-l3') (CA'-C'A)-\-(iy-y') (AB'-A'B)=0. 
 
 A result which might have been obtained from the last 
 article by the consideration that if two straight lines intersect 
 their shortest distance vanishes. 
 
 If the two straight lines be given by the equations 
 
 Ax + Bi/ + Cz = D) 
 
 A'x-^B'y + C'z = D\ 
 
 Px + Qy + Rz = S 
 
 P'a, + Q'y + R'r; = S' 
 
 the condition of intersection is obtained from the considera- 
 tion that these four ecjuations must be able to be satisfied 
 by the same values of x, y, z. The condition for this is 
 
 A B G D 
 
 .(8), 
 (9), 
 
 A' B' 
 
 c 
 
 D' 
 
 1' Q 
 
 R 
 
 S 
 
 P' Q' 
 
 R' 
 
 S' 
 
 = 0. 
 
 v1 
 
THE STRAIGHT LINE AND PLANE. 33 
 
 -^ 32. To find the equation of a plane passing through three 
 given points and the volume of a tetrahedron whose angular 
 points are given. 
 
 Let x^, y^, z^; ^^,y^,z^\ ^z^ Vz^ ^^ be the co-ordinates of 
 the given points, and let 
 
 Ax + By + Cz = D 
 
 be the equation of the plane passing through them. 
 
 We must have therefore 
 
 Ax^ -}■ %, + Cz^-D = 0, 
 
 Ax^ + By^ + 0^, - D = 0, 
 
 Ax^^By^-^Gz^-D = 0, 
 
 and if x, y, z be the co-ordinates of any point in the plane, 
 we have also 
 
 Ax + By^-Gz-D = 0. 
 
 From these four equations, eliminating A, B, C, D, we 
 obtain 
 
 = (1), 
 
 which when expanded is the equation required. 
 
 If the three points be P, Q, R, and S be any fourth point 
 whose co-ordinates are x^, y^, z^, the volume of the tetra- 
 hedron PQRS is one-third of the product of the area of the 
 triangle PQB, into the length of the perpendicular from S 
 I on the plane PQR. 
 
 The coefficients of x, y, z in the determinant in (1), by a 
 well-known formula of plane co-ordinate geometry, are equal 
 to double the projections of the area of PQR on the planes of 
 yz, zx and xy, that is if A, //, p be the direction-cosines of the 
 perpendicular on (1) and the area of PQR be represented 
 
 by A, 
 
 2XA = A, 2fjLA = B, 2vA = C, 
 
 since the area of the projection of a plane figure on a second 
 plane is obtained by multiplying by the cosine of the angle 
 between the planes (Art. 9). 
 
 A. G. 3 
 
 X 
 
 y z 
 
 i 
 
 x^ 
 
 2/i ^1 
 
 1 
 
 ^2 
 
 2/2 ^2 
 
 1 
 
 ^3 
 
 2/3 ^3 
 
 1 
 
34 
 
 EXAMPLES. CHAPTER 11. 
 
 Hence, squaring and adding, 
 
 But if j; be the perpendicular from x^, y^, z^ on the plane 
 
 PQR, 
 
 Ax, + By,+ Cz^-D 
 Hence the volume of the tetrahedron required which 
 
 = + 
 
 6 
 
 ^4 
 
 Va ^4 
 
 i 
 
 ^1 
 
 y. ^I 
 
 1 
 
 ^2 
 
 2/2 ^2 
 
 1 
 
 ^3 
 
 2/3 ^3 
 
 1 
 
 since A, B, C, —D are the coefficients of x, y, Zy 1 in the 
 determinant (1). 
 
 The result may be written more symmetrically 
 
 X, y^ z^ 1 
 1 
 1 
 
 F= + 
 
 ^2 2/2 ^ 
 
 6 
 
 ^3 2/3 ^' 
 
 ,(2). 
 
 ^■4 Va ^4 
 
 If the fourth point be the origin, this formula reduces to 
 
 -4 
 
 ^1 2/1 ^1 
 ^2 2/2 ^2 
 
 ^3 ^3 ^3 
 
 .(3). 
 
 EXAMPLES. CHAPTER II. 
 
 1. Find tlie equations of a straight line passing through 
 the point (1, 2, 3)_and whose direction-cosines are proportional 
 to ^/3, 1 and 2 ^3. 
 
 2. Find the equations of the straight line joining the 
 two points whose co-ordinates are (1, 2, 3) and (3, 2, 1) re- 
 spectively. 
 
EXAMPLES. CHAPTER II. 35 
 
 3. Find the equations of the sides of the triangle formed 
 by joining the points (1, 2, 3), (3, 2, 1), (2, 3, 1). Deduce the 
 ' values of the angles of the triangle. 
 
 *y 4. Find the equation of the plane which passes through 
 the three points in the last question, and the length of the 
 perpendicular on it from the origin. 
 
 5. Find the ecjuations of a straight line which passes 
 through the point (1, 2, 3), and is perpendicular to the plane 
 
 X -\- 2y + Sz = 6. 
 
 6. Find the equations of a straight line which passes 
 through the point (1, 2, 3), and is perpendicular to the two 
 straight lines in questions (1) and (2). 
 
 7. Find the equation of a plane passing through two 
 given points and perpendicular to a given plane. 
 
 8. Find the equations of a straight line passing through 
 ' the point (1, 2, 3) and parallel to the plane in question (4) 
 
 and to the plane of xi/. 
 
 9. Find the equation of a plane passing through the 
 point (2, 3, 4) and the straight line in question (1). 
 
 10. Find the equations of a straight line drawn from 
 the origin of co-ordinates at right angles to one given straight 
 line, and making a given angle with another. If the given 
 .straight lines be at right angles to each other and the given 
 
 angle be j , shew that there are two solutions, and that the 
 
 two straight lines so found are at right angles to each other. 
 
 11. Find the equation of a plane which passes through 
 a given point, and is perpendicular to each of two given 
 planes. 
 
 12. Shew that the equation of a plane in oblique co- 
 ordinates can be put in the form 
 
 X cos a + 2/ cos ^ -{- z cos y = p, 
 
 where p is the length of the perpendicular on the plane from 
 the origin, and a, /S, y the angles which it makes with the 
 axes. 
 
 3—2 
 
36 EXAMPLES. CHAPTER II. 
 
 18. Shew that if ct, /5, 7 be the angles between any 
 straight line and the axes of co-ordinates, I, m, n the direc- 
 tion-ratios of the line, and X, /-t, v have the meanings given 
 in Art. 13, 
 
 cos 0L = l + m cos v + n cos ji, 
 
 cos 13 = '>n + n cos \ + l cos v, 
 cos y = n-\- 1 cos /jl -\- m cos \. 
 
 14. Deduce the conditions that in oblique co-ordinates 
 the straicrht line 
 
 X _ y _z 
 
 I 111 n 
 ma}'' be perpendicular to the plane 
 
 Ax + By ■\-Gz = D. 
 
 15. Shew that the locus of a point which moves so as 
 always to be equidistant from two given points, is a plane 
 which bisects at right angles the straight line joining the two 
 points. 
 
 16. What loci are rej)resented by each of the equations 
 
 fix) = ; /(r) = 0; f{e) = 0; /(^) = 0; 
 where r, 6, </> are the usual polar co-ordinates ? 
 
 17. Interpret the equations : 
 
 [6 = a, ^^^ |(^ = 0, 
 
 \r =a. 
 
 (1) ^ = 0; (2) 1^^^^ (3) 
 
 18. Find the polar equation of a plane. 
 
 19. Find the angle between the two lines given by 
 
 o A r T£.( (1)» and x = y=z (2). 
 fix -h 4fy -^ 5z = 12] ^ ^ -^ ^ ^ 
 
 20. Three planes are at perpendicular distances p^yJ^^^Ps 
 from the origin ; three planes are drawn through the lines of 
 intersection of any two perpendicular to the third ; shew that 
 the last three planes will intersect in a straight line passing 
 through the origin if 
 
 2\ cos A =p,^ cos B=p^ cos C, 
 where A, B, C are the angles between the first three planes. 
 
EXAMPLES. CHAPTER II. 37 
 
 21. Shew that through two given points (a, h, c), (a, b\ c), 
 two planes may be drawn cutting olf from the axes intercepts 
 whose sum is zero ; and these two planes will be at right 
 angles to each other if 
 
 1 1 1 n 
 
 , + T r, + > = 0. 
 
 a — a — c — c 
 
 22. Find the cosine of the angle between the two straight 
 lines represented by 
 
 3 5 8^ 
 + - =0. 
 
 y — 2 z — x sc — y 
 
 23. Find the condition that the two straight lines whose 
 direction-cosines are given by the equations 
 
 Al + Bm-\-Cn = 0, 
 
 may be at right angles to each other. 
 
 24. If the co-ordinates of four points be a — b, a — c, 
 a- d ; h — c, b — d, b — a ', c — d, c — a, b — b ; d — a, d — b, 
 d — c, respectively, prove that the straight line joining the 
 middle points of any two opposite edges of the tetrahedron 
 formed by joining the points, will pass through the origin. 
 
 25. Shew analytically that the least distance between 
 two straight lines is perpendicular to each of them. 
 
 26. The shortest distance between the lines 
 
 x-cL_ y-l3 _z-y x-a _y-fi' _ z-y 
 
 b m n L m n 
 
 intersects the latter in the point whose co-ordinates are 
 
 a + V cosec^ 6 (lo + u cos 6), 
 
 and two similar expressions where is the angle between the 
 lines and 
 
 11 = I {a'- a) + m (/3' -(3)+ n (y - 7), 
 
 u' = l'{a- a!) + m' (^ - p') + n (7 - 7). 
 
38 EXAMPLES. CHAPTER II. 
 
 27. Prove that the straight lines joining the middle 
 points of opposite edges of a tetrahedron all meet in a point 
 and bisect one another. 
 
 28. If X, y be the lengths of two of the straight lines 
 joining the middle points of opposite edges of a tetrahedron, 
 CO the angle between these lines, and a, a those edges of the 
 tetrahedron which are not met by either of the lines, prove 
 that 
 
 cos ft) = 
 
 2 '2 
 
 a ~ a 
 
 4txy 
 
 29. Find the shortest distance between the diagonal of 
 a cube and any edge which it does not meet. 
 
 30. Find the area of the triangle formed by joining the 
 three points where the plane 
 
 X y z ^ 
 
 + f + - = 1 
 a c 
 
 cuts the axes. 
 
 31. From the origin are drawn three equal straight lines 
 of length p, such that the inclinations of the first to the axes 
 of Xy y, z respectively, are the same as those of the second to 
 y, z, X, and of the third to z, x^ y. A plane is drawn perpen- 
 dicular to each of them through its extremity. Find the co- 
 ordinates of the point of intersection of these three planes 
 and the equations of the line joining it with the origin. 
 
 32. A straight line is drawn from the oricjin to meet 
 ■I • • 
 
 the straight line 
 
 X — a _y — b z — c 
 I m n 
 
 at right angles. Shew that its equations are 
 
 X 
 
 - x__ 
 
 a — It h — Dit c — nt* 
 
 ^ ^ al-h hm -I- C7i 
 where t = -,^ — ., , . 
 
EXAMPLES. CHAPTER II. 39 
 
 33. Shew that by a proper choice of axes the equations 
 of any two straight lines can be put in the forms 
 
 z = c, y = mx ; z = — c, y — — inx. 
 
 34. If the co-ordinates of the points {x, y, z) and (f, 77, f) 
 be connected by the equations 
 
 f^^ i = y. i=^ 
 
 y ^ ' 7 z ' y z' 
 
 where c and 7 are given lines ; shew that if (x, y, z) be a 
 point in a straight line whose direction-cosines are I, m, n, 
 and which cuts the plane of xy at a point (a, h, 0), then (f, 77, 5") 
 will be a point in a straight line whose direction-cosines are 
 X, ya, V, and which cuts the plane of ^r] in the point (a, /S, 0), 
 where 
 
 a = lh, (3 = mS, 7 = nS, 
 
 a = \d, h = fjid, c = vd, 
 
 d being the distance of the point (0, 0, c) from the point 
 (a, b, 0), and 8 that of (0, 0, 7) from (a, /S, 0). Shew also that 
 if the lengths of the two lines from the points where they 
 cut the planes of xy and ^r] respectively be r and p, 
 
 rp = dS. 
 
 35. Prove that the four planes my + nz = 0, nz +lx = 0, 
 Ix -\- my = 0, Ix -\- my + nz = p, form a tetrahedron whose 
 
 volume is ,, . 
 oimn 
 
CHAPTER III. 
 
 ON CERTAIN SURFACES OF THE SECOND ORDER. 
 
 33. We have shewn that the general equation of the 
 first degree represents a plane. Before proceeding to the 
 discussion of the general equation of the second degree, we 
 shall find the equations of certain special surfaces included in ■ 
 the class represented by the equation of the second degree. I 
 
 The Sphere. 
 
 A sphere is a surface every point of luliich is at a constant 
 distance from a fixed point called the centre The constant 
 distance is called the radius. 
 
 Let a, h, c be the co-ordinates of the centre, r the radius, 
 00, y, z the co-ordinates of any point on the surface. Then 
 the distance of the point {x, y, z) from the centre is equal to 
 
 J^x-af-\-{y-hr+{z-c)\ 
 
 But this distance must equal the radius r. Hence for all 
 points on the surface 
 
 J{x - af + (3/ - by -h (2 - cy = r, 
 
 or (^j,.ay-\-iy-by + {,-cy=r' (1), 
 
 which is the equation required. 
 
 Conversely any equation of the form 
 
 w' + y^ -^z'^Ax-\- By ■^Cz + D = 
 represents a sphere. For it can be put into the form 
 
 ( 
 
 , A\' t B\' ' CV A' + B' + C „ 
 
ON CERTAIN SURFACES OF THE SECOND ORDER. 41 
 
 and, comparing this with (1), we see that it represents a 
 sphere whose centre is at a point (— ^~ , — ^ , —^j and whose 
 radius is 
 
 y 
 
 A' -^-B' + C _ ^ 
 
 ; 
 
 34. The Cone. 
 
 A cone is a surface generated by a straight line which al- 
 ways passes through a fixed point called the vertex, and through 
 a fixed curve. 
 
 We shall only discuss in this and the next Article the 
 case when the fixed curve is a plane curve of the second 
 degree. 
 
 Take the plane of the curve as the plane of xy, and let 
 the equation of the curve be 
 
 Ax'' -\- Cy'' + Ex = (1), 
 
 to which form the equation of any conic section can be re- 
 duced ; and let a, /3, 7 be the co-ordinates of the vertex. 
 
 The equations of any straight line through the vertex are 
 
 —r-= — - = — - (2); 
 
 I fa n 
 
 when this meets the plane of xy we have ^ = 0, and therefore 
 
 I m 
 
 These values of x and y must satisfy the equation (1), 
 since the line always passes through some point in the curve 
 represented by (1). Hence we have 
 
 or, multiplying by v^, 
 
 A {noL - lyf -f C (72/3 - rnyf + En {noL - ly) = 0. 
 
 |i 
 
42 ON CERTAIN SURFACES 
 
 This is a relation which must be satisfied by I, m, n if the 
 straight line (2) meet the curve (1). But if {x, y, z) be any 
 point in (2) we have 
 
 Z _ 771 _ 11 
 
 X — a y — [S z — 'y' 
 
 Consequently, if (x, y, z) be any point in any straight line 
 joining (a, y5, 7) with some point of the curve (1), we must 
 have 
 
 + ^ (^ - 7) {a (^ - 7) - 7 (^ - a)l = ; 
 or reducing, 
 
 A {oLZ - r^xy+ C {0z - yyy+ E (z - 7) (2^ - 7^) = 0. . .(3), 
 
 which is therefore the equation of the cone. 
 
 If we transfer the origin to the point (a, /3, 7) we must 
 put 
 
 x = X -\- :i, y = y' + 0, z = z + 7, 
 
 and the equation becomes 
 
 A {iz - yxj + C {fiz - yy'Y + Ez (iz - yx') = 0, 
 
 of which every term is of the second degree in x, y\ z . The 
 equation of a cone of the second degree whose vertex is at 
 the origin is therefore homogeneous. Conversely every homo- 
 geneous equation of the second degree represents a cone 
 whose vertex is at the origin. For let 
 
 Fx'^ + Qy" + Uz' + Fyz + ^zx + Kxy = (4), 
 
 be the equation. And let x^,y^, z^ be the co-ordinates of any 
 point on the locus. Then the equations of the straight line 
 joining {x^, y^, z^ with the origin are 
 
 ^ = ^=£ (5). 
 
 ^1 Vx ^1 
 
 But, since {x^, y^, z^ is a point in (4), 
 
 and therefore by (5), if {x, y, z) be any point in (5), 
 7 V + Qy' + Rz" -f Fyz + Q zx + R'xy = 0. 
 
OF THE SECOND ORDER. 43 
 
 Hence every point on the straight line joining the origin with 
 (x^, y^, z^ lies on the surface. Thus, the surface is generated 
 by a straight line which always passes through the origin, and 
 is therefore a cone. 
 
 35. The Cylinder. 
 
 A cylinder is a surface generated by a straight line which 
 aliuays passes through a fixed curve and remains parallel to 
 itself. 
 
 Let the plane of the curve be taken as the plane of xy, 
 and let its equation be 
 
 Ax' ->r Cy'' -{■ Ex = Q (1). 
 
 Also let Z, m, n be the direction-cosines of the straight 
 line to which the generating line always continues parallel. 
 Let 7, yS, be the co-ordinates of the point in the curve (1) 
 through which any generating line passes. The equations of 
 this line will therefore be 
 
 x-a^y-£^._ 
 
 C in n 
 
 Iz ^ mz 
 
 n "^ n 
 
 But a, yS are the co-ordinates of some point in (1), and 
 therefore we have by substitution 
 
 or A (nx - Izf + C (ny - mzf + nE {nx — lz) = (3), 
 
 which, being a relation satisfied by the co-ordinates of any 
 point in any one of the generating lines, is the equation of 
 the surface. 
 
 36. The Ellipsoid. 
 
 The ellipsoid is a surface generated by a variable ellipse 
 luhich ahuays moves parallel to itself, and has its vertices on 
 tiuo ellipses tuhose planes are perpendicular to each other and 
 to the plane of the moving ellipse, and which have one axis 
 common. 
 
44 
 
 ON CERTAIN SURFACES 
 
 Let the planes of the fixed ellipses be taken as the planes 
 of zx and xy, and the direction of their common axis as the 
 
 
 
 >w' 
 
 
 
 
 
 
 
 ^ 
 
 R 
 
 N 
 
 
 
 
 / 
 
 / 
 
 V 
 
 \ 
 
 M 
 
 V 
 
 
 
 / 
 
 / . 
 
 / 
 
 X 
 
 
 \ } 
 
 N, 
 
 / 
 
 
 
 B 
 
 axis of X. The plane of the moving ellipse will be parallel to 
 the plane of yz. 
 
 Let COA, AOB be the fixed elhpses, OA=a, OB = b, 
 OC = c. And let EPS be any position of the moving ellipse, 
 MR, MS its semi-axes, P any point in it. 
 
 Draw PN parallel to Oz to meet MS in N'. 
 
 Let OM = X, MN=y, NP = z. 
 
 From the ellipse RPS, 
 
 'MP ^ MS' 
 
 From the ellipse COA, 
 
 Rj\P 
 
 = 1- 
 
 X 
 
 a' 
 
 From the ellipse AOB, 
 
 MS" 
 
 a 
 
 (!)• 
 
 .(2). 
 
 (3). 
 
OF THE SECOND ORDER. 45 
 
 Whence substituting in (1) 
 
 c^ '^ h' a' ' 
 
 a he ^ ^ 
 
 If the two semi -axes 00 and OB be equal, it can be 
 seen from (2) and (3) that MR and MS are also equal. 
 Now an ellipse Avhose axes are equal is a circle. Hence 
 the surface in this case would be generated by the revo- 
 lution of the ellipse BOA round OA, and its equation 
 becomes 
 
 a^"^ h' 
 
 The surface is called an oblate or prolate spheroid ac- 
 cording as the semi-axis a is less or greater than h. If 
 all the three semi-axes OA, OB, OC he equal, the equation 
 becomes 
 
 w +y + z = a", 
 
 which shews that the surface in that case becomes a sphere 
 whose centre is at 0. 'hts^*^^^^ 
 
 37. The Hyi^erholoid of one Sheet. lu^^c/L •»- 
 
 The hyjjerholoid of one sheet is generated hy a variable 
 ellipse luhich moves parallel to itself and has its vertices on 
 two hyperbolas luhose planes are peiyendicidar to each other 
 and to the plane of the nfioving ellipse, and which have a com- 
 mon conjugate axis. 
 
 Let AQ he one hyperbola in the plane of zx, BR the 
 other in the plane of yz, and RPQ any position of the 
 moving ellipse, RM and QM its semi-axes, and P any point 
 on it. Let OA=a, OB = b, and OC, the common conju- 
 gate semi-axis, = c. Draw PJV parallel to MR to meet 
 I MQ in K Let OM = z, MN = x, NP = y. Then from the 
 ellipse RPQ, 
 
 2 2 
 
 "^ ~+ =1 
 
 MR' ' MQ' 
 
46 
 
 ON CERTAIN SURFACES 
 
 from the hyperbola AQ, 
 
 iW 
 
 a 
 
 = 1 + 
 
 IT D2 
 
 from the hyperbola BR, —j-^ 
 
 = 1 + 
 
 ,2 5 
 
 ,2 ) 
 
 or 
 
 
 1, 
 
 the equation required. 
 
 38. The Hyperholoid of tu'o Sheets. 
 
 This is generated as the last surface except that the 
 hyperbolas have a common transverse aais. 
 
 Take the direction of the common axis as axis of x, the 
 planes of the hyperbolas as the planes of zx, xy, and the 
 plane of yz parallel to that of the moving ellipse. Let 
 OA = a be the common transverse semi-axis, and OB = h, 
 OC = c, the two conjugate semi-axes. Let QPR be any 
 position of the moving ellipse, J\IQ, MR its semi-axes, and 
 P any point in it. Draw FN parallel to Q2I to meet 
 RM in iV. 
 
OF THE SECOND ORDER. 
 
 47 
 
 Let OM = ^, M]Sr= y, NP = z. 
 From the ellipse QFR, ^^^ + ^^, 
 
 from the hyperbola AQ,, —^ = —^ — 1^ 
 
 = 1, 
 
 C 
 
 from the hyperbola AR, ,.^ = -^ — 1 ; 
 
 6^ "^ c^ ~ a^ ' 
 
 or 
 
 ^ 
 d' 
 
 7,2 ^2 -•^J 
 
 ,the equation required. 
 
 These three surfaces, the ellipsoid, the hyperboloid of one 
 sheet, and the hyperboloid of two sheets, are all included in 
 'he equation 
 
 Ax^ + By"" + Cz' = 1. 
 
 39. The Elliptic Paraboloid. 
 
 The elliptic paraboloid is generated by a parabola which 
 .noves with its vertex in a fixed parabola, the planes of the two 
 
48 
 
 ON CERTAIN SURFACES 
 
 parabolas being at right angles, their axes jmrallel, and their 
 concavities turned in the same direction. 
 
 Take the plane of the fixed parabola as j^lane of xy, its i 
 vertex as origin, and its axis as axis of x. Then the plane of 
 the moving parabola is parallel to that of £x. 
 
 Let PQ be any position of the moving parabola, P any 
 point in it, I' its latus rectum, and let Z be the latus rectum 
 of the fixed parabola. Draw PM parallel to O2 to meet the 
 
 axis of the moving parabola in 31, and draw QH and MN 
 parallel to the axis of y. 
 
 Then from the parabola PQ, 
 
 P3P = z' = r. QM, 
 
 and from the parabola QO, 
 
 QH' = f = 1 . OH =lx-l . QM 
 
 . / ^'_ 
 
OF THE SECOND ORDER. 
 
 49 
 
 40. The Hyperbolic Paraboloid. 
 
 This is generated in the same manner as the last sur- 
 face except that the concavities are turned in opposite 
 directions. 
 
 Let OQ be the fixed parabola in the plane of xy, PQ any 
 position of the moving parabola parallel to the plane of zx, 
 
 P any point in it. Draw P3I parallel to O2, MN and QH 
 parallel to Oy. Let I and I' be the latera recta of the two 
 parabolas OQ, PQ. 
 
 From the parabola PQ, 
 
 PM' = z' = V . QM, 
 
 I rom the parabola OQ, 
 
 QH^=^f = l.OH 
 
 = l.{x + QM) 
 
 — ox -{• J, ', 
 
 • 'J. rp 
 
 "I l'~ 
 
 The two paraboloids are both included in the equation 
 
 Bf + Gz' = x, 
 
 A. G. 4 
 
50 ON CERTAIN SURFACES 
 
 We shall shew hereafter that any equation of the second 
 degree in x, y, z can be reduced to that of one of the surfaces 
 whose equations we have considered in this chapter. 
 
 41. Asymptotic surfaces. 
 
 The equation of the hyperboloid of one sheet is 
 
 —+^--=1 m 
 
 which can be put into the form 
 
 c ~ U' ^7 V aY + h'xV 
 
 where the remaining terms contain higher powers of 
 ay + ^^^^ ill the denominator. 
 
 Hence, if we increase x or y, or both, indefinitely, the 
 value of z approaches indefinitely near to 
 
 And if we construct the surface 
 
 Z^ 01? v^ 
 
 6'^~^''^¥ ^^^' 
 
 (which by Art. 84 represents a cone whose vertex is the 
 origin), the ordinate of this surface parallel to Oz, corre- 
 sponding to any given values of x and y, approaches indefi- 
 nitely near to equality with the ordinate of the hyperboloid 
 corresponding to the same values of x and y, when these 
 values are increased indefinitely; that is, the cone (2) is 
 asymptotic to the hyperboloid. 
 
 Similarly the cone whose equation is 
 
 is asymptotic to the surface 
 
 x' 
 a' 
 
 *> 
 
 r 
 b' 
 
 -$-». 
 
 jrfa( 
 
 x' 
 
 a' 
 
 je 
 
 -•:-■ 
 
OF THE SECOND ORDER. 51 
 
 42. The equation of the hyperbolic paraboloid is 
 
 2 2 
 
 \-7-^ W; 
 
 /I /, l'x\i 
 
 \ 
 
 =±yj^(i+^.+...) 
 
 Now if z be increased indefinitely and x be not very large, 
 the second and all the succeeding terms of the series on the 
 : right will diminish indefinitely. Hence the equations 
 
 y = ±J> (2). 
 
 ' represent two planes which are asymptotic to the surface (1) 
 at points for which y and z are increased indefinitely while x 
 remains finite. 
 
 EXAMPLES. CHAPTER III. 
 
 1. Find the polar equation of a sphere, any point not the 
 centre being the pole. Shew that if through a fixed point 
 any chord OPQ be drawn meeting a sphere in P and Q, the 
 rectangle OP . OQ is invariable. 
 
 2. From any point a straight line is drawn to meet a 
 given plane in P. In OP a point Q is taken so that the rect- 
 angle OP . OQ is equal to a given constant F. Find the 
 locus of Q. 
 
 3. From any point a straight line is drawn to meet a 
 given sphere in P. In OP a point Q is taken so that the 
 rectangle OP .OQ is equal to a given constant F. Find the 
 locus of Q. 
 
 4—2 
 
52 EXAMPLES. CHAPTER III. 
 
 4. Shew that if through any point of a sphere a plane 
 be drawn perpendicular to the straight line joining the 
 centre with that point, the plane will only meet the sphere 
 in that one point. 
 
 5. A and B are two fixed points, P a point which moves 
 so that FA is to FB in a constant ratio. Find the locus 
 of P. 
 
 6. A and B are two fixed points, P a point which moves 
 so that the angle AFB is a right angle. Find the locus 
 of P. 
 
 7. Find the surface generated by the line of intersection 
 of two planes which pass each through a fixed straight line 
 and are at right angles to each other. 
 
 8. Shew that all the points of intersection of two spheres 
 lie on a circle whose plane is perpendicular to the straight 
 line joining the centres of the spheres. 
 
 9. About three fixed points as centres, spheres are 
 described having variable radii which are always in the same 
 ratio to each other. Shew that they always intersect two 
 and two on three fixed spheres, and that these three spheres 
 have one circle common. 
 
 10. Prove that the planes of the three circles in which 
 three spheres intersect each other two and two, all intersect 
 in a straight line which is perpendicular to the plane con- 
 taining the centres of the three spheres. 
 
 11. Prove that the six planes of intersection of four 
 spheres two and two have one point common to them all. 
 
 12. Sliew that if each of six equal spheres intersects all 
 the rest but one, so that the radii at the line of intersection 
 are inclined at GO'', the portion of space common to all will 
 have eight solid angles coinciding with those of a cube whose 
 
 side is . of the diameter of the sphere. 
 v/l8 ^ 
 
 l.S. A straiglit line moves so that three given points of it 
 lie respectively in three planes at right angles to each other. 
 
 I 
 
EXAMPLES. CHAPTER III. 53 
 
 Shew that a fourth point in the straight line, whose distances 
 from the other three are respectively a, b, c, traces out an 
 ellipsoid. 
 
 14. The two straight lines 
 
 £c ±a _^ ±y _ z 
 
 cos a sm a 
 
 ! meet the axis of x in 0, 0', and P, F' are points on the two 
 lines such that OF . O'F' = c^ ; shew that the surface traced 
 [■out by the straight line FF' is the hyperboloid 
 
 x^ 1? z^ 
 ^ u = 1 
 
 I a c cos a c sm a 
 
 P, F' being taken on the same side of the plane xy, 
 
 15. Find the surface generated by a straight line which 
 ' revolves round a fixed straight line which it does not meet. 
 
 16. Find the surface which is the locus of the family of 
 curves defined by the equations 
 
 0^ -\r y^ -^ z^ = d^ and ^/^ + / = r^a^ — c^ 
 
 where a is a variable parameter and c an absolute constant ; 
 and discuss its form for different values of n. 
 
 17. A perpendicular FN is let fall from a point P in a 
 right cone on a plane through the vertex perpendicular to 
 the axis, and a point P' is taken in FN or FN produced 
 such that FN . FN is constant. Find the locus of P'. \y^ 
 
CHAPTER IV. 
 
 \f^\ TRANSFORMATION OF CO-ORDINATES. 
 
 43. Many of the equations which we shall have occasion 
 to employ will be much simplified by a proper choice of axes. 
 It is necessary therefore to investigate the relations which 
 hold between the co-ordinates of any point when referred to 
 two different sets of axes. 
 
 The simplest case is that in w^iich the directions of the 
 two sets of axes are identical, the origin only being different. 
 
 Let X, y, z be the co-ordinates of P referred to the old set % 
 of axes ; x , y, z , the co-ordinates of the same point referred 
 to the new set. Let a, ^, 7 be the co-ordinates of the new 
 oriofin referred to the old axes. Then the distance of P from 
 the old plane of yz is equal to the distance of P from the 
 new plane of yz together with the distance between these 
 two planes, or 
 
 x = x' -\- a. 
 
 Similarly y = y-^^> 
 
 z = z +'y. 
 
 These results will hold whether the axes be oblique or 
 rectangular. 
 
 44. To find the co-ordinates of a jyoint P referred to one 
 set of rectangidar axes, in terms of the co-ordinates of the 
 same point referred to another set of axes, also rectangidar, 
 with the same origin. 
 
 Let Ox, Oy, Oz be the old axes ; Ox, Oy, Oz the new. 
 Let x, y, z be the co-ordinates of P referred to the old axes ; 
 
TRANSFOR:irATION OF CO-ORDINATES. 
 
 55 
 
 ■ x\ y\ z the co-ordinates of the same point referred to the 
 i new axes. Let l^, m^, n^ be the direction-cosines of Ox' re- 
 
 ferred to Ox, Oy, Oz\ l^, m^, n^ those of Oy, and l^, m^, n^ 
 those of Oz. 
 
 Through P draw PM parallel to Oz to meet the plane 
 Oxy' in M, and through M draw MN parallel to Oy to meet 
 Ox in N. Then ON = x, NM= y', MP = z' . 
 
 Also the projection of OP on Ox \^ x. And the projec- 
 tions of ON, NM, MP on Ox are \x\ l^y , l^z, respectively, 
 since \, l^, l^ are the cosines of the angles between Ox and 
 ON, NM and MP, respectively. But the projection of OP 
 on any straight line is equal to the sum of the projections of 
 ON, NM and MP on the same line. Hence 
 
 X = l^x + l^y' + I/. 
 
 Similarly by projecting on the lines Oy and Oz we get 
 
 y = m^x + m^y -V m^z , 
 
 z = n^x + n^y + n^z\ 
 
 The nine quantities l^, m,, n^, l^, m^, n^, l^, m^, n^ are not 
 independent, but are connected by six relations. For since 
 l^, m^, Tij are the direction-cosines of Ox, we have 
 
 Similarly Z/ + m./ -\-nJ^ = 1, 
 
 «/ + < + < = !• 
 
56 
 
 TRANSFORMATION OF CO-ORDINATES. 
 
 Also the cosine of the angle between Oy and Oz is equal 
 to l,Jl^-\-m^m^-\-n^n^\ but this angle being a right angle, its 
 cosine is equal to zero ; 
 
 .'. Lk + mjii, + nji, = 0. 
 Similarly 
 
 "2"3 
 
 2 3 
 
 ^3^^ + m^m^ + n^n^ = 0, 
 These relations may be replaced by the six equations 
 
 i: +v +c =1. 
 
 < +< +^V 
 
 = 1, 
 
 nj^ ^-nj.^ +nj^ =0, 
 l^m^ + l^m^ -f /gWg = 0. 
 
 These equations can be algebraically deduced from the 
 previous set, but they can be more easily proved independ- 
 ently thus : 
 
 Zj, TTij, n^ are the cosines of the angles between Ox and 
 Ox^ Oy, Oz; l^, m,^, n„ those of the angles between Oy' and 
 Ox, Oy, Oz; and l^, m^, n^ of the angles between Oz and Ox, 
 Oy, Oz. Consequently \, l^, l^ are the cosines of the angles 
 between Ox and Ox, Oy, Oz ; m,, m^, m^ those of the angles 
 between Oy and Ox, Oy, Oz; and 7i^, n^, n^ those of the angles 
 between Oz and Ox, Oy' , Oz . Considering Ox, Oy , Oz as 
 axes, and remembering that Ox, Oy, Oz are mutually at right 
 angles, we obtain the above formulae at once. 
 
 45. The formulae given in the last Article are extremely 
 useful, and from their symmetrical character are easy to re- 
 member. They are liable to the objection that nine con- 
 stants are introduced of which six are superfluous, and other 
 formulae have been proposed which employ only three con- 
 stants. 
 
 Let Ox, Oy, Oz be the old axes ; Ox! , Oy', Oz the new 
 ones. Let the plane of x'y' cut the plane of xy in Ox^, and 
 let a plane through Oz and Oz, which is therefore by Euclid, 
 XI. 18, perpendicular to the planes of xy and xy\ cut these 
 planes in Oy^, Oy^, respectively. 
 
TRANSFORMATION OF CO-ORDINATES. 
 
 57 
 
 Then since Oz is perpendicular to the plane of xy it is 
 perpendicular to Ox^, and since Oz is perpendicular to the 
 
 plane of x'y', it also is perpendicular ' to Ox^. Hence Ox^ is 
 perpendicular to the lines Oz and Oz\ and is therefore per- 
 pendicular to the plane in which they lie, and therefore 
 perpendicular to Oy^, Oy^. Hence by Euclid, xi. Def. 6, the 
 angle yfiy^ is the angle between the planes of xy and x'y . 
 Let this angle be called 6, and let the angle between Ox 
 and Ox^ be called </>, and the angle between Ox^ and Ox be 
 called -v/r. 
 
 ' Let X, y, z be the co-ordinates of any point P referred to 
 the axes Ox, Oy, Oz. Then if we take Ox^, Oy^, and Oz as 
 axes, the ordinate z will be unaltered, and if x^ , y^ be the new 
 co-ordinates parallel to Ox^, Oy^, we have by the ordinary 
 formulse of transformation in plane co-ordinates, 
 
 x = x^ cos <i> — y^ sin <^, 
 
 y = x^ sin </) + 3/i cos (f>. 
 
 Again, if we take Ox^, Oy^, Oz as axes, the x^ will be un- 
 altered, and if y^, z be the new co-ordinates parallel to Oy^, 
 ^Jz\ we have 
 
 Vx = 2/2 *^^s Q - z sin 0, 
 z = 2/2 sin ^ -I- z cos 6. 
 
58 TRANSFORMATION OF CO-ORDINATES. 
 
 And lastly, taking Ox , Oy, Oz as axes, the z will be un- 
 altered, and we get 
 
 x^ = X cos '^ — y' sin yjr, 
 
 2/2 = x' sin yjr + y' cos ^fr. 
 
 And, making the substitutions for x^^y^^y^, we get finally 
 
 x=-x' (cos ^ cos -v^r — sin (^ sin -v^r cos &) 
 
 — y (sin -v/r cos (/> + cos ^ cos a/t sin c^) + z sin <^ sin ^, 
 y = x' (sin cos i/r -|- cos c^ sin ^/r cos 6) 
 
 — y (sin (^ sin \/r — cos </> cos i/r cos ^) — z sin ^ cos (/>, 
 2r = a?' sin ^/r sin 6 + y' cos i/r sin ^ + / cos 6. 
 
 These are called Euler's Formulae. They are useful in 
 discussing the nature of the sections of surfaces, but their 
 unsym metrical character renders them difficult to remember. 
 
 46. If we wish to change both the origin and the direc- 
 tion of the axes we have only to combine the formulae of 
 Arts. 43 and 44. For changing the origin to a point whose 
 co-ordinates are a, ^, 7, and keeping the direction of the axes 
 unchanged, we get x = x^-\- a, y = y^-\- /S, z = z^ + y. And 
 then changing the directions of the axes we get 
 
 ^, = l,x + l^y -\- 1/, 
 
 or a; = l^x + l^y' + l^z + ^. 
 
 Similarly y = m^x 4- m^y' + 7n^z' + y8, 
 
 z = n^x + n^y' + n^z -\- 7. 
 
 47. The formulae for transformation of co-ordinates in 
 Art. 44 hold also when the axes are oblique if l^, ?7i,, ?z, denote 
 the direction-ratios of the new axis of x with respect to the 
 old axes. The six relations which bold between the nine 
 constants involved, which can be obtained from Art. 13, are 
 in general very cumbrous. 
 
 48. A proof exactly similar to that given in Todhunter's 
 Conic Sections, Art. 87, will shew that the degree of any ex- 
 pression involving x, y, z is unaltered by transformation of 
 co-ordinates. 
 
TRANSFORMATION OF CO-ORDINATES. 59 
 
 49. The following proposition is useful in many ques- 
 tions of transformation of co-ordinates. 
 
 The condition that the expression 
 
 Ax^ -f Bf + Cz' + 2A'yz + 2B'zx -}- ^Cxy (1) 
 
 should be the product of two linear expressions in x, y, z, is 
 
 ABC + lA'B'C - AA" - BB" - CO" = 0. 
 For if one of the factors be 
 
 Xx -{- fiy -\- vz (2), 
 
 it is evident, by considering the coefficients of x\ y^ and z"^ in 
 (1), that the other factor must be 
 
 ABC 
 
 x^'^jLy^'v' (^>- 
 
 Multiplying (2) by (3) and equating the coefficients of yz, 
 zx and xy in the product, to those of the same terms in (1) 
 we have 
 
 B--\-C^ = 2A\ 
 
 /Jb V 
 
 C-+A^ = 2B', 
 
 V \ 
 
 A^-\-B^=2C\ 
 whence by multiplication we get 
 
 2 2\ ' ■\2 i!\ 
 
 SA'B'C = 2ABC + A iB'^- + C'^j +B [C^ +A-- ^, 
 
 = 2ABC+A{^A'^-2BC) + B {4<B"-2CA) + C(4<C"-2AB), 
 or transposing and dividing by 4, 
 
 2A'B'C' -f- ABC - AA''- BB"' - CC = 0. 
 
 The expression 2A' B' C + ABC - AA' - BB" - CC" is 
 called the discriminant of the expression (1). 
 
 • 
 
 • 
 
60 TRANSFORMATION OF CO-ORDINATES. 
 
 50. It is evident that in any transformation of co-ordi- 
 nates from one set of axes to another, the origin being un- 
 changed, the expression x^ + y^ -\- z^ will be transformed into 
 x"^ + y"^ + z"^ if both sets of axes be rectangular ; or the ex- 
 pression 
 
 a? + y' + -2"^ + ^yz cos \ + ^zx cos ya -f- ^xy cos V 
 
 will be transformed into 
 
 x"^ + y"^ + z"^ + ^yz cos X' -H ^z'x cos yu + ^x!y cos i/' 
 
 if the axes are oblique, the expressions in each case repre- 
 senting the square of the distance of the point whose co- 
 ordinates are considered, from the common origin. 
 
 Thus if the axes are rectangular, and the expression 
 
 Ax"" + Bif + a/ H- "lA'yz + Wzx -h "IG'xy (1) 
 
 become by transformation 
 
 Fx"" + qy"" + Rz' + 2P'2/V -h 2Q'^V + "LBlxy'. . . (2) ; 
 
 we shall have also the expression 
 
 Aa? -h By" + Gz"^ 2A'ijz + 2B'zx + 2C'xy - X (x'+ y'+ /). . .(3), 
 
 where \ is any constant, transformed into 
 
 P^"+ Qy"-^ Rz"'-\- 2Fyz^ 2Q'zx+ 2Rxy-\{x"+y-'-\-z^) . , .(4). 
 
 But if, for any values of X the expression (3) be the pro- 
 duct of two linear expressions in x, y, z, the expression (4) 
 must, for the same values of X, be the product of the two 
 expressions in x , y\ z into which the former two would be 
 reduced by the transformation. Hence the discriminant of 
 (3) is identical with that of (4), or the two equations 
 
 (^i -X){B- X) {C -X)- A'\A - X) - B\B - X) - C (C - X) 
 
 -^2A'B'C' = (5), 
 
 (P -X)(Q- X) {R - X) - P" (P - X) - (r{Q - X) - R' {R- X) 
 
 ■\-2F'Q'R' = (6), 
 
 are identical, and satisfied by the same values of X. Thus 
 
TRANSFORMATION OF CO-ORDINATES. 61 
 
 the coefficients of the different powers of X in these equations 
 must be equal, and we have 
 
 A+B + C = P + Q + R, 
 BG+CA+AB- A" - B" - C" 
 
 = QR + RP + PQ-r'-Q"'-E'\ y (7). 
 ABC + 2A'BV' - A A" - BB' - CC" 
 
 = PQR + 2P'Q'R' ~PP''- QQ" - RR'\ 
 
 The expressions on the left-hand side of the equations 
 'J) are called invariants of the expression (1). 
 
 51. As a particular case of the foregoing, let us suppose 
 if possible, as it will be proved to be hereafter, that the ex- 
 pression (1) is transformed into an expression of the form 
 
 The equation (6) then becomes 
 
 (P - X) (Q - X) {R-X) = 0, 
 
 md the roots of this equation are P, Q, R, the coefficients of 
 ' ", y~, z'^ in the transformed expression. These coefficients 
 ire therefore the roots of the equation (5) with w^hich (6) is 
 dentical, namely, 
 
 A -X){B- X) (C-\)- A'\A -\)- B'\B - X) - G'\G - X) 
 
 -f 2A'B'G' = 0. 
 
 A.nother proof of this result will be given hereafter (Art. 86). 
 
 EXAMPLES. CHAPTER IV. 
 
 1. The co-ordinates of a point are (1, 2, 3). Find its co- 
 )rdinates relative to new axes whose equations are x — y — z\ 
 Ix = — y = 2z ; X = — z, y = i). 
 
 2. Transform the expression xy + yz + zx to the new 
 ixes in the last question. 
 
62 EXAMPLES. CHAPTER IV. 
 
 3. Shew that x^ -\- y"^ -\- z^ + yz + zx -\- xy can be reduced 
 by transformation of co-ordinates to the form 
 
 A (x' + y'') + Bz\ 
 
 4. From the formulae in Art. 44 prove that 
 
 5. Find the values of P, Q, R when the expression 
 
 x^ + y"^ + z"^ — 4*xy — 4tyz — 4!zx 
 is transformed into the form Px^ + Qf/'^ + Pz'^. 
 
 6. Shew that if the expression 
 
 Ax" + By' + Cz' + 2A'yz + 2B'zx + 2C'xy 
 
 be transformed into Px' + Qy'^ + i^/^ where the first axes 
 are inclined at angles X, /n, v, and the new axes are 
 rectangular, P, Q, R will be the values of k given by the 
 cubic equation 
 
 (A-k){B-k)(C-k)-(A'-kcosXy{A-k) 
 
 - (F - k cos ^if {B - k) -(C'-k cos vf (C - k) 
 + 2{A' -k cos X) {B' - k cos ^) {C - k cos v) = 0. 
 
 7. Prove that the equation 
 
 Jx + Vy -\- Jz = 
 represents a cone of revolution round the line 
 
 x = y = z, 
 whose semi-vertical angle is cot~^ J 2. 
 
CHAPTER V. 
 
 ox GEXERATIXG LIXES AXD SECTIOXS OF QUADRICS. 
 
 52. We have seen (Arts. 34, 35) that the cone and 
 !? cylinder admit of being generated by the motion of a 
 
 straight Hne. This is also the case with the hyperboloid 
 of one sheet and with the hyperbolic paraboloid, but not 
 with any other surfaces whose equations are of the second 
 degree in ^, y, z. 
 
 Surfaces whose equations are of the second degree in 
 {x, y, z) are called Quadrics, or, following the analogy of the 
 ' terms ellipsoid, &c., Conicoids. 
 
 53. On the generating lines of the hyperboloid of one 
 sheet. 
 
 The equation of the hyperboloid of one sheet is 
 
 or ^-^ = 1-1 .. m 
 
 This equation is satisfied by all values of x, y, z which 
 satisfy either of the pairs of equations 
 
 X _z_ / _y\^ 
 
 a c~^\ bj 
 
 X z 1 /_ y\ \ 
 - + - = - 1 + f 
 a c fM\ bJ J 
 
 X z ( ^ y\ 
 
 or =yu, 1 + f 
 
 a c \ b) 
 
 a' c fjb\ bj 
 
 (2), 
 
 (3). 
 
64 
 
 ON GENERATING LINES 
 
 whatever be the vahie of ^. Each of these pairs represents a 
 straight line. There are thus two systems of straight lines 
 lying wholly on the surface. We shall first prove that all 
 the straight lines of one system intersect all the straight 
 lines of the other; and secondly, that no two lines of the 
 same system intersect one another. 
 
 54. The equations of any two straight lines of opposite 
 systems are 
 
 1 
 
 X z 
 
 - + - 
 
 a c 
 
 ■-I 
 
 (1). 
 
 
 a 
 
 X 
 
 c 
 
 z 
 
 hi 
 
 a c fi'\ h, 
 
 (2). 
 
 And if the straight lines represented by these equations 
 meet, these four equations must be satisfied by the same 
 values of x, y, z. But the four equations are all satisfied 
 
 if we take 
 
 X z _ 
 a c 
 
 = M 1- 
 
 X 2 _1 
 a c fjb 
 
 1 + 
 
 y 
 
 and /Lt' 
 From which we obtain 
 
 i.|) = . 
 
 h. 
 
 (3). 
 
 1-m/ 
 
 (4). 
 
 y fJ^ — fJ'' X _1 -^ /xfji' z 
 
 b fjb-\- fji" a ii-\- fx ^ c /^ + /^' 
 
 Hence any two generating lines of opposite systems meet 
 in a point. 
 
