AN ELEMENTARY TEEATISE ON SOLID GEOMETKY. By the same Author. Price 2s. A CHAPTER ON FRESNEL'S THEORY OF DOUBLE REFRACTION. Second Edition. Price 3s. 6d. AN ELEMENTARY TREATISE ON GEOMETRICAL OPTICS. AN ELEMENTARY TREATISE ON SOLID GEOMETRY BY W. STEADMAN ALDIS, M.A. TRINITY COLLEGE, CAMBRIDGE, PRINCIPAL OP THE UNIVERSITY OP DURHAM COLLEGE OP PHYSICAL SCIENCE AT NEWCASTLE-UPON-TYNE, AND PROFESSOR OF MATHEMATICS IN THE SAME. THIRD EDITION, REVISED. B R A n OF THt UNIVERSITY ' CAMBRIDGE. DEIGHTON, BELL, AND CO. LONDON : BELL AND SONS. 1880 '(X U XV Camfirrtrge: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. PREFACE TO THE FIRST EDITION. The present work is intended as an introductory text-boo for the use of Students reading for the Mathematics Tripos. Many of the higher applications of the subje< are therefore either omitted entirely or treated very brief!; At the same time the Author believes that the book ii eludes as much as the great majority of Cambridge Stud en have time to master thoroughly, while those who ai desirous of making farther acquaintance with the subje< will perhaps find a work like the present not unsuitable i an introduction to the more complete treatises of Salmc and others. The Author begs to thank those of his friends who ha^ kindly assisted him by revising the manuscript and proo sheets, and will feel obliged to any one who will offer corre tions or improvements. PREFACE TO THE FIRST EDITION. Examples, selected chiefly from recent College and Uni- srsity Examination Papers, will be found at the end of each hapter. Cambridge, August, 1865. SECOND EDITION. The present Edition has been revised and re-arranged id somewhat enlarged. Newcastle-on-Tyne, September, 1873. THIRD EDITION. The Third Edition has been revised and farther olarged, chiefly by the addition of hints for the solution of le Examples. Nkwcastle-on-Tyne, September, 1879. I CONTENTS. CHAPTER I. VA Introductory Theorems. Co-ordinates. Direction-cosines. Projec- tions. Polar Co-ordinates CHAPTER II. The Straight Line and Plane CHAPTER III. The Sphere. The Cone and Cylinder. The Ellipsoid. The Hyper- boloids. The Paraboloids. Asymptotic Surfaces CHAPTER IV. Transformation of Co-ordinates CHAPTER V. On Generating hues and sections of Quadrics CHAPTER VI. Diametral Planes. Conjugate Diameters. Principal Planes CHAPTER VII. On the Surfaces represented by the General Equation of the Second Degree ........... Vlll CONTENTS. CHAPTER VIII. PAGK On Tangent Lines and Planes. The Normal. Enveloping Cones and Cylinders 112 CHAPTER IX. On Curves in Space. The Tangent. Normal and Osculating Planes. Line of Greatest Slope. Curvature of Curves . . . 132 CHAPTER X. On Envelopes, with one and two parameters. Reciprocal Polars . 158 CHAPTER XL On Functional and Differential Equations of Families of Surfaces. Conical, Cylindrical, and Conoidal Surfaces. Developable Sur- faces. Surfaces of Revolution 167 CHAPTER XII. On Foci and Confocal Quadrics . 184 CHAPTER XIII. On Curvature of Surfaces. Meunier's Theorem. Indicatrix. Prin- cipal Sections. Lines of Curvature. Umbilici. Geodesic Lines 194 Answers to the Examples 224 \ brTJ> OF THE UNIVERSITY _ OF ID GEOMET 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 drawn 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, which 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 K 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 INTRODUCTORY THEOREMS. Hence PIl is equal to OM, and similarly PL to OB, PK to ON. If therefore we measure off from Ox, Oy, Oz, respec- tively, lengths OM, ON, OR equal to the given distances of P from the co-ordinate planes, and through M, N, R draw H * R ) K /' / / / M Of •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 x 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 opposite. The positive directions for the three axes are usually taken to be those represented in the figure by Ox, Oy, Oz, and the negative directions to be Ox', Oy, Oz. It will be seen that the whole of space is divided by the co-ordinate planes into eight compartments, and the signs of INTRODUCTORY THEOREMS. 3 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, /3, 7 are represented by (a, A, 7), (—a, A> 7), (a, -A 7), (a, A -y),( (-«, "A 7), (— a, — A — 7), according as the point lies in the compart- ment Oxyz, Oxyz } Oxy'z, Oxyz, Oxy'z', Oxyz , 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 o, 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 2 =OL 2 + PL\ 1—% INTRODUCTORY THEOREMS. z R K / H , P / / T\ M .2f /N But OL 2 =OM* + MU 0P* = 0M 2 + ML 2 + PL 2 = x 2 + y 2 + .(1). 5. Let a, /3, 7 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 J Similarly, ' • > 9 ^ ON = OP cos PON = OP cos 0; or y = rcos/3 ['•**'* Oft = OPcos POP = OP cos 7; or z = r cos 7 J x, y, z being the co-ordinates of P, and OP being denoted byr. Squaring and adding, we get OM 2 + ON 2 + OP 2 = OP 2 (cos 2 a + cos 2 /3 + cos 2 7), or taking account of (1), 1 =cos 2 a + cos 2 /3+cos 2 7 (3). The letters I, m> n are frequently used to denote cos of, cos ft cos 7, which are called the direction-cosines of the line OP. It is usual to denote by a, ft, 7 the angles which OP makes with the positive directions of the axes, in which case the formulae (2) hold for all positions of the point P. INTRODUCTORY THEOREMS. 6. To find the distance between two points whose co-ordi- nates are given. Let P and Q be the two points; w 19 y xt z x \ x 2 , y 2 , 2 2 their co-ordinates. Join PQ, and through P and Q draw planes L K / / Q P / H M i / X ">> 1 E / /f parallel to each of the co-ordinate planes, thus forming a par- allelepiped whose edges are parallel to the co-ordinate axes, and are equal in length to x 2 — x lf y 2 — y t , z 2 — z x , respectively. As in Art. 4, we obtain PQ> = PH* + HN 2 + NQ* = K-^) 2 + (y 2 -^) 2 + fe-^) 2 (*>.' We have also formulae similar to those of equation (2), or, ft, 7 being the angles between PQ and the lines drawn * through P parallel to the axes, viz. PH— w^ — % l = PQ cos a = Ir \ PM= Vi-y t = PQ cos ft = mr> (5), PL — z 2 — z x — PQ cos y = nr' where r represents the length of PQ, and I, m, n are the direction-cosines of PQ. 6 INTRODUCTORY THEOREMS. 7. To find the co-ordinates of a point which divides the straight line joining two 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 x to n 2 . Let a %i y v z be the co-ordinates of P, a? 2 , o£R. 2/ 2 , z 2 t^ose of Q, x, y\ z those Draw PM, RH, QK parallel to the axis of z to meet the plane of xy in M, H, 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 MHK to meet RH in E and QK in F. Then PM=z x , RH=z' y QK=z Also RE PR n x QF~ PQ n t + fij or z -z x _ n x z 2 — z x n x + n 2 ' whence 2'K + w 2 ) = w 1 2; 2 + 7? 2 1 ; . „'_¥ 2 +¥. n x +n 2 Similarly it may be shewn that x r = *W + *W , = n x y 2 + n 2 y x Y - INTRODUCTORY THEOREMS. If R be the middle point of PQ, n x =n 2 , and we have / x + x x = 1 CT 2 8. To ^M the angle between two 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 0P } OQ be the two lines; I, m, n the direction-cosines of OP\ l', m, n those of OQ. Let x l9 y tt z x , be the co-ordi- nates of P any point in OP; x 2 , y 2 , z 2 those of Q any point in OQ. Then by Art. (6) - < + yt + < + < + y? + <~ 2 0% + m* + *i*$ Bat by Art. (4) x? + y? + z?=OP\ <+y:+*:=0Q\ 8 INTRODUCTORY THEOREMS. And by Art. (5) x x = OP.l, y x = OP.m, z 1 = OP.n, x, = OQ.l\ y^OQ.m', z 2 =0Q.ri; and .'. x x x 2 + y x y 2 + z x z 2 = OP . OQ (IV + mm + nri). Hence PQ> m OP' + OQ 2 -2 OP. OQ (IV + mm + nn). But by Trigonometry we have from the triangle OPQ PQ* = OP 2 + OQ 2 -20P.OQ. cos POQ. Comparing these two expressions for PQ 2 , we get cos PO Q = W + mm' + nn (6). The formula (1), (3), (4) and (G) are of very frequent use, and should be carefully remembered by the student. From (6) we can deduce sin 2 POQ = 1 - (IV + mm + nn)* m [f + w 2 + w 2 ) (r + m ' 2 + w /2 ) - (Jf + mm' + nn')* = (win' - mnf + (nV - n'l) 2 + (Im - I'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 i?and F, EF is the projection of PQ on the plane of xy, and is equal and parallel to PN. Also Pl\ r = PQ cos QPK But QPN is equal to the angle which PQ makes with the plane of xy. Hence we derive the theorem : The projection of a straight line of limited length on a given plane is equal to the length of the line multiplied by the cosine of the angle between the line and plane. 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 projec- tion of PQ on the fixed line, and the following theorem holds: INTRODUCTORY THEOREMS. 9 The projection of a straight line of limited length on a second straight line, is equal to the length of the first line mul- tiplied by 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 which do meet. This theorem is proved as follows : Let PQ be the line of limited length, and AB the line on which 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 RB Q, PR 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 PR Q 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 an}' length, from R a straight line R8, and join SQ; and from P, R, #and Q draw perpendiculars PA, R C, SB, QB on AB ; A C, CD and DB will be the projections of PR, R8 and SQ on AB ; and as long as A, G, I), B fall in the order represented in the figure, the arithmetic sum of these projections is equal to 10 INTRODUCTORY THEOREMS. 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 G, 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 we pass from one point to a second, is equal to the projection on the same line, of the straight line joining the two points." This statement may be illustrated thus. Suppose a point to move from P to Q along PR, RS, SQ, and from each INTRODUCTORY THEOREMS. 11 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 is 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. , If, in the figure of that Article,. QN~ be drawn parallel to the axis of z to meet the plane of xy in N, and NM drawn 12 INTRODUCTORY THEOREMS. 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, MN and NQ on OP, that is, if 6 be the angle POQ, and I, m, n; V , m\ ri be the direction-cosines of OP and Q respectively, OQcoa0=OM.l + MIT.m + NQ.n = 0Q.r.l+0Q.m'.m + 0Q.n'.n; .*. cos 6 = IX + mm + nri. 13. To find the distance of a point from the origin when 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, K, R ; and join OP. The ratios of OM, ON and OR to OP 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, m, n. We then get formulae corresponding to those of Art. (5), x = LOP, y = m.OP, z = n.OP. INTRODUCTORY THEOREMS. 13 Again, let A,, ft, v be the angles between [Oy, Oz), ( Oz, Ox), {Ox, Oy). Then we have, if PL be the edge of the paral- lelepiped through P parallel to Oz, OL 2 = OM 2 + ML* - 2 OM . ML cos OML = x 2 + y 2 + 2#y cos v. And OP 2 =0L* + PL 2 -20L.PL cos OXP. But the projection of OL on 072 is equal to the sum of the projections of OM and ML on OR, or by Art. 9, OL cos BOL =0M cos fju + ML cos\=- OLcos OLP; and therefore OP 2 = x 2 + y 2 + z 2 + 2yz cos \ + 22# cos //, + 2#y cos i\ Combining this with the formulas x = I . OP, y = m . OP, z = n. OP, we get 1 = I 2 + m 2 + n 2 + 2mn cos \ + 2nl cos p + 2lm cos v. . .(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 x ,z x ', x 2 , y % , z 2 is te - x ^ + (#i - y*Y + (*i - ^) 2 + 2 (2/1 - # 2 ) (*i - *J cos ^ + * (V" * a ) fo - sj cos /a + 2 (^ - a? a ) (ff % - y 2 ) cos 1/: And as in (8) that the cosine of the angle between two lines whose direction-ratios are I, m, n; V, m, ri is IV + mm + nri + (mri + rn'n) cos \ + (nl' + ril) cos //, + (lm f + Z'???) cos v. . .(2). 14. The volume of the parallelepiped of which OP is the diagonal is evidently equal to the product of the. area of the parallelogram OMLN into the perpendicular from J? on the plane of xy. If 6 be the angle between OR and a line per- pendicular to the plane of xy, this volume would equal OM. ON sin vx OR cos 6 = xyz . sin v . cos 6. But if l' , rri, n be the direction-ratios of the line through perpendicular to the plane of xy, since it is perpendicular 14 INTRODUCTORY THEOREMS. to Ox and Oy whose direction-ratios are (1, 0, 0), (0, 1, 0) respectively, we have, by formula (2) of the last Article, X + m cos v + ri cos /j, = (1), X cos v + m' +n cosX = (2). And since it makes an angle 6 with Oz whose direction-ratios are (0, 0, 1) we have ri + X cos fi + rri cos X — cos 6 (3). From these, since by formula (1) of the last Article X (V + rri cos v + ri cos /x) -\- rri (rri + ri cos X + X cos 7') + n (ri + X cos fi + m cos X) = X 2 + m* + ri 2 + 2m ri cos X -1- 2riX cos /^ + 2ZW cos v = 1, we have »' cos = 1 (4). And from (1) and (2) we have X _ m _ ri COS /Jb — COS X COS V COS X — COS fJb cos j/ cos 2 v — 1 = cosfl COS 2 X+ COS 2 /L6 + cos 2 z/ — 2 COS X COS /X COS V— 1 ^ ^ '* whence we get cos 2 6 sin 2 v=l— cos 2 X — cos 2 fi — cos 2 v + 2 cos X cos jjl cos //. And the volume of the parallelepiped becomes xyz Vl — cos 2 X — cos 2 /j, — cos 2 1/+2 cos X . cos /j, . cos za The volume of the tetrahedron cut off from the co-ordi- nate axes by a plane through It, 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 tt RODUCTORY THEOREMS. 15 plane of zx. These are called the polar co-ordinates of P and are usually denoted by the letters r, 0, . 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 PN parallel to Oz to meet the plane of xy in N, and NM parallel to Oy to meet Ox in M. Join ON. Then x = OM = ON cos — OP sin cos cj> = r sin cos , y — MN= ON sin = OP s'm sin cj> = r sin sin <£, z = PN=OP cos0 = r cos 0, from which we can obtain the equivalent system r 2 = a? 2 + y 2 + 2 2 , tan# = vV + # 2 tand)=^- : -Sly r a? which give r, 0, (j> in terms of x, y, z. ( 16 )r> u* 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, C are three points on the axes of co, y, z respectively ; if OA = a, OB = b, OG — c, find the co-ordi- nates of the middle points of AB, BG and GA respectively. 7. In the last question find the co-ordinates of the centre of gravity of the triangle ABG and the distances of this point from A, B, C respectively. 8. Shew that if D, E be the middle points of BG, GA in the last question, DE=^BC. t 9. Find the distance between two points in terms of their polar co-ordinates. 10. The co-ordinates of a point are (V3, 1, 2 V.3) ; find its polar co-ordinates. 7T 7T 11. The polar co-ordinates of a point are (4, -^ , -J; find its rectangular co-ordinates. CHAPTER II. THE STRAIGHT LINE 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 *>,**) -0 (1). Solving with respect to z we obtain 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, b we get one or more values of z, that is, the straight line drawn through the point (a, b) 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 1 (x,y,z) = 0, F 2 {x,y,z)=0, considered as simultaneous will be satisfied by the co-ordi- nates of all the points of intersection of the two surfaces F 1 (x,y,z) = 0, F 2 (x,y,z) = 0, 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 A. g. 2 18 THE STRAIGHT LINE AND PLANE. 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 without 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 I, m, n be the direction-cosines of the straight line, a, /3, 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 x — a = lr, y — ft = mr, z — ( i). I m n K ' These are the symmetrical equations of a straight line. If A, B, G be any quantities which are proportional to I, m, n, we can replace these equations by %-z_y-0_z-y A B G W> 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, n } we have I m A~~B n == ~G == V? + m* + n* _ 1 *JA* + B»+G* V^ 2 + B i +C*' The equations (2) can be also written thus : y = "|f+g» B A «)• z = aV G A ')• ()] r writing B A m, /3 - B -A a= l>> G A :«, 7 G THE STRAIGHT LINE AND PLANE. 19 y= mx +p\ (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 z 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, /3, 7 ; a, /3', 7' be the co-ordinates of the two given points. By the last article the equations of any straight line through (a, /3, 7) can be written in the form x — a y — (S z— 7 I m n (!)• But if the line also pass through the point (a, ft', 7') we must have _^_ I m n Dividing each member of (1) by the corresponding mem- ber of (2), we get as the equations required x— a _y — /3 _ z— 7 a — a # ' — fi 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 I, 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 parallel to Oy to meet Ox in N. Then the projec- tion of OP on OD is the sum of the projections of 0N y NM 2—2 20 THE STRAIGHT LINE AND PLANE. and MP on OD. But these are Ix, my, nz, respectively, and the projection of OP on OD is p. Hence Ix -J- 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 n = 0. Hence the equation in that case be- comes Ix+my—p (2), and does not contain the variable z. If the plane is perpendicular to two of the co-ordinate ,planes, as those of xy and zx, 1 = 1, ra = 0, n = 0, and the equation becomes «=P (3). These results are geometrically evident. 20. To find the equ atio n of the plane in terms of its in- tercepts on the axes. m^ This can be deducedirom the equation (1) in the last article, but may also be obtained independently thus. Let the plane cut the axes in A, B, G; and let any plane parallel to that of yz cut the co-ordinate planes of zx, xy in THE STRAIGHT LINE AND PLANE. 21 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 BO, respectively. Draw PM parallel to RN to meet QN in M. Let ON=x, NM = y, MP=z, 0A = a, OB=b, OC = c. Then by similar triangles Also Hence PM MQ NM RN~NQ~ NQ' RN AN NQ C0~ A0~ BO' PM RN AN NM RN X CO~AO NQ K NQ BO' PM MN AN " C0 + B0~ A0~~ ON A0 ; or a o c (4). 21. All these forms of the equation of the plane are in- cluded in the form Ax + By+ Gz^B (5). Conversely we can shew that any equation of the form (5) represents a plane. 22 THE STRAIGHT LINE AND PLANE. For let a, /3, y ; a', /3', 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 g-g y-ft «-7 / ft v a' -a /3'-/3 7-7 w ' But since (a, ft, 7), (a', /3', 7') lie on (5) we have Aa+Bj3 + Gy=J), AJ + Bp+Gy' = D. Subtracting, A (a - a') + B (j3 - J3') + C(y-y')= 0. And therefore by (6) A {x -a) + B (y - /3) +€ (z-y) = 0, where x y y f z are the co-ordinates of any point in the line (6), or Ax + By^Cz=Aa+Bl3+Cy = D. Hence x, 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 or x = -j ; or if this distance be called a, -j = a. Similarly -^ = b, -jt = c ; and substituting for A, B, G in (5) we get THE S ;TRi JG BT lt: *E AND PLANE. Da a ' + By b + Dz c -A or X a + y b + z c = 1, 23 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 -f my + nz=p, where I, m, n are the direction-cosines, and p the length, of the perpendicular froju-ih^--o*igin on the plane. But \Ax + Bi l±JC*^£> — 4 represents a plane. Hence if these represent the same plane, we have I _ m _ n _ p A~~B~U ~~D' Also P+m 2 + n 2 = l; ,i A ' l! m = n = J A' + !?+&' B J'A* + B*+G* K G JA' + JP+G 1 and p = ft Ja*+b* + c 2 ' Thus the direction-cosines of the perpendicular from the origin on the plane Ax + By + Gz = D are proportional to A, B, C, and the length of the perpendi- cular is . . sjA* + B l + G* 24. The angle between any two planes whose equa- tions are ' Ax + By + Gz = D, A'x + B f y + C'z = D' } 1 24 THE STRAIGHT LINE AND PLANE. 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 C s/A' + B'+C 2 ' Ja< + B 2 +U 2 ' JA 2 + B 2 +C 2 ' A' B' C JA' 2 + B' 2 +C 2 ' JA'* + B'* + C r *' JA' 2 + B'* + C ,2> and the cosine of the angle between the planes is therefore equal to AA' + BB'+CC J A 2 + B 2 +C 2 JA" + B' 2 +C' 2 ' The condition that the two planes should be at right angles is therefore AA' + BF + CC = 0. The conditions that they should be parallel may be obtained by equating ihe- cosine of the angle between them to unity. It will be found that this leads to the con- ditions A_B C_ A' B'~ G" These may be also obtained independently from the con- sideration that the direcik>n-cosines of the perpendicular on the one plane are proportional to A, B, (7, and those of the perpendicular on the other to A', B\ & ; and if the planes be parallel, and consequently the perpendiculars from the origin on them coincident, we must have A, B, C pro- portional to A', B\ C' y or A B C A'~ B'~G" 25. The equation of a plane through a point (a, (3, z) is on the same side of the plane as the origin. If the ex- pression be denoted by a, the length of the perpendicular from any point on the plane a=0 is + a or — a, according as the point and the origin are on the same or opposite sides of the plane. 26. If we take four planes forming a tetrahedron, whose equations are a = 0, = 0, 7 = 0, 3 = 0, 26 THE STRAIGHT LINE AND PLANE. all expressed in the form Ix + my + nz —p = 0, any other plane may be represented by the equation la. + m/3 + ny + q& = 0. For this represents some plane, being of the first degree in x f y, z, and since it contains three arbitrary constants, namely, the ratios of three of the quantities I, m, n, q to the fourth, it may be made to satisfy three conditions, and may therefore be made to represent any plane. This method of representing planes may be developed in a similar manner to that used for straight lines in Plane Co- ordinate Geometry (Todhunter's Conic Sections, Chap. IV.). Thus the equations of the two planes bisecting the angles between the planes a = 0, /3 = 0, will be a-/3 = Oanda + /3 = 0, the former bisecting that angle within which the origin lies, and the latter the supplementary angle. Any equation which is not homogeneous in a, /3, y, S, can be rendered so by means of the relation where V is the volume of the tetrahedron, and A, B, C, D the areas of its faces. This equation merely states that the algebraic sum of the four tetrahedra whose vertices are at the point (a, ft, 7, B) is equal to the fundamental tetrahedron. 27. If a straight line x-a. y-ft s-7 m a~ b a [) is parallel or perpendicular to a plane A'x + B'y+Cz = D (2), it is perpendicular or parallel respectively to the perpen- dicular on that plane, whose direction-cosines are proportional to A, F, C. The condition that (1) may be parallel to (2) is therefore AA' + BF + CC = 0, Mf * THE STRAIGHT LINE AND PLANE. 27 and the conditions that (1) may be perpendicular to (2) are A_B_G A'~ B'~~C" 28. It is often requisite to know the length of the per- pendicular on a given straight line from a given point. Let the equations of the straight line be x — ol y — $ z — 7 a b a and let a, j3\ y be the co-ordinates of the given point. The equation of .any plane through (a', $', y) is (1), If this plane be perpendicular to (1) we have \ fJb v A=B~~C' and its equation becomes A {x-a)+B (y-0) + C{z - y) = (3). The point where this plane meets the line (1) is evidently the foot of the perpendicular from (a, y6', 7') on (1). Let then P be the point (a, £, 7), F the point (a , /3', 7), and Q the foot of the perpendicular from P' on the line (1) therefore PQ is the perpendicular from P on the plane (3), 28 THE STRAIGHT LINE AND PLANE. and we have PQ - ^±$ ±* W ~ ® + °M^) , *JA* + £* + 0* and PQ 2 = PF*-PQ 2 by the right-angled triangle P'QP; .