Y OF GUIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OO I ur uALIfORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY '*&^: CJS c:^^S y^r.V --wVJB^ ^^ >v OF CUiFOR ffi^ OF CUiruHMI O ^^ AN 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 = 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, 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 — y^ sin <^, y = x^ sin . 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 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 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 — 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' + 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^ = ; 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 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' 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 = (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 sec 6, z = c tan 6 ; and find the equations of the generating lines through that 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, 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 = {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'- (0 ^ _y'^:^irj) _ z-x(t) ^(t + T)-(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 + 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 ,(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,{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 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

^ (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 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 = {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, ^, (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(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, ) = 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 + 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 — 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 + 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 + 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 = tan (3 log tan -^ + C ; r, 0, 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^ 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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