 Conversely, through any point of a hyperboloid of one 
 sheet two straight lines can be drawn lying wholly on the 
 surface. For if we assume the co-ordinates of the point to 
 be X, y, z; from equations (3) we can determine /j, and /x', 
 
AND SECTIONS OF QUADRICS. 
 
 65 
 
 and therefore the equations of the two generating lines 
 through the point in question. 
 
 55. Secondly, no two lines of the same system intersect. 
 For let their equations be 
 
 a c \ b 
 
 - + - = - 1 + ^ 
 
 y, 
 
 
 a i 
 
 X z 1 f^ \f 
 - + - = - 1 + f , 
 
 From the first and third we get by subtraction, 
 
 (^-/.')(l-f) = 0; 
 
 I therefore /^ = Z^'? o^ 2/ = ^• 
 
 From the second and fourth we get by subtraction, 
 
 y 
 
 1 + f =0 
 
 .*. fJb = fJb', or y = — h. 
 
 Hence since we cannot have y equal both to b and — b we 
 joaust have /jl = fjf, or the lines must coincide. Therefore no 
 5W0 lines of the same system intersect. 
 
 oQ. The equation of the Hyperbolic paraboloid is 
 
 y 
 
 z 
 
 I r~^' 
 
 ^hich will be satisfied by all values of Xy y, z^ which satisfy 
 jdther of the pairs of equations 
 
 
 Jl Jl 
 
 A. G. 
 
66 
 
 ON GENERATING LINES 
 
 Jl 
 
 X 1 
 
 or -,_ r- = - 
 
 Jl' f^ 
 
 y ^ 
 
 Hence in this case also there are two systems of straight 
 lines lying wholly on the surface. A proof similar to that of 
 the last two articles will shew that all the straight lines of 
 one system intersect all those of the other, and that no two 
 straight lines of the same system intersect one another. 
 
 It may be noticed that from the form of the second equa- 
 tion in each set, it follows that all the lines of each system 
 are parallel to a fixed plane. 
 
 57. We have shewn in the preceding articles that the 
 h3r[3erboloid of one sheet and the hyperbolic paraboloid ad- 
 mit of rectilinear generators ; we shall now shew that these 
 are the only surfaces among those which we have considered, 
 besides the cone and cylinder, with which this is the case. 
 
 Let us first take the equation 
 
 Aw' + Bif -\- Cz' = 1 (1), 
 
 which includes the ellipsoid and the two hyperboloids ; and 
 if possible let the line whose equations are 
 
 ~ir-~^~~^~' ^^^' 
 
 lie wholly on the surface (1). 
 
 From (2), 
 
 X = Oi-\- Ir, y = (3 -h mr, z = y -{- nr ; 
 and if the straight line (2) lies wholly on (1) the equation 
 
 A{oL + Irf -\-B{^+ mrf + C (7 + mf = 1 
 must be satisfied for all values of r. 
 The conditions for this are 
 
 Ah-\-Bmfi+Cny =0 
 Al' + Bm' +6V =0 
 
 .(3), 
 .(4), 
 .(5). 
 
 The first of these equations merely expresses the condi-, 
 
AND SECTIONS OF QUADRICS. 67 
 
 tion that the point (a, (3, 7) may lie on the surface. The 
 second and third are the conditions which I, m, n must 
 satisfy. They will in general give two values for the rati(^ 
 I : m : n. It remains to examine whether these values are 
 real or not. 
 
 From (4) we have 
 
 ^ AloL + BmB 
 
 (Jn = . 
 
 7 
 
 Substituting in (5) we get 
 
 CAiy + CBmy + (^^a + Bm^f = 0, 
 
 which is a quadratic in — . • 
 
 m 
 
 The roots of this quadratic will be possible or impossible 
 according as 
 
 (ACy' + A'ol'XBGj' + B'/3') < or > A'B'a'^\ 
 
 or as ABCy + A'BGdy + B'A G/Sy < or > 0, 
 
 or as ABGy' {A a' + B/3' + Gy') < or > 0, 
 
 or as ABG < or > 0. 
 
 Hence that the generating lines may be real we must 
 have ABG a negative quantity; thus one or three of the 
 quantities A, B, G must be negative. If they are all three 
 negative, the surface is impossible, so that the only possible 
 surface is the hyperboloid of one sheet in which one is nega- 
 tive. In this case we may take 
 
 11 1 
 
 and the equations which determine the directions of the 
 generating lines are 
 
 loL m/S 7iy _ 
 
 58. It may be noticed that since for either of the gene- 
 rating lines we have 
 
 Aloi + BmP + Gny = 0, 
 
 5—2 
 
68 ON GENEKATING LINES 
 
 and for any point in either line we have 
 
 x — aL_y — IB_z — y 
 I m n ' 
 
 we must also have the equation 
 
 AoL {x-a) + BP (3/- /3) + Cy{z - 7) = 0, 
 
 satisfied for any point in either of the straight lines through 
 the point (a, y&, 7). But this is the equation of a plane : 
 it is therefore the equation of the plane containing the two 
 straight lines. 
 
 The equation can be ^vritten 
 
 Aax + B(3y + Gyz = Ao? + B^'+ (77' = 1, 
 
 and it may be noticed that whether the lines themselves be 
 real or not, this plane is a real plane. We shall prove here- 
 after that it is the tangent plane to the surface at any point 
 
 («, A 7)- 
 
 59. The equation of the projection of either line on the 
 plane of xy is 
 
 x—a_y—^ 
 I Til ' 
 
 or 2/ = -ya; + /3--ya (1), 
 
 Z *^ ' ^ I 
 
 m 
 
 the values of -j being deduced from the quadratic equation 
 
 given in Art. 57. 
 
 AV (Cy + Aa') + 2ABa0 Im + Bm"" (Cy' + Bl3') = 0, 
 or AP (1 - B/3') + 2AB2^ Im + Biiv" (1 - Aol^) = ; 
 
 .-. AV + Brn^ = AB {1/3 - may. 
 Hence the equation (1) can be written 
 
 _m /l 1 m^ 
 
 which is a w^cll-known form of the equation of the tangent 
 to the curve 
 
 Ax'-hBy' = l. 
 
AND SECTIONS OF QUADRICS. 69 
 
 But this curve is the ellipse in which the given surface is 
 cut by the plane of xy. Hence the projections of the gene- 
 rating lines on the plane of xy are tangents to the curve in 
 which the surface is cut by that plane. 
 
 The same is true for the planes of yz and zx. 
 
 60. The equations of the two paraboloids are both in- 
 cluded in the equation 
 
 Bf+G2^=:X (1). 
 
 The conditions that a straight line 
 
 ^-g ^ 2/-^ ^ ^-7 /g) 
 
 I in n 
 
 should lie wholly on the surface (1) are found by a process 
 similar to that of Art. 57 to be 
 
 B^' + Grf = a (3), 
 
 Bm' -\- Gre == (4), 
 
 25^^ + 2(7717-^ = (5). 
 
 The first equation indicates that the point (a, /3, 7) lies 
 on the surface (1). The second and third give the values 
 of the ratios I : m : n. These values will be real if B and G 
 have opposite signs, so that the surface must be the hyper- 
 bolic paraboloid. 
 
 61. The equation of the projection of one of the gene- 
 rating lines on the plane of xy is 
 
 2/ = j^ + (^P- jaj (6). 
 
 But from (5) 
 
 (25m/3 - ly = 4^C'ny 
 
 = — 4>BGy^m^ from (4) ; 
 .-. 4>Bm' {BP' + Crf) - 4<Blm0 +l' = 0', 
 .'. ^Bm\ - 4<Blml3 + 1^ = from (3) ; 
 
 or Ip — moL = -7-r» • — • 
 
70 ON GENERATING LINES 
 
 And the equation (6) becomes 
 
 _ m 11 
 
 a well-known form of the equation of the tangent to the 
 curve By^ = x. i 
 
 Hence the projection of the generating line on the plane if 
 of 007/ is a tangent to the curve in which that plane is cut by 
 the surface. A similar proof holds for the projection on the 
 plane of zx. 
 
 The equation of the projection on the plane of yz is 
 
 in n ' 
 or 2/ = -^ + ^--7 (7). 
 
 But JBm''+Cn' = 0- .'.- = + a/- ^, \ 
 
 n \ B 
 
 and the equation (7) becomes ^ 
 
 ■4 
 
 Hence the projections of the generating lines on the 
 plane of yz are parallel to the two straight lines in which 
 the surface is cut by that plane. 
 
 G2. The sections of the ellipsoid 
 
 (X^ v^ z^ 
 
 made by planes parallel to either of the co-ordinate planes 
 are ellipses. For taking the equation of a plane parallel to 
 that of a:y to be 
 
 ^ = 7 (2). 
 
 we get for the points where this meets (1) 
 
 2 2 S 
 
 ^ 4- 'C - 1 _ :l 
 

 AXD SECTIONS OF QUADRICS. 71 
 
 This is the equation of the projection of the curve of 
 
 section on the plane of xy. But since the cutting plane is 
 
 parallel to the plane of xy^ the projection of the curve of 
 
 section on that plane is equal and similar to the curve itself. 
 
 "i Hence this curve is an ellipse. And it may be noticed that 
 
 this ellipse is always similar to the ellipse — 2 + t2 = 1j iii 
 
 which the surface is cut by the plane of xy. 
 
 In a similar manner the sections by planes parallel to the 
 other co-ordinate planes may be shewn to be ellipses. 
 
 The sections of the hyperboloid of one sheet 
 
 x^ y^ / ^ 
 
 2 T^ 7 2 2 ■*■> 
 
 a b c 
 
 by planes parallel to that of xy are ellipses, and those by 
 planes parallel to the planes of yz or zx are hyperbolas. 
 
 The sections of the hyperboloid of two sheets 
 
 2 2 2 
 
 a' If & ' 
 
 by planes parallel to those of zx or xy are hyperbolas, and 
 by planes parallel to that of yz are ellipses, which are im- 
 possible if the value of x for points in the cutting plane is 
 numerically less than a. 
 
 The sections of the two paraboloids 
 
 f ^_ 
 
 " __ /yi 
 
 by planes parallel to those of zx or xy are parabolas whose 
 latera recta are r and I respectively. 
 
 Their sections by planes parallel to that of yz are 
 respectively ellipses and hyperbolas, the former being impos- 
 sible when the cutting plane is to the left of the origin. 
 
 To find the nature of the sections of these surfaces by 
 planes not parallel to the co-ordinate planes it is no longer 
 
72 ON GENERATING LINES 
 
 'IV 
 
 sufficient to find the equations of the projections of the curve 
 of section on the co-ordinate planes, since the projection vdW 
 not in general be similar to the curve itself The simplest 
 method is to transform the co-ordinates so that the plane 
 of xy shall be parallel to the cutting plane, and then the 
 nature of the section will be given as above by its projection 
 on the plane of ocy. For this transformation the formulae 
 of Art. 45 are very useful. We may in general avoid the 
 third substitution, and since we wish to find merely the nature 
 of the sections by planes parallel to that of xy, which we 
 shall prove in the next article to be always similar to the 
 section by the plane of x'y' itself, we may before substitu- 
 tion put z' = 0. The required substitutions will then be 
 derived from the formulae in Art. 45 by putting yjr = and 
 z' = 0. We thus get ' 
 
 x = x COS <f> — y' cos 6 sin </>, 
 
 y = x sin ^ -\- y' cos 6 cos </>, 
 
 z =y sin 6. 
 
 If the equation of the cutting plane be given in the form 
 
 Ix -f my ■\-nz=p, we have tan 6 = , and cos 6 = n. The 
 
 m 
 
 above substitutions then become 
 
 _ mx' -f- Iny _ mny — Ix 
 
 where we assume that F + m^ + n^ = 1. 
 
 ^ 63. We shall first prove the following general propo- 
 sition. 
 
 All sections of surfaces of the second order made by 
 parallel jylanes are swiilar and similarly situated. 
 
 Take the plane of xy parallel to the system of cutting 
 planes. The equation of the surface can be put into the form 
 Aa^ -f By'' + Cz"" + 2A'yz + 2B'zx + 20 'xy 
 
 + 2A"x+2B"y + 2C"z -f F= (1). 
 
 The curve in which this is cut by the plane 
 
 ^ = 7 (2), 
 
AND SECTIONS OF QUADRICS. 73 
 
 is given by the equation 
 
 Ax' + By' + 2C'xy + (2^V + ^^") ^ + (2^7 + 25") y 
 
 And whatever be the value of 7 this curve is always 
 similar and similarly situated to the curve 
 
 Ax'^ + Bf + ^G'xy + ^A"x -f ^B"y + i^= 0, 
 
 in which the surface is cut by the plane of xy. 
 
 Hence in discussing the form of the sections of surfaces 
 jby a series of planes, we need only consider planes through 
 the origin. 
 
 This method will not fail even if the curve of section by 
 a plane through the origin become impossible, since the 
 terms of the second degree in the equation of this curve are 
 the same as in the equations of the possible curves formed 
 by the intersection of parallel planes with the surface. 
 
 64. We shall consider first the equation 
 
 Ax'-^By''+Gz' = \, 
 which includes the three central surfaces. 
 
 Making the substitutions suggested in Art. 62, we get as 
 the equation of the curve of section 
 
 x' {A cos^^ + B sin^^) + 2xy (5 — ^) cos <^ sin <^ cos 6 
 
 + y' {A cos'(9 sin'^c/) + B cos'^' cos'0 + G sin'(9) = 1. 
 
 And the section will therefore be an ellipse or hyperbola 
 according as 
 
 B — AJ cos- 6 cos^cf) sin^<^ 
 
 - (A cos'</) + B sin'(j)) (A cos'<9 sin'c^ + B cos-6' cos'^ + Csin'^^') 
 
 is negative or positive. This expression can be reduced to 
 the form 
 
 - {BGsin'e sin'0 + GA sin'O cos'cf> + AB cos'6>}. 
 
 In the case of the ellipsoid A, B and G are all positive, 
 md this expression is therefore always negative. All sec- 
 ions of the ellipsoid are therefore ellipses. The investigation 
 )f the nature of the sections in the other surfaces is long and 
 
74 ON GENERATING LINES 
 
 the results uninteresting, except in the particular case in 
 which the section becomes a circle. I 
 
 The conditions that this may be the case are, that the co- 
 efficient of xy should vanish and the coefficients of x"^ and 
 y- should be equal. We have therefore 
 
 {B — A) cos Q sin cos (/> = 0, 
 
 A cos^cj) + B sin^^ = A cos^6 sin^<^ + B cos^^ cos^<^ + C sin^d. 
 
 From the first equation we must have either B=A, in 
 which case it is already obvious that all sections parallel to 
 the plane of xy are circles, or 
 
 cos . sin (f) . cos (j> = 0. 
 
 If cos 6 = 0, we have 6 = 90*^, and the second equation gives 
 
 A cos^^ + B sin^(/) = G=C (cos^</) + sin^</)) ; 
 
 ^ , 2 f ^ — A 
 
 . . xan u) — "^ p^ , 
 
 and if the values of tan (p be real, we get circular sections by 
 two planes through the axis of z. 
 
 If we take cos (/> = ; we have cp = 90^ or the plane 
 passes through the axis of y, and the second condition gives 
 
 B = Acos'e+Gsm'e; 
 
 A — B 
 
 and therefore tan'^^ = - n _ p , 
 
 and if the values of tan 6 be real, we get circular sections by 
 planes through the axis of y. 
 
 Similarly from the condition sin (j) = 0, we get circular 
 sections by planes through the axis of x inclined to the 
 plane of xy at angles given by the equation A| 
 
 tan"" 6 = p^^A • 
 
 In all cases the circular sections are made by planes 
 passing through one of the axes. It only remains to examine | 
 in what cases they are real. 
 
AND SECTIONS OF QUADRICS. 75 
 
 Only one of the three quantities 
 
 C-A A-B A-B 
 B-C B-C C^A 
 
 can be positive, consequently there are only two real central 
 circular sections, and they pass through the axis of 2, y or x, 
 iccording as the first, second, or third of these expressions is 
 positive. 
 
 (1) In the ellipsoid A, B, G are all positive, and if we 
 take them in order of magnitude, the second of the above 
 expressions is positive. Consequently the central circular 
 sections of an ellipsoid are made by planes through the mean 
 axis. 
 
 (2) In the hyperboloid of one sheet G is negative, and 
 if we suppose A>B, it is again the second of the above ex- 
 pressions that is positive, and the circular section is made by 
 a plane through the greater real axis, since 
 
 A-^ B = ^ 
 
 'and A being > B, a <b. 
 
 (3) In the hyperboloid of two sheets, B and G are 
 aegative, and if we suppose B numerically greater than C, 
 or b<c, B — G wdll be negative, the first of the above ex- 
 pressions is positive, and the circular section is made by a 
 plane through the greater impossible axis. 
 
 65. We have shewn in the last article that the onli/ 
 planes which give circular sections of central quadrics are 
 certain planes through one of the axes. It is easy to shew 
 without transformation that these planes do give circular 
 sections. 
 
 Thus the equation of the ellipsoid can be written in the 
 form 
 
 
 
 or 
 
 [c a ) \c a ) 
 
76 ON GENERATING LINES 
 
 which shews that either of the planes 
 
 -JW^' --J^r^' = (1), 
 
 c d 
 
 or i^5^IV + -7^^^^=0 (2), 
 
 c a ^ ^ 
 
 cuts the ellipsoid in the same points in which it cuts the] 
 sphere 
 
 But every plane section of a sphere is a circle. Henceff 
 the planes (1) and (2) and consequently by Art. 63 all planes] 
 parallel to them cut the ellipsoid in circles. 
 
 The circular sections of the hyperboloids of one and two' 
 sheets can be deduced in a similar manner. 
 
 66. The two paraboloids are included in the equation 
 
 By' + Gz' = X. 
 
 Making the same substitutions as in Art. 64 we obtain 
 for the equation of the curve of intersection, 
 
 ^sin'^ (^x"^ + 2B sin <^ cos </> cos 6 xy 
 
 + y^ {B cos'^ 6 cos^<^ + Osin^ 0) = x cos <^ — y cos sin 0, 
 
 which will represent an ellipse, parabola, or hyperbola, ac- 
 cording as 
 
 B"" sin'</) cos'^i/) cos'^ - B sin'^^c^ {B cos'<9 cos^^ + C sin'^^) 
 
 is negative, zero, or positive. That is, according as 
 
 BG sin'^ (/) sin^ (9 
 
 is positive, zero, or negative. 
 
 The sections of both paraboloids are therefore parabolas 
 if </> or Q vanish, that is, if the cutting plane pass through the 
 axis of X or coincide with the plane of xy. In all other 
 cases the sections of the elliptic paraboloid are ellipses, and 
 of the hyperbolic paraboloid, hyperbolas. 
 
AND SECTIONS OF QUADRICS. 77 
 
 The conditions that the section may be a circle are 
 
 B sin (^ cos (^ cos ^ = 0, 
 
 B sin'^ = ^cos^(/) cos'6' + G sin'6'. 
 From the first equation 
 
 sin = 0, cos (^ = 0, or cos ^ = 0. 
 
 If sin (f) = 0, the coefficient of cc'^ vanishes, and the section 
 reduces to a straight line or parabola. 
 
 If cos (j) = 0,we have from the second equation B = Csin^O, 
 
 and if B and C are of the same sign and B < G this gives 
 
 G 
 two possible values of 6. If cos 6 = 0, we get sin^0 = -^, and 
 
 B 
 
 this gives two possible values of (/> if (7 < 5, and B and (7 have 
 the same sign. Thus we get real circular sections of the 
 elliptic paraboloid passing through the axis of y or 2, accord- 
 ing as j£> < or > G, that is as Z > or < l'. 
 
 If B and G have opposite signs, there are no real circular 
 sections. 
 
 j 67. The equation of the elliptic paraboloid can be put 
 into the form 
 
 x^-\-y^ + z^ 0^ 'if' y^ _ 
 
 x' + y^+z\ f /I 1 x\ / /I 1 ^ \ 
 Thus each of the planes 
 
 and therefore all planes parallel to them will cut the surface 
 in circles. These planes are real if I' >l If T <l we can 
 shew similarly that the planes 
 
78 ON GENERATING LINES 
 
 cut the surface in circles. 
 
 C8. We shall conclude this chapter with the investiga-^ 
 tion of the position and magnitude of the axes of the section 
 of an ellipsoid by a plane through its centre. 
 
 (x^ li^ z^ 
 
 / Let -2 + ^2+^2 = 1 (1) 
 
 a he ^ 
 
 be the equation of the ellipsoid, 
 
 Ix + my -\-nz = {) (2) 
 
 the equation of the cutting plane. 
 
 r J. X y z ,^. 
 
 Let -='l = - = r (3) 
 
 \ fJb V 
 
 be the equations of any straight line in the plane (2), and let 
 r be the distance from the origin of the point where it meets 
 the ellipsoid ; therefore 
 
 r'~d''^Jf'^e ^^^' 
 
 and l\ -f mfx ■\-nv = Q (5), 
 
 since the line (3) lies in the plane (2). 
 
 Also if r be the length of one of the semiaxes of the 
 section of (1) by (2), we must have r a maximum or mini- 
 mum by the variation of X, fi, v, which are connected by the 
 relation (5) and also by the relation 
 
 X^ + /.»+i^' = l (6). 
 
 Differentiating (4) we get when r is a maximum or mini- 
 mum 
 
 ,. \d\ iidii vdv 
 
AND SECTIONS OF QUADRICS. 79 
 
 And from (5) and (6) respectively, 
 
 = ld\ + mdiJb + ndv, 
 = Xf/X + ^diJL 4- vdv. 
 
 Whence by indeterminate multipliers, 
 
 X 
 
 I 
 
 a 
 
 + U + h'\ = (7), 
 
 ^ + A;m + A;> = (8), 
 
 h 
 
 1 
 
 -2 + ^7^ + ^'V = o (9). 
 
 Multiplying (7) by X, (8) by /i, (9) by v, and adding, we 
 :get 
 
 J. + ^ = o, I 
 
 i 
 and therefore i 
 
 X I — s ^ I = — rd) •*• X = 
 
 .(1.-1.)=-.,. .= 
 
 klr^aj^ ( 
 
 V (-2 — :,] = — kn 
 \c r ' c" — r 
 
 a'- 
 
 r'' 
 
 kmr^b^ 
 
 ¥- 
 
 r'' 
 
 knr 
 
 'c' 
 
 2 ^'^ I """"' ^ ^2 ^2 
 
 And therefore from (5), 
 
 + 7:2-— -2 + ;:^— -2 = (10), 
 
 a^ — r^ ¥ - r^ c^ — r^ 
 
 which is a quadratic equation and gives two values of r^ 
 
 The product of these two values 
 
 a'b'c' 
 
 I'd' + 7n'¥ + 7iY ' 
 and the area of the section is therefore 
 
 Trabc 
 
 Jra' + mV + u'c"- 
 
 (11). 
 
r^ 
 
 80 
 
 EXAMPLES. CHAPTER V. 
 
 The directions of the two axes may be obtained by elimi- 
 nating k and Tc from equations (7), (8) and (9) ; we then get 
 
 a" 
 
 f^ 
 
 h' 
 
 I \ 
 
 m fjb 
 
 V 
 
 -^ n V 
 
 = 0, 
 
 or ^/..(J-^,)+m.x(i-y+n\^(^^-p) = 0...(12), 
 
 which united with (5) and (6) gives two sets of values of 
 X, II, V. 
 
 The expression for the area of a section of an ellipsoid by 
 a plane not passing through the centre will be given in a 
 future article. (Art. 79.) 
 
 EXAMPLES. CHAPTER V. 
 
 1. Shew that the two generating lines of the surface 
 
 = ±<^J 
 
 
 are at 
 
 drawn through a point for which z 
 right angles to each other. 
 
 2. Shew that all the points on the surface 
 
 for which the generating lines are inclined at an angle a, lie 
 in one or other of two fixed planes. 
 
 3. Find the angle between the two generating lines of 
 the surface 
 
 
 at the point a, /S, 7. 
 
EXAMPLES. CHAPTER V. 81 
 
 4. If the surface 
 
 {x^ +f^ zj = aV + Iff + c'z' 
 . be cut by a central circular section of the ellipsoid 
 
 x^ if z^ ^ 
 ■ — I- — -I — = 1 
 a" ^ b' ^ e ' 
 
 the sum of the squares on any two perpendicular radii vec- 
 tores of the curve of section is constant. 
 
 5. The equation of a surface can be put into the form 
 x^ + 2/^ + -s^ + (^^ + 'f^y + "i^z —p) Qfx + my + nz — p) = 0, 
 
 find the planes which give circular sections. 
 
 6. Prove that the sections of the surface 
 
 xy + yz + zx = 1, 
 by planes parallel to x -{-y + z = 0, are circles. 
 
 7. If the two generators drawn from a point on the 
 surface 
 
 x^ y^ / , 
 
 — ^- = 1 
 
 a c 
 
 intersect the principal ellipse in points P, P' at the ends of 
 .conjugate diameters, then will 
 
 OF' + OP'' =a' + h' + 2c\ 
 
 8. Find the circular sections of the surface 
 
 a' ¥ "^ c' 
 
 9. Prove that if the section of the surface 
 
 yz zx ^y _ -, 
 a c 
 
 fby the plane Ix + my + w^ = be a rectangular hyperbola, 
 
 — — — = 
 
 la' mb^ nd^ 
 
 A. G. 6 
 
82 EXAMPLES. CHAPTER V. 
 
 10. The angle between the generating lines of 
 
 — J. -Z- 4- - = 1 at the point {x, y, z) is cos r-^ — ~ 
 a c ^"^ \ 
 
 where X, and \ are the two roots of 
 
 a^ v^ z^ 
 
 Ill 
 
 11. Prove that the foci of all centric sections of the 
 
 surface 
 
 ax' + hif-{-cz'' = l 
 
 lie on the surface 
 
 (^2_^_5/2+/)(l_a^^-6/-c/)ja(c-6)V^^^+6(a-c)V^^+c(Z^-a)V/} 
 =(a^.^+62/^+c/)[(c-6)VV+(a-c)Va;'^+(6-a)V2/'^}.|j 
 
 12. Find the equation of a right circular cylinder whose 
 axis is the line 
 
 oc _y _ z ■' 
 
 I m n ' 
 and whose radius is a. 
 
 13. Find the condition that the cone 
 Ax^ + -B/ + Cz" + 2A'yz + 2B'zx + 2C'xy = 
 
 may have three generating lines mutually at right angles. 
 
 14. Find the equation of the right cone which has a 
 centric circular section of the ellipsoid 
 
 x^ y"^ z^ ^ 
 — 4- — -I — = 1 
 a h c 
 
 for its base and its altitude equal to 6. 
 
 15. Find the equation of a right circular cone referred to 
 rectangular axes, having its vertex at the origin, and meeting 
 each of the co-ordinate planes in one line onh\ 
 
 IG. Find the equation of a right circular cone whose 
 
 X II z 
 
 axis is the line t = — = - , and semi-vertical ansrle a. 
 
 I Hi n ° 
 
 17. Find the equation of a right circular cone which 
 contains three given straight lines passing through the 
 origin. 
 
EXAMPLES. CHAPTER V. 83 
 
 18. Find the locus of the points at which the two gene- 
 rating lines of the surface 
 
 Ax' -{- Bi/' + Cz' = 1 
 
 are at right angles. 
 
 19. If a plane be drawn through the straight line 
 
 X _ y _z 
 I 111 n ' 
 
 the two other straight lines in which it cuts the cone 
 [B — G)yz {inz - ny) + (G-A ) zx {iix — Iz) -\-{A—B) xy {ly — mx) = 
 iwill he at right angles to each other. 
 
 20. Shew that any point on the hyperboloid of one sheet 
 I may be represented by the equations 
 
 X = a cos cf) sec 6, 
 
 y = h sin <f> sec 6, 
 
 z = c tan 6 ; 
 
 and find the equations of the generating lines through that 
 <point. 
 
 21. Shew that if the two generating lines at any point of 
 the surface 
 
 be at right angles respectively to those of opposite systems 
 through a second point, the two points are either in a plane 
 through the axis of z or equally distant from the plane of xy. 
 
 22. If two planes be drawn passing respectively through 
 two generating lines of the same system at the extremities of 
 the major axis of the principal elliptic section of a hyperboloid 
 of one sheet and intersecting in any third generating line, the 
 traces of these planes on either of two fixed planes will be at 
 right angles. 
 
 23. If a = 0, /3 = 0, 7 = 0, 8=0 be the equations of the four 
 faces of a tetrahedron expressed as in Art. 26, the equation 
 of a hyperboloid of one sheet passing through two opposite 
 edges is 
 
 Po.p + Q7S + Myi + SjS^ = 0. 
 
 6—2 
 
84 
 
 EXAMPLES. CHAPTER V. 
 
 24. Find the equation of an ellipsoid referred to the 
 planes of its central circular sections and a central plane 
 at right angles to them. If these are rectangular axes, 
 prove that the squares of the axes are in harmonical pro- 
 gression, and that the equation takes the form 
 
 {x + zY + 3/' {x - z") 4- f _ 
 9 ~r 
 
 = 9 
 
 a 
 
 25. Prove that the two generators of the hyperboloid 
 
 x^ if f^ _ -, 
 
 through the point {x, y, z) will meet the principal elliptic 
 section at the ends of diameters at right ang^les if 
 
 ii' + b' " & 
 
 V 
 
CHAPTER VI. 
 
 DIAMETRAL PLANES. 
 
 69. It will be useful to commence the chapter with the 
 I following definitions. 
 
 1. The centre of a surface is a point such that all chords 
 ''passing through it are bisected by it 
 
 2. The locus of the middle jjoints of a system of parallel 
 chords of a surface is called the diametral surface of the 
 system. 
 
 We shall shew that if the original surface be a quadric, 
 the diametral surface of any system of parallel chords is a 
 plane. In this case we shall require the following definition. 
 
 3. A principal plane of a quadric is a plane perpen- 
 dicular to the chords which it bisects. 
 
 We shall shew hereafter that such a plane can always be 
 found. 
 
 70. If a quadric have a centre and be referred to a 
 system of axes luith the centre as origin, the equation will not 
 'Contain any terms of the first degree. 
 
 For the general equation of the second degree is 
 
 \Ax'' + By'' + Cz'' + 2A'yz + 2B'zx + IC'xy 
 
 + ^A"x + 2B"y + 2&Z + i^= (1). 
 
 Then if x^, y^, z^ be the co-ordinates of any point on the 
 fsurface, —x^y ~ Vv ~^i ^^^st also satisfy the equation (1), 
 since the origin is the centre. Hence w^e have 
 
If 
 
 86 DIAMETRAL PLANES. 
 
 Ax^ + By^ + Cz^' + 2A'y^z^ + 2B'z^x^ + iC'x^y^ 
 
 + 2A"x^ + 25"2/i + 2C"2, + i^= 0, 
 
 ^< + %i' + C'^x' + 2^'2/i^i + 25'^,^^ + 20X2/1 
 
 - 2^X - 2^'Vx - 2C"0, + i^= 0. 
 Subtracting we obtain 
 
 4(yiX + ^>i+C''X) = (2). 
 
 This equation must be satisfied for all values of x^, y^, z^ 
 consistent with (1). But unless A" = 0, B" = 0, C" = 0, equa- 
 tion (2) can only be satisfied by the co-ordinates of points 
 lying in the plane 
 
 A"x + B"y + C"z = 0. 
 
 Consequently we must have 
 
 A" = 0, B" = 0, C" = 0, 
 
 or the equation (1) does not involve the first powers of 
 X, y, z. 
 
 Conversely, if the equation of a quadric do not involve 
 the first powers of x, y, z, the origin is the centre of the sur- 
 face. Moreover, if the equation can be put in the form 
 
 Ax' + By''+ Cz' = F (3), 
 
 the axes being rectangular, the co-ordinate planes will be 
 principal planes. For if x^, y^, z^ satisfy the equation (3), so 
 do — x^, 3/j, z^. Hence the plane of yz bisects all ordinates 
 parallel to the axis of x, and similarly for the other co- 
 ordinate planes. 
 
 Conversely, if each co-ordinate plane bisect all chords 
 parallel to the corresponding axis the equation must assume 
 the above form. ^ 
 
 71. To find the locus of the middle points of a system of 
 parallel chords drawn in an ellipsoid. 
 
 Let the eipiation of the ellipsoid be 
 
 X y z* ^ ,_. 
 
 a be 
 
 i 
 
DIAMETRAL PL AXES. 87 
 
 and let the equations of any one of the system of parallel 
 chords be 
 
 x-OL y-^ z-y 
 
 —j—=- = -=^ (2), 
 
 where I, m, n are direction-cosines. 
 
 To find the points where (2) meets (1) we have 
 
 v2 
 
 (g + IrY { 13 + mrf {y + nr)'' _ 
 a' '^ If '^ & " ' 
 
 ovr -1 + 77 
 \a h 
 
 This equation gives two values of ?' wdiich are the distances 
 from the point (a, /3, 7) of the two points where the straight line 
 (2) cuts the ellipsoid. If (a, /3, y) be the middle point of the 
 chord these two values must be equal, and opposite in sign ; the 
 coefficient of r in the equation (3) must therefore vanish, or 
 
 It. niS ny 
 a c 
 
 Hence (2, (3, 7) always lies in the plane 
 
 Ix my nz ^ ... 
 
 -i3+-Tf +^ = (4), 
 
 which is therefore the equation of the locus of the middle 
 [points of the system of chords. . 
 
 72. If x^, 2/p z^ be the co-ordinates of the point in which 
 
 the line V = = ~ meets the ellipsoid, that is, the co-ordi- 
 i m n 
 
 nates of the extremity of the diameter drawn parallel to the 
 
 system of parallel chords, we have 
 
 and the equation (4) of the last article may be written 
 
88 DIAMETEAL PLANES. 
 
 Also if x^, y^, z^ be the co-ordinates of any point in the 
 curve in which this plane cuts the ellipsoid, we have 
 
 ■ — a" + TT "i — r — ^y 
 a c 
 
 which shews that the point (^,, y^, z^ lies ia the plane 
 which bisects all chords parallel to the diameter through 
 
 1*^2' 2/2' ^i)' 
 
 The planes which bisect chords parallel to the two diame- 
 ters through (.r^, y^, z^, {x.^^ y.^, z^ will intersect in a straight 
 line. Let the co-ordinates of the point where this line meets 
 the ellipsoid be x^, 3/3, z^. Then since {x^, y^, z^ lies in the 
 plane which bisects chords parallel to the diameter through 
 i^v Vv ^^ we have 
 
 ~8 ' 7,2 ' 2 ^> 
 
 a c 
 
 and since it lies in the plane which bisects chords parallel to 
 the diameter through (x^, y^, z^, we have 
 
 These last equations shew that {x^, y^, z^, (x^, y^, z^ both 
 lie in the plane which bisects all chords parallel to the diame- 
 ter through (^3, 2/3, z^. 
 
 Hence the three diameters have this property, that the 
 plane through any two of them bisects chords parallel to the 
 third. 
 
 The three diameters are called conjugate diameters. 
 
 73. The equation of the ellipsoid luhen referred to a sys- 
 tem of three conjugate diameters as axes assumes tlie form 
 
 x' y' z' -, 
 u • - -I = 1 
 
 where a', b', c' are the lengths of the conjugate semi-diameters. 
 
 For the equation must be of the second degree by Art. 48, 
 and since each co-ordinate plane bisects chords parallel to 
 the corresponding axis, by Art. 70 the equation must assume 
 the form 
 
 Ax'-^Bf-+ Cz' = F. 
 
DIAMETRAL PLANES. 
 
 89 
 
 When the axis 
 
 of X meets the surface we have 
 
 
 x = a , y = 0, 2 = 0, 
 
 and therefore 
 
 F 
 a =~^ . 
 A. 
 
 Similarly 
 
 F 
 
 ^, 
 
 
 F 
 
 c ^. 
 
 And the equation 
 
 becomes 
 
 - 
 
 x' y' z' , 
 4-4- =1 
 
 74. The co-ordinates of the extremities of three conju- 
 gate diameters are connected by the relations 
 
 x^ v'^ 2^ "^ 
 
 ^2+^4-^-1 = 
 a: b' c 
 
 ^l , 2/2! , ^' _ 1 ^ 
 2 -t- ^. + ^2 
 
 a 
 
 ■^ "^ 6-^ "^ c' ~ ^ 
 
 
 
 .(1), 
 
 
 y,v^ ^ M? ^ 
 
 n 
 
 2/32/1 
 
 c 
 
 -T^-" + -^2 + jr — ^ r 
 
 a 
 
 55 , 2/1I2 4. 55 = 
 
 a^ "^ 6-^ "^ e 
 
 J 
 
 .(2). 
 
 Squaring all these equations, and adding twice the squares 
 of the second three to the squares of the first three, we get 
 
 I 
 
 J2^''^^^ 4. 2/22/3. VsV 4.9^^3^^ 4- Ml 4_55V+2f55+.M^+55y=o 
 
 * 7/^ Z^ V 
 
 
 fx^ y^ , ^„^ 
 V a 6 c 
 
\ci 
 
 00 DIAMETRAL PLANES. 
 
 Expanding, and rearranging the terms we get 
 
 V ic be ' be J '^ \ ca ca ca J \ ab ab ab J 
 I Whence x^^ + a:/ + o-g^ = a^\ 
 
 yr'^u: + ys' = ^'\ (3), 
 
 ^1^1 + ^2^2 + ^3^3 =" I (4). 
 
 ^i2/i + ^22/2 + ^32/3 = ^ ) 
 This transformation can be easily seen to be equivalent to 
 
 that affected in Art. 44, usins: -^ for L and so on. And the 
 
 ° a ^ 
 
 method of that article may be employed to deduce (3) and 
 
 (4) from (1) and (2). 
 
 Similar relations exist between the direction-cosines of 
 the normals to the three planes, each of which bisects chords 
 parallel to the intersection of the other two. For if l^, m^, n^ 
 be the direction-cosines of the normal to the plane bisecting 
 chords parallel to the line 
 
 wc have 
 
 or 
 
 X _y 
 
 z 
 
 ^1 2/1 
 
 h' 
 
 L -!i 
 
 ='ii 
 
 ^1 2/1 
 
 a' b' 
 
 c' 
 
 al. bm. 
 
 en. 
 
 K^ = ^BB^ 
 
 
 5i .^i 
 
 a b 
 
 c 
 
 and similar relations for l^, m^, n^. Whence equations (2) 
 easily give 
 
 ■^ a%/3 + Z^^;y??3 + c-/?,??3 = 0l (5), 
 
 d'lj^ + b'm^m.^ + c' 
 
 »,??3 - u > 
 n^n^ = Oj 
 
DIAMETRAL PLANES. 91 
 
 and obviously also 
 
 l^' + m^^ + n^^ = 1] 
 
 C + < + < = l (^)- 
 
 75. From equations (3) of the last article we obtain by 
 addition 
 
 a" + h"-\-c" = a'-\-h' + c' (1), 
 
 iwhere a, h', c are the lengths of the conjugate semi-diameters. 
 
 Let \, fji, V be the angles between {h\ c), (c, a) and 
 {a, h'), respectively. 
 
 Then since the direction-cosines of a referred to the prin- 
 cipal axes of the ellipsoid are — J,^,-!, and similarly for 
 ^ ^ a a a "^ 
 
 those of h\ c, we have, by Art. (8), 
 
 smX- ^^, , 
 
 But we have ^ . -^ + f-^ f^ + -V -^ = 0, 
 
 a a b b c c 
 
 .-Ti'i ~r • ^5 
 
 a a c> t> c c 
 
 
 ^1 
 
 2/i 
 
 
 
 ^ 
 
 
 
 a 
 
 6 
 
 
 
 c 
 
 
 y^^ 
 
 3 - 2/3^3 
 6c 
 
 ^2-^3 - 
 
 ca 
 
 ^3^2 
 
 ^2^. 
 
 a6 
 
 ^33/2 
 
 2 2 VS 
 
 Vlz^Zl^S^ + /fl^Jlis^y' + /^ 23/3 - ^3^2 
 
 a^ "^ i-^ "^ c^ 
 
 = + 1 
 
 by equations (1) and (2) of the last article. 
 
 / 
 
92 DIAMETRAL PLANES. 
 
 ' Hence 6'V^ sin^X = 6V^' + cVf^V a'Z> 
 
 a b" c 
 
 27,2 ^1 
 
 Similarly cV s'm'/i = hV '-^ + cV |V + a'^i' 
 
 
 a b' c 
 
 Adding, we get 
 (b'c sin \y + (c'a sin ya)^+ (a'6' sin vy= bV + cV+ a^Z>\ . .(2). 
 
 Again, if _/? be the peipendicular from the point (^3,2/3, z^ 
 on the plane which contains a' and b', whose equation is 
 
 ' L2 "T ^^ '^J 
 
 d' ' b' 
 
 we have p = 
 
 ^' 4. .^^ 4. - 
 g'^ "^ b' "^ c'^ 
 
 .a^ "^ 6^ "^ c^ 
 1 
 
 .a' b^ c 
 
 4 
 
 Hence squaring and multiplying by the value previously 
 obtained for a%'^ sin" v we jret 
 
 2 "27 '2 ' 2 ?7 2 2 /0\ 
 
 p a b sm v = a b c (3). 
 
 But db' sin v is the area of the parallelogram whose edges 
 are a and b\ and pcib' sin v is the volume of the parallel- 
 epiped whose base is this parallelogram and whose altitude is 
 2), that is, the volume of the parallelepiped whose three edges 
 are a, b', c. 
 
 By Art. 14 this volume can be expressed in the form 
 db'c J\ — cos^ \ — cos' jjb — cos'^ v+ 2 cos \ cos /i, cos p. 
 Hence this expression is equal to abc. 
 
 70. Another method of obtaininq- these relations is 
 afforded by the consideration that the expression 
 
 ~, + f + '^. + H-^' + r + ^) 
 
DIAMETRAL PLANES. 93 
 
 is transformed by taking three conjugate diameters as axes to \. 
 the expression 
 
 a? v"^ z^ o . ^■ 
 
 — + r^a + "^2 + ^^ (^' + 3/^ + -^'"^ + ~]}^ cos A, + 2ir.r cos/x + ^xAj cos v). ' 
 
 Consequently, if for any value of k the first expression split 
 up into two linear factors, the second expression will do so 
 likewise for the same value of h. 
 
 By Art. 49 the requisite values of h for the two exjDres- 
 sions are given respectively by the equations 
 
 '^ + S ^ + 6~=) ^ + ?) = *^' 
 
 and 
 
 1^ (^+A-.)(fc + l)(^-+-l) 
 
 — ^'" COS" \ [h -\- -,A — W" COS- ^[h ^r -j-izA—h'^ cos^i/ f k + — ^ j 
 
 + 2^^^ cos \ COS /A cos z^ = 0, I 
 
 which when cleared of fractions and expanded become re- 
 spectively, 
 
 a'h'c'k^ + {a%^ + ¥0" + cV) ^^' + (a' + 5' + c') ^ + 1 = 0, 
 
 and 
 
 a%V (1 — cos^ \ — cos^ fz — cos^ z^ + 2 cos X cos fju cos z^) F 
 
 + (6''c'' sin' X + c'a' sin' /^ + a'b" sin' z;) ^'' 
 
 + (a'' + 6'' + c") k+l = 0. 
 
 And since these equations are identical we get the rela- 
 tions (1), (2) and (3). 
 
 They can also be obtained geometrically by a series of 
 transformations ; or by finding the values of the maximum 
 radius vector of the surface when referred to three conjugate 
 diameters as axes. The result will be a cubic equation in r', 
 and the three values of r^ will be a', ¥, c"; whence the 
 values of 
 
 a'bV, a'6' + cV+6V, a'+b' + c' 
 are known in terms of a, b', c', ^^ 
 
94 DIAMETRAL PLANES. 
 
 The formulae obtained in Arts. 71 — 76 hold for the othei 
 central surfaces if the proper changes be made in the signl 
 of a^ If and cl 
 
 77. The equation of the plane -which bisects all chords 
 of the ellipsoid parallel to the line 
 
 ^=^=^ (1) ' 
 
 a^i »/, ^i 
 
 is xA,+y.ll + zA, = Q (2). 
 
 Conversely the chords which are bisected by the plane 
 
 Ix + my -\- nz = (3) 
 
 are parallel to the line 
 
 — = -1— = -- (4.') 
 
 d'l b'm chi ^ ^' I 
 
 The line (4) is said to be conjugate to the plane (3). 
 
 By Art. 72 every system of chords parallel to any line 
 which lies in the plane (3) is bisected by some plane passing 
 through (4). 
 
 Hence the plane passing through the origin which bisects' 
 any system of parallel chords of the section of the ellipsoid 
 by a plane 
 
 lx-\- my -\- nz — p = (5) 
 
 parallel to (3), must contain the straight line (4). Whence 
 it easily follows that the point Avhere (4) meets (5) is the 
 centre of the section of the ellipsoid made by (5). The co- 
 ordinates of this centre are therefore given by 
 
 X ^ y__ z^_ _lxj\-7ny -\-nz ^ p . 
 
 dH ~ I'm ~ c'n ~ a'F+bW + c V " ^^FT6W"TcV " ' ' ^ ^' 
 
 78. The co-ordinates of the centre of the section of the 
 ellipsoid : 
 
 SL^ 11^ z^ 
 
 -. + f. + ^ = l (1) 
 
 a he ^ ^ 
 
 by the phmc Ix + my + nz = p (2) 
 
 can also be obtained in the foUowincr manner. 
 
DIAMETRAL PLANES. 95 
 
 Let a, /8, 7 be the co-ordinates required, and let \ 
 
 oc-cc^y-^^z-y^^^ \ 
 
 \. be the equations of any straight line drawn in the plane (2) 
 I to meet the ellipsoid, r being the length of the radius vector. 
 
 Then if x, y, z be the co-ordinates of the point where (3) 
 
 meets (1), we have from (3) 
 
 cc = a + Xr, 2/ = /3 4- ^r, ^ = 7 -F it, 
 and therefore from (1) by substitution 
 
 ^ But if a, /3, 7 be the co-ordinates of the centre of the 
 f'' section of (1) by (2), the two values of r given by (4) must 
 
 be equal in magnitude and of opposite sign for all straight 
 
 lines lying in (2) ; that is, we must have 
 
 A,a /^^ 1^7 
 
 -^ + 70 + rr = ^ \S>) 
 
 U a- Ir jc' ^ ^ 
 
 for all values of \, [i, v consistent with the equation 
 
 \l -i- /jLVi + vn = (6), 
 
 hich is the condition that (3) may lie in (2). 
 
 Hence the equations (5) and (6) must be identical, or we 
 have 
 
 Id' mb'' 7ic" 
 ti and as in the last article each of these fractions 
 
 ^ P _ 
 
 \\ w 
 
 a'P -b b'TTi' + c'n 
 
 ••^^,2 * 
 
 79. The equation (4) of the last article, when the values 
 of a, /3, 7 are substituted in it, becomes 
 
 1 
 
 2 2 2\ A^2 
 
 a^ y^' v'\ . p' 
 
 Comparing this with equation (4) of Art. 68 we see that 
 
96 
 
 DIAMETEAL PLANES. 
 
 if Tj be the central radius vector which is parallel to r, w( 
 have 
 
 
 Consequently, since the areas of similar figures are pro- 
 portional to the squares of any corresponding lines in the 
 figures, if A be the area of the section of (1) by (2), and A\ 
 the area of the parallel central section, . 
 