-. P'Q 2 . («- aT + 08 - py + (7 - 7 7 {A(a'-a)+B(ff-0)+C(y'-y)Y A' + B'+G* 29. To ^mo! the conditions that a straight line may lie wholly in a given plane. be the equations of the line, A'x + By + Ce = D (2) the equation of the plane. Put each of the fractions in (1) equal to k. Therefore x = ol + Ah, s y ~fi+.Bk, z = 7 + Ck, and if the line (1) lies wholly in (2), these values of x, y, z must satisfy (2) whatever be the value of k. Hence the equation A a + B'/3 +C'y-D+ (A A + BE + CC) k = 0, must be satisfied independently of fe? This gives us the two conditions A'a + B'/3 + C'y-J)~(}, AA' + BB'+ CC' = 0; The first of these equations denotes that the point (a, /9, 7) lies in the plane (2), and the second that the angle between the line (1) and the perpendicular on the plane (2) is a right angle. These are evidently necessary and sufficient conditions. 30. To find the shortest distance between two straight •lines whose equations are given. We must first prove that the shortest line between two given straight lines is perpendicular to each of them. THE STRAIGHT LINE AND PLANE. 29 Let BC, AD be the two straight lines, and AB a line perpendicular to each of them. Then AB is clearly shorter than the line joining A with any other point of BC, and also than the line joining B with any other point of AD. Let P be any point in AD, and Q any point in BC. Then PA and QB are both perpendicular to AB, and therefore AB is the projection of PQ on AB, and is equal to the length of PQ multiplied by the cosine of the angle bet we an them, and is therefore less than PQ, since the cosine of a)^ angle is less than unity. U * U t~** o ° +*> +**oM^& *f 7/ Let ABC x—ol _ y—fi _ z — y •(2), A' B' be the equations of the two straight lines. Let the equation of any plane through (1) be P(x-a)+Q(y-l3) + B(z-ry)=0 (3). Then we have, since (3) contains (1), PA + QB + BC=0 (4). And if we take the plane through (1) to be also parallel to (2), we have PA' + QB' + BC = (5). From (4) and (5) we have P_ Q _ B BO'-EC~ CA'-CA AE-A'B' 30 THE STRAIGHT LINE AND PLANE. The equation of a plane through (1) parallel to (2) is therefore (BC'-BC)(x-CL)+(CA'-GA)(y-/3)+(AB'-A'B){z-y)=0 (6). * Similarly the equation of a plane through (2) parallel to (BC-B'G)(x-a.')+(CA'-GA)(y-/3')+(AB'-A'B) {z-y)=0 (7). The length of the perpendicular from the origin on (6) is (BC'-B'C )ol+( GA' - C A) + (AB - A'B) y ^(BCT^BGf + (GA' - O Af + (AB - A'Bf ' ' and the length of the perpendicular on (7) is (B C -BG)a!+( CA - GA) ff + (AB - A'B) y' *J(BG - BGf + (GA' - G'Af +^AB ~ ^W The difference of these, or (BG'-B'C) (a-q') + (GA'~ GA) Q3-ff) + (AB-A'B) ( 7 -V) J(BG'-BGf + (C A' -CAf + (AB-A'B)* is^ clearly the perpendicular distance between the two given lines. The equations of the line AB can be obtained by finding the equations of two planes, one of which contains the straight line BG and is perpendicular to the plane (6), and the other contains the line AD and is perpendicular to the same plane. Each of these planes evidently contains the straight line AB, and their equations considered as simultaneous determine the line. The requisite conditions for the two planes will be found in Articles 24 and 29. 31. To find the condition that two straight lines whose equations are given may intersect. Let the equations of the straight lines be x-a _y-& _z-y A B G {i) > x -a! _y-/3' _z-y A' " B " G" " {Z) ' THE STRAIGHT LINE AND PLANE. 31 Then if they intersect, a plane can be made to pass through both of them. Let this plane be Px + Qy + Rz = D. Since this contains the line (1) we have, by Art. 29, Poi + Q/3 + Ry = D (3), PA + QB + RC=0 (4). And since it contains the line (2) we have Pa'+Qff + &/=D (5), PA' + QI? + BC' = (6). From (3) and (5) we have {P(a-ot) + Q(P-l3)+R{y-y') = (7). And eliminating P, Q, R from (4), (6) and (7) we get with the usual notation of determinants, ABC A' B a = 0, a-a' P-p 7-7 or (a-a'liBC'-B'Cj + i/S-^iCA'-C'AWv-y'XAB'-A'B^Q. A result which might have been obtained from the last article by the consideration that if two straight lines intersect their shortest distance vanishes. If the two straight lines be given by the equations Ax + By+Cz = D\ ( A'x + B f y+C'z=D') W ' Px + Qy + Rz = S\ (9), Fx + Q'y + R'z the condition of intersection is obtained from the considera- tion that these four equations must be able to be satisfied by the same values of x, y, z. The condition for this is A B G D A' B' a k P Q R 8 F Q E S' = 0. ( 32 ) EXAMPLES. CHAPTER II. ^. Find the equations! of a straight line passing through the point (1, 2, 3) and whose direction-cosines are proportional to V3, 1 and 2 V3. ^2. Find the equations of the straight line joining the two points whose co-ordinates are (1, 2, 3) and (3, 2, 1) re- spectively. l/ S. Find the equations of the sides of the triangle formed by joining the points (1, 2, 3), (3, 2, 1), (2, 3, 1). Deduce the values of the angles of the triangle. 4. Find the equation of the plane which passes through the three points in the last question, and the length of the perpendicular on it from the origin. ^o. Find the equations of a straight line which passes through the point (1, 2, 3), and is perpendicular to the plane x + 2y + Sz = 6. 6. Find the equations of a straight line which passes through the point (1, 2, 3), and is perpendicular to the two straight lines in questions (1) and (2). 7. Find the equation of a plane passing through two given points and perpendicular to a given plane. 8. Find the equations of a straight line passing through the point (1, 2, 3) and parallel to the plane in question (4) and to the plane of xy. 9. Find the equation of a plane passing through the point (2, 3, 4) and the straight line in question (1). 10. Find the equations of a straight line drawn from the origin of co-ordinates at right angles to one given straight line, and making a given angle with another. If the given straight lines be at right angles to each other and the given 77° angle be - , shew that there are two solutions, and that the two straight lines so found are at right angles to each other. EXAMPLES. CHAPTER II. 33 11. Find the equation of a plane which passes through a given point, and is perpendicular to each of two given planes. 12. Shew that the equation of a plane in oblique co- ordinates can be put in the form x cos a + y cos /3 + z cos 7 = p, where p is the length of the perpendicular on the plane from the origin, and a, fi y 7 the angles which it makes with the axes. 13. Shew that if a, /3, 7 be the angles between any straight line and the axes of co-ordinates, I, m, n the direc- tion-ratios of the line, and X, fi, v have the meanings given in Art. 13, cos a = I + m cos v + n cos /j,, cos /3 = m + n cos X + 1 cos v> cos 7 = n + 1 cos fi + m cos X. 14. Deduce the conditions that in oblique co-ordinates the straight line x _ y _z I m n may be perpendicular to the plane Ax + By + Cz=D. y 15. Shew that the locus of a point which moves so as always to be equidistant from two given points, is a plane which bisects at right angles the straight line joining the two points. 16. What loci are represented by each of the equations /»=O;/(r) = O;/(0) = O;/(4>)=O; where r, 6, (j> are the usual polar co-ordinates ? 17. Interpret the equations: ft e =°-> w {J:?; < 8 > {?:«: A. G. 3 34 EXAMPLES. CHAPTER II. 18. Find the polar equation of a plane. 19. Find the angle between the two lines given by x+y+z=3 \ b : *4Hrfa-i«J (1) ' and *=^* W- 20. Three planes are at perpendicular distances p t , p 2 , p H 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 p x cos A = p 2 cos B —p s cos C, where A, B, (7. are the angles between the first three planes. 21. Shew that through two given points (a, b, c), (a, b', c T ), two planes may be drawn cutting off from the axes intercepts whose sum is zero; and these two planes will be at right angles to each other if a — a b — b 22. Find the cosine of the angle between the two straight lines represented by + y + z = 0, 3 t iXi y — z z — x x — 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; b — c, b — d, b — a; c — d, c — a, c — 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. EXAMPLES. CHAPTER II. 35 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-a. 2/-/3 z-y _ A x-a! _ y-& _z-y j — j — ctLLU. jft -, — — 7 — , intersects the latter in the point whose co-ordinates are a + V cosec 2 (u +u cos 0), and two similar expressions where 6 is the angle between the lines and w = I (a - a) + m (£' - £) + n (y -y), u' = t (a - a) + m (J3 - /3') + n' {y- y). 27 y 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 a?- a'* COS ft) = 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 a b g 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 x, 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- 3—2 36 EXAMPLES. CHAPTER II. 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 origin to meet the straight line x—a_y—b_ z — c I m n at right angles. Shew that its equations are x y z a — It b — mt G — nt y , al 4- bm + en where t = -n~ i — ri — *• 33. Shew that by a proper choice of axes the equations of any two straight lines can be put in the forms — c, y = — mx. CHAPTER III. ON CERTAIN SURFACES OF THE SECOND ORDER. 32. 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. 33. The Sphere. A sphere is a surface every point of which is at a constant distance from a fixed point called the centre. The constant distance is called the radius. Let a, b, c be the co-ordinates of the centre, r the radius, #, 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-a,y + (y-by+(z-cy. But this distance must equal the radius r. Hence for all points on the surface J(x-ay+(y-by+(z-cy = r, or (a>-a)*+(y-by+{z-cy = i* (1), which is the equation required. Conversely any equation of the form x 2 + f + z 2 + Ax + By + Cz + D = represents a sphere. For it can be put into the form 38 ON CERTAIN SURFACES and, comparing this with (1), we see that it represents a sphere whose centre is at a point ( — ~ , — •=; , — — j and whose radius is j A* + B* + C* B 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 2 + Cf + JEx = (1), to which form the equation of any conic section can be re- duced ; and let a, /3, y be the co-ordinates of the vertex. The equations of any straight line through the vertex are ~t~~ ^m~-~^r W ' when this meets the plane of xy we have z = 0, and therefore I m x = a y, V = p 7. n" u ■ n ' 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 n 2 , A (na - ly) 2 +C(nj3- my) 2 + En (na - ly) = 0. OF THE SECOND ORDER. 39 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 I m n x — a. y — ft z — y' Consequently, if (%> y, z) be any point in any straight line joining (a y ft 7) with some point of the curve (1), we must have + E{z-y){ and therefore by (5), if (x, y, z) be any point in (5), IV + Qy* + Bz 2 + Fyz + Q'zx + B'xy = 0. 40 ON CERTAIN SURFACES Hence every point on the straight line joining the origin with {x v y x ,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 always 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 2 +Cy 2 + Ex=0 (1). Also let ?, m, n be the direction-cosines of the straight line to which the generating line always continues parallel. Let a, @, 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-p _z I m ~n w ' Iz n mz n' 9 n But a, /3 are the co-ordinates of some point in (1), and therefore we have by substitution '(-*M»-=M~3-* or A (nx - Iz) 2 +C(ny- mz) 2 + nE (nx - Iz) = (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 which always moves parallel to itself, and has its vertices on two ellipses whose planes are perpendicular to each other and to the plane of the moving ellipse, and ivhich have one axis common. OF THE SECOND ORDER. 41 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 axis of x. The plane of the moving ellipse will be parallel to the plane of yz. Let CO A, A OB be the fixed ellipses, OA = a, OB = b, OG= c. And let EPS be any position of the moving ellipse, MB, MS its semi-axes, P any point in it. Draw PN parallel to Oz to meet MS in JSf. Let OM = x, MN = y y NP = z. From the ellipse EPS, ** - f _-, . BM 2 From the ellipse COA } BM 2 c 2 From the ellipse A OB, MS 2 + MS" (1). = 1- .(2). = 1 or ■(»)■ 42 ON CERTAIN SURFACES Whence substituting in (1) z = 1- a? a?' or X* a" + b* z* + 7^ = 1 (4). If the two semi-axes OC and OB be equal, it can be seen from (2) and (3) that MR and MS are also equal. Now an ellipse whose axes are equal iff 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- ? + b< ~ U The surface is called an oblate or prolate spheroid ac-, cording as the semi-axis a is less or greater than b. If all the three semi-axes OA, OB, OC be equal, the equation becomes x*+y* + z*^a*, which shews that the surface in that case becomes- a sphere whose centre is at 0. 37. The Hyperboloid of one Sheet. The hyperboloid of one sheet is generated by a variable ellipse which moves parallel to itself and has its vertices on two hyperbolas whose planes are perpendicular to each other and to the plane of the moving ellipse, and which have a com- mon conjugate axis. Let AQ be one hyperbola in the plane of zx, BR the other in the plane of yz, and RPQ any position of the moving ellipse, MM and QM its semi-axes, and P any point on it. Let OA—a, OB = b, and OG, the common conju- gate semi-axis, = c. Draw PN parallel to MR to meet MQ in iV. Let OM= z, MN = x, NP=y. Then from the ellipse RPQ, MR?^MQ 2 V OF THE SECOND OEDER. 2 43 MQ 2 from the hyperbola AQ, — J£- —1+ -3, i/E 2 s 2 from the hyperbola BR, — rr- = 1 + -3 ; C 2 ' 22 ' 2 > o 6 c or * 1, the equation required. 38. The Hyperboloid of two Sheets. This is generated as the last surface except that the hyperbolas have a common transverse axis. 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 = b, OG—c, the two conjugate semi-axes. Let QPR be any position of the moving ellipse, MQ, MR its semi-axes, and P any point in it. Draw PN parallel to QM to meet RMinN. 44; ON CERTAIN SURFACES Let OM=x, MN=y, NP=z. z* From the ellipse QPR, j^+ jj^ = h from the hyperbola A Q, MQ* (V -3--1. MR 2 a? from the hyperbola AR, — ^- = — — 1 f . z* x 2 " 6 2 + c 2 -?~* or a 2 6 2 c 2 ! the equation required. These three surfaces, the ellipsoid, the hyperboloid of one sheet, and the hyperboloid of two sheets, are all included in the equation 39. The Elliptic Paraboloid. The elliptic paraboloid is generated by a parabola which moves with its vertex in a fixed parabola, the planes of the two OF THE SECOND ORDER. 45 parabolas being at right angles, their axes parallel, and their concavities turned in the same direction. Take the plane of the fixed parabola as plane of xy, its vertex as origin, and its axis as axis of x. Then the plane of the moving parabola is parallel to that of zx. Let PQ be any position of the moving parabola, P any point in it, V its latus rectum, and let I be the latus rectum of the fixed parabola. Draw PM parallel to Oz to meet the axis of the moving parabola in if, and draw QH and MN parallel to the axis of y. Then from the parabola PQ, PM 2 = z 2 = l'.QM, and from the parabola Q 0, QH 2 =y 2 = l.OH = lx-l.QM h: £ — iy ■ I ■ V f 40 ON CERTAIN SURFACES 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, N H P any point in it. Draw PM parallel to Oz, MN and QH parallel to Oy. Let I and V be the latera recta of the two parabolas OQ, PQ. From the parabola PQ, PM 2 = z 2 =l'.QM, from the parabola OQ, QH 2 ~y*=l.OH = L(x + QM) lx + p 7 x "* The two paraboloids are both included in the equation By*+ Cz* = x. OF THE SECOND ORDER. 47 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 ifi-i-i a). s + ^T. which can be put into the form c w by \ ay + 6V. _ («? , yy ab V«* bV 2{ay + b 2 x 2 ) h '"' where the remaining terms contain higher powers of ay 4-ftV* in the denominator. Hence, if we increase % or y, or both, indefinitely, the value of z approaches indefinitely near to z And if we construct the surface *-s+f » (which by Art. 34 represents a cone whose vertex is the or-igin)t,, 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 ?/, when these values are increased indefinitely ; that is, the cone (2) is asymptotic to the hyperboloid. Similarly the -cone whose equation is a? ¥ c 2 ' is asymptotic to the surface 48 ON CERTAIN SURFACES OF THE SECOND ORDER. 42. The equation of the hyperbolic paraboloid is t-f=* to , 1 1 (-, t'x Vr( 1+ S + -) Now if 2; be increased indefinitely and on be not very large, the second and all the succeeding terms of the series on the right will diminish indefinitely. Hence the equations y = ±Vr w> 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 IIL 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 k\ 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 Jc 2 . Find the locus of Q. EXAMPLES. CHAPTER III. 49 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 PA is to PB 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 APB 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 containing 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. Shew 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 60°, 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. 13. A straight line moves so that three given points of it lie respectively in three planes at right angles to eacli other. A. G. 4) 50 EXAMPLES. CHAPTER III. 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 x ± a ±y z cos a sin a meet the axis of x in 0, 0\ and P, P' are points on the two lines such that OP . O'P' = c 2 ; shew that the surface traced out by the straight line PP is the hyperboloid a 2 c 2 cos 2 a c 2 sin 2 a ' P, P' 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 x 2 + y 2 + z 2 = a 2 and y 2 + z 2 — n 2 a 2 - c 2 , where a is a variable parameter and c an absolute constant; and discuss its form for different values of n. 17. A perpendicular PJV 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 PN or PiV" produced such that PN. P'N is constant. Find the locus of P'. CHAPTEK IV. 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 which 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, /3, 7 be the co-ordinates of the new origin 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+j3, z—z+ but this angle being a right angle, its cosine is equal to zero ; Similarly lj, x + m 3 7^ 1 + n a n x = 0, kh + m i m 2 + n i n 2 = 0. These relations may be replaced by the six equations m^ + ra 2 2 + ra 3 2 , = l, ^ m 1 w 1 + ^ 2 + m 3 n 3 = 0, n t l t +n 2 l 2 +n z l 3 = 0, ^-t^ m 2 +^ m 3 =0 - These equations can d^ algebraically deduced from the previous set, but they can be more easily j)r(£vfed independ- ently thus : v ^ft* r L, m,, n t are the cosines of the angles between Oaf and Ox, Oy, Oz; Z 2 , ra 2 , n 2 those of the angles between Oy and 0#, Oy, Oz; and Z 3 , m 3 , w 3 of the angles between 0/ and Ox, Oy, Oz. Consequently l x , l 2 , l 3 are the cosines of the angles between Ox and Ox', Oy', Oz \ m,, ra 2 , m 3 those of the angles between Oy and 0#', Oy', Oz; and 7^, n 2 , n 3 those ,of the angles between Os and 0#', Oy , Oz. Considering Ox, Oy, Oz' as axes, and remembering that Ox, Oy, Oz are mutually at right angles, we obtain the above formulas at once. 45. The formulas 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 formulas 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 xy cut the plane of xy in 0x x , 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 0y 1} 0y 2 , respectively. 54 .TRANSFORMATION OP CO-ORDINATES. Then since Oz is perpendicular to the plane of xy it is perpendicular to 0x x , and since Oz' is perpendicular to the plane of xy', it also is perpendicular to 0x x . Hence 0x x is perpendicular to the lines Oz and 0z\ and is therefore per- pendicular to the plane in which they lie, and therefore perpendicular to 0y x , 0y 2 . Hence by Euclid, XI. Def. 6, the angle yfiy 2 is the angle between the planes of xy and x'y. Let this angle be called 0, and let the angle between Ox and 0x x be called , and the angle between 0x x and Ox be called -x/r. Let x, y, z be the co-ordinates of any point P referred to the axes Ox, Oy, Oz. Then if we take 0x x , 0y v and Oz as axes, the ordinate z will be unaltered, and if x v y t be the new co-ordinates parallel to 0x v 0y x , we have by the ordinary formulas of transformation in plane co-ordinates, x b= x t cos — y x sin <£, y = x x sin cf> + y x cos $. Again, if we take 0x v 0y 2 , Oz' as axes, the x x will be un- altered, and if y 2 , z be the new co-ordinates parallel to 0y i} 0z\ we have ! Vi — V* cos ^ ~ z ' sm ft 2 = y 2 sin + 2' cos 0. TRANSFORMATION OF CO-ORDINATES. 55 - And lastly, taking Ox\ Oy, Oz' as axes, the z will be un- altered, and we get x t — x cos "fy — y sin i/r, y t — x sin ^ + y cos \jr. : And, making the substitutions for oc^y^y^ we get finally x = x (cos cos sfr — sin + cos cos ty sin <£) 4- z sin <£ sin 0, y = x (sin $ cos ijr -f cos (/> sin i|r cos 6) — 2/' (sin sin >/r — cos <£ cos -vfr cos 0) — z sin cos (£, 2 = #' sin ^ sin + y' cos ^ sin # + z cos 0. These are called Euler's Formulae. They are useful in discussing the nature of the sections of surfaces, but their unsymmetrical 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 formulas of Arts. 43 and 44. For changing the origin to a point whose co-ordinates are a, /3, 7, and keeping the direction of the axes unchanged, we get x = x 1 + ol, y = y 1 + j3 i z = z x -f 7. And then changing the directions of the axes we get x x m \x + Iff + l z z\ or x m l x x + l 2 y' + 1/ + or. Similarly y = m x x+m 2 y+m z z' + l3 J z — n t x + n 2 y + njf + 7. 47. The formulae for transformation of co-ordinates in I Art. 44 hold also when the axes are oblique if l lt m lt n x denote the direction-ratios of the new axis of x with respect to the old axes. The six relations which hold 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. ...... 56 TRANSFORMATION OF CO-ORDINATES. 49. The following proposition is useful in many ques- tions of transformation of co-ordinates. The condition that the expression Ax 2 + By 2 + Gz 2 +2A'yz + 2B'zx + 2G'xy (1) should be the product of two linear expressions in x, y, z, is ABC + 2A'BG' - A A' 2 - BB' 2 - GC' 2 = 0. For if one of the factors be Xx + fiy + vz (2), it is evident, by considering the coefficients of x 2 } y 2 and z 2 in (1), that the other factor must be ABC x* + ^ + r (ft 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 fX V V A A /M whence by multiplication we get SA'FCr m 2ABG + A (b 2 - 2 + G 2 £\ + B ((J 2 ^ + A 2 ~) + c H+ B i 2 ) = 2ABG+ A {4A' 2 - 2BG) + B(m 2 - 2GA) + G (4(7'* - 2AB), or transposing and dividing by 4, 2A'B'G' + ABG - A A' 2 - BB' 2 - GG' 2 = 0. The expression 2A'B'G f + ABG - AA' 2 -BB' 2 - GG' 2 is called the discriminant of the expression (1). TKANSFOEMATION OF CO-ORDINATES. 57 50. It is evident that in any transformation of co-ordi- nates from one set of axes to another, the origin being un- changed, the expression x 2 + y 2 + z 2 will be transformed into x ' 2 + y' 2 + z 2 if k° tn se * s 9? axes ^ rectangular; or the ex- pression #* + y 2 + ^ 2 + 2;*/2 cos \ + 2.z# cos ft + 2#y cos v 11 be transformed into so 2 + y' 2 + s' 2 + 2yz cos V + 2/a/ cos ft + 2xy cos v if the axes are oblique, the expressions in each case repre- senting the square of the distance of the point whose co- ordinates are considered, from the common origin. Thus if the axes are rectangular, and the expression Ax 2 + Bif + Cz 2 +2A'yz + 2B'zx+2C'xy (1) become by transformation Px' 2 + Qy 2 + Rz 2 + 2P'y f z' + 2Q'z V + 2Rlx'y'.. . . (2) ; we shall have also the expression Ax 2 + Btf+Cz 2 + 2A'yz + 2B r zx + 2G'xy - \ {x 2 + f +z 2 ) . . . (3), where X is any constant, transformed into ; P^+Qy ,2 +Rz 2 +2P f y , z , +2Q f z , x r +2R , x , y , ~X(x /2 +y ,2 +z ,i ) . . . (4). But if, for any values of \ the expression (3) be the pro- duct of two linear expressions in x, y, z, the expression (4) must, for the same values of \ be the product of the two expressions in x\ y, z into which the former two would be reduced by the transformation. Hence the discriminant of (3) is identical with that of (4), or the two equations (A-\){B-\){C-\)-A' 2 (A-\)-B' 2 (B-X)-C ,2 (C-X) + 2A'B'C' = (5), (P-*)(Q-*)(R-\)-P*(P-\)-Qr(Q-\)-Br t (R-\) + 2P'Q'R' = (6), are identical, and satisfied by the same values of \. Thus (7). 