 A = A^\1 
 
 irahc 
 
 P' 
 
 ciH- + b'm' + c'li 
 
 •^^.^\ 
 
 1 - 
 
 F 
 
 80. The result of the last article can also be obtained iri| 
 the followinfij manner. 
 
 Let a, j3, y be the co-ordinates of the centre of the sec-| 
 tion. Then the equation 
 
 " "^ b^ "^ c" ~ 
 
 a 
 
 (1) 
 
 represents an ellipsoid whose centre is at (a, /?, 7), and whos( 
 semi-axes are ka, kh, kc. 
 
 At the points where this cuts the given ellipsoid we have 
 by subtraction 
 
 Or, putting for a, fi, 7 their values from equation (6) of| 
 Art. 77, 
 
 2 {Ix -f- my 4- nz) 
 
 a'P+hV+c'n'' 
 
 f 
 
 +1-^1, 
 
 and if this ecjuation be identical with 
 
 Ix + my + nz = J) (2), 
 
 the sections of the two ellipsoids by this latter plane willj 
 V coincide. 
 
DIAJMETRAL PLANES. 97 
 
 The condition for this is 
 
 l-k' = 
 
 F 
 
 ar + h%i' 4- c'n' 
 
 2 7 
 
 .'.¥ = 1- ^' 
 
 ■ But the area of the section of (1) by the plane (2) whicli 
 passes through its centre, by Art. (68) 
 
 _ irh^ahc _ nrahck^ 
 
 irahc f^ p^ 
 
 ,2^2 i 
 
 
 Jeer + 6W + cV I ci'l" + &'^?i' + cV 
 which is therefore the area required. 
 
 81. It can be shewn by an investigation similar to that 
 in Art. 71, that the locus of the middle points of a system 
 of parallel chords of the surface 
 
 By'' + (7/ = X, 
 
 ^Yhose direction-cosines are I, m, n, is 
 
 2Bmy + 2Gnz = I. 
 
 Also the equation of the surface, when two diametral 
 planes and a plane through the point where their line of 
 ntersection cuts the surface, parallel to the two systems of 
 chords bisected by them, are taken as planes of zoo, xy and yz 
 •espectively, will assume the form 
 
 By + V = X, 
 
 :yhere B' and C have the same or opposite signs according 
 IS B and G have. 
 
 We shall however at once proceed to the more general 
 problem. 
 
 82. To find the locus of the middle jjoints of a system of 
 Parallel chords in any quadric. 
 
 Let the equation of the surface be 
 Ax" + B^f + Cz' + 2A'yz + 2B'zx + 20' xy 
 
 ■\-2A"x + 2B"y + 2C"z + F=0 (1), 
 
 vhich we will denote by F [x, y, z) = 0. 
 
 A. G. 7 
 
98 
 
 And let 
 
 DIAMETKAL PLANES. 
 
 I m n 
 
 (2), 
 
 or 
 
 i^(a,A7) + |z-^-+m 
 
 n^J\r + Pr^ = 0...{^), 
 
 d^ 
 
 be the equations of any one of the system of parallel chords.! 
 
 To find the points where (2) meets (1) we must substitute; 
 a + h\ /3 -f- mr, y + nr for x, y, z in (1). We thus get 
 
 i^ (a 4- h\ p + mr, y + nr) = 0, 
 dF 
 dl3 
 
 where -r- , -ttsj -i- are the partial differential coefficientsj 
 do. dp dy j 
 
 of F (a, /3, 7) with respect to a, /3, 7 respectively and P is] 
 
 some function of I, m, w. 
 
 The equation (3) gives two values of r, which are the! 
 distances from (a, /3, 7) of the two points where the line (2)J 
 cuts the surface (1). If (a, /3, 7) be the middle point of th( 
 chord these two values must be equal and opposite in sign,! 
 and the coefficient of r in the above quadratic must vanish ; 
 
 ,dF dF dF ^ 
 i -^ — h 111 -T7^ -\- n -^ — 0, 
 
 d 
 
 a 
 
 d^ dy 
 
 or writing out the values ^^ -f- y -fo and -v- , and rearrang-] 
 
 a{Al+C'm + B'n) + ^(C'l-\-Bm + A'n) + y{El + A'm-\-Gn) 
 
 + A"l-\-B"m + G"n = 0,\ 
 
 which shews that the locus required is a plane. 
 
 83. The diametral plane wdll not in general be perpen- 
 dicular to the chords which it bisects. There are however 
 certain directions of the chords for which this is the case- 
 Let us suppose I, m, n to be the direction-cosines of any chord 
 of the system. 
 
 The equation of the diametral plane is therefore by the 11 
 last article, 
 
 x{Al-{- Cm + Bn) + y [C'l + Bm + An) +z{Bl-\- Am + On) 
 
 + A"l + B"m+ C'n^O. 
 
DIAMETRAL PLANES. 
 
 99 
 
 If this plane be perpendicular to the system of chords we 
 must have, by Art. 23, 
 
 j^l + C'm + B'7i C'l + Bm + A'n B'l + A'm + On 
 
 I 
 
 m 
 
 n 
 
 (1). 
 
 Let each of these fractions be put equal to some quantity 
 s. We have then 
 
 (A-s)l + C'm + B'n = 0\ 
 
 C'l'\-(B-s)m-]-An = ol 
 
 B'l + A'm + (C-s)n = 0] 
 
 Whence eliminating I, m, n, we get 
 
 [A-s), G\ B' 
 
 G\ {B-s), A' 
 B\ A', (C-s) 
 
 or (A - s) (B - s) (C -s)- A'' (A -s)-B"{B-s)-C'(G-s) 
 
 + 2A'EG' = (2). 
 
 This cubic equation will certainly give one real value of s, 
 and the corresponding values of I, m, n are known from any 
 two of the three equations (1). From the second and third 
 we get 
 
 = 0, 
 
 m 
 
 n 
 
 (3), 
 
 A'B' -a [C- s) AV -B'{B-s)' 
 
 or m [A' C -B'(B- s)} = n {A'B' - C {C - s)] 
 
 = l[BC'-A'{A-s)]... 
 by symmetry. 
 
 And when the value of s is known, equations (3) give the 
 corresponding values of I, m, n. 
 
 In Todhunter's Theory of Equations, Art. 176, it is shewn 
 ithat all three roots of the cubic are real. 
 
 The equation (2) is frequently called the discriminating 
 3iibic of the quadric (1). 
 
 o 
 
 •^ 
 
( 100 ) 
 
 EXAMPLES. CHAPTER VI. 
 
 1. If A^, A^, A^ be the areas of the sections of the 
 ellipsoid 
 
 x^ f z" , 
 — V — -\ — = 1 
 
 made by planes perpendicular to any three generators of the 
 cone 
 
 cc' (a' - d') 4- y' {h' - d') + z' (c^ - cZ') = 0, 
 
 and if p^, 2\, p^ be the perpendiculars on the planes from the 
 origin, then 
 
 A (p: -Pi) + ^. ivi-p") + ^s (p: -Pi) = 0. 
 
 2. Find the locus of the centres of sections of an ellip- 
 soid, the areas of which are always in a constant ratio to the 
 areas of the parallel central sections. 
 
 8. OX, OM, ON are conjugate semi-diameters of an 
 ellipsoid ; x^^ y^, z^ the co-ordinates of L ; x,^, y^, z^ and 
 x^, 2/s' -^3 those of M and N respectively. Prove that the 
 equation of the plane LMN is 
 
 ^2 K + x^ -f x^) + y, (y^ -{- y^ + 7/3) + ^2 {z^ + ^^ -f ^3) = 1 . 
 
 4. Find the area of the section of the ellipsoid by the 
 plane LMN in the last example. 
 
 5. OL, OM, ON are conjugate semi-diameters of an ellip- 
 soid ; a perpendicular is drawn from on the i^lane LMN 
 meeting it at Q ; and a diametral plane is drawn parallel 
 to the plane LMN. Shew that the cone which has its vertex 
 at Q and for its base the section of the eUipsoid by the, 
 diametral plane, is of constant volume. 
 
 '4 
 
 G. Find the locus of the directrices of all sections of an 
 ellipsoid made by planes passing through the least axis. 
 
 I 
 
EXAMPLES. CHAPTER VI. 101 
 
 7. Shew that a straight line parallel to the least axis of 
 an ellipsoid will be the directrix of two plane sections of the 
 ellipsoid, provided the straight line be situated between two 
 definite cylindrical surfaces. 
 
 8. Find the locus of the centres of sections of an ellipsoid 
 made by planes at a constant distance from the origin. 
 
 9. If A, B, C be the areas of any three conjugate dia- 
 metral sections of an ellipsoid ; X, Y, Z those of the sections 
 made by planes respectively parallel to them and intersecting 
 in a point on the surface, prove that 
 
 X Y Z ^ 
 
 ■ — I 1--=2. 
 
 ABC 3 i \'> > \ ->' 
 
 10. Any generating line of the cone 
 
 Psc^ + Qif + Rz^ = 
 
 being taken, a plane is drawn diametral to it with respect to 
 the surface 
 
 Ax"" + Bif + Cz" = 1. 
 
 Shew that the principal axes of the sections of the latter 
 surface by such planes all lie on the surface 
 
 -^ {{A -B)f+{A- C) zf + -^- {[B - G) z'^ + {B - A) ^^ 
 
 + ?^,[{G-A)x'+{G-B)ff = 0. 
 
 11. Find the co-ordinates of the centre of the section of 
 the surface 
 
 Bf -\-Gz^ = x 
 
 ■made by the plane Ix + my + nz = p. 
 
 Find the locus of the centres of all sections made by 
 planes passing through a fixed point. 
 
 12. If in question 3, the point L remain fixed, shew 
 that the perpendicular from the origin on the plane LMN 
 describes the cone 
 
 a/x' + hY + c'^' = 3 [xx^ + yy^ + ^^J^ 
 
102 EXAMPLES. CHAPTER VI. 
 
 13. If the plane Ix + my + nz =p cut the surface 
 
 2 "• 7,2 T5 -*• 
 
 in a parabola, prove that 
 
 a^V" + Wm"" - cV = 0. 
 
 14. Corresponding points on an ellipsoid of semi-axes 
 a, 6, c and a sphere of radius r, being defined by 
 
 X _x y _y ^ ^^' 
 a r ^ h r ^ c r ' 
 
 then, if, .61P, and Op be corresponding radii of the ellipsoid 
 •: •* /and^ '■t'he. -slphere, Oq and Or any two radii of the sphere 
 
 '. perpendicular, to '0P, prove that Oj) will be perpendicular to 
 : *..•'. QQ, 'a:ad pfi^, ,ih.(iX'^dii of the ellijDsoid corresponding to Oq 
 
 ' and'0?\ 
 
^ 
 
 CHAPTER VII. 
 
 V 
 
 THE GENERAL EQUATION OF THE SECOND DEGREE. 
 
 84. The general equation of the second degree can be 
 written 
 
 Ax' + Bf + Cz' + 2A'y2 + 2B'zx + 2Cxij 
 
 -{-2A"x + 2B''y + 2C''z + F = (1), 
 
 which we will denote by F {x, y, 2^) = 0. 
 
 The object of the present chapter is to examine the 
 nature of the different surfaces represented by (1), and the 
 conditions that it may represent any particular kind of 
 surface. 
 
 We shall first examine whether the locus represented by 
 (1) has a centre. 
 
 If it has a centre and this point be taken for origin we 
 ' know, by Art (70), that the terms of the first degree must 
 disappear. 
 
 Assume a, ft 7 as the co-ordinates of the centre. The 
 equation w^hen the origin is transferred to this point is ob- 
 tained by substituting in (1) x -\-(i,y -\- ^, z -\-^ for x, y, Zy 
 respectively (Art. 43), and is therefore 
 
 F{x'-\-a, y' + (3, / + 7) = 0, 
 which can be written 
 
 rr/ a N 'dF ,dF ,dF 
 F(a,^,y)-,x^^y^^ + z^ + ...=0, 
 
104 THE GENERAL EQUATION 
 
 the remaining terms being of the second order in a/, y\ z . 
 
 and -;— , -T7., ^- navmsf the same meaning^ as m Art. 82. 
 da' dl3' dy ^ ^ 
 
 If the coefficients of x, y\ z vanish, we have 
 
 ^^=0 ^^=0 ^=0 
 cZa ^' fZ/3 ^' dy ' 
 
 or writing them out at length, 
 
 Aci+ C'p +B'y + A" = 0\ 
 
 C'a+Bp +A'y + B" = o\ (2). 
 
 B'oL-\-A'^+Cy +C"=0j 
 
 These equations determine a, yS, 7. We get from them 
 
 ___ ^ A"{A"-BC) ■^B"{GG'-A'B') + C"{BB'- C'A') 
 "~ -^— - ABG + 2A'BG'-AA"-BB"-GG'^ 
 
 A" 
 
 G^ E 
 
 A" 
 
 B A' 
 
 G" 
 
 A' G 
 
 A 
 
 G' B' 
 
 a 
 
 B A' 
 
 E 
 
 A' G 
 
 A"{GG'-A'B') + B" jB'^- GA) + G'\AA'-B'G ') 
 femiilarly (:i - ABG+2A'B'G'-AA"-BB"-GG" 
 
 ^ A"(BB'-G'A')+B"{AA'-BV') + G" jG" - AB) 
 ^~ ABG-\-2A'B'G'-AA''-BB"-GG" 
 
 We can therefore always obtain finite values of a, (B, 7 except 
 when 
 
 ABG + 2A'B'G' -AA"- BB'- - GG" = 0, 
 
 in which case the surface has not a centre unless the nume- 
 rators of the above three fractions vanish, when the values 
 of a, /S, 7 become indeterminate ; the reason of such inde- 
 terminateness being that the three equations (2) are not all | 
 independent. (Todhunter's Algebra, Arts. 214, 215.) 
 
 If the denominator do not vanish the surface has a centre 
 whose co-ordinates are given by (3). 
 
 It may be noticed that the equations (2) are the con- 
 ditions tliat the point (a, /S, 7) shall lie in the diametral plane 
 to all systems of chords. (Art. 82.) 
 
OF THE SECOND DEGREE. 105 
 
 85. We see from the last article that it is not always 
 
 : possible to get rid of the terras involving x, y, z. We shall 
 
 now shew that it is always possible to simplify the equation 
 
 by transformation so as to get rid of the terms involving yz, 
 
 zx and xy. 
 
 By Art. 83 we know that there is at least one system of 
 parallel chords which is perpendicular to its diametral plane. 
 
 Let a straight line parallel to these chords be taken as the 
 axis of z and let the transformed equation be 
 p^2 ^ g^2 _j_ ^^2 ^ ^-p>y^ ^ 2 Qzx + "iExy 
 
 + "iF'x + 2Q'V + 2i2"^ + ^ = 0. 
 
 The direction-cosines of the chords which are perpen- 
 dicular to their diametral plane are given by the equations 
 
 Fl + -R'm + qn = si, 
 
 R'l + Qm + P'n = sm, 
 
 Q'l + P'm + Rn = sn. 
 
 But since these chords are parallel to the axis of z, these 
 equations must be satisfied by 
 
 1 = 0, m = 0, n = 1. 
 
 Whence we get Q' = 0, P' = ; and the equation of the 
 surface is 
 
 P^' + Q/ + Rz' + 2Exy + 2P"x + 2Q"y + 2R"z -i-F=0. 
 
 Turning the axes of x and y in their own plane through 
 an angle 6 given by the equation 
 
 2R' 
 
 tan 26 = p — -^ (Todhunter's Conic Sections, Art. 271), 
 
 the term involving xy disappears, and the equation assumes 
 the form 
 
 Px' + Qy' + Rz' + 2P"x + 2 Q'y + 2E'z + F=0. 
 
 The equations which determine the directions of the 
 principal diametral planes are now satisfied by Z = 1, m = 0, 
 n = 0, or by l=zO,m = l,n = 0. Consequently each of the 
 axes of X and y as well as that of z is parallel to one of the 
 three lines determined by equations (1) of Art. 83. 
 
106 THE GENERAL EQUATION 
 
 We thus have an independent proof that these three] 
 directions are all real and at right angles to each other. 
 
 86. We have now shewn that by a proper choice ofl 
 axes the terms involving yz, zx and xy can be made to-' 
 disappear. It remains to explain how the coefficients of 
 the different terms in the resulting equation can be de-^ 
 ter mined. 
 
 Let Zj, m^, n^ ; l^, m^, n^ ; l^, m^, n^ be the direction- 
 cosines of the new axes. These values all satisfy the equa-l 
 tions (1) of Art. 83. Let s^, s^, s^ be the corresponding] 
 values of s. 
 
 By Art. 44 the required transformation will be effected, 
 by substituting for co, y, z the expressions 
 
 respectively. If therefore the original equation be 
 
 Ax^ + By^ + Cz"" + 2A'yz + 2B'zx + 20' xy 
 
 + 2A"x + 2B'\j + 20" z + F=0, 
 
 the coefficient of x'"^ in the result will be 
 
 Ai; + Bm^' + On^' + 2A'm^n^ + 2B'7iJ^ + 20\m^. 
 
 But from Art. 83 we have 
 
 Al^-\- Cm^ + B'n^ = sJ^, 
 O'l^ + Bm^ -{■ A'n^=^ s^m^, 
 B'l^ 4- A'm^ + Cn^ = s^n^. 
 
 Multiplying these equations by l^, ii\, n^, respectively, 
 and adding, we get 
 
 Al^' + Bm^' + On; + 2A'm^n^ + 2B'nJ^ + 20'l^m^ = 5,. 
 
 Hence P the coefficient of x'^ is 5^. Similarly Q = s^, 
 R = S3, or P, Q, 11 are the three roots of the discriminating 
 cubic. 
 
 It follows from this that the coefficients of the discrimi- 
 nating cubic remain unaltered in value however the axes 
 may be turned about the origin. 
 
I 
 
 OF THE SECOND DEGREE. 107 
 
 The results of this article have been already obtained by 
 a different method in Art. 51. 
 
 87. It is easy to verify that the coefficients of yz\ z'x 
 and x'y disappear ; since \, 7ii^, 7\ ; l^, m^, n^ ; l^, m^, n^ are 
 the direction-cosines of lines such that any one is parallel to 
 each of the planes which bisect chords parallel to either 
 of the others, and thus l^, m^, n^, l^, m^, n^, satisfy the 
 relation 
 
 AIJ^ + Bm^m^ + Cn^n^ + A' {m^% + m^nj 
 
 + B' {nj.^ + 7^,y + C {l^m, + l^m^) = 0, 
 
 and the expression on the left-hand side of this equation is 
 the coefficient of x'y' in the transformed equation. 
 
 The coefficients of x, y' and z in the transformed equa- 
 tion will be 
 
 ^ and 2 {A'\ -t- F'm^ + G'\), 
 
 respectively, and the constant term remains unchanged. 
 
 88. The equation when transformed to 
 Pcc^ + Qif + R^^ + 2P"x -h 2Q"y -F 2R"z + F=0 
 
 can be farther simplified by a change of origin. 
 
 \ Suppose first that none of the quantities P, Q, R vanish, 
 i| that is, that none of the roots of the discriminating cubic 
 l| vanish, which will be the case if the constant term of the 
 i! cubic, or 
 
 ABC + 2A'B'C' - AA" - BB" - CO", 
 be different from zero. 
 
 In this case the equation can be written 
 
 I V ' PJ ' ^V ' QJ ' ^'V ' R 
 
 12 
 
 F" Q'" R'" ^ ^, 
 F ^ Q ^ R ' 
 
108 THE GENERAL EQUATION 
 
 (- 
 
 and transferring the origin to the point whose co-ordi 
 nates are 
 
 P" _q'_ _Br 
 
 P' Q' R 
 
 this becomes 
 
 This represents an ellipsoid, a hyperboloid of one or two 
 sheets, or an impossible locus, respectively, according as the 
 
 w w w 
 
 quantities p , -^ , -^ are all positive, two positive and one 
 
 negative, one positive and two negative, or all negative. 
 
 Thus unless 
 
 ABC + ^A'B'C -AA"- BB" - CC" 
 
 vanish, the surface has a centre and is one of the surface" 
 whose equations we have already investigated. 
 
 Now if we had first changed the origin to be the centre, 
 we should have got rid of the terms of the first degree, and 
 the equation would have been 
 
 Ax^ + Bif + Cz' + 2A'yz + 2B'zx + IC'xy = F' (1), 
 
 which by turning round the axes would become 
 
 1^0? + qif + Rz^ = P', 
 
 and consequently, if F be positive the surface (1) will re- 
 present an ellipsoid, a hyperboloid of one or two sheets, 
 or an impossible locus according as the roots of the dis- 
 criminating cubic are all positive, two positive and one 
 negative, one positive and two negative, or all negative. 
 If F' be negative the order of the statement must be 
 reversed. 
 
 89. If F vanish the surface is a cone. Now returning 
 to Art. 84 we see that F = — F {a, j3, 7), where a, /3, 7 are 
 determined from the equations 
 
 ^a + C"/3+P'7 + ^" = 
 
 0'a+ 7i/3 +.4'7 + i^" = o[ (2). 
 
OF TBE SECOND DEGREE. 109 
 
 Multiplying the first of these by a, the second by p, the 
 ihird by 7 and adding, we get 
 
 ^a^ + B^' + Cy + 2.4 '/37 -h 2B'r^ + 2G'ol(3 
 
 + A"oL + B"^ + C"y = 0. 
 3ut 
 
 ^'/ + BIB^ + Cy^ + 2J.'y57 + 25'7a + 2C"^/3 
 
 + 2^ "a + ^B"13 + 2(7"7 + i^ = i^ (a, A 7) = - i^'. 
 
 Subtracting the first of these from the second, we get 
 
 -F =A"a + B"^ + C"y + F. 
 Hence if the surface be a cone 
 
 And eliminating a, /3, 7 between this equation and the 
 ihree equations (2), we get as the condition that the surface 
 epresents a cone 
 
 A C B' A" 
 C B A' B" 
 B A' C C" 
 A" B" G" F 
 
 = 0. 
 
 90. Suppose, secondly, that one of the quantities P, Q, R 
 vanishes, as P. From this it follows that the constant term 
 }i the cubic in s must vanish, or 
 
 ABC + 2A'EC' - AA" - BB" - CC' = 0, 
 
 vhich we saw in Art. 84 indicated that there was not a 
 lefinite centre. 
 
 The equation becomes 
 
 Qif + Rz' + 2P"^' + 2Q''y + 2R"z + P = 0, 
 
 md by changing the origin we can get rid of the terms 
 i n y and z, and the constant term ; the equation thus 
 becomes 
 
 I Qif + Rz' 4- 2P"^ = 0, 
 
 vhich represents an elliptic or hyperbolic paraboloid ac- 
 
110 THE GENERAL EQUATION i 
 
 cording as Q and R have the same or opposite signs, o 
 according as 
 
 which is the coefficient of s in the cubic, and therefore 
 equal to the product of the two finite roots, is positive 0]| 
 negative. 
 
 i/{^ 91. Thirdly, let two of the quantities P, Q, R vanish 
 which necessitates the two conditions, 
 
 ABC + ^A'B'C - AA" - BB" - CC" = 0, 
 
 BG + GA+AB-A"-B"-C' = 0. 
 
 The equation now becomes 
 
 Rz^ + 2P''a) + 2 g'y + 2R"z + F=0. 
 
 And by changing the origin, the term involving z and the 
 constant term may be removed, and we get 
 
 Rz' + 2P"a) + 2Q'y = 0. 
 
 By turning the axes of oo and y round in their own plane, 
 the equation can be reduced to the form 
 
 Rz' + 2P'''a) = 0, 
 
 which represents a parabolic cylinder whose generating lines 
 are parallel to the axis of y. 
 
 The two conditions 
 
 ABC + 2A'B'C' - A A'' - BB" - CC = 0, 
 BC + CA -^ AB - A" - B" - a' = 0, 
 
 can be replaced by simpler ones. For the first equation is 
 equivalent to either of the forms 
 
 {GA - B") (AB - C"-') = {B'C - AA')\ 
 
 (AB - C") {BC - A") = {C'A' - BB')\ 
 
 {BC - A"') {CA - B") = {A'R - CGJ, 
 
 whence it foHows that the three quantities AB — C'^, GA — B'^y 
 BC — A'^ have all the same sign, and therefore if their sum 
 vanishes they must vanish separately, and we must have 
 
 BC-A" = 0, CA-B" = 0, AB-C"' = 0, 
 
i 
 
 OF THE SECOND DEGREE. Ill 
 
 We must also have 
 BV'-AA' = 0, C'A'-BB' = 0, A'B'-CG' = 0, 
 but these are included in the former. 
 
 I 92. If only one of the quantities P, Q, R, as P, vanish, 
 1 and P" also vanish, the equation becomes 
 
 Qf + Rz' + 2q'y + 2R"z + F = 0, 
 which can be reduced to the form 
 
 Qf + Rz' + i^' = 0, 
 
 I and therefore represents an elliptic or hyperbolic cylinder ac- 
 cording as Q and R have the same or opposite signs, that is, 
 according as 
 
 BG-A" + CA -B" + AB-C'' 
 
 is positive or negative. 
 
 If Q, R and F' have all the same sign the locus is an 
 impossible one. 
 
 The condition that P" may vanish is, that 
 
 A'\+B"m^ + C'\ 
 
 should vanish, where l^, m^, n^ are the values of I, m, n de- 
 rived from equations (1) of Art. 83 by putting 5 = 0. But 
 these values are proportional to 
 
 1 1 1 
 
 B'C'-AA" CA'-BB" A'B-CG" 
 
 I so that we get 
 
 A" B" C" 
 
 B'C - A A' ' C'A' - BB' ' A'R - GO' 
 
 This condition may be obtained in another form from the 
 consideration that the equations 
 
 A"l^ + B%i^-i-C"n^ = 0] 
 A\ +Cm^ +B\ =0 
 G\ +Bm^ +A\ = 
 B\ +A'm^ + Gn^ =0 
 
 •(1) 
 
112 THE GENERAL EQUATION 
 
 must be all satisfied by the same values of l^, m^, 7i^, and the 
 requisite conditions that this may be the case are 
 
 ABC 4- 2A'B'C - AA" - BB" - CC" = 0, 
 united with any one of the set, 
 
 A'' (CC - A'E) -i- B" (B" - GA) + C" {AA' - B'C) = 0, ) 
 A'' {A" -BC) + B" {CC - A'B') + C" {BE - C'A') = 0, > 
 A" {BE - C'A') + B" {A A' - EC) + C" {C" - AB) = 0. ^ 
 
 The equations (1) are evidently the conditions that the 
 three equations (2) of Art. 84 should not be independent, and 
 consequently there is a line of centres. 
 
 93. If two of the roots of the discriminating cubic as P 
 and Q vanish, and P", Q" also vanish, the locus reduces to 
 
 which represents two parallel planes. The conditions for the 
 two roots vanishing are 
 
 BC-A" = 0, CA-E'=^0, AB-C" = 0.^....{2\ 
 
 and I^, m^, n^ are only restricted by the equation 
 
 ^Z, + OX + ^X = (3), 
 
 with which the other two equations in (1) Art. 83 become 
 identical. 
 
 If we have also A"l^ + 5"m, + C'n^ = 0, for all values of 
 Zj, TWj, ?ij consistent with (3) we must have 
 
 A" B" C 
 
 or from (2) 
 
 / 
 
 A C E ' 
 A/^ ^E^^C^ 
 J A JB JC ' 
 
 f^ 94. On the whole then we have the following results. 
 
 I. If ABC + 2A'EC -AA"- BE' - CC be not zero, 
 the equation represents an cllijosoid, a hyperboloid, or an 
 impossible locus, with the cone as a variety of the hyi^er- 
 boloids. 
 
OF THE SECOND DEGREE. 113 
 
 II. If ABC + 2A'B'C' - AA" - BB" - CC" vanishes, 
 the equation in general represents an elliptic or hyperbolic 
 paraboloid according as 
 
 BC-{-GA-^AB- A" - B'^ - G" 
 
 is positive or negative; which may degenerate into an 
 elliptic or hyperbolic cylinder, with an impossible locus, a 
 straight line or two intersecting planes, as particular cases. 
 
 III. If BC-A", GA-B'\ AB-G'^ all vanish, the 
 equation represents a parabolic cylinder which may degene- 
 
 I rate into two parallel or coincident planes. 
 
 The conditions that the equation may represent a surface 
 
 of revolution may be obtained from the consideration that 
 
 ' two roots of the cubic in s are equal. This is discussed in 
 
 Todhunter's Theory of Equations^ Art. 179, to which the 
 
 reader is referred. 
 
 The reduction of the equation in the particular case 
 when 
 
 ABG + 2A'B'G' - A A'' - BB" - GG" = 
 
 may be effected by writing it in the form 
 
 (Ax + Gy + B'zf + (AB - G'^) f + 2 '{AA! - B'G) yz 
 
 + {GA - B') z'-\-A {2A"x + 2B"y + 2G" z + i^) = 0, 
 
 AA' - B'G GA - B" 
 
 or puttmg ^^ _ G'^ =V = aA'-B'G' ' 
 
 {Ax + Gy + B'zf + {AB - G") {y + pz)' 
 
 + A {2A"x + 2B"y + 2G"z -\-F)=0. 
 
 And if we take as co-ordinate planes the planes 
 
 Ax + G'y+B'z = 0, 
 y-\-pz = 0, 
 
 2A"x-\-2B"y + 2G"z-¥F=0, 
 
 this equation will in general assume the form 
 
 Py^+Qz' + Rx=0, 
 
 which represents one of the paraboloids. The axes are not 
 tiowever rectangular. The exceptional cases can be deduced 
 
 A.G. 8 
 
114 EXAMPLES. CHAPTER VII. 
 
 I 
 
 from the consideration that the reduction fails when any 
 two of the three planes are parallel, or when one of them 
 is parallel to the intersection of the other two. i 
 
 We shall conclude this chapter with the following general 
 proposition. 
 
 95. If two surfaces of the second degree intersect in one 
 plane curve, all their other points of intersection lie in another 
 plane curve. 
 
 For let S = and B' = be the equations of the two 
 surfaces, and Ix 4- my +nz — p=^, ox a = 0, the equation of 
 the plane of intersection. Then the curve in which a = 
 cuts the surface >Sf = coincides with the curve in which it 
 cuts the surface S' = 0. So that the three equations S = 0, 
 S' = 0, a = are satisfied by an indefinite number of values 
 of X, y and z. 
 
 Consequently the expression 8 must be identical with 
 k8' + a^, where A; is a constant and /3 a linear function of 
 X, y, z. 
 
 Hence when S=0 and 8' = 0, we have a = or /S = 0, 
 that is, all the points of intersection lie in one of the two 
 planes a = 0, or /S = 0. 
 
 EXAMPLES. CHAPTER VII. 
 
 1 . Investigate the nature of the surfaces, 
 
 (1) ^x" + ^xf + 3^' + 2yz - Szx - 2xy -1=0. 
 
 (2) ic' + 42/' -z"- ^yz -zx+ ^xy + 22; = 0. 
 
 2. Interpret the equations : 
 
 (1) yz-\-zx + xy- x-2y-^z -\-2 + a = 0. 
 
 (2) a;' + 2 ?/' - 3;3' -f 27/2; - 4^zx - Ixy + 3a; = 0. 
 
 (3) x^ + 9/ - ^xy + 2y-4!Z = 0. ■ 
 
 (4) x' + ?/'-/ + 2yz -f 2zx - 2xy -{■2x+2y + 22 = d\ 
 
• 
 
 EXAMPLES. CHAPTER VII. 115 
 
 3. Shew that the two surfaces whose equations are 
 [K" -f 6 V c') x" + {h^ + c^ + a^) y" + (^' + a^ + 6')^' 
 
 — 26ci/2 — '^cazx — 2ahxy = 1, 
 
 and (c?/ — 5^;)^ + {as — cxY + (6^ — ayY = 1, 
 
 have their axes coincident in direction. What kind of sur- 
 face are they respectively ? 
 
 4. Discuss the surfaces obtained by giving different 
 values to fju in the equation 
 
 5. Find the nature of the surface 
 
 a b c DC ca ab a b G 
 and shew that it touches the co-ordinate planes. 
 
 6. If one of the angles between the co-ordinate axes be 
 a right angle and the other two be supplementary, prove that 
 the sum of the squares of the axes of the surface 
 
 xy -\- yz -\- zx + (f = 
 
 is 126^^ (Ex. 7, Chap. iv.). 
 
 7. Shew that if two generators of a hyperboloid of one 
 sheet be taken as two of the axes of co-ordinates, the equa- 
 tion is of the form 
 
 z"^ + az = lyz + mzx ■\- nxy. 
 
 8. Find by the method of Art. 68 the position and mag- 
 nitude of the axes of the section of the surface 
 
 Ax" + By^ + Gz'' -h 2A'yz + Wzx 4- 2G'xy = 1 
 
 by the plane 
 
 Ix -f my -f nz = 0. 
 
 9. Find by the methods of Arts. 68 and 78 the centre 
 and axes of the section of the surface 
 
 Jx + Jy+Jz^O 
 by the plane 
 
 lx + my + nz = 1. 
 
 10. If the equation 
 
 ax^ + by^ -h cz"" + 2b' zx + 2c' xy 4- 2a" x + 2b" y + 2c" z + cZ = 
 represent a paraboloid of revolution, prove that c = ^ + a. If 
 
 8—2 
 
116 EXAMPLES. CHAPTER VII. 
 
 the upper sign be taken, prove that the equations to the 
 axis are 
 
 cz + c" = 0, (ex + a) Ja + (cy + b") Jh = 0, 
 
 and find the condition that the paraboloid may reduce to a 
 circular cylinder. 
 
 11. Find the equation of a surface of the second degree 
 which contains two given straight lines at right angles, and 
 the condition that it may be a hyperboloid of one sheet. 
 
 Take the shortest distance between the lines as axis of z, 
 the middle point of it as origin, and the axes of x and y 
 parallel to the two lines. 
 
 12. Find the equation of the surface generated by a 
 straight line which meets three straight lines which are 
 mutually at right angles, but which do not intersect. 
 
 13. Shew that the section of the surface 
 
 Ax^ + By^ + G£' -h 2A'yz + 2B'zx + ^G'xy = 1, 
 by the plane Ix + my -\-nz = 0, will be a circle if 
 Bn' + Giit'-'lA'mn _ CP + An'-2B'nl ^ Am' ■hBl'-2C'lm 
 ;;?T^? n'-^l' I'+m' 
 
 14. Shew that the axes of the surface 
 
 Ax' + By' + Cz' + 2A'yz + 2B'zx + 2C'xy = 1 
 
 lie on the two cones 
 
 C (x' - f) - B'yz + A'zx -(A-B)xy= 0, 
 jl' ^,f _ 2') -{B-C)yz- C'zx + B'xy = 0. 
 
 15. A cone whose equation referred to its principal 
 
 axes is 
 
 aV + /Qy = (a' + ^') z\ 
 
 is thrust into an elliptical hole whose equation is 
 
 a' ^ b' 
 
 Shew that when the cone fits the hole its vertex must lie 
 on the ellipsoid 
 
 x' y' 2/l,l^_^ 
 
CHAPTER VIII. 
 
 ON TANGENT LINES AND PLANES. 
 
 96. The straight line joining any point P on a surface 
 to another point Q on the surface, is called a chord. If the 
 point Q be made to approach indefinitely near to P, the 
 limiting position of the chord PQ is said to be a tangent line 
 to the surface at the point P. 
 
 In general all the tangent lines at the point P lie in a 
 plane, which is called the tangent plane at P. This we will 
 now prove. 
 
 Let X, y, z be the co-ordinates of any point P on a surface 
 whose equation is 
 
 P(^,2/,^)=0 (1). 
 
 And let the equations of any straight line through P be 
 x' — xy'—yz—z ,^. 
 
 where oo', y\ z are current co-ordinates. 
 
 To find the points where (2) meets (1) we must substitute 
 x-vlr.y -\- mr, z + nr for x, y, z in (1) ; 
 we thus get the equation 
 
 F{x -\-lr, y -\- mr, z + nr) = ; 
 
118 ON TANGENT LINES AND PLANES. 
 
 f,dF dF dF\ 
 
 r^{jd d dVj^, . 
 
 + 
 
 + ^l^i + ™| + 4r^<"'2/.^) = 0...(3), 
 
 supposing F (x, y, z) to be of the p^^ degree in x, y, z. 
 
 This equation gives the distances from P of the different 
 points in which (2) cuts (1), and since {x, y, z) is a point on 
 the surface (1), F {x, y, z) vanishes and the equation (3) is 
 satisfied by one value of r equal to zero. 
 
 If I, m, n be such as to satisfy the equation 
 
 ,dF dF dF ^ ,^. 
 
 ^^+^^5^ + ^^ = ^ W' 
 
 two values of r are zero, and the line (2) meets the surface in 
 two coincident points, and is therefore a tangent line to the 
 surface at {x, y, z). Equation (4) is therefore a condition 
 which must be satisfied by the direction-cosines of all tangent 
 lines at the point P. 
 
 But for all points in any such tangent line we have 
 X —X _y —y _z' — z 
 I m n ' 
 
 Consequently for all points in any such tangent line we 
 have 
 
 , , . dF , , s dF , , V dF - ,^. 
 (.'-a.)^^+(2/-y)^^ + {.-.)^ = 0...(o), 
 
 whence it follows that all the tangent lines in general lie in 
 a plane whose equation is (5J. 
 
 97. It may happen that at a given point of a surface the 
 
 three quantities -^ ,-r- and -y- all vanish. 
 ^ ax ay az 
 
ON TANGENT LINES AND PLANES. 
 
 119 
 
 If this be the case, the equation (3) of the last article 
 always gives two values of r equal to zero, and all lines 
 through the point P meet the surface in two coincident 
 points. The vertex of a cone is such a point. If we take 
 I, m, n such as to satisfy the condition 
 
 d?F , d'F , d'F 
 
 dx' 
 
 ■i-m 
 
 dij 
 
 + n 
 
 dz" 
 d'F 
 
 + 2nl f^ + 2Zm -^ = ... (1), 
 
 dydz ' *"'" dzdx dxdy 
 
 three values of r mil be zero, and the straight lines whose 
 direction-cosines satisfy this equation meet the surface in 
 three coincident points ; eliminating I, m, n, we have as the 
 equation of the locus of all such straight lines 
 
 d'F 
 
 dx' 
 
 , 2 d^F , , .2 ^^ 
 
 y) ^2-^(^ - ^) 
 
 dy' 
 
 dz' 
 
 ^2{y'-y){z'-z)^^+2{z'-z){x'-x) ^'^ 
 
 dy dz 
 + 2(^'-^)(y'-y);^=0 
 
 dzdx 
 ••(2). 
 
 dxdy 
 
 which is the equation of a cone of the second degree whose 
 vertex is at the point {x, y, z). See Art. 34. 
 
 A point at which -r- , -,— and -^r all vanish is called a 
 ^ ax dy dz 
 
 singular point on the surface, and the cone (2) is called the 
 
 tangent cone at that point. 
 
 98. In the case of Art. 96 we see that all straight lines 
 whose direction-cosines satisfy (4) meet the surface in two 
 coincident points. If we take I, m, n such as to satisfy both 
 the conditions 
 
 ,dF dF dF ^ 
 dx dy dz 
 
 d'F 
 
 dx 
 
 5+m' 
 
 d'F . d'F 
 
 ,2 
 
 4-n 
 
 dz' 
 
 dy' 
 
 d'F ^ ^ , d'F 
 + 2mn - — - + 2no 
 
 dy dz 
 
 dz dx 
 
 + 21771 
 
 fF_ 
 
 dxdy 
 
 = 
 
 \ ...(1)> 
 
120 
 
 ON TANGENT LINES AND PLANES. 
 
 the straight lines whose direction-cosines are obtained from 
 these equations meet the surface in three coincident points. 
 They are therefore tangents to the curve in which the tan- 
 gent plane meets the surface. This curve, therefore, has a 
 double point at the point of contact, since the above equa- 
 tions in general give two values of the ratios Z : m : n, which 
 values may be possible or impossible. 
 
 If the surface be of the second degree, the two straight 
 lines given by (1) lie wholly on the surface, and are possible 
 if the surface be a hyperboloid of one sheet or a hyperbolic 
 paraboloid, and impossible in other cases. 
 
 99. The equation of a surface is often given in the form 
 
 ^ =/ (^> y)> or z -f {X, y) = 0. 
 
 -r , . dF , df dz dF , dz 
 
 in this case -r- becomes — /-or — 7- , -r- becomes — r- , 
 ax ax ax ay ay 
 
 dF 
 and -y- becomes unity. The equation of the tangent plane 
 
 becomes therefore 
 
 dz dz 
 It is usual to denote the quantities -1- and -j- by the 
 
 letters p, g, and the quantities -r-^ , x^ ' ;7~7r ^^ *^^ letters 
 r, t, s, respectively. 
 
 100. The equation of the tangent plane being 
 , , ,dF , , -.dF , , .dF ^ 
 
 the length of the perpendicular on it from the origin is 
 
 dF dF dF 
 
 dx "^ dy dz 
 
 /(IF 
 \dx. 
 
 +f'^v+(Sy 
 
 (!)• 
 
 dy) 
 
ON TANGENT LINES AND PLANES. 121 
 
 The letters U, V, W are frequently used to denote 
 
 dF dF dF 
 dx ' dy ' dz ^ 
 
 and the letters u, v, w, u\ v\ w to denote 
 
 ^ d^F d'F dlF_ ^F_ d^F 
 dx^ ' dy^ ' dz^ ' dydz' dz dx ' dxdy' 
 
 respectively. With this notation the above expression be- 
 comes 
 
 JJx^Vy^Wz 
 
 JU'+V'+ W 
 
 (2). 
 
 If we take the form of the equation in Art. 99, the length 
 of the perpendicular is 
 
 z-px-qy 
 
 Jl+p' + q' ^' 
 
 101. As an example take the tangent plane at any point 
 (x, y, z) of an ellipsoid whose equation is 
 
 £C^ V^ ^ 
 
 -2+f:. +-2 = 1 (1). 
 
 a ^ ^ 
 
 Here ^=??, V =% . W=^; 
 
 a b c 
 
 and the equation of the tangent plane is 
 
 (^' - ^) f 2 + (y - 2/) I + (^' - ^) ^2 = 0, 
 
 XX yy zz_x y ^ _-, 
 
 i 
 
 The equation of every plane can be expressed in the form 
 
 Xx + fjuy' + vz'=p (3), 
 
 where p is the length, and \, /jl, v are the direction-cosines, of 
 ithe perpendicular on it from the origin. 
 
122 ON TANGENT LINES AND PLANES. 
 
 If we suppose (2) identical with (3), we get 
 
 (4), 
 
 \ _fJb _v _p 
 
 X y z L 
 ^ P ? 
 
 And the equation of the tangent plane becomes 
 
 Xa;' + ixy -f vz = Ja'X' + 6 V + c'v' (5), 
 
 a form which is often useful. 
 
 The length of the perpendicular on (2) from the origin 
 
 1 
 
 V f z^ ' 
 
 The values of X, fju, v the direction-cosines of this perpen- 
 dicular are —^,-^,-^ by (4), and the co-ordinates of the 
 
 9 2 2 
 
 foot of this perpendicular are consequently ^--^ , -jr , —^ • 
 102. The equation of a paraboloid being 
 
 T + ? = ^ ^^>' 
 
 the equation of the tangent plane at {x, y, z) becomes 
 (x-x)-'^{y'-y)-j(z-z)=0, 
 
 or x—-j-.y-Y.z=x — ^ y = — x, 
 
 or • ^,y' + ^.z' = x+x (2). 
 
ON TANGENT LINES AND PLANES. 123 
 
 This can be put into another form, for comparing it with 
 
 \x + fiy + vz = p, 
 
 Lb V \ p 
 
 p III I'v 
 
 and therefore from (1), 
 
 ' .2 
 
 
 and the equation of the tangent plane becomes 
 
 4\ 
 
 Xx'-\-^y' + vz' = - ^ ,^ (3). 
 
 103. The normal to a surface at any point is the straight 
 line drawn through that poi?it perpendicular to the tangent 
 plane. 
 
 The equation of the tangent plane at {x, y, z) is 
 . , . dF . , . dF . , . dF 
 
 and the equations of a straight line through the point {x, y, z) 
 perpendicular to this plane are 
 
 x—x_y—y_z—z , . 
 
 dx dy dz . 
 
 These are therefore the equations of the normal. 
 
 The equations of the normal to an ellipsoid at the point 
 (a?, y, z) are 
 
 g^ {x - x) ^ h^y'- y) ^ & (z - z) 
 
 X y z ' 
 
124 ON TANGENT LINES AND PLANES. 
 
 If we take the equation of the surface to be 
 
 the equation of the tangent plane is 
 
 z-z-p{x'-x)-q{y' -y)=0, 
 
 and the equations of the normal are therefore, 
 
 X -x+p[z-z) = 0\ 
 
 y-y + q[z-z)=0] ^-^' 
 
 104. The equation of the tangent plane to a surface 
 
 F{x,y,z) = Q (1) 
 
 at the point (^, y, z) is 
 
 , , . clF , , , dv , , . dF - 
 
 If this plane pass through a point whose co-ordinates are 
 a, yS, 7, we have 
 
 , . dF . ^ , dF , . dF ^ ,-v 
 
 (a-.)^ + (^-2,)^. + (T,-.)^^=0 (2). 
 
 This relation is satisfied by the co-ordinates of all points, 
 the tangent planes at which pass through a given point 
 (a, /3, 7). It is the equation of a surface which by its inter- 
 section with (1) determines the points of contact of tangent 
 planes to (1) drawn through (a, y&, 7). 
 
 105. We can shew that all these points of contact lie on 
 a surface of the degree next below that of the original surface. 
 For let F {x, y, z) be of the p^^ degree, and let us assume 
 
 F {x, y, z) = 11^ -j- 'Up_^ + Wp_2 + . . . 4- 2^2 + z/j -h Mo, 
 
 where u^, u^_^... denote the terms of the y^, {p — Vf^... de- 
 grees respectively. 
 
 Then the points of contact are determined by (1) and (2), 
 and the latter may be written 
 
 ^dF dF dF_^dF ^ dF ^dF 
 dx dy dz dx dy dz ' 
 
ON TANGENT LINES AND PLANES. 125 
 
 But by a well-known theorem (see Todliunter's Diff. Calc. 
 Chapter viii. Ex. 3), 
 
 du„ du^ du„ 
 
 dF dF dF , \. 
 
 But for all the points of contact we have 
 
 F {x, y,z) = 0; 
 
 therefore =pUp -\-pUp_i + • • • +pu^ +pu^ +P'^a (4). 
 
 Subtracting (4) from (3) we get 
 dF dF dF ^ / ox / -.N 
 
 ^n^^y'd^'^^jz =~^^- ~^v2-----(p-2K-(p-i)^-p% 
 
 and equation (2) becomes 
 dF L. dF dF _, / -1 \ /-v / K \ 
 
 Now ^- , ^— , ^— are of the ( » — 1)*^ degree, conse- 
 dx dy dz x/^ / 8 > 
 
 quently (5) represents a surface of the (p — Vf^ degree. 
 
 If the original surface be of the second degree, all the 
 points of contact lie in a plane. 
 
 106. The equation of the tangent plane to an ellipsoid 
 at the point {x, y, z) is 
 
 x'x y'y zz _ - 
 If this pass through a point (a, P, y), we must have 
 
 ^ + F"^?--^ ^^^' 
 
126 ON TANGENT LINES AND PLANES. 
 
 a relation which is satisfied by the co-ordinates of all the 
 points of contact, and which is therefore the equation of the 
 plane of contact. 
 
 The plane (1) is called the polar plane of the point (a, j3, 7) 
 with respect to the ellipsoid : and (a, yS, 7) is called the pole 
 of the plane (1). 
 