58 TRANSFORMATION OF CO-ORDINATES. the coefficients of the different powers of \ in these equations nlust be equal, and we have A + B+C = P+Q + R, BC + CA + AB - A' 2 - B 2 - C* = QR + RP+PQ-P' 2 -Q' 2 -R' 2 i ABC + 2A'B'C - A A' 2 - BB' 2 - CC 2 =PQR + ZP'Q'R' - PP' 2 - QQ' 2 - BR' 2 } The expressions on the left-hand side of the equations (7) are called invariants of the expression (1). 51. As a particular case of the foregoing, let us suppose if possible, as it will be proved to be hereafter, that the ex- pression (1) is transformed into an expression of the form Px ,2 +Qy' 2 + R2f\ The equation (6) then becomes {P-X)(Q-\)(R-\)=0, and the roots of this equation are P, Q, R, the coefficients of a?' 2 , y' 2 , z' 2 in the transformed expression. These coefficients are therefore the roots of the equation (5) with which (6) is identical, namely, (A-\)(B-\){C-\)-A' 2 {A-\)-B 2 {B-\)-C' 2 (C-\) + 2A'B'C'=0. Another proof of this result will be given hereafter (Art. 86). EXAMPLES. CHAPTER IV. 1. The co-ordinates of a point are (1, 2, 3). Find its co- ordinates relative to new axes whose equations are x = y = z\ 2x = —y — 2z; x = — z, y = 0. - 2. Transform the expression xy+yz + zx to the new axes in, the last question. EXAMPLES. CHAPTER IV. 59 3. Shew that x 2 + y 2 + z 2 + yz + zx + xy can be reduced by transformation of co-ordinates to the form A(x' 2 + y' 2 )+Bz'\ 4. From the formulas in Art. 44 prove that 5. Find the values of P, Q y B when the expression x* +y 2 ■+ z 2 — 4#j/ — 4?^ — 4 iWjg 1 , or as ABCy + A*BCa* 0, or as ABGy* (Az* + B^ + Cfy 2 ) < or > 0, or as ABC < or > 0. */■ Hence that the generating lines may be real we must have ABC 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 A-- ai , JJ- ¥ , 0--j, and the equations which determine the directions of the generating lines are ? m 2 n* a 2+ b* W = 0, Iol mj3 ny = 0. 58. It may be noticed that since for either of the gene- rating lines we have Ala + BmP + Cny=Q t AND SECTIONS OF QUADKICS. C5 and for any point in either line we have x—VL_y — ft^z — y I m n ' we must also have the equation Aol {x - a )+B/3(y-j3) + Cy (z-y) = 0, satisfied for any point in either of the straight lines through the point (a, /?, 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 written Aax+B/3y + Cyz = A* 2 + Bj3 2 + C 7 * = 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 k A 7). 59. The equation of the projection of either line on the plane of xy is I m > Or y = j X + @-jCt (1), the values of -j- being deduced from the quadratic equation given in Art. 57. A r ( Cy 2 + Aol 2 ) + 2ABrf Im + Bm 2 (Cy 2 + B,8 2 ) = 0, or Af (1 - Bj3 2 ) + 2ABxj3 Im + Bm 2 (1 - Ao?) = ; /. Al 2 + Bm 2 = AB (1/3 - ma) 2 . Hence the equation (1) can be written m /l 1 T x± Vb + a 'A' Y' which is a well-known form of the equation of tli3 tangent to the curve Ax 2 + By 2 = l. A. G. 5 GO ON GENEKATING LINES But this curve is the ellipse in which the given surface ia 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 By 2 +Cz 2 = x (1). The conditions that a straight line x — ol _y~ ft _z — 7 n l m < 2 )> should lie wholly on the surface (1) are found by a process similar to that of Art. 57 to be Bj3*+Cy 2 = a (3), Bnf + Cn^O.... ....\ (4), 2Bm{3 + 2Cny-l = .. (5). The first equation indicates that the point (a, fl, 7) lie: on the surface (1). The second and third give the value; of the ratios I : m : n. These values will be real if B and ( 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 • m ( n m \ ,2 But from (5) (2Bmj3-l) 2 =4 ! G 2 ny = -4Em 2 x - 4Blmj3 + l 2 = Q from (3) ; or fp~mx=- rb . - , 4iB m AND SECTIONS OF QTJADKICS. 67 And the equation (6) becomes m 11 a well-known form of the equation of the tangent to the curve Bif=%. Hence the projection of the generating line on the plane of xy is a tangent to the curve in which that plane is cut by the surface. A similar proof holds for the projection on the plane of zx. The equation of the projection on the plane of yz is m n * P r y=n Z+ @~n y W " But Bm* + Cn* = 0; .\- = ±\/-„, and the equation (7) becomes 2/ = ±/ V /-J.^ + (^ + / V /-J.7). Hence the projections of the generating lines on the plane of yz are parallel to the two straight lines in which the surface is cut by that plane. 62. The sections of the ellipsoid 5+? + ? =1 «■ made by planes parallel to either of the co-ordinate planes are ellipses. For taking the equation of a plane parallel to that of xy to be * = 7 (2), we get for the points where this meets (1) „2 ' 1, 2 X -2 5—2 68 ON GENERATING LINES This is the equation of the projection of the curve of section on the plane of xy. But since the cutting plane is parallel to the plane of xy, the projection of the curve of section on that plane is equal and similar to the curve itself. Hence this curve is an ellipse. And it may be noticed that x 2 y 2 this ellipse is always similar to the ellipse -2+^ = 1, in which the surface is cut by the plane of xy. In a similar manner the sections by planes parallel to the other co-ordinate planes may be shewn to be ellipses. The sections of the hyperboloid of one sheet 2 2 2 a* T b 2 & ~ ' 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 a 2 V c 2 ~ ' 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 I + f ~ X > i r ~ X} by planes parallel to those of zx or xy are parabolas whose latera recta are X and I respectively. Their sections by planes parallel to that of yz are re- spectively 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 AND SECTIONS OF QUADMCS. 69 sufficient to find the equations of the projections of the curve of section on the co-ordinate planes, since the projection will 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 xy. 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 xy itself, we may before substitu- tion put z — 0. The required substitutions will then be derived from the formula} in Art. 45 by putting yfr = and z = 0. We thus get x — x cos 4> — y' cos 6 sin <£, y = x sin (f> + y cos 6 cos <£, z = y sin 6. If the equation of the cutting plane be given in the form Ix + my + nz =p, we have tan cf> = , and cos = n. The above substitutions then become mx' + Iny ran?/ — laf , . x = — * . y= , J ^--, z — y VZ'4-w 2 where we assume that Z 2 + ra 2 + n 2 = 1. 63. We shall first prove the following general propo- sition. All sections of surfaces of the second order made by 'parallel planes are similar 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 Ax 2 + By 2 + Cz 2 + 2A'yz + 2Bzx + 2 C'xy + 2A"x + 2B"y+2C"z+F=0 (1). The curve in which this is cut by the plane * = Y (2), 70 ON GENERATING LINES is given by the equation Ax 2 + By 2 + 2 G'xy + (2B'y + 2 A") X + (2A'y + 2B f ) y + Cy 2 + 2G"y+F=0. And whatever be the value of 7 this curve is always similar and similarly situated to the curve Ax' + By 2 +2 G'xy + 2A!'x + 2B'y + F= 0, in which the surface is cut by the plane of xy. Hence in discussing the form of the sections of surfaces by 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 Aa?+By 2 +Cz 2 = l } which includes the three central surfaces. Making the substitutions suggested in Art. 62, we get as the equation of the curve of section x 2 (A cos 2 cj> + B sin 2 )'+ 2xy (B — A) cos <£ sin <£ cos 9 + y' 2 {A cos 2 6 sin 2 $ ■+ B cos 2 6 cos 2 + G sin 2 <9) = 1. And the section will therefore be an ellipse or hyperbola according as "''.'' (5-^) 2 cos 2 0cos 2 <£sin 2 <£ -{A cos 2 cj> + B sin 2 $) {A cos 2 6 sin 2 cf> + B cos 2 cos 2 <£+ <7sin 2 0) is negative or positive. This expression can be .reduced to the form - {BCsm 2 sin 2 <£ + CA sin 2 6 cos 2 <£ +AB cos 2 0}. In the case of the ellipsoid A, B and G are all positive, and this expression is therefore always negative. All sec- tions of the ellipsoid are therefore ellipses. The investigation of the nature of the sections in the other surfaces is long and AND SECTIONS OF QUADRICS. 71 I the results uninteresting, except in the particular case in which the section becomes a circle. The conditions that this may be the case are, that the co- efficient of xy should vanish and the coefficients of x' 2 and y 2 should be equal. We have therefore (B — A) cos 6 sin <£ cos $ = 0, A cos 2 $ + B sin 2 = A cos 2 # sin 2 + B cos 2 6 cos 2 <£ 4- G sin 2 0. From the first equation we must have either B=A, in which case it is already obvious that all sections parallel 10 the plane of xy are circles, or cos 6 . sin = 0. : If cos = 0, we have Q— 90°, and the second equation gives A cos 2 cf> + B sin 2 <£ = G = G (cos 2 + sin 2 ) ; 2 , L> — A .-.tan <£ = ;#—- q> and if the values of tan <£ be real, we get circular sections by two planes through the axis of z. If we take cos = ; we have (f> = 90°, or the plane passes through the axis of y, and the second condition gives £ = ^lcos 2 <9+Osin 2 0; A-B and therefore tan 2 6 — B-G* 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 (f> = 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-B tan 2 = G-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. 72 ON GENEKATING LINES Only one of the three quantities G-A A-B A-B B-C B-C G-A can be positive, consequently there are only two real central circular sections, and they pass through the axis of z, y or x, according 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 f 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- 1 B- 1 and A being >B, ax 2 + 2B sin cos 6 xy + y 2 (B cos 2 6 cos 2 cj> + G sin 2 6) — x cos — y cos 6 sin , which will represent an ellipse, parabola, or hyperbola, ac- cording as B 2 sin 2 <£ cos 2 <£ cos 2 - B sin 2 <£ (B cos 2 cos 2 <£ + sin 2 0) is negative, zero, or positive. That is, according as BG sin 2 sin 2 6 is positive, zero, or negative. The sections of both paraboloids are therefore parabolas if or 6 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. 74 , ON GENERATING LINES The conditions that the section may be a circle are B sin cos <£ cos 6 = 0, B sin 2 <£ = B cos 2 <£ cos 2 + C sin 2 0. From the first equation sin

— 0, cos $ = 0, or cos 6 = 0. If sin <£ = 0, the coefficient of a/ 2 vanishes, and the section reduces to a straight line or parabola. If cos = 0, we have from the second equation B =C sin 2 6, and if B and C are of the same sign and B = -^ , and this gives two possible values of (j> if G < B, and B and O have the same sign. Thus we get real circular sections of the elliptic paraboloid passing through the axis of y or z, accord- ing as B < or > C, that is as I > or < V. If B and C have opposite signs, there are no real circular sections. 67. The equation of the elliptic paraboloid can be put into the form a? + y* + z 2 a? y 2 y 2 l' + 'l ~l'- X > of + tf + z* , or 1 + Thus each of the planes y\/]-T +x s/i =0 ' and therefore all planes parallel to them will cut the surface in circles. These planes are real if I' > I If I' < I we can shew similarly that the planes AND SECTIONS OF QUADRICS. 75 3ut the surface in circles. 68. We shall conclude this chapter with the investiga- tion of the position and magnitude of the axes of the section of an ellipsoid by a plane through its centre. Let •§+F+7- 1 •••••- (1) be the equation of the ellipsoid, Ix + my + nz = (2) the equation of the cutting plane. Let ? = y = * = r ...... (3) A ./A V be the equations of any straight line in the plane (2), and let r be the distance from the origin of the point where it meets the ellipsoid ; therefore - V a 2+ 6 2 c 2 K)i and l\ + m/ju + nv = (5), since the line (3) lies in the plane (2). Also if r be the length of one of the semiaxes of the section of (1) by (2), we must have r a maximum or mini- mum by the variation of X, //,, v, which are connected by the relation (5) and also by the relation X 2 + ^ +J , 2 =l (6). Differentiating (4) we get when r is a maximum or mini- mum a — ^^ A^A* vdp a b c 76 ON GENERATING LINES And from (5) and (6) respectively, = ld\ + md/ub + ndv, = \d\ + pdfjL +" vdv. Whence by indeterminate multipliers, \+M+k'\ = (7), j&+fm+Kl*~b (8), V + kn+k'v = (9). Multiplying (7) by X, (8) by fi, (9) by v, and adding, w< get and therefore x(i,4)=-«, ,.*.« a — f 7cmr*b 2 Jcnr*c c — r And therefore from (5), which is a quadratic equation and gives two values of r 2 . The product of these two values ~W + m 2 & 2 + rcV' and the area of the section is therefore wabc . (11 ) AND SECTIONS OF QUADRICS. 77 The directions of the two axes may be obtained by elimi- nating k and M from equations (7), (8) and (9) ; we then get a* v -o n v c or ly.v (i - J) + mvX g - 1) + r»V (J - p) = . . . (12), which united with (5) and (6) gives two sets of values of K ft, v. The expression for the area of a section of an ellipsoid by % 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 drawn through a point for which s = ±Ca/ — % , right angles to each other. 2. Shew that all the points on the surface are at °? + y" -- = i a" 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, fi } y. d i+ b f I = 1 78 , EXAMPLES. CHAPTER V. ' 4. If the surface (^ 2 + 2/ 2 + ^) 2 = aV + 6y + cV be cut by a central circular section of the ellipsoid a b g 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 a? + y 1 + z 2 + {loo + my + nz —p) (I'x + my + nz—p) = 0, find the planes which give circular sections. 6. Prove that the sections of the surface. xy + yz + zx m 1, by planes parallel to x + y + z = 0, are circles. 7. If the two generators drawn from a point on the surface intersect the principal ellipse in points P, F at the ends of conjugate diameters, then will OP 2 + 6>P' 2 -a 2 + 6 2 +2c 2 . 8. Find the circular sections of the surface , ; a 2 + r c 2 * 9. Prove that if the section of the surface yz ,w^ al *•?■#. TV j by the plane Za? + my + n^ = 0bea rectangular hyperbola, n + ^ + ^-o. EXAMPLES. CHAPTER Y. 79 10. The angle between the generating lines of x 2 y 2 z 2 - ., ,- . , . N . _, X, + X„ — + j- + - = 1 at the point (x, y, z) is cos r- 1 — — 2 , where \ and \ are the two roots of a (a + X) J (6 + X) c (c + X) 11. Prove that the foci of all centric sections of the surface ax 2 +by 2 + cz 2 = l lie on the surface z 3 +/+s 2 )(l-az 2 -% 2 -c^ - (ax 2 + by 2 + cz 2 ) {(c - b) 2 y 2 z 2 + (a - c)'W+(& - a) 2 x 2 y% 12. Find the equation of a right circular cylinder whose axis is the line m _ y _ ^ I m n? and whose radius is a. 13. Find the condition that the cone Ax 2 + By 2 + Cz 2 + tA'i/M + 2#z# + 2 G'xy m 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 c? + b* + (? for its base and its altitude equal to b. 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 only. 16. Find the equation of a right circular cone whose axis is the line T = *- m - , and semi- vertical angle a. 17. Find the equation of a right circular cone which contains three given straight lines passing through the origin. . ' . 80 EXAMPLES. CHAPTER V. 18. Find the locus of the points at which the two gene- rating lines of the surface Ax 2 + By'+Cz 2 = l are at right angles. 19. If a plane be drawn through the straight line x _ y _z I m n y the two other straight lines in which it cuts the cone (B-C) yz (mz — ny) + (G— A) zx (nx — Iz) + (A--B) xy (ly — mx) = will be at right angles to each other. 20. Shew that any point on the hyperboloid of one sheet may be represented by the equations x = a cos sec 0, y = b sin <£ sec 0, z—c tan ; and find the equations of the generating lines through thai point. 21. Shew that if the two generating lines at any point o] the surface rf*y »» i be at right angles respectively to those of opposite system! through a second point, the two points are either in a plant through the axis of z or equally distant from the plane of xy 22. If two planes be drawn passing respectively througl two generating lines of the same system at the extremities o the major axis of the principal elliptic section of a hyperbolok of one sheet and intersecting in any third generating line, th( traces of these planes on either of two fixed planes will be a right angles. 23. If a=0, /3=0, 7=0, 8=0 be the equations of the fou faces of a tetrahedron expressed as in Art. 26, the equatioi of a hyperboloid of one sheet passing through two oppositi edsjes is Pa/3 + Q 7 S + jRSa + ^7 = 0. CHAPTER VI. DIAMETRAL PLANES. 69. It will be useful to commence the chapter with the bllowing definitions. 1. The centre of a surface is a point such that all chords )assing through it are bisected by it. 2. The locus of the middle points of a system of parallel hords of a surface is called the diametral surface of the ystem. We shall shew that if the original surface be a quadric, he diametral surface of any system of parallel chords is a )lane. In this case we shall require the following definition. 3. A principal plane of a quadric is a plane perpen- Ucular to the chords which it bisects. We shall shew hereafter that such a plane can always be ound. 70. If a quadric have a centre and be referred to a, ystem of axes with the centre as origin, the equation will not ontain any terms of the first degree. For the general equation of the second degree is ia?+ By*+Cz* + 2A'yz + 2B'zx + 2G'xy + 2A"x + 2B"y + 2G"z + F= (1). Then if x^y v z x be the co-ordinates of any point on the urface, — x 1} —y v —z x must also satisfy the equation (1), ince the origin is the centre. Hence we have A. G. 6 82 ' DIAMETRAL PLANES. Ax?+ By?+Cz*+ %£y t z t + 2B'z 1 x 1 +2C'x 1 y 1 + 2A"x t + 2B"y t + 2C'z 1 +F= 0, Ax*+ By?+Cz?+ 2A'y 1 z l +2B'z 1 x 1 + Wx$ t - 2A"x x - 2B"y- 2C\+F= 0. Subtracting we obtain ^{A"x 1 + B"y^C' f z 1 )^0 (2). This equation must be satisfied for all values of x t , y lt I consistent with (1). But unless A" = 0, B" — 0, 0" = 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 i''=0, £"=0, C" = 0, or the equation (1) does not involve the first powers oi 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 2 +Cz 2 = F (3), the axes being rectangular, the co-ordinate planes will be princijDal planes. For if x t , y v z x satisfy the equation (3), so do — x xy y v z x . 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 equation of the ellipsoid be * + £.!* a 2± b 2± c + C + ^ = 1 (1), DIAMETRAL PLANES. 83 aid let the equations of any one of the system of parallel chords be x — aLy — ftz — y ... — r- =^— - = r =r (2), I m n v vhere I, m, n are direction-cosines. To find the points where (2) meets (1) we have (* + lr) 2 (/3 + mr) 2 (y + nr) 2 ^,2 JL2 Ji ~~ •*•> »/P m 2 n 2 \ a fh m@ ny\ a 2 £ 2 7 2 _ A /ox This equation gives two values of r which are the distances rom the point (a, /?, 7) of the two points where the straight line '2) cuts the ellipsoid. If (2, j3, 7) be the middle point of the bhord these two values must be equal, and opposite in sign; the 3oefficient of r in the equation (3) must therefore vanish, or lot m& wi a a 2 + b 2 + c 2_U * 6 .lence (a, /3, 7) always lies in the plane Ix^rrvy nz__ which is therefore the equation of the locus of the middle points of the system of chords. 72. If a?!, y v z x be the co-ordinates of the point in which X 11 z the line T = — = - meets the ellipsoid, that is, the co-ordi- aates of the extremity of the diameter drawn parallel to the system of parallel chords, we have l m n' and the equation (4) of the last article may be written a 2+ v + e u (ij - 6—2 I 84 DIAMETRAL PLANES. Also if # 2 , y 2) z 2 be the co-ordinates of any point in tli curve in which this plane cuts the ellipsoid, we have a 2 + 6 2 c 2 ~ U ' which shews that the point (x lf y v z t ) lies in the plan which bisects all chords parallel to the diameter throug The planes which bisect chords parallel to the two diam* ters through \x %i y v zj, (x 2 , y 2 , z 2 ) will intersect in a straigl: line. Let the co-ordinates of the point where this line meet the ellipsoid be x 3 , y 3> z 3 . Then since (x 3> y 3 , z 3 ) lies in tli plane which bisects chords parallel to the diameter throug K> v%* z i) we nave a' c and since it lies in the plane which bisects chords parallel t the diameter through (x v y 2 , z 2 ), we have Vs , M* 4. Z l* - a b c These last equations shew that (x v y lf #,), (x 2 , y 2 , z 2 ) bot lie in the plane which bisects all chords parallel to the diame ter through (x 3 , y 3 , z 3 ). Hence the three diameters have this property, that th plane through any two of them bisects chords parallel to th third. The three diameters are called conjugate diameters. 73. The equation of the ellipsoid when referred to a syi tern of three conjugate diameters as axes assumes the form x 2 f z 2 , a /2 %' 2i e* ' where a', b', c' are the lengths of the conjugate semi-diameters For the equation must be of the second degree by Art. 4$ and since each co-ordinate plane bisects chords parallel t the corresponding axis, by Art. 70 the equation must assum the form Ax* + By 2 + Cz 2 = F. DIAMETRAL PLANES. When the axis of x meets the surface we have x = a' y y = 0,.z = 0, F A' F B' c - c . And the equation becomes 85 and therefore Similarly b' 2 x 2 r v 2 74. The co-ordinates of the extremities of three conju- gate diameters are connected by the relations a 2 + 6 2 + c 2 x l+.yLi. z l a 2 ^ b 2 ^ c 2 a 2 + b 2 + c 2 1 = 1 = (1): ^3 , v*y* , v. a 2 "*" 6 2 ■*" c 2 = a 9 * 6 2 + c 2 = a 2 + 6 2 + c 2 = .(2). 1 Squaring all these equations, and adding twice the squares of the second three to the squares of the first three, we get 86 DIAMETEAL PLANES. Expanding, and rearranging the terms we get 2 ~ 2 « 2 \ 2 (2+2+2 A\(y2+y2 + y2- 1 )\(2^+^- 1 c c V 6c be be J \ ca ca ca) \ ab ab ab J 1 Whence w* + x* 4- # 3 2 = a 2 1 y, ? +tf**V#| (3), 2/l S i + ^2 + 2/3^3 = ) ^ + ^ 2 + ^3=0 (4). ^1 + ^2 + ^3=° I This transformation can be easily seen to be equivalent to that effected in Art. 44, using — * for l x , and so on. And the 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 x , m x , n x be the direction-cosines of the normal to the plane bisecting chords parallel to the line we have or and similar relations for l 2 , m 2 , »,. Whence equations (2) easily give a\l 3 + b 2 m 2 m 3 + c\n 3 = 0> (5), a\l x + b*m 3 m x + c\n x = J X _y _z x x Vi V I m x __n x x x a? V c 2 al x bm x cr\ a Vx *i ' b c DIAMETRAL PLANES. 87 jind obviously also C + < + < = l (6). £*+«/ + <- 1 1 75. From equations (3) of the last article we obtain by addition a' 2 + &' 2 + c' 2 = a 2 + & 2 + c 2 (1), I where a', b ', c' are the lengths of the conjugate semi-diameters. Let X, fi, v be the angles between (b' y c), (c\ a) and (a, b') y respectively. Then since the direction-cosines of a referred to the prin- SC 1/ z cipal axes of the ellipsoid are —,,—,, -\ , and similarly for those of b', c' } we have, by Art. (8), sin a — £,2^,2 , /. bV sin 2 X - (y A - ^ 2 ) 2 + (^ 3 - ^ 2 ) 2 + foy 8 - ^ 2 ) 2 . But we have oc i.°^ + hh + ?i5 ==0 a a o o c c a a b b c c a .&■ ! ■& a 6 c yfi-ytP* ¥ 3 -¥ 2 %^Mi 6c ca a& = / a 2 + 6 2 + c 2 V U 2 S 2 " cVU 2 + 6 2 + c 2 ; U 2 & 2 c 2 J by equations (1) and (2) of the last article. 