 If all the points in which (1) cuts the ellipsoid be joined 
 with (a, ^, 7) the joining lines will form a cone, and will all 
 touch the ellipsoid, since each of them lies in the tangent 
 plane at the point where it meets the surface. This cone is 
 
 called an enveloping cone. 
 
 • 
 
 Conversely, if at all points at which any plane cuts an 
 ellipsoid, tangent planes be drawn, these planes will all meet 
 in one point, which is the pole of the cutting plane. 
 
 If a series of planes be drawn passing through a fixed 
 point and cutting an ellipsoid, the poles of these planes will 
 all lie in a fixed plane which is the polar of the fixed point. 
 Let (a, /S, 7) be the fixed point, and (x, y, z) the pole of any 
 plane through (a, yS, 7). 
 
 The equation of the polar of {pc, y, z) is 
 
 XX yy z'z _'t 
 
 If this plane pass through (a, /S, 7) we must have 
 
 ^2 + J2 + ^2 - -L> 
 
 which shews that {x, y, z) lies on the polar of (a, /3, 7). 
 
 If a series of planes be drawn passing through two fixed 
 points and therefore through a fixed straight line, the poles 
 of these planes will all lie in each of two fixed planes which 
 are the polar planes of the two fixed points, that is, they will 
 all lie in a fixed straight line. 
 
 Similar results hold for all the surfaces of the second 
 degree. 
 
ON TANGENT LINES AND PLANES. 127 
 
 107. The equation of the enveloping cone can be found 
 by a process similar to that adopted in Art. 34. The equa- 
 tions of any generating line can be written 
 
 ''-^ = y^ll = l^ = r (1), 
 
 L ra n ^ 
 
 and the equations of the curve of contact are 
 
 By substituting for w, y, z from (1) in the equations (2) 
 their values a -f h\ /3 + mr, 7 + nr and eliminating r, we 
 obtain a relation which I, m, n must satisfy in order that the 
 line (1) may pass through some point of the curve (2). 
 
 The equations (2) can be reduced to one equation of the 
 ^tii degree, and one of the {p — 1)*^, and the result of substi- 
 tuting for X, y, z from (1) will therefore be 
 
 where A^ is a homogeneous function in I, m, n of the ]f^ de- 
 gree, J.^_i and Bp_^ are homogeneous functions of the (j9 — Xf^ 
 degree, and so on. 
 
 The equations (3) can therefore be expressed in the form 
 
 a; {nrf -f- A\_lnrY-^ + . . . + ^>r + ^^ = 0, 
 
 ^Vi (^^)""' + B\_lnry-' -h . . . + B;nr -{-B,=0, 
 
 where AJ, A'^_^, ---A^, A^, -S'^-i> •••^/j ^0 ^^^ functions 
 
 of - , — , and the result of eliminating^ nr between them will 
 n n ° 
 
 be of the form 
 
 \n nj 
 and the equation of the cone is therefore 
 
 ^ \Z - ^ Z - r^J 
 
128 ox TANGEXT LINES AND PLANES. 
 
 108. In the case of an ellipsoid the equation of the plane 
 of contact is 
 
 — + Tr + -^-l = (1), 
 
 a c ^ ' 
 
 and we have to substitute % + h\ y5 + mr, 7 4- ni\ for x, y, z 
 in (1), and in the equation of the ellipsoid 
 
 / 
 
 -2 + 71 + ^=1 (2). 
 
 We thus ofet 
 
 O' 
 
 (al /3m yn\ a^ /^^ 7^ -, ^ 
 J fl' m^ n\ 2 - /Oil l3m 7? A 
 
 + -UC + ^'-l = (4); 
 
 a b' c 
 
 and substituting for ?- from (3) in (4) we obtain 
 
 This is the relation which /, m, n must satisfy in order 
 that the strai^rht line 
 
 I m n 
 
 may pass through some point in the curve of intersection of 
 (l)\and (2). 
 
 The equation of the enveloping cone is obtained by sub- 
 stituting x — a, y — ^, 2 — 7 for I, m, n, and is therefore 
 
 ^ f(£-a)» , (y-m^ , (^-7)7 ]' 
 
 f(^-^^(^^^(^-7)7r (6). 
 
 [a b c J 
 
ON TANGENT LINES AND PLANES. 129 
 
 109. This equation can be obtained in another form b}^ 
 the aid of the following proposition. 
 
 Let >Sf = be the equation of any surface of the second 
 degree, and let ^t = 0, ?; = be the equations of two planes. 
 Then the equation 
 
 S + \uv = (1), 
 
 where X is some constant, w^ill represent any surface of the 
 second degree passing through the curves of intersection of 
 S = with u = and v = 0. For if >S' = be the equation of 
 any such surface, it is evident that S' cannot assume any 
 other form than k (S + \uv) consistently with the suppositions 
 that it is of the second degree, and is satisfied by all values 
 of X, y, z which make S and u vanish simultaneously, and 
 also by all values which make >S^ and v vanish. 
 
 Again, if we suppose the plane u — O to change its position 
 so as to coincide wdth v = 0, the equation (1) represents any 
 surface touching S = along the curve in which the latter is 
 cut by v = 0, and becomes 
 
 Hence the equation 
 
 -. + p + ^.-l+X^^ + ^.+^-lj -0 (2), 
 
 represents any surface of the second degree touching the 
 ellipsoid at all the points of contact of tangent planes through 
 {a, 13, 7). If we take X such that (2) shall pass through 
 (a, P, <y) it must represent the enveloping cone. Substituting 
 a, 13, <y for X, y, z, we get 
 
 a c \a c ) 
 
 Whence the equation of the enveloping cone becomes 
 
 This equation can of course be deduced from that of the 
 last Article. 
 
 A. G. 9 
 
130 ON TANGENT LINES AND PLANES. 
 
 110. If we suppose the point (a, /?, 7) to recede from 
 the origin to an infinite distance, the cone will ultimately 
 become a cylinder whose generating lines are parallel to the 
 line joining {a, /S, 7) with the origin. This is called an en- 
 veloping cylinder, and the equation of any such cylinder can 
 be found from that of the cone, by putting a = \k, /3 = fiJc, 
 7 = vk, where \, /jl, v are the direction-cosines of the generat- 
 ing lines, dividing by the highest power of k, and then mak- 
 ing k infinite. The equation of the enveloping cylinder of 
 an ellipsoid deduced in this manner from either of the equa- 
 tions in Arts. 108, 109 is 
 
 (: 
 
 x^ y^ z^ -, \ /X^ y^ v\ (\x iiv vzV 
 
 111. The equation of the cylinder which envelopes a 
 
 given surface 
 
 i^(^,2/,^) = (1) 
 
 can however be obtained independently of the enveloping 
 cone. 
 
 For let \, fjL, V be the direction-cosines of one of the gene- 
 rating lines ; x, y, z the co-ordinates of the point where it 
 touches (1). Then since this generating line of the cylinder 
 is a tangent line to (1) at {x, y, z), we must have 
 
 dx dy dz 
 
 This equation combined with (1) gives the locus of the 
 points at which the enveloping cylinder touches the surface, 
 and we have only to find the equation of a C3dinder with its 
 generating lines in a given direction, and passing through the 
 curve given by (1) and (2), which can be done as in Art. 35. 
 
 \i X y y\ z be the co-ordinates of any point in the gene- 
 rating line which touches (1) at the jDoint {x, y, z), we have 
 
 X — X 11' — ^1 z — z 7 
 
 = = —— k suppose, 
 
 A, ^ V 
 
 or X = X 4- \k, y = y' + p-k, z = z -\- vk. 
 
 Substituting these values of x, y, z in the equations (1) 
 and (2), and eliminating k between the two equations, we get 
 
ON TANGENT LINES AND PLANES. 131 
 
 a relation between x\ y' , z which is the equation of the en- 
 , veloping cylinder, 
 
 112. In the case of the ellipsoid, the curve of contact is 
 determined by the equations 
 
 2 2 2 
 
 1- — -I — =1 
 
 a c 
 
 \X LLII VZ 
 
 h^-^ -^ — = 
 
 a c 
 
 Putting x' + \7c, y + jik, z + vh for x, y, z we get 
 
 — + 72 + ^-1 + 2— + 9I-+^U'+- + 72 + -2^=^^ 
 a G \a c J \a b cj 
 
 Substituting for Iz from the second in the first we get 
 
 '2 '2 '1 \ /-v 2 2 2\ /-v ' ' '\ 2 
 
 y z \ \ yiT v\ _ l\x ixy vz y 
 
 :he same equation as we obtained in Article 110. 
 
 + -F+0'' 
 
 113. Let the equation of a surface be given in the form 
 
 .^ ('A yS, 7, S) = (1), 
 
 where a, /3, 7, S are the lengths of the perpendiculars from 
 liny point on the four faces of a tetrahedron, and let any 
 jtraight line be drawn through the point {a, /3, 7, S). Then 
 f a, /3', 7', 8' be the values of a, 13, 7, S for any other point in 
 ihe line we shall have by obvious geometry 
 
 a -a ^ ^'-l3 ^ y -y ^ 8' -8 ^j^ . 
 
 I 7)1 n q 
 
 vhere I, m, n, q are the cosines of the angles between the 
 iine and the perpendiculars on the four faces of the tetrahe- 
 Iron, and k is the distance between the two points. 
 
 We obtain the value of 7c for the points where the line (2) 
 Ineets (1) from the equation 
 
 <^ (a + Ik, 13 + mk, 7 + nk, 8 + qk) = 0, 
 
 9—2 
 
132 ON TANGENT LINES AND PLANES. 
 
 or .^(.,/3,./,S) + Z;(^g + mg + 4n4g 
 
 + Ak' + Bk' + ...=0 (3). ' 
 
 This equation gives as many values of Jc as the degree of 
 the equation (1). 
 
 Smce (oL, /5, y, 5) is a point on (1), ^ (a, /S, 7, 8) vanishes, 
 and one value of k is zero. If I, m, n, q be restricted by the 
 relation 
 
 defy ^* , ,, ^* , „ f^* _ 
 
 two values of k vanish, and the line (2) is a tangent line to 
 (1) at (a, yS, 7, B). Hence eliminating I, m, n, q by means 
 of (2) the equation of the locus of the tangent lines at 
 (a, ft 7, B) is 
 
 or 
 
 , d6 ^, d(b , ,d6 , ^, d6 d6 , add) ^ (^^ , ^ f?</> 
 
 But the expression (j) {% /S, 7, 8) may be supposed homo- 
 geneous, since if it be not, it can be made so by means of 
 the relation given in Art. 26; and if it be of the ^^'' degree, 
 we have by a well-known formula 
 
 since the point (a, /?, 7, 3) is on the surface (1). Hence the 
 equation of the tangent plane at (a, /3, 7, S) becomes 
 
( 133 ) 
 
 EXAMPLES. CHAPTER VIII. 
 
 , 1. Find the locus of the point of intersection of three 
 
 i tangent planes to an ellipsoid which are mutually at right 
 angles. 
 
 2. Find the locus of a point which moves so that the 
 locus of the centre of the section of an ellipsoid by its polar 
 plane W'ith respect to that ellipsoid is a similar and similarly 
 situated ellijDsoid whose axes are each half of the correspond- 
 ing axis of the original ellipsoid. 
 
 8. Shew that the polar equation of the locus of the foot 
 of the perpendicular from the origin on the tangent plane to 
 an ellipsoid is 
 
 r^ = o? sin^ Q cos^ ^ + 6^ sin^ 6 sin^ <^ + c^ cos^ 6. 
 
 4. Find the equation of the locus of the foot of the per- 
 pendicular from a point (a, /3, 7) on the tangent planes of the 
 ellipsoid 
 
 5. Find the equation of the locus of the poles of all 
 tangent planes of the ellipsoid 
 
 222 
 
 X If z 
 V— 4- - =1 
 
 with respect to a sphere whose centre is at the point (a, /3, 7) 
 and whose radius is h. 
 
 6. Shew that in general six normals can be drawn 
 through a given point to an ellipsoid, and that these six all 
 lie on a cone of the second degree, three of whose generating 
 lines are parallel to the axes of the ellipsoid. 
 
 j 7. If normals be drawn to an ellipsoid 
 
 0^ ip- z^ ^ 
 V - A — = 1 
 
134 EXAMPLES. CHAPTER VIII. 
 
 at the points v/here it is cut by the cone 
 
 I 111 n ^ 
 
 - + - + - = 0, 
 
 X y z 
 
 prove that these normals all pass through a diameter of the 
 ellipsoid. 
 
 8. In an ellipsoid whose semi-axes are a, h, c, plane 
 sections are drawn so as always to touch a confocal ellipsoid 
 (see Art. 160). Shew that the centres of these sections 
 always lie on a surface of the fourth degree which intersects 
 the ellipsoid in the cone 
 
 2 2 2 
 
 30 y z 
 
 a' + 6' + ? = 0- 
 
 9. Prove that through any central radius of an ellipsoid 
 one plane can be drawn cutting the ellipsoid in a curve of 
 which that radius is a semi-axis. Shew that if it be so for 
 more than one section it must be so for all such sections. 
 
 10. On a plane section of a given ellipsoid as base two 
 cones are constructed of which the vertices are the centre 
 of the surface and the pole of the section. If the ratio of 
 the volumes of these cones is constant, prove that each of 
 them is constant ; and find the volume when the ratio is one 
 of equality. 
 
 11. Find the locus of a luminous point, in order that the 
 boundary of the shadow of an ellipsoid cast by it upon a given 
 principal plane may be circular. 
 
 12. Prove that the right circular cylinders described 
 about the ellipsoid 
 
 ^' f ^' -, 
 
 — -4- "^ -I = 1 
 
 a^ ¥ c 
 h being the mean semi-axis, are represented by the equation 
 
 (h'- c'jaf- (c'- a') if+ {a'- h')z' ± 2 (a'^- h') ' (b'- c') ' zx = (a=- &)¥. 
 
 13. The shadow of a ball is cast by a candle on an in- 
 clined plane in contact with the ball ; prove that as the 
 candle burns down, the locus of the centre of the shadow is 
 a straight line. 
 
EXAMPLES. CHAPTER VIII. 135 
 
 14. Find the equation of the tangent plane to the sur- 
 face 
 
 ccyz = a^, 
 ■ and the volume cut off by this plane from the axes. 
 
 15. Find the equation of the tangent plane at any point 
 of the surface 
 
 Z 2. 2. 2. 
 
 oc''^ + y^ + z^ = a^. 
 
 Find also the length of the perpendicular on it from the 
 origin, and the area of the triangle intercepted on the tangent 
 plane by the co-ordinate planes. Shew that the sum of the 
 - squares of the intercepts on the axes of co-ordinates is con- 
 stant. 
 
 16. Find the equation of the enveloping cone of the sur- 
 face By"^ + Cz^ = x, whose vertex is at a point (a, /3, 7). 
 
 I 
 
 17. Find the length of the normal at any point of an 
 ellipsoid cut off by the plane of xy. Find also the co-ordi- 
 nates of its point of intersection with the plane of xy. 
 
 18. Find the equations of the normal at any point of the 
 surface 
 
 By'' -f Gz"" = X. 
 
 Find the locus of the points in which the normals to the 
 surface drawn at all points of its intersection with the plane 
 x = a cut the plane of yz. 
 
 19. Shew that the points on the surface 
 
 xyz = c^ 
 
 at which the normals intersect a fixed line 
 
 X — a _y — _z — y 
 I m n 
 
 ill lie on the surface 
 
 ^ [my — nz) + /3y [nz — Ix) -\- yz (Ix — my) = x^ [my — nz) 
 
 -h 2/^ {iiz — Ix) -f- z"^ (Ix — my). 
 
136 EXAMPLES. CHAPTER VIII. 
 
 20. Find the locus of the point of intersection of three 
 tangent planes to a paraboloid which are mutually at right 
 angles. 
 
 21. Find the equation of a surface of the second degree 
 which passes through all the points of contact of tangent 
 planes drawn through an external point (a, /3, 7) to the 
 surface 
 
 x^ -\-y^ -^-z^ — oxyz = c^ 
 
 and discuss its nature for different positions of (a, /3, 7). 
 
 22. Find the equation of a surface of the second degree 
 which passes through all the points of contact of tangent 
 planes drawn through an external point (a, yS, 7) to the 
 surface 
 
 xi/z = a^, 
 
 and discuss its nature for different positions of {a, /3, 7). 
 
 23. Find the equation of the locus of the foot of the 
 perpendicular from the origin on the tangent planes of the 
 surface 
 
 Bf + Cz" = X. 
 
 24. Shew that the plane 
 
 Ix + my + ?2 J = 
 will touch the cone 
 
 Ax'' + Bif-^Cz'' = 
 if I, m, n satisfy the condition 
 
 r m' n^ ^ 
 
 25. Shew that the axes of a central section of the ellip- 
 
 3 a 2 
 
 X 11 z 
 
 sold -2 + ;7 + "i = 1 hy a plane parallel to the tangent plane 
 at (a, P, 7) are given by the equation 
 
 r' - {a- + h' + c'-a'-/3'- r) r + '''^^' = 0, 
 
 where p is the perpendicular from the centre on the tangent 
 plane. 
 
EXAMPLES. CHAPTER VIII. 137 
 
 26. If a line cut two similar and co-axial ellipsoids in 
 P, P ; Q, Q' ; respectively, prove that the tangent planes to 
 
 I the former at P, P' cut those to the latter at Q or Q' in pairs 
 of parallel straight lines equidistant res]Dectively from Q 
 or Q'. 
 
 27. Find the condition that two spheres may intersect 
 at right angles. 
 
 28. Four spheres whose radii are a, h, c, d intersect at 
 right angles, shew that the volume of the tetrahedron whose 
 ano'les are their centres is 
 
 i i 1 1 
 
 -2 + ^-^ + ^^ + ^ 
 
 JaMA/-2+75+:7^+ 72 
 
 29. The centre of a sphere bisects the shortest distance 
 between two given straight lines and a tangent line to the 
 sphere passes through each of the lines: shew that the point 
 of contact lies on a hyperbolic paraboloid. 
 
CHAPTER IX. 
 
 ON CURVES IN SPACE. 
 
 114. We have seen (Art. 16) that any two equations 
 
 since they are satisfied by the co-ordinates of all the points of 
 intersection of the surfaces represented by each equation, will 
 in general represent a curve. 
 
 These equations can be reduced to the form 
 
 ^=^^!'^'!1 (2). 
 
 by eliminating y and z in turn between the two equations 
 (1). It may be noticed that the two equations (2) will in 
 some cases represent a curve not included iu (1). For in- 
 stance, if the two equations (1) were of the first and second 
 degrees respectively, by eliminating ?/ and z in turn we 
 should get two equations of the second degree, and the first 
 two equations would represent one plane curve, while the 
 second pair would represent the original curve, and another 
 plane curve besides. (See Art. 95.) 
 
 Assuming x to be any arbitrary function of a new vari- 
 able t, the equations (2) can be replaced by the three 
 
 x = <f>{t), y==^lr{t), z = xif) (3). 
 
 This third form possesses many advantages from its sym- 
 metrical character, and we shall in general use it. 
 
ON CURVES IN SPACE. 139 
 
 115. As an example the pair of equations 
 
 Ax-vBy+ Gz = D\ ,.s 
 
 A'x + B'y + Cz = D'] ^ ^ 
 
 represent a straight line. 
 
 Eliminating y and z in turn we get the two equations 
 
 A'B-AB' B'D-BD'^ 
 
 ^ BG-BG'^'^ BG-BC 
 
 ^ G'A - GA' GU - CD 
 y R'/^ Tin' "^ "^ 
 
 y (2), 
 
 > ... (3), 
 
 B'G - BG' ' B'G - BG j 
 which correspond to the form (2) in the last Article. 
 
 Lastly, assuming x = (B'G — BG') t, we get 
 
 CD' — G' J) 
 x=[EG- BC) t,y = [G'A - CA') t + j^f^, _ ^^ 
 
 z = {A'B - AE) t + 5§^' 
 
 which correspond to the form (3) in the last Article. 
 
 116. The curves of the most frequent occurrence and 
 greatest importance are plane curves, the discussion of which 
 properly belongs to plane geometry. As an instance of a 
 curve not plane we may take the helix. 
 
 This is the curve formed by the thread of a screw. It 
 may be produced by wrapping a right-angled triangle round 
 a circular cylinder, the base of the triangle being at right 
 angles to the axis of the cylinder. 
 
 Take the axis of the cylinder as axis of z, a plane through 
 the base of the triangle as plane of xy, and a line through the 
 • acute ancrle at the base of the trianofle as axis of x. 
 
 Let be the origin ; x, y, z the co-ordinates of any point 
 \P in the curve, a the radius of the cylinder, 9 the angle AOM 
 between the axis of x and DM the projection of OP on the 
 
140 
 
 ON CURVES IN SPACE. 
 
 H r 
 
 plane of xy, and a the acute angle at the base of the triangle. W' 
 "We obtain without difficulty, 
 
 X = ON = OM cos 6 = a cos 6, 
 y — MN = OM sin ^ = a sin ^, 
 z = PM = arc AM x tan ol = a6 tan a, 
 
 Whence 
 
 or if a tan a = c, 
 a; = a cos 0, y = asinO, z = c9 
 
 z z 
 
 x — a cos - , y = a sin - 
 
 ,(1). 
 
 (2). 
 
 Either (1) or (2) may be considered as the equations of 
 the helix. 
 
 117. The limiting position of the straight line joining 
 two points of a curve when the second point moves up in- 
 definitely near to the first, is called the tangent to the curve 
 at that point. 
 
 Let the equations of the curve be 
 
 ^ = ^{0> 2/ = tW, ^ = %(0 (!)> 
 
 and let t and i + t be the values of t for two points on the 
 curve. The equations of the straight line joining these are 
 
 'o'- <j> (0 ^ _y'^:^irj) _ z-x(t) 
 
 ^(t + T)-<j>(t) ^Ir it + t) - ^|r (t) % U + TJ - % W ' 
 
ON CURVES IN SPACE. 1-il 
 
 x\ y , z being current co-ordinates ; 
 
 
 or 
 
 T 
 
 But when r is diminished indefinitely the two points 
 coincide and the straight line joining them becomes the 
 
 tangent at (^, y, z). Also the limit of — lAJ {^ 
 
 T 
 
 doc 
 i <^' (t) or -J- , and similarly for the other denominators. 
 
 Hence the equations of the tangent at {x, y, z) are 
 
 X — X _ y' — y _ z' — z 
 dx dy dz 
 
 dt dt dt 
 
 (2). 
 
 118. The length of the chord joining two jDoints {x,y, z) 
 and {x^,y^, zj is 
 
 J(x^-xy+{y^-yy + {z^-zy. 
 
 But by Newton (Section I. Lemma vii.) when the two points 
 approach indefinitely near to each other, the ratio of the arc 
 to the chord becomes ultimately a ratio of equality. Hence 
 if s and 5 + 85 be the lengths of the arcs measured from some 
 ifixed point up to the points {x, y, z), (x^, y^, z^) respectively, 
 the fraction 
 
 J{x, - xf + (y, - yf + {z^ - zf 
 becomes ultimately equal to nnity, or 
 
 ■■■e)"=e/-s)"-(i)" (■'■ 
 
142 ox CURVES IN SPACE. 
 
 From tliis result we see that the cosine of the angle which W 
 the tangent at (x, y, z) makes with the axis of x, which by J? 
 Art. 17 is 
 
 dx 
 
 "eft 
 
 j^ 
 
 dx^ fdy^ /dzV 
 Jt) "^ \dt) "^ \dt) 
 
 dx 
 
 1 , dt dx 
 
 is equal to — or ,- . 
 
 ^ ds^ ds 
 
 di 
 And similarly, the cosines of the angles which the tan- 
 gent makes wdth the axes of y and z are -y and -j- re- 
 spectively. 
 
 fdsV 
 Dividing hy I -y^J the equation (1) reduces to the form 
 
 ©■+(S)"^(S)'=' »■ 
 
 119. Any straight line through the point (x, y, z) per- 
 pendicular to the tangent is called a normal line. All 
 such lines lie in a plane through {x, ?/, z) perpendicular to 
 the tangent, which is called the normal plane. Its equation 
 is at once seen to be 
 
 (-'-)5+(y-2/)S+(^'-)J=o. 
 
 120. It is always possible to draw a plane through any 
 three points of a curve. The limiting position of this plane 
 when two of the points move up indefinitely near to the 
 third is called the osculating plane at that point. 
 
 Let the equations of the curve be 
 
 x = ^{t), y = f{t), z = xii) (1)' 
 
 and let t, t-hr, t + 2T be the values of t corresponding 
 
ON CURVES IN SPACE. 143 
 
 to three points on the curve. Let the equation of any 
 plane be 
 
 Ax+By+Cz=D (2). 
 
 If this plane pass through the three points t,t + T,t-{- 2t, 
 we have 
 
 Acl>(t) + B^lr{t) + Cxii) = D (3), 
 
 A(l>{t-{-r) + Bf{t + T) + Cx{t+T)=I) (4), 
 
 A<t> (t + 2t) + Bf(t + 2t) + Cx{t + 2t) = I) (5). 
 
 Subtracting the first of these equations from the second 
 we have 
 
 A {cp (t + T)-cj,{t)]+B{f{t + T)-ylr (t)} 
 
 + O{x{t + T)-x[t)}=-0, 
 Or, dividing by r, 
 
 ^^ cj, (t + t) - (f, (t) ^ ^ ^/^(^ + T)-^/.(0 
 
 T T 
 
 T 
 
 Subtracting twice the second from the sum of the first 
 and third and dividing by t^ we get 
 
 ^ (^(^+2T)-20(^-fT)+(^ffl ^^ ^/rft + 2T)-2^/r(^ + T)4-^/r(O 
 T T 
 
 ^ ^, X(t + 2r)-2x(t^r) + x(t) _^ 
 
 But if we make the three points coincide, r vanishes, 
 md these two equations become (Todhunter's l)iff. Calc. 
 irt 127) 
 
 . dx ^dy ^dz ^ 
 d'x dSj d'z _ 
 
 ^W^^W'^^df-^^ 
 
 ABC 
 
 ' ' dy d^z dz d^y dz d^x dx d^z dx d^y dy die 
 ^ df'Jtdt^ di~dt'~~dtde dtd^'^dtdf 
 
144 ON CURVES IN SPACE. 
 
 And subtracting (3) from (2) we have 
 
 Whence the equation of the osculating plane at the point' 
 (x, y, z) becomes 
 
 , , (dy d^z dz (Ty) , . [dz d^x dx d^z) 
 
 {X - X) j-^^ ^p - j^ ^,| + (i/ - 2/; 1^^ df~JtW] 
 
 ^^' ^^{dtdf dtdiy^- 
 
 121. The osculating plane is sometimes defined as the 
 plane which lies closer to a curve at a given point than 
 any other plane, and its equation is obtained in the fol- 
 lowing manner. 
 
 Let A{x'-x) + B{y-y) + C{z-z) = (1) 
 
 be the equation of any plane through (x, y, z). The perpen- 
 dicular on this from a point (x^, y^, z^) is 
 
 A {x^-x)+ JH^,-y ) + G{z^-z) 
 J A' ■\-B'-\- C 
 
 But if {x^, ?/j, z^ be a point on the curve correspondiDg 
 to a value ^ + t of ^, 
 
 _ dx t" d^x 
 
 '"'^'"'^'''dt^Xldf^ 
 
 dy T" fZ^/ . 
 dz T^ d'^z 
 
 Hence the length of the perpendicular becomes 
 
ON CURVES IN SPACE. 145 
 
 And when r is diminished indefinitely, the succeeding 
 : terms are very small compared with the first and second, 
 , and the smallest value which this fraction can assume will be 
 t when A, B, G are determined by the equations 
 
 dt dt dt 
 
 . d^CG -r, d\ . ^ d^z ^ 
 
 's whence we obtain the same result as in the last Article. 
 
 122. All straight lines drawn through the point (x, y, z) 
 . perpendicular to the tangent at that point are normals. That 
 : normal which lies in the osculating plane may be considered 
 as the normal drawn in the plane of the curve, and is called 
 the principal normal. The equations of the normal plane 
 and the osculating plane considered as simultaneous are the 
 equations of this line. 
 
 dx d^x 
 Writing for shortness x, x for -^ , -^ , and similarly 
 
 replacing the other differential coefficients, these are 
 
 (x ~x)x + {y-y)y + {z -z)z = 0, 
 
 {x - x) {yz - ijz) + {y - y) (zx - zx) + {z - z) {xy - xy) = 0. 
 
 If we put these equations in the form 
 X — X _y' — y _z — z 
 
 the value of P is 
 
 y {xy — xy) — i (zx — zx) 
 
 = x(yy + zz)-x(f+2^). 
 
 But by Art. 118, 
 
 s' = x' + f-\-z'', 
 
 therefore differentiating, 
 
 ss = XX + yy + zz. 
 A. G. 10 
 
146 ON CURVES IN SPACE. 
 
 Hence P = d) {ss — xx) — x (s^ — x^) 
 
 = s {xs — xs) ; 
 
 and similar values may be found for Q and R. Hence the 
 equations of the principal normal are 
 
 X — X _ y —y _ z — z 
 
 xs — xs ys ^ ys zs — zs 
 which may be written in either of the forms 
 
 X — X _ y — y _ z — z 
 d ldx\ d (dy\ d fdzX " 
 dtVdsl dt [dsj dt Vds) 
 
 (1), 
 
 X — X 
 
 or 
 
 ^-^ (2). 
 
 d^x d^y d^z 
 
 ds:^ ds^ ds^ 
 
 123. The equations (2) of the last Article can also be 
 obtained as follows. 
 
 If \ fi,v; X\ [X , V be the direction-cosines of two straight 
 lines, the direction-cosines of the two straight lines which 
 bisect the angles between them are proportional to A, -i- X', 
 /z- H- yLt', V -\-v and X — \\ /x — /j!, v — v. 
 
 For planes through the origin perpendicular to the two 
 given straight lines have their equations 
 
 \x -\- fiy + vz =^ (1) 
 
 and l^x + iJiJy + vz — (2) respectively. 
 
 By Art. 26 the equations of two planes which bisect the 
 angles between (1) and (2) are 
 
 {\ + X)x+{tL+ii)y+{v-\-v)z = 0, 
 (\-\') X -[- (fjL - fji') y + {v - v') z = 0. 
 
 And the direction-cosines of the normals to these planes, 
 which are evidently parallel to the bisectors of the angles 
 between the two original straight lines, are proportional to 
 X -f V, A6 + /Lt', 1^+1'' and \ - V, fi- fi', v - v respect- 
 ively. 
 
ON CURVES IN SPACE. 147 
 
 If I, m, n be the actual values of the direction-cosines of 
 the latter line, we have 
 
 , A/ — A, 
 
 V2-2cos^ ^ ^ 
 
 if 6 be the angle between the two straight lines. 
 
 124. Let now X, fju, v be the direction-cosines of the 
 tangent to a curve at the point {pc, y, z), and X', fj!, v their 
 values at an adjacent point on the curve distant hs from the 
 former. Then ultimately if the two points be made to 
 approach indefinitely near to each other and coincide, of 
 the two bisectors considered in the last Article, the one 
 will coincide with either tangent, and the other will be 
 the principal normal. The former will evidently have its 
 direction-cosines proportional to X -|- V, yu< -j- yu,', v + v , and 
 the latter must have its direction- cosines proportional to 
 X — \', fjj — fjLy V — v . 
 
 dX 
 
 But X' = X+-T- Bs + terms involving (Ssf 
 
 ^=^ + _8, + 
 
 / dv ^ 
 
 V = V + -r OS + 
 
 as 
 
 Hence the direction-cosines of the principal normal are 
 
 proportional to 37- §5, -J^Ss, -^ Ss, or to -^ , -^ . —^ and 
 
 as as as ds ds ds 
 
 doc du n z 
 putting for X, yu,, v their values -^ , -^ , -- the equations 
 
 of the principal normal become as before 
 
 x —X _y —y _z' —z 
 
 "Wx IFy d'z ' 
 ds^ ds^ ds^ 
 
 10—2 
 
 / 
 
148 
 
 ON CURVES IN SPACE. 
 
 125. If the curve be a plane curve, the equation of the 
 osculating plane must reduce to the equation of the plane in 
 which the curve lies. Hence the ratios 
 
 'dyd^z dz d^y" 
 Mde~~dt'dej 
 
 dzd^x dxd^z\ ^ (dxdj^y dy dj^x^ 
 
 dy d x\ 
 dt df dt dtV ' \dt df ~didf) 
 
 must be constant for all points on the curve. 
 
 0^( 
 
 We may therefore assume 
 
 yz-yz = \v (1), 
 
 zx — zx = fxv (2), 
 
 xy — xy—vv (3), 
 
 where \, /jl, v are constants, and v some function of t. 
 
 Eliminating X, ^ and v from (1) and (2) by differentiating, 
 we get 
 
 {zx — zx) {yz — yz) — {yz — yz) (zx — zx) - • 
 
 or reducing and dividing by i, 
 
 X (yz — yz) + y (zx - zx) + z {xy - xy) = 0, 
 
 which may be written 
 
 dx 
 
 dy 
 
 dz 
 
 dt' 
 
 dt' 
 
 dt 
 
 d^x 
 
 d^y 
 
 d'z 
 
 df 
 
 df 
 
 df 
 
 d'x 
 
 d'y 
 
 d'z 
 
 df 
 
 df 
 
 df 
 
 = 
 
 (4). 
 
 The symmetry of this relation shews that we should get 
 the same result by eliminating fi, v and v from equations (2) 
 and (3). 
 
 This relation may be also obtained from Art. 121, since ' 
 if the curve lie in the plane (1), the pei-pendicular on this 
 plane from any point in the curve must vanish. We must 
 therefore have 
 
ON CURVES IN SPACE. 149 
 
 whence the relation (4) follows. We must also have 
 
 for all values of n. But this will be the case if equation (4) 
 is satisfied for all points in the curve, as may be seen by 
 differentiating. 
 
 126. If a curve he dratun on a given surface such that 
 the inclination of its tangent to a given fixed plane is always 
 greater than that of any other tangent line to the surface at 
 the same point, the curve is called a line of greatest slope to 
 the given plane. 
 
 Let F(x, y,z) = (1) 
 
 be the equation of the given surface, and let 
 
 Ax + By + Cz = D (2) 
 
 be the equation of the given plane. 
 
 The direction-cosines of the tangent line to the curve at 
 
 . , . . dx dy dz 
 
 anypomt(^,2/,^)are^,^,^. 
 
 The equation of the tangent plane to (1) at {x, y, z) is 
 
 , , > dF , , . dF / , ^ dF ^ 
 (^'— )5-+(2/-2/)^ + {^-)^=0, 
 
 and the direction-cosines of the line of intersection of this 
 iplane with the plane (2) are proportional to 
 
 dF_^dF^ (jdF_^dF j^dF_^dF 
 dz dy ' dx dz' dy dx' 
 
150 ON CURVES IN SPACE. 
 
 and it is evident that the tangent line to the curve of greatest 
 slope must be perpendicular to the intersection of the tangent 
 plane with the plane (2), whence we get 
 
 as \ dz ay J ds\ ax dzj ds\ dy dxj 
 
 The integral of this equation united with (1) gives the 
 curves required. The integration will introduce one arbitrary 
 constant which is determined if one point on the curve be 
 known. Hence, a line of greatest slope can be drawn through 
 any point on the surface. 
 
 If the given plane be the plane of xy^ -4 = 0, 5 = 0, and 
 the equation (3) becomes 
 
 dF dy _dF dx _ ^ 
 dx ds dy ds 
 
 dFdy_dF_ . 
 
 dx dx dy 
 
 As an example of the last case take the equation of the 
 ellipsoid 
 
 ^2 yi £2 
 
 -2 + f2+-2 = l (5). 
 
 a DC 
 
 Equation (4) becomes 
 
 a'dx ¥ ' 
 •*• ~2 ^^S y ~ T? ^^o ^ ~ constant ; 
 
 ,\ y = mx^' (6). 
 
 This equation united with (5) gives the lines of greatest 
 slope. 1£ a = b, (6) becomes 
 
 y = 7}ix, 
 
 so that in the case of a spheroid the meridians are the lines 
 of greatest slope to the plane of circular section. 
 
ON CURVES IN SPACE. 
 
 151 
 
 127. We shall devote the remainder of this Chapter to 
 the discussion of the curvature of curves in space. This is of 
 two kinds, the first being the curvature of the curve con- 
 sidered as lying in its osculating plane, and the second, the 
 curvature by which it leaves the osculating plane, which is 
 called the curvature of torsion. On this account curves in 
 space are called curves of double curvature. 
 
 Before proceeding to the formulae relating to the two 
 kinds of curvature at any point of a curve some geometrical 
 explanations and definitions must be given. 
 
 Let PQ, QR, BS, ST, ... be a series of lines of equal 
 length, which when their length is diminished indefinitely 
 become ultimately small portions of a continuous curve. Let 
 p, q, r, s ... be their middle points. 
 
152 ON CURVES IN SPACE. 
 
 Through p let a plane be drawn perpendicular to PQ and 
 through q, r, s ... planes perpendicular to QR, RS, ST, . . . 
 respectively. These will ultimately be normal planes to the 
 curve at consecutive points. Let the planes through p, q 
 intersect in a line AE, and the planes through q, r in a line 
 BF which cuts AE in some point A, and so on. 
 
 Let the plane which passes through P, Q, R meet AE 
 in Oj, and the plane through Q, R, 8 meet BF in 0^. It is 
 evident that the point 0^ is equidistant from P, Q and R, 
 and a circle with centre 0^ and radius O^P will pass through 
 Q and R. This circle will ultimately pass through three 
 consecutive points of the curve, and lies in the plane PQRO^, 
 which is ultimately the osculating plane at Q. Hence it is 
 the circle of curvature of the curve considered as a plane 
 curve lying in the osculating plane. It is called the circle of 
 absolute or circular curvature, and the point 0^ is called the 
 centre of absolute or circular curvature. 
 
 Again, all points in the straight line AE are equidistant 
 from the three points P, Q and R. All points in the straight 
 line BF are equidistant from Q, R and S. Hence the point 
 A, where AE and BF meet, is equidistant from the four 
 points P, Q, R, and S, and a sphere with centre A and radius 
 AP will ultimately pass through four consecutive points of 
 the curve. The point A is called the centime of spherical 
 curvature, and the length AP the radius of sphencal curva- 
 ture. 
 
 The lines AE, BF, GG ... ultimately generate a surface 
 which is touched by the normal planes of the curve, and the 
 ultimate intersections of these lines produce a curve which 
 is called the edge of regression of this surface. 
 
 128. The locus of the centres of absolute curvature is 
 not an evolute, but an infinite number of evolutes can be 
 drawn on the surface generated by the lines AE, BF,... For 
 let Oj be any point in AE, and let pO^, qO^ be joined and 
 qO^ be produced to meet BF in u ; join ru and produce it 
 to meet CG in v ; join sv and produce it to meet DH in w, 
 and so on. We have 
 
 0,P = O^q ; 
 
ON CURVES IN SPACE. 153 
 
 vu + uO^ + 0,p = vu-{- uq = vu+ ur = vr, 
 
 Hence if a string be laid along the curve luvuO^ and its 
 end be at p, as it is unwrapped this extremity will pass 
 through qrst... and describe ultimately the original curve. 
 
 An evolute can thus be found passing through any point 
 of any one of the lines AE, BF... 
 
 129. The centre of absolute curvature may be defined as 
 the point where the line of intersection of two consecutive 
 normal planes meets the osculating plane. 
 
 Let the equation of the normal plane at a point {x, y, z) 
 be denoted by 
 
 ^© = (1). 
 
 Any other normal plane can be represented by 
 
 -P(O = (2), 
 
 where t^ is the corresponding value of t 
 
 At the points where (1) and (2) meet, we have 
 
 F{t,)-F{t) = 0, 
 
 F{Q-F(t) „ 
 or — ^ ^ = 0. 
 
 And this latter equation when ^^ — ^ is indefinitely diminished 
 becomes 
 
 f=» «. 
 
 Hence the line of intersection of two consecutive normal 
 planes is given by the two equations 
 
 Fit) = 0, f =0. 
 But ^(0 = (.'-.)J + (,'-,)|+(/-.)g, 
 
154 ON CURVES IN SPACE. 
 
 dF , , .(Fee , , .d^y , , .^z fdxV fduV fd: 
 
 2 
 
 , , . d'^x , , . d^ij , , .dj^z /ds\'^ 
 
 Hence tlie line of intersection of two consecutive normal 
 planes is given by 
 
 (x'-x)x+{y'-y)y+{z-z)z = (4), 
 
 (x' — x)x + (y -y)p -^(z -z)z = ^^ (5). 
 
 The point where this line meets the osculating plane is 
 given by (4) and (5) united with 
 
 {x-x)(yz-yz) + (y-y)(zx-zx) + (z'-z){xy-xy) = 0.,.{6). 
 
 From (4) and (6) we obtain as in Art. 122 
 
 X -X ^y' -y ^ z -z ^ 
 
 x's — xs ys — ys zs — zs 
 
 and by equation (5) each of these fractions is equal to 
 
 s" 
 
 2 
 
 {x^ + y^-\-'z^)s-ss 
 
 m 
 
 J. 
 
 2 
 "2 
 
 Also each of them is equal to 
 
 (xs — xsY + {'ys — ys)'^ + {zs — zs) 
 
 P 
 
 Sjx^ + 7/' + ^ - S' 
 
 where p is the radius of absolute curvature. 
 Hence 
 
 <ds\' 
 
 p 
 
 r'(S)"HS)'HS)"-{S)' <»'• 
 
ON CURVES IN SPACE. 155 
 
 or if s be taken as independent variable, 
 
 i fd'x\' fdyV fd'zV 
 
 Equation (9) or (10) gives p, and equations (7) and (8) 
 give a)\ y\ z the co-ordinates of the centre of absolute cur- 
 vature. 
 
 130. The results of the last Article can be obtained by 
 means of the formulae proved in Article 123. 
 
 For if Zs be the arc between two points of a curve, Q the 
 angle between the tangents at those points, and X, yu,, v\ 
 X, /jl\ V the direction-cosines of those tangents, we have 
 
 p = limit -J , 
 
 when Ss is indefinitely diminished. 
 
 Also if I, m, n be the direction-cosines of the principal 
 normal, we have 
 
 1 d 
 
 I = limit of ^ (\ - V) cosec - 
 
 6 
 = limitof ^.^.— ^ = pg-=p^-^. 
 
 Similarly m =Pds^Pdf' 
 
 dv d?z 
 
 But also by Art. 6 if og', y\ z be the co-ordinates of the 
 centre of absolute curvature, 
 
 X —x=lp=p 
 
 ^/-y=mp=p^ 
 
 ds'' 
 
 d^y 
 
 ds'' 
 
156 ON CURVES IN SPACE. 
 
 z-z=np = p-^,, 
 whence squaring and adding and dividing by p*, we get 
 
 1 _ /^y /^y (cVz-^ 
 
 131. To find the centre and radius of spherical curva- 
 ture. 
 
 The centre of spherical curvature is the point in which 
 three consecutive normal planes intersect. 
 
 If F {t) = be the equation of any normal plane, the line 
 of intersection of this with the consecutive plane is given by 
 
 i^(0 = 0, and i^' (0 = (1). 
 
 And if F{t^ = 0, F' (t^) = be the equations of any other 
 line of intersection of consecutive normal planes, at the point 
 of intersection of these two lines, 
 
 m^) = (2). 
 
 And ultimately when t = t^ the four equations give 
 
 F(t) = 0, F'{t) = 0, 
 
 and from (2) 
 
 r'{t) = (3). 
 
 But 
 
 J"(0 = (^-)5 + (F-,)J+(^-.)J-gy=o 
 
 ■■■{*)■ 
 
 These equations determine X — cc, Y— ?/, Z — 2, where 
 X, Y, Z are the co-ordinates of the centre of spherical curva- 
 
 ture. We get from them 
 
— x = 
 
 ( 
 
 ON CURVES m SPACE. 157 
 
 ds d?s (dy d^z dz dy\ 
 
 dsV fdz d^y _ dy d^z\ 
 di) [Jt'df'dtdfr' 
 
 - 
 
 dtdf \dt df dt dfj 
 
 dx dy dz 
 'di' 'di' ~di 
 
 d^x d^y d^z 
 If' df' df 
 
 d^x d^y d^z 
 di" df' df 
 
 and similar values for Y—y^Z — z. The radius of spherical 
 curvature R, which 
 
 is then known. 
 
 132. To find the angle and radius of torsion at any point 
 of a curve. 
 
 The angle of torsion (Se) is the angle between two con- 
 secutive osculating planes. 
 
 Let \ fx, V be the direction-cosines of the normal to one 
 osculating plane; then those of the normal to the osculating 
 plane corresponding to the value ^ + r of the independent 
 variable will be 
 
 d\ dfjL dv 
 
 And the sine of the angle between these lines is (Art. 8) 
 dv dLbV f dx ^ dvV / da d\^^ 
 
 ,(4) 
 
 But 
 
 dt] V dt '^ dt. 
 
 dt 
 
 dz d^x dx d^z 
 
 + i^*-^* 
 
 z\ _ 7 fdx d^y dy d'^x\ 
 
 V' ""' [dtdf'lidf)' 
 
 4t df dt df 
 
 , 1 _ I'dy d^z dz d^yV fdz d^x _ dx d'^zV 
 
 where p - l^^"^ "J^^^ j ^ \Jt~df ~ dt~df) 
 
 'dx d^y dy d^xV ^ 
 
 { 
 
 dt df dt df 
 
158 
 
 ON CURVES IN SPACE. 
 
 dv dfjL , o /dz d^x dx d^z\ (dx d^y dy d 
 
 x^ 
 
 V 
 
 dt 
 
 ,dt df dt dfj \dt df 
 
 - P(' 
 
 dz d' 
 
 X 
 
 dt df 
 
 = * di 
 
 dx 
 di 
 
 d^x 
 
 d^x 
 
 dy 
 dt' 
 
 d\ 
 df ' 
 
 dx d^z' 
 dtdf, 
 
 dz 
 
 dt 
 
 d^z 
 df 
 
 d'z 
 
 dt dfJ 
 
 dx d^y dy d^x^ 
 'dtdf~'dtdfj 
 
 df df df 
 
 .(1). 
 
 Whence the sine of the ang-le of torsion becomes 
 
 equal to 
 
 
 dx 
 
 dy 
 
 dz 
 
 
 dt' 
 
 dt' 
 
 dt 
 
 
 d'x 
 df 
 
 d\j 
 df 
 
 d'z 
 df 
 
 
 d'x 
 
 d'y 
 
 dz 
 
 
 df 
 
 df 
 
 df 
 
 .(2), 
 
 and the radius of torsion p^ which is equal to the limit of the 
 small element of arc hs divided by the angle of torsion is 
 given by the equation 
 
 -/j 
 
 .2 
 
 Pi 
 
 dx 
 
 dy 
 
 dz 
 
 dt' 
 
 dt' 
 
 dt 
 
 d'x 
 
 dy 
 
 dz 
 
 df 
 
 df 
 
 df 
 
 d'x 
 
 dy 
 
 dz 
 
 df 
 
 df 
 
 df 
 
 .(3). 
 
 133. It may be noticed that both the radius of spherical 
 curvature and the radius of torsion become infinite if 
 
ON CURVES IN SPACE. 
 
 159 
 
 dx 
 
 dy 
 
 dz 
 
 dt' 
 
 dt' 
 
 dt 
 
 d'x 
 
 d'y 
 
 d'z 
 
 df 
 
 df 
 
 df 
 
 d'x 
 
 d'y 
 
 d'z 
 
 df 
 
 df 
 
 df 
 
 = 0. 
 