88 DIAMETEAL PLANES. Hence &'V 2 sin 2 \ m Vc* -^ + cV |^ + a 2 b 2 % . a b c Similarly c'W sin 2 p = bV ^ + cV ^ + a 2 b 2 ^ , a' 2 6' 2 sin 2 1/ = 6 V ^ + cW ^ + a 2 6 2 % . a 2 6 2 c 2 Adding, we get (b'c sin X) 2 + (cV sin /*) 2 + {ijfV sin z/) 2 = 6V + cV + a 2 6 2 . . . (2). Again, if p be the perpendicular from the point (a? 8> y s> zl on the plane which contains a and &', whose equation is ft2 + 6 2 i" C 2 v, we have p i , 2/3 , 5l a 2 + 6 s " ■*" c 2 /fe 2 + ^-+^ ^ Venus' Hence squaring and multiplying by the value previously obtained for a' 2 6' 2 sin 2 v we get ^V 2 6 ,2 sin 2 i/ = a 2 6V (3). But aV sin v is the area of the parallelogram whose edges are a' and &', and pa'b' sin v is the volume of the parallel- epiped whose base is this parallelogram and whose altitude is p, 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 ab'c v 1 — cos 2 X — cos 2 fj, — cos 2 v + 2 cos X cos ja cos v. Hence this expression is equal to abc. 76. Another method of obtaining these relations is afforded by the consideration that the expression DIAMETEAL PLANES. 89 is transformed by taking three conjugate diameters as axes to the expression x 2 y 2 z 2 -7g + jt 2 + — 2 + k {x 2 + y 2 + z 2 + 2yz cos X + 2zx cos fi+2xy cos v). d c 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 k. By Art. 49 the requisite values of h for the two expres- sions are given respectively by the equations (*4)(*+J)(*-4)-o, and p+*X*+*).(*+*) - F COS 2 X [h + —A - F COS 2 fl(k + prA - k 2 cos 2 v(k + —J + 2& 8 cos X cos yu cos */ = 0, which when cleared of fractions and expanded become re- spectively, a'&W + {a 2 b 2 + b 2 c 2 + cV) h 2 + {a 2 + b 2 + c 2 ) & + 1 - 0, and a z b' 2 c' 2 (1 — cos 2 X — cos 2 //, — cos 2 v + 2 cos X cos //, cos v) k 3 + (&' V 2 sin 2 X + c'V 2 sin 2 ^ + a' 2 b' 2 sin 2 *) F + {a 2 + b' 2 + c' 2 )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 2 , and the three values of r 2 will be a 2 , b 2 , c 2 ; whence the values of aW, a 2 6 2 +cV + 6 2 c 2 , a 2 + b 2 + c 2 are known in terms of a', b\ c. 90 DIAMETKAL PLANES. The formulae obtained in Arts. 71 — 76 hold for the other central surfaces if the proper changes be made in the signa of a 2 , b 2 and c 2 . 77. The equation of the plane which bisects all chords of the ellipsoid parallel to the line * = y = z - .....a) *i fx *% ^ «^+y.J»+*.J=q (2). Conversely the chords which are bisected by the plane Ix + my + nz = (3) are parallel to the line a 2 l Vm c 2 n * W 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 Ix + my +nz — p = (5) parallel to (3), must contain the straight line (4). Whence it easily follows that the point where (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 Ix -f my + nz _ p . x aH ~Vm~~fri ~ar + b*m 2 + c*n 2 ~ a 2 T 2 T^ 2 m 2 + c 2 n 2 "" ( h 78. The co-ordinates of the centre of the section of the ellipsoid x 2 v 2 z 2 a° + V + ? =1 (1) by the plane lx + my + nz = p (2) can also be obtained in the following manner. DIAMETKAL PLANES. 91 Let a, /3, 7 be the co-ordinates required, and let a? — ay— B z — 7 ,-; -- — m* — £» l= r (3) be the equations of any straight line drawn in the plane (2) to meet the ellipsoid, r being the length of the radius vector. Then if w, y, z be the co-ordinates of the point where (3) meets (1), we have from (3) x = a + \r, 2/ = /3 + fir, z = y+vr, and therefore from (1) by substitution LV' + ^+Sr^ + ^ + ^ + ^ + ^ + SLl- But if a, /?, 7 be the co-ordinates of the centre of the 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 (1); that is, we must have tf + lf + tf- (5) for all values of \, fi, v consistent with the equation Xl + fjum + vn = (6), which is the condition that (3) may lie in (2). Hence the equations (5) and (6) must be identical, or we have a m j$_ = ^y la 2 mb 2 n& ' and as in the last article each of these fractions P aH 2 + b 2 m 2 + c 2 n 2 ' 79. The equation (4) of the last article, when the values of a, /3, 7 are substituted in it, becomes Comparing this with equation (4) of Art. 68 we see that 92 DIAMETRAL PLANES. if r, be the central radius vector which is parallel to r, we 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 x '- ' the area of the parallel central section, A=A fl p" 1 wabc L p* \ ~ 7a¥+ bV+cV \ " a'F + bW + cVj ' 80. The result of the last article can also be obtained in the following manner. Let a, /3, 7 be the co-ordinates of the centre of the sec- tion. Then the equation represents an ellipsoid whose centre is at (a, /3, 7), and whose semi-axes are ha, kb, Jcc. At the points where this cuts the given ellipsoid we have by subtraction 2ax 2/3y 2yz _o^ S 2 A t , vj + 1 - #> , r ' p [a 2 l 2 + b 2 m 2 + c 2 n j and if this equation be identical with Ix + my + nz =p (2), the sections of the two ellipsoids by this latter plane will coincide. DIAMETRAL PLANES. 93 The condition for this is l-& 2 = f aJV + bW + tfn* '. &' = 1- K a*?+b*m*+ p it Ps ^ e tne perpendiculars on the planes from the origin, then A (pi - Ps) + A b>.* -pi) + A (pi -Pi) = o. 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. 3. OL, OM, ON are conjugate semi-diameters of an ellipsoid; m v y lt z x the co-ordinates of L; x %y y 2 , z 2 and x s , y 3 , z 3 those of M and N respectively. Prove that the equation of the plane LMN is | 2 K + ^ 2 + ^ 3 ) +f 2 (3/1 + 3/2 + 3/3) + 35 (^1 + ^ + ^3) = !• 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 plane 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 ellipsoid by the diametral plane, is of constant volume. 6. Find the locus of the directrices of all sections of an ellipsoid made by planes passing through the least axis. 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. EXAMPLES. CHAPTER VI. 97 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 pnade by planes respectively parallel to them and intersecting in a point on the surface, prove that A + B + C 2 " 10. Any generating line of the cone Pa?+ Qy* + Bz 2 = :>eing taken, a plane is drawn diametral to it with respect to the surface Ax 2 + By* + Cz 2 = l. 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)zJ + 9y* {{ B-C)z>+(B-A)xr ~Rz 2 + ™ J -{{C-A)x' + (C-B)yJ=Q. 11. Find the co-ordinates of the centre of the section of the surface By* + Cz* = x made by the plane lx + 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 aV + by + cV = 3 (xx i + yyi + zz x )\ 13. If the plane lx + my + nz =p cut the surface a b* c in a parabola, prove that a 2 l 2 + bW-c 2 n 2 =0. A. G. 7 CHAPTER VII. THE GENERAL EQUATION OF THE SECOND DEGREE. 84. The general equation of the second degree can be written Ax' + By" + Cz* + 2A'yz + 2B'zx + 2G'xy + 2A"x + 2B"y + 2C"z + F=0 (1), which we will denote by F (a?, y, z) = 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 sur- face. 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, y as the co-ordinates of the centre. The equation when the origin is transferred to this point is ob- tained by substituting in (1) x + a, y 4- ft, z +y for x, y, z, respectively (Art. 43), and is therefore F(x+a, y' + ft, z+y)=0, which can be written -, . t ,dF ,dF ,dF THE GENERAL EQUATION OF THE SECOND DEGREE. 99 the remaining terms being of the second order in x\ y\ /, ,dF dF dF, . .. . • A . co and -j- , -v^ , -T- having the same meaning as in Art. 82. If the coefficients of x', y', z' vanish, we have dx Uj d/3 ' dy U ' or writing them out at length, C'a + Bj3 + A'y + B" = ol (2). 0m+A'fi+Oy±.Cr.-o\ These equations determine a, ft 7. We get from them A' a B B" B A' G" A' C A a B a B A' B' A' C _ A" (A'*-BC)+B" (CC'-A'B'W {BB-C'A') ABC+2A'B'C'-AA' 2 -BB U -CC" 2 A" (CC'-A'B')+B" (B 2 - CA)+ G" (AA ' -B C) nlarly p - ABC+2A'B'C'-AA ,2 -BB'*-CC" = A" (BB'-C 'A') +B"(AA'-B'C')+ C"(C'*-AB) 7 ABG+2A! B G'-AA 2 -BB 7% -GG' 2 We can therefore always obtain finite values of a, ft 7 except when ABC + 2A'BC - A A' 2 - BB' 2 - GG' 2 = 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, ft 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 that the point (a, ft 7) shall lie in the diametral plane to all systems of chords. (Art. 82.) 7—2 (3). 100 THE GENERAL EQUATION 85. We see from the last article that it is not always possible to get rid of the teims 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 iV + Qif + Rz* + 2P'yz + 2 Q'zx + 2R'xy + 2P"x + 2Q"y + 2R"z + F=0. The direction-cosines, of the chords which are perpen- dicular to their diametral plane are given by the equations Pl+R'm+ Qn = sl, 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, 7n= 0, n — 1. Whence we get Q = 0, P' = ; and the equation of the surface is p x * + Qtf + Ez 2 + 2Rxy + 2P"x + 2Q"y + 2E'z +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 — tj (Todhunter's Conic Sections, Art. 271), P — Q, the term involving xy disappears, and the equation assumes the form Pj? + Q } f + fa? + 2P"x + 2 Q'y + 2R"z + F=0. The equations which determine the directions of the principal diametral planes are now satisfied by 1 = 1, m = Q, n = 0, or by 1=0, 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. i^Rl OF THE SECOND DEOftSE. 101 We thus have an independent proof lhat fhes4 'tlire'e directions are all real and at right angles to each other. 86. We have now shewn that by a proper choice of 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- termined. Let Zj, m x , n x \ l 2 , m % , n 2 ; Z 3 , m 3 , n 3 be the direction- cosines of the new axes. These values all satisfy the equa- tions (1) of Art. 83. Let s x , s 2 , s 3 be the corresponding values of s. By Art. 44 the required transformation will be effected by substituting for x, y, z the expressions l x x + l 2 y + I/, m x x + ra.y + m 3 z, n x x + n % y + n 3 z, respectively. If therefore the original equation be Ax* + Bif + Cz* + 2A'yz + 2B'zx + 2G'xy + 2A"x + 2B"y + 2C"z + F = 0, the coefficient of x 2 in the result will be Al* + Bm* + fa? + 2A'm x n x + 2B\l l + 2G\m x . But from Art. 83 we have Al x + C'm x + B'n x =s x l xi G'l x + Bm l + A'n x = ^m,, B'l x + A'm l + Ga x = s x n x . Multiplying these equations by l %% m li n x , respectively, and adding, we get All + Bm i + Cn i + 24'*»i n i + 2B\l x + 2 G'l x m x = s x . Hence P the coefficient of x* is s x . Similarly Q=s 2 , R = g p or P, Q, R 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. 102 THE GENERAL EQUATION The result's^ tMs 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 l lt m,, n x ; Z 2 , ra 2 , n 2 \ l 3 , m 3 , n 3 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 lt m lt n lt l 2 , m 2 , r? 2 , satisfy the relation Al t l t + Bm x m 2 + Cnji 2 + A' (m t n % + mjij + B' (4& + nJJ + C (Z x m 2 + \m x ) = 0, and the expression on the left-hand side of this equation is the coefficient of x^ in the transformed equation. The coefficients of x, y and z in the transformed equa- tion will be 2 {A'% + B"m x + C"nJ, 2 (A% + B"m, + C"n) and 2(A"l 3 + B"m s + C"ri B ), respectively, and the constant term remains unchanged. 88. The equation when transformed to pj + Qf + Bz* + 2P"x + 2Q"y + 2R'z + F=0 can be farther simplified by a change of origin. Suppose first that none of the quantities P, Q, R vanish, that is, that none of the roots of the discriminating cubic vanish, which will "be the case if the constant term of the cubic, or ABC + 2A'B'C - AA' 2 - BB 2 - CG'\ be different from zero. In this case the equation can be written ■■ OF THE SECOND DEGREE. 103 and transferring' the origin to the point whose co-ordi- nates are \ P' Q* RJ' this becomes This represents an ellipsoid, a hyperboloid of one or two sheets, or an impossible locus, respectively, according as the F' F F quantities -p , -^ , -p are all positive, two positive and one negative, one positive and two negative, or all negative. Thus unless ABC+ 2A'FC -AA' 2 - BB' 2 - CC' 2 vanish, the surface has a centre and is one of the surfaces 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 Aa?+Bif+C2? + 2A'yz + 2Kzx+2C'xy = F (1), which by turning round the axes would become Px 2 + Qy 2 + Rz 2 = F\ 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, 0, 7), where a, p, 7 are determined from the equations Jt + fffi+By + A* -V C'a+B0+A'y+ff' = O} B'2+A'/3+Cy + C" = 104 THE GENEKAL EQUATION Multiplying the first of these by a, the second by 0, the third by 7 and adding, we get Aa 2 + Bj3 2 + CV+ 2A'/3y + 2B 'ya + 2G'a/3 + A"(x + B"{3 + C"v = 0. But Aol 2 + B& 2 + C7 2 + %A'fy + 25' 7 a + 2C'a/3 + 2A"a + 2B"/3 + 2 0" 7 + F= F (a, ft 7 ) = - F, Subtracting the first of these from the second, we get rJTm A"a + g'0 + G"y + F. Hence if the surface be a cone A"ol + B"{3+C"v+F=0. And eliminating a, /?, 7 between this equation and the three equations (2), we get as the condition that the surface represents a cone A C B' A" C B A' B" B A' G G" 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 of the cubic in s must vanish, or ABC + 2A'B'G'-AA'*-BB' 2 - CG' 2 = 0, which we saw in Art. 84 indicated that there was not a defi- nite centre. The equation becomes Qy 2 + Rz 2 + 2P"x + 2 Q"y + 2R"z + F= 0, and by changing the origin we can get rid of the terms in y and t, and the constant term; the equation thus becomes Qf + Rz 2 +2P"w = 0, which represents an elliptic or hyperbolic paraboloid ac- OF THE SECOND DEGREE. 10o cording as Q and R have the same or opposite signs, or according as BC+CA + AB-A ,2 -B' 2 -C' 2 , which is the coefficient of s in the cubic, and therefore equal to the product of the two finite roots, is positive or negative. 91. Thirdly, let two of the quantities P, Q, R vanish, which necessitates the two conditions, ABG + 2A'B'G f - AA' 2 - BB' 2 - GO' 2 = 0, BC+CA + AB-A' 2 - B' 2 -C' 2 = 0. The equation now becomes Rz 2 + 2P "x + 2 Q"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 2 + 2P"x + 2Q"y = 0. By turning the axes of x and y round in their own plane, the equation can be reduced to the form ik 2 +2P'"a = Q, which represents a parabolic cylinder whose generating lines are parallel to the axis of y. The two conditions ABC+ 2A'B'C - A A' 2 - BB' 2 - CC' 2 = 0, BC+CA + AB-A' 2 -B 2 -O' 2 = 0, can be replaced by simpler ones. For the first equation is equivalent to either of the forms (CA - B' 2 ) (AB - G' 2 ) = (B'G' - AA') 2 , (AB-C' 2 )(BG-A' 2 ) = (G'A'-^BB') 2 , (BG - A?) (GA - P' 2 ) = (A'B - GGJ, whence it follows that the three quantities AB — G' 2 , CA—B' 2 , BG — A' 2 have all the same sign, and therefore if their sum vanishes they must vanish separately, and we must have BG-A' 2 = } CA-B ,2 = 0, AB-C' 2 = 0. 106 THE GENERAL EQUATION We must also have 0C'-AA' = Q t CA'-BP-mO, A'B'-CC' = Q, but these are included in the former. 92. If only one of the quantities P, Q, B, as P, vanish, and P" also vanish, the equation becomes Qy* + Bz l + 2Q"y + 2R"z + F m 0, which can be reduced to the form Qf + Rz^ + F'^O, and therefore represents an elliptic or hyperbolic cylinder ac- cording as Q and li have the same or opposite signs, that is, according as BC-A'*+CA-B' 2 + AB-C* is positive or negative.' If Q, B and F' have all the same sign the locus is an impossible one. The condition that P" may vanish is, that A!'l x + B"m x + C"n x should vanish, where l t> m v n x 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" C'A' - BB' ' A'B' - CC ' so that we get I g | Q" -o B'C -AA! ' C'A' - BB' ' A'B' - CC This condition may be obtained in another form from the consideration that the equations A"l l + B"m 1 +C"n 1 = Al x + C'm l + B'n x ="0 C% + Bm 1 + A\ = B'l x + A'm x + Cn x =0 (1) OF THE SECOND DEGREE. 107 must be all satisfied by the same values of l x , m x , n x , and the requisite conditions that this may be the case are ABC+ 2A!BC - A A' 2 - BE 2 - CC 2 = 0, united with any one of the. set, A" {CC - A'F) + B"(B" - CA) + C" (A A' - EC) - 0, A" [A' 2 -BC) + B"(CC - A'B) + G" [BE - CA') = 0, A" [BE - CA') + B 1 (A A' - B C) -f G" {C 2 - AB) M 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 tne discriminating cubic as P and Q vanish, and P", Q" also vanish, the locus reduces to Bz 2 + 2R"z + F=0, which represents two parallel planes'. The conditions for the two roots vanishing are BC-A' 2 = 0, (Li-E ,2 =0, 4&-tf*«0. : ....(2), and l x , m lt n x are only restricted by the equation Al x +Cm x + B'n x = I (3), with which the other two equations in (1) Art. 83 become identical. If we have also A"l x + B"m x + C"n x = 0, for all values of l XJ m x> n x consistent with (3)' we must have A^_B^_CT A ~ C~W> , m A" W G" or from (2) — - ^- ^ 94. On the whole then we have the following results. I. If ABC + 2A'BC - AA' 2 - BE 2 - CC 2 be not zero, the equation represents an ellipsoid, a hyperboloid, or an impossible locus, with the cone as a variety of the hyper- boloids. A Rp\ or the X NIVER8ITY / 108 THE GENERAL EQUATION II. If ABC + 2A'B'C - A A' 2 - BB' 2 .- CC' 2 vanishes, the equation in general represents an elliptic or hyperbolic paraboloid according as BG + CA + AB - A! 2 - B' 2 - C' 2 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' 2 , CA-B' 2 , AB-C' 2 all vanish, the equation represents a parabolic cylinder which may degene- 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 ABC + 2A'B'C - AA! 2 - BB' 2 - CC' 2 = may be effected by writing it in the form (Ax + C'y+B'z) 2 +(AB- C' 2 )y 2 +2(AA'- B'C')yz + {CA-B' 2 ) z 2 + A (2A"x + 2B"y + 2G" z + F) = 0, AA'-B'C CA-B' 2 or putting JB^C* =P = A4.rrBff ' (Ax + C'y + B'z) 2 -f {AB - C' 2 ) {y + pz) 2 + A (2A"x + 2B"y + 2 C" z + F) = 0. And if we take as co-ordinate planes the planes Ax + C'y -4- B'z = 0, y+pz = 0, 2A"x+2B"y + 2C"z+F = 0, this equation will in general assume the form Py 2 +Qz 2 + Bx=0, which represents one of the paraboloids. The axes are not however rectangular. The exceptional cases can be deduced OF THE SECOND DEGREE. 109 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. 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=0 and $'=0 be the equations of the two surfaces, and lx + my + nz — p = 0, or a=0 ; the equation of the plane of intersection. Then the curve in which a = cuts the surface S = 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 S must be identical with kS' + olj3, where k is a constant and /3 a linear function of x, y, z. Hence when S = and S' = 0, we have a = or j3 = () t that is, all the points of intersection lie in one of the two planes a = 0, or /3 = 0, EXAMPLES. CHAPTER VII. 1 . Investigate the nature of the surfaces, (1) 1x % 4- 5y* + Sz 2 + 2yz - 8zx - 2xy -1=0. (2) x 2 + 4y 2 - z* - 2yz - zx + kxy + 2z = 0. • 2. Interpret the equations : (1) yz + zx + xy — x — 2y — 3z + 2 + a — 0. (2) x 2 + 2y 2 - Sz 2 + 2yz - izx - 2xy + dx = 0. (3) x 2 + 9y 2 -6.ry + 2y- 4.3=0. (4) x 2 + f - z 2 + 2yz + 2zx - 2xy + 2x +2y + 2z = a" 110 EXAMPLES. CHAPTER VII. 3. Shew that the two surfaces whose equations are {h 2 + b 2 + c 2 )x 2 +{h 2 + c 2 + a 2 )y 2 +(h 2 + a 2 + b 2 )z 2 — 2bcyz — 2cazx — 2abxy = 1, and (cy — bz) 2 -f (az — ex) 2 -f (bx — ay) 2 = 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 fi in the equation x 2 + 2y 2 + 2z 2 - (2 - 2/m) yz - 2zx = c 2 . 5. Find the nature of the surface a 2 b l c z be ca ab a b c 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 + d 2 = is 12d 2 (Ex. I, 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 2 + 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 2 + By 2 + Cz 2 + 2A'yz + 2B'zx + 2 G'xy = 1 by the plane Ix -f- my + nz = 0. 9. Find by the method of Art. 78 the axes of the section of the surface Jx + Jy + Jz = by the plane lx + my + nz = l. 10. If the equation ax 2 + by 2 + cz 2 + 2b' zx + 2c xy + 2a"x + 2b"y + 2c" z + d= represent a paraboloid of revolution, prove that c — b±a* If El EXAMPLES. CHAPTER VII. Ill the upper sign be taken, prove that the equations to the axis are cz + c" = 0, {ex + a") Ja+ (cy + b") Jb = 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 2 + By 2 + Cz 2 + 2A'yz + 2B'zx + 2C'xy = 1, by the plane Ix -f my + nz = 0, will be a circle if Bn 2 +Cm 2 - 2A'mn Cl 2 +An 2 - 2B'nl _ Am 2 +Bl 2 - 2C'lm m 2 + n 2 n 2 + l 2 F + m* ~~* 14. Shew that the axes of the surface Ax 2 + By 2 + Cz 2 + 2A'yz + 2B'zx + 2C'xy = 1 lie on the two cones C'O 2 - 2/ 2 ) - B 'y z + ^ zx ~(A-B)xy = 0, A\y 2 - z 2 ) -(B-C)yz- Uzx + B'xy = 0. 15. A cone whose equation referred to its principal axes is a V + /3y=(a 2 + /3 2 )s 2 , is thrust into an elliptical hole whose equation is Shew that when the cone fits the hole its vertex must lie on the ellipsoid a> + b i + * W W~* * CHAPTER VIII. ON TANGENT LINES AND PLANES. 9G. 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 F{x,y,z) = (1). And let the equations of any straight line through P be x— x y' — y z—z ,_, —7— = ^ — *«• --r (2), where x, y\ z' are current co-ordinates. To find the points where (2) meets (1) we must substitute x + lr 7 y + mr, z + nr for x, y, z in (1) j we thus get the equation F (x + lr, y + mr, z + nr) = 0; >io ON TANGENT LINES AND PLANES. 113 dF dF dF^ r d d d + rd* -r+™in. + n ji\ F ( x > v> z ) 2 \ dx dy dz\ + supposing F (x, y, z) to be of the p th 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 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 n l dx- +m Wj + n dz-° 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 _i/ — y _z' — z I m n Consequently for all points in any such tangent line we have ., .dF dF , , .dF . whence it follows that all the tangent lines in general lie in a plane whose equation is (5). 97. It may happen that at a given point of a surface the ., .... dF dF .dF „ three quantities -y- , -— and -j- all vamsh. ax dy dz A. G. 8 114 ON TANGENT LINES AND PLANES. 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 n tfF. ,d 2 F ^ 2 d 2 F F da? +m W + n ^ ■ d 2 F a , d 2 F , 07 r d 2 F A n . + 2mn - T -^ r +2nl- i — i -+2lm^ — =- =0 ... (1), dydz dzdx dxdy w three values of r will be zero, and the straight lines whose direction-cosines satisfy this equation meet the surface in three coincident points ; eliminating I, on, n, we have as the equation of the locus of all such straight lines d 2 F . , , so d 2 F . , , „d 2 F ^ -<»+«-»■» #***-« dz* + 2(y'-y)(z'-z)^ + 2(z'-z)(x' dydz K * * dzdx +tW^V-f)££mQ (2), which is the equation of a cone of the second degree whose vertex is at the point (a?, y, z). See Art. 34. A point at which -=- , -j- and -j- all vanish is called a (XX Ctlj CLZ 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 2 F ,d 2 F , ,d 2 F dx 2+m df +n M d 2 F a 7 d 2 F , oz d?F + 2mn - 7 — =- + 2nl -, — r + 2lm dy dz dz dx dx dy a). Kl ON TANGENT LINES AND PLANES. 115 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 I: mm, 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 z m f(& y)> or * -f( x > y) ■ o. T ,,. dF, df dz dF, dz In this case -y- becomes •*- -f- or — r* * -f- becomes — -y- , ax ax ax dy dy and -=- becomes unity. The equation of the tangent plane becomes therefore It is usual to denote the quantities -y- and ,- by the letters p, q, and the quantities -3-5 , j^ , ■ by the letters r, t, s, respectively. 100. The equation of the tangent plane being the length of the perpendicular on it from the origin is dF dF dF dx * dy dz ._. (1). A dF\ 2 (dFV (dF\* dx) + {dy) + \dz) 8—2 116 ON TANGENT LINES AND PLANES. The letters U, V, W are frequently used to denote dF dF dF dx ' dy y dz ' and the letters u, v, w, u', v, w to denote #F d*F d*F d 2 F d 2 F d 2 F dx 2 ' dy 2 ' dz 2 ' dydz' dz dx ' dxdy' respectively. With this notation the above expression be- comes Ju 2 + v 2 +w 2 \ } ' If we take the form of the equation in Art. 99, the length of the perpendicular is z-vx-qy Jl+M 101. As an example take the tangent plane at any point (x, y, z) of an ellipsoid whose equation is a b (i). TT 7T 2iZr -IT 2« TJr 2z Here U=- 2> F=/ ; JF= ? ; and the equation of the tangent plane is x'x ■ yy , zz x 2 y 2 z 2 * The equation of every plane can be expressed in the form \x + py' -f vz = p (3), where p is the length, and \, fi, v are the direction-cosines, of the perpendicular on it from the origin. ■i ON TANGENT LINES AND PLANES. 117 If we suppose (2) identical with (3), we get w, x y 1 aX b/j, cv Ja?\* + &V -t- cV / Kx , , 72 2 =-* or^ = — = — = — = * — , i ■— ; = VaV + 6 V + c V * 3? 5 /a? 7 i a 6 c Vo , + F + c J And the equation of the tangent plane becomes W + ^ + **' P JaW + 6 V + cV (5), a form which is often useful. The length of the perpendicular on (2) from the origin 1 7 X l + V - + - a* + 6 4 c 4 The values of \, /i, v the direction-cosines of this perpen- dicular are *5 , ~ , ^ by (4), and the co-ordinates of the (X c ^2^. ^2,,, „%. foot of this perpendicular are consequently 1 ~^ , ^ , ^-g- . 102. The equation of a paraboloid being f * J + T = x the equation of the tangent plane at (x, y, z) becomes { x >-x)-*f{y'-y)- 2 *(z'-z) = Q i , 2y , 2z , 2y 2 2z* x -T' y -T' z=x -l-T = -*> 2y , 2z , , ftts or . -j- .y'+ j.z =x+x (2). f+rr (1) - 118 ON TANGENT LINES AND PLANES. This can be put into another form, for comparing it with \x + fiy' -p vz —p, ii v X j»p l V p , III Vv »"*•;»"*»'?— *L« and therefore from (1), W + IV _ ~p . Itf + l'v* . 4A. 2 ~ X' />i? ~ 4X ' and the equation of the tangent plane becomes \x +fiy + vz = — - — (3). 103. The normal to a surface at any point is the straight line drawn through that point 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 tf- y _ z'-z m dF " dF dF { } ' dx dy dz These are therefore the equations of the normal. The equations of the normal to an ellipsoid at the point (x, y, z) are o > M = y(y , -y) jfiJ-z) x y ' z ON TANGENT LINES AND PLANES. 119 If we take the equation of the surface to be *»/(*, y)i 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) = Q\ y-y + q{z-z) = o]"" W 104. The equation of the tangent plane to a surface F(x,y,z) = (1) at the point (x, y, z) is . , . dF . , . dF , s N dF n If this plane pass through a point whose co-ordinates are a, ft, 7, we have <— )5 + (^-»)f+^-)J-o (2). This relation is satisfied by the co-ordinates of all points, the tangent planes at which pass through a given point ( a > A 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, ft, 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 th degree, and let us assume F(x, y, z)=u p + 1^-1 + m m + ••• + u * + u i + u o> where u p , u p _ x ... denote the terms of the p th , (p — l) th ... de- grees respectively. Then the points of contact are determined by (1) and (2), and the latter may be written dx dy ' dz dx ^ dy dz ' 120 ON TANGENT LINES AND PLANES. But by a well-known theorem (see Todhunter's Biff. Calc. Chapter VIII. Ex. 3), du n . du du p •af+itf+'a?-^ du„_, du„_. du, , _ N dF dF dF , _ 6 •'' W fa' Jry ~d& + Z di = pUp + (j) ~ 1 >p-i + '" + 2w t +w x ...( 3). But for all the points of contact we have therefore = ^w p + pty,| + . . . + pu 2 + pu x +pu Q (4). . Subtracting (4) from (3) we get dF dF , dF a , ON % - X fa +y dy + *dz = ~ u *-~ 2u *-~' "~if-% V" (P-^-P 1 ^ and equation (2) becomes AT dF dF dF . ., , 1Nth , Now -T- , t- , -7- are oi the y? — l) m degree, conse- quently (5) represents a surface of the {p — l) th 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 #5 , yy , zz_ a? + b* * e If this pass through a point (a, J3, 7), we. must have a 2 + 6 8 c* {) ' ON TANGENT LINES AND PLANES. 12t 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, ft, 7) with respect to the ellipsoid; and (a, ft, 7) is called the pole of the plane (1). If all the points in which (1) cuts the ellipsoid be joined with (a, ft, 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, ft, 7) be the fixed point, and (x, y, z) the pole of any plane through (a, ft, 7). The equation of the polar of (x, y, z) is ar 6 c If this plane pass through (a, ft, 7) we must have a* "*" 6* + c* ' which shews that (x, y, z) lies on the polar of (a, ft, 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. (1), 122 X)N TANGENT LINES AND PLANES. 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 I m n and the equations of the curve of contact are F(x,y, z)=0 By substituting for x, y y z from (1) in the equations (2) their values a + lr, &+mr, -i - m a be We thufe get w \_ ltd 8m ynX a* 8* 7* f - .,« l^t"^ 7 ' ^ ^ ? : ^\ 444-^ ui, and substituting for r from (3) in (4) we obtain This is the relation which £, w, w must satisfy in order that the straight line 'O' a;— a _y~ $ _z—y _ I m n may pass through some point in the curve of intersection of (1) and (2). The equation of the enveloping cone is obtained by sub- stituting x — a, y — ft z — 7 for I, m, n, and is therefore m f («-a)« (y-8)ff (3-7)7 a , ...(G). 124 ON TANGENT LINES AND PLANES. 109. This equation can be obtained in another form by the aid of the following proposition. Let 8 = be the equation of any surface of the second degree, and let u = 0, v = be the equations of two planes. Then the equation S+\uv = (1), where \ is some constant, will represent any surface of the second degree passing through the curves of intersection of 8 = with u = and v = 0. For if 8' = be the equation of any such surface, it is evident that 8' cannot assume any other form than k (8+Xuv) consistently with the suppositions that it is of the second degree, and ;is satisfied by all values of x, y, z which make 8 and u vanish simultaneously, and also by all values which make 8 and v vanish. Again, if we suppose the plane u = to change its position so as to coincide with v = 0, the equation (1) represents any surface touching 8=0 along the curve in which the latter is cut by v = 0, and becomes 8±\v*=0. Hence the equation - 2 + p+ c ,-l+X^ ¥ +- r -lJ -0 (2), represents any surface of the second degree touching the ellipsoid at all the points of contact of tangent planes through (a, /3, 7). If we take X such that (2) shall pass through (a, ft, 7) it must represent the enveloping cone. Substituting a, fa, 7 for x, y, z, we get ^ * + g + 2_ 1+x g + | + V_ 1 Y =0 . a 2 ¥ cr \a c J Whence the equation of the enveloping cone becomes [a' + ¥ + ?~ x )w + v + 7- 1 ) - w + v + ? V -™ This equation can of course be deduced from that of the last article. *_ • M - ^ , o ^ Mi-"- ! ON TANGENT LINES AND PLANES. 125 110. If we suppose the point (a, ft, 7) to recede from the origin to an infinite distance, the cone will ultimately become a cylinder whose generating lines are parallel to the line joining (a, ft, 7) with the origin. This is called an en- veloping cylinder, and the equation of any such cylinder can be found from that of the cone, by putting a = \k, ft = fik, 7 = vk, where X, fi, v are the direction-cosines of the generat- ing lines, dividing by the highest power of k, and then mak- ing k infinite. The equation of the enveloping cylinder of an ellipsoid deduced in this manner from either of the equa- tions in Arts. 108, 109 is W + P. + e V W & 2 J) w + v + cv • 111. The equation of the cylinder which envelopes a given surface F(x,y,z) = (1) can however be obtained independently of the enveloping cone. For let \, fi, v be the direction-cosines of one of the gene- rating lines; x, y, z the co-ordinates of the point where it touches (1). Then since this generating line of the cylinder is a tangent line to (1) at (x, y, z), we must have ; dF dF dF A , as X dx^ fl d^ + V dz- =0 ' (2) * Vjth tvu U/Z This equation combined with (1) gives the locus of the points at which the enveloping cylinder touches the surface, and we have only to find the equation of a cylinder with its generating lines in a given direction, and passing through the curve given by (1) and (2), which can be done as in Art. 35. If x, y', z be the co-ordinates of any point in the gene- rating line which touches (1) at the point (x, y, z), we have x — x y —y z ' — z , ~-r — = » = = — k suppose, A /JL V or x — x + \Jc, y — y' + /*k, z — z + vk. Substituting these values of x, y, z in the equations (1) and (2), and eliminating k between the two equations, we get 126 ON TANGENT LINES AND PLANES. a relation between x, y, z which is the equation of the en- veloping cylinder. 112. In the case of the ellipsoid, the curve of contact is determined by the equations a 2 + 6 2 c 2 ' Putting x' + \k, y + fik, z' + vk for x, y, z we get Substituting for k from the second in the first we get a* + ¥ + ? X ) W + V + cV V a 2 + 6' cV ' the same equation as we obtained in Article 110. 113. Let the equation of a surface be given in the form *(a,A 7 ,8) = 0.. (1), where a, & 7, S are the lengths of the perpendiculars from any point on the four faces of a tetrahedron, and let any straight line be drawn through the point (a, ft, 7, 8). Then if a', ft', 7', B' be the values of a, /3, 7, S for any other point in the line we shall have by obvious geometry <^ = ^ = ^ = ^ = jt (2); I m n q where I, m, n, q are the cosines of the angles between the line and the perpendiculars on the four faces of the tetrahe- dron, and k is the distance between the two points. We obtain the value of k for the points where the line (2) meets (1) from the equation ((z + lk, ft + mk, 7 + n/j, S + ^) = 0, ON TANGENT? LINES AND PLANES. 127 + Ak 2 + Bk* + ... = (3). This equation gives as many values of h as the degree of the equation (1). Since (a, /3, 7, 8) is a point on (1), (a, ft, 7, 8) vanishes, and one value of k is zero. If l t m. n, q be restricted by the relation two values of k vanish, and the line (2) is a tangent line to (1) at (a, /3, 7, 8), Hence eliminating Z, m, n, q by means of (2) the equation of the locus of the tangent lines at (a, A % S) is or , dd> &d& . ,dd> , ~ dd> dd> a d& , d6 t ~ d But the expression ^ (a, /3, 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 p th degree, we have by a well-known formula since the point (a, /9, 7, 8) is on the surface (1). Hence the equation of the tangent plane at (a, ft, 7, 8) becomes + fc 2 sin 2 sin 2 <£ + c 2 cos 2 0. 4. Find the equation of the locus of the foot of the per- pendicular from a point (a, j3, 7) on the tangent planes of the ellipsoid x* f z* , ■ — \- — -\ — = 1 5. Find the equation of the locus of the poles of all tangent planes of the ellipsoid a 2 + 6 2+ c 2 with respect to a sphere whose centre is at the point (a, /?, 7) and whose radius is k. 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. 7. If normals be drawn to an ellipsoid d i + t>' + c' x EXAMPLES. CHAPTER VIII. 120 at the points where it is cut by the cone I m 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, b, 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 a* b* c* 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 x 2 y 2 z % , — v — ^ — =1 a 1 T V + c 2 ' b being the mean semi-axis, are represented by the equation (b 2 -c 2 )x 2 -(c 2 -a?) f+(a 2 -b 2 )z 2 ± 2 (a 2 -b 2 )*(b 2 -c 2 )hx=(a 2 -c 2 )b 2 . 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. A. G. 9 130 EXAMPLES. CHAPTER VIIL 14. Find the equation of the tangent plane to the sur- face xyz = a s , 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 oo -f y + z s = 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 1 + Cz 2 m x, whose vertex is at a point (a, /3, 7). 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 Bf+Gz 2 = 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 all lie on the surface ax (my - nz) + /3y (nz - Ix) + yz (Ix - my) = x* (my - nz) + 2/ 2 (nz — Ix) + z 2 (Ix — my), EXAMPLES. CHAPTER VIII. 131 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, ft, y) to the surface . a?-\ tf +z* - Sxyz = c 3 , 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, /3, 7) to the surface xyz — a 3 , 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 By 2 +Cz* = x. 24. Shew that the plane Ix + my + nz = will touch the cone Ax* + By 2 +Cz* = if I, m, n satisfy the condition l 2 m 2 7i 2 \ A + B + C=°' 25. Shew that the axes of a central section of the ellip- x 2 y 2 z* soid -2 + Ta + -2 = l by a plane parallel to the tangent plane a c at (a, /3, 7) are given by the equation r * _ (a 1 + 6 2 + c 2 - a 2 -/3 2 - 7 2 ) r* + ~f = 0, where p is the perpendicular from the centre on the tangent plane. 9—2 CHAPTER IX. ON CURVES IN SPACE. 114. We have seen (Art. 16) that any two equations f 2 (x, v ,z)=o] y>> 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 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 in (1). For in- stance, if the two equations (1) were of the first and second degrees respectively, by eliminating y 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 = + (t), z = X (t) (3). This third form possesses many advantages from its sym- metrical character, and we shall in general use it. ON CURVES IN SPACE. 133 115. As an example the pair of equations Ax + By+Gz=B A'x + By + C'z = D' (1) represent a straight line. Eliminating y and z in turn we get the two equations A'B-AB B'D-BD'^ B'C-BC ' B'G-BG' = a a - ga' an - cd \ (2), BC-BC ' BC-BC j which correspond to the form (2) in the last Article. Lastly, assuming x = {B'G — BC) t, we get CTY — C'D~\ x = (EC- BC) t,y = (C'A- GA') t + ^ _ B(J . \ (••••(3), z = { A'B-AB*)t + *^\ 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 angle at the base of the triangle 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, the angle AOM between the axis of x and OM the projection of OP on the 134 ON CURVES IN SPACE. plane of xy, and a the acute angle at the base of the triangle. We obtain without difficulty, x= OiV= OM cos = a cos } y =MJSf= OM sin = a sin 0, z — PM = arc AM x tan a = a0 tan a, Whence or if a tan a = c, x — a cos 0, y = asm0, z = c0 z . z x = a cos - , y = a sin - , C G (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 * = £(*), y = f(t), z = %(0 (1), and let t and t + r be the values of t for two points on the curve. The equations of the straight line joining these are OX CURVES IN SPACE. 135 x, y\ z being current co-ordinates ; x'-${t) j/-jr(t) Z'-X(t) 4>(t + r)-(t) jr(t + r)-f(t) x (t + r)- X (t) - T T T But when t is diminished indefinitely the two points coincide and the straight line joining them becomes the tangent at (x, y, z). Also the limit of -^ ' — zA± j s T dx ' (t) or -j- , and similarly for the other denominators. Hence the equations of the tangent at (x, y, z) are x'-xy'-y z'-z dx dy dz * * dt di dt 118. The length of the chord joining two points (x, y, z) and [x ti y lt z x ) is J\x x - x y + (y x -yy + (z t - z) \ But by Newton (Section I. Lemma vn.) 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 s + 8s be the lengths of the arcs measured from some fixed point up to the points (x, y, z), (x v y lt z x ) respectively, the fraction becomes ultimately equal to unity, or -(th&Ht) » 136 ON CURVES IN SPACE. From this result we see that the cosine of the angle which the tangent at (#, y, z) makes with the axis of x, which by Art, 17 is dx di JWWW dx , , dt dx is ecjual to -r or ^-. It And similarly, the cosines of the angles which the tan- gent makes with the axes of y and z are -¥■ and -7- re- & 9 ds ds spectively. Dividing by (-77) the equation (1) reduces to the form ©MtHI)'- 1 »• 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, y, z) perpendicular to the tangent, which is called the normal plane. Its equation is at once seen to be (rf-.)g+V-*)| + ('-.)£-a 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 oscidating plane at that point. Let the equations of the curve be *=*(<). r-jHft 9m x$ CO* and let t, t + r, t+2r be the values of t corresponding ON CURVES IN SPACE. 137 to three points on the curve. Let the equation of any plane be Ax +By'+Gz=D (2). If this plane pass through the three points t, t + r, t + 2r, we have A+ (0 + Bf {t) + C x (t) =2> (3), A(f>{t + T)+Byjr(t + r) + C X (t + r)^D (4), A (t + 2r) + Byjr (t + 2r) + C X {t + 2r) = D (5). Subtracting the first of these equations from the second we have A{4>(t+T)-(t)}+B{+(t+T)-+(t)} + C{ x (t+T)- X (t)}=0. Or, dividing by r, , g X(^T)- X (<) _ ft T Subtracting twice the second from the sum of the first and third and dividing by r 2 , we get A (t+2r) -2 (t+r) +{t) B f(t+2r)-2yjr(t-hr)+ylr(t) T T ., c X(t + ^)-2 x {t + r)+ X (t) ft But if we make the three points coincide, t vanishes, and these two equations become (Todhuriter's Biff. Gale. Art. 127) A dx t ■„ dy t „dz A A B G dy d 2 z dz d 2 y dz d 2 x dx d 2 z dx d 2 y dy d 2 x didf~diW di ~df " ~dt ~d?. di W ~ di W 138 ON CURVES IN SPACE. And subtracting (3) from (2) we have A (V - x )+B(y'-y) + C(z' - z) = 0. Whence the equation of the osculating plane at the point (a, y, z) becomes (x' - or) [%L d * z _ dz &, 4. tfJ-\ [ d ± d * x __ ^ d * z \ r x) \dt df dt dt*\ + {y y) \dt df dt de] ; f , Adx d?y dy d 2 x) 121. The osculating plane is sometimes denned 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 x , y x , z x ) is A (x x -x)+B (y x -y) + C(z x -z) JA* + B*+G? But if (x x> y x , z x ) be a point on the curve corresponding to a value t + r of t, dx t 2 d 2 x X > = x + T di + £W + , dy , t 2 d*y , y^y +r dt^J + dz t 2 d*z z ^ z + T Tt + ]2de + Hence the length of the perpendicular becomes JA 2 + B 2 + C* ON CURVES IN SPACE. 130 And when t 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 when A, B, G are determined by the equations 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. These are (-*)t + (/- y )i + (*'-)g=o, {x '\dt df dt df] + Ky y) \dtdt* dt df) If we put these equations in the form x — x _y — y _z' — z * * q b : the value of P is dy (dx d 2 y dy d 2 x\ dz (dz d 2 x dx d 2 z\ di (di W ~ di~d?) " di [dt ~d? " di ~dt 2 ] _ dx (dy d 2 y dz d 2 z\ d 2 x (/dy ~dt\didf~*~di df]~"df \{di. 2 'dz + \dt But by Art. 118 'ds\ dt) ~\dt) ! \dt) ' \dt ds\ 2 _fdx\ 2 fdyV fdz^ 2 ~\dt) + {dt) " 140 ON CURVES IN SPACE, therefore differentiating, ds d 2 s _ dx d 2 x dy d?j£ dz d 2 z p- -p_dx (ds d 2 s dx d 2 x\ d 2 x {fds\ 2 Hence j?-—^ ^ -^ jp|- ^ _ cfe fcfaj JV cfo d?x\ ~~dt\dt de~~dt~Ey v((ds\ 2 _(dx^ F |W \dt) and similar values may be found for Q and R. Hence the equations of the principal normal are . x —x y —y ds dfx d' l s dx ds d 2 y d?s dy ds arz d 2 s dz * dtWdfdi di'WWdi lft~d?~df ~dt which may be written in either of the forms X — X _ y -( dt\ -y . 'dy\ K ds) z' -z d dt fdcc\ {ds~J d fdz\ dt \ds) x — x ^y '-y z — z (1), or ^ r ^=^_^ = L»-l (2). d 2 x d l y d 2 z v ' ds 2 ds 2 ds 2 123. The equations (2) of the last article can also be obtained as follows. If X, /n, v; X', fi t 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 X-f-X', fjL + /j,', v + v and X — X', p—p, v — v\ For planes through the origin perpendicular to the two given straight lines have their equations \w + /j,y + vz = ..(1) and X'x + fiy + v'z — 0. (2) respectively. ON CURVES IN SPACE. 141 By Art. 26 the equations of two planes which bisect the angles between (1) and (2) are (X - V) x + (/* - //) 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 + X\ fM + fiy v + v and X — V, p — p, v — v respect- ively. If I, m, n be the actual values of the direction-cosines of the latter line, we have x-V z = V(\ - xy + o*-y) 2 + {v-yy X — X f l , Vv = /o o d = 9 ( X ~* X ) C0SeC 9 » V2 - 2 cos 2 ^ if be the angle between the two straight lines. 124. Let now X, fi, v be the direction-cosines of the tangent to a curve at the point (%, y, s)> and V, fi t v their values at an adjacent point on the curve distant 8s 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 + X f , fi + p, v + v, and the latter must have its direction-cosines proportional to X — X' } /jl — fj! , v— v . But X r — X + -T- 8s + terms involving (8s)* H= /J , + ls &s + dfj, ds dv ' = v + — 8s + 142 ON CURVES IN SPACE. Hence the direction-cosines of the principal normal are , , dX /dx d 2 y dy d 2 x\ dtWdi'dt 2 ) : [dt di 2 ~~~dt df) '' [di'df "didfj must be constant for all points on the curve. We may therefore assume dy d 2 z dz d 2 y dtdf'Jt d?~ Xv (1) ' dz d 2 x dx d 2 z _ . di'dt 2 ~~dtdi z ~ fiv {Z) ' dx d 2 y dy d 2 x _ r ~dtdf~~dtdf~ VV W' where X, \i, v are constants, and v some function of t. Eliminating X, fi and v from (1) and (2) by differentiating, we get dz d?x dx d?z\ (dy d s z dz d\ didf di WJ \di~df~"dt 3? /dy d?z dz_ d 2 y\ /dz d 5 x dx d 3 z\ _ U* df * dt df) \did?~didf)~ °' ON CURVES IN SPACE. 143 dz or reducing and dividing by -r. , 'dy d?z dz d 2 y\ d z y /dz cfx _ doc d 2 z\ ^W^Ttdf) + df\dtdf didf) d 3 z (dx d?y dy cfx d»x df drz /dx df\dt df dt df 0, which may be written dx dt' dy dt' dz ~dt d 2 x df d?y df d 2 z df = d 3 x df d?y df d*z df (4). The symmetry of this relation shews that we should gat 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 perpendicular on this plane from any point in the curve must vanish. We must therefore have j dx r>dy ~dz dt dt dt 0, A df +B df+ C df-°> A d :* + B^ + C-, = 0, df ' ~ dt' whence the relation (4) follows d?z df We must also have #+»»♦<« 0, 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. 14-4 ON CURVES IN SPACE. 126. If a curve be drawn 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(fr$,g)mQ (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 . , , s dx dy dz any point (x,y,z) are ^, £,■£-. The equation of the tangent plane to (1) at (x, y, z) is .dF , , .dF /f s dF A and the direction-cosines of the line of intersection of this plane with the plane (2) are proportional to B dF_ c dF c dF_ A dF_ A dF_ B dF dz dy ' dx dz' dy dx ' 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 ds \ dz dy ) ds\ dx dz ) ds\ dy dx) '"^' 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, A = 0, B = 0, and the equation (3) becomes dF dy_dF dx _ dx ds dy ds ' dF^y__dF_ () (4 s dx dx dy~ " ^ '" ON CURVES IN SPACE. 145 As an example of the last case take the equation of the ellipsoid & f £ i Equation (4) becomes a 2 dx V > r • ~i l°g 2/ — p log # — constant ; .*. y = mx b * (6). This equation united with (5) gives the lines of greatest slope. If a = b, (6) becomes y = mx, so that in the case of a spheroid the meridians are the lines of greatest slope to the plane of circular section. 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. Through p let a plane be drawn perpendicular to PQ and through q, r, s ... planes perpendicular to QR, R8, ST,... respectively. These will ultimately be normal planes to the curve at consecutive points. Let the planes through p, q A. G. 10 146 ON CURVES IN SPACE. 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 O t , and the plane through Q, B, 8 meet BF in 2 . It is evident that the point 0. is equidistant from P, Q and i£, and a circle with centre 0. and radius O x P will pass through Q and JR. This circle will ultimately pass through three consecutive points of the curve, and lies in the plane PQBO v 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 O x 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 B. All points in the straight ON CURVES IN SPACE. 147 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, B, 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 centre of spherical curvature, and the length AP the radius of spherical curva- ture. The lines AE, BF, CO ... 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 t be any point in AE, and let pO iy qO t be joined and qO t 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 = iq ; .*. u0 1 + O l p = uq, vu + uO x + x p = vu + uq = vu + ur = vr, Hence if a string be laid along the curve wvuO t 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 f(0 = o (i). Any other normal plane can be represented by J(OH> • ( 2 )> where t t is the corresponding value of t, 10—2 148 ON CURVES IN SPACE/ At the points where (1) and (2) meet, we have F( tl )-F(t)=0, ^W-fQ-o. t x -t And this latter equation when t t — t is indefinitely diminished becomes g-° % Hence the line of intersection of two consecutive normal planes is given by the two equations F«>-«, f-0. Bit i-<<).