 This is in fact the condition that four consecutive points 
 of the curve should lie in one plane (see Art. 125). 
 
 We have also 
 ¥ \[dt ^ 
 
 dy]' (d_z^ 
 dt) '^UtJ 
 
 Jf) "^ [df. 
 
 + 
 
 
 dx d^x dy d^y dz d'z 
 
 + 
 
 + 
 
 = r^vi^^v + ^^ -^^ 
 
 ^d'z' 
 
 xltJ \\df 
 
 4f. 
 
 ■^ [dfj [df, 
 
 /ds\' 
 [dtJ 
 
 dt df dt df dt dt 
 'd's' 
 
 0' 
 
 (1), 
 
 where p is the radius of absolute curvature. 
 
 Hence if we denote the above determinant by the symbol 
 A we have 
 
 1 p^A 
 
 \dt> 
 
 ,(2). 
 
 EXAMPLES. CHAPTEK IX. 
 
 1. Find the equations of the osculating plane at any 
 point of the curves 
 
 (1) x = a cos 6, y = a sin 6, z = cO. 
 
 (2) x^-]-y'-ry = 0, z' + ry = r\ 
 
160 EXAMPLES. CHAPTER IX. 
 
 Find also the length of the arc of (1) between two points 
 whose co-ordinates are given. 
 
 2. The equations of a curve of double curvature being 
 given, find the equation of the surface formed by making it 
 revolve round the axis of z. 
 
 Ex. x = a cos Q, y = asm6, z = c6. 
 
 3. A helix joins two points, the distance between which 
 is h, the angle between the tangent lines and the axis of the 
 generating cylinder being a given angle a ; prove that if the 
 lenofth of the helix is a maximum, the helix has a constant 
 
 7, 
 
 lenofth, and that the radius of the generating circle is -^ — - , 
 where n is a positive integer. 
 
 4. A curve is traced on a right circular cylinder of radius 
 a, such that if the cylinder were unrolled into a plane the 
 curve would become a catenary whose axis formed one of the 
 generating lines, and directrix the base, of the cylinder. 
 Shew that 
 
 _ az^ _ az^ 
 
 p, /3j being the radii of absolute curvature and torsion, z the 
 ordinate, s the arc measured from the vertex, and c the con- 
 stant of the catenary. 
 
 5. Find the equation of the osculating plane at any 
 point of the curve given by the equations 
 
 x + y ■{■ z = 1, 
 
 ax^ -f hy^ + cz^ = 1. 
 
 6. Find the equations of a curve traced on a sphere so 
 as to cut all the great circles passing through a fixed point 
 at the same angle. 
 
 7. Find the equations of the lines of greatest slope to 
 the plane of xy on the surfaces 
 
 (1) xyz = a^ 
 
 /o\ a , x^ + y^ 
 
EXAJklPLES. CHAPTER IX. 161 
 
 8. Shew geometrically and analytically that if a sphere 
 be described concentric with a given ellipsoid, the tangent 
 line to the curve of intersection of the sphere and ellipsoid is 
 parallel to one of the principal axes of the central section of 
 the ellipsoid which passes through that tangent line. 
 
 9. Find the equations of a curve traced on a sphere, such 
 that the sum of the arcs of great circles joining any point on 
 it with two fixed points on the sphere, the arc joining which 
 is a quadrant, is constant. 
 
 10. Find the equations of a curve traced on a sphere by 
 a point which moves with constant velocity along the arc of 
 a great circle w^hile the great circle revolves with constant 
 velocity round a fixed diameter. 
 
 11. A point moves on an ellipsoid so that its direction of 
 motion always passes through the perpendicular from the 
 origin on the tangent plane to the ellipsoid at that point. 
 Shew that the curve traced out by the point is given by the 
 intersection of the ellipsoid with the surface 
 
 ^m-n yu-i ^i-m ^ constant, 
 
 I, m, n being inversely proportional to the squares of the axes 
 of the ellipsoid. 
 
 12. Find the equation of a curve traced on a right cone 
 which cuts all the generating lines at a constant angle. 
 
 Find the length of the curve measured from the vertex. 
 
 13. A straight line is drawn on a plane which is then 
 wrapped on a cone. Shew that if p be the radius of absolute 
 curvature of the curve on the cone at a distance r from the 
 vertex 
 
 r' = + a^p, 
 
 where a is a constant. 
 
 14. Find the values of the radii of absolute and spheri- 
 cal curvature at any point of a helix. 
 
 15. Find the locus of the centres of spherical curvature 
 of a helix 
 
 A. G. 11 
 
162 EXAMPLES. CHAPTER IX. 
 
 16. If, at any point of a curve, equal lengths Ss be 
 measured along the curve and its circle of curvature, the dis- 
 tance between the extremities of these lengths is ultimately 
 equal to 
 
 Bl /^ l/c/py 
 6/>V a' '^ p' \ds I ' 
 
 p being the radius of curvature and cr the radius of torsion at 
 the point. 
 
 17. Shew that the normal plane at any point to the 
 locus of the centres of circular curvature of any curve bisects 
 the radius of spherical curvature at the corresponding point 
 of the original curve. 
 
 18. If a curve be drawn on a right circular cone cutting 
 all the generating lines at a constant angle /?, shew that the 
 radius of absolute curvature at any point is to the correspond- 
 ing radius of curvature when the cone is developed in the 
 ratio of sin a to ^ sin^ a cos^ ^ + sin'' /S. 
 
 19. Shew that the curves represented by the equations 
 are circles of radius a. 
 
 a' 
 
CHAPTER X. 
 
 ON ENVELOPES. 
 
 134. Let the equation of any surface be 
 
 F{x,y,z,a) = (1), 
 
 where a is a constant. If a be changed to a we obtain the 
 equation of another surface 
 
 F{x,y,z,a!) = (2), 
 
 differing from (1) in magnitude or position or both, but of the 
 same general nature. 
 
 These two equations will both be satisfied by the co- 
 ordinates of all points in the curve of intersection of the two 
 surfaces, and if we suppose the value of d to approach indefi- 
 nitely near to that of a, this curve of intersection approaches 
 some limiting position. The locus of all such limiting positions 
 for different values of a is a surface which is called the envelope 
 of the surface (1). Its equation can be found in the following 
 manner. 
 
 At all points for which (1) and (2) are satisfied, we have 
 
 F{x,yy z, a) = 0, 
 
 F{x, y, z, a) - F {x, y, ^> ^) ^ q 
 a — a 
 
 But ultimately when a' becomes equal to a these equa- 
 tions reduce to 
 
 F(x,y,0,a) = O (3), 
 
 ^^F{x,y,z,a) = .(4), 
 
 11—2 
 
164 ON ENVELOPES. 
 
 which are therefore the equations of the ultimate position of 
 the curve of intersection of (1) and (2). Ehminating a be- 
 tween (3) and (4) we obtain the equation of the locus of such 
 curves, or the envelope of the surface (1). 
 
 135. The curve given by the two equations (3) and (4) 
 of the last article is called the characteristic of the envelope. 
 
 If we take the equations of two consecutive characteristics 
 and treat them as in Art. 131 we get, to determine their point 
 of intersection, the three equations 
 
 F {oo, y, z, a) = 0, 
 
 F (x, 2/, z, a) = 0, 
 
 F' (x, y, z, a) = 0. 
 
 If between these three equations we eliminate a we shall 
 get two relations between x, y, z which are the equations of 
 the locus of ultimate intersections of two consecutive charac- 
 teristics. The curve so obtained is called the edge of regres- 
 sion of the envelope, or sometimes simply the edge of the 
 envelope. 
 
 Thus the line given by equations (4) and (5) of Art. (129) 
 is the characteristic of the envelope of the normal planes to 
 the curve, while the locus of the centres of spherical curvature 
 is the edge of regression of the same envelope. 
 
 136. We will now shew that the envelope obtained in 
 Article 134 touches each of the series of intersecting surfaces. 
 
 For suppose from equation (4) of that article we obtain a 
 
 value of a, 
 
 a = 4>{x,y, z). 
 
 Substituting in (3), the equation of the envelope becomes 
 
 F[x,y,z,<i>{x,y,z)} = (1). 
 
 dz dz 
 
 The values of -r- and , at any point of this surface are 
 
 given by the equations 
 
 dF dF d^ ^F/d<f> dcf) dz\ 
 
 dx dz dx d4> \dx dz dxj 
 
 dF dF(h dF/dcf) d^d^\ 
 
 dy dz dy d(f) \dy dz dyj 
 
 (2). *■ 
 
 i 
 
(3). 
 
 ON ENVELOPES. 165 
 
 At any point of the surface F {x, y, z, a) = the values of 
 -J- and -^ are given by the equations 
 
 cLoc ^y 
 
 dF clFdz ^ ^"^ 
 
 doc dz dx 
 
 dF dF dz _ 
 
 dy dz dy 
 
 But at the points where the envelope meets the surface 
 
 Fix, y, z, a) = 0, 
 ■we have 
 
 a = (l>(x, y, z) and j^ = 0- 
 
 7 77? tl TP 
 
 'Now Tj only differs from -j- in having (/> {x, y, z) instead of 
 d<p da 
 
 a, consequently at all the points of intersection of the surface 
 
 F(x,y, z,a) = 
 
 with the envelope, ^-r = 0, and the equations (2) become iden- 
 
 d(p 
 
 d z dz 
 
 tical with equations (3). The values of j- and ^ being the 
 
 same for the surface and its envelope, the two surfaces touch. 
 
 137. If the equation of a surface be 
 
 F(x,y,z,a,h) = \ (1), 
 
 -when a and h are constants, any two other surfaces formed by 
 ■giving new values to a and h will intersect (1) in a point or 
 points, which assume a limiting position when the new values 
 of a and h approach indefinitely near to their first values. 
 ^The locus of such limiting positions is called the envelope 
 of the surface (1). 
 
 Let a and h become a-^h, h + k respectively. The equa- 
 tion of the corresponding surface is 
 
 F (x, y, z, a-\-h, b + k)=0, 
 
 or F{x, y, z, «, h) + hF' (a + Oh) + kF' (h + Ok) = 0. - .(2), 
 
166 ON ENVELOPES. 
 
 where ^ is a proper fraction and F' (a + Oh) means that 
 F(a), y, z, a, h) has been differentiated with respect to a, and 
 a -f Oh put in the result for a. 
 
 At the points of intersection of (1) and (2) we have 
 
 F{x,y,z,a,h) = ] ,^. 
 
 JiF {a + Oh) +kr (h + Ok) =0 J ^ ^' 
 
 and whatever be the ratio of h to k, when h and k are di- 
 minished indefinitely all the curves of intersection given by 
 (3) pass through the points given by 
 
 F{a),y,z,a,h) = 0, F'(a)=0, F' (b) = 0. 
 
 By eliminating a and b between these equations we obtain 
 the equation of the envelope. 
 
 138. The envelope in this case also touches each of the 
 series of intersecting surfaces. For let the equation of one 
 of the surfaces be 
 
 F(x,y,z,a,h) = (1). 
 
 The corresponding point on the envelope is given by (1) com- 
 bined with 
 
 dF ^ dF 
 
 From (2) we can obtain by solving for a and b 
 
 and the equation of the envelope becomes 
 
 F[x, y, z, (t>^ (x, y, z), </>, {x, y, 3)] = 0. 
 
 The values of -p and -^ for any point of the envelope 
 are given by the equations 
 
 dF dFdz_ dF M, dcp.dzX dF/dJ) d^^d_z\^ 
 dx dz dx d(f)^\dx dz dx) d(f>^\dx dz dx) 
 
 dF dFdz d]^/d4^ , d(j)^ch\ dF /d(p^ 
 
 dy dz dy d<f)^ \ dy "^ dz dy) d^^ \ dy dz dy) 
 
ON ENVELOPES. 167 
 
 But at the points where (1) meets the envelope 
 
 consequently at those points -yj- = 0, -,— = 0, and the above 
 
 equations become 
 
 dF dFdz_ 
 dx dz dx 
 
 dF dFdz^Q 
 
 dy dz dy 
 
 dz d z 
 
 which are the same as the equations which give -y- and -^ 
 
 for the point {x, y, z) of the surface (1). Hence at the points 
 
 where (1) meets the envelope the values of -7- and -7- are 
 
 the same for the surface and the envelope, which therefore 
 touch one another at those points. 
 
 139. If the equation of a family of surfaces contains n 
 arbitrary constants connected by (n — l) equations there is 
 really oJie independent constant, and the envelope can be 
 found by substituting for {n — l)oi the constants their values 
 in terms of the ?^^^ It is better in general to consider 
 (n — 1) of the constants to be functions of the n^^, and dif- 
 ferentiating all the equations to eliminate by undetermined 
 multipliers. 
 
 If the n constants be connected by (?i — 2) equations, two 
 of the constants are arbitrary, and the envelope falls under 
 the second class. The method of undetermined multipliers 
 can be used in this case also. 
 
 For examples of the solution of problems the reader is 
 referred to Todhunter's Differential Calculus, Chapter xxv., 
 the methods employed there being equally applicable to the 
 problems of Solid Geometry. 
 
168 ON ENVELOPES. 
 
 140. The polar plane of any point (a, /3, 7) with respect 
 to any quadric can be obtained as in Art. 106. If the point 
 (a, ^, 7) be constrained to lie on any given surface 
 
 f{x,y,z)^(i (1), 
 
 the equation of its polar plane will contain three parameters 
 a, y8, 7 connected by one relation 
 
 /(a,/3,7) = 0. 
 
 The equation of its envelope can therefore be found by 
 the methods of Art. 137. 
 
 Suppose this equation to be 
 
 .p{x,y,z) = Q (2). 
 
 Then any point {a\ ^'^ 7') in (2) is the limiting position of 
 the point of intersection of the polar plane of some point 
 i'X, y8, 7) on (1) with the polar planes of points on (1) adja- 
 cent to (a, JS, 7). Hence by Art. 106 the polar plane of 
 (a', /3', 7') with respect to the given quadric must pass 
 through the point (a, /3, 7) and other points on (1) contiguous 
 to (a, /3, 7), that is the polar plane of (a", yS', 7') is a tangent 
 plane to (1) at (a, /3, 7). Thus the surface (1) is the en- 
 velope of the polar planes of all points on (2) with respect 
 to the same quadric. The two surfaces are from this pro- 
 perty called reci2:)rocal polar s. 
 
 Each surface may be also defined as the locus of the 
 poles of the tangent planes to the other with respect to the 
 given quadric. 
 
 141. Let the quadric with respect to which the polars 
 are taken be the sphere, 
 
 aJ2 + / + ^2 = ^.2 (J) 
 
 The equation of the polar plane of any point (a, yS, 7) with 
 respect to this sphere is 
 
 ax-^(3y + r^z = k'' (2). 
 
 Let the surface to be reciprocated be the ellipsoid 
 
 a^^ y"^ ^^ 1 /ox 
 
ON ENVELOPES. 169 
 
 Hence we have to get the envelope of the plane (2), a, /5, 7 
 being parameters connected by the relation 
 
 a^ /3' rJ- 
 
 a^+F + ?=l W- 
 
 Using the method of undetermined multipliers we get to 
 determine the envelope, 
 
 whence 1 + \k^ = ; 
 
 _a^x o_^^y _ ^^^ 
 
 and substituting in (2) the envelope becomes 
 
 aV + 6y + cV = ^* (5). 
 
 The surface represented by (5) is often called the reci- 
 procal ellipsoid of that represented by (1). 
 
 EXAMPLES. CHAPTER X. 
 
 1. Find the envelope of the series of planes 
 
 where a, ^, 7 are parameters connected by the relations 
 
 a,' + ^' + y' = l, 
 la + mjB + ny = 0. 
 
 2. Find the envelope of a sphere of constant radius 
 which moves with its centre on a fixed circle. 
 
 3. Find the envelope of central sections of an ellipsoid 
 of which one axis is constant and equal to k. 
 
170 EXAMPLES. CHAPTER X. 
 
 4. Find the envelope of planes which are the polars of 
 points on the ellipsoid 
 
 x^ if z" ^ 
 — I- - ^ — = 1 
 
 with respect to the ellipsoid 
 
 5. Find the envelope of a sphere of constant radius which 
 moves with its centre in a fixed plane. 
 
 6. Find the envelope of an ellipsoid whose axes are 
 given in direction and the product of whose axes is constant 
 and equal to 8^^ 
 
 7. Find the envelope of the series of planes 
 
 Ix + my + nz = v^ 
 
 where I, m, n, v are parameters connected by the relations 
 
 "1 2 72 "T" ~o ~o V/« 
 
 v^ — a^ v^ — If v^ — & 
 
 8. Find the envelope of a sphere whose centre is at a 
 point (a, /3, 0), and radius is 7 where a, yS, 7 are connected ; 
 by the relation _^ 
 
 h being a constant. 
 
 9. Find the envelope of the surface 
 
 ?'+ll + £^ = i *> 
 
 a /S 7 
 where a, yS, 7 are parameters connected by the relations 
 
 a, 6, c being constants. 
 
EXAMPLES. CHAPTER X. 171 
 
 10. Find the envelope of all planes whicli cut off a 
 constant volume from the co-ordinate axes. 
 
 11. Find the envelope of a series of planes which move 
 so that the perpendicular on them from the origin is constant 
 in length. 
 
 12. Find the envelope of a series of planes which move 
 so that the area of the section of an ellipsoid made by any 
 one is in a constant ratio to the area of the parallel section 
 through the centre of the ellipsoid. 
 
 13. Find the envelope of a sphere of constant radius 
 which moves with its centre on a fixed sphere. 
 
 14. Find the envelope of the plane 
 
 ax ^ Py I 7^ 
 
 _ _L ^ J_ ZZ — 1 
 ^2 "•" L2 "T ^2 ■*■' 
 
 when a, P, y are connected by the relations 
 
 a' fi' y' , 
 
 la + mff + ny= 1. 
 
 15. Through a given point (a, /3, 7) a series of chords 
 are drawn to an ellipsoid whose equation is 
 
 oc^ if z^ - 
 — \-— A — = 1 
 
 ^2 + ^2 + ^2 -L, 
 
 in such directions that the line of intersection of the tangent 
 planes at the extremities of each chord is perpendicular to 
 that chord. Prove that the envelope of the lines of inter- 
 section of the tangent planes is a parabola which is the 
 intersection of the polar plane of (a, (3, 7) with the cone whose 
 equation is 
 
 ^{V - c') ax , 7(o' - a') py , J{a' - 6') 7^ _ ^ 
 a c 
 
CHAPTER XI. 
 
 ON FUNCTIONAL AND DIFFERENTIAL EQUATIONS OF 
 FAMILIES OF SURFACES. 
 
 142. To find the general equation of conical surfaces. 
 
 A conical surface is generated by a straight line luhich 
 ahuays passes through a fixed point and meets a fixed curve. 
 
 Let {a, /3, 7) be the fixed point, and let the equations of 
 any generating line be 
 
 ^=2/_^ = i^ (1). 
 
 I m n 
 
 Let the equations of the curve through which (1) always 
 passes be 
 
 y = <t>{x\ z = ylr(x) (2). 
 
 Since (1) always meets (2) we have 
 
 m 
 
 ^ + j{x-a) = (j){x), 
 
 f 
 
 7 + -^ (a: — a) = -v/r {x). 
 
 And eliminating x between these equations, we shall get 
 
 n 1)1 
 
 a relation between -j and y, which can be put into the 
 
 form 
 
EQUATIONS OF FAMILIES OF SURFACES. 173 
 
 whence the equation of the cone becomes 
 
 '-^^f(V^) (3). 
 
 This is the functional equation of conical surfaces. In 
 all cases it is clear that the equation is homogeneous in 
 x — a,y — ^,z — y; in fact the result we have obtained is 
 the analytical statement of the fact that the equation of 
 any conical surface whose vertex is at a point (a, ^, <y) is 
 homogeneous in x — a, y — ^, z — y ; an extension of the 
 result of Art. 34. 
 
 A differential equation holding for all such surfaces can 
 be deduced thus. 
 
 From (3) differentiating with respect to sc, 
 
 ax \x — a/ \x — aj \x— aj 
 and with respect to y, 
 
 dy \x — a) ' 
 
 dz _z — y y — ^ dz 
 ' ' dx X —OL X — a dy' 
 
 143. To find the general equation of cylindrical surfaces. 
 
 A cylindrical surface is generated by a straight line luhich 
 moves always parallel to itself and meets a fixed curve. 
 
 Let I, m, n be the direction-cosines of any one of the 
 generating lines, and 
 
 X—x Y—y Z—z .^. 
 
 — j—= ^ = =r (1) 
 
 m n 
 
 the equations of the line. Let the equations of the directing 
 curve be 
 
 Y=4>(X), Z=^{X) (2). 
 
 Since (1) meets (2), we have 
 
 y + mr = (p{x + Ir), z -{■ nr = yjr (x + Ir), 
 
174 ON FUNCTIONAL AND DIFFERENTIAL EQUATIONS 
 
 and by eliminating r between these two equations we get a 
 relation between x, y, z, the co-ordinates of any point in any 
 one of the generating lines, which is therefore the equation 
 of the surface. 
 
 The general form of the result is obtained thus. 
 From (1) mX — Z F = mx — ly, 
 
 nT— mZ= ny — mz. 
 
 But from (2) mX — lY and nY—mZ can ordinarily both 
 be expressed as functions of X, and we can therefore deduce 
 a relation of the form 
 
 mX-lY=F{nY-mZ)', 
 
 .'. mx — ly = F {ny — mz) (3), 
 
 which is the general functional equation of cylindrical 
 surfaces. 
 
 The differential equation can be deduced. For from (3), 
 differentiating with respect to x, 
 
 dz 
 m = — mF' {ny — mz) -^ , 
 
 and differentiating with respect to y 
 
 dz\ 
 dy 
 
 — I = ( n — m^ ] F' {ny — mz), 
 
 , J dz dz 
 
 whence t ^- —n - m ^ , 
 
 ax dy 
 
 idzdz ... 
 
 dx + "'dy = " (*>• 
 
 If the direction of the generating line of the cylindrical 
 surface be parallel to the axis of y we have 1 = 0, m=l, 
 n = 0, and equation (3) becomes 
 
 x = F{-z) or f{x,z) = (5). 
 
 Any equation of this form represents therefore a cylin- 
 drical surface whose base is the curve of which (5) is the 
 equation regarded as an equation restricted to the plane 
 of zx. 
 
OF FAMILIES OF SURFACES. 175 
 
 Similarly the equations 
 
 f(?=, y) = 0, 
 fdi, ^) = 0, 
 
 represent cylindrical surfaces whose generating lines are 
 parallel to the axes of z and x. 
 
 These results are obvious also from general consider- 
 ations. 
 
 144. To find the general equation of conoidal surfaces. 
 
 A conoidal surface is a surface generated hy a straight 
 line luhich always meets a fixed straight line, is parallel to a 
 fixed plane, and meets a fixed curve. 
 
 Let the equations of the fixed line be 
 
 ^ = l^ = '-^ = r (1), 
 
 l m n 
 
 ^and let the equation of the fixed plane be 
 
 I'x +^ m'y -\-7iz = (2). 
 
 The co-ordinates of any point in (1) can be represented 
 by oL + lr, /8 + mr, y + nr, and the equations of any straight 
 line through this point are 
 
 X — a — lr _y — — mr _z — y — nr 
 
 m^ ^^ ^^ ••••••••yO/a 
 
 A fJb V 
 
 If this be parallel to (2), we have 
 
 \l' + iJLni + vn =0 (4). 
 
 From (3) and (4) 
 V {x — a) + ni (y - I3) + n (z -y)= (W + mm' + nn) r. . .(5), 
 
 and from (3) eliminating r 
 
 n\ — ly_ n (x — a) — l(z — 7) 
 
 n/Jb—mv n(y — ^)— m(z — y) ^ ^' 
 
 Now the condition that the straight line (3) may meet the 
 |fixed curve, combined wdth (4), will ordinarily enable us to 
 
176 ON FUNCTIONAL AND DIFFERENTIAL EQUATIONS 
 
 express - and - as functions of r, and consequently we can 
 arrive at a result of the form 
 
 — = F \(ll' 4- mm + nn') r], 
 
 TlfM—mv ^ ^ 
 
 or 
 
 „(a; a)-;(^-7) ^j, ^^ ^ ^^^^ 
 
 n(y — p) — m{z — y) ^ "^ #/j x / 
 
 which is the general functional equation of conoidal surfaces. 
 
 If the fixed plane be taken as the plane of ocy, and the 
 point where the fixed line meets it as the origin, we have 
 
 r =0, m' = 0, n=l, a = 0, /3 = 0, 7 = 0, 
 
 and the equation (7) becomes 
 
 71X-IZ ^^ 
 
 7iy — mz 
 If the fixed line be perpendicular to the fixed plane 
 
 Z = 0, m=0, ?i = l, 
 and the equation of the surface becomes 
 
 In this case the surface is called a right conoid. 
 
 145. The differential equation of conoidal surfaces can be 
 deduced from (7) ; for differentiating it with respect to a?, we 
 have 
 
 ' dz \ dz 
 
 n - I ^J {n iy-13)- m (^ - ry)l + m ^ [n (x-cl)-1(z- y)} 
 
 {n{y-l3)-m{z-y)}'' ' 
 = {l'+n^^F'[l'{x-a)^m'{y-^) + n'{z-y)]; 
 
 "^■^ 
 
OF FAMILIES OF SURFACES. 177 
 
 and differentiating with respect to y, we have 
 
 dz ( dz\ 
 
 ^ dy ^''^^ " ^^ " ^^^^ ~ '^^' ^ V ~ ^^^ dy) {^^(^"°') - ^ (^ - 7)} 
 
 [n (2/ - /3) - m (^ - 7)}' 
 
 = ^m' + n' -^) i?'' {^' (a; - a) + m (y - ^) + n' {z - 7)}, 
 
 and reducing and eliminating 
 
 F {V {x-0L) + m! {y - ^) +n' {z -r^)] 
 
 we obtain 
 
 (dz\ dz 
 
 + (^' + "' £) L"(^-") - H^ - 7) + |<l«(y--8)-™(c8-^)}] = 0, 
 
 or 111 [n (2/ — /3) — w (^ — 7)} + 1' [n (cc— a) — l(z — 7)} 
 
 dz 
 
 + -J- [m {m {x — OL) — l{y — p)] + n [n{x — a) — l{z — 7)}] 
 
 + jK{n(2/-/3)-m(^-7)}+r{K2/-yS)-m(^-a)}] = 0...(10). 
 
 The differential equation corresponding to equation (8) is 
 obtained by putting 
 
 a = 0, /3 = 0, 7 = 0, r = 0, m=0, n =1, 
 Land is therefore 
 
 {nx-lz)^-\-[ny-mz)-^ = ^^^^' 
 
 The differential equation of a right conoid is obtained 
 from (11) by putting 
 
 1 = 0, m=0, n = l, 
 
 and is therefore 
 
 4^4=" ^''^- 
 
 A. a 12, 
 
178 ON FUNCTIONAL AND DIFFERENTIAL EQUATIONS 
 
 The forms (11) and (12) can of course be obtained di- 
 rectly from (8) and (9) by differentiation. 
 
 For instance from (9), differentiating witli respect to x, 
 we have 
 
 dx y^ \y)' 
 and differentiating with respect to y, 
 
 dz _ X , fx\ 
 
 whence eliminating i/r' ( - J , we have 
 
 dz dz _^ 
 
 dx ^ dy ' ' 
 
 146. The three classes of surfaces we have considered 
 are all included in the general class of rided surfaces, that is, 
 surfaces which can be generated by the motion of a straight 
 line. The first and second differ from the third in this, that 
 any two consecutive generating lines in any surface of the 
 first or second classes lie in one plane, whereas this is not 
 in general the case with the third class. Ruled surfaces 
 in which consecutive generating lines lie in one plane are 
 called developable surfaces, while all other ruled surfaces are 
 called skew surfaces. Thus the surface generated by the 
 ultimate intersections of the normal planes to a given curve 
 is developable. 
 
 Developable surfaces are so named for the following c 
 reason. Let a series of consecutive generating lines be 
 drawn. The plane which passes through the first and 
 second line intersects the plane which passes through the 
 second and third line in the second line. The first plane 
 may be turned round the second line till it coincides with the j: 
 second plane, and thus three generating lines of the surface 
 can be made to lie in one plane. Again, this plane can be 
 turned round the third line till it coincides with the plane _| 
 which passes through the third and fourth lines, and so four 
 consecutive lines can be made to lie in one plane. In this 
 
OF FAMILIES OF SURFACES. 179 
 
 manner the whole surface can be developed so as to lie in 
 one plane without tearing. 
 
 Since any two consecutive generating lines of a develop- 
 able surface lie in one plane, any such surface may be pro- 
 duced by the ultimate intersections of a series of planes, and 
 since any two consecutive planes intersect in a line on the 
 surface, the equation representing any one of the series can 
 only involve one arbitrary constant (Art. 134). 
 
 147. Let the equation of one of the planes be 
 
 Ax + By^-Cz-D:=0 (1). 
 
 Then since the equation only involves one arbitrary con- 
 stant, A, B, C, D must be functions of one constant which 
 we may call a. Thus equation (1) may be written 
 
 #i(«) + #.W + #3 ('>')- </>*« = (2), 
 
 and the envelope is found by eliminating a between (2) 
 and the equation obtained by differentiating it with respect 
 to a, viz. 
 
 x<l>; {a) + y4,:{0L) + z4>; (a)- <^;(a) = (3). 
 
 To obtain the general differential equation of developable 
 surfaces we must differentiate (2), considering a as a function 
 of X, y, z determined from (3). 
 
 Differentiating with respect to x, we get 
 dz 
 dx 
 
 <i>M + -j-^^A^) 
 
 , , , . , {doL doL dz \ 
 — 6. (oL)\ W 
 
 + [.i.; (a) + #; (a) + z^: (.) - <!>: («)} 1^ + ^ ^4 = 0, 
 
 dz 
 or by (3), ,^.(a) + ^<^3(a) = (4). 
 
 Similarly, differentiating with respect to y, we get 
 
 '^«W+J<^sW = (5)- 
 
 12—2 
 
180 ON FUNCTIONAL AND DIFFERENTIAL EQUATIONS 
 EliminatiDg a between (4) and (5), we get 
 
 £=•^(1) ^6>' 
 
 and differentiating again with respect to x and y in turn, 
 we get 
 
 d^z _ J., (dz\ d^z 
 
 dx' "^ \dy) ' dxdy' 
 
 d^z _xf(dz\ d^z 
 
 dx dy "^ \dy) ' dy^ ' 
 
 And eliminating /M-r-J , we get 
 
 d^ d'z / d'z V^ ^ 
 dx^ ' dy^ \dx dy) ' 
 
 which is the differential equation of developable surfaces. 
 
 148. To find the general equation of surfaces of re- 
 volution. 
 
 A surface of revolution is the surface produced by the 
 revolution of a plane curve round a fijxed straight line in its 
 plane called the axis of revolution. 
 
 Let the equations of the axis of revolution be 
 
 Xj-a^y_-^^zj-y . 
 
 I m n 
 
 And let y =f{oc) be the equation of the revolving curve 
 when the axis of revolution is taken as the axis of x, and 
 the point (a, /3, 7) as origin. Let P be any point on the 
 surface, PR perpendicular on the line (1), and Q the point 
 {1, 13, 7). Then from the definition of a surface of revo- 
 lution, 
 
 PR=f{RQ) (2). 
 
 But RQ = I (x - a) + m (y - ^) + n (z - 7), 
 
 since it is the perpendicular from Q on a plane through P 
 perpendicular to (1), and 
 
 PR' = {x-OLy + {y-fiy + (2 - 7/ 
 
 - {l(x - a) -\- m {y - ^) -{- n{z - ry)Y. 
 
 
 i 
 
OF FAMILIES OF SURFACES. 181 
 
 Hence 
 
 ov{x-ay^{y-Pr + {z-r^r 
 
 = cl>{l(w-a) + m{y-^) + n(z-y)} (3), 
 
 which is the functional equation of surfaces of revolution. 
 The differential equation can be thus deduced. 
 Differentiating (3) with respect to cc we get 
 
 2{(.-a) + (.-,)|} 
 
 " {^ "^ ""S}^' 1^(^" ^) + ^(3/ -/3) + n(z - y)}, 
 and differentiating with respect to y 
 
 = \m+nj-y (f)' [I (x-a.) + m (y - ^) + n (z - y)]. 
 
 Eliminating <^' and reducing, we get 
 
 dz 
 m(x-OL)-l(y-^)+ {m (z - y) - n (y - p)] ^ 
 
 dz 
 + [n{x-a)-l{z-^)]'f^ = (4), 
 
 which is the differential equation required. 
 
 149. The conditions that the general equation of the 
 second degree should represent a surface of revolution, can 
 be obtained either from the functional or differential equa- 
 tion of the last Article. We will obtain them from the func- 
 tional equation. 
 
182 ON FUNCTIONAL AND DIFFEKENTIAL EQUATIONS 
 
 Let the equation be 
 Ax^ + By"" + 0/ + 2A'yz + 2B'zx + 2G'a;y 
 
 + 2A"x + 2B"y + 2G"z + F=0 (1). 
 
 If this equation represents a surface of revolution it can 
 be put into the form 
 
 (x - af + (y-^y+(z - yf = P(lx + my-hnzy 
 
 + Q(lx i- my + nz) + R (2), 
 
 where P, Q, R are constants. This is evident from the 
 considerations that the right-hand member must be some 
 function of 
 
 l(x-a) + m{y-^)-\-n(z-y\ 
 or of Ix + my -i-nz — (Icl + mff -\- ny), 
 
 and that it cannot contain x, y, z to a higher degree than 
 the second. Making the equations (1) and (2) identical, 
 we obtain from the terms of the second degree 
 
 PP-l=kA (3), Pmn = kA' (6), 
 
 Pm'-l=kB (4), Pnl = kB' (7), 
 
 Pn'-l^kC (5), Plm^kC (8), 
 
 where k is some constant. 
 
 Multiplying (7) by (8) and dividing the product by (6), 
 we obtain 
 
 PP = k^=kA-[-l; 
 A ' 
 
 B'C , 1 G'A' ^ A'B' ^ ,_, 
 
 These are the conditions which must be satisfied by the 
 coefficients of the equation. . Jk 
 
 The relations which must subsist between a, /5, 7 are 
 obtained by equating the coefficients of the terms of the first 
 degree in (1) and -(2), We thus obtain 
 
 Ql + 2a = 2kA'\ 
 
 Qm-\-2fi=2kB", 
 
 Qn + 27 = 2kC". 
 
OF FAMILIES OF SURFACES. 183 
 
 Whence kA::p^^'^:K^^^:2:i^ (10). 
 
 But ■^, = ^7j, = ^T^ , and k is given by (9). 
 ~T~ ~W "C"" 
 
 The three equations (10) being the relations which 
 a, P, <y must satisfy are the equations of the axis of re- 
 i volution. 
 
 150. The preceding investigation fails if the quantities 
 -jr-A, -^-B, -^-Cvanish, 
 
 for then h is required to be infinite. 
 
 We know that the equation (1) in this case represents a 
 parabolic cylinder, or two parallel planes (Art. 91), conse- 
 quently the surface cannot be a surface of revolution. 
 
 The investigation also fails if A\ B\ or C vanish. Sup- 
 pose A' = 0. From equation (6), mn = ; .*. m = or ?i = 0, 
 and therefore, B or C' must vanish also. Suppose n = 0, 
 land therefore B = 0, we get then 
 
 Plm = kC, 
 
 kC = -l, 
 
 and (1 + kA) (1 + kB) = PHW = k'G" ; 
 
 .-. {C-A){G-B)=C'\ 
 
 which with B' = is the condition required. The other 
 exceptional cases can be treated in the same way. 
 
 151. The differential equations of the different classes 
 of surfaces can be put into a more symmetrical form by the 
 •substitutions 
 
 dF dF^ 
 
 dz _ dx dz _ dy 
 
 d~x^~dF' dy~~dF' 
 
 dz dz ^ 
 
184 EQUATIONS OF FAMILIES OF SURFACES. ^ 
 
 and corresponding substitutions for the second differential 
 coefficients of z, the equation of the surface being assumed 
 to be 
 
 F {x, y, z) = 0. 
 
 Thus the differential equation of cylindrical surfaces 
 becomes 
 
 ,dF dF dF ^ 
 
 I ^ — l-m -7- + ?i-y- = (1). 
 
 ax ay az 
 
 The equations can be more conveniently used in this I 
 form to discover whether a surface whose equation is given 
 belongs to the peculiar class considered. 
 
 For instance, if the surface be cylindrical, there must be 
 some values of I, m, n which shall make the expression 
 
 ,dF dF dF ,„, 
 
 ^T-+??z ,-+72 ^T- (2) 
 
 ax ay az 
 
 vanish identically for all values of x, y, z corresponding to 
 any point on the surface. 
 
 The conditions that this may be possible will be that the 
 coefficients of the several powers and products of x,y, z in 
 (2) must vanish for the same values of I, m, n. 
 
 The differential equations can be found independently of 
 the functional. For instance, equation (1) is the algebraical 
 statement of the fact that at all points of the surface 
 
 F(x,y,z)^0, 
 
 a straight line whose direction-cosines are I, m, n is a tangent 
 line to the surface, a condition obviously satisfied by cylin- 
 drical surfaces only. 
 
 In the case of conical surfaces we at once obtain the dif- 
 ferential equation 
 
 . .dF , o^dF , ^dF ^ 
 (^-«)^ + (2/-^)^^+(--7)^-=0, 
 
 from the consideration that the straight line joining any 
 point {x, y, z) with the vertex is a tangent line to the surface 
 at the point {x, y, z). 
 
( 185 ) 
 
 EXAMPLES. CHAPTER XI. 
 
 1. Shew how to find the functional and differential 
 equations of a tubular surface, that is, a surface which is the 
 envelope of a sphere of constant radius which moves with its 
 centre on a fixed curve. 
 
 2. Prove that the surface 
 
 x^ ■\- y^ -\- z^ — Zxyz = c^ 
 
 is a surface of revolution round the line x = y — z. Find the 
 equation of the generating curve. 
 
 3. Find the equation of a conoidal surface of which the 
 generating lines pass through the axis of z and are parallel to 
 the plane of xy, and whose directing curve is a circle with its 
 centre in the axis of x and its plane parallel to that of yz. 
 {The Cono-Guneus) 
 
 4. Find the equation of the surface generated by a 
 straight line which passes through two fixed straight lines 
 at right angles to each other, and also through a circle 
 whose plane is parallel to each of the straight lines and 
 whose centre is at the middle point of the shortest distance 
 between them. 
 
 5. Find the equation of the surface generated by a 
 : straight line which always passes through the axis of z and 
 ; some point of the curve 
 
 x = a cos 6, y = a sin 6, z = c6; 
 
 and is parallel to the plane of xy. 
 
 6. Find the equation of the surface generated by the 
 tangent lines of the curve 
 
 x= a cos 6, y = ci sin 6, z = cd. 
 
 7. Find the equation of a conical surface whose vertex 
 is at any point on the surface of a sphere, and whose base is 
 a small circle of the sphere. 
 
186 EXA3IPLES. CHAPTER XI. 
 
 Find also the curve in which the cone is cut by a plane ] 
 through the centre of the sphere perpendicular to the dia- I 
 meter through the vertex. 
 
 1 
 
 I 
 
 8. Find the equation of the surface generated by the 
 revolution of a circle round a straight line in its own plane 
 which does not cut it. 
 
 9. Prove that all tangent planes to the surface in the 
 last question which pass through its centre cut it in two 
 circles. 
 
 10. A fixed straight line AB meets a fixed plane in A. 
 A straight line AP moves so that the sine of the angle which 
 it makes with AB bears a constant ratio to the sine of the 
 angle which it makes with the fixed plane. Find the surface 
 generated by AF. 
 
 11. Find the conditions that the surface 
 Ax^ + By^ + Cz' + 2A'yz + 2B'zx + 2C'a:i/ 
 
 + 2A"x + 2B''y + 2C"2 + F = 
 may be a cylindrical surface. 
 
 12. Shew that with the notation of Art. 100 the con- 
 dition that the surface F (x, y, z) — Q may be develop- 
 able is 
 
 W' {vw - ii") + F' {wu - v") + }Y^ (uv - tu") 
 •h2VW(v'w- im) + 2WU(w'u'- vv) + 2 UV(2cv'- iuw')=0. 
 
 Deduce the conditions that the surface in (11) may be 
 developable. 
 
 13. Find the equation of the surface generated by all 
 the normals drawn to an ellipsoid 
 
 at the points where it is cut by the cone 
 
 a b c ^. 
 
 --l--+-=0. ^ 
 
 X y z I 
 
EXAMPLES. CHAPTER XI. 187 
 
 14. A surface is generated by a straight line which 
 ;passes through the axis of z, and the line x = a, z=0; re- 
 maining parallel to the plane y = kz. Shew that its equa- 
 tion is 00 (y — kz) = ay. 
 
 15. Describe the general nature of the surfaces repre- 
 sented by the several equations 
 
 (1) f(r, 6) = 0. (2) f{r, 4.) = 0. (3) /{$, 4>) = 0. 
 
 16. Examine the nature of the surfaces represented by 
 (1) r' = a^cos2^. (2) r'=a'cos2<^. 
 
 17. Find the equation of a cylindrical surface having 
 one central circular section of an ellipsoid for its guiding 
 curve, and its axis perpendicular to the other circular 
 section. 
 
 18. With the axis of z as axis a series of helices are 
 described, all intersecting two given curves; prove that the 
 functional equation of the surfaces generated is 
 
 and that the differential equation is 
 
 2 d^z d^z 2 d^^ _ dz dz 
 
 ^ dd ^ dxdy dy^~^ dx ^ dy' 
 
 19. A candle is placed at a given distance in front 
 of a plane vertical circular mirror on a line perpendicular 
 to the plane of the mirror through the extremity of its 
 horizontal diameter; shew that the boundary of the re- 
 flected light which falls on a wall of which the plane is per- 
 pendicular to that of the mirror is a parabola, and deter- 
 mine its latus rectum. 
 
 20. A straight line AB moves on two fixed straight 
 lines not in the same plane so that the portion between 
 the lines subtends a right angle at a fixed point 0. Prove 
 that the locus of this line is a skew surface of the second 
 order. 
 
188 EXAMPLES. CHAPTER XI. 
 
 21. Obtain the differential equation of surfaces of revo- 
 lution from the consideration that at every point of such 
 surfaces one tangent line is perpendicular to the plane con- 
 taining that point and the axis of revolution. 
 
 22. Shew that if a section of a right conoid whose 
 generating lines are parallel to the plane of xy be made 
 by any plane parallel to that of xy, the normals at points in 
 the lines of section will meet the plane of xy in concentric 
 hyperbolas. 
 
 23. Prove that the general functional equation of the 
 surfaces generated by a circle which always touches the axis 
 of z at the origin may be written in the form 
 
 a;'' + y^ + ^ = 2c^/(|), 
 
 and that the differential equation is J 
 
 
 IE 
 
 foe: 
 
 24. Shew that the equation I 
 z" (2z-x- yf + 2z{a- z) {x - yj - 2a' {z-x){z-y) = 
 
 represents a conoidal surface. . 
 
 25. Describe the form of the surface whose equation is * 
 
 • -^z ^ _i y 
 sm ~ = n tan - . 
 c X 
 
 If n = 2, prove that through any point an infinite number 
 of planes can be drawn, each of which shall cut the surface 
 in a conic section. 
 
 f 
 
 H 
 
CHAPTER XII. 
 
 ON FOCI AND CONFOCAL QUADRICS. 
 
 152. A FOCUS of a conic section is a point such that the 
 iistance of any point on the curve from it can be expressed 
 IS a linear function of the co-ordinates of that point. 
 
 There are certain points which have analogous properties 
 .n reference to quadrics, and which may therefore be called 
 bci of quadrics. 
 
 153. For instance the equation of the ellipsoid is 
 
 x^ if' z^ , 
 
 a^ + P+? = l (1). 
 
 vhere we will suppose a, b, c in descending order of magni- 
 ;ude. Also let e^, e^, e^ be the excentricities of the sections 
 if (1) by the planes of yz, zx, xy respectively. 
 
 The co-ordinates of the focus of the section by the plane 
 )f xy are ae^^ 0, 0. The square of the distance of any point 
 Xy y, z) in (1) from this focus 
 
 = {x- ae^Y +f + z^ 
 
 z" 
 
 = ^M 1 - -J - 2ae^x + a' - "^ " -^ 
 
 aV ' c' 
 
 x2 hVz' 
 = {e^x — a) — 
 
 c' 
 
 = (e^x — ez — a) (e^x + e'z — a), ii e = —^ 
 
 c 
 
190 ON FOCI AND CONFOCAL QUADRICS. 
 
 Hence the square of the distance of any point on (1) 
 from the focus of the section of (1) by the plane of xy is 
 equal to the product of two linear functions of the co-ordi- 
 nates of the point. 
 
 Or, geometrically, we may say that the square of the 
 distance of any point on the quadric from the focus of the 
 section of the quadric by the plane of xy, is proportional to 
 the product of the distances of the point from two planes 
 whose equations are 
 
 e^x — ez — a = (2), 
 
 ejJc-\-e'z — a = .. (3j. 
 
 ■J 
 
 These two planes intersect in a line whose equations are 
 £r = 0, e^x — a = 0, that is in the directrix of the section of 
 the quadric by the plane of xy. 
 
 Similar properties hold for the foci of the sections of the 
 quadric by the planes of yz and zx, but in these cases the 
 two planes corresponding to (2) and (3) are impossible, 
 though their line of intersection is real. 
 
 154. These points are not however the only points which 
 have the same property. We will examine the conditions 
 which must be satisfied by the co-ordinates of any point, in 
 order that the square of its distance from any point on a 
 given central quadric, may be proportional to the rectangle 
 contained by the distances of the latter point from two 
 planes, real or impossible. 
 
 If a, /8, 7 be the co-ordinates of such a point, we must 
 have the expression (x — of -f (?/ — PY + (^ — 7)*^ identically 
 equal to 
 
 {l(x-a')+m(y-/3')-\-n(z-y)] {l\x-a)-^m(y-^')+n(z-y% t 
 
 for all values of x, y, z which satisfy the equation of the 
 quadric ; a\ yS', 7' being the co-ordinates of any point in the 
 line of intersection of the planes. 
 
ON FOCI AND CONFOCAL QUADRICS. 191 
 
 Let the equation of the quadric be 
 
 Ax'' + By''-\-Gz^=l (1). 
 
 Then the equation 
 
 must be satisfied by all values of x, y, z which satisfy (1). 
 
 This can only be the case when the two equations are 
 identical, and as first conditions for this the coefficients of 
 yz, zx and xy in (2) must vanish. We thus get 
 
 Tfin 
 
 ' + mn — 0, nV + nl = 0, hii! + Vm ~ 0, 
 
 which can only be satisfied by one of the sets of conditions 
 
 or 
 
 m = 0, m 
 n = 0, n 
 
 m 
 
 o> -=-- 
 
 = 0, 
 
 m 
 
 r 
 
 n 
 n 
 
 «.'z 
 
 n 
 n 
 I 
 I 
 
 m 
 m 
 
 (3). 
 
 If we take the second set of these equations and put 
 
 I' 
 
 J =Jc, the equation (2) becomes 
 
 {x-ay-V{y-l3y+{z-r^f-U\x-oLy+hn\z-iY=0 (4). 
 
 Comparing the remaining terms of the second degree 
 with those in (1) we obtain 
 
 LtA^' _ 1 _ 1 + '^'^^ 
 A ~B~ 
 
 or 
 
 hr = 1 - 
 
 G 
 
 ^ 7 2^1 
 
 (5). 
 