(«'-«)J +( y- 5 )4 + (/-«)|, f=(«'-.»S + z ) a) = (1), where a is a constant. If a be changed to a we obtain the equation of another surface i^,y,s,a')==0 (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 a! 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(aj,y,*,a)-0, F(x, g, z, a) - F Q, y, z, a) = a' — a But ultimately when a' becomes equal to a these equa- tions reduce to F{x,y t z,a) = (3), ~Ffa%z,a)-0 (4), ON ENVELOPES. 159 which are therefore the equations of the ultimate position of the curve of intersection of (1) and (2). Eliminating 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{x, y, z, a) = 0, F(x,y ) 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 = (j>(x } ?/, *). Substituting in (3), the equation of the envelope becomes F{x > y,z > {x ) y,z))^0 (1). The values of -v- and -7- at any point of this surface are given by the equations dF dFdz dF/a\j) d^_dz\_ dx dz dx d(f> \dx dz dx) ^,^ \dy dz dyj . J .(2). (3). 1G0 ON ENVELOPES. At any point of the surface F (x, y, z y a) = the values of -, and -r- are given by the equations dF dFdz doc dz dx dF dFdz dy dz dy But at the points where the envelope meets the surface F(x } y, z, a) = 0, we have Now Tr only differs from -7- in having <£ (#, y, z) instead of a, consequently at all the points of intersection of the surface dW with the envelope, tt = 0, and the equations (2) become iden- n z (1 z tical with equations (3). The values of -7- and -5- being the same for the surface and its envelope, the two surfaces touch. 137. If the equation of a surface be F(x, v ,z,a,b) = (1), when a and b are constants, any two other surfaces formed by giving new values to a and b will intersect (1) in a point or points, which assume a limiting position when the new values of a and b approach indefinitely near to their first values. The locus of such limiting positions is called the envelope of the surface (1). Let a and b become a + h, b-\-h respectively. The equa- tion of the corresponding surface is F(x, y } z,a+h,b + k)=z0, or F{x, y, z, a, b) + hF (a + 6h)+kF' (b + 6k) = 0...(2), ON ENVELOPES. 1G1 where is a proper fraction and F' (a + 6h) means that F (x, y, z, a, b) has been differentiated with respect to a, and a+ Oh put in the result for a. At the points of intersection of (1) and (2) we have , F(x,y,z,a ) b) = \ hF'(a + 6h) + kF'(b+dk) = 0) K °' 9 and whatever be the ratio of h to Jc, when h and Jc are di- minished indefinitely all the curves of intersection given by (3) pass through the points given by Fix, y, z,a, 6) = 0, F'(a) = 0, F'{b) = 0. By eliminating a and b between these equations we obtain the equation of the envelope. 138. The envelope in this case also touches each of the series of intersecting surfaces. For let the equation of one of the surfaces be Ffay,z,a,b) = (1). The corresponding point on the envelope is given by (1) com- bined with '^ = dF = (2). da ' db U {) From (2) we can obtain by solving for a and b a = & fa y> z )> h = & fa y> z ) ; and the equation of the envelope becomes F{x, y, z, fa fa y, z), fa (x, y, z)\ = 0. d z d z The values of -r- and -r- for any point of the envelope are given by the equations dF d^dz dF_/dfa dfadz\ dF_fdfa dfa dz\ _ dx dz dx dcp t \ dx dz dxj d(j> 2 \ dx dz dxj ' dF dFdz dF/dfa dfadz\ dF_ /dfa d(f> 2 dz\ _ dy dz dy dcf) 1 \ dy dz dy) dfa \ dy dz dy) A. G. 11 162 ON ENVELOPES. But at the points where (1) meets the envelope jl t \ dF n h = (j> 2 (oc,.y,z), rg =0; 'AW dF consequently at those points -j-r- = 0, -j-r- = 0, and the above equations become dF dF dz >dx dz dx ' dF dF dz_ dy dz dy ' which are the same as the equations which give -z- and -r- for the point (a?, y, z) of the surface (1). Hence at the points where (1) meets the envelope the values of -j- and -j- are the same for the surface and the envelope, which therefore touch one another at those points. 139. If the equation of a family of surfaces contains n arbitrary constants connected by (n — 1) equations there is really one independent constant, and the envelope can be found by substituting for (n — 1) of the constants their values in terms of the n th . It is better in general to consider (n — 1) of the constants to be functions of the n ih , and dif- ferentiating all the equations to eliminate by undetermined multipliers. If the n constants be connected hy.(n — 2). equations, two of the constants are arbitrary, and the envelope falls under the second class. The method of undetermined multipliers can be used in this case also. For examples of the solution of problems the reader is referred to Todhunter's Differential Calculus, Chapter xxv., the methods employed there being equally applicable to the problems of Solid Geometiy. ON ENVELOPES. 163 140. The polar plane of any point (a, /?, 7) with respect to any quadric can be obtained as in Art. 106. If the point (a, /3, 7) be constrained to lie on any given surface /(*,y.*)-o (i), the equation of its polar plane will contain three parameters a, ft, 7 connected by one relation /(*,/3,7) = 0. The equation of its envelope can therefore be found by the methods of Art. 137. Suppose this equation to be f(*,SM)-0 (2). Then any point (a', /3', 7') in (2) is the limiting position of the point of intersection of the polar plane of some point (a, (B, 7) on (1) with the polar planes of points on (1) adja- cent to {a, /3, 7). Hence by Art. 106 the polar plane of (a, ft', 7') with respect to the given quadric must pass through the point (a, j3, 7) and other points on (1) contiguous to (a, /3, 7), that is the polar plane of (a', 0', 7') is a tangent plane to (1) at (a, /3, 7). Thus the surface (1) is the en- velope of the polar planes of all points on (2) with respect to the same quadric. The two surfaces are from this pro- perty called reciprocal polars. Each surface may be also denned as the locus of the poles of the tangent planes to the other with respect to the given quadric. 141. Let the quadric with respect to which the polars are taken be the sphere, x 2 +y 2 + 2 2 = Jc 2 (1). The equation of the polar plane of any point {a, /5, 7) with respect to this sphere is ax + Py+vz = k* (2). Let the surface to be reciprocated be the ellipsoid £+£+£ = 1 . (3) x \°J- 11—2 164 ON ENVELOPES. Hence we have to get the envelope of the plane (2), a, /3, 7 being parameters connected by the relation M+$-' * Using the method of undetermined multipliers we get to determine the envelope, ft c whence 1 + \Jc* = ; and substituting in (2) the envelope becomes ftV + &y + cV = & 4 (5). The .surface represented by (5) is often called the reci- procal ellipsoid of that represented by (1). EXAMPLES. CHAPTER X. 1. Find the envelope of the series of planes ax + /3y + yz = 1, where a, ft, 7 are parameters connected by the relations a 2 + /3 2 + 7 2 = l, leu -f m/3 + wy = 0. 2. Find the envelope of a sphere of constant radius which moves with its centre on a fixed circle. 3. Find the envelope of central sections of an ellipsoid of which one axis is constant and equal to h EXAMPLES. CHAPTER X. 1G5 4. Find the envelope of planes which are the polars of points on the ellipsoid a 2 + ¥ + c 2 ' with respect to the ellipsoid 5. Find the envelope of a sphere of constant radius which moves with its centre in a fixed plane. 6. Find the envelope of an ellipsoid whose axes are given in direction and the product of whose axes is constant and equal to 8k 3 . 7. Find the envelope of the series of planes Ix + my + nz = v, where I, m, n, v are parameters connected by the relations P + »»' + w" = l, 2 2 m n ~^ „.2 7,2 ~t" „,2 JZ — U. v 2 — a 2 v — o~ v — or 8. Find the envelope of a sphere whose centre is at a point (a, j3, 0), and radius is 7 where a, /3, 7 are connected by the relation k being a constant. 9. Find the envelope of the surface where a, /3, 7 are parameters connected by the relations °L - & - t- a 2 F " 7 2 ' a, b, c being constants. 166 EXAMPLES. CHAPTER X. 10. Find the envelope of all planes which 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 a 2 r V "*" c 2 ' when a, /3, 7 are connected by the relations d l + ¥ ^ c 2 ' h + m/3 + wy = 1. 15. Through a given point (a, /9, 7) a series of chords are drawn to an ellipsoid whose equation is x* y* z* , a b c in such directions that the line of intersection of the tangent jDlanes 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 *J(K 2 -&)ax \V-a 2 )/fy V(a 2 -6 2 )7^ _ n 1 7 "\ " — "• a c CHAPTER XL 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 which always 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 I m n ^ '* Let the equations of the curve through which (1) always passes be y = ${x), z = yjr(x) (2). Since (1) always meets (2) we have 7+ j(x-a)-^(x). And eliminating x between these equations, we shall get a relation between -j and -7 , which can be put into the form n „(m\ r 168 ON FUNCTIONAL AND DIFFEKENTIAL EQUATIONS whence the equation of the cone becomes g-7 _ v(y-P *=f(*^S) (3). a \x — aj ' This is the functional equation of conical surfaces. In all cases it is clear that the equation is homogeneous in % — a, y — ft, 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, ft, 7) is homogeneous in x — a, y — ft, z — y; an extension of the result of Art. 34. A differential equation holding for all such surfaces can be deduced thus. From (3) differentiating with respect to x, dx \x - a/ \x — aj \x — aj and with respect to y, dy \x — oJ dz _ z — 7 =' ■ ' y — ft dz dx x — a x — a'dy' ox ( x -^% +( y-®% =z -i w- 143. To find the general equation of cylindrical surfaces. A cylindrical surface is generated by a straight line which moves always parallel to itself and meets a fixed curve. Let I, m, n be the direction-cosines of any one of the generating lines, and *-£_Z=*_*=f_ P I m n v ' the equations of the line. Let the equations of the directing curve be r=*(j), z=^{x) (2). Since (1) meets (2), we have y + mr = (x + Ir), z + nr = y{r(x + Ir), OF FAMILIES OF SURFACES. 1G0 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 —IY= mx — ly, n Y— mZ— ny — mz. But from (2) mX — IY and n Y— 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, m = -mF'(ny-mz)-^, and differentiating with respect to y -l=^n-m-^)F' (ny-mz), , 7 dz dz whence I -y- = n — m -j-~ , ax ay ? dz , dz ... l dx +m dy = n <*)• If the direction of the generating line of the cylindrical surface be parallel to the axis of y we have I = 0, m = 1, n = 0, and equation (3) becomes x = F(-z) or /(a?,«) = (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. 170 ON FUNCTIONAL AND DIFFERENTIAL EQUATIONS Similarly the equations /(«,y)-0, /(y,*)=6, 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 by a straight line which 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 x — a.y — Sz—y ,, x —7—=- — L = r (1), I m n w and let the equation of the fixed plane be l'x + my + w'2 = 0...». (2). The co-ordinates of any point in (1) can be represented by OL+lr, /3 + mr, 7 + nr, and the equations of any straight line through this point are x — a. — Ir _y — /3 — mr z — 7 — nr ._. \ jM V If this be parallel to (2),. we have \T + pm'+im'=Q (4). From (3) and (4) . I' (x — a)+ m (y — J3) -Hi'(* "~ t) = (^' + mm ' + nn ') r -"(5)i and from (3) eliminating r n\ — lv_ n(x — a)—l(z — y) nfx — mv ~ n (y — J3) — m (z — 7) ^ '" Now the condition that the straight line (3) may meet the fixed curve, combined with (4), will ordinarily enable us to OF FAMILIES OF SURFACES. 171 express - and - as functions of r, and consequently we can arrive at a result of the form = F {{11! + mm! + nri) r), 11/jl — mv or which is the general functional equation of conoidal surfaces. If the fixed plane be taken as the plane of xy, and the point where the fixed line meets it as the origin, we have J' = 0, m' = 0, n'=l, a = 0, £=0, 7 = 0, and the equation (7) becomes J^^=F(z) (8). ny — mz v ' v ' If the fixed line be perpendicular to the fixed plane I = 0, m = 0, w = 1, and the equation of the surface becomes y v * = *© (°)- In this case the surface is called a W^Ai conoid, 145. The differential equation of conoidal surfaces can be deduced from (7) ; for differentiating it with respect to x> we have / dz\ dz v~ l dx) {" (y-®- m ( z -ri}+ m dx { n ( x - a )- 1 (*-w = (l> + n'^)F'{l'(x-a)+m(y-/3)+n'(z- 7 )} ; 172 ON FUNCTIONAL AND DIFFERENTIAL EQUATIONS and differentiating with respect to y, we have {^(2/-/3)-m(s-7)} 2 = ( m ' +w ' s) F l r ( ^~ a) + m '(y~® + »'(*- v)T- and reducing and eliminating F [1! (x-z) + m'(i,-/3) + *! (*-?)} we obtain ( m ' + n ' fi/J* (y-&~ m ( z -v) + -£{ m (x-z) - i(y-P)\l or ??i' {n [y — /3) — m(z — 7)} + Z'{w (x — a) - Z (z — 7)} + j- [m [m {x - a) - 1 (y - ft) } + ri [n (x - a) - I (z - 7) }] + jK^(2/-/3)-m(^-7)} + r^(2/-^)-m(^- a )}] = 0...(10). The differential equation corresponding to equation (8) is obtained by putting a = 0, 13=0, 7=0, JT-ft m' = 0, »'*1, and is therefore ( ^~^^ + ^~ m ^^ ==0 (11 )* The differential equation of a right conoid is obtained from (11) by putting I = 0, m = 0, ?i = 1, and is therefore 4* . &* r, /ion •»**$* °^ W OF FAMILIES OF SURFACES. 173 The forms (11) and (12) can of course be obtained di- rectly from (8) and (9) by differentiation. For instance from (9), differentiating with respect to x } we have dx y ^ \y) * y \y. and differentiating with respect to y, dz _ x , /x\ whence eliminating yjr' f-J, we have dz dz _ n dx ** dy 146. The three classes of surfaces we have considered are all included in the general class of ruled surfaces, that is, surfaces which can be generated by the motion of a straight line. The first and second differ from the third in this, that any two consecutive generating lines in any surface of the first or second classes lie in one plane, whereas this is not in general the case with the third class. Kuled surfaces in which consecutive generating lines lie in one plane are called developable surfaces, while all other ruled surfaces are called skew surfaces. Thus the surface generated by the ultimate intersections of the normal planes to a given curve is developable. Developable surfaces are so named for the following reason. Let a series of consecutive generating lines be drawn. The plane which passes through the first and second line intersects the plane which passes through the second and third line in'the- second line. The first plane may be turned round the second line till it coincides with the second plane, and thus three generating lines of the surface can be made to lie in one plane. Again, this plane can be turned round the third line till it coincides with the plane which passes through the third and fourth lines, and so four con- secutive lines can be made to lie in one plane. In this 174 ON FUNCTIONAL AND DIFFERENTIAL EQUATIONS manner the whole surface can be developed so as to lie in one plane without tearing. Since any two consecutive generating lines of a develop- able surface lie in one plane, any such surface may be pro- duced by the ultimate intersections of a series of planes, and since any two consecutive planes intersect in a line on the surface, the equation representing any one of the series can only involve one arbitrary constant (Art. 134). 147. Let the equation of one of the planes be Ax + By+Cz-D = (1). Then since the equation only involves one arbitrary con- stant, A, B, C, D must be functions of one constant which we may call a. Thus equation (1) may be written *fc to + y& W + **. to -&(*) = o (2), and the envelope is found by eliminating a between (2) and the equation obtained by differentiating it with respect to a, viz. #/ to + yfc'to +#/to - &' to = o (3). To obtain the general differential equation of developable surfaces we must differentiate (2), considering a as a function of x, y, z determined from (3). Differentiating with respect to x } we get or by (3), fc-(«) + g*b(«)-0 (4). Similarly, differentiating with respect to y, we get dz &«+g&(«) = (5). OF FAMILIES OF SUEFACES. 175 Eliminating a between (4) and (5), we get £=/©• <«>■ and differentiating again with respect to x and y in turn, we get d?z _„ fdz\ oW "~* \dy) dx* J \dyj ' dxdy* d?z _ f , /dz\ we get *" «*'* ( d * z V - q dx* ' dy* \dx dy) ~~ ' which is the differential equation of developable surfaces. 143. To find the general equation of surfaces of re- volution. A surface of revolution is the surface produced by the revolution of a plane curve round a fixed straight line in its plane called the axis of revolution. Let the equations of the axis of revolution be I m n ^ '" And let y=f(a) be the equation of the revolving curve when the axis of revolution is taken as the axis of x, and the point (a, ft, 7) as origin. Let P be any point on the surface, PR perpendicular on the line (1), and Q the point (a, @, 7). Then from the definition of a surface of revo- lution, PR=f(RQ) (2). But BQ =* l(& — a) + m (y- £) + n («- 7), since it is the perpendicular from Q on a plane through P perpendicular to (1), and PE 2 = (*-«) 2 + (y-/3) 2 + ( S - 7 ) 2 ~{l(x-a) + m (y-fi+n(fi-y)}\ 176 ON FUNCTIONAL AND DIFFEKENTIAL EQUATIONS Hence ( a; - a ) 2 +( 2 /-/9) !! +(^- 7 ) !! ={Z(a ! -a) + TO(y-/3)+»(^- 7 )P + [f{l (x - a) + m (y - P) + n {z - 7 )}] 2 , or ( fl ._ a )« + (y_ j 8)» + (,_ 7 )i = {*(*-*) +m(y- / 8) + »(*- 7 )} (3), which is the functional equation of surfaces of revolution. The differential equation can be thus deduced. Differentiating (3) with respect to x we get 2{(*-a) + (*~ 7 )J} = { J + n £}*' t^-«)+™(2/-/3)+»(*-7)l. and differentiating with respect to y 2{(j-ffl+fr-7)|} ! w + n ~r \ $' P (^ ~ a ) + m (y ~ £) + n ( z ~ ?) J • dy Eliminating and reducing, we get dz m(x-a)-l(y-/3) + [m {z-y)-n(y- £)} ^ + {n(x-«)-l(z-y)}^ = (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. OF FAMILIES OF SURFACES. 177 Let the equation be Ax 2 + By 2 + Cz 2 + 2A!yz + 2B'zx + 2G'xy + 2A"x + 2B"y + 2C"z + F = (1). If this equation represents a surface of revolution it can be put into the form (x -a) 2 +(y-ft) 2 +(z- 7) 2 = P 9* + my + nz) 2 + Q(lx+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 I (x - a) + m (y - j3) + n (z - 7), or of Ix + my + nz — (la + ra/3 -f 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 Pl 2 -l=kA (3), Pmn = kA' (G), Pm 2 -\=kB (4), Pnl = kF (7), Pn 2 -l=kC (5), Plm = kC (8), where k is some constant. Multiplying (7) by (8) and dividing the product by (6), we obtain jyrn These are the conditions which must be satisfied by the coefficients of the equation. The relations which must subsist between a, ft, 7 are ob- tained by equating the coefficients of the terms of the first degree in (1) and (2). We thus obtain Ql + 2a = 2kA'\ Qm+2j3 = 2kB" t I Qn + 2y = 2kC". A. G. 12 178 ON FUNCTIONAL AND DIFFERENTIAL EQUATIONS Whence *£^J*^J*b=l (10). I 2 m 2 n 2 But -grjj, = ^tj ^-jrg, , and h is given by (9). The three equations (10) being the relations which a, )3, 7 must satisfy are the equations of the axis of re- volution. 150. The preceding investigation fails if the quantities B'C A C'A' D A'B' n .. -j7--A, -g--B, -Q-, G vanish, for then k is required to be infinite. We know that the equation (1) in this case represents a parabolic cylinder, or two parallel planes (Art. 91), conse- quently the surface cannot be a surface of revolution. The investigation also fails if A\ B\ or C vanish. Sup- pose A' = 0. From equation (6), mn = 0; .*. m = orw = 0, and therefore, B' or C must vanish also. Suppose n = 0, and therefore B' — 0, we get then Pirn = hC\ JfeC=-l, and (1+M)(1 + JcB) = PTm 2 m k 2 G' 2 ; .-. (C-A)(C-B)=C 2 , which with B' = is the condition required. The other exceptional cases can be treated in the same way. 151. The differential equations of the different classes of surfaces can be put into a more symmetrical form by the sub- stitutions dF dF dz _ dx dz _ dy dx~~dF_' ty~~2F> dz dz OF FAMILIES OF SURFACES. 179 and corresponding substitutions for the second differential coefficients of z, the equation of the surface being assumed to be Thus the differential equation of cylindrical surfaces becomes jdF , dF t dF ft m l fa +m ~d^ + n ~dz- = ° (1) ' LLJu LIU ULZ The equations can be more conveniently used in this form to discover whether a surface whose equation is given belongs to the peculiar class considered. For instance, if the surface be cylindrical, there must be some values of I, m, n which shall make the expression ,dF dF dF /sn l ^ + m -di + n di (2) vanish identically for all values of x, y, z corresponding to any point on the surface. The conditions that this may be possible will be that the coefficients of the several powers and products of x, y> z in (2) must vanish for the same values of I, m, n. The differential equations can be found independently of the functional. For instance, equation (1) is the algebraical statement of the fact that at all points of the surface F(x,y,z) = 0, a straight line whose direction-cosines are I, m, n is a tangent line to the surface, a condition obviously satisfied by cylin- drical surfaces only. In the case of conical surfaces we at once obtain the dif- ferential equation .dF , a , dF ■ .dF n from the consideration that the straight line joining any point (x, y, z) with the vertex is a tangent line to the surface at the point (x, y } z), 12—2 ( 180 ) EXAMPLES. CHAPTER XL 1. Shew how to find the functional and differential equations of a tubular surface, that is, a surface which is the envelope of a sphere of constant radius which moves with its centre on a fixed curve. 2. Prove that the surface x 3 + y 3 + z 3 - Sxyz m a 3 is a surface of revolution round the line x = y — z. Find the equation of the generating curve. 3. Find the equation of a conoidal surface of which the generating lines pass through the axis of z and are parallel to the plane of xy, and whose directing curve is a circle with its centre in the axis of x and its plane parallel to that of yz. (The Cono-Cuneus.) 4. Find the equation of the surface generated by a straight line which passes through two fixed straight lines at right angles to each other, and also through a circle whose plane is parallel to each of the straight lines and whose centre is at the middle point of the shortest distance between them. 5. Find the equation of the surface generated by a straight line which always passes through the axis of z and some point of the curve x = a cos 0, y — ci sin 0, z = c0; and is parallel to the plane of xy. 6. Find the equation of the surface generated by the tangent lines of the curve x = a cos 0, y = a sin 0, z — c0. 7. Find the equation of a conical surface whose vertex is at any point on the surface of a sphere, and whose base is a small circle of the sphere. EXAMPLES. CHAPTER XI. 181 Find also the curve in which the cone is cut by a plane through the centre of the sphere perpendicular to the dia- meter through the vertex. 8. Find the equation of the surface generated by the revolution of a circle round a straight line in its own plane which does not cut it. 9. Prove that all tangent planes to the surface in the last question which pass through its centre cut it in two circles. 10. A fixed straight line AB meets a fixed plane in A. A straight line AP moves so that the sine of the angle which it makes with AB bears a constant ratio to the sine of the angle which it makes with the fixed plane. Find the surface generated by A P. 11. Find the conditions that the surface Ax' + Bif + Cz 2 + 2A'yz + 2B'zx + 2G'xy + 2A"x + 2B"y + 2C"z + F= may be a cylindrical surface. 12. Shew that with the notation of Art. 100 the con- dition that the surface F(x,y, z) = may be develop- able is U 2 (vw - u" z ) + V 2 (wu - v 2 ) + TP (uv - w 2 ) + 2 VW{v'w - uu)+2WU(w'u - vv) + 2 UV(u'v-icw) = 0. Deduce the conditions that the surface in (11) may be developable. 13. Find the equation of the surface generated by all the normals drawn to an ellipsoid a 2 + 6 2 c 2 ' at the points where it is cut by the cone a b c A -+-+- = 0. x y z 182 EXAMPLES. CHAPTER XI. 14. A surface is generated by a straight line which passes through the axis of z, and the line x = a, z — ; re- maining parallel to the plane y = hz. Shew that its equa- tion is x (y — kz) = ay. 15. Describe the general nature of the surfaces repre- sented by the several equations (1) f(r, 6) = 0. (2) f(r, ) = 0. (3) f{0, ) = 0. 16. Examine the nature of the surfaces represented by (1) r 2 = a 2 cos2<9. (2) r 2 = a 2 cos 2<£. 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 tan -1 2 = z . F(a? + f) +f(x 2 + y"), x and that the differential equation is „ d 2 z - d 2 z o d 2 z dz , dz u dx- J dx dy dy i dx 9 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. EXAMPLES, CHAPTER XL 183 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 x* + y* + z* = 2cxf(^), and that the differential equation is CHAPTER XII. ON FOCI AND CONFOCAL QUADRICS. 152. A FOCUS of a conic section is a point such that the distance of any point on the curve from it can be expressed as a linear function of the co-ordinates of that point. There are certain points which have analogous properties in reference to quadrics, and which may therefore be called foci of quadrics. 153. For instance the equation of the ellipsoid is ^ + t + ? = 1 (1) where we will suppose a, b, c in descending order of magni- tude. Also let e v e 2 , e 3 be the excentricities of the sections of (1) by the planes of yz, zx, xy respectively. The co-ordinates of the focus of the section by the plane of xy are ae 3 , 0, 0. The square of the distance of any point {x, y f z) in (1) from this focus = (x-ae 3 Y + y*+z 2 -2as 3 x+a\* + b*(l-^-?) + z* b 2 — c 2 2ae if e = — 1 c ON FOCI AND CONFOCAL QUADRICS. 185 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 3 x — ez — a = (2), e 8 x + ez — a = (3). These two planes intersect in a line whose equations are z = 0, e z 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, /?, 7 be the co-ordinates of such a point, we must have the expression (x — a) 2 + (y — ftf + (z — y) 2 identically equal to {l(x-anMy-^+n(z-y , )}{r(x-oL , )+m(y-^)+n f (z-y% for all values of x, y, z which satisfy the equation of the quadric ; a', ft', y being the co-ordinates of any point in the line of intersection of the planes. 186 ON FOCI AND CONFOCAL QUADKICS. Let the equation of the quadric be Ax* + Btf+Cz 2 =l (1). Then the equation (*-a) 2 +(2/-/5) 2 +(*-7) 2 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 mri + rim = 0, nV + ril = 0, Im + I'm = 0, which can only be satisfied by one of the sets of conditions m ri i or z = o, r = o, m = 0, m = 0, n = 0, ri = 0, 7 = 111 n ri X — = — — n I X m — = __ — , / m (3). If we take the second set of these equations and put X Y — k, the equation (2) becomes (^-a) 2 +(2/-^) 2 +(^-7) 2 -^ 2 (^- a ') 2 +^ 2 (^-77=0 (4). Comparing the remaining terms of the second degree with those in (1) we obtain 1-fcP 1^ 1 + Jcn* A ~ B~ G ' or £Z 2 = 1-^, fcf-g-l (5). ON FOCI AND CONFOCAL QUADRICS. 187 And by comparing the terms involving x, y, z % and the constant term in (4) with the corresponding terms in (1) we have a-£ZV=0, /3 = 0, 7 + /™y = (G), a* + & + y* -7d* a!* + hi 2 y 2 =- ± (7). And substituting for a , y from (6) in (7) we obtain by help of (5), 2 9 tVtV' » 2 £ 2 The equation (8) combined with /3 = 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 y cor- responding to any particular focus (a, ft 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 tVt^t-' <»»• A G B G T7T + T7T= 1 < 10 >; BACA 188 ON FOCI AND CONFOCAL QUADRICS. and 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, C> one will be an ellipse, another an hyper- bola, and the third an impossible locus. 157. For instance, in the ellipsoid whose equation is x* f £ -, 2 « £» » 2 — J -f a b c the equations of the focal conies will be + jr- — a = 1 in the plane of xy, zx, + -2--2 = l yz. a 2 cr - And if we assume a, b, 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 original ellipsoid by the planes of yz and zx: the second a hy- perbola 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- boloids. 158. The focal conies of a cone Ax 2 + By 2 + Cz 2 = (1) can be deduced from those of a central quadric Ast+Bf+ Cz 2 = \ (2), by putting X equal to zero. ON FOCI AND CON FOCAL QUADRICS. 180 The focal conies of (2) would be, writing — , c - > t in- stead of A, B, G in the formulas (8), (9), (10) of Articles 154 and 156, A A A A A~B G~B or + /3 2 A A A A A~G B~C A_ A ^_^: ^ 4 -4 Or, multiplying these equations by A and then making A to vanish, the focal conies of the cone (1) become a 2 7 2 1 1 ' 1 1 A B G B a 2 i P 1 i ' 1 l A c B G £ 2 7 2 = 0, 1 1 l_I =a B A G A Of these, whatever be the signs of 11 11 l_.l B C G A' A 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 By*+Gz 2 = ce (1), 190 ON FOCI AND CONFOCAL QUADMCS. we must as in Art. 154 make the equation (1) identical with (x-*y+(y-f3y+(z- y y -[^- a 04-m(2/~/30+w(^-70)^X^-O+^(3/-/3>^(^-7)i=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 j = k, equation (2) becomes as in that Article (x - a) 2 + (y - /3) 2 + (z - 7 ) 2 - W (x - a') 2 + kn' (z - 7 ') 2 = 0. And since (1) contains no term involving x 2 and no con- stant term, we get 1 - M 2 = 0, or + j3 2 + 7 2 - hi 2 a! 2 + bfy' 2 = ; and by comparing the remaining terms in the two equations, we have C ~B~ 1 and thus we get for the locus of the foci the two equations / 8 = Oanda 2 + 7 2 -(«- 2 -^) 2 + ^g=0 ) or cy i/ _ jly and j3=o\ 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-C~ C\ 4C7* The first of the conditions (3) of Art. 154 is in this case inadmissible inasmuch as (1) contains no term involving x 2 . ON FOCI AND CONFOCAL QUADRICS. 191 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* = l, A'af + By+C'^^l, will have the same focal conies if A B~ A' B" 1 1 1 1 1 1 1 1 B~ ' G~ ~~ B' d" 0" 'A' ~ C" ~ A or as we may write the conditions, if i 1 1 1 1 A ' A'~~ ' B~ B~~ a l G" Two quadrics whose equations satisfy these conditions are called confocal quadrics. Thus if the equation of an ellipsoid be ? + H- 3 (1) > all surfaces whose equations are of the form a* + k ' b' + k ' c' + k = 1 (2), 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, 0, 7) and confocal with (1) by determining k from the equation ■ ^ +^-1 = (3), cf + k b^ + 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 192 ON FOCI AND CONFOCAL QUADRICS. quadrics confocal to (1) which pass through the point (a, /3, 7), of which one can be shewn to be an ellipsoid, 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 A^ B^ G l {l)t *V , ■■'/•' , "*■ m i (2 ) A+k + B+k^C+k~ Vf ; ' The equation of the tangent plane to (1) at the point (w, y, z) is xx yy zz ., ,«.. A + B + C- 1 ^- And the equation of the tangent plane to (2) at the same point is xx yy z'z _ ( . A^k + TTk + C + k~ W * But from (1) and (2) by subtraction we obtain at all their points of intersection x 2 f z* A{A + k) + B(B+k) + C(G + k) ' which is the condition that (3) and (4) should be at right angles to each other, ( 193 ) EXAMPLES. CHAPTER XII. 1. Find the equations of the focal conies of the quadric 2. Find the equations of the quadrics confocal with the quadric 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 x to a series of con- focal surfaces whose axes coincide with the axes of co-ordi- nates. 4. Shew that the surfaces a ax — a ax-^b + y+ ** , -i r Q~ /J.. . ~2 Z.2 x > 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. A. G. 13 CHAPTER XIII. ON CURVATURE OF SURFACES. 163. Two surfaces are said to have contact of the firs1 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 anc y the values of z, -=- and -7- shall be the same for the two surfaces. Two surfaces are said to have contact of the n th order at 1 point where they meet when the sections of the two surface: by every plane passing through that point have contact of th< n th. or( j er# This we will prove to be the case if the sections o the surfaces by all planes which contain any given straigh line through the point of contact not lying in the tangen plane have contact of the n th order. For let the common point be taken as origin and th< given line as axis of z. Let the equations of the two surface be *«/<*, y) (i), *~*{«,v) (2). Expanding (1) and (2) we obtain -©•+(D» + ~+K-s + 4)' /+ * =S)«(f)^-+-K + 4)"* + ""< ft ON CURVATURE OF SURFACES. 195 where z x and 2 2 are the ordinates of the two surfaces corre- sponding to the same values of x and y\ and in the quantities ■4- , -j- , ... x 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 ?i th order, the difference of z x and z 2 must be of the (n + If 1 degree in x and y. Hence we have df^dF df = dF dV = d*F d*f = d*F dx dx ' dy dy ' dx* dx* * dxdy dxdy ' ' dy^d^F d n f d n F dx n dx n> '" dafdy"-* dafdf^ 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 r+ *z' differential coefficient of the form , , r , IS can be expressed in terms of the differential coefficients of z with respect to x and y of orders up to but not exceeding the (r + s) th . Hence if the differential coefficients of z with respect to x and y for one surface, up to those of the n th 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 n th order, so will their sections by all planes through the common point. The conditions that two surfaces should have contact of the n tli order at a given point are therefore that the values of dx' d^ y '"d^ } dx n - x dy ,% "dy ni should be the same for the two surfaces for the given values of x and y. 13—2 196 ON CURVATURE OF SURFACES. 164. If two surfaces touch at a given point and the sections by a plane through the normal and any tangent line have contact of the second order, then all sections by 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 =f{x, y), z = F(x, y) being the equa- tions of the two surfaces, the values of -— , -J- , -j- , -*? ax ay ax dy vanish at the origin and the equations of the surfaces can be put in the form z — ax* + hxy + cy* -f (1), z=Aa?+Bscy+Cy*+ , (2), where a, b, c are the values of \ ~~ , , •* , } -~~ at the origin, and A, B, G those of \ ^ , ^- , \ ^ . 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 x of x, in the one surface z } = ax* -*- hx 1 y 1 + cy? + . . . , and in the other z, = Ax? + Bx 1 y 2 +Cy * + ...; •'• Si-s a = ffi (% 1 -% 2 )+ c 2/i 2 -^ 2 2 + ••• But z v z 2 being of the second degree in x v y x and y 2 are so also by (3), and therefore x (by x — By 2 ) is of the third degree, and therefore z x — z % is of the third degree in a?,, 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 — l) th order at a given point, and the sections by any plane ON CURVxYTURE OF SURFACES. 197 through the normal and a given tangent line have contact of the n th order, then all sections by planes through this tangent line have contact of the n th 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 is the radius of curvature of an oblique section through the same tangent line inclined at an angle 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 ; 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 xy 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 2 + bxy + cy 2 + (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 = #taa# (2). The radius of curvature of this section is the limit of x 2 + v 2 — — — when the values of x and y are diminished indefi- 2-z nitely. Hence if p be this radius, we have I ' . ax* + hxy 4- cy* 4- Ax % 2p ' x*+tf a -f b tan -f c tan*fl -f Ax *• l+tan 2 = aeos 2 0+ &sin#cos0 + csin 2 (3). 198 ON CURVATURE OF SURFACES. If we construct the conic section whose equation is ax 2 + b%y + cy 2 — 1 (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 2 + bay + cy* (5). The curve in which this surface is cut by a plane z = h parallel to the plane of xy is similar and similarly situated to (4). Hence the indicatrix at any point of a surface may be denned 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 Aa*+Cy*=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 a? and y in equation (4) of the last article. Then 1 1 ON CURVATURE OF SURFACES. 199 But A + C = a + c. (Todhunter's Conic Sections, Art. 274.) And therefore 1 + 1 = 1 + 1 (2 ). Also if the section whose radius of curvature is p be in- clined at an angle to the principal section whose radius is R, we have from (1) l^cos^+Csin 2 ^; Zp .\i=icos 2 + -Lin 2 (3). p it J.C 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)=0 (1), and let x, y, z be the co-ordinates of the given point P. Let I, 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 + /3,z+y be the co-ordinates of a point Q in the curve of section near to P. Let QR be drawn perpendicular on the tangent plane. Then, by Newton, the radius of curvature of the section is the limit of -775 when Q is made to ap- proach indefinitely near to P. But the equation of the tangent plane is <*'-*>f + ^>f + c'-*>§= < 2 >- 200 Otf CURVATURE OF SURFACES. Hence QR . dF dF dF a dx + P dy 1 dz aU+pV+yW {fdFV fdFV fdFV) VZ7 2 +F 2 +JF 2 with the notation explained in Art. 100. And P dX =P > dZ 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 for the sphere and surface must coincide as far as terms of the second degree in x — x, y r — y for points lying in the plane (1), whence we obtain d*Z d*Z ,d 2 Z — v + %ms + tm 2 . dX* dXdY dY* We deduce from (3) x—a^y—b z—c P p q -i and from (4) Jl+p* + q*' d 2 Z 1+/ d*Z dX 2 " c-z' dXdY~ pq d 2 Z l + . 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 -f Wn —0, or ft = — -^ m, and the expression (1) becomes 72 ■ i , V * 2 2u ' V 2 , o7 ( , Vv\ ul 2 4- vm* +.w.yp.n? ^ ra 2 + 2Zm f w - -=^J , which must remain constant for all values of m and w con- sistent with the relation r + m- (l+jP) = l. Hence WW-W-0, F 2 w 2aT" v + lP J7 V 2 w + W 2 v-2VWu' and tt=»- p: = -p 2 + jp (6). 1 + -fp 206 ON CURVATURE OF SURFACES. Similarly if V=0 or TT=0 the requisite conditions may- be deduced. In these cases, three conditions have to be satisfied by oo, 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 ra. 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 2 + vm 2 + wn 2 + 2umn+ 2v'nl + 2wlm -h(l 2 + m 2 + n 2 ) (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- efficients of ran, nl and Im as in Art. 49, we have W V and two similar equations, whence W*v+V 2 w-2u'VW km V* + w* V*u+U i v-2w'UV u*+v* U*w+W 2 u-2v'WU . $~tp by symmetry. ON CURVATURE OF SURFACES. 207 174 Lines of Curvature. A line of curvature on any surface is a curve such that the tangent line to it at any point coincides with the tangent line to one of the principal sections at that point. The differential equation of such lines is obtained by substituting -=- , -^ , -=- for I, ra, 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'm+ v'n, I, U w'l+ vm +u'n, m, V v'l + u m + wn, n, W = .(1), and replacing I, ra, n by -7- , — , -,- , respectively, we get the differential equation of the lines of curvature. The differential equation of the projection of the lines of curvature on the plane of xy is obtained by writing -M- 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 x —x y —y z —z (2). are U V The equations of the normal at a point (x + a, y + /3, z + 7) y-y-P- x X U+UQL + wp + vy + V + w'a + v0 + u 7 + . . . z — z — 7 W + va + u'/3 + 107 + . (3), 208 ON CURVATURE OF SURFACES. 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, 27+ ui +w'/3 + v'y+ ..., U, a. V+w'a + V0 + uy + ..., V, £ W+ va +u'/3 + wy+ ..., W , 7 uol + w'/3 + v'7, U, a whence ma. + v/3 + w'7, F, /3 =0 (4), v'a + u'fi + ^7, TF, 7 but ultimately a, j3, 7 are proportional to dx dz dy ds ' ds ' ds' 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 formula?. 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 tie 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 giye circular sections. The equation, of the tangent plane at any point (a, /3, 7) to an ellipsoid whose equation is c 2 1" 'ti "fc "3 — *i qr 6 2 c 8 ON CURVATURE OF SURFACES. 209 If this plane be parallel to either plane of circular section we have « -g- ^=, by Art. 65, and since (a, /3, 7) is a point on the ellipsoid, each of these ratios = + , — T Hence the ellipsoid has four umbilici whose co-ordinates are given by a= ± a V^c 2 ' /3sas0 ' r/=±c V^^' 177. If a tangent 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 » 2 -■» z 2 x y and let a sphere be described with the origin as centre and any radius k. The equation of this sphere is a? + tf + z 2 = k 2 (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 '(HMJ-iWG-p)- --^ 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- A. G. 14 210 ON CURVATURE OF SURFACES. dently a tangent plane to the cone (3). Hence if we draw a tangent plane to (3) along any generating line 0P V OP 1 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 0P X 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 Q OR which is diame- tral to 0P X . Let OP, 0P X , 0P 2 be three consecutive generating lines of the cone (3); OQ, 0Q X the lines conjugate to the planes POP v P x OP 2 which are ultimately consecutive tangent planes to the cone (3). Then since 0P X lies in a plane which is dia- metral to OQ, and also in a plane diametral to OQ v the plane Q0Q x is diametral to 0P X and therefore coincides with QOR, and the line joining QQ X 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 h is constant for all points on the line of curvature whose tangent at Q is parallel to r. But ii 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 -f- my' + nz' = (4), where I, m, n are connected by the relation n Li u (0); a 2 k? v \t e e (See Chapter vm. Ex. 24.) ON CURVATURE OF SURFACES. 211 rid the equation of a tangent plane to (1) at the point (x, y, z) a 2 + > + a'""" 1 (G). Hence if (6) be parallel to (4) r from (5) x _ y z a 2 l ~ b 2 m ■ c z n * 2 (l 7.2 2 a ~F b ~T? c ~k* nd subtracting this from the equation a 2 + ^+ c 2 A> ve get P + F- + ^ tf = 1, vhich shews that the lines of curvature on an ellipsoid are its urves of intersection with confocal surfaces. 179. In the ellipsoid rr_^ F _% w _* a" ' ~ b 2 ' ~ c* ' 2 t; = f0 = -i t*' = 0, 0, w' = 0. a" ¥' c Hence the differential equation of the lines of curvature is 1 dx x dx cfds' a 2 ' ds ldy y dy Fds' V* ~ds 1 dz z dz = 0, .4£*£«]D (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 1 a? tf / ... ?"? + t + ? (0) - Also from the equation of the ellipsoid, by differentiation (4) - Multiplying (2) by x, (3) by y, (4) by z, and adding, we get c 1 x + c 2 y + c 3 z = (5), shewing that all geodesic lines are great circles. 184. As a second example take the ellipsoid i44 =i «■ The differential equations of the geodesic lines become d 2 x d 2 y d 2 z ds 2 _ d£_ _ ds^ {S> , x y z ' ^ a 2 P ? Now let p be the perpendicular from the centre on the tangent plane to (1) at the point (x, y, z), 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). Then -^ = — A + f« + - i} p a b c I = l { Ts =Z UjO UjO 0; whence pr = 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) v Hence er+ew* i p +£ a* + b i + c 4 p - p r 2 ' - or /> = + —. ■(6); P .*. p = kr 3 , where k is some constant. 218 ON CURVATURE OF SURFACES. 185. We shall conclude this subject with the following proposition, known as Dupin's Theorem. If there be 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 intersection 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 v 8 2 , S 3 be the surfaces touched by the planes of yz, zx, xy, respectively, and let P, Q, R be points near in the curves of intersection of S v S 3 ; S 3 , /S^; 8 V $ 2 , respectively. Then since the surfaces 8 V $ 3 cut at right angles, the normals at Pto 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 Q 3 , 2 be the angles which the normals at P to S 3 , S 2 , respectively make with the planes of zx, xy, respectively; 3 those which the normals at Q to S v S n make with the planes of xy, yz, respectively, and ^r 2 , fy x those which the normals at R to S 2 , S t make with the planes of yz, zx, respectively. Let the lengths of OP, OQ, OR be a, /3, 7, respectively. Since the normal to $ 2 lies in the tangent plane to S 3 , the tangent of the angle which the normal to $ 2 at makes with the plane of xy is (-r- ) > the suffix denoting the surface from which the differential coefficient is obtained. Hence the tan- gent of the angle which the normal to # 2 at P makes with the plane of xy is dz\ d (dz\ dy) 3 dx\dy) 3 ©.-»• whence e *= a dx (e^), ultimate1 ^ ON CUKVATUKE OF SURFACES. 210 Similarly, therefore Similarly, But since the normals to S v S 3 at P are at right angles, Similarly, ± + 3 = 0, ^ 2 + ^ 3 = 0, whence 2 = 0. Hence the normals to S 9 at and P both lie in the plane of xy and therefore intersect one another, and therefore OP is the tangent to the line of curvature on $ 2 at 0. Whence the theorem follows. *»= _ d fdz^ dy\dxj )> 3 a = ±3. ±J = " 7 ' 7 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 3 , . I ! I a 6 c 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 = a 3 , on the plane of xy. 220 EXAMPLES. CHAPTER XIII. 4. Deduce the formulae 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* + By*+Cz* = 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 °? + y* + fc* (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 f.i-i p+ -i k : -i a ax— b ax — c are circles, and that the plane of any one of them contains a fixed straight line 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. EXAMPLES. CHAPTER XIII. 221 11. U and V are 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 PU andPF, 12. If a point Pmove 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 tfd* = a* + c 2 -r 2 , where 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 cylin- der. 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. 222 EXAMPLES. CHAPTER XIII. 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 sec- tions at right angles is the same at every point of the surface generated. 21. If two surfaces cut each other at right angles, and R be the radius of curvature of the curve of intersection at any point, p v p 2 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 i 22. If r, r' be the principal radii of curvature at any point of an ellipsoid on the line of intersection with a concen- f rr ')k trie sphere, shew that the expression J , is invariable. 23. If a geodesic line be drawn on a developable surface and cut any generating line of the surface at any angle ty and at a distance t from the edge of regression measured along the generator, prove that ^ + tcotf = p, 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 geo- desic from the origin, p the radius of absolute curvature, and p the perpendicular from the origin on the tangent plane to the surface, . j. i *vi EXAMPLES. CHAPTER XIII. 223 25. The centres of curvature of plane sections of a sur- face at any point lie on the surface Vi rr the axes being the tangents to the lines of curvature at that point and the normal, and p v p 2 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- *- =a z sin' a. Pi P* ANSWEES TO THE EXAMPLES. CHAPTER I. 1. J$, 2^3 and J3. 2. The length of each side is J6. 1 _2_ _3_ 2 3 6 26 JW JW JTi' ' 7' V V 7 JU' 5 90° 6 - - 0- - -• a ° 7 ' 3' 3' 3 ; 8 . ^.g-oy + (o-|y + (£-iy=^=^, 9. If !*., 0,, 4>n T 2> @ 2 > $2 ^ e * ne P°l ar co-ordinates of the points, the (dist.) 2 between them by Arts. (6) and (15) = (r x sin 6 X cos ^ — r 2 sin 2 cos <£ 2 ) 2 + (r x sin t sin <^> 1 - r 2 sin 6 2 sin <£ 2 ) 2 + (r x cos 0, - r 2 cos 2 f » r x 2 (sin 2 X cos 2 £, + sin 2 ^ sin 2 ^'+ cos 2 0,J + r 2 2 (sin 2 2 cos 2 <£ 2 + sin 2 6 2 sin 2 <£ 2 + cos 2 6 2 ) - 2^^ {sin 6 X sin 2 (cos j cos 2 + sin x sin <£ 2 ) ^ + cos 0j cos 0J ■■ r," + r 2 - 2r 1 r g {cos 6 X cos 2 + sin Q x sin 2 cos (<^ - 2 )}. 10. r = 4, ».| f |=|. 11. 0-I, y=J3, s = 2 v /3. ANSWERS TO THE EXAMPLES. 225 CHAPTER II , aj-1 . z-3 - _ 3 a+z=4) aj + y=5\ _ 1 _ 2-3 2/=z/ * = 1\J * i_2/ "~ 2 J' 6' 2' 3* 4. a + 2 / + *=6; 2V3. 5. «-f«|. 6. *-*»-?^U-3. 3^3 7. Let (a, /?, y) j (a , f3', 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, G must satisfy the two conditions A{a'-a) + B{P-P) + C(y'-y) = O i Al + Bm + Cn = 0, whence A : B : C :: m (V - y) - n (/5' - /?) : rc (a'-a) - Z(y'-y) : l(P-j3)-?>i(a'-a). 8. s = 3, a; + 2/= 3. 9. Let yl (x - 2) + i? (y - 3) + (7 (z - 4) = represent the plane required; ... A (I - 2) + B (2 -3) + (7(3-4) =0, or ^+£ + (7=0, .4. V3 + .B+ (7.2^3=0, whence i : B : (7 :: 2 J3- 1 : - J3 : 1 - v/3, and the plane becomes (2 V3 - 1) {x - 2) - n/3 (y - 3) + (1 - &) (*- 4) = 0. 10. Let I, m, n ; l\ m', n' be the direction-cosines of the given lines ; A., fi, v those of the required one ; .*. \l + fxm + vn = 0, \l' + fim' + vn! = cos a. The latter equation gives (XI + fxm + vnf = cos 2 a (X 2 + /x 2 + v 2 ), A. G. 15 22 C ANSWEKS TO THE EXAMPLES. which combined with the former will give two values of the ratios A : //. : v, as in Art. 57. For the latter part put cos a = -j— and find the value of \ } \ 2 + fi l fi 2 + v 1 v 2 J remembering that IV + mm + 7i7i vanishes this will also be found to vanish. 11. Let (a, /3, y) be the given point, I, m, n; V, m', n! the direction-cosines of the perpendiculars on the two planes. The required plane is {mn' - m'n) (x-a) + {nV - n't) (y-fi) + (lm' - I'm) (g- ^ = 0. (See Art. 30.) 12. The proof of Art. 19 holds when the axes are not rectangular if I, m, 7i mean the cosines of the angles between OB and the axes. 13. Draw the oblique co-ordinates of the point D, and pro- ject OD on the axes in succession. I + m cos v + n cos fx m + n cos X + I cos v 14. A B n + I cos jx + m cos A C 15. The condition is (X - af + sin 6 + C cos 6 = — . 19. s -. r 2 ANSWERS TO THE EXAMPLES. 227 20. Let l x , m l , n x ) l 2 , m a , n 2 ; l 3 , m a , n a ; be the direc- tion-cosines of the normals to the three planes. Then the equa- tion of any plane through the line of intersection of the first and second is (l x + \l 2 ) x+(m l + Xm a ) y + (n i + Xn 2 ) %**p x + Xp 2 , where X is a constant, and if this is perpendicular to the third, K (h + K) + m a K + Xr) \) + n s ( n i + Xn *) = °> or cos B + X cos A = 0. Also if the plane passes through the origin p x + \p a = ; .'. p x cos A =p i cosB i and the plane becomes (IjX + m x y + n^) cos A - (l a x + m 2 y + n 2 z) cos B = 0. If in addition p 2 cos B - p 3 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 lx + my + nz=p be the equation of one of the planes ; .-. from the data £+^+^ = 0, O r T +- + - = (1), and l(a-a) + m(b'-b)+n(c'-c) = (2) ; . \ substituting for n out of the second in the first 1 _1 c'-c _ Q I m I (a' -a)+m (b' - b) ? . -. p {a' - a) + Pirn + m 2 (b' -b)=Q, which gives two values of — , and corresponding to each of these m from (2) we can set one value of — . If — , — 2 - be these two values, ■ ' 2 = —. . Similarly — — — = —r — - . m x m 2 a -a m l m 2 c —c Hence if the lines be at right angles 1 + AL + JV!*-,0; m 1 m 2 m^n 2 b'-b b'-b ' 1 1 1 .-. 1 + -7 + -=0j .*. + 77-^; +-7— =0. a —a c — c a — a 6—6 c — c 15—2 228 ANSWERS TO THE EXAMPLES. 22. 5. This and Example 23 are to be solved as the last m example. 23. P (B 2 + C 2 ) + Q (G 2 + A 2 )+R (A 2 + B 2 ) = 0. 24. The co-ordinates of the middle points of the lines joining 1, 2 and 3, 4 are, Art. (7), i(a-c), ^(a+b-c-d), i(b-d), and J (c - a), J (c + d-a — b) i \(d — b), whence the result follows. 25. The co-ordinates of any point on one of the lines may be represented by a + It, b + mt, c + nt ; and those of any point on the other by a' + l't', b' + m't', c' + n't'. The square of the dis- tance between these points is (a-a' + lt- l't') 2 + (b-b' + mt- m't') 2 + ( G -c'+nt- n't') 2 . The conditions that this may be a minimum by the variation of t and t' are (a-a' + lt- l't') l+(b-b' +mt- m't') m + (c-c' + nt- n't') n = 0, and (a-a' + lt- l't') l'+(b-b' + mt- m't') m' + (c-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, l(a-a) + m(l3- /3') + n (y - y) + t - t' cos 6 = 0, I' (a - a) + m'((3 - j3') + n(y -y) + t cos - 1' = 0, whence t' sin 2 = u' + u cos 6. 27. Taking x v y , » , &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 opj)osite edges are 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 \a and \d and cu is the angle between the dia- gonals, whence by Trigonometry the result follows. ANSWERS TO THE EXAMPLES. 229 29. — j= if c is the edge of the cube. 30. J Jb 2 c 2 + c*a* + a 2 b 2 . 31. The equations of the planes are &c f my + nz =p, mx + ny + Iz =p, nx + ly + mz = p ; P ,\ x-y = z l+m + n' 32. Any point on the given line can have its co-ordinates expressed by a — It, b-mt, 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 III. 1. r 2 + r (A sin 6 cos + B sin 9 sin + C cos 0) + D = 0. This equation gives two values of r the product of which is D. 2. The polar equation of any plane is A sin 6 cos d> + B sin sin + C cos 6 = — . T- T- r k 2 Hence if this be the equation of the locus of P, since OP — n ^ the equation of the locus of Q is Br A sin 6 cos <£ + B sin 6 sin <£ + G cos 6 = -p- , which is the polar equation of a sphere. 3. If the locus of P be r 2 + r (A sin 6 cos + B sin $ sin <£ + G cos 6) + D = 0, that of Q is ¥ + k 2 r (A sin 9 cos <£ + B sin sin + (7 cos 9) + Dr 2 = 0, which is another sphere. 230 ANSWERS TO THE EXAMPLES. 4. The plane in question is xx' + yy' + zz' = x' 2 + y' 2 + z' 2 = c 2 , also where it meets the sphere x 2 + y 2 + z 2 = c 2 , whence x 2 + y 2 + z 2 - 2 (xx' + yy' + zz') + x' 2 + y' 2 + z' 2 = 0, or (x - x') 2 +(y~ y') 2 + (z- z') 2 = j .-. x=x', y = y\ z = z'. 5. Take A as origin and AB (=a) as axis of x, the equation of the locus is x 2 + y* + z 2 -m 2 {(x -a) 2 + y 2 + z 2 }, 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 2 + z 2 — 0, a sphere, on AB as diameter. 7. Take for the equations of the fixed straight lines those given in Ex. 33, Chap. n. The equations of the two planes can then be written y - mx + \ (z - c) = and y + mx + fi(z + c) = where A and //, are constants. The condition of perpendicularity gives 1 —m* + \fi = and by substituting for X and /x out of the first two in the third we get (1 - m*) (z 2 - c 2 ) + y 2 -m 2 x 2 = 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 must cut each sphere in a circle. 9. The equations of the spheres can be written S-kr 2 = 0, S'-k'r 2 = 0, S"-k"r 2 = 0, where h, k\ k" are constants and r changes. The first and second a a/ intersect on the sphere j - -y = 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. ANSWERS TO THE EXAMPLES. 231 13. Take the three planes as co-ordinate planes, and let I, m, 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 s accession x = la, y = mb, z = nc; x 2 y 2 z 2 a 2 b* e 14. "We have nr. 4- n. n/ and 6 may be. Substitute in the equations of Art. 57, and we shall get finally x - a cos <£ sec 6 _ y - b sin <& sec 6 _ z - c tan 6 a sin (<£ =*= 0) b cos ( ± 6) ± 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 \b cj a \o 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 b y c * 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, (3, y, 8. Find the conditions that this may be satis- fied by either of the pairs a = 0, 8 = 0, and /? = 0, y = 0. CHAPTER VI. 1. If X, /a, v be the direction-cosines of any generator of the given cone a 2 A 2 + b 2 fi 2 + cV = d*, whence by Art. 79 the result follows. 2. Use equations (6) of Art. 77, and in the given case by Art. 78, ^ =k\ a*P + b 2 m? + c*»« and the locus becomes ANSWERS TO THE EXAMPLES. 237 3. Use formulae of Art. 74 to shew that the plane passes through the three given points. 4. ^ — ^k> where p is the perpendicular from the origin on the plane LMN. m . irabc 5. volume^. 6. A cylinder whose axis is parallel to Oz and whose trace on the plane of xy is given by ab I * • *u u i/j/i c 2 (a 2 sin 2 fl- f& 2 cos 2 fl)) — = Jot sin J + b 2 cos 2 < 1 jTi > . 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 x 2 y 2 angle 6 to Ox to cut the ellipse -^ + ^ = 1 in two points. If r x , r 2 be the distances of these two points from a, j3, the square of the eccentricity of the vertical section through a straight line x = a, y = P supposed to be its directrix must = - , but it also equals _ c 2 (a 2 sin 2 + 6 2 cos 8 0) , . . M J •". - 1 5 oT ^ - by Art. bo, whence since r. and r are a 2 6 2 J l 2 expressed in terms of 6 we can get a quadratic equation in tan 2 the roots of which must be real. 8. Use Art. 77, jp being a constant : Jx 2 y* z 2 \ (x 2 y 2 z 2 V 9. If a, V, c' be the conjugate semi-diameters, and x', y', z' the co-ordinates of the point in which the three planes meet X _x[ 2 by similar triangles and Art. 79. 10. We have to find the directions of the axes of the section of Ax 2 + By 2 + Cz 2 = 1 by the plane Alx + Bmy + Cnz = 0, where PI 2 + Qm 2 + En 2 = 0. See Art. 68, Equations 5 and 12 and elimi- nate I, m, n. 238 ANSWERS TO THE EXAMPLES. n /i\ p m n m n 1L (1) a=z l + W£ + Wd> P—WS> y = ~2lC' (2) %BfiQ-fy + *Cy(i-y)=*-m if a, b, c be co- ordinates of the fixed point. 12. If x, y, z be the co-ordinates of any point on the perpen- dicular, ax by cz oc 1 + x 2 + x 3 y } +y 3 + y 3 z x +z 2 + z 3 = Ja 2 x 2 + b 2 y* + c V a b c J3 by Art. 74, - 2 - y - m - xx ^ + y y\ + zZi ~ x,+x 2 + x~ Vl +y 2 + y s ~ z,+z 2 + zf ~T~ a 2 b 2 c s 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 3 - 10s 2 + 13s + 55 - 0. This has two 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 < 2. (2) Co-ordinates of centre */, §, * ¥ 5 ; hyperboloid of two sheets. (3) A parabolic cylinder. (4) A hyperboloid of one or two sheets as a 2 > or < 3. 3. The two equations merely differ by h 2 (x* + y 2 + z 2 ) which remains unaltered by any transformation round the origin. The second is a right circular cylinder, the first a spheroid. ANSWERS TO THE EXAMPLES. 239 4. An ellipsoid if 1 - /x < J'2, a hyperboloid of one sheet if l- f t> /v /2. 5. An ellipsoid whose centre is at the point — , — , — - : the equation when z = can be put into the form (H--Hi-'HM*-°- 6. See Example G, 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 x = 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 if c = a + b, b" = 0, c' 2 = ab, and the equation can be written (xja + yjbf + c («+ -) + 2a"x+2b"y + d- — = 0, which can be again written (xJa + yJb + k) 2 + c(z + C -\ + 2(a"-kja)x + 2{b" -k Jb)y + d- v — - #• = (). And if A- be so chosen that x J a +y Jb + h-0 i and the line 2x (a" - k Jo) + 2y {b" -Jcjb) = 0, c" are at right angles, the former united with z + — — must give c the axis. 11. z 2 + cxy = k 2 . 1 2. Take for the fixed straight lines x=0, y-0; x — a,z = 0; y-b, z=*c; and take the equations (3) of Art. 17 as the gene- rating line : the equation becomes ayz + bxz — cy{x — a). 13. The condition required is that A\ 2 + £fx 2 + Gv 2 + 2A'fxv + 2B'v\ + 2(7 V shall retain an invariable value for all values of \, /x, v consistent with l\ + m/x + nv - 0. See Art. 173. 14. Eliminate s between the equations (1) of Art. 83. 240 ANSWERS TO THE EXAMPLES. 15. If x, y', z' be the co-ordinates of the vertex, the equation of the cone is (x'z-xz'f (y'z-yz'Y — ^— + — j5 — («-*?. And by Art. 50 equation (7) it follows that S + S^ ,2 (i + p)- 1=a2 ^ 2 -( a2+ ^> a CHAPTER VIII. 1. x 2 +y 2 + z 2 = a 2 + b 2 + c 2 . 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- /?) + z (*- y)} 2 = a 2 x 2 + b 2 y 2 + c 2 z 2 . 5. a 2 x* + b 2 y 2 + c 2 z 2 =k\ 6. The conditions that the normal to the ellipsoid at (x, y, z) shall pass through (a, /3, y) are a 2 (x-a) _ b 2 (y-(3) _c 2 (z-y) x y and these combined with = *, x* y* z 2 , • a 2 b 2 c 2 ' give an equation of the sixth degree in h. All six lines lie on the cone (i'-c g )q | (, the equation of the osculating plane can be written 2x cos 3 <£ - y' sin <£ (1 + 2 cos 2 <£) - 2s' + r sin <£ (2 + cos 2 <£) = 0. 2. Length of arc = J a 2 + c 2 . (0 X - 6 2 ). From the equations of the curve obtain x 2 + y 2 as a function of z : let as 2 + y 2 =f(z). This is the equation required. Ex. x 2 + y 2 =■ a 2 . 3. We easily get, if a be the radius of the generating cylinder, b 2 = 4a 2 sin 2 °^ + a 2 cot 2 a ($ l - 2 ) 2 = ia 2 sin 2 ^^ + I 2 cos 2 a, # — # if £ be the length. Hence, when I is a maximum, sin * 2 = j m and the maximum length = . But this maximum length cos a ° bb a cosec a (0 X - 2 ) and 6 l — s = 2mr ; 2?ms 6 6 tan a .*. . — = ; .*. a = — . sin a cos a £nir .ANSWERS TO THE EXAMPLES. 243 4. The equations of the curve are x = acos0, y = «sin 0, z = -~ (e c ' + c *). 5. x + y + z = l. a 6. r = a t 6 = tsLii fi log ta,n-: + C ; r t 0, being polar co- m ordinates, 7. (1) y*-x* = c. (2) y + atan" 1 ^. 8. Analytically. Differentiate the equations of the sphere f//y fiti fl2Z and ellipsoid, and find the ratios ~y '- ~r~ '- ~T • The equat'cn of the plane can then be found, and then the equation (12) of Art. 68 can be used. 9; (1) cos -1 (cos sin 6) + cos -1 (sin sin 6) = const. which can be transformed into (2) cos <£ sin J 1 - sin 2 sin 2 + sin cj> sin J 1 - cos* sin* 6 = const. or (3) x J a 2 - y 2 + y J a 2 -x* = const. 10. = a. 11. By Art. 101, -=-, ■—■ , -s~ will be proportional to whence m* en* fl-lM^ /I.IM^ fI_!M^=o \6 3 c 2 / a; ds + V Also the equation of a plane through the origin and the line required is *(*-*') , y(fi-P) , ^{y-y r ) _r } a 2 b* * c 2 ~ ' the envelope of which treating a — a, /3 — /?', y — y as parameters gives us the cone required. That the curve is a parabola can be shewn because a plane through the origin parallel to the polar plane of (a, j$, y) can easily be shewn to touch the cone. CHAPTER XI. 1. If x^f^t), y=f 2 (t), z=f 3 (t) be the equations of the curve, we have to find the envelope of & -A (t)T + \y -/, W + {* -/. (t)Y - A where t is the parameter. The envelope is obtained from the intersection of the sphere with the normal plane to the curve at the point t. 2. The equation can be put in the form f (x + y + z ){(^ + »v(^)p, and if the line x = y=-z be taken as axis of z' this becomes, 3 /3 by Arts. 25 and 28, — £- z' (x 2 + y' 2 ) = c 3 , which is a surface 2c 3 formed by the revolution of the curve z'x' 2 = ——^ round the axis ofz'. Or, apply Art. 148. 3. a?y 2 + x 2 z 2 = c 2 x 2 ; x=a, y 2 + z 2 =c 2 being the equations of the circle. 4. See Ex. 11, Chap. vn. for choice of axes, x 2 y 2 __ a 2 (c + z) 2+ (c-z) 2 ~~7' ANSWERS TO THE EXAMPLES. 247 . z z x$m- = y cos - c 6. x cos v + y sm = a, where V = — — . c a 7. (1) (x 2 +y 2 ) (k-na) + 2a(z- a) (Ix + my) + (k + na) (z-a) 2 , tlie vertex being at the point (0, 0, a) and the plane of the small circle being Ix + my + nz=k. (2) Put z = in the above. 8. Jx 2 + y 2 + Jc 2 -z' = a, or (x 2 + y 2 + z 2 + a 2 -c 2 ) 2 =4:a 2 (x 2 + y 2 ) .(1). 9. The points at which the tangent plane passes through the origin are given by » = ± — J a 2 — c 2 , that is, they lie in two CO horizontal rings. Take one of these points in the plane of zx. The tangent plane at this point has for its equation * = s^^ (2). c Also the equation (1) can be put into the form {x 2 + y 2 + z 2 - (a 2 - c 2 )} 2 = 4c 2 y 2 + 4c V - 4 (a 2 - c 2 ) z 2 = ic 2 y 2 + 4:(cx-z J a 2 - c*) (ex + z J a" - ?), whence at the points of intersection of (2) with (1) x 2 + y 2 + z 2 -(a 2 -c 2 ) = ±2c?j. Hencs (2) cuts (1) in two circles. From the symmetry of the surface the same will be true for all the points. 10. The fixed plane being the plane of yz, and I, m, n the direction-cosines of AB, the equation of the surface is (mz - nyf + (nx - Izf + (ly - mx) 2 = k 2 (y 2 + z 2 ). 11. The conditions are given in Art. 92. See Art. 151. 13 <*V b 2 y 2 c 2 z 2 (a*-*7 (b 2 -k 2 y (c 2 -k 2 ) 2 ' , 7 o 7 ciyz + bzx + cxy where tr = abc . 7 — ^ j— . bcyz + cazx + aoxy 248 ANSWERS TO THE EXAMPLES. 14. Take y=px and y — kz + q as equations of the gener- ating line. 15. (1) A surface of revolution round Oz. (2) A surface such that all sections by planes through Oz are circles. (3) A cone whose vertex is 0. 16. (1) A surface produced by the revolution of the lem- niscate in the plane of zx round Oz. (2) A surface produced by the motion of a circle whose centre is and radius is any radius of the same lemniscate placed in the plane of xy. „. gy^_^.}'(l-^ sy (2,>_|,)' 18. The equations of any helix can be written x = a cos 6 } y = a sin 0, z = cO + y, and by virtue of the given conditions y and c must be expressible as functions of a. Hence since a z = x 2 + y 2 and = tan _I - , and x also = — - , we get c c ° tan" 1 ¥- = zF(x* + y 2 ) +/(x 2 + y 2 ). 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 generating lines. 20. If a?, y., t.\ x 2 , y 2l z 2 be the co-ordinates of the points A, B; being the origin, the condition that AB subtends a right angle at is x x x a + y x y 2 + s x *y = 0. Also the equations of AB are x-x } _ y-y x _ z — z x and from the equations of the straight lines x 2 , y 2 can be expressed in terms of z 2 and x lf y } in terms of z . Then eliminating z lt z a , between these equations we get a relation between x, y, z. ANSWERS TO THE EXAMPLES. 249 21. Equation (4) of Art. 148 is evidently the required con- dition. x 22. If -=f(z) be the equation of the surface, the locus if required is (*'-y'/)(y+*'/)=*-^A where f f f are the values of f(z) suidf'(z) for the given value of z. 23. The equations of any such circle are x 2 + y 2 + z* = 2ax and y = mx, also a must be expressible as a function of m, = 2cf(m) say. The differential equation can be easily deduced. CHAPTER XII. 1. 6a 2 -12y 2 = l, /3=0; 4a a + 12/2 3 =l, r = 0; impossible locus. 2x* Zy 2 4z 9 2. If ^ — T + OT - + -j-, — =- = 9 be either of the surfaces, 2&+1 3&+1 4&+1 ' the two values of k are the roots of the quadratic 72 3. 29 n 3. Let a be the distance of the point along the axis of x, and x 2 v 2 z % -» + tt + -5 = 1 one of the surfaces : the locus required is a 2 b 2 C * x if z 2 , _ + ^_ 1 sa 1. a ax - (a 2 —b 2 ) ax— (a 2 — c 2 ) 4. At the points of intersection we easily get ax = /3y + a 2 . Also the direction-cosines of the normal to the first surface at any such point are easily proved to be proportional to la az 2 2 2z a F~ (ax-b 2 ) 2 ' 0' 'ax~^¥' while those of the normal to the second are proportional to 2 I ft f& 2z a' fi a 2 (ax-b 2 ) 2 ' ax-b 2) and these lines are therefore perpendicular to each other since their direction-cosines satisfy the requisite condition. 250 ANSWERS TO THE EXAMPLES. 5. If the two quadrics be By 2 +Cz 2 = x and B'y 2 + C'z 2 =x + h, the coincidence of the foci involves ±B~±B f *> 4(7 ~ 46" ' whence also the focal conies will coincide, since B-G B'-C BC ~ B'C ' 6. At the points where the two quadrics in (5) cut, we have (B-B')y 2 + (C-C')z 2 + h = 0, or iBB'y* + iCG'z 2 + 1 = 0, which is the condition that the tangent planes to the two quad- rics at (x, y } z) should be at right angles. CHAPTER XIII. b 2 c 2 x 2 cW a 2 b 2 z 2 . 1. 2 + ^+- 2 = pp — a pp- o pp - c where p is the perpendicular from the centre on the tangent plane. This can be reduced to a 2 b 2 c 2 p 2 p 2 - (a 2 + b 2 + c 2 - r 2 ) pp + — 2— = 0, where r 2 = x 2 + y 2 + z 2 . For the umbilici the two roots will be equal. This will require one of the quantities x, y or z to vanish. 2. (1) x=y = z = a. (2) When xa? = ±yb* = ± z&. 3 - (|) 2 (^y-^)^+i^y(2/ 2 -^ 2 )-2/ 2 (^y-^)-o. 4. (1) Eliminate m between equations (6) and (7) of Art. 171, writing p = hjl+p 2 + q 2 . (2) The coefficients of the several powers of m in the equation (7) of Art. 171 must vanish. ANSWERS TO THE EXAMPLES. 251 5. The two values of h in (9) of Art. 1 69 must be equal and of opposite sign ; \ U 2 (v + w)+V 2 (w + u)+W 2 (u+v)-2u f rir-2v f WU-2w , UV=0. The points of intersection of the surface with the sphere . 2 2 1 1 1 6. Take the general equation of a quadric and determine the conditions that it may touch the plane of xy 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 + by + h (z - a) - z = 0. 8. See Ex. 4, Chap. xn. The surface in the question and the two surfaces x 2 y z 2 , os? y 2 z ~ /^y + 6 2 /8 fiy + tf-c* ' yz + c 2 yz + c 2 -b 2 y can be shewn to cut always at right angles, where /? 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 — /3y + b 2 a plane ; and by combining this with the given equation, that can be written - (ax- c 2 ) + | ((3y + b 2 -c 2 ) + z 2 = ax- c 2 , a jS which is the equation of a sphere. Hence the lines of curvature are circles : and the plane of any one of them being ax = fiy + b 2 always contains the line ax = b 2 , 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 curvature. 1 1 . The value of pr is the same for the two geodesies through P since they each pass through an umbilicus. Hence the value of r is the same. The tangents to these two geodesies are there- 252 ANSWERS TO THE EXAMPLES. 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 urn bilicus at right angles. Hence if d, d' be the semidiameten parallel to the tangent to the geodesic circle and the line througl the umbilicus, and p, p be the semi-axes of the central sectioi parallel to the tangent plane at the point 1 JL JL 1 _ p*(a 2 + b 2 +c 2 -r 2 ) d 2 + d'* ~ p* + p'* a 2 b a c a Ex. 25, Chap. vm. But p 2 d' 2 = a V as can be ascertained from the known co-ordinate of the umbilici. 14. At any point in the principal section by the plane of y, the two roots of the equation in (1) can be shewn to be c 2 and — . The former root is the radius of curvature of the principa section : the latter gives the distance along the normal of tin point whose locus is required which can then be worked out b] plane geometry. 15. Taking x 2 + y 2 = a 2 as the equation of the cylinder w d 2 z dz easily get for the geodesies ^- 2 = 0; therefore -r=c, whence th curves are helices. 16. s = Jz 2 sec 2 a-c 2 f where a is the semi-vertical angle o the cone, and s the length of the arc from the nearest point to th vertex. 17. If x 2 -vy 2 =f(z) be the equation of the surface it easil; follows from (1) of Art. 182 that for all points in any geodesic lin< dy dx ANSWERS TO THE EXAMPLES. 253 knd it can easily be proved that the sine of the angle required c M+?' 18. If f ~f(x) be the equation of the surface r 2 being y* + z 2 ;he required expressions are {■ ♦ m / 3 and r art dx 2 Mil- 19. With the usual notation for an ellipse the product required is 20. The radii of curvature of the principal sections are r 2 f and : — -, where r is the focal radius of the point on r-psm the angle between that radius ind the tangent, and p the radius of curvature of the ellipse (Besant on Glisaettes, &c.). Hence the sum of the curvatures _2 psin