192 ON FOCI AND CONFOCAL QUADRICS. 
 
 And by comparing the terms involving x, y, z, and the 
 constant term in (4) with the corresponding terms in (1) we 
 have 
 
 a-Wa: = 0, ^ = 0, 7 + A.-7iV = (6), 
 
 a^ + /3^ + 7"-^'^'a^ + ^W = -5 (7). 
 
 And substituting for a, 7' from (6) in (7) we obtain by help 
 of (5), 
 
 T:T+r^ = i («)• 
 
 A B G B 
 
 The equation (8) combined with ^ = gives a conic 
 section in the plane of zx, all the points on which may be 
 considered as foci of the quadric. This curve is called a focal 
 conic of (1). 
 
 155. The equations (6) give values of a and 7' cor- 
 responding to any particular focus (a, (3, 7). These values 
 determine the position of a straight line which we may call 
 the directrix corresponding to that particular focus. 
 
 The directrices corresponding to the different foci lying 
 on the conic (8) all lie on a cylinder whose equation will be 
 found by eliminating a and 7 between (6) and (8), to be 
 
 156. The other conditions in (3) will similarly give us 
 two other focal conies in the planes of xy and yz whose 
 equations are 
 
 r7T+TS = i (^)' 
 
 A G B G 
 
 B' 7' 
 
 Y^ + Ti =^ (^^^' 
 
 B~A G~A 
 
ON FOCI AND CONFOCAL QUADKICS. 193 
 
 land corresponding to any focus there will be a directrix per- 
 pendicular to the plane in which the focal conic lies. 
 
 Of these conies, whatever be the signs and relative mag- 
 nitudes of A, B, G, one will be an ellipse, another an hyper- 
 bola, and the third an impossible locus. 
 
 157. For instance, in the ellipsoid whose equation is 
 
 of if ^ - 
 
 the equations of the focal conies will be 
 
 o^ if 
 
 ^^3^2 + ^23^ = 1 ill the plane of xy, 
 
 ^' ^ -1 
 
 "T" ^2 r:i~ ^ ^^y 
 
 if- z^ 
 
 W'ir^^'^~&^^^^^ ^^' 
 
 And if we assume a, 6, c to be in descending order of mag- 
 nitude, the first of these is an ellipse the extremities of 
 Whose axes are the foci of the sections of the orio^inal 
 ellipsoid by the planes of yz and zx\ the second a hy- 
 uperbola with its real axis in the axis of x, the extremities 
 of this real axis being the foci of the section of the ellipsoid 
 by the plane of xy : while the third is altogether an im- 
 ■possible locus. 
 
 Similar results may be obtained for the two hyper- 
 iboloids. 
 
 158. The focal conies of a cone 
 
 Ax^ + BfA-Cz'^O (1) 
 
 can be deduced from those of a central quadric 
 
 Ax' + By' + Cz'=.\ (2), 
 
 by putting \ equal to zero. 
 
 A. G. 13 
 
194 ON FOCI AND CONFOCAL QUADRICS. 
 
 ABC 
 The focal conies of (2) would be, writing t- , -r- , — in- i 
 
 A. A. A 
 
 Stead of A, B, C in the formulae (8), (9), (10) of Articles 154 
 and 156, 
 
 a' 
 
 
 *T^ 
 
 \ 
 
 = 1, 
 
 X 
 
 \ 
 
 A 
 
 B 
 
 c 
 
 B 
 
 
 a' 
 
 
 -x^ 
 
 T" 
 
 = 1, 
 
 \ 
 
 \ 
 
 A 
 
 C 
 
 B 
 
 c 
 
 
 ^ 
 
 T 
 
 I ^^ 
 
 
 = 1. 
 
 \ 
 
 ' X 
 
 \ 
 
 .m 
 
 BACA 
 
 Or, multiplying these equations by \ and then making \ 
 to vanish, the focal conies of the cone (1) become 
 
 a« 
 
 
 *T^ 
 
 1 
 
 - 
 
 1 
 
 1 
 
 ^> 
 
 A 
 
 B 
 
 G 
 
 B 
 
 
 a' 
 
 
 -/ 
 
 T" 
 
 0, 
 
 1 
 
 1 
 
 A 
 
 G 
 
 B 
 
 G 
 
 
 /3^ 
 1 
 
 T 
 
 -r^ 
 
 T" 
 
 :0. 
 
 IB 
 
 A 
 
 G 
 
 A 
 
 
 Of these, whatever be the signs of 
 
 i_I 1_1 1_^ 
 
 5 C C ^* ^ B' 
 
 one will give two straight lines, and the other two give a 
 point, the vertex. 
 
 159. To find the focal conies of the paraboloid 
 
 Bif^Gz'^x (1), 
 
 i 
 
ON FOCI AND CONFOCAL QUADKICS. 195 
 
 we must as in Art. 154 make the equation (1) identical with 
 
 7(a;-a')+^?i(3/-/3')+^K^-7)H^X^-«')+^'(2/-^0+?^'(^-7)} = 0(2). 
 
 The first conditions for this identity are the same as 
 
 equations (3) of Art. 154, and if we take the second of those 
 
 V 
 conditions and put y = A?, equation (2) becomes as in that 
 
 Article 
 
 {x - of + {ij- ^y + {z- yf - kr {x - aj + W {z - jj = 0. 
 
 And since (1) contains no term involving x^ and no con- 
 stant term, we get 
 
 l-kl' = 0, a'^ + ^' + j'- kPa!' + knY = ; 
 
 and by comparing the remaining terms in the two equations, 
 we have 
 
 1 + kn' ^ 1 ^ 2 (a - kPa) 
 G ~B~ 1 
 
 y3=0, 2(7 + Z;wV) = 0; 
 and thus we get for the locus of the foci the two equations 
 
 or ■ * 
 
 Grf _ 1/ _ 1\\ 
 
 C-B BV ^Bji. 
 
 and /8 = 0l 
 
 By taking the third of the conditions (3) of Art. 154 we 
 shall similarly get another focal conic in the plane of xy 
 whose equations are 
 
 7 = 0, 
 
 B-G C\ 4(77 • 
 
 The first of the conditions (3) of Art. 154 is in this case 
 inadmissible inasmuch as (1) contains no term involving x^. 
 
 13—2 
 
196 ON FOCI AND CONFOCAL QUADRICS. 
 
 Thus in this case the focal conies are two parabolas whose 
 vertices are the foci of the sections of the surface (1) by the 
 planes of xy and zx. 
 
 160. Two central quadrics 
 
 Ax"" + By"" + Cz'' = 1, 
 A'x' + B'f + OV = 1, 
 
 will have the same focal conies if 
 
 1111 
 
 111 
 
 1 
 
 1111 
 
 B G~B' G" 
 
 G A~G' 
 
 A" 
 
 A B~ A' B' 
 
 or as we may write the conditions, if 
 
 A A'~B B'~G G" 
 
 Two quadrics whose equations satisfy these conditions are 
 called confocal quadrics. 
 
 Thus if the equation of an ellipsoid be 
 
 ^+f!+^'=i (1), 
 
 a be 
 all surfaces whose equations are of the form 
 
 + 7:^+3^ = 1 (2), 
 
 a +k h' + k c' + k 
 
 where k is any quantity positive or negative, are confocal 
 with the ellipsoid. 
 
 161. If a, /3, 7 be the co-ordinates of any point, we can 
 find the equation of a surface passing through (a, /3, 7) and 
 confocal with (1) by determining k from the equation 
 
 OL^ 13' , y' 
 
 + 7X^7+7^-1 = (3), 
 
 a' + k ' ¥ + k c' + k 
 
 which is the condition that (2) should pass through the point 
 (a, /?, 7). This equation is a cubic in k, of which it can be 
 shewn that the roots are all real. There are therefore three 
 
ON FOCI AND CONFOCAL QUADRICS. 197 
 
 quadrics confocal to (1) which pass through the point (a, /3, 7), 
 of which one can be shewn to be an elHpsoid, and the others 
 to be hyperboloids of one and two sheets respectively. 
 
 162. Any two confocal quadrics intersect at right angles 
 at all points where they meet. 
 
 For let X, y, z be the co-ordinates of any point common 
 to the two quadrics 
 
 01? iP' z^ 
 
 Z-^l + a = i (^)' 
 
 ^' 4._l!_4-^!_-i m 
 
 A+k'^B + k'^C + k''- ^"^* 
 
 The equation of the tangent plane to (1) at the point 
 {x, y, z)is 
 
 A'^ B ^ G ^ ^' 
 
 And the equation of the tangent plane to (2) at the same 
 point is 
 
 x'x yy zz _ , . ^ 
 
 'A^'^B^k'^'CVk' ^ ^* 
 
 But from (1) and (2) by subtraction we obtain at all their 
 points of intersection 
 
 Ai^A^k)^ B{B-vk)^ 0{C-Vk) ' 
 
 which is the condition that (3) and (4) should be at right 
 angles to each other. 
 
( 198 ) 
 
 EXAMPLES. CHAPTER XII. 
 
 1. Find the equations of the focal conies of the quadric 
 
 2^' + 32/' + 4^' = 9. 
 
 2. Find the equations of the quadrics confocal with the 
 quadric 
 
 2^' + 31/' + 4^' = 9, 
 which pass through the point (1, 1, 1). 
 
 3. Find the locus of the points of contact of tangent 
 planes drawn from a point in the axis of a; to a series of con- 
 focal surfaces whose axes coincide with the axes of co-ordi- 
 nates. 
 
 4. Shew that the surfaces 
 
 - -4 ^C 1- = 1 
 
 1^ 2 "^ 7 2 > 
 
 a ax — a ax — b 
 
 x^ V ^ 
 
 + I + 7T— — 2 7-2 = 1, 
 
 intersect everywhere at right angles. 
 
 5. Shew that if the foci of the principal sections of two 
 paraboloids coincide, their focal conies will also coincide. 
 
 6. Extend the proposition of Art. 162 to the case of two 
 confocal paraboloids. 
 
 7. If from any point a tangent cone be di^awn to an 
 ellipsoid shew that the axes of the cone are the three 
 normals that can be drawn at the vertex to the three 
 confocals through the vertex. 
 
 8. Prove that if a, a , a\ a" be the transverse axes of 
 an ellipsoid, and of the ellipsoid and hjrperboloids of one and 
 two sheets which can be drawn through any point confocal 
 to the first ellipsoid ; and if a'^ -}- a""^ -F a "' = 3a', then three 
 tangent planes can be drawn from the given point to the 
 given ellipsoid mutually at right angles. 
 
CHAPTER XIII. 
 
 ON CURVATURE OF SURFACES. 
 
 163. Two surfaces are said to have contact of the first 
 order at any point where they meet when they have a com- 
 mon tangent plane at that point. The necessary and suffi- 
 cient conditions for this are that for the same values of x and 
 
 y the values of z, -j- and -r- shall be the same for the two 
 '^ ax ay 
 
 surfaces. 
 
 Two surfaces are said to have contact of the in}^ order at a 
 point where they meet when the sections of the two surfaces 
 by every plane passing through that point have contact of the 
 ii}^ order. This we will prove to be the case if the sections of 
 the surfaces by all planes w^hich contain any given straight 
 line through the point of contact not lying in the tangent 
 plane have contact of the n^^ order. 
 
 For let the common point be taken as origin and the 
 given line as axis of z. Let the equations of the two surfaces 
 be 
 
 ^=f{^,y) (1), 
 
 z = F(x,y) (2). 
 
 Expanding (1) and (2) we obtain 
 
 "• = (f)'^+©2'+- + f»(^i + 2'|)"/+ (3), 
 
 Kax. 
 
 2 
 
 '\ , fdF\ I f d d\" „ 
 
 )^^[^-^)y+...+ [,- + y^^yF+ (4), 
 
200 ON CURVATURE OF SURFACES. 
 
 where z^ and z^ are the ordinates of the two surfaces corre- 
 sponding to the same values of x and y, and in the quantities 
 
 df (JW 
 
 4- , -J- , ...w and y are put equal to zero after the differen- 
 tiations are performed. 
 
 Now since all sections of (1) and (2) by planes which con- 
 tain the axis of z have contact of the n^^ order, the difference 
 of z^ and z^ must be of the {n -\- ly^ degree in oc and y. Hence 
 we have 
 
 ^^dF d£^dF dy^^d^ d^^d^ -^ 
 
 dx dx' dy dy ' dx^ dx^ ' dxdy dxdy' '" \ 
 
 > •••(5). 
 
 dJ]f^_dJ^ dj _ d^'F 
 
 dx"" ~ dx""' '" dx^'dy''-'' ~ dx^'dy'"' 
 
 r > 
 
 J 
 
 If now the axes be changed in position, the origin remain- 
 ing the same, since the new co-ordinates x, y\ z of any point 
 are linear functions of the old co-ordinates, it is clear that any 
 
 d^'^'z 
 differential coefficient of the form , . .. . can be expressed in 
 
 dx dy 
 
 terms of the differential coefficients of z with respect to x and 
 y of orders up to but not exceeding the (r -I- sY". Hence if the 
 differential coefficients of z with respect to x and y for one 
 surface, up to those of the w*^ order inclusive, be respectively 
 equal to the corresponding quantities for a second surface, the 
 same will be true of the differential coefficients of z with 
 respect to x and y \ that is, if conditions (5) be satisfied for 
 two surfaces with any one set of axes, they will be also satis- 
 fied with any other set of axes. ^^"\ 
 
 Thus if the sections of the two surfaces (1} and (2) by all 
 planes through the axis of z have contact of the tS'^ order, so 
 will their sections by all planes through the common point. 
 
 The conditions that two surfaces should have contact of 
 the 71^^ order at a given point are therefore that the values of 
 
 dz^ dz dr£ d^z d^'z 
 
 ^' Tx' d^' '" d^' d^T^dy' '" dy^'' 
 
 should be the same for the two surfaces for the given values 
 of X and y. 
 
ON CUKVATURE OF SURFACES. 201 
 
 164. If two surfaces touch at a given point and the 
 sections hy a plane through the normal and any tangent line 
 have contact of the second order, then all sections hy planes 
 through the same tangent line have contact of the second order. 
 
 Take the common point as origin and the common normal 
 as axis of z. Then, z =/(iC, y), z = F {x^ y) being the equa- 
 tions of the two surfaces, the values of -^ , ~- , -7- , -r- 
 
 aoc ay ax ay 
 
 vanish at the origin and the equations of the surfaces can be 
 put in the form 
 
 z = ax^ + hxy + cy'^+ (1), 
 
 z = Ax'+Bccy-\-Gy'+ (2), 
 
 cZy d^f d^f 
 where a, h, c are the values of J —^ , , ^ , J -~^ at the 
 
 d^'F d^F d^F 
 ongm, and A, B, C those of § ^ , d^' i ^ • 
 
 Also if the given tangent line be the axis of x, the sec- 
 tions by the plane of zx have contact of the second order, and 
 we have a = A. 
 
 Consider now the sections by a plane through the axis 
 of X whose equation is 
 
 y=-rnz (3), 
 
 we have for a given value x^ of x, in the one surface 
 
 z^ = ax^ + hx^y^ + cy^ + . . . , 
 
 and in the other 
 
 ••• ^1 -^2 = ^1 (%i - %2) + ^Vi -Gy^ + "' 
 
 But z^ , z^ being of the second degree in x^ , y^ and y^ are so 
 also by (3), and therefore x^ {hy^ — By^ is of the third degree, 
 and therefore z^ — z^ is of the third degree in x^, and the 
 sections of the two surfaces by (3) have contact of the second 
 order. 
 
 Similarly if two surfaces have complete contact of the 
 {n — Xf^ order at a given point, and the sections by any plane 
 
202 ON CURVATURE OF SURFACES. 
 
 through the normal and a given tangent line have contact of 
 the n^^ order, then all sections by planes through this tangent 
 line have contact of the if" order. 
 
 165. From the proposition proved in the last Article it 
 follows that if R be the radius of curvature of any normal 
 section of a surface, R cos 6 is the radius of curvature of an 
 oblique section through the same tangent line inclined at an 
 angle 6 to the normal section. For if a sphere whose radius 
 is R be described touching the surface at the given point, 
 the normal sections of this sphere and the surface through 
 the given tangent line have contact of the second order and 
 therefore also any oblique sections. 
 
 But the radius of curvature of the oblique section of the 
 sphere is obviously R cos 6 ; hence the radius of curvature of 
 the oblique section of the given surface is also R cos 6. ' This 
 proposition is called Meunier's Theorem. 
 
 166. If the tangent plane at any point be taken as the 
 plane of ccy and the point of contact as the origin, we have 
 seen that the equation of the surface can be put into the 
 form 
 
 z = ax^ + hxy -{-cy^-^- (1), 
 
 where the remaining terms are of a higher degree than the 
 second. 
 
 Consider the section of this surface by a plane through 
 the axis of z whose equation is 
 
 y = X tan 6 (2). 
 
 The radius of curvature of this section is the limit of 
 
 x^ + V^ 
 
 ' — ^r-— when the values of x and y are diminished indefi- 
 
 nitcly. Hence if p be this radius, we have 
 1 _ , ax^ + hxy + c?/ + Ax^ 
 
 ¥p~ ' ZTl^^ 
 
 _ . a + h tan 6 + c i^w^O + Ax 
 
 = a cos^^ + h sin ^ cos ^ -h c sin^^ (3). 
 
ON CURVATURE OF SURFACES. 203 
 
 If we construct the conic section whose equation is 
 
 ax^ + hxy + cy^=l (4), 
 
 it is evident from (3) that the square of any radius vector of 
 this conic represents the diameter of curvature of the section 
 of (1) by a normal plane passing through this radius vector. 
 This conic section is called the indicatrix of the surface at 
 the given point. 
 
 If in (1) we suppose x and y so small that the terms on 
 the right hand after the third may be neglected, we get 
 
 z = ax^ + hxy + cy'^ (5). 
 
 The curve in which this surface is cut by a plane z = k 
 parallel to the plane of xy is similar and similarly situated 
 to (4). Hence the indicatrix at any point of a surface may be 
 defined as a curve similar and similarly situated to the limit 
 of the curve in which the surface is cut by a plane indefi- 
 nitely near to the tangent plane at the given point. 
 
 167. By choosing the axes of x and y so as to coincide 
 with the principal axes of the indicatrix the equation (4) of 
 the last Article assumes the form 
 
 Ax^-^Cf=l (1). 
 
 Also the radii vectores drawn in the directions of the 
 principal axes are respectively the least and greatest radii 
 of the curve. Hence the normal sections for which the 
 radius of curvature is least and greatest respectively, pass 
 through the principal axes of the indicatrix. The radii of 
 curvature of these sections are called the principal radii of 
 curvature at the given point, and the sections themselves, 
 the principal sections. 
 
 Let R and R' be the principal radii of curvature, p and p 
 the radii of curvature of any other sections at right angles, 
 which we may take to be the sections through the axes of x 
 and y in equation (4) of the last Article. Then 
 
 ^ -A 
 2B~ ' 
 
 ^ -G 
 
 1 
 
 2p = ''' 
 
 1 
 
 2/)' ~ ^- 
 
204 ON CURVATURE OF SURFACES. 
 
 But A + C = a -^ c. (Todhunter's Conic Sections, Art. 
 274.) 
 
 And therefore 
 
 i+i=^7 ^'^- 
 
 Also if the section whose radius of curvature is p be in- 
 clined at an angle 6 to the principal section whose radius 
 is M, we have from (1) 
 
 ^=^cos'6'+asin'^; 
 Zp 
 
 .*. - = ^ cos^ ^ + TV sin^ ^ (3). 
 
 p K K 
 
 We can thus obtain the radius of curvature of any normal 
 section if we know those of the principal sections, and by 
 Art. 165 we can deduce that of any oblique section. Hence, 
 if we know the principal radii of curvature at any point of a 
 surface, the curvatures of all sections of the surface at that 
 point are known. 
 
 168. To find the radius of curvature of any normal 
 section of a surface at a given point. 
 
 Let the equation of the surface be 
 
 F{x,y,z) = (1), 
 
 and let x, y, z be the co-ordinates of the given point P. Let 
 ?, m, n be the direction-cosines of the tangent line at {x, y, z) 
 through which the cutting plane passes. Also let x + a, 
 y + ^, z + <y he the co-ordinates of a point Q in the curve of 
 section near to P. Let QR be drawn perpendicular on the 
 tangent plane. Then (Frost's Newton, Art. 78) the radius of 
 
 curvature of the section is the limit of ^^^ when Q is made 
 
 to approach indefinitely near to P. 
 
 But the equation of the tangent plane is 
 
 . , . dF , , . dF . , . dF ., ._,. 
 
 (^— )^ + (y-2/)rf^+(^--)^ = (2). 
 
ON CURVATURE OF SURFACES. 205 
 
 Hence 
 
 dF dF dF 
 
 with the notation explained in Art. 100. 
 And PQ'=.a' + 0' + y\ 
 
 The radius of curvature is therefore the limit of 
 
 But since the point (cc -{■ a, y + jS, 2+y) is on the surface (IX 
 F(x + a,y + ^,z + y) = 0, 
 or expanding, and remembering that F(a), y, z) = 0, 
 
 aU+pV+yW^\{a\i^fih-Vy^w+2l3yu'-^2yav'+2a^w')+...=0. 
 Whence the above expression becomes 
 
 ■*■ d'u + P^v + 7% + ^Pyu 4- 27a?;' + 2oi(3w + . . . 
 
 where the remaining terms in the denominator are of higher- 
 dimensions than the second in :i, /3, 7. 
 
 Now, by Newton, Lemma YI. the angle between PR and 
 PQ diminishes indefinitely as Q approaches P. Hence we 
 have ultimately 
 
 I 111 n' 
 
 And making these substitutions and diminishing a, /3, 7 in- 
 definitely, we obtain for the radius of curvature 
 
 ^ TjU'-\-V'+W' 
 
 " ul^ + vni^ + wn' + 2itmn + 2v'nl + 2w'lm 
 
206 
 
 ON CURVATURE OF SURFACES. 
 
 169. The principal sections are those for which p is a 
 maximum or minimum. Hence we have to make the ex- 
 pression 
 
 ul"^ + vm^ 4- lun^ + 2umn + ^v'nl + liu'lm (1) 
 
 a maximum or minimum by the variation of I, m, n, which are 
 connected by the relations 
 
 l' + m' + 7i' = l (2), 
 
 m+ Vm+ Wn = (3), 
 
 the latter expressing the fact that the line whose direction- 
 cosines are I, m, n lies in the tangent plane at the point 
 (oo, y, z). We shall denote the expression (1) by the symbol h. 
 
 Differentiating (1), (2) and (3) and using undetermined 
 multipliers, we obtain 
 
 ul-\-ivm-\-vn + kl + k'U=0 (4), 
 
 lul + vm + u'n + hm + /:' F = (5), 
 
 vl + itm + wn -{■ kn -\- ¥ W = 
 
 .(6). 
 
 Multiplying (4) by I, (5) by m, and (6) by n, and adding, 
 we get 
 
 h-\-k = (7). 
 
 And the three equations (4), (5), and (6), become 
 
 (u — h)l + %vm + v'n = — M U, 
 
 wl-\-{v —h)m+u'n = — k'V, 
 
 v'l -\- u'm + {lu — h)n = — k' W, 
 whence 
 
 ' ""' " .(8). 
 
 u, 
 
 w , 
 
 v' 
 
 
 V. 
 
 v — h, 
 
 1 
 u 
 
 
 w. 
 
 w , 
 
 tu — 
 
 h 
 
 
 m 
 
 
 u, 
 
 V , 
 
 u — h 
 
 V, 
 
 u , 
 
 w 
 
 w, 
 
 W — hy 
 
 v' 
 
 
 n 
 
 
 u. 
 
 u — h, 
 
 w' 
 
 V, 
 
 w' , 
 
 v — h 
 
 w, 
 
 V , 
 
 u' 
 
 Substituting in (3), and reducing 
 U^ [{v -h)(w-h)- u"} + V {(w - h) (u - h) - v"} 
 
 + 2 F F [viu - II {u -hy^^'L'WJJ [ivii -v{v- h)] 
 
 + Wy\xiv -w'{w- h)} = 0...(9). 
 
ON CURVATURE OF SURFACES. 207 
 
 From (9) we obtain two values of h and therefore of 
 p, and from (8) we deduce the corresponding values of 
 /, m, 11. 
 
 170. The formulae of the last two Articles are somewhat 
 simplified if we take the unsymmetrical form of the equation 
 of a surface 
 
 or f{x,y)-z = 0. 
 
 The reductions may be effected by the substitutions 
 ?7 = p, F=g, W=-\, 
 u = r, v = t, w = 0, 
 
 u =0, V = 0, w' = s. 
 
 Moreover, instead of determining the tangent line through 
 which the section is made by its direction-cosines, it is usual 
 to determine it by its projection on the plane of xy, whose 
 equation we may assume to be 
 
 y —y — m{x' — x) (1). 
 
 The direction-cosines of the line of intersection of this 
 plane with the tangent-plane at (ic, y, z)y whose equa- 
 tion is 
 
 p{x -x) + q (y' -y)- {z -z) = 0, 
 are proportional to 1, m,p + qm, respectively. 
 
 The value of p becomes with these substitutions equal to 
 
 J 1 + p^ -\- q^ {1 + p^ -\- 22jqm -f- (1 -f q^) m^] 
 r -\- 2sm -f tm^ 
 
 171. The result of the last Article can be obtained inde- 
 pendently. Let a sphere be described having contact of the 
 first order with the given surface at (x, y, z), and let the 
 sections of the surface and the sphere by the plane (1) have 
 contact of the second order. Then the sections of the sur- 
 face and the sphere by a normal plane through the line 
 in which (1) cuts the tangent plane will, by Meunier's 
 Theorem, have contact of the second order with each other, 
 
208 
 
 ON CURVATUEE OF SURFACES. 
 
 and the radius of the sphere is therefore the radius of 
 curvature of the section required. 
 
 Let the equation of the sphere be 
 
 {X-af-{-{Y-hf + {Z-cf = p' 
 
 - , fdZV .^ .d'Z ^ 
 
 .(2); 
 
 (3), 
 
 (4). 
 
 But at the point {x, y, z) 
 
 dZ 
 
 X=a7, Y=y, Z = z, ^ = P 
 
 dZ_ 
 dY 
 
 = <1> 
 
 since the sphere and surface have a common tangent plane. 
 Also since their sections by the plane (1) have contact of the 
 second order, the values of z in terms of x —x, y —y for the 
 sphere and surface must coincide as far as terms of the second 
 degree in x' ^ x, y —y for points lying in the plane (1), 
 whence we obtain 
 
 d'Z ^ d'Z ,d'Z 
 
 dX^^^'^dXdY^''' jY-^ = r+^-ms + tm. 
 
 We deduce from (3) 
 
 X — a _y — h _z — c _ p 
 
 .(5). 
 
 -1 71 + 
 
 p' + q^' 
 
 P 9 
 
 and from (4) 
 
 d^ = 1+/ d'Z ^ ^q_ drZ _ 1+q ' 
 dX' c-z' dXdY c-z' dY'~ c-z 
 
ON CURVATURE OF SURFACES. 209 
 
 Whence from (5) 
 
 ^ 1 +/ + 2pqm + (1 + q^) m^ 
 r + 2s7n + tm^ ' 
 
 and 
 
 The equation which gives the sections of greatest and 
 least curvature at any point is obtained by making this 
 expression for p a maximum or minimum by the variation 
 of m. Whence 
 
 {pq + (1 +q^)m} {r + 2sm + t7n^} 
 
 -(s + till) [I +/ + 2pqm + (1 + q^) m'] = 0, 
 
 or m' {s (1 + q') - pqt] + m {r (1 +q')-t(l+ /)} 
 
 + {pqr-s(l+f)] = (7). 
 
 172. It may happen that at certain points of a surface 
 the two principal radii of curvature become equal. It follows 
 from Art. 167 that the radii of curvature of all normal sec- 
 tions at that point are equal, the indicatrix in this case beino- 
 a circle. Such a point is called an umbilicus. 
 
 The conditions for the existence of an umbilicus can be 
 deduced from the consideration that at such a point the 
 expression 
 
 tW^ + vm^ + tuif + 2u'mn + 2vnl + 2wlni (1), 
 
 nust retain the same value for all values of I, m, n consistent 
 ivith the conditions 
 
 Ul+Yiii-\-Wn = (2), 
 
 r -\- m^ + 7z' =1 (3). 
 
 From (2) TPr + V^m'' + 2 UVlm = FV ; 
 
 2lm = 
 
 UV 
 
 Dimiiarly, 2ni = ^r^y , 
 
 ^''''' = vW • 
 
 A. G. 14 
 
+ n' ■{w + 
 
 210 ON CURVATURE OF SURFACES. 
 
 Whence, substitating in (1), the expression 
 
 TFy_ W_ W" 
 
 Vv V u 
 
 must have the same value for all values of I, m, n consistent 
 with (3). 
 
 This gives the conditions 
 u+ ^{Uu -Vv' - Wio}=v + ^{Vv' - Wtu - Uic} 
 
 = w-\-^{WW- Uu'-Vv'} (4). 
 
 If the equation of the surface be of the form 
 
 u' = 0, V = 0, w' = 0, and equations (4) become 
 
 u = v = w (5). 
 
 If U, V or W vanish the investigation fails. Suppose 
 
 V 
 Then Vm 4- Wn = 0, or n = - jr^m, 
 
 and the expression (1) becomes 
 
 id + vm -\- w . "tt^^ . m — ^ m + zlm ( lu — -j^ J , 
 
 which must remain constant for all values of m and n con- 
 sistent with the relation 
 
 P^,n^ ^1 + ^^ = 1. 
 
 Hence Ww' - Vv = 0, 
 V'w_2u'V 
 '" + W W ' Vhu +W'v-2 VWit 
 and u = pr, = jn~fW^ ^^^• 
 
ON CURVATURE OF SURFACES. 211 
 
 Similarly if F= or TT^ the requisite conditions may 
 be deduced. In these cases, three conditions have to be 
 satisfied by x, y, z besides the equation of the surface, which 
 will not generally be consistent. 
 
 The conditions for an umbilicus when the unsymmetrical 
 form of the equation of a surface is used may be deduced 
 from the consideration that the value of p in Art. 171 must 
 be independent of m. We thus get 
 
 r s t ' 
 
 173. The conditions for an umbilicus can be obtained 
 in a slightly different form. 
 
 If h is the value of the expression (1) for all values of 
 I, m, n consistent with (2), it is evident that the ex- 
 pression 
 
 ul^ + vm^ + lun^ + 2u'm7i + 2v'nl + 2w'lm 
 
 -h{P-\-m' + n') (1) 
 
 must vanish for all values of I, m, n consistent with (2). 
 Hence Ul + Vm + Wn must be a factor of (6). The other 
 factor must be 
 
 u — h^ V — h w — h 
 
 and multiplying these factors together and equating co- 
 eflScients of mn, nl and Im as in Art. 49, we have 
 
 W V 
 
 and two similar equations, whence 
 
 W'v + V'w - 2iL VW 
 
 h = 
 
 y\, + jjs - 2io' UV 
 U'w+Whi-2vWU 
 
 by symmetry. 
 
 W 4- U'' 
 
 U— 2 
 
212 ON CURVATURE OF SURFACES. 
 
 174. Lines of Curvatm^e. 
 
 A line of curvature on any surface is a curve such that 
 the tangent line to it at any point coincides luith the tangent 
 line to one of the principal sections at that point. 
 
 The differential equation of such lines is obtained b}' 
 
 substituting -7-, -^, y for Z, m, n respectively, in the 
 
 equations which determine the directions of the principal 
 sections in Art. 169. From the equations (4), (5) and (6) of 
 that Article we have, eliminating k and k', 
 
 ul + w'ni + v'n, I, U 
 
 wl + vni + u^'n, m, V =0 (1), 
 
 v'l + u'ln + wn, n, W 
 
 ox dii dz 
 and replacing /, 771, ?z by ^ , -^ , ^, respectively, we get 
 
 the differential equation of the lines of curvature. 
 
 The differential equation of the projection of the lines of 
 
 dv 
 curvature on the plane of xy is obtained by writing -^ for m 
 
 in the equation (7) of Art. 171. 
 
 175. A line of curvature is sometimes defined as a curve 
 such that the normals to the surface drawn at any two con- 
 secutive points of the curve intersect each other. This defi- 
 nition leads at once to the equation (1) of the last Article. 
 For the equations of the normal at a point {x, y, z) are 
 
 {^' -0^ y -y _z' -z /ON 
 
 -^(T^ 'v~-yr ^^^' 
 
 The equations of the normal at a point (a? -f a, 3/ + yS, 2" + 7) 
 are 
 
 X — X — a y —y - ^ 
 
 IJ -V itoL -f w'l3 + v'y + ... V + w'l -^vfi -\- u'y + ... 
 
 2' — z — y 
 
 W + v'a + u'^ + wy -{- ... 
 
 •(3), 
 
ON CURVATURE OF SURFACES. 
 
 213 
 
 where the remaining terms in the denominators contain 
 higher powers of a, /3, 7. 
 
 The condition that (2) and (3) should intersect is by 
 Art. 31, 
 
 U-\- u'x+iuP-\-vy + ..., U, a 
 
 V + wa+ vl3 +u'^+..., F, /5 =0, 
 
 ua. + w'^ + vy, U, a I 
 
 whence wa+v^+uy, V, ^ = (4), 
 
 v'a + 10^ + wj, Wj y 
 
 but ultimately a, /3, y are proportional to ';r , -j- , -r , 
 
 CLs as cts 
 
 respectively, and the equation (4) reduces to the same 
 
 as (1). 
 
 176. The radii of curvature at any point of a quadric 
 can be obtained from the preceding formulae. Some of the 
 results are so simple and important that they deserve a 
 separate consideration. 
 
 Since all parallel sections of a quadric are similar, it 
 follows that the indicatrix at any point of such a surface is 
 similar and similarly situated to the section of the quadric 
 by a plane through the origin parallel to the tangent plane 
 at the given point. Hence the tangents to the lines of cur- 
 vature at any point are parallel to the axes of the section 
 by this plane, and the umbilici are the points at which 
 tangent planes can be drawn parallel to the planes which 
 give circular sections. 
 
 The equation of the tangent plane at any point (a, /3, 7) 
 to an ellipsoid whose equation is 
 
 222 
 — I- — -I — =1 
 
 xcL yp zy_ 
 
214 ON CURVATURE OF SURFACES. 
 
 If this plane be parallel to either plane of circular section 
 we have 
 
 a ^ 7 
 
 a 
 
 Ja^-W ±cJYZr^' 
 
 , by Art. Qb, 
 
 and since (a, y8, 7) is a point on the ellipsoid, each of these 
 ratios = + , . 
 
 Hence the ellipsoid has four umbilici whose co-ordinates 
 are given by 
 
 a= + ay^^— ^2. P = 0, 7 = ±c 
 
 177. If a tang^ent line be drawn to a surface of the second 
 degree at the extremity of the axis of any plane section of 
 that surface and lying in the cutting plane, the axis of the 
 section and this tangent line are at right angles. This tan- 
 gent line to the quadric is therefore also a tangent line to a 
 sphere described with the origin as centre, and the length of 
 the semi-axis of the section as radius. 
 
 Let the equation of an ellipsoid be 
 
 x^ v^ z^ 
 
 a^ + i? + c^='' «■ 
 
 and let a sphere be described with the origin as centre and 
 any radius k. The equation of this sphere is 
 
 x^-\-7/-{-z^ = k^ (2). 
 
 The equation of the cone formed by straight lines joining 
 the origin with all the points of intersection of (1) and (2) is 
 therefore 
 
 For this equation does represent a cone whose vertex is 
 the origin and being satisfied by all values of x, y, z which 
 satisfy both (1) and (2) represents some surface passing 
 through their intersection. 
 
 Now every plane which passes through the origin and any 
 tangent line to the curve of intersection of (1) and (2) is evi- 
 
ON CURVATURE OF SURFACES. 215 
 
 dently a tangent plane to the cone (3). Hence if we draw a 
 tangent plane to (3) along any generating line OP^, OP^ is one 
 axis of the section of (1) made by this plane. Let OR be 
 the other axis and Q be the point of (1) at which a tangent 
 plane can be drawn to (1) parallel to this section, then OQ 
 is conjugate to the cutting plane and OP^^ is conjugate to the 
 plane through OQ and OR. 
 
 The tangent to one line of curvature at Q is parallel to 
 OR, and consequently lies in the plane QOR which is diame- 
 tral to OP^. 
 
 Let OP, OP^y OP^ be three consecutive generating lines 
 of the cone (3) ; OQ, OQ^ the lines conjugate to the planes 
 POP^, PfiP^ which are ultimately consecutive tangent planes 
 to the cone (3). Then since OP^ lies in a plane which is dia- 
 metral to OQ, and also in a plane diametral to OQ^, the 
 plane QOQ^ is diametral to OP^ and therefore coincides with 
 QOR, and the line joining QQ^ is ultimately parallel to OR 
 and therefore is the tangent line to one line of curvature 
 which passes through Q. Hence one line of curvature through 
 the point Q is the locus of the points at which tangent planes 
 can be drawn to (1) parallel to the tangent planes to (3). 
 
 Hence if Q be any point on an ellipsoid, and r, k the 
 semi-axes of the central section which is parallel to the tan- 
 gent plane at Q, the axis k is constant for all points on the 
 line of curvature whose tangent at Q is parallel to r. But if 
 p be the perpendicular on the tangent plane at Q, 
 
 prk = abc Art. 75, equation (3), 
 and therefore pr = -j- = constant. 
 
 178. The equation of any tangent plane to (3) is 
 
 Ix +my' -\-nz — (4), 
 
 where I, m, n are connected by the relation 
 
 m^m^^n^ ^'^' 
 
 ci^ k^ W k"" c" k' 
 (See Chapter viii. Ex. 24.) 
 
216 
 
 ON CURVATURE OF SURFACES. 
 
 and the equation of a tangent plane to (1) at the point 
 
 XX y y z'z _-i 
 a c 
 
 Hence if (6) be parallel to (4) 
 
 X _ y _ z 
 
 (U). 
 
 2„, » 
 
 or from (5) 
 
 aH h^m c^n 
 
 ^" f ^ r. 
 
 -4 + -^ + 74=0, 
 
 k^ k^ ¥ 
 
 and subtracting this from the equation 
 
 2 2 2 
 
 X y z 
 1- - A =1 
 
 ce ^ ¥ ^ c' ' 
 
 we get 
 
 •2 
 
 X y z 
 
 •> ^^ 7 V 7 '2 1^ 
 
 d'-k' ' h'-k' ' d'-U' 
 
 = 1, 
 
 which shews that the lines of curvature on an ellipsoid are its 
 curves of intersection with confocal surfaces. 
 
 179. In the ellipsoid 
 
 
 W = 
 
 2z 
 
 ,2 } 
 
 U = - 
 
 2 > 
 
 v = 
 
 2 
 
 w= -^^ u =0, v =0, id = 0. 
 
 a- V ' c 
 
 Hence the differential equation of the lines of curvature is 
 
 \ dx X dx 
 
 d^ ds ' a' ' ds 
 
 1 dy y dy 
 
 b'ds' 1?' ds 
 
 1 dz z dz 
 
 c'ds' c" ds 
 
 = 0, 
 
 ds ds 
 
 dsds 
 
ON CURVATURE OF SURFACES. 
 
 217 
 
 180. Taking the equation 
 
 X 
 
 + 
 
 f 
 
 + 
 
 ^ 
 
 oj" + k h' + k c' + k 
 
 = 1 
 
 .(1), 
 
 we have at the points where it meets the ellipsoid 
 
 2 2 2 
 
 X y z ^ 
 
 — I- - + - = 1 
 
 21^7 2'^ 2 — -^ 
 
 a c 
 
 by subtraction 
 
 X 
 
 + 
 
 f 
 
 + -ir 
 
 = 0. 
 
 d" {a' + k) ¥ (¥ + k) ^ & {& + k) 
 Also by differentiating (1) and (3) we obtain 
 
 X 
 
 dx 
 ds 
 
 y 
 
 + 
 
 dy 
 ds 
 
 + 
 
 dz 
 
 ds 
 
 ct" + k b- -\-k c^ + k 
 
 = 0. 
 
 X 
 
 dx 
 ds 
 
 y 
 
 + r 
 
 dy 
 
 ds 
 
 + 
 
 dz 
 
 ds 
 
 a' (a' + k) ' h' {W + k) & {& + k) 
 
 = 0. 
 
 (2), 
 
 (3). 
 
 .(4), 
 
 (5). 
 1 
 
 And from (3), (4) and (5) eliminating -^^-^ , ^-^, ^,^^, 
 we obtain 
 
 X 
 
 a 
 
 dx 
 
 2 » 
 
 yl 
 
 dy 
 
 dz 
 
 = 0. 
 
 .(6), 
 
 ^ds' ^ ds' ^ ds 
 X dx y dy z dz 
 d^ ds ' P ds ' c^ds | 
 
 which is the same as equation (1) of the last Article. 
 
 Thus we obtain an independent proof of the fact that the 
 lines of curvature on an ellipsoid coincide with its curves of 
 1 intersection with a series of confocal quadrics. 
 
218 ON CURVATURE OF SURFACES.. 
 
 181. If we denote by I, m, n the direction-cosines of the 
 tangent to either line of curvature at the point (oc, y, z) on the 
 ellipsoid they must satisfy the equations 
 
 ^ + \Z + /x4 = (1), 
 
 | + \m + /t^, = (2), 
 
 ^, + \7i + /.^, = (3), 
 
 which are obtained from the equations (3), (4) and (5) of the 
 last Article by the use of undetermined multipliers X and fi. 
 
 But if r be the central radius vector of the ellipsoid paral- 
 lel to the tangent line considered, and p the perpendicular 
 from the centre on the tangent plane to the ellipsoid at the 
 point {x, y, z), we have 
 
 \ _x^ f z^ 
 
 ~~h — ~j ~r fx -\ 4 («^j' 
 
 p^ or }y c ^ ' 
 
 Also from the equation of the ellipsoid, by differentiation 
 
 _ Ix my nz ,^. 
 
 Differentiating (6), we have by means of (4) 
 
 dl dm dn 
 1 ds "^ ds ds ^ 
 
 ? + — + —+—=« <7>- 
 
 Multiplying (1) by 4 , (2) by f-^ and (3) by ^ and adding 
 we have 
 
 1 fix my nz\ 
 
 p \a c ' 
 
ON CURVATUKE OF SURFACES. 219 
 
 or using the result obtained by differentiating (5), 
 
 p^ p^ ds 
 
 Again, multiplying (1) by ^^ , (2) by ^-£ , (3) by ^-£ and 
 adding, we have by (7) and (4) since also P-{- 'Jif + 7i^ = 1, 
 
 1 A*- ^^ _ A 
 
 r^ r^ ds~ 
 
 Thus we obtain 
 
 1 dp _ 1 dr 
 p ds r ds ' 
 
 dp , dr 
 '~ds+Pd's = ^' 
 
 .'. pr = constant. 
 
 182. A few propositions must be added concerning a 
 class of lines of great importance, namely geodesic lines. 
 These may be defined as follows : 
 
 A geodesic line on a surface is such that every small ele- 
 ment PQ is the shortest line that can he drawn on the surface 
 hetiueen P and Q. 
 
 The general equation of geodesic lines on a surface 
 
 F (x, y, z) = 0, 
 can be obtained by the help of Meunier's Theorem. 
 
 For if PQ be two points on a geodesic line, so near to one 
 another that the arc between them may be considered as a 
 jjlane curve, the length of PQ will be least when the curva- 
 ture of the curve is least, or when the radius of curvature of 
 the small arc PQ is greatest. But this will be the case when 
 the section of the surface by a plane through the element 
 PQ is a normal section. Hence the osculating plane at any 
 point of the curve must contain the normal to the surface at 
 
220 
 
 ON CURVATURE OF SURFACES. 
 
 that point. But the direction-cosines of that normal to the 
 curve which lies in the osculating plane are proportional to 
 
 d^x d^y d^z 
 d?' dl' ds'' 
 
 and the direction-cosines of the normal to the surface are pro- 
 portional to 
 
 dF dF dF 
 
 
 dx 
 
 • dy' 
 
 dz 
 
 
 Hence for al . 
 
 points ir 
 
 d'x 
 ds' 
 dF 
 dx 
 
 I a geodesic 
 
 d'y d'z 
 ds' ds' 
 dF dF" 
 dy dz 
 
 ine 
 
 
 
 (1). 
 
 These equations can be also deduced by the Calculus of 
 Variations. (Todhunter's Int. Calc. Art. 351.) 
 
 183. The equations of the last Article can be applied to 
 discover the forms of the geodesic lines on any surface. In 
 the case of developable surfaces, this object can often be more 
 simply effected by the consideration that when the surface is 
 developed, the geodesic must become a straight line. Thus 
 the geodesic lines on a right circular cylinder are easily seen 
 to be helices. 
 
 As an example in the case of a surface not developable, 
 take the sphere 
 
 x' + 7/-^z' = a' (1). 
 
 The differential equations of the geodesic lines become 
 
 d^x d^y d^z 
 ds' (/? 
 y ~ z ' 
 
 d£ 
 
 X 
 
 z 
 
 ds' 
 
 d'z . 
 
 dy dz 
 
 ds 
 
 V , = constant = c, 
 ^ ds ' 
 
 (2). 
 
ON CURVATURE OF SURFACES. 221 
 
 Similarly, ^ 
 
 dz dx 
 
 "ds-'dS^'^ (3)' 
 
 dx di/ ,,. 
 
 ^ds-'^ds^'^ ('^^ 
 
 Multiplying (2) by x, (3) by y, (4) by z, and adding, vre 
 get 
 
 c,x+c.^7j+c^z = (5), 
 
 shewing that all geodesic lines are great circles. 
 
 184. As a second example take the ellipsoid 
 
 sf V^ 2^ 
 
 ^ + f^+? = l (!>• 
 
 The differential equations of the geodesic lines become 
 
 (fx cly d^ 
 ds' _ ds^ _ ds^ 
 X y z 
 
 '^^ w e 
 
 Now let J) be the perpendicular from the centre on the 
 tangent plane to (1) at the point {x, y, 2), and let r be the 
 central radius of the ellipsoid drawn parallel to the tangent 
 to the geodesic line at the point (x, y, z). 
 
 ,2 „ ,2 „2 
 
 Then -2 = -4 + fi + -4 , 
 
 l_l(dx\\l (dy\\l(dz\\ 
 r'~a'[dsj '^¥ [dsj '^c'\dsJ ' 
 
 1 dp _x dx y dy zdz 
 
 p^ds~a^ds h* ds c^ ds ^* "^^ 
 
 1 dr _1 dx d^x 1 dy d^y 1 dz d^z 
 r^ ds ~ d^ ds ds^ h^ ds ds^ c^ ds ds'^ 
 
 _ ( X dx y dy z dz\ , ,.. 
 
 -Wdl'^¥ds'^?'ds)'' ^*^' 
 
 if k be put for each of the fractions in (2). 
 
222 ON CURVATURE OF SURFACES. 
 
 Now each of the fractions in (2) 
 
 X dj^x y d^y z dj^z 
 a'd?~^b'd?'^?d? 
 
 
 x' f z^ 
 
 
 Lch by equation 
 
 (7) of Article 181 
 1 
 
 
 
 7^ 
 
 p' 
 
 
 X- if z' 
 
 r'- 
 
 Hence from (3) 
 
 and (4) 
 
 
 
 1 dp r^ \ dr 
 
 p^ ds p^ ' r^ ds ~" 
 
 0. 
 
 dp dr _ 
 
 Avhence 
 
 j9r = constant (5). 
 
 This property is the same as that proved for lines of cur- 
 vature, but the two systems of lines do not coincide. 
 
 Let p be the radius of absolute curvature of the geodesic 
 at any point. Then each of the fractions in (2) 
 
 = + 
 
 Hence 
 
 V 
 
 yd'xV /d'yV /d'z\' 
 
 [dsV "^ W) ^ w) 
 
 1 
 
 1 p 
 p 
 
 x' y' z' 
 
 -p r'' 
 
 
 or p = + - .... 
 
 (6); 
 
 .'. p = kr^, where 
 
 k is some constant. 
 
ON CURVATURE OF SURFACES. 223 
 
 185. We shall conclude this subject with the following 
 proposition, known as Dupin's Theorem. 
 
 If there he three series of surfaces such that all the sur- 
 faces of each series intersect those of the other series at right 
 angles, then the lines of interseciion of the surfaces of different 
 series are the lines of curvature on the surfaces. 
 
 Let be the point of intersection of three surfaces, one of 
 each system. Take as origin, and the tangent planes of 
 the three surfaces as co-ordinate planes. Let >S^^, S^, S^ be 
 the surfaces touched by the planes of y2, zw, xy, respectively, 
 and let P, Q, J? be points near in the curves of intersection 
 oiB^, S^; S^, S^; S^, S^, respectively. Then since the surfaces 
 S^, S^ cut at right angles, the normals at P to these surfaces 
 are at right angles. Also since OP is ultimately a tangent 
 line to both of them at P, the normals at P are both perpen- 
 dicular to OP which is ultimately the axis of x. Let 0^, 0^ 
 be the angles which the normals at P to S^, S^, respectively 
 make with the planes of 2x, xy, respectively; 0^, (p^ those 
 which the normals at Q to S^, S^ make with the planes of 
 xy, yz, respectively, and -v/r^, -v/r^ those which the normals at 
 R to S^, 8^ make with the planes of yz, zx, respectively. Let 
 the lengths of OP, OQ, OR be a, jS, y, respectively. 
 
 Since the normal to S^ lies in the tangent plane to S^, the 
 tangent of the angle which the normal to >S^2 at makes with 
 
 the plane of xy islj-j , the suffix denoting the surface from 
 
 which the differential coefficient is obtained. Hence the tan- 
 gent of the angle which the normal to S^ at P makes with 
 the plane of xy is 
 
 /dz\ d_ /dz\ 
 
 \dy) ^ dx \dyj 
 
 + 
 
 3 
 
 But 
 
 whence 6^ — ol-t- l-^-j ultimately. 
 
224 ON CURVATURE OF SURFACES. 
 
 Similarly, *> = ^5^(^)3' 
 
 therefore - = n • 
 
 a p 
 
 Similarly, ^ - T7 > ^. ~ R ' 
 
 a 7 ' 7 
 
 But since the normals to S^, S^ at P are at right angles, 
 
 Similarly, </>i + </>3 = 0, "^2 + '^i = ^' whence 6^ = 0. 
 
 Hence the normals to S^ at and P both lie in the 
 plane of ooy and therefore intersect one another, and therefore 
 OP is the tangent to the line of curvature on >S^2 at 0. 
 Whence the theorem follows. 
 
 EXAMPLES. CHAPTER XIII. 
 
 1. Find the quadratic equation which gives the principal 
 radii of curvature at any point of an ellipsoid. 
 
 Deduce the position of the umbilici. 
 
 2. Find the umbilici of the surfaces 
 
 (1) xyz = a\ 
 
 2 2 2 
 
 and find the value of the radius of curvature at the umbilicus 
 in each case. 
 
 3. Find the equation of the projection of the lines of 
 curvature of the surface xyz = (^^ on the plane of xy. 
 
examples' chapter XIII. 225 
 
 4. Deduce the formulse for an umbilicus 
 
 r s t ' 
 
 first, from the consideration that the two principal radii of 
 curvature are equal at an umbilicus ; secondly, from the con- 
 sideration that the directions of the lines of curvature at an 
 umbilicus are indeterminate. 
 
 5. Find the condition that the two principal radii of cur- 
 vature at any point of a surface may be equal in magnitude 
 but opposite in sign. 
 
 Find the points on the surface 
 
 Ax'' -f- By^ + Gz^ = l 
 for which this is the case. 
 
 6. Shew that if the origin be at an umbilicus and the 
 normal at that point the axis of z, the equation of an ellipsoid 
 may be put into the form 
 
 a^ + y^ -{- kz (z — a) + hyz + czx = 0. 
 
 7. Any chord is drawn through an umbilicus of an ellip- 
 soid, and its extremity is joined with the extremity of the 
 normal at the umbilicus. Prove that the locus of the inter- 
 section of the joining line with the plane through the umbili- 
 cus perpendicular to the chord is a plane. 
 
 8. Prove that the lines of curvature of the surface 
 
 a ax — b ax — c 
 
 are circles, and that the plane of any one of them contains 
 one of two fixed straight lines lying wholly on the surface. 
 
 9. Shew that pr is constant for all lines of curvature 
 which pass through the same umbilicus of an ellipsoid. 
 
 10. Shew that pr has the same value for all geodesic 
 lines on an ellipsoid which touch the same line of curvature. 
 
 A. G. 15 
 
226 EXAIVIPLES. CHAPTER XIII. 
 
 11. Z7and Fare two adjacent umbilici of an ellipsoid, 
 P is any point on the surface which is joined by geodesic arcs 
 with U and V. Shew that the lines of curvature which pass 
 through P bisect the interior and exterior angles between 
 Ptr and PF. 
 
 12. If a point P move on an ellipsoid so that the sum or 
 difference of the geodesic arcs PU, PV joining it with two 
 adjacent umbilici of the ellipsoid is constant, shew that the 
 locus of P is a line of curvature. 
 
 13. Shew that at every point of a geodesic circle round 
 an umbilicus of an ellipsoid 
 
 d C 2 1 2 2 
 
 w^here a, h, c are the semi-axes of the ellipsoid, r the central 
 radius to the point, p the central perpendicular on the tan- 
 gent plane, and d the semidiameter parallel to the tangent to 
 the circle at that point. 
 
 14. The 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 intersection is an ellipse having four real or imagi- 
 nary contacts with the evolute of the principal section. 
 
 15. From the differential equation of geodesic lines in- 
 vestigate the nature of the geodesies on a right circular 
 cylinder. 
 
 16. Find the equations of the geodesic lines on a right 
 circular cone ; first, from the differential equations, and 
 secondly from the consideration that when the cone is 
 developed the geodesies become straight lines. 
 
 17. Shew that the distance of any point of a geodesic 
 traced on a surface of revolution from the axis varies inversely 
 as the sine of the angle between the geodesic and the meri- 
 dian of the surface which passes through that point. 
 
EXAMPLES. CHAPTER XIII. 227 
 
 18. Find expressions for the principal radii of curvature 
 at any point of a surface of revolution round the axis of x. 
 
 19. Prove that the product of the principal radii of cur- 
 vature at any point of a prolate spheroid varies as the product 
 of the squares of the distances of the point from the foci of 
 the generating ellipse. 
 
 20. Shew that the locus of the focus of an ellipse rolling 
 along a straight line is a curve such that if it revolve about 
 that line, the sum of the curvatures of any two normal 
 sections at right angles is the same at every point of the 
 surface generated. 
 
 21. If two surfaces cut each other at right angles, and It 
 be the radius of curvature of the curve of intersection at any 
 point, /3^, p,^ the radii of curvature of the normal sections of 
 the two surfaces through the tangent line to the curve at 
 that point, prove that 
 
 i_-i_ JL 
 
 ^ 9x P2 
 
 22. If r, 7^' be the principal radii of curvature at any 
 point of an ellipsoid on the line of intersection with a concen- 
 tric sphere, shew that the expression -^^ , is invariable. 
 
 23. If a geodesic line be drawn on a developable surface 
 and cut any generating line of the surface at any angle -yjr 
 and at a distance t from the edge of regression measured 
 along the generator, prove that 
 
 where p is the radius of curvature of the edge of regression at 
 the point where the generator touches it. 
 
 24. Prove that if r be the distance of any point of a 
 geodesic from the origin, p the radius of absolute curvature, 
 and p the perpendicular from the origin on the tangent 
 plane to the surface. 
 
 1 d' (r-) 
 
 15—2 
 
228 EXAMPLES. CHAPTER XIII. 
 
 25. The centres of curvature of plane sections of a sur- 
 face at any point lie on the surface 
 
 the axes being the tangents to the lines of curvature at 
 that point and the normal, and p^, p^ being the principal radii 
 of curvature. 
 
 If these sections touch a right cone of semi-vertical angle 
 a, about the axis of z, the centres lie on the elliptic paraboloid 
 
 — I- — = <^ sm a. 
 
 Pi P2 
 
CHAPTER XIY. 
 
 ON VECTORS AND QUATERNIONS. 
 
 186. The student is familiar with the word radius 
 vector employed in plane co-ordinate geometry to denote the 
 distance of any point from the origin. In this case the 
 direction of the line is determined by another quantity 
 called the vectorial angle. 
 
 We shall now define the word vector as meaning the 
 transference, or step, of a point from one position to another. 
 A vector is determined when the direction and magnitude 
 of this transference or step are both known. 
 
 Thus if two straight lines AB and CD be equal and 
 parallel, A and C being towards the same parts, the vector 
 J. 5 is equal to the vector CD. 
 
 If a point move from A to B and then from B to C, the 
 resulting step is one from A to C. This may be expressed 
 algebraically 
 
 vector AB -\- vector BC = vector AG ; 
 
 or, if we agree to represent the vector AB by the symbol AB, 
 
 AB + BG = AG (1). 
 
 This result is true whatever be the directions of the lines 
 AB and BG. 
 
230 ON VECTORS AND QUATERNIONS. 
 
 187. The symbol AB in equation (1) does not merely 
 represent the magnitude of the line joining the points A 
 and B, but also includes the determinate direction in which 
 AB is drawn. In conformity with the use of the sign — 
 already adopted in measuring co-ordinates, a step from B 
 to A would be represented by the same symbol as one from 
 ^ to jB with the sign — prefixed. Thus 
 
 vector BA = — vector AB ; 
 
 or, with the notation agreed upon, 
 
 BA = -AB (2). 
 
 188. By means of (1) and (2) it easily follows that, if 
 A, B, C be any three points in space, 
 
 AB-\-BC+CA = (3); 
 
 the meaning of this equation being that if a point travel 
 completely round a triangle, its final displacement is zero. 
 
 Obvious extensions of (1) and (3) are that if A,B, C, D, E 
 be any number of points, 
 
 AB^BG■\■GD^-I)E = AE, 
 
 AB + BG+GD-\-DE + EA = 0. 
 
 189. The symbol for a vector defines not merely the 
 length of the step but its direction. Thus if the symbol a 
 be employed to denote the vector AB, the symbol 2a will 
 reasonably denote a step in the same direction AB oi double 
 the amount, and so on. 
 
 It will be convenient then to represent vectors by a com- 
 pound symbol, one factor representing a step of a unit length 
 in the given direction, and the other factor giving the number 
 of units in that direction over which the particular trans- 
 ference takes place. This second factor is called the tensor 
 of the vector; thus if a denote a vector in any direction, 
 Ta the number of units of length in a, and Ua a unit vector 
 in that direction, we may write 
 
 a= UoL.Ta=Ta.Ua (1). 
 
I ON VECTORS AND QUATERXIOXS. 231 
 
 190. Referring now to the figure of Article 4, let us 
 suppose that unit steps parallel to the axes of x, y, z re- 
 spectively are denoted by the symbols i, j, k. Then the 
 vector OM will be denoted by xi, the vector 01^, or ML 
 which is equal and parallel to ON by yj, and the vector OR 
 or LP by zk. Hence, since 
 
 OP = OM-^ML + LP (Art. 188), 
 
 we have OP = xi -\- yj -\- zk (2). 
 
 Thus any vector in space can be expressed in terms of 
 unit vectors parallel to the three co-ordinate axes. 
 
 191. If Ca denote a unit vector along OP and Ta the 
 number of units contained in OP, or the tensor of OP, the 
 equation (2) may be written 
 
 T:i.Ua = xi -\- yj + zk, 
 
 or ^'^^'Wx'^'^'hi'^'^ioL'^' 
 
 where {Toif = x' -^ y' + z' (Art. 4); 
 
 or, if I, m, n represent the direction-cosines of OP, 
 
 UoL = li + mj + nk (3). 
 
 The quantities x, y, z, Tql, &c., which merely represent 
 the number of units of length, on some given scale, contained 
 in vectors, are called scalar quantities. 
 
 192. If /3 be any vector, a step u^, where u is a scalar, 
 represents a step parallel to /S, the magnitude of which 
 depends on u. Thus r, a being any other vectors connected 
 with l3 by the relation 
 
 r = a + njS (4), 
 
 it follows that the end of the vector r always lies on a vector 
 drawn through the extremity of a parallel to /3. This equation 
 therefore may be regarded as the equation of a straight line 
 drawn through the extremity of one vector a parallel to a 
 second vector ^ ; u being the variable quantity which deter- 
 mines different points in this line. 
 
232 ON VECTORS AND QUATERNIONS. 
 
 193. Let three given vectors a, yS, 7 be connected with 
 a fourth vector r by the relation 
 
 r = a + w/3 + 2^7 (4), 
 
 where u and v are scalar quantities which may assume any 
 values. 
 
 The step r is ejBfected by moving to the end of the vector 
 a, and from that point moving over u unit steps parallel to 
 the vector /3, and from the point now reached moving over 
 V unit steps parallel to the third vector 7. It is geometri- 
 cally evident that by changing u and v the extremity of the 
 vector r may be made to lie at any point of a plane passing 
 through the extremity of the vector a and parallel to the 
 two vectors ^ and 7. 
 
 Equation (4) may thus be considered the equation of this 
 plane, u and v being the variables. 
 
 194. If be any fixed point, and the vectors OAy OB 
 be represented by a and /3, the vector AB will be represented 
 by /8-a. For {Art. 186, (1)} 
 
 OA+AB=OB, 
 
 or AB = OB-OA 
 
 = l3-a. 
 
 Hence if C be any point in the straight line AB, such 
 that the length of AC=ux length of AB, 
 
 OC=OA+u.AB, 
 
 or y = a-]-u{/3- a), 
 
 7 denoting the vector OC. 
 
 This equation may be written 
 
 7 + a(ti-l)-Wy5 = 0, 
 or poL + qfi -\- ry = (5), 
 
 where j[), (/, r are any numbers in the same ratios as u— 1, 
 — 2^, 1. 
 
ON VECTORS AND QUATERNIONS. 233 
 
 The only condition which p, q, r need satisfy independent 
 
 of u is 
 
 p-hq + r = (6). 
 
 Hence if (6) be satisfied and a, (3, y represent any three 
 conterminous vectors, the other extremities of these vectors 
 will lie in a straight line. 
 
 195. Again, let four conterminous vectors OA, OB, OC, 
 OD represented by a, /3, 7, S, be such that the points A, B, 
 C, D all lie in one plane. 
 
 Then in equation (4) of Article 193 we may replace ^ 
 by ^ — a, 7 by 7 — a and r by 8. We thus obtain 
 
 g = a + li (yS — a) + V (7 — a) 
 
 or 0.(1 — u — v) +u^-\-V'y — h = 0, 
 
 which mav be written as 
 
 _pa + 5^/3 + r7 + 58 = (7), 
 
 provided jp, q, r, s satisfy the relations 
 
 P _?_''_ ^ 
 1 —u — V u V — 1 * 
 
 The only restriction on the values of p, q, r, s is that they 
 satisfy the relation 
 
 JJ + q + ?' + 5 = (8). 
 
 If this condition be satisfied the outer extremities of any 
 four conterminous vectors a, /S, 7, B connected by the relation 
 (7) must lie in a plane. 
 
 196. As an instance of the use of some of the preceding 
 theorems we may take the following proof of a well-known 
 elementary theorem. 
 
 Let ABC be any triangle, A\ B', C the middle points of 
 the sides BG, CA, AB respectively. Then with the notation 
 of Articles 186 — 189 we have 
 
 BA' = A'C=\BG, 
 CB' =B'A = iCA, 
 AG'=G'B = \AB, 
 A A' = AB + BA\ BE = BG + GB', GG' = GA + AG'. 
 
234 ON VECTORS AND QUATERNIONS. 
 
 Hence 
 AA' -^ BB' -h CC = AB + BO -^ GA + BA' -\- GB' + AC 
 
 = (AB + BG+ GA) + \{BG +GA + AB) 
 = 0. {Art. 188 (3).} 
 
 Hence AA\ BB', GG' are steps such that, if taken in 
 succession, the point returns to its original position, that is, 
 a triangle can be formed whose sides are equal and parallel 
 to AA\ BB', GG'. 
 
 197. A vector OA may be changed into another vector 
 OB in the same direction by a process of multiplication. 
 Thus if a represent the unit vector in this direction and X7, 
 yoL represent the two vectors OA, OB; OA becomes OB by 
 
 the process of multiplication by the fraction - . 
 
 If the vectors OA and OB be not in the same direction 
 no arithmetical multiplication will convert one into the other. 
 
 Geometrically the conversion can be effected by two 
 processes ; first a rotation of the vector OA in the plane 
 containing both vectors into the direction of OB, and secondly 
 an extension or contraction of OA until its length becomes 
 the same as that of OB. The latter is a process of arith- 
 metical multiplication, and it is convenient to use the alge- 
 braical notation for multiplication to denote the whole of the 
 process so that if a and ^ denote the two vectors OA and 
 OB, we write 
 
 q. OA = OB (1) 
 
 q denoting the combined operation of rotation and extension. 
 
 198. The quantity, or more strictly speaking the entity, 
 q, is called a quaternion, ^y analogy with algebraical 
 phraseology since q is an entity whicli when multiplied into 
 OA gives OB, q may be said to be the quotient of OB by 
 
 07? 
 
 OA, and may be represented by the symbol prr ' 
 
 The equation (1) may be thus replaced by the equation 
 
 OB 
 OA 
 
 <1 = ^ (2) 
 
ON VECTORS AND QUATERNIONS. 235 
 
 It is important to notice that, while the equation 
 
 OA 
 has a meaning, and with that meaning is true, the equation 
 
 OA .^,=0B 
 OA 
 
 has no meaning at present and is not therefore in any sense 
 true. 
 
 199. If the two vectors OA and OB be of equal length 
 the change of one into the other is merely an ojDeration of 
 rotation. In this case the quaternion is called a versor. 
 
 Since the operation denoted by a quaternion consists of 
 two parts, one of rotating OA into the position of OB and 
 the other of extending OA into the length OB, a quaternion 
 may be properly represented as the product of two factors, 
 one the versor of the quaternion and the other a scalar factor 
 which is called the tensor of the quaternion. 
 
 Thus if q denote any quaternion, and its versor and tensor 
 respectively be denoted by the symbols Uq and Tq we have 
 
 q=Tq. Uq= Uq . Tq (3), 
 
 since the order of the operations is clearly a matter of in- 
 difference. 
 
 200. The most important class of quaternions is that in 
 w^hich the angle between the two vectors is a right angle. 
 In this case the quaternion is called a right quaternion and 
 the corresponding versor a i^ight versor. 
 
 201. A right quaternion can be properly represented by 
 a vector. For a right versor is completely determined if the 
 plane in which the rotation takes place and the direction of 
 the rotation in that plane be known. A right quaternion 
 requires the additional element of the tensor. 
 
 The plane and direction of rotation are completely de- 
 termined by a vector drawn perpendicular to that plane so 
 that the rotation round this vector should be in the direction 
 
236 
 
 ON VECTORS AND QUATERNIONS. 
 
 of the hands of a clock. Thus the vector will be drawn on 
 one side or the other of the plane according to whether the 
 rotation be clockwise or counterclockwise, or, as we may more 
 mathematically phrase it, be positive or negative. A unit 
 vector will thus completely represent a right versor and a 
 vector of length represented by the tensor will represent any 
 right quaternion. 
 
 202. Any quaternion whatever can be represented by 
 the sum of a scalar quantity and a vector. 
 
 For let OA be the vector which when multiplied by the 
 quaternion q becomes OB. Draw OC in the plane A OB 
 
 perpendicular to OA. and BC parallel to OA to meet OC 
 
 in a 
 
 Then {Art. 197, (1)) 
 
 q.OA = OB 
 
 = OG+GB. 
 
 But the operation required to be performed on OA to 
 produce OC is a right quaternion, and that required to change 
 OA into CB is a scalar multiplication. Hence, if these 
 multiples be denoted by Vq and Sq respectively, we have 
 
 q.OA=^Vq.OA-\-Sq.OA, 
 
 or q = Sq + Vq (4). 
 
 But Vq may be represented by a vector, whence the pro- 
 position follows. 
 
 203. We have seen that if i, j, k represent unit vectors 
 in the directions of three co-ordinate axes, any other vector 
 can be represented by the expression 
 
 xi + yj 4- zk, 
 
ON VECTOES AND QUATERNIONS. 237 
 
 where x, y, z are scalar quantities. Hence any quaternion 
 can be represented by the expression 
 
 u -\- xi -\- yj -\r zh (5), 
 
 where u, x, y, z are scalar. 
 
 Thus a quaternion in its most general expression consists 
 of four terms. 
 
 204. The vectors ^, j, k represent right versors, the 
 directions of rotation being in the planes of yz from y to z, 
 of zx from z to x, and of xy from x to y. (See figure of 
 Article 3.) 
 
 Thus i^ must mean a rotation of the vector on which i 
 operates in the plane yz, twice through a right angle. The 
 effect of this rotation is to reverse the direction of the operand 
 vector, or to multiply it by — 1. 
 
 Hence i^ = — 1. 
 
 A similar proof applies to any right versor, and we there- 
 fore have 
 
 i^=/ = fe' = -l (1). 
 
 Again, the product i.j denotes the effect of rotating a 
 vector Oy round Ox through a right angle. This brings it 
 into the position Oz. Hence 
 
 i .j = k. 
 
 The product J*. 2 on the other hand denotes the result of 
 rotating the vector Ox round Oy into the position Oz. 
 
 Hence j .i = — k = — i.j (2). 
 
 Thus the commutative law does not hold good in the 
 multiplication of two vectors. 
 
 In a similar way we can obtain the relations 
 
 ik = — j = — ki (3), 
 
 kj = -i = -jk (4). 
 
 All these results are combined in the one statement 
 
 e=f = k' = ijk = -l (5). 
 
238 ON VECTORS AND QUATERNIONS. 
 
 205. The product of two unit vectors which are at right 
 angles can, as shown in the last article, be represented by 
 a unit vector perpendicular to each of them. The product 
 of any two mutually perpendicular vectors can similarly be 
 represented by a vector perpendicular to each of them whose 
 length — or tensor — is the product of the lengths of the two 
 given vectors. The product of two vectors not mutually 
 perpendicular is not a vector but may be represented as a 
 quaternion. 
 
 Let the two vectors a and ^ be represented, as in Article 
 190, by xi + yj + zk and xi + y'j + z'k respectively. 
 
 Then a . y8 = {xi + yj + zk) . (xi + yj + z'k). 
 
 Assuming that the operation of multiplication by a vector 
 is distributive both as regards the operator and operand, this 
 gives 
 
 ay5 = XX i"^ + xy'ij + xzik + yxji + yyj"^ + yzjk 
 
 + zxki + zy'kj + zzU\ 
 which by means of the relations of the last article becomes 
 a . /3 = — (xx + yy + zz) 
 
 H- (yz — y'z) i + {zx — zx)j + {xy — x'y) k (1). 
 
 Thus the product of two vectors assumes the quaternion 
 form of Article 203. 
 
 If the two original vectors be parallel it follows from 
 Article 191 that 
 
 % = l-^. (2). 
 
 X y z 
 
 In this case a^ = — {xx + yy + zz), which, by an easy 
 algebraical transformation, taking account of (2), reduces to 
 
 oi^ = - J{x' + f + /) (x"' + y" + z") 
 
 = -Ta.Tl3 (8). (Art. 191.) 
 
 Thus the product of two parallel vectors is a negative 
 scalar, the product of the tensors of the vectors. 
 
 If a and /3 be identical this reduces to 
 
 a' = -(Taf (4). 
 
ON VECTORS AND QUATERNIONS. 239 
 
 From this we deduce 
 
 a {Taf ^ ^' 
 
 or the reciprocal of a vector may be regarded as a vector in 
 the opposite direction to the original one, its length being 
 that of the original one divided by the square of its tensor. 
 
 If the two vectors be perpendicular we have by Article 
 191, XX ■\-yy' -\- zz =0. Hence in this case the product a^ 
 reduces to a vector. 
 
 206. With the notation of (4) of Article 202, it follows 
 that 
 
 S(a^)=-{xx' + yy'-Vzz'), 
 
 F (a/3) = [yz — y'z) i + {zx — zx)j + {xy — xy) k. 
 
 By working out the value of /3ol in a manner similar to 
 that of the last Article it is easily seen that 
 
 V{^.a) = -V(a./3). 
 Thus, since l3a=S {(3ol) + V (/3a), 
 
 and ay8 = >Sf(a/3)+F(a/3), 
 
 it easily follows that 
 
 S(^a) = S(c/3)="^" (1), 
 
 F(c/3) = -F(^a) = ?^^ (2); 
 
 results of considerable importance in the theory of quaternions. 
 
 207. There is a close relation between the quaternion g, 
 
 B 
 which represents the quotient — , and that which represents 
 
 the product ^ol. 
 
 Assuming the expressions for a and ^ of Article 205, let 
 
 B 
 u+pi + qj + rk represent the quotient -. Then (Articles 
 
 198, 199) it follows that 
 
 (u + pi + qj + rk) (xi + yj + zk) = xi + y'j + z'k. 
 
240 ON VECTORS AND QUATERNIONS. 
 
 Multiplying out the two factors on the left-hand side and 
 attending to the laws of Article 204, we have, since the ex- 
 pressions on the two sides represent identical vectors, 
 
 px-{-qy + rz=0 (1 ) , 
 
 wo + qz —ry = x (2), 
 
 uy -{■ rx — pz = y (3), 
 
 uz -\-py — qx = z (4). 
 
 From (2), (3) and (4) we easily obtain 
 
 px + qy -{• rz' = (5). 
 
 The geometrical interpretation of (1) and (5) is that the 
 vector part of the quaternion q is perpendicular to each of 
 the vectors a and /3, a result agreeing with that of Article 
 202. 
 
 From (1) and (5) we obtain 
 
 P ^ 9 ^ ^ my 
 
 yz —yz zx —zx xy —xy 
 
 S . 
 which shows that the vector part of - is parallel to the vector 
 
 part of ^.a (Art. 205). 
 
 Multiplying (3) by x and (2) by y, and subtracting, we 
 obtain 
 
 r {x' + 2/') - {p)x -\-qy)z = xy - x'y, 
 
 or r {x^ -\-y^-\- z') = xy' - xy, by (1). 
 
 \ 
 
 X'' + y' + z^ 
 
 Again, multiplying (2) by x, (3) by y, and (4) by z, and 
 adding, we obtain 
 
 u (x^ -\-y^-\- z^) = XX + yy' + zz. 
 
 S 
 On the whole, then, a or becomes 
 
 ^ a 
 
 {xx + yy' + zz) + {yz - y'z) i + {zx - zx)j + {xy — x'y) k 
 
 x' + y' + z' 
 
 Hence each of the fractions in (6) is equal to — ^., ^ ^.^ . 
 
 I 
 
ON VECTORS AND QUATERNIONS. 241 
 
 Comparing this result with (1) in Article 205, we have 
 with the notation of Articles 189 and 202, 
 
 V 
 
 
 12 
 
 208. The multiplication of a vector a by a second vector 
 13 at right angles to it, means that the first vector is rotated 
 through a right angle round the second, and then suitably 
 extended or contracted. 
 
 The product /3a, when a and /5 are not mutually perpendi- 
 cular, can be only interpreted geometrically by supposing it to 
 represent a compound operation on some third vector suitably 
 assumed. Thus the product ij in Article 204 may be regarded 
 as denoting a double operation to be performed on a third vec- 
 tor, which must first be in the plane zx, and after being rotated 
 positively through a right angle must lie in the plane yz. 
 Referring to the figure of Article 3 this vector must evidently 
 be either Ox or Ox. If we take the former and perform in 
 succession the two operations j and i it will pass successively 
 into the positions Oz and Oy : on the whole it will have 
 passed from Ox to Oy, that is, the operation denoted by k 
 will have been performed upon it. With this interpretation 
 of the product of the two vectors, we have, as before, 
 
 ij = L 
 
 209. It is evident that, in the general case, the inter- 
 mediate position of the operand vector must be perpendicular 
 to both a and ^. 
 
 Thus in the figure of Article 45 if Oz and Oz represent 
 the vector a and jS respectively and the angles cj), yjr be each 
 a right angle, the operation /Sol performed on the vector Ox 
 will turn it from the position Ox to Ox^, and then from Ox^ 
 to Ox ; or on the whole turn it from Ox to Ox and multiply 
 its length by the product of the tensors of a and yS. 
 
 This operation is equivalent to that effected by multipli- 
 cation by a quaternion. The connection between the plane 
 
 A. G. 16 
 
242 
 
 ON VECTORS AND QUATERNIONS. 
 
 of this quaternion and that of the original vectors can be 
 investigated either by Spherical Trigonometry or the formulae 
 of Analytical Geometry. 
 
 210. The multiplication of a vector by a quaternion 
 whose plane contains the vector can be geometrically re- 
 presented by a rotation of that vector through some angle 
 in that plane, combined with a suitable extension or con- 
 traction. 
 
 The interpretation of the product of a quaternion into a 
 vector not lying in its own plane can only be made by assum- 
 ing the vector to represent a right quaternion, and that the 
 product represents a compound operation to be performed on 
 some suitably chosen operand vector. 
 
 The product of two quaternions in general can be similarly 
 interpreted geometrically. 
 
 211. If there be three vectors or, /9, 7, the scalar part of 
 their product, or S(oi^j) has an important geometrical in- 
 terpretation. 
 
 Let the vectors be denoted by wi + yj + zJc, x'i + y'j + z'k, 
 x'i -t- y"j + z"h respectively. 
 
 Then we have 
 a/i?7 = {xi + yj + zk) [x'i + y'j + z'k) {x'i -\- y'j -}- z"k). 
 
 The terms in the product will be of one of the three type 
 forms ai^y ai^j, aijk where a is some scalar. Terms of the 
 first two types will, by Article 204, reduce to a vector form ; 
 the scalar jmrt of a/37 ^"^^^^ ^® entirely deducible from the 
 third type. 
 
 The first term of this type is xy'z'ijk or —xy'z", the others 
 can be deduced from this by interchanges of the letters xyz 
 combined with corresponding interchanges of ^, j, k. By 
 Article 204 it follows that any interchange of two of these 
 latter changes the sign of the product. Hence the whole 
 assemblage of these terms or B {p^/Sy) will be properly re- 
 presented by the determinant 
 
 xyz 
 
 X y 
 
 z 
 
 'f 
 
 n 
 
 X y 
 
ON VECTORS AND QUATERNIONS. 24 
 
 o 
 
 From the theory of determinants it easily follows that 
 
 S(al3y) = S{l3ya) = S{ya/3) = -S{ay/3} =- S{yl32) = - S{ffoiy), 
 
 and from the formula (3) of Article 32, the numerical value 
 of each of these is six times the volume of the tetrahedron of 
 which the three vectors a, ,/3, y are conterminous edges, 
 
 212. Again, since 
 
 affy = [xi -f yj + zh) {xi + y'j + z Jc) {x"i + y"j + z"h) 
 
 = {xi + yj + zh) [ — {x'x" + y'y" + zz") + (yz" — y"z) i 
 
 + {zx" - z'x) j + (x'y" - x"y') h] (Art. 205), 
 
 the vector part of a^y, or V((x^y), will consist of two portions, 
 the first of which may be denoted as aS {^y), and the second 
 is the vector part of 
 
 (xi + yj + zk) {(y'z" — y'z) i + {z'x' — z"x')j + {x'y" — x'y) k], 
 
 w^hich may be written V[a . V{0y)]. 
 
 Working out this product in accordance with the rules of 
 Article 204, the coefficient of i in the vector part becomes 
 
 yyxy —xy) — z{zx —zx), 
 which = x {xx" + yy" + zz") — x" {xx + yy' -\- zz) 
 
 = - xS {ay) + xS (a/3). (Art. 205.) 
 
 Hence on the whole 
 V[0LV{^y)] = - {xi + y'j + z'k) S {ay) + {x"i + y"j + z"k) S (a^) 
 = -/3S {ay) + 7>Sf («/3). 
 Thus finally 
 
 V {-xfiy) = aS {I3y) - ^S i^i) + r^S (^-xP) (1), . 
 
 and therefore 
 
 apy = S{a^y) + V{oil3y) 
 
 = S (oilSy) + aS {^y) - ^S (^7) + 7>Sf (a/3) (2). 
 
 The product of three vectors is thus expressed as a qua- 
 ternion. 
 
 It may be noticed that the necessary condition that the 
 product of three vectors may be a vector, is that 
 
 S{a^y) = 0. 
 
 16—2. 
 
244 ON VECTORS AND QUATERNIONS. 
 
 From this it follows that the volume of the tetrahedron of 
 which the vectors a, ft 7 are three conterminous edges must 
 vanish, or the three vectors must lie in one plane. 
 
 213. From Article 205 it follows that if two vectors a 
 and /3 be parallel 
 
 rm = (1); 
 
 while if they be perpendicular 
 
 S{a^) = (2). 
 
 Hence if p be a variable vector and a a fixed vector, and 
 the condition 
 
 >Sf(pa) = (3) 
 
 be satisfied, the extremity of the vector p must lie somewhere 
 in a plane through the origin perpendicular to the vector a. 
 The equation (3) may therefore be regarded as the equation 
 of this plane. 
 
 214. The equation 
 
 S{p-/3}a = () {i) 
 
 indicates that the vector p — /3 is perpendicular to the vector 
 a. But p — 13 is the vector which joins the extremities of the 
 vectors p and /3. Hence if these points be called P and B, 
 and OA be the vector a, the equation denotes that BP is 
 perpendicular to OA. Hence, if a and /3 be fixed vectors 
 and p a variable one, (4) is the equation of a plane through B 
 perpendicular to OA. 
 
 215. To find the equation of a plane passing through 
 three given points. 
 
 Let the three given points A, B, bo determined by the 
 vectors a, /B, 7 and let p be the vector to any point P of the 
 plane. Then the vectors AP, BP, CP are represented by 
 p — CL, p — /3, p — <y (Art. 18G), and these are co-planar vectors. 
 Hence (Art. 212) 
 
 S[(p-a){p-l3){p-ry)]=0 (5). 
 
 This may be regarded as the equation of the plane re- 
 (juired, but it can be reduced into a simpler form. 
 
ON VECTORS AND QUATERNIONS. 245 
 
 If the multiplications indicated be performed we obtain 
 eight terms, of which four, namely, those which contain three 
 or two factors p, are the product of co-planar vectors. These 
 all vanish by Article 212, and we have left 
 
 S {ppy + ap7 -f a/3p) - S (a/By) = 0, 
 
 or as we may write it by Article 211, 
 
 Sp (I3y + 7a + c(/3) - >Sf {a^y). 
 
 But since /3y = S {^y) + V (I3y) 
 
 Sp^y = 8 ipSfiy) + S (pV^y) 
 = S(pV^y), 
 since pS/3y is a vector. 
 
 Hence this equation may be written 
 
 S {p {V^y + Vyx + Va^)} - S2l3y = (6). 
 
 Let the vector V/Sy + F7a + Fa/3 be denoted by S. 
 Then SiS = S{a(Vl3y+VyoL+Va^)} 
 
 = S (al3y) + S (ayx) + S {aoi/S) 
 
 ==S(a^y), 
 since the other two terms vanish. (Art. 212.) 
 
 Hence (6) can be written 
 
 SpS = S2S, or S(p-a.)B = 0. 
 
 Comparing this with (4) of Article 214 we see that S, or 
 V(l3y)+ F(7a) + V{al3), is a vector perpendicular to the 
 plane ABC. 
 
 216. The equation of a straight line coinciding in direc- 
 tion with a given vector a can be written as p = ic7. where 
 1^- is a variable scalar. 
 
 Hence the equation of the straight line through the origin 
 perpendicular to the plane (6) can be written 
 
 p = uS (7). 
 
 Hence, at the point where (G) and (7) meet, we have 
 
246 ON VECTORS AND QUATERNIONS. 
 
 But (Art. 205) uh^ is entirely a scalar. Hence this gives 
 
 uh' =-- S («/37), 
 and dividing these equals by S we obtain 
 
 ^ 8 Vl3y + Vyoi + Val3 ' 
 
 which gives the value of the perpendicular on (6) from the 
 origin in direction and magnitude. 
 
 217. The equation 
 
 Tp = To^ (1) 
 
 signifies that the absolute length of the variable vector p is 
 equal to that of ot. Hence (1) may be regarded as the equa- 
 tion of a sphere whose centre is the origin and radius is'T(a). 
 
 The equation gives, by (4) of Article 205, 
 
 p~ = a". 
 Now (p —a){p-\-a) = p^ — a^ -\- py. — ap 
 
 = p'-a'+2 Vp2. {Art. 206 (2).} 
 Hence S (p — a.) (p + a) = 0, if p satisfy (1). 
 
 Thus the two vectors p — a and p + a are mutually perpen- 
 dicular: but these are the vectors which join the extremities of 
 the diameter in the direction of a with the variable extremity 
 of p. We thus obtain the well-known geometrical property, 
 that the straight lines joining any point of a sj^here with the 
 extremities of a diameter are at right angles. 
 
 218. The space at our disposal will not allow further 
 discussion of this subject or of its applications. It is hoped 
 that this slight sketch of the ali)habet of quaternions may 
 enable students to understand some references to it which 
 they may meet with in their reading, and possibly incite 
 them to study works specially devoted to it. Among these 
 may be mentioned Tait's Elementary Treatise on Quaternions, 
 
 Houel's Tlieorie Elenientaire des Quantites Complexes, 4™^ 
 Partie, and, chief of all, that stupendous monument of the 
 powers of the human intellect, Hamilton's Elements of Qua- 
 ternions. 
 
ANSWERS TO THE EXAMPLES. 
 
 CHAPTER I. 
 
 1. JS,2j3andj3. 
 
 2. The length of each side is ^6. 
 .,123 , 2 3 6 26 
 
 ^' ^^- ^' 2' 2' "^ '2' 2' 2' ' 2* 
 
 7. 
 
 a 6 c ^ 
 3' 3' 3' 
 
 1^ 
 
 3 
 
 9. If r^, 0^, ^,, r^, ^2, ^., be the polar co-ordinates of the 
 points, the (dist.)^ between them by Arts. (6) and (15) 
 
 = (r^ sin 6^ cos c}>^ - r^ sin 0^ cos <^y 
 
 + (r^ sin ^1 sin ^^ — j'^ sin ^^ sin <^y + (7*^ cos 0^ - r^ cos ^g)^ 
 
 = T^ (sin^ ^j cos" <^j + sin^ ^j sin^ ^^ + cos^ ^J 
 + r^ (sin^ ^„ cos^ c^, + sin" ^^ sin^ <^., + cos^ 0^ 
 — 2i\7'^ {sin 0^ sin ^., (cos ^^ cos ^., -i- sin cji^ sin <^,) 
 
 + cos 0^ cos ^J 
 
 = r^^ + r/ - 2rjr2 {cos 0^ cos ^^ + sin 0^ sin ^^ cos ((/)j - cft,^)}. 
 
 10. r^4, ^=|, c^ = ^. 
 
 11. x=l, y = J'^, z = 2j'6. 
 
248 ANSWERS TO THE EXAMPLES. 
 
 CHAPTER 11. 
 
 1. — -^=^-2 = — T^. 2. x + z = 4:. ?/ = 2. 
 
 ^- 2/ = 2j z = lj ^-y--- 2 j' 6'2'a- 
 
 4. x + ij + z=Q; 2j3. 5. x = ^=^. 
 
 2 3 
 
 6. aj-l = -^^ = s-3. 
 
 3^/3 
 
 7. Let (a, /3, y) ; (a', p', y), be the two points, Ix + my + nz=p 
 the given plane. Then the required plane can have its equation 
 in the form 
 
 A{x-a)+-B(y-P)+C{z-y) = 0, . 
 
 and A, B, C must satisfy the two conditions 
 
 A(a'-a)+B {P' - yS) + C (y' - y) = 0, Al + Bm + Cn = 0, 
 
 whence 
 
 A : B : G :: m (y -y)-n{(S' - (3) : n {a' -a) - l{y - y) 
 
 : l(^'-l3)-m(a-a). 
 
 8. z=3, x + y = 3. 
 
 9. Let A (x - 2) + B {y - 3) + C (z - i) = rejjresent the 
 plane required ; 
 
 .-. ^(l-2) + i?(2-3) + (7(3-4) = 0, or A + B+C = 0, 
 A . J3 + B + C . 2^3 = 0, 
 whence A : B : C :: 2 Js - 1 :-j3 : 1 -^3, 
 
 and the plane becomes 
 
 (2 V3 - 1) (a; - 2) - V3 (^ - 3) + (1 - Js) {z--i) = 0. 
 
 10. Let I, 7)1, n; l', m', oi be the direction-cosines of the 
 given lines ; \ fJi, v those of the required one ; 
 
 .*. kl + jxni + v7i = 0, XI' + ixni + vn — cos a. 
 
 The latter equation gives 
 
 (A.^ + ixm + vn)' = COS" a (/V" + fx" + v'). 
 
ANSWEKS TO THE EXAMPLES. 249 
 
 which combined with the former will give two values of the 
 
 ratios A. : /a : v, as in Art. 57. For the latter part put cos a = — -- 
 
 and find the value of X^X^ + ^^/x^ + v-^v^ ; remembering that as 
 W + mm + nn vanishes this will also be found to vanish. 
 
 11. Let (a, P, y) be the given point, I, m, n ; l', m', n' the 
 direction-cosines of the perpendiculars on the two planes. The 
 required plane is 
 
 (77171 — m'n) (x — a)-¥ [nV - n'l) (y - P) + {I'm' — I'm) (x — a) = 0. 
 
 (See Art. 30.) 
 
 12. The proof of Art. 19 holds when the axes are not 
 rectangular if I, m, n mean the cosines of the angles between 
 OD and the axes. 
 
 13. Draw the oblique co-ordinates of the point D, and pro- 
 ject OD on the axes in succession. 
 
 l + m cos V + n cos u m + n cos X + l cos v 
 
 ^^- 1 = B 
 
 n + 1 cos fx + m cos A, 
 
 15. The condition is 
 
 {X - a)' + {y- Pf + {z- yf = (x - a'f + {y - ji'f + (« - y'f, 
 
 which reduces to 
 (a'-a){.-'^}.(/3'-^){,-^^'}.(y-,){.-I±l'} = 0. 
 
 16. (1) A series of planes parallel to that of yz; iovf{x) =0 
 gives a series of equations £c = «,, x = a„, tfec. (2) A series of 
 spheres with the origin as centre. (3) A series of right circular 
 cones with Oz as axis. (4) A series of planes passing through 
 the line Oz. 
 
 17. (1) The axis of z. (2) A straight line OP through 
 inclined at an angle a to Oz, and such that the plane zOP makes 
 an angle ^ Avith zOx. (3) A circle whose radius is a in the 
 plane of zx and with its centre at the origin. 
 
 D Ti- 
 
 ls. ^ cos ^ sin^ + i>sinrf) sin ^ -I- Ccos^= — . 19. -. 
 
 r ^ 2 
 
250 ANSWERS TO THE EXAMPLES. 
 
 20. Let Zj, m^, n^ ; l^, m^, n^ ; ^3, m^, n^ ; be the direction- 
 cosines of the normals to the three planes. Then the equation 
 of any plane through the line of intersection of the first and 
 second is 
 
 where A is a constant, and if this is perpendicular to the third, 
 
 or cos ^ + A. cos ^ = 0. 
 
 Also if the plane passes through the origin p^ + Xp,^ = ; 
 
 . '. p^ cos A = J) 2 ^^^ ^' 
 and the plane becomes 
 
 {l^x + m^y + n^z) cos A — {l^x + m,,y + n,^z) cos 5 = 0. 
 
 If in addition jp^ cos 5 = ^9^ cos C, the other two planes will 
 have equations of a similar form and all three planes will inter- 
 sect in one straight line through the origin. 
 
 21. Let Ix + my + oiz ^ 2^ ^® ^^^^ equation of one of the 
 planes ; 
 
 .-. from the data ^ + — +- = 0, or - + — + - = (1), 
 
 I m n I m n ^ ' 
 
 and l{a -a) + m{h'-h) + n{c -c)-^0 (2) ; 
 
 .*. substituting for n out of the second in the first 
 
 11 c'-c 
 
 -- -\. = (J • 
 
 I 1)1 l{a —a) + in{b' — b) ' 
 . •. I' {a' -a) + rim + m' (b' - 6) = 0, 
 
 which gives two values of — , and corresponding to each of these 
 
 n II 
 
 from (2) we can get one value of — . If -^ , --be these two 
 ^ ' ^ TYi m^ m^ 
 
 values, - ' ^ =-7- . Simihirly - ' ^ = , . 
 •))i{ni^ a —a 'm^ni^ c — c 
 
 Hence if the lines be at right angles 
 
 - IJ. 7i,n„ 
 
 1 + — ^^ -I- -i -^ = : 
 
 h'-b b'-b ^ 1' 1 1 ^ 
 
 .-. 1 +- -I- -, =0j .'. -; +J-, 7 + -, = 0. 
 
 a —a c - c a -a —b c —c 
 
ANSWERS TO THE EXAMPLES. 251 
 
 22. -. This and Example 23 are to be solved as the last 
 example. 
 
 23. P {B' + C) 4- Q iC' + A') + B (A' + B') = 0. 
 
 24. The co-ordinates of the middle points of the lines joining 
 1, 2 and 3, 4 are, Art. (7), 
 
 and i(c-a), J(c + cZ-«-5), |-(fZ-&), 
 
 whence the result follows. 
 
 25. The co-ordinates of any point on one of the lines may be 
 represented hj a + lt, b + mt, c + nt ; and those of any point on 
 the other by a + I't', h' + mt\ c + ri't'. The square of the dis- 
 tance between these points is 
 
 (a - a' + U- I'tJ + {b-b' + mt - m'ty + (c - c' + nt - n'tj. 
 
 The conditions that this may be a minimum by the variation 
 of t and t' are 
 
 (a-a'+U- It') l+{b-b' ->tmt- mt') m+{c-c+nt- n't') n = 0, 
 
 and 
 
 (a-a' + U- I't') r+{b-b' + mt - m't') m' +(G-c' + nt- n't') n = 0, 
 
 which shew that the line joining the two points is perpendicular 
 to both the given lines. 
 
 26. By the solution of the last question, 
 
 I (a - a') + m (/5 - /?') + n {y-y) + t- 1' cos <9 - 0, 
 I' {a- a') + m' (13 -/3') + n'{y-y') + t cos e-t' = 0, 
 whence t' sin^ O — u' + u cos 0. 
 
 27. Taking x^, y^, z^, &c. as the co-ordinates of the angles of 
 the tetrahedron it is easily shewn that the co-ordinates of the 
 middle point of the line joining the middle points of two 
 opposite edges are 
 
 ^ (x^ + x^-^ x^ + x^), &c. 
 
 28. By the help of a figure and the last question it is easily 
 seen that the two lines x, y are the diagonals of a parallelogram 
 whose sides are ^ct and ^a and w is the angle between the dia- 
 gonals, whence by Trigonometry the result follows. 
 
252 
 
 ANSWERS TO THE EXAMPLES. 
 
 29. —=. if c is the ed^e of the cube. 
 
 30. J sjb'c' + c'a' + a%'. 
 
 31. The equations of the planes are 
 
 lx + my + nz=p, mx + ny + Iz - j:), 7ix + Iy + mz=p ; 
 
 - P 
 « , cc — 2/ — ^ — 7 • 
 
 t + m + 7i 
 
 32. Any point on the given line can have its co-ordinates 
 expressed by a — It, b - 7)it, c — nt; the value of t is obtained 
 from the condition of perpendicularity. 
 
 33. Take the shortest distance between the lines as axis 
 of z, its middle point as origin, and the plane of zx to bisect 
 the angle between the lines. 
 
 CHAPTER TIL 
 
 1. r^ + r {A sin 6 cos <ji + B sin sin ^ + C cos 6) + I) ~ 0. 
 This equation gives two values of r the product of which is D. 
 
 2. The polar equation of any plane is 
 
 A sin 6 cos cf> + B sin sin ^ + C cos 9 = — . 
 
 Hence if this be the equation of the locus of F, since OP = 
 
 the equation of the locus of Q is 
 
 Br 
 
 A sin 6 cos cfi + B sin sin ^ + C cos - ,.j , 
 
 which is the polar equation of a sj^jhere. 
 
 3. If the locus of B be 
 
 r" + r (A sin 6 cos <f> + B sin 6 sin ^ + C cos 6) + I) = 0, 
 that of (? is 
 
 /j' + k'r (A sin cos ^ + ^ sin 6 sin ^ 4- C cos ^) + Br = 0, 
 which is another sphere. 
 
 OQ' 
 
ANSWERS TO THE EXAMPLES. 253 
 
 4. The plane in question is 
 
 XX + yy + zz = X + y + z = c , 
 
 also where it meets the sphere x^ + y^ + z^ = c^, 
 whence x^ + y^ -{- z' — '2 (xx + 7jy' + zz) + x' + y'^ + z"^ = 0, 
 or {x - x'f + {y- y'Y ^{z- z'f = ; 
 
 / / _^ / 
 
 • • yU tXy J IJ — y 5 "^ — " ^ • 
 
 5. Take A as origin and AB (=ci) as axis of x, the equation 
 of the locus is 
 
 £c- + 2/^ + 2;^ = m^ {{x - of + y^ ■\- z^], 
 which reduces to the equation of a sphere. 
 
 6. With the same axes as in the last question the two lines 
 whose direction-cosines are proportional to x, y, z and x — a, y, z 
 must be at right angles. Hence x (x — a) + y~ + z' = 0, a sphere, 
 on ^^ as diameter. 
 
 7. Take for the equations of the fixed straight lines those 
 given in Ex. 33, Chap. 11. The equations of the two planes can 
 then be written y — mx + A, (2; — c) = and y + mx + /x (2; + c) = 
 where X. and fx are constants. The condition of perpendicularity 
 gives 1 —7)f + Xfx = and by substituting for A. and /a out of the 
 Urst two in the third we get (1 — m^) (z^ — c^) + y^ — m"ic^ ^ as 
 the locus. 
 
 8. If >S' = 0, /S" = be the equations of two spheres in their 
 simplest form, the equation S' — /S = is easily seen to be a plane 
 perpendicular to the line joining their centres, which muse cut 
 each sphere in a circle. 
 
 9. The equations of the spheres can be written 
 
 S - kr' := 0, >S" - h'r' = 0, S" - k'V = 0, 
 
 where k, k', k" are constants and r changes. The first and second 
 
 »,S' iS" 
 intersect on the sphere ^ — -,7 = 0, whence the rest will follow. 
 
 10 and 11. These follow easily from (8). 
 
 12. The six centres of the spheres must lie at the angular 
 points of a regular octahedron the edge of which is the radius. 
 
254 ANSWERS TO THE EXAMPLES. 
 
 13. Take the three planes as co-ordinate planes, and let 
 I, 7)1, n be the direction-cosines of the straight line, x, y, z the 
 co-ordinates of the point. Then by projecting on the axes in 
 succession x = la, y = mb, z = nc ; 
 
 2 2 2 
 
 X y z , 
 
 • u — -}--= 1 
 
 a 0^ c 
 
 14. We have 
 
 cr. -4- n. Qi >>' 
 
 -OF, 
 
 X 
 
 ■ha 
 
 
 y 
 
 
 f 
 Z 
 
 
 
 
 
 cos a 
 
 
 sin a 
 
 x" 
 
 — a 
 
 
 -y" 
 
 
 z" 
 
 and '^—^ = -^ = ^^ = O'F, 
 
 U cos a sm a 
 
 if x, y', z are the co-ordinates of P and x", y\ z" those of P'. 
 Also if {x, y, z) be any point in PP' 
 
 X — x' y — y' z — z 
 
 X —X y —y z —z 
 
 between these equations and OP . O'P' = c^ we have to eliminate 
 x\ y\ z, x", y\ z\ OP, O'P'. 
 
 15. Take the line round which the line revolves as axis of z 
 and the point where the shortest distance between the lines meets 
 it as origin, and let c be the length of this shortest distance and 
 the angle it makes with Ox. Let also a be the angle between 
 the fixed and revolving lines. The equations of the revolving 
 line are 
 
 X — c cos 6 y — c sin 6 z . ,., o . o 
 
 J = = , where L + iir = sm" a, 
 
 t m cos a 
 
 X 7/ ''' 
 
 and since this is perpendicular to the line ;: = — '-. — >, = k» '^ve 
 
 c cos B c sin 6 
 
 have I cos 6 + m sin ^ = 0, whence eliminating I, m and 6 we get 
 
 " . n 2 ~ 
 
 •^ cos^ a 
 
 16. n'x' + {n'-\){y' + z') = c\ 
 
 1 7. Let y- -irz^ = x^ tan^ a be the right cone. Then the equa- 
 tion required is 
 
 x' (y' + z') = k' tan' a. 
 
ANSWERS TO THE EXAMPLES. 255 
 
 CHAPTER IV. 
 
 1. 2^3, 0, -n/2: 2. x"-'^^^, 
 
 3. Take x — y — z and any two straight lines perpendicular 
 to it as axes : the axes in the last question will do. 
 
 4. From the last two of the second set of relations the ratios 
 of l^y ^2' ^3 ^^^^ ^® deduced, and their absolute values from the 
 first, with the help of the other three. 
 
 5. 3, 3, — 3, use Art. 51. 
 
 6. The proof is exactly similar to Art. 50 with the excep- 
 tion that 
 
 0^ -Vy^ -{■ %^ ■\- lyz cos X + ^zx cos /x, + ^xy cos v 
 is transformed into x'^ + y'' + z^, 
 
 7. Transform so as to take the line x — y = z as axis of x 
 and any two lines perpendicular to it and each other as axes 
 of y and z : as in Examples 1 and 2. 
 
 CHAPTER V. 
 
 1. The direction-cosines of the generating lines through any 
 point (a, yS, y) are given by 
 
 V ni^ n^ la . m^ ny 
 
 a a G a a c 
 
 The condition that these shall be at right angles is obtained as 
 in Examples 21 — 23, Chapter ii., and gives by the help of the 
 
 2 . 02 2 
 
 relation r, -^ = 1, a value of y^. 
 
 2 and 3. The direction-cosines of the generating lines are 
 given by 
 
 la. mB ny ^ , T 7}f r^ _ 
 
 From these we easily get, eliminating mi. 
 r /o^ ^\ _ 2lnay n 
 
 2 I „2 + 7,2 ) ^,2^2 + ^2 ( ^2 ^2 ) " ^' 
 
256 ANSWERS TO THE EXAMPLES. 
 
 Whence 
 
 c \c / c \a 
 
 ^^ c \c b J c \a- J a -a^ 
 
 n^n^ 2 /^' . ^'\ ^ f^ , Y^ v" + c 
 
 J a \ c J 
 
 2 ) 
 
 2 I 2 2 2 
 
 a \a 
 
 2ay 
 
 • 2~2 
 
 Whence by symmetry we get 
 
 ^7? " ^^^P " /T? " 2^8^^ " ~^27a" " 2a^"~ ' 
 
 and if ^ be the angle between the two straight lines, each of these 
 ratios 
 
 cos B 
 " (a^ - a') + (/3^ - b') + (y' + c') 
 
 sin 9 
 
 \/4 {/3y - (^" - b') (y" + c")} + . . . similar terms 
 
 .•.cot^= a- + ^- + y--a--6- + c- 
 
 2 7y=^ (a=^'+ 6^) + p" {cr - c') + a' (b' - c') + b'c' + cr? - a'b' ' 
 The solution of (2) easily follows by putting b = a, and 6 = a. 
 
 4. If I, m, n ; l\ m\ oi' be the direction-cosines of the two 
 
 radii vectorcs, these Avith - - - — , 0, ,— — form a set of 
 
 b J a' -c' b Ja^ - c 
 
 nine quantities satisfying the conditions of Art. 44. Also if r, r 
 
 be the two radii 
 
 r' = aT + ¥m^ + cSi\ r" = aH" + b'm" + cV ; 
 .-. r"- + r" = a' (f + l") + b' {nr + m") + c' (rr + n") 
 
 _ a* jb' - c') + b* jar - c"-) + c* (a' - b') 
 b'{a'-c') 
 
ANSWERS TO THE EXAJ^IPLES. 257 
 
 5. Planes parallel to Ix + my ^nz = and Vx + my + n'z — 0. 
 
 (\2 2 " o 
 X + y + z) -X -y -z" . , , 
 
 u. -^tj-tij^-f,.^- '—-^ , whence the result 
 
 follows. 
 
 7. By Art. 59, the projections of OP, OF' on the plane of xy 
 are tangents to the principal ellipse at the ends of conjugate 
 diameters. The sum of the squares of these projections is there- 
 fore cr 4- y^. Also the height of above the plane of xy can be 
 easily shewn to be c, whence the result follows. 
 
 8 and 9. If X, /x, v be the direction-cosines and r the length 
 of any radius vector in the plane Ix + m?/ ■\-nz — ^\ 
 
 r a c 
 
 while XI + /xm + v?i = ... (2). If the section be a rectangular 
 
 1 hyperbola two direction? at right angles make - vanish. By the 
 
 methods of Ex. 21 — 23, Chap. ii. the condition for this is found. 
 
 The condition for a circular section is that -g shall be invariable 
 
 r 
 
 for all values of X, /x, v consistent with (2) ; whence \l + fxin + vn 
 
 must be a factor of 
 
 flV V\ XfX 
 
 where h is the constant value of —, . For the rest see Art. 49. 
 
 10. By Art. 63 the generating lines at any point (x, y, z) 
 must be parallel to the asymptotes of a section by a plane through 
 the centre parallel to that which cuts the surface in these two 
 lines. The equation of such a plane is by Art. 58 
 
 ax By yz - 
 — + V^ + ^ = 0, 
 a c 
 
 if (a, (3, y) be the point : the semi-axes of the section by this 
 plane are given {Art. 68 (10)} by the equation 
 
 a {a- r") 6 (6 — r') c{c~ rf 
 A. G. 17 
 
258 ANSWERS TO THE EXAMPLES. 
 
 Also if r^^ T^ be the two values of r^ in tliis equation and 10 
 the angle between the asymptotes of the curve 
 
 tan^(9 = _^; 
 .-. COS 2^ = ^ ' • 
 
 7. 2 _ 2 ^ 
 '2 ' I 
 
 which is the required result. 
 
 11. The square of the distance of the focus of any section from 
 the centre is the difference of the squares of the semi- axes of that 
 section. Hence if p, X, fi, v be the radius vector and direction- 
 cosines of any point in the locus, p^ = i\^ ~ r^^ where r^, r^ are the 
 two values of r in equation (10) of Art. G8, and A, /x, v are de- 
 termined by equation (12) of that article; between these equations 
 and (5) we have to eliminate I, m, n^ and for Xp, yap, vp to substi- 
 tute x^ y, z. 
 
 12. X' ^if -vz^ - {Ix + my 4- nzf = or. See Art. (28). 
 
 13. A+B + C=^0. See Arts. 34 and 44. 
 
 14. If lx + 7)iy + nz~0 ... (1) be the equation of the plane 
 
 base, the co-ordinates of the vertex are sjiven by 7 = — = - = 5. 
 
 C m n 
 
 Let then ' — r — =- = ^ - - =r... (2) be the equations of 
 
 A fX V 
 
 the generating line ; substitute for x, y, z from these equations 
 in (1) and the equation of the ellipsoid, and eliminator. Thus we 
 get a relation between A, /x, v and then from (2) the equation of 
 the cone as in Art. 34. 
 
 15. x^ -vy' ^z^ -1y% — 1zx-1xy = ^. 
 
 16. a;^ 4- 2/^ -I- 2;^ = (Ix -\- my + nz)' sec' a. See Art. 2%. 
 
 17. Determine I, m, n and a in the last question so as to 
 make the cone contain the given lines. 
 
 18. Is solved in question (2). 
 
 19. Assume \x + /my -hvz^ 0...(1) the plane; .'. Xl + iJim + v)i = 0. 
 Eliminate z between (1) and the given cone. We get a cubic 
 
 equation in - one value of which must be -j ; the product of the 
 X c 
 
 other values is easily obtained. See Ex. 23, Chapter 11. 
 
ANSWERS TO THE EXAMPLES. 259 
 
 20. These values satisfy the equation of the hyperboloid 
 whatever <^ and 6 may be. Substitute in the equations of Art. 57, 
 and we shall get finally 
 
 x — a cos <^ sec 9 y — h sin <^ sec 9 z — c tan 9 
 
 a sin (cfi^O) — b cos (cfi^O) ± c 
 
 21. Use the equations in the last question. 
 
 22. Any planes through the two generating lines in question 
 may have their equations written 
 
 a \o cj a \b cj 
 
 The condition that the line of intersection of these should be a 
 generating line is easily found to be kk' = — 1. 
 
 It can be shewn that the intersections of these planes with 
 either of the planes 
 
 - Va^T7±|Vf7^^=0 
 c b 
 
 are always at right angles to each other. These are the planes 
 which give circular sections. 
 
 23. Take the general homogeneous equation of the second 
 degree in a, p, y, 8. Find the conditions that this may be satis- 
 fied by either of the pairs a = 0, y = 0, and ^ = 0, 8 = 0. 
 
 24. Substitute x^{x' - z) cos 9, z- {x + z) sin 9^ 
 
 where tan 9 = 
 
 ajb'-c'' 
 
 CHAPTER YI. 
 
 1. If X, /x, V be the direction-cosines of any generator of the 
 given cone a^X^ + b'fx^ + cV^ = d', whence by Art. 79 the result 
 follows. 
 
 2. Use equations (6) of Art. 77, and in the ^iven case by 
 Art. 78, 
 
 2 
 
 P _^,^ 
 
 2 , 2 2 — "'3 
 
 dH' + b^m^ + cr 71 
 and the locus becomes 
 
 8 2'' 
 
 17—2 
 
260 ANSWERS TO THE EXAMPLES. 
 
 3. Use formulae of Art. 74 to shew tliat tlie plane passes 
 through the three given points. 
 
 4. „ — T^ , where p is the perpendicular from the origin on 
 the plane LMN. 
 
 Trahc 
 
 5. volume 
 
 3^3* 
 
 , 6. A cylinder whose axis is parallel to Oz and whose trace 
 on the plane of xy is given by 
 
 ah / .,... ;. T-n ( 1 c^(^^^sin^^ + &--cos^^) ) h 
 
 — = J a- sin- + b cos ^ -^ 1 ^^ ^7^ \ . 
 
 r ^ ( a'W j 
 
 7. Let a, ^ be the co-ordinates of the point where the straight 
 line cuts the plane of xy^ and let a line be drawn inclined at an 
 
 2 2 
 
 X 11 
 
 angle B to Ox to cut the ellipse -^ + ^ = 1 in two points. If r^, r., 
 
 be the distances of these two points from a, /?, the square of the 
 eccentricity of» the vertical section through a straight line x — a^ 
 
 (r ~tY 
 y-p supposed to be its directrix must = I -^ ^ > , but it also equals 
 
 C^ (a^sin^^ + ft^ COS"^) , K ^ nn ^ ' • ^ 
 
 1 ^^ STT— — by Art. oo, whence since ?•, and ?% are 
 
 a b' ' 
 
 expressed' in terms of 6 we can get a quadratic equation in tan'^ 
 
 the roots of which must be real. 
 
 8. Use Art. 77, 2^ being a constant : 
 
 /x^ tr' '^\ /x^ v^ '^\' 
 
 9. If a', h\ c' be the conjugate semi-diameters, and x\ y, z 
 the co-ordinates of the point in which the three planes meet 
 
 X - x'^ 
 A a 
 
 by similar triangles and Art. 79. 
 
 10. We have to find the directions of the axes of the section 
 of Ax^ + ]hf + Cz" -- 1 by the j)lane Alx + Bmy + Cnz = 0, where 
 PI- + Qm^ + Rif = 0. See Art. QS, Equations 5 and 12 and elimi- 
 nate Z, m, n. 
 
ANSWERS TO THE EXAMPLES. 261 
 
 n /I \ _ ■?^ J^^ _^ /3 _ _ ^ ^ ^ 
 
 li. (i; ''- 1 + 2l-2B'^2l'C' ^~ 21B''^~ 21C' 
 
 (2) 2BI3{b-/3) + 2Cy{c-y) = a-a if a, 6, c be co- 
 ordinates of the fixed point. 
 
 1 2. If X, y, z be the co-ordinates of any point on the perpen- 
 dicular, 
 
 ax by cz 
 
 x^ + x^ + x^ ~ y^+ y, + y^ ~ ^,+ z, + gg _ ^aV -f- 6y + c V 
 a b G J3 
 
 by Art. 74, . 
 
 X y z xx^ -t- 2/2/, + zz^ 
 
 ~ x^+ r e, + a?3 ~ 2/i + 2/2 + 3/3 ~ ^1 + ^2 + '^ a 
 
 3 
 
 2 12 2 
 
 whence the result follows. 
 
 13. If the curve be a parabola the line joining its centre 
 to the origin must be parallel to the plane, whence the result 
 follows. 
 
 CHAPTER VII. 
 
 1. (1) The discriminating cubic is s^- 10/ + 13s + 47 = 0. 
 This has tw^o positive roots and one negative root by Descartes' 
 rule of signs, all the roots being real. Hence the equation repre- 
 sents a hyperboloid of one sheet. 
 
 (2) A hyperbolic cylinder. 
 
 2. (1) Hyperboloid of revolution whose centre is at the 
 point (2, 1,0); of one or two sheets according as a > or < 0. 
 
 (2) Co-ordinates of centre - f i - if, yV ; hyperboloid of one 
 sheet. 
 
 (3) A parabolic cylinder. 
 
 (4) A hyperboloid of one sheet. 
 
 3. The two equations merely differ by Ir (x^ + y^ + ^') which 
 remains unaltered by any transformation round the origin. 
 
 The. second is a right circular cylinder, the first a spheroid. 
 
262 ANSWERS TO THE EXAMPLES. 
 
 4. An ellipsoid if 1— /x<^2, a liyperboloid of one sheet 
 
 5. An ellipsoid whose centre is at the point -j- , — , — : 
 the equation when z = can be put into the form 
 
 6. See Example 6, Chapter iv. Wrong reference in question. 
 
 7. Take the general equation of the second degree and find 
 the conditions that it may be satisfied when a; = and z=0, and 
 also when y = and z = 0. 
 
 10. See Art. 150 for the condition that the equation repre- 
 sents a surface of revolution, and Art. 90. These conditions give 
 ii c = a+b, b' = 0, c' = ah, and the equation can be written 
 
 "\ 2 -,"2 
 
 (xja + y Jhf + c(z-\--\ + 2a"x + 2h"y + c^ - ^ = 0, 
 
 which can be again written 
 
 z -h —\ +2 {a" - h J a) 
 
 X 
 
 + 2{b"-kjb)y + d ^-'-0. 
 
 c 
 
 And if k be so chosen that x J a + y Jb + /t = 0, and the line 
 2x {a" -kja) + 2y (b" - k Jb) - 0, 
 
 c" 
 are at right angles, the former united with z -\ — =0 must give 
 
 c 
 
 the axis. 
 
 11. z^ + cxy ^- W. 
 
 12. Take for the fixed straight lines a;=0, ?/ = 0;aj = CT, 2; = 0; 
 ji/ = &, z = c\ and take the equations (3) of Art. 1 7 as the gene- 
 rating line : the equation becomes 
 
 — ayz -\- hxz = cy (x — a). 
 
 13. The condition required is that 
 
 Ak' + Bfx' + Cv' + 2A'fxv + 2B'v\ + 2C'XfJL 
 
 shall retain an invariable value for all values of A, /x, v consistent 
 with Ik + nifx + nv = 0. See Art. 173. 
 
 14. Eliminate s between the equations (1) of Art. 83. 
 
ANSWERS TO THE EXAMPLES. 2G3 
 
 15. If x\ y'j z be the co-ordinates of tlie vertex, the equation 
 of the cone is 
 
 And by Art. 50 equation (7) it follows that 
 
 CHAPTER YIII. 
 
 1. x^ + y^ + z^ = a^ +h^ -\- c^. Use equation 5 of Art. 101. 
 
 2. A similar and similarly situated ellipsoid whose axes are 
 double those of the first. 
 
 3. Use Art. 101. 
 
 4. {x{x-a)+y(y-l^)+z{z-y)Y=a\x-aY+h\y-py+c\z-yf. 
 
 5. a\x-aY+h''{y-PY-¥c'{z-yy 
 
 ^{a(x-a) + p(y-P) + y{z-y) + hJ. 
 
 6. The conditions that the normal to the ellipsoid at {x, y, z) 
 shall pass through (a, p, y) are 
 
 ^^ {x - a) ^ b^y-P) ^ c^z-y) ^ ^ 
 X y z ^ 
 
 and these combined with 
 
 2 2 2 
 
 X y z ^ 
 
 1- — H = 1 
 
 2 LJ ^^ 2 — i 
 
 a c 
 
 give an equation of the sixth degree in k. All six lines lie on 
 the cone 
 
 X — a y ~ P Z — y 
 
 7. Obtain the condition that the normal at the point {x, y, z) 
 may intersect a given diameter - = - = - . By properly choosing 
 
 Xj /Lt, V this condition can be made identical with 
 
 I TYi n ^ 
 - + — + - = 0. 
 X y z 
 
264 ANSWERS TO THE EXA:\[PLES. 
 
 8. The tangent plane to any such ellipsoid can have its 
 equation written as 
 
 Ix + my + nz = Ja^l^ + h^m^ + c^n~ — k^, 
 
 whence by Art. 77 (6) the result can be obtained. 
 
 9. There will be one straight line in the tangent plane at the 
 extremity of the radius, perpendicular to the radius. 
 
 10. If lx + m7/ + nz=2y be the equation of the cutting plane, 
 the first volume is given by multiplying the area of the section 
 given in Art. 80 by ^p. The co-ordinates of the pole of the sec- 
 tion can be obtained from Art. 106, and the perpendicular on the 
 
 2J2 7 2 2 2 2 2 
 
 plane from this point is found to be — : whence 
 
 o TO 7 «' 2 •"> t> o 
 
 - - - . a'l + o'm + c'W — p 1 • r. 1 . 1 
 
 the ratio or the volumes is r, ^~, and it this be 
 
 constant it easily follows that either volume is constant. 
 
 11. The shadow is the section by the ]Dlane, of the envelop- 
 ing cone whose vertex is the luminous point. 
 
 12. Use Arts. 149, 150. 
 
 13. Take the centre of the ball as origin, a plane parallel 
 to the inclined plane as plane of xy, and a- It, (3 - mt, y - nt as 
 the co-ordinates of the luminous point at any time. 
 
 1 4. yzx + zxy' + xyz — 3a^ ; -— . 
 
 15. x'x~^ + y'y~^ + zz~^* = a^. 
 
 16. (B(3'+Cy'-a) {By'-^Cz'-x)-{Bfty + CyZ-i {x + a)Y = 0. 
 
 X — a y — 1^ _z — y By^ Cz' _ 
 
 • "^T " ~lB^ " Wy ' (1 + 'IJJaf "^ (1 -f 'ICaf ~ ""' 
 
 19. The equations of the normal at (x, y, z) are 
 
 x{x -x) = y{y' - y) = z{z'- z). 
 Use the condition of Art. 31. 
 
 20. Ax + I' + 1=^0. Use equation (3) of Art. 1 02. 
 
 21 . a («' - yz) + 13 (y- - zx) + y {z' - xy) = c' : a hyperboloid 
 of one or two sheets according as a + ^ -f y is positive or negative. 
 
ANSWERS TO THE EXAMPLES. 265 
 
 22. ayz + pzx + yxy = 3a^: a hyperboloid of one or two sheets 
 according as aj8y is negative or positive. 
 
 2 2 
 
 23. ix{x' + 7/ + z') + ^^+^^ = 0. 
 
 24. The equation of any tangent plane to the cone can be put 
 into the form Axx' + Byy + Czz = 0, where Ax"^ + By'' + Cz"' = 0, 
 and if I m _^ n 
 
 we get the required result. 
 
 25. Use (10) of Art. QS putting — , , ^^, X, for l, m, n and 
 
 reducing. Or else use (1) and (3) of Art. 75. 
 
 27. {a — aY + {h — h'Y-\-{G — c')~ = r' + r'' where r, r are the 
 radii and (a, b, c) ; (a, h\ c) the co-ordinates of the centres of the 
 two spheres. 
 
 29. Taking the equations of Example 33 of Chapter ii., if 
 a be the radius of the sphere, the locus required is 
 
 (m' + 1 ) cxy — m {a" — c') z = 0. 
 CHAPTER IX. 
 
 2 
 
 1. (1) x'y-yx + -{z'-z) = 0. 
 
 (2) If we assume z=r sin <j5), the equation of the osculating 
 plane can be written 
 
 2x' cos^ (^ - ?/' sin </) (1 + 2 cos^ <^) -2z +r sin <^ (2 + cos" (^) = 0. 
 
 2. Length of arc = J a' + c' . {6^ — 0^). From the equations 
 of the curve obtain x" + y" sl^ a function of z: let x' + y' =/ (z). 
 This is the equation required. Ex. x' + y' = a^. 
 
 3. We easily get, if a be the radius of the generating cylinder, 
 h' = ia' sin=^ ^V^ + «' cot^ a {6^ - O^f = ia' sin^ ^^-^ + I' cos= a, 
 
 0—6 
 if I be the length. Hence, when Hs a maximum, sin -^ — ^ + ; 
 
 and the maximum lensrth = . But this maximum lenofth 
 
 cos a ° 
 
 = a cosec a {0^ - 9^ and 0^-0^= 2n7r ; 
 
 2mra b b tan a 
 
 sm a cos a Znir 
 
266 ANSWERS TO THE EXAMPLES. 
 
 4. The equations of the curve are 
 
 a9 aO 
 
 c — — — \ 
 x = a cos 0, y = a ain 0, z = ^ (€ ^ + € ^)- 
 
 5. xA-y + z-\. 
 
 g 
 
 6. r = a, <f> = tan (3 log tan -^ + C ; r, 0, <f) being polar co- 
 
 ordinates. 
 
 -i2/ 
 
 7. (1) 9/-x' = c. (2) y + a tan"' ''- = c. 
 
 X 
 
 8. Analytically. Differentiate the equations of the sphere 
 and ellipsoid, and find the ratios —r'~r'-^~' ^^^ equation 
 
 Ct^ Cto CvS 
 
 of the plane can then be found, and then the equation (12) 
 of Art. 68 can be used. 
 
 9. (1) cos~^ (cos c^ sin 6) + cos~^ (sin </> sin 0) -- const, 
 which can be transformed into 
 
 (2) cos^ sin Jl- sin^^sin'6^ + sin<5!) sin^ ^1 — cos^0 sin"^ = const. 
 
 or (3) X Ja^ -y^ + y V""" "" ^^ ^ const. 
 
 10. e=acl>. 
 
 11. By Art. 101, py- , -j-^, -t^ will be proportional to 
 
 \Cl^ Q/b Clb 
 
 whence 
 
 (S-i). (j-i)^ (S-i). 
 
 /i_2\if^-« fl_l\l^I^ /^i_^V^^_o 
 
 \^^ c'v X ds \c^ a" J y els \d' V) z ds 
 
 12. 2 = ^ €^ sin acot s^ wlicrc z is the distance of the point from 
 the vertex, a the semivertical angle of the cone, (3 the fixed angle, 
 and 6 the angle made by the plane through the point on the curve 
 and the axis of the cone, with some fixed i)Uine. The length of curve 
 
 , , , -, c n -^^ I floSinacotfl e,sinacot)3, . 
 
 between any two values oi t; = ^ W — € }. At 
 
 •^ cos^ ^ ^ 
 
 the vertex ^= — oo . 
 
 13. From the method (>f producing the curve we easily see 
 that if 8 be tlie arc measured from the point nearest to the vertex, 
 
ANSWERS TO THE EXAMPLES. 267 
 
 r' = c' + s^ Also if the axis of the cone be the axis of a:, x = r cos a : 
 
 70 o 
 
 ClOC C COS ot 
 
 whence -7-, = 5 — . Also the principal normal to the curve 
 
 as' r"* 
 
 is the normal to the cone at that point (Art. 182). Whence 
 
 p — T = ± sin a, and . '. r^ = ap. 
 
 as' 
 
 U. (1) «^\ (2) ^1±£!. 
 
 2 2 
 
 15. X = cos 0, y= sin 0, z^ cO. 
 
 a a 
 
 16. Take the common tangent to the two curves as axis of x 
 and the plane of the circle as plane of xy. Then if x^, y^, z^ be the 
 co-ordinates of a point on the circle at the end of the arc Ss, 
 and p the radius of the circle 
 
 p p \ p/ zp \^p 
 
 and if x , y^, z^ be the co-ordinates of the point on the curve 
 
 we get 
 
 dx - , d^x ^2 1 ^^^ 
 
 -8. + 1^8.+^^ 
 
 X„=-^Ss+i -jjSs'+ r^-^Ss^+ ... 
 
 and similar values for y^, z^. But it can easily be shewn by 
 Arts. 119, 130 that 
 
 dx ^ dy ^ dz ^ d^x _ d^y 1 d^z „ 
 ds ds ' ds ' ds^ ' ds^ /o ds^ 
 
 whence the square of the distance required becomes 
 fhs^U/dJ'x IV /d'yy /d'zV) 
 yiUVV ^\dsO^[ds^)\' 
 
 And by differentiating the formula (10) of Art. 129, and (2) 
 of Art. 118 the required result may be obtained. 
 
 17. Prove geometrically from the figure in Art. 127. 
 
 18. By Ex. 12 the equations of the curve may be written 
 
 x = A tan aec« cos 0, y=^ A tan ae^^ sin $, z= ^c*'^, 
 
 where c = sin a cot /3 ; whence p can be obtained by (9) in Art. 129. 
 When developed the curve is an equiangular spiral. 
 
268 ANSWERS TO THE EXA]MPLES. 
 
 CHAPTER X. 
 
 1. x^ ^ rf ^ z^ - i^.x + my + nz)' = 1. 
 
 2. ^x' + 2/' + Vc' -z- ^ a. 
 
 aV 5"y^ cV ^ 
 a* /3^ 7^ 
 
 5. cr = c", c being the radius, and tlie plane of xy the fixed 
 plane. 
 
 c - ^" 
 
 0. xyz - 3^3 . 
 
 2 2 7 2 2 2 2 
 
 . —^ i + T]r^—. + -o — o = 0, where r' = a" + ?/- + 2;-. The 
 
 a^-r"^ }f-T^ <?-r 
 
 Wave Surface. See Chapter on Fresnel's Theory of Double 
 
 Refraction. 
 
 8. The surface x" ^\f -v 2z^ = Ik" \ and the curve x- + 7/ = k% 
 
 
 10. xyz = ^ , where ^Jc^ is the given volume. 
 
 2 2 ^2 1 
 
 11. aj- + 2/ + ^ = ^ • 12. ^ + p + - = -p 
 
 13. a;V 2/' + »^ = (c ± 0^ 
 
 .2 „ ,2 
 
 14. ( ^, + |_„ + ^ 1 ) («=^Z^ + h"-m' 4- c^^^ - 1 ) = (^o: + my -^nz-l )\ 
 Xa" 0' c J 
 
 15. By Art. lOG all the lines in question lie in the polar 
 plane of (a, /?, y). If a , /8', y' be tlic co-ordinates of one point 
 in which any chord meets the ellipsoid, the line required will be 
 given by the two equations 
 
 xa. yB zy ^ , xa yfS' Zy - 
 
 . o + 7^ + 4 = 1, and -, +•-/;-+;= 1. 
 
 Tlic condition that this line may be perpendicular to the line 
 joining (a, ^, y) to (a, (3', y) can bo reduced to the form 
 
 (i'-c')<. ^ (c'-«°)|8, («'-6')y ft 
 
ANSWEES TO THE EXAMPLES. 2G9 
 
 Also the equation of a plane through the origin and the line 
 required is 
 
 a:(a-a) ?/(/?-/?-) Z (y - Y) 
 
 a c 
 
 the envelope of which treating a- a, (3 - (3', y - y as parameters 
 gives us the cone required. That the curve is a parabola can be 
 shewn because a plane through the origin parallel to the polar 
 plane of (a, ^, y) can easily be shewn to touch the cone. 
 
 CHAPTER XI. 
 
 1. If x=f^{t), y=f^{t), z=f^{t) be the equations of the 
 curve, we have to find the envelope of 
 
 {^ -/. {t)Y + {!/ -A (tW + {~~ -/. (or = <^, 
 
 where t is the parameter. The envelope is obtained from the 
 intersection of the sphere with the normal plane to the curve 
 at the point t. 
 
 2. The equation can be put in the form 
 
 _ A' + y + a V) _ 
 
 i{x + y + z)\{x^ + f + z') - f^^7j 
 
 = c 
 
 and if the line x = 7/ = z be taken as axis of z this becomes, 
 
 3 /3 
 by Arts. 25 and 28, —'X— z {x^ + y") = c^, which is a surface 
 
 2c^ 
 
 formed by the revolution of the curve zx'^ = S' ,^ round the axis 
 of ^. Or, apply Art. 148. 
 
 3. a'y^ + Qi?%^ = c'x' ; x = a, y~ + z^ = c^ being the equations of 
 the circle. 
 
 4. See Ex. 11, Chap. vii. for choice of axes, 
 
 
 x^ , y^ a^ 
 
 
 (c 4- z)^ ' (c — z)- c' 
 
 5. 
 
 . z z 
 X sm - = y cos - . 
 c -^ c 
 
 6. X cos 6' + 2/ sm 6' = <x, where V = — 
 
 c a 
 
270 ANSWERS TO THE EXAMPLES. 
 
 7. (1) {x^ + y^)(k-na) + 2a{z-a){lx + my) + {k + na){z-a)-, 
 the vertex being at the point (0, 0, a) and the plane of the small 
 circle being Ix + my + nz = k. 
 
 (2) Put 2; = in the above. 
 
 8. Jx^ + y^ + Jc^ -z^ = a, or 
 
 (x' + y' + z' + a'-cy = W {x' + y') (1). 
 
 9. The points at which the tangent plane passes through 
 
 the origin are given by z = ^- s/a^ — c', that is, they lie in two 
 
 ct 
 
 horizontal rings. Take one of these points in the plane of zx. 
 
 The tangent plane at this point has for its equation 
 
 c' 
 
 x = z (2). 
 
 c ^ 
 
 Also the equation (1) can be put into the form 
 
 {2. 2. 2 / 2 2\12 ^22./422 A / 2 2\2 
 
 X + y + z - {a — c)Y = 4c y + 4c a; — 4 (cir — c) z 
 
 — ^c'y'^ + 4 (ex — z Ja^ — c") {ex + ;:; J a' — c'), 
 
 whence at the points of intersection of (2) with (1) 
 
 X +y +z — (c& - c") = ± 2cy. 
 
 Hence (2) cuts (1) in two circles. From the symmetry of the 
 surface the same will be true for all the points. 
 
 10. The fixed plane being the plane of yz, and I, m, n the 
 direction-cosines of AB, the equation of the surface is 
 
 (mz — ny)' + {iix — Iz)' + {ly — mxf = k'x^. 
 
 11. The conditions are given in Art. 92. See Art. 151. 
 
 {a' - kj ^ {b' - ky "^ {c' -ky ' 
 
 - 7 9 7 ciyz + bzx + cxy 
 
 wliere k = aoc . ■= — ' j — . 
 
 ocyz + cazx + aoxy 
 
 1 4. Take y=px and y ^kz-\- q as equations of the gener- 
 atiug line. 
 
 15. (1) A surface of revolution round Oz. (2) A surface 
 such that all sections by planes through Oz are circles. (3) A 
 cone whose vertex is 0. 
 
 16. (1) A surface produced by the revolution of the lem- 
 niscate in the plane of %x round Oz. (2) A surface produced 
 
ANSWERS TO THE EXAMPLES. 271 
 
 by the motion of a circle whose centre is and radius is any 
 radius of the same lemniscate placed in the plane of xy. 
 
 17. \-slb' -c^ - - J a' - 6'> I -2 :, ) + h-y"" ( -, + -, - ,-, ) 
 
 tc a j \c a J ^ \c- a' by 
 
 \c a b / 
 
 18. The equations of any helix can be written 
 
 x = a cos 0, y = a sin 0, z = c9 + y, 
 and by virtue of the given conditions y and c must be expressible 
 
 as functions of a. Hence since a' = ic^ + ?/' and 6 = tan~^ - , and 
 
 -^ X ' 
 
 7 
 
 also = — - , we ffet 
 c c 
 
 tan-^ ^^zF{x' + y') +f{x' + y'). 
 
 The second part easily follows by differentiation. 
 
 19. The reflected light forms a cone of the second order, and 
 the wall on which it falls is parallel to one of its generatino- 
 lines. 
 
 20. If x^, y^, z^; x^, y^, z_^ be the co-ordinates of the 
 points A, B ; being the origin, the condition that AB subtends a 
 right angle at is x^x^ + y^y^ + z^z^ = 0. Also the equations of 
 AB are 
 
 x-x, y-y, z- 
 
 Al 
 
 '^'2 5 
 
 ^,-^2 3/1-2/2 =^l--2' 
 
 and from the equations of the straight lines a?,, y^ can be expressed 
 in terms of z^ and x^ , y^ in terms of z^ . Then eliminating z^ , 
 between these equations we get a relation between x, y, z. 
 
 21. Equation (4) of Art. 148 is evidently the required con- 
 dition. 
 
 X 
 
 22. If - — / (z) be the equation of the surface, the locus 
 required is 
 
 where /, /' are the values of /{z) and /' (z) for the given 
 value of z. 
 
272 ANSWERS TO THE EXAMPLES. 
 
 23. The equations of any such circle are x^ -^ if a- z^ =1ax 
 and y = mx, also a must be expressible as a function of m, = -cf{m) 
 say. The differential equation can be easily deduced. 
 
 25. A right conoid, whose directing curve is formed by 
 folding the curve of sines with its base horizontal round a vertical 
 cylinder. In the case of qi = 2, the equation becomes 
 
 z {x^ + y') = 2cxy ; 
 
 and any plane 
 
 z — mx + ny + p, 
 
 will cut this in a conic section, if 
 
 2? (m^ + n^) + Icmn — 0. 
 
 The projection of the curve on the plane of xy is 
 
 ix^ + 2/^) (wi^ + n') — 2cmn {nx + my) = 0. 
 
 CHAPTER XII. 
 
 1. 6a--12y- = 9, P^O; 4a' +12/5' = 9, y = 0; impossible 
 locus. 
 
 2x' ?>y^ iz' 
 
 2k + T "^ 3^ +~1 "^ 4/j + 1 
 the two values of k are the roots of the quadratic 
 
 If ^^^^^^^ — -+ „7 -V + .7 . T = 9 be either of the surfaces, 
 
 S 29 
 
 3. Let a be the distance of the point along the axis of x, and 
 
 Q O O 
 
 _ + ^- + — = 1 one of the surfaces : the locus required is 
 a' c 
 
 2 2 
 
 X 'ij rJ ^ 
 |_ -^ ^ . = \^ 
 
 a ax — (a^— h) ax — {a' — c^) 
 
 4. At the points of intersection we easily get ax = /3y + a°. 
 Also the direction-cosines of the normal to the tirst surface at any 
 such point are easily proved to be proportional to 
 
 1_ a a^ 2 2z 
 
 a P' {ax -by ^' a.r-b'' 
 while those of the normal to the second are proportional to 
 
 2 1 _ ^ _ jgg" 2z 
 
 a' ^~^'~ {ax-by ax-b'' 
 and these lines are therefore perpendicular to each other since their 
 direction-cosines satisfy the requisite condition. 
 
ANSWERS TO THE EXAMPLES. 273 
 
 5. If the two quadrics be B\f + C^ — x and B if + C^ - x + h^ 
 the coincidence of the foci involves 
 
 _1 _±__j ±^ 1 _; 
 
 4:B 4:B ' 4(7 IC ' 
 
 whence also the focal conies will coincide, since 
 
 B-C B'-C 
 
 'BC ^ ^'W ' 
 
 6. At the points where the two quadrics in (5) cut, we have 
 
 {B-B')i/+{C-C')z' + h^O, 
 or 4:BB'y' + 4.CG'z'+ 1=^0, 
 
 which is the condition that the tangent planes to the two quad- 
 rics at {x, 7/, z) should be at right angles. 
 
 7. Use Article 161 and equations (2) and (3) of Article 83 
 applied to the surface of Article 108. 
 
 CHAPTER XIII. 
 
 Z2 2 2 2 2 2 212 2 
 
 OCX cay a o z ^ 
 
 2Jp — a" 2^9 ~ ^^ l^P ~ ^' 
 
 where p is the perpendicular from the centre on the tangent 
 plane. This can be reduced to 
 
 212 2 
 
 jy^p'^ — {a^ + lf + c^ - r^) 2?p H ^— = 0, where r^ = x^ + y^ + z\ 
 
 For the umbilici the two roots will be equal. This will require 
 one of the quantities x*, y ov zto vanish. 
 
 2. (1) x—y = z — a. 
 
 (2) When xa^ = =*= yW = ^ zc^. 
 
 4. (1) Eliminate m between equations (6) and (7) of Art. 
 171, writing p = h Jl +p^ + q-. 
 
 (2) The coefficients of the several powers of m in the 
 equation (7) of Art. 171 must vanish. 
 
 A. G. 18 
 
274 ANSWERS TO THE EXAMPLES. 
 
 5. The two values oi h in (9) of Art. 169 must be equal and 
 of opposite sign ; 
 
 .-. U'(v+w)+V'{w + u) + W'{u+v)-2u'VW-2v'WU-2iv'UV=0. 
 
 The points of intersection of the surface with the sphere 
 
 111 
 ^ ABC 
 
 6. Take the general equation of a quadric and determine 
 the conditions that it may touch the plane of xij at the origin, 
 and that sections by planes parallel to that plane may be circles. 
 
 IS 
 
 7. Using the equation in the last question the locus required 
 
 ex + hy + h (z — a) — z = 0. 
 
 8. See Ex. 4, Chap. xii. The surface in the question and 
 the two surfaces 
 
 2 2 *> 9 
 
 X y z X y ^ ^ 
 
 can be shewn to cut always at right angles, where /3 and y are 
 any constants. Hence the intersections of these surfaces with the 
 given one are its lines of curvature. 
 
 At the points of intersection of the first with the given 
 surface we have ax = /5?/ + 6' a plane ; and by combining this with 
 tlie given equation, that can be written 
 
 - {ax - c") + o {(^U + ^' - c^) + z'' = ax - c", 
 
 which is the equation of a sphere. Hence the lines of curvature 
 are circles : and the plane of any one of them being ax = jSy + b' 
 always contains the line ax = b^, y = 0. 
 
 9. The result follows from the fact that r has the same 
 value for all tangent lines at the umbilicus. 
 
 10. At the points of contact pr has the same value for the 
 geodesic and the line of cuiwature. 
 
 11. The value of pr is the same for the two geodesies through 
 J* since they each pass through an umbilicus. Hence the value 
 of r is the same. The tangents to these two geodesies are there- 
 
ANSWERS TO THE EXAMPLES. 275 
 
 fore parallel to the equal radii of the indicatrix, and the tangents 
 to the lines of curvature being parallel to the axes bisect the 
 angles between these. 
 
 12. Can be proved from 11 by the method of infinitesimals. 
 
 13. The geodesic circle cuts all geodesies through the um- 
 bilicus at right angles. Hence if d, d' be the semidiameters 
 parallel to the tangent to the geodesic circle and the line through 
 the umbilicus, and p, p be the semi-axes of the central section 
 parallel to the tangent plane at the point 
 
 1 JL _ i ■ 1 - P' ^^' + h' + c'- r") 
 a a " p p' a'o'c 
 
 Ex. 25, Chap. viii. 
 
 27,2 2 212 2 
 
 ab c a c <, ., 
 .*. ., ,.> - + ., ,,., =a^ + o^ + c^ - Q". 
 p'd" p'd 
 
 But jt>^<:r^ = aVas can be ascertained from the known co-ordinates 
 of the umbilici. 
 
 14. At any point in the principal section by the plane of yz 
 the two roots of the equation in (1) can be shewn to be - 
 
 a* 
 
 P 
 
 and — . The former root is the radius of curvature of the prin- 
 
 -?^ . . 
 
 cipal section : the latter gives the distance along the normal of 
 
 the point whose locus is required which can then be worked out 
 
 by plane geometry. 
 
 15. Taking x^ + y^ = a^ as the equation of the cylinder we 
 easily get for the geodesies -^ = 0; therefore -^ = c, whence the 
 
 Cl/S CIS 
 
 curves are helices. 
 
 16. s = ^^ sec^ a — c^ , where a is the semi- vertical angle of 
 the cone, and s the length of the arc from tlie nearest point to the 
 vertex. 
 
 17. If x^ -^rif =f{z) be the equation of the surface it easily 
 follows from (1) of Art. 182 that for all points in any geodesic line 
 
 dy dx 
 
 X -y - y -J- = c. 
 
 ds '^ ds 
 
276 ANSWERS TO THE EXAMPLES. 
 
 And it can easily be proved tliat tlie sine of the angle required 
 
 _ c 
 
 18. If r =/ {x) be the equation of the surface t^ being if + z' 
 the required expressions are 
 
 t \dx] j ^ f (dl 
 
 err V \djc 
 
 19. With the usual . notation for an ellipse the product 
 required is 
 
 CD"' GD^ BC"- _CD^ 
 
 PY ^ ^^~ pb" • pf~ AC '^^ • ^ ^ • 
 
 20. The radii of curvature of the principal sections are 
 
 r' 
 
 7* and : , where r is the focal radius of the point on 
 
 r — p sni (f> 
 
 the ellipse which is in contact, </> the angle between that radius 
 
 and the tangent, and p the radius of curvature of the ellipse 
 
 (Besant on Glissettes, &c.). Hence the sum of the curvatures 
 
 _ 2 psin<^_ 2 r(2a-r)_ 1 
 r r^ T ar'' a ' 
 
 21. By Meunier's Theorem. 
 
 22. Use the quadratic equation in question (1) of this chapter, 
 r Ijeing a constant. 
 
 23. Prove geometrically from the fact that when the surface 
 is developed the geodesies become straight lines. 
 
 24. Differentiate r^ = x^ + if + z^ twice and use the formulaj (1 ) 
 of Art. 182, (10) of Art. 129, 'and (1) of Art. 100. 
 
 25. Use Meunier's Theorem, and (3) of Art. 167. 
